This is TeX, Version 3.14159 (Web2C 7.3.1) (format=discheckediv 2002.10.5) 5 OCT 2002 11:23 **&discheckediv _region_.tex (_region_.tex LaTeX2e <2000/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. CUSTOMISED FORMAT. Preloaded files: . discheckediv.tex article.cls 2000/05/19 v1.4b Standard LaTeX document class size11.clo 2000/05/19 v1.4b Standard LaTeX file (size option) fancybox.sty 2000/09/19 1.3 macros.tex amsmath.sty 2000/07/18 v2.13 AMS math features amstext.sty 2000/06/29 v2.01 amsgen.sty 1999/11/30 v2.0 amsbsy.sty 1999/11/29 v1.2d amsopn.sty 1999/12/14 v2.01 operator names amsfonts.sty 1997/09/17 v2.2e amssymb.sty 1996/11/03 v2.2b amsthm.sty 2000/10/26 v2.08 umsa.fd 1995/01/05 v2.2e AMS font definitions umsb.fd 1995/01/05 v2.2e AMS font definitions . \openout2 = `macros.aux'. (_region_.aux) \openout1 = `_region_.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 15. 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(/usr/share/texmf/tex/latex/preview/preview.sty Package: preview 2002/08/05 preview-latex 0.7.3 (/usr/share/texmf/tex/latex/preview/prauctex.def \hbadness=\count101 \hfuzz=\dimen113 (/usr/share/texmf/tex/latex/preview/prauctex.cfg)) \pr@snippet=\count102 \pr@box=\box30 No auxiliary output files. Preview: Fontsize 10.95pt ) !name(discheckediv.tex) !offset(-4) ! Preview: Snippet 1 started. <-><-> l.22 We study $ p$-divisibility of discriminant of Hecke algebras Not a real error. ! Preview: Snippet 1 ended.(282168+127431x329728). <-><-> l.22 We study $p$ -divisibility of discriminant of Hecke algebras Not a real error. [1] ! Preview: Snippet 2 started. <-><-> l.24 cusp forms of weight bigger than~$ 2$, we are are led to make a Not a real error. ! Preview: Snippet 2 ended.(422343+0x327681). <-><-> l.24 cusp forms of weight bigger than~$2$ , we are are led to make a Not a real error. [2] ! Preview: Snippet 3 started. <-><-> l.26 ..., if true, implies that there are no mod~$ p$ congruences Not a real error. ! Preview: Snippet 3 ended.(282168+127431x329728). <-><-> l.26 ...if true, implies that there are no mod~$p$ congruences Not a real error. [3] ! Preview: Snippet 4 started. <-><-> l.27 between non-conjugate newforms in $ S_2(\Gamma_0(p))$. Not a real error. ! Preview: Snippet 4 ended.(491520+163840x2748650). <-><-> l.27 ...n-conjugate newforms in $S_2(\Gamma_0(p))$ . Not a real error. [4] ! Preview: Snippet 5 started. <-><-> l.30 \section {Introduction} Not a real error. ! Preview: Snippet 5 ended.(655359+0x28311480). <-><-> l.30 \section{Introduction} Not a real error. [5 ] ! Preview: Snippet 6 started. <-><-> l.36 \section {Discriminants of Hecke Algebras} Not a real error. ! Preview: Snippet 6 ended.(655359+183500x28311480). <-><-> l.36 \section{Discriminants of Hecke Algebras} Not a real error. [6 ] ! Preview: Snippet 7 started. <-><-> l.37 Let~$ R$ be a ring and let~$A$ be an~$R$ algebra that is free as an~$R$ Not a real error. ! Preview: Snippet 7 ended.(490372+0x550424). <-><-> l.37 Let~$R$ be a ring and let~$A$ be an~$R$ algebra that is free as an~$R$ Not a real error. [7] ! Preview: Snippet 8 started. <-><-> l.37 Let~$R$ be a ring and let~$ A$ be an~$R$ algebra that is free as an~$R$ Not a real error. ! Preview: Snippet 8 ended.(490372+0x538215). <-><-> l.37 Let~$R$ be a ring and let~$A$ be an~$R$ algebra that is free as an~$R$ Not a real error. [8] ! Preview: Snippet 9 started. <-><-> l.37 Let~$R$ be a ring and let~$A$ be an~$ R$ algebra that is free as an~$R$ Not a real error. ! Preview: Snippet 9 ended.(490372+0x550424). <-><-> l.37 Let~$R$ be a ring and let~$A$ be an~$R$ algebra that is free as an~$R$ Not a real error. [9] ! Preview: Snippet 10 started. <-><-> l.37 ...$A$ be an~$R$ algebra that is free as an~$ R$ Not a real error. ! Preview: Snippet 10 ended.(490372+0x550424). <-><-> l.37 ...$ be an~$R$ algebra that is free as an~$R$ Not a real error. [10] ! Preview: Snippet 11 started. <-><-> l.38 module. The trace of an element of~$ A$ is the trace, in the sense of Not a real error. ! Preview: Snippet 11 ended.(490372+0x538215). <-><-> l.38 module. The trace of an element of~$A$ is the trace, in the sense of Not a real error. [11] ! Preview: Snippet 12 started. <-><-> l.39 ...f left multiplication by that element on~$ A$. Not a real error. ! Preview: Snippet 12 ended.(490372+0x538215). <-><-> l.39 ...left multiplication by that element on~$A$ . Not a real error. [12] ! Preview: Snippet 13 started. <-><-> l.42 Let $ \omega_1,\ldots,\omega_n$ is a~$R$-basis for~$A$. Then the Not a real error. ! Preview: Snippet 13 ended.(308974+139537x3170138). <-><-> l.42 Let $\omega_1,\ldots,\omega_n$ is a~$R$-basis for~$A$. Then the Not a real error. [13] ! Preview: Snippet 14 started. <-><-> l.42 Let $\omega_1,\ldots,\omega_n$ is a~$ R$-basis for~$A$. Then the Not a real error. ! Preview: Snippet 14 ended.(490372+0x550424). <-><-> l.42 Let $\omega_1,\ldots,\omega_n$ is a~$R$ -basis for~$A$. Then the Not a real error. [14] ! Preview: Snippet 15 started. <-><-> l.42 ...ga_1,\ldots,\omega_n$ is a~$R$-basis for~$ A$. Then the Not a real error. ! Preview: Snippet 15 ended.(490372+0x538215). <-><-> l.42 ..._1,\ldots,\omega_n$ is a~$R$-basis for~$A$ . Then the Not a real error. [15] ! Preview: Snippet 16 started. <-><-> l.43 {\em discriminant} of~$ A$, denoted $\disc(A)$, is the determinant of Not a real error. ! Preview: Snippet 16 ended.(490372+0x538215). <-><-> l.43 {\em discriminant} of~$A$ , denoted $\disc(A)$, is the determinant of Not a real error. [16] ! Preview: Snippet 17 started. <-><-> l.43 {\em discriminant} of~$A$, denoted $ \disc(A)$, is the determinant of Not a real error. ! Preview: Snippet 17 ended.(538214+179404x2296385). <-><-> l.43 ...m discriminant} of~$A$, denoted $\disc(A)$ , is the determinant of Not a real error. [17] ! Preview: Snippet 18 started. <-><-> l.44 the $ n\times n$ matrix $(\tr(\omega_i\omega_j))$, which is well Not a real error. ! Preview: Snippet 18 ended.(418611+59802x1738566). <-><-> l.44 the $n\times n$ matrix $(\tr(\omega_i\omega_j))$, which is well Not a real error. [18] ! Preview: Snippet 19 started. <-><-> l.44 the $n\times n$ matrix $ (\tr(\omega_i\omega_j))$, which is well Not a real error. ! Preview: Snippet 19 ended.(538214+209587x3080505). <-><-> l.44 ...\times n$ matrix $(\tr(\omega_i\omega_j))$ , which is well Not a real error. [19] ! Preview: Snippet 20 started. <-><-> l.45 defined modulo squares of units in~$ A$. Not a real error. ! Preview: Snippet 20 ended.(490372+0x538215). <-><-> l.45 defined modulo squares of units in~$A$ . Not a real error. [20] ! Preview: Snippet 21 started. <-><-> l.47 When $ R=\Z$ the discriminant is well defined, since the only units are Not a real error. ! Preview: Snippet 21 ended.(494359+0x1985658). <-><-> l.47 When $R=\Z$ the discriminant is well defined, since the only units are Not a real error. [21] ! Preview: Snippet 22 started. <-><-> l.48 $ \pm 1$. Not a real error. ! Preview: Snippet 22 ended.(462465+59802x916960). <-><-> l.48 $\pm 1$ . Not a real error. [22] ! Preview: Snippet 23 started. <-><-> l.51 Suppose~$ R$ is a field. Then~$A$ has discriminant~$0$ if and only Not a real error. ! Preview: Snippet 23 ended.(490372+0x550424). <-><-> l.51 Suppose~$R$ is a field. Then~$A$ has discriminant~$0$ if and only Not a real error. [23] ! Preview: Snippet 24 started. <-><-> l.51 Suppose~$R$ is a field. Then~$ A$ has discriminant~$0$ if and only Not a real error. ! Preview: Snippet 24 ended.(490372+0x538215). <-><-> l.51 Suppose~$R$ is a field. Then~$A$ has discriminant~$0$ if and only Not a real error. [24] ! Preview: Snippet 25 started. <-><-> l.51 ...$ is a field. Then~$A$ has discriminant~$ 0$ if and only Not a real error. ! Preview: Snippet 25 ended.(462465+0x358810). <-><-> l.51 ...is a field. Then~$A$ has discriminant~$0$ if and only Not a real error. [25] ! Preview: Snippet 26 started. <-><-> l.52 if~$ A$ is separable over~$R$, i.e., for every extension $R'$ of $R$, Not a real error. ! Preview: Snippet 26 ended.(490372+0x538215). <-><-> l.52 if~$A$ is separable over~$R$, i.e., for every extension $R'$ of $R$, Not a real error. [26] ! Preview: Snippet 27 started. <-><-> l.52 if~$A$ is separable over~$ R$, i.e., for every extension $R'$ of $R$, Not a real error. ! Preview: Snippet 27 ended.(490372+0x550424). <-><-> l.52 if~$A$ is separable over~$R$ , i.e., for every extension $R'$ of $R$, Not a real error. [27] ! Preview: Snippet 28 started. <-><-> l.52 ...able over~$R$, i.e., for every extension $ R'$ of $R$, Not a real error. ! Preview: Snippet 28 ended.(551690+0x734289). <-><-> l.52 ...e over~$R$, i.e., for every extension $R'$ of $R$, Not a real error. [28] ! Preview: Snippet 29 started. <-><-> l.52 ...r~$R$, i.e., for every extension $R'$ of $ R$, Not a real error. ! Preview: Snippet 29 ended.(490372+0x550424). <-><-> l.52 ...$R$, i.e., for every extension $R'$ of $R$ , Not a real error. [29] ! Preview: Snippet 30 started. <-><-> l.53 the ring $ A\tensor R'$ contains no nilpotents. Not a real error. ! Preview: Snippet 30 ended.(551690+59802x2149590). <-><-> l.53 the ring $A\tensor R'$ contains no nilpotents. Not a real error. [30] ! Preview: Snippet 31 started. <-><-> l.56 If~$ A$ contains a nilpotent then that nilpotent is in the kernel of Not a real error. ! Preview: Snippet 31 ended.(490372+0x538215). <-><-> l.56 If~$A$ contains a nilpotent then that nilpotent is in the kernel of Not a real error. [31] ! Preview: Snippet 32 started. <-><-> l.57 the trace pairing. If~$ A$ is separable then we may assume that~$R$ is Not a real error. ! Preview: Snippet 32 ended.(490372+0x538215). <-><-> l.57 the trace pairing. If~$A$ is separable then we may assume that~$R$ is Not a real error. [32] ! Preview: Snippet 33 started. <-><-> l.57 ...$A$ is separable then we may assume that~$ R$ is Not a real error. ! Preview: Snippet 33 ended.(490372+0x550424). <-><-> l.57 ...$ is separable then we may assume that~$R$ is Not a real error. [33] ! Preview: Snippet 34 started. <-><-> l.58 algebraically closed. Then~$ A$ is an Artinian reduced ring, hence Not a real error. ! Preview: Snippet 34 ended.(490372+0x538215). <-><-> l.58 algebraically closed. Then~$A$ is an Artinian reduced ring, hence Not a real error. [34] ! Preview: Snippet 35 started. <-><-> l.59 ... a ring to a finite product of copies of~$ R$, since~$R$ Not a real error. ! Preview: Snippet 35 ended.(490372+0x550424). <-><-> l.59 ... ring to a finite product of copies of~$R$ , since~$R$ Not a real error. [35] ! Preview: Snippet 36 started. <-><-> l.59 ...a finite product of copies of~$R$, since~$ R$ Not a real error. ! Preview: Snippet 36 ended.(490372+0x550424). <-><-> l.59 ...finite product of copies of~$R$, since~$R$ Not a real error. [36] ! Preview: Snippet 37 started. <-><-> l.60 ...raically closed. Thus the trace form on~$ A$ is nondegenerate. Not a real error. ! Preview: Snippet 37 ended.(490372+0x538215). <-><-> l.60 ...ically closed. Thus the trace form on~$A$ is nondegenerate. Not a real error. [37] ! Preview: Snippet 38 started. <-><-> l.62 \subsection {The Discriminant Valuation} Not a real error. ! Preview: Snippet 38 ended.(546132+0x28311480). <-><-> l.62 \subsection{The Discriminant Valuation} Not a real error. [38 ] ! Preview: Snippet 39 started. <-><-> l.63 Let $ \Gamma$ be a congruence subgroup of $\SL_2(\Z)$, e.g., Not a real error. ! Preview: Snippet 39 ended.(490372+0x448513). <-><-> l.63 Let $\Gamma$ be a congruence subgroup of $\SL_2(\Z)$, e.g., Not a real error. [39] ! Preview: Snippet 40 started. <-><-> l.63 Let $\Gamma$ be a congruence subgroup of $ \SL_2(\Z)$, e.g., Not a real error. ! Preview: Snippet 40 ended.(538214+179404x2195055). <-><-> l.63 ...a$ be a congruence subgroup of $\SL_2(\Z)$ , e.g., Not a real error. [40] ! Preview: Snippet 41 started. <-><-> l.64 $ \Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$. For any integer $k\geq 1$, let Not a real error. ! Preview: Snippet 41 ended.(538214+179404x3084348). <-><-> l.64 $\Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$. For any integer $k\geq 1$, let Not a real error. [41] ! Preview: Snippet 42 started. <-><-> l.64 $\Gamma=\Gamma_0(p)$ or $ \Gamma_1(p)$. For any integer $k\geq 1$, let Not a real error. ! Preview: Snippet 42 ended.(538214+179404x1679015). <-><-> l.64 $\Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$ . For any integer $k\geq 1$, let Not a real error. [42] ! Preview: Snippet 43 started. <-><-> l.64 ...0(p)$ or $\Gamma_1(p)$. For any integer $ k\geq 1$, let Not a real error. ! Preview: Snippet 43 ended.(498346+97575x1711816). <-><-> l.64 ... $\Gamma_1(p)$. For any integer $k\geq 1$ , let Not a real error. [43] ! Preview: Snippet 44 started. <-><-> l.65 $ S_k(\Gamma)$ denote the space of holomorphic weight-$k$ cusp forms Not a real error. ! Preview: Snippet 44 ended.(538214+179404x1783489). <-><-> l.65 $S_k(\Gamma)$ denote the space of holomorphic weight-$k$ cusp forms Not a real error. [44] ! Preview: Snippet 45 started. <-><-> l.65 ...$ denote the space of holomorphic weight-$ k$ cusp forms Not a real error. ! Preview: Snippet 45 ended.(498346+0x396186). <-><-> l.65 ...denote the space of holomorphic weight-$k$ cusp forms Not a real error. [45] ! Preview: Snippet 46 started. <-><-> l.66 for $ \Gamma$. Let Not a real error. ! Preview: Snippet 46 ended.(490372+0x448513). <-><-> l.66 for $\Gamma$ . Let Not a real error. [46] ! Preview: Snippet 47 started. <-><-> l.67 $ $ Not a real error. ! Preview: Snippet 47 ended.(1070694+0x28311480). <-><-> l.69 $$ Not a real error. [47] ! Preview: Snippet 48 started. <-><-> l.70 be the associated Hecke algebra. Then~$ \T$ is a commutative ring Not a real error. ! Preview: Snippet 48 ended.(494359+0x478414). <-><-> l.70 be the associated Hecke algebra. Then~$\T$ is a commutative ring Not a real error. [48] ! Preview: Snippet 49 started. <-><-> l.71 that is free and of finite rank as a $ \Z$-module. Also of interest is Not a real error. ! Preview: Snippet 49 ended.(494359+0x478414). <-><-> l.71 that is free and of finite rank as a $\Z$ -module. Also of interest is Not a real error. [49] ! Preview: Snippet 50 started. <-><-> l.72 the image $ \T^{\new}$ of~$\T$ in $\End(S_k(\Gamma)^{\new})$. Not a real error. ! Preview: Snippet 50 ended.(494359+0x1478307). <-><-> l.72 the image $\T^{\new}$ of~$\T$ in $\End(S_k(\Gamma)^{\new})$. Not a real error. [50] ! Preview: Snippet 51 started. <-><-> l.72 the image $\T^{\new}$ of~$ \T$ in $\End(S_k(\Gamma)^{\new})$. Not a real error. ! Preview: Snippet 51 ended.(494359+0x478414). <-><-> l.72 the image $\T^{\new}$ of~$\T$ in $\End(S_k(\Gamma)^{\new})$. Not a real error. [51] ! Preview: Snippet 52 started. <-><-> l.72 the image $\T^{\new}$ of~$\T$ in $ \End(S_k(\Gamma)^{\new})$. Not a real error. ! Preview: Snippet 52 ended.(538214+179404x4627268). <-><-> l.72 ...ew}$ of~$\T$ in $\End(S_k(\Gamma)^{\new})$ . Not a real error. [52] ! Preview: Snippet 53 started. <-><-> l.74 Let $ \Gamma=\Gamma_0(243)$, which is illustrated on my T-shirt. Not a real error. ! Preview: Snippet 53 ended.(538214+179404x3799726). <-><-> l.74 Let $\Gamma=\Gamma_0(243)$ , which is illustrated on my T-shirt. Not a real error. [53] ! Preview: Snippet 54 started. <-><-> l.75 Since $ 243=3^5$, experts will immediately deduce that $\disc(\T) = Not a real error. ! Preview: Snippet 54 ended.(598293+0x2703360). <-><-> l.75 Since $243=3^5$ , experts will immediately deduce that $\disc(\T) = Not a real error. [54] ! Preview: Snippet 55 started. <-><-> l.75 ...5$, experts will immediately deduce that $ \disc(\T) = Not a real error. ! Preview: Snippet 55 ended.(538214+179404x3552214). <-><-> l.76 0$ . A computation shows that Not a real error. [55] ! Preview: Snippet 56 started. <-><-> l.77 $ $ Not a real error. ! Preview: Snippet 56 ended.(1070694+0x28311480). <-><-> l.79 $$ Not a real error. [56] ! Preview: Snippet 57 started. <-><-> l.80 which reflects the mod-$ 2$ and mod-$3$ intersections all over my Not a real error. ! Preview: Snippet 57 ended.(462465+0x358810). <-><-> l.80 which reflects the mod-$2$ and mod-$3$ intersections all over my Not a real error. [57] ! Preview: Snippet 58 started. <-><-> l.80 which reflects the mod-$2$ and mod-$ 3$ intersections all over my Not a real error. ! Preview: Snippet 58 ended.(462465+0x358810). <-><-> l.80 which reflects the mod-$2$ and mod-$3$ intersections all over my Not a real error. [58] ! Preview: Snippet 59 started. <-><-> l.86 Let~$ p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. ! Preview: Snippet 59 ended.(308974+139537x361052). <-><-> l.86 Let~$p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. [59] ! Preview: Snippet 60 started. <-><-> l.86 Let~$p$ be a prime and suppose that $ \Gamma=\Gamma_0(p)$ or Not a real error. ! Preview: Snippet 60 ended.(538214+179404x3084348). <-><-> l.86 ...rime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. [60] ! Preview: Snippet 61 started. <-><-> l.87 $ \Gamma_1(p)$. The {\em discriminant valuation} is Not a real error. ! Preview: Snippet 61 ended.(538214+179404x1679015). <-><-> l.87 $\Gamma_1(p)$ . The {\em discriminant valuation} is Not a real error. [61] ! Preview: Snippet 62 started. <-><-> l.88 $ $ Not a real error. ! Preview: Snippet 62 ended.(1100877+0x28311480). <-><-> l.90 $$ Not a real error. [62] ! Preview: Snippet 63 started. <-><-> l.95 \section {Motivation and Applications} Not a real error. ! Preview: Snippet 63 ended.(655359+183500x28311480). <-><-> l.95 \section{Motivation and Applications} Not a real error. [63 ] ! Preview: Snippet 64 started. <-><-> l.96 Let~$ p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. ! Preview: Snippet 64 ended.(308974+139537x361052). <-><-> l.96 Let~$p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. [64] ! Preview: Snippet 65 started. <-><-> l.96 Let~$p$ be a prime and suppose that $ \Gamma=\Gamma_0(p)$ or Not a real error. ! Preview: Snippet 65 ended.(538214+179404x3084348). <-><-> l.96 ...rime and suppose that $\Gamma=\Gamma_0(p)$ or Not a real error. [65] ! Preview: Snippet 66 started. <-><-> l.97 $ \Gamma_1(p)$. The quantity $d_k(\Gamma)$ is of interest because it Not a real error. ! Preview: Snippet 66 ended.(538214+179404x1679015). <-><-> l.97 $\Gamma_1(p)$ . The quantity $d_k(\Gamma)$ is of interest because it Not a real error. [66] ! Preview: Snippet 67 started. <-><-> l.97 $\Gamma_1(p)$. The quantity $ d_k(\Gamma)$ is of interest because it Not a real error. ! Preview: Snippet 67 ended.(538214+179404x1716960). <-><-> l.97 $\Gamma_1(p)$. The quantity $d_k(\Gamma)$ is of interest because it Not a real error. [67] ! Preview: Snippet 68 started. <-><-> l.98 measures mod~$ p$ congruences between eigenforms in $S_k(\Gamma)$. Not a real error. ! Preview: Snippet 68 ended.(308974+139537x361052). <-><-> l.98 measures mod~$p$ congruences between eigenforms in $S_k(\Gamma)$. Not a real error. [68] ! Preview: Snippet 69 started. <-><-> l.98 ...od~$p$ congruences between eigenforms in $ S_k(\Gamma)$. Not a real error. ! Preview: Snippet 69 ended.(538214+179404x1783489). <-><-> l.98 ...uences between eigenforms in $S_k(\Gamma)$ . Not a real error. [69] ! Preview: Snippet 70 started. <-><-> l.100 Suppose that $ d_k(\Gamma)$ is finite. Then the discriminant Not a real error. ! Preview: Snippet 70 ended.(538214+179404x1716960). <-><-> l.100 Suppose that $d_k(\Gamma)$ is finite. Then the discriminant Not a real error. [70] ! Preview: Snippet 71 started. <-><-> l.101 valuation $ d_k(\Gamma)$ is nonzero if and only if there is a mod-$p$ Not a real error. ! Preview: Snippet 71 ended.(538214+179404x1716960). <-><-> l.101 valuation $d_k(\Gamma)$ is nonzero if and only if there is a mod-$p$ Not a real error. [71] ! Preview: Snippet 72 started. <-><-> l.101 ...s nonzero if and only if there is a mod-$ p$ Not a real error. ! Preview: Snippet 72 ended.(308974+139537x361052). <-><-> l.101 ...nonzero if and only if there is a mod-$p$ Not a real error. [72] ! Preview: Snippet 73 started. <-><-> l.102 ...gruence between two Hecke eigenforms in $ S_k(\Gamma)$ (note that Not a real error. ! Preview: Snippet 73 ended.(538214+179404x1783489). <-><-> l.102 ...een two Hecke eigenforms in $S_k(\Gamma)$ (note that Not a real error. [73] LaTeX Warning: Reference `prop:separable' on page 1 undefined on input line 106 . ! Preview: Snippet 74 started. <-><-> l.107 $ d_k(\Gamma)>0$ if and only if $\T\tensor \Fpbar$ is not separable. Not a real error. ! Preview: Snippet 74 ended.(538214+179404x3032590). <-><-> l.107 $d_k(\Gamma)>0$ if and only if $\T\tensor \Fpbar$ is not separable. Not a real error. [74] ! Preview: Snippet 75 started. <-><-> l.107 $d_k(\Gamma)>0$ if and only if $ \T\tensor \Fpbar$ is not separable. Not a real error. ! Preview: Snippet 75 ended.(637879+209587x2107227). <-><-> l.107 ...mma)>0$ if and only if $\T\tensor \Fpbar$ is not separable. Not a real error. [75] ! Preview: Snippet 76 started. <-><-> l.108 The Artinian ring $ \T\tensor\Fpbar$ is not separable if and only if Not a real error. ! Preview: Snippet 76 ended.(637879+209587x2107227). <-><-> l.108 The Artinian ring $\T\tensor\Fpbar$ is not separable if and only if Not a real error. [76] ! Preview: Snippet 77 started. <-><-> l.109 the number of ring homomorphisms $ \T\tensor\Fpbar \ra \Fpbar$ is Not a real error. ! Preview: Snippet 77 ended.(637879+209587x3975245). <-><-> l.109 ...omomorphisms $\T\tensor\Fpbar \ra \Fpbar$ is Not a real error. [77] ! Preview: Snippet 78 started. <-><-> l.111 $ $ Not a real error. ! Preview: Snippet 78 ended.(1265247+0x28311480). <-><-> l.113 $$ Not a real error. [78] ! Preview: Snippet 79 started. <-><-> l.114 Since $ d_k(\Gamma)$ is finite, the number of ring homomorphisms Not a real error. ! Preview: Snippet 79 ended.(538214+179404x1716960). <-><-> l.114 Since $d_k(\Gamma)$ is finite, the number of ring homomorphisms Not a real error. [79] ! Preview: Snippet 80 started. <-><-> l.115 $ \T\tensor\Qpbar \ra \Qpbar$ equals $\dim_\C S_k(\Gamma)$. Using Not a real error. ! Preview: Snippet 80 ended.(637879+254316x4214453). <-><-> l.115 $\T\tensor\Qpbar \ra \Qpbar$ equals $\dim_\C S_k(\Gamma)$. Using Not a real error. [80] ! Preview: Snippet 81 started. <-><-> l.115 $\T\tensor\Qpbar \ra \Qpbar$ equals $ \dim_\C S_k(\Gamma)$. Using Not a real error. ! Preview: Snippet 81 ended.(538214+179404x3510543). <-><-> l.115 ... \ra \Qpbar$ equals $\dim_\C S_k(\Gamma)$ . Using Not a real error. [81] ! Preview: Snippet 82 started. <-><-> l.117 eigenforms, we see that $ \T\tensor\Fpbar$ is not separable if and Not a real error. ! Preview: Snippet 82 ended.(637879+209587x2107227). <-><-> l.117 eigenforms, we see that $\T\tensor\Fpbar$ is not separable if and Not a real error. [82] ! Preview: Snippet 83 started. <-><-> l.118 only if there is a mod-$ p$ congruence between two eigenforms. Not a real error. ! Preview: Snippet 83 ended.(308974+139537x361052). <-><-> l.118 only if there is a mod-$p$ congruence between two eigenforms. Not a real error. [83] ! Preview: Snippet 84 started. <-><-> l.122 If $ \Gamma=\Gamma_0(389)$ and $k=2$, then $\dim_\C S_2(\Gamma) = Not a real error. ! Preview: Snippet 84 ended.(538214+179404x3799726). <-><-> l.122 If $\Gamma=\Gamma_0(389)$ and $k=2$, then $\dim_\C S_2(\Gamma) = Not a real error. [84] ! Preview: Snippet 85 started. <-><-> l.122 If $\Gamma=\Gamma_0(389)$ and $ k=2$, then $\dim_\C S_2(\Gamma) = Not a real error. ! Preview: Snippet 85 ended.(498346+0x1711816). <-><-> l.122 If $\Gamma=\Gamma_0(389)$ and $k=2$ , then $\dim_\C S_2(\Gamma) = Not a real error. [85] ! Preview: Snippet 86 started. <-><-> l.122 ... $\Gamma=\Gamma_0(389)$ and $k=2$, then $ \dim_\C S_2(\Gamma) = Not a real error. ! Preview: Snippet 86 ended.(538214+179404x5159497). <-><-> l.123 32$ . Let~$f$ be the characteristic polynomial of $T_2$. One can Not a real error. [86] ! Preview: Snippet 87 started. <-><-> l.123 32$. Let~$ f$ be the characteristic polynomial of $T_2$. One can Not a real error. ! Preview: Snippet 87 ended.(498346+139537x428580). <-><-> l.123 32$. Let~$f$ be the characteristic polynomial of $T_2$. One can Not a real error. [87] ! Preview: Snippet 88 started. <-><-> l.123 ...$f$ be the characteristic polynomial of $ T_2$. One can Not a real error. ! Preview: Snippet 88 ended.(490372+107642x730659). <-><-> l.123 ...be the characteristic polynomial of $T_2$ . One can Not a real error. [88] ! Preview: Snippet 89 started. <-><-> l.124 check that~$ f$ is square free and $389$ exactly divides the Not a real error. ! Preview: Snippet 89 ended.(498346+139537x428580). <-><-> l.124 check that~$f$ is square free and $389$ exactly divides the Not a real error. [89] ! Preview: Snippet 90 started. <-><-> l.124 check that~$f$ is square free and $ 389$ exactly divides the Not a real error. ! Preview: Snippet 90 ended.(462465+0x1076430). <-><-> l.124 check that~$f$ is square free and $389$ exactly divides the Not a real error. [90] ! Preview: Snippet 91 started. <-><-> l.125 discriminant of~$ f$, so $T_2$ generated $\T\tensor \Z_{389}$ as a Not a real error. ! Preview: Snippet 91 ended.(498346+139537x428580). <-><-> l.125 discriminant of~$f$ , so $T_2$ generated $\T\tensor \Z_{389}$ as a Not a real error. [91] ! Preview: Snippet 92 started. <-><-> l.125 discriminant of~$f$, so $ T_2$ generated $\T\tensor \Z_{389}$ as a Not a real error. ! Preview: Snippet 92 ended.(490372+107642x730659). <-><-> l.125 discriminant of~$f$, so $T_2$ generated $\T\tensor \Z_{389}$ as a Not a real error. [92] ! Preview: Snippet 93 started. <-><-> l.125 discriminant of~$f$, so $T_2$ generated $ \T\tensor \Z_{389}$ as a Not a real error. ! Preview: Snippet 93 ended.(494359+107642x2702278). <-><-> l.125 ..., so $T_2$ generated $\T\tensor \Z_{389}$ as a Not a real error. [93] ! Preview: Snippet 94 started. <-><-> l.126 ring. (If it generated a subring of $ \T\tensor\Z_{389}$ of finite Not a real error. ! Preview: Snippet 94 ended.(494359+107642x2702278). <-><-> l.126 ...enerated a subring of $\T\tensor\Z_{389}$ of finite Not a real error. [94] ! Preview: Snippet 95 started. <-><-> l.127 index, then the discriminant of~$ f$ would be divisible by $389^2$.) Not a real error. ! Preview: Snippet 95 ended.(498346+139537x428580). <-><-> l.127 index, then the discriminant of~$f$ would be divisible by $389^2$.) Not a real error. [95] ! Preview: Snippet 96 started. <-><-> l.127 ...scriminant of~$f$ would be divisible by $ 389^2$.) Not a real error. ! Preview: Snippet 96 ended.(598293+0x1387730). <-><-> l.127 ...nant of~$f$ would be divisible by $389^2$ .) Not a real error. [96] ! Preview: Snippet 97 started. <-><-> l.129 Modulo~$ 389$ the polynomial~$f$ is congruent to Not a real error. ! Preview: Snippet 97 ended.(462465+0x1076430). <-><-> l.129 Modulo~$389$ the polynomial~$f$ is congruent to Not a real error. [97] ! Preview: Snippet 98 started. <-><-> l.129 Modulo~$389$ the polynomial~$ f$ is congruent to Not a real error. ! Preview: Snippet 98 ended.(498346+139537x428580). <-><-> l.129 Modulo~$389$ the polynomial~$f$ is congruent to Not a real error. [98] ! Preview: Snippet 99 started. <-><-> l.130 $ $\begin{array}{l} Not a real error. ! Preview: Snippet 99 ended.(3630692+0x28311480). <-><-> l.136 $$ Not a real error. [99] ! Preview: Snippet 100 started. <-><-> l.137 The factor $ (x+175)^2$ indicates that $\T\tensor \Fbar_{389}$ is Not a real error. ! Preview: Snippet 100 ended.(598293+179404x3233105). <-><-> l.137 The factor $(x+175)^2$ indicates that $\T\tensor \Fbar_{389}$ is Not a real error. [100] ! Preview: Snippet 101 started. <-><-> l.137 The factor $(x+175)^2$ indicates that $ \T\tensor \Fbar_{389}$ is Not a real error. ! Preview: Snippet 101 ended.(637879+107642x2662410). <-><-> l.137 ...2$ indicates that $\T\tensor \Fbar_{389}$ is Not a real error. [101] ! Preview: Snippet 102 started. <-><-> l.138 not separable since the image of $ T_2+175$ is nilpotent (its square Not a real error. ! Preview: Snippet 102 ended.(490372+107642x2684175). <-><-> l.138 not separable since the image of $T_2+175$ is nilpotent (its square Not a real error. [102] ! Preview: Snippet 103 started. <-><-> l.139 is~$ 0$). There are $32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. ! Preview: Snippet 103 ended.(462465+0x358810). <-><-> l.139 is~$0$ ). There are $32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. [103] ! Preview: Snippet 104 started. <-><-> l.139 is~$0$). There are $ 32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. ! Preview: Snippet 104 ended.(462465+0x717620). <-><-> l.139 is~$0$). There are $32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. [104] ! Preview: Snippet 105 started. <-><-> l.139 is~$0$). There are $32$ eigenforms over~$ \Q_2$ but only $31$ Not a real error. ! Preview: Snippet 105 ended.(494359+119603x869450). <-><-> l.139 ...). There are $32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. [105] ! Preview: Snippet 106 started. <-><-> l.139 ...re $32$ eigenforms over~$\Q_2$ but only $ 31$ Not a real error. ! Preview: Snippet 106 ended.(462465+0x717620). <-><-> l.139 ...$32$ eigenforms over~$\Q_2$ but only $31$ Not a real error. [106] ! Preview: Snippet 107 started. <-><-> l.140 mod-$ 389$ eigenforms, so there must be a congruence. Let~$F$ be the Not a real error. ! Preview: Snippet 107 ended.(462465+0x1076430). <-><-> l.140 mod-$389$ eigenforms, so there must be a congruence. Let~$F$ be the Not a real error. [107] ! Preview: Snippet 108 started. <-><-> l.140 ...ms, so there must be a congruence. Let~$ F$ be the Not a real error. ! Preview: Snippet 108 ended.(490372+0x561139). <-><-> l.140 ..., so there must be a congruence. Let~$F$ be the Not a real error. [108] ! Preview: Snippet 109 started. <-><-> l.141 $ 389$-adic newform whose $a_2$ term is a root of Not a real error. ! Preview: Snippet 109 ended.(462465+0x1076430). <-><-> l.141 $389$ -adic newform whose $a_2$ term is a root of Not a real error. [109] ! Preview: Snippet 110 started. <-><-> l.141 $389$-adic newform whose $ a_2$ term is a root of Not a real error. ! Preview: Snippet 110 ended.(308974+107642x690626). <-><-> l.141 $389$-adic newform whose $a_2$ term is a root of Not a real error. [110] ! Preview: Snippet 111 started. <-><-> l.142 $ $ Not a real error. ! Preview: Snippet 111 ended.(1070694+0x28311480). <-><-> l.145 $$ Not a real error. [111] ! Preview: Snippet 112 started. <-><-> l.146 Then the congruence is between~$ F$ and its Not a real error. ! Preview: Snippet 112 ended.(490372+0x561139). <-><-> l.146 Then the congruence is between~$F$ and its Not a real error. [112] ! Preview: Snippet 113 started. <-><-> l.147 $ \Gal(\Qbar_{389}/\Q_{389})$-conjugate. Not a real error. ! Preview: Snippet 113 ended.(637879+179404x4891270). <-><-> l.147 $\Gal(\Qbar_{389}/\Q_{389})$ -conjugate. Not a real error. [113] ! Preview: Snippet 114 started. <-><-> l.151 The discriminant of the Hecke algebra $ \T$ associated to Not a real error. ! Preview: Snippet 114 ended.(494359+0x478414). <-><-> l.151 The discriminant of the Hecke algebra $\T$ associated to Not a real error. [114] ! Preview: Snippet 115 started. <-><-> l.152 $ S_2(\Gamma_0(389))$ is Not a real error. ! Preview: Snippet 115 ended.(538214+179404x3703883). <-><-> l.152 $S_2(\Gamma_0(389))$ is Not a real error. [115] ! Preview: Snippet 116 started. <-><-> l.153 $ $ Not a real error. ! Preview: Snippet 116 ended.(891290+0x28311480). <-><-> l.157 $$ Not a real error. [116] ! Preview: Snippet 117 started. <-><-> l.159 ...nstra. Using the Sturm bound I found a~$ b$ Not a real error. ! Preview: Snippet 117 ended.(498346+0x307978). <-><-> l.159 ...tra. Using the Sturm bound I found a~$b$ Not a real error. [117] ! Preview: Snippet 118 started. <-><-> l.160 such that $ T_1,\ldots,T_b$ generate $\T$ as a $\Z$-module. I then Not a real error. ! Preview: Snippet 118 ended.(490372+139537x3015786). <-><-> l.160 such that $T_1,\ldots,T_b$ generate $\T$ as a $\Z$-module. I then Not a real error. [118] ! Preview: Snippet 119 started. <-><-> l.160 such that $T_1,\ldots,T_b$ generate $ \T$ as a $\Z$-module. I then Not a real error. ! Preview: Snippet 119 ended.(494359+0x478414). <-><-> l.160 such that $T_1,\ldots,T_b$ generate $\T$ as a $\Z$-module. I then Not a real error. [119] ! Preview: Snippet 120 started. <-><-> l.160 ...hat $T_1,\ldots,T_b$ generate $\T$ as a $ \Z$-module. I then Not a real error. ! Preview: Snippet 120 ended.(494359+0x478414). <-><-> l.160 ... $T_1,\ldots,T_b$ generate $\T$ as a $\Z$ -module. I then Not a real error. [120] ! Preview: Snippet 121 started. <-><-> l.161 found a subset~$ B$ of the $T_i$ that form a $\Q$-basis for Not a real error. ! Preview: Snippet 121 ended.(490372+0x580324). <-><-> l.161 found a subset~$B$ of the $T_i$ that form a $\Q$-basis for Not a real error. [121] ! Preview: Snippet 122 started. <-><-> l.161 found a subset~$B$ of the $ T_i$ that form a $\Q$-basis for Not a real error. ! Preview: Snippet 122 ended.(490372+107642x641785). <-><-> l.161 found a subset~$B$ of the $T_i$ that form a $\Q$-basis for Not a real error. [122] ! Preview: Snippet 123 started. <-><-> l.161 ...d a subset~$B$ of the $T_i$ that form a $ \Q$-basis for Not a real error. ! Preview: Snippet 123 ended.(494359+119603x558150). <-><-> l.161 ... subset~$B$ of the $T_i$ that form a $\Q$ -basis for Not a real error. [123] ! Preview: Snippet 124 started. <-><-> l.162 $ \T\tensor_\Z\Q$. Next, viewing $\T$ as a ring of matrices acting Not a real error. ! Preview: Snippet 124 ended.(494359+119603x2295942). <-><-> l.162 $\T\tensor_\Z\Q$ . Next, viewing $\T$ as a ring of matrices acting Not a real error. [124] ! Preview: Snippet 125 started. <-><-> l.162 $\T\tensor_\Z\Q$. Next, viewing $ \T$ as a ring of matrices acting Not a real error. ! Preview: Snippet 125 ended.(494359+0x478414). <-><-> l.162 $\T\tensor_\Z\Q$. Next, viewing $\T$ as a ring of matrices acting Not a real error. [125] ! Preview: Snippet 126 started. <-><-> l.163 on $ \Q^{32}$, I found a random vector $v\in\Q^{32}$ such that the Not a real error. ! Preview: Snippet 126 ended.(598293+119603x1147982). <-><-> l.163 on $\Q^{32}$ , I found a random vector $v\in\Q^{32}$ such that the Not a real error. [126] ! Preview: Snippet 127 started. <-><-> l.163 on $\Q^{32}$, I found a random vector $ v\in\Q^{32}$ such that the Not a real error. ! Preview: Snippet 127 ended.(598293+119603x2398660). <-><-> l.163 ...}$, I found a random vector $v\in\Q^{32}$ such that the Not a real error. [127] ! Preview: Snippet 128 started. <-><-> l.164 set of vectors $ C=\{T(v) : T \in B\}$ is linearly independent. Then Not a real error. ! Preview: Snippet 128 ended.(538214+179404x6263885). <-><-> l.164 set of vectors $C=\{T(v) : T \in B\}$ is linearly independent. Then Not a real error. [128] ! Preview: Snippet 129 started. <-><-> l.165 I wrote each of $ T_1(v),\ldots, T_b(v)$ as $\Q$-linear combinations Not a real error. ! Preview: Snippet 129 ended.(538214+179404x4879274). <-><-> l.165 I wrote each of $T_1(v),\ldots, T_b(v)$ as $\Q$-linear combinations Not a real error. [129] ! Preview: Snippet 130 started. <-><-> l.165 ...rote each of $T_1(v),\ldots, T_b(v)$ as $ \Q$-linear combinations Not a real error. ! Preview: Snippet 130 ended.(494359+119603x558150). <-><-> l.165 ...e each of $T_1(v),\ldots, T_b(v)$ as $\Q$ -linear combinations Not a real error. [130] ! Preview: Snippet 131 started. <-><-> l.166 of the elements of~$ C$. Next I found a $\Z$-basis~$D$ for the Not a real error. ! Preview: Snippet 131 ended.(490372+0x564226). <-><-> l.166 of the elements of~$C$ . Next I found a $\Z$-basis~$D$ for the Not a real error. [131] ! Preview: Snippet 132 started. <-><-> l.166 of the elements of~$C$. Next I found a $ \Z$-basis~$D$ for the Not a real error. ! Preview: Snippet 132 ended.(494359+0x478414). <-><-> l.166 ...the elements of~$C$. Next I found a $\Z$ -basis~$D$ for the Not a real error. [132] ! Preview: Snippet 133 started. <-><-> l.166 ...ents of~$C$. Next I found a $\Z$-basis~$ D$ for the Not a real error. ! Preview: Snippet 133 ended.(490372+0x614063). <-><-> l.166 ...ts of~$C$. Next I found a $\Z$-basis~$D$ for the Not a real error. [133] ! Preview: Snippet 134 started. <-><-> l.167 $ \Z$-span of these $\Q$-linear combinations of elements of~$C$. Not a real error. ! Preview: Snippet 134 ended.(494359+0x478414). <-><-> l.167 $\Z$ -span of these $\Q$-linear combinations of elements of~$C$. Not a real error. [134] ! Preview: Snippet 135 started. <-><-> l.167 $\Z$-span of these $ \Q$-linear combinations of elements of~$C$. Not a real error. ! Preview: Snippet 135 ended.(494359+119603x558150). <-><-> l.167 $\Z$-span of these $\Q$ -linear combinations of elements of~$C$. Not a real error. [135] ! Preview: Snippet 136 started. <-><-> l.167 ...$\Q$-linear combinations of elements of~$ C$. Not a real error. ! Preview: Snippet 136 ended.(490372+0x564226). <-><-> l.167 ...Q$-linear combinations of elements of~$C$ . Not a real error. [136] ! Preview: Snippet 137 started. <-><-> l.169 of~$ D$, and deduce the discriminant by computing the determinant of Not a real error. ! Preview: Snippet 137 ended.(490372+0x614063). <-><-> l.169 of~$D$ , and deduce the discriminant by computing the determinant of Not a real error. [137] ! Preview: Snippet 138 started. <-><-> l.170 .... The most difficult step is computing~$ D$ Not a real error. ! Preview: Snippet 138 ended.(490372+0x614063). <-><-> l.170 ... The most difficult step is computing~$D$ Not a real error. [138] ! Preview: Snippet 139 started. <-><-> l.171 from $ T_1(v),\ldots,T_b(v)$ expressed in terms of~$C$, and this Not a real error. ! Preview: Snippet 139 ended.(538214+179404x4879274). <-><-> l.171 from $T_1(v),\ldots,T_b(v)$ expressed in terms of~$C$, and this Not a real error. [139] ! Preview: Snippet 140 started. <-><-> l.171 ...v),\ldots,T_b(v)$ expressed in terms of~$ C$, and this Not a real error. ! Preview: Snippet 140 ended.(490372+0x564226). <-><-> l.171 ...,\ldots,T_b(v)$ expressed in terms of~$C$ , and this Not a real error. [140] ! Preview: Snippet 141 started. <-><-> l.172 explains why we embed $ \T$ in $\Q^{32}$ instead of viewing the Not a real error. ! Preview: Snippet 141 ended.(494359+0x478414). <-><-> l.172 explains why we embed $\T$ in $\Q^{32}$ instead of viewing the Not a real error. [141] ! Preview: Snippet 142 started. <-><-> l.172 explains why we embed $\T$ in $ \Q^{32}$ instead of viewing the Not a real error. ! Preview: Snippet 142 ended.(598293+119603x1147982). <-><-> l.172 explains why we embed $\T$ in $\Q^{32}$ instead of viewing the Not a real error. [142] ! Preview: Snippet 143 started. <-><-> l.173 elements of $ \T$ as vectors in $\Q^{32^2}$. This whole computation Not a real error. ! Preview: Snippet 143 ended.(494359+0x478414). <-><-> l.173 elements of $\T$ as vectors in $\Q^{32^2}$. This whole computation Not a real error. [143] ! Preview: Snippet 144 started. <-><-> l.173 elements of $\T$ as vectors in $ \Q^{32^2}$. This whole computation Not a real error. ! Preview: Snippet 144 ended.(698853+119603x1421045). <-><-> l.173 elements of $\T$ as vectors in $\Q^{32^2}$ . This whole computation Not a real error. [144] ! Preview: Snippet 145 started. <-><-> l.177 \subsection {Literature} Not a real error. ! Preview: Snippet 145 ended.(546132+0x28311480). <-><-> l.177 \subsection{Literature} Not a real error. [145 ] LaTeX Warning: Reference `thm:disc' on page 1 undefined on input line 178. ! Preview: Snippet 146 started. <-><-> l.181 \item Ribet: {\em Torsion points on $ J_0(N)$ and Galois Not a real error. ! Preview: Snippet 146 ended.(538214+179404x1922205). <-><-> l.181 \item Ribet: {\em Torsion points on $J_0(N)$ and Galois Not a real error. [146] ! Preview: Snippet 147 started. <-><-> l.189 $ X_{\split}(p)$} Not a real error. ! Preview: Snippet 147 ended.(538214+218855x2601825). <-><-> l.189 $X_{\split}(p)$ } Not a real error. [147] ! Preview: Snippet 148 started. <-><-> l.193 \section {Data About Discriminant Valuations} Not a real error. ! Preview: Snippet 148 ended.(655359+0x28311480). <-><-> l.193 \section{Data About Discriminant Valuations} Not a real error. [148 ] ! Preview: Snippet 149 started. <-><-> l.195 \subsection {Weight Two} Not a real error. ! Preview: Snippet 149 ended.(546132+152916x28311480). <-><-> l.195 \subsection{Weight Two} Not a real error. [149 ] ! Preview: Snippet 150 started. <-><-> l.197 The only prime $ p<60000$ such that $d_2(\Gamma_0(p))>0$ is $p=389$. Not a real error. ! Preview: Snippet 150 ended.(462465+139537x3111922). <-><-> l.197 The only prime $p<60000$ such that $d_2(\Gamma_0(p))>0$ is $p=389$. Not a real error. [150] ! Preview: Snippet 151 started. <-><-> l.197 The only prime $p<60000$ such that $ d_2(\Gamma_0(p))>0$ is $p=389$. Not a real error. ! Preview: Snippet 151 ended.(538214+179404x4237606). <-><-> l.197 ... $p<60000$ such that $d_2(\Gamma_0(p))>0$ is $p=389$. Not a real error. [151] ! Preview: Snippet 152 started. <-><-> l.197 ...0000$ such that $d_2(\Gamma_0(p))>0$ is $ p=389$. Not a real error. ! Preview: Snippet 152 ended.(462465+139537x2394302). <-><-> l.197 ...such that $d_2(\Gamma_0(p))>0$ is $p=389$ . Not a real error. [152] ! Preview: Snippet 153 started. <-><-> l.198 (Except possibly $ 50923$ and $51437$, which I haven't finished Not a real error. ! Preview: Snippet 153 ended.(462465+0x1794050). <-><-> l.198 (Except possibly $50923$ and $51437$, which I haven't finished Not a real error. [153] ! Preview: Snippet 154 started. <-><-> l.198 (Except possibly $50923$ and $ 51437$, which I haven't finished Not a real error. ! Preview: Snippet 154 ended.(462465+0x1794050). <-><-> l.198 (Except possibly $50923$ and $51437$ , which I haven't finished Not a real error. [154] ! Preview: Snippet 155 started. <-><-> l.204 theorems about $ d_2(\Gamma_0(p))$. The rest of this proof describes Not a real error. ! Preview: Snippet 155 ended.(538214+179404x2921976). <-><-> l.204 theorems about $d_2(\Gamma_0(p))$ . The rest of this proof describes Not a real error. [155] ! Preview: Snippet 156 started. <-><-> l.208 below took about one week using $ 12$ Athlon 2000MP processors. In Not a real error. ! Preview: Snippet 156 ended.(462465+0x717620). <-><-> l.208 below took about one week using $12$ Athlon 2000MP processors. In Not a real error. [156] ! Preview: Snippet 157 started. <-><-> l.209 ...ed the result stated above but only for $ p<14000$ Not a real error. ! Preview: Snippet 157 ended.(462465+139537x3111922). <-><-> l.209 ...esult stated above but only for $p<14000$ Not a real error. [157] ! Preview: Snippet 158 started. <-><-> l.213 operators on spaces of dimensions up to~$ 5000$. Not a real error. ! Preview: Snippet 158 ended.(462465+0x1435240). <-><-> l.213 ...tors on spaces of dimensions up to~$5000$ . Not a real error. [158] ! Preview: Snippet 159 started. <-><-> l.215 The aim is to determine whether or not~$ p$ divides the discriminant Not a real error. ! Preview: Snippet 159 ended.(308974+139537x361052). <-><-> l.215 The aim is to determine whether or not~$p$ divides the discriminant Not a real error. [159] ! Preview: Snippet 160 started. <-><-> l.216 of the Hecke algegra of level~$ p$ for each $p < 60000$. If~$T$ is Not a real error. ! Preview: Snippet 160 ended.(308974+139537x361052). <-><-> l.216 of the Hecke algegra of level~$p$ for each $p < 60000$. If~$T$ is Not a real error. [160] ! Preview: Snippet 161 started. <-><-> l.216 ...the Hecke algegra of level~$p$ for each $ p < 60000$. If~$T$ is Not a real error. ! Preview: Snippet 161 ended.(462465+139537x3111922). <-><-> l.216 ...algegra of level~$p$ for each $p < 60000$ . If~$T$ is Not a real error. [161] ! Preview: Snippet 162 started. <-><-> l.216 ... of level~$p$ for each $p < 60000$. If~$ T$ is Not a real error. ! Preview: Snippet 162 ended.(490372+0x519029). <-><-> l.216 ...f level~$p$ for each $p < 60000$. If~$T$ is Not a real error. [162] ! Preview: Snippet 163 started. <-><-> l.218 $ \disc(T)$ for $\disc(\charpoly(T))$, which also equals Not a real error. ! Preview: Snippet 163 ended.(538214+179404x2277199). <-><-> l.218 $\disc(T)$ for $\disc(\charpoly(T))$, which also equals Not a real error. [163] ! Preview: Snippet 164 started. <-><-> l.218 $\disc(T)$ for $ \disc(\charpoly(T))$, which also equals Not a real error. ! Preview: Snippet 164 ended.(538214+179404x5538384). <-><-> l.218 $\disc(T)$ for $\disc(\charpoly(T))$ , which also equals Not a real error. [164] ! Preview: Snippet 165 started. <-><-> l.219 $ \disc(\Z[T])$. We will often use that Not a real error. ! Preview: Snippet 165 ended.(538214+179404x3154291). <-><-> l.219 $\disc(\Z[T])$ . We will often use that Not a real error. [165] ! Preview: Snippet 166 started. <-><-> l.220 $ $\disc(T)\!\!\!\!\mod{p} = \disc(\charpoly(T)\!\!\!\!\mod p).$$ Not a real error. ! Preview: Snippet 166 ended.(1070694+0x28311480). <-><-> l.220 ...p} = \disc(\charpoly(T)\!\!\!\!\mod p).$$ Not a real error. [166] ! Preview: Snippet 167 started. <-><-> l.222 Most levels~$ p<60000$ were ruled out by computing characteristic Not a real error. ! Preview: Snippet 167 ended.(462465+139537x3111922). <-><-> l.222 Most levels~$p<60000$ were ruled out by computing characteristic Not a real error. [167] ! Preview: Snippet 168 started. <-><-> l.227 $ \disc(T_q)$ modulo~$p$ for several primes~$q$, and in most cases Not a real error. ! Preview: Snippet 168 ended.(538214+209587x2477815). <-><-> l.227 $\disc(T_q)$ modulo~$p$ for several primes~$q$, and in most cases Not a real error. [168] ! Preview: Snippet 169 started. <-><-> l.227 $\disc(T_q)$ modulo~$ p$ for several primes~$q$, and in most cases Not a real error. ! Preview: Snippet 169 ended.(308974+139537x361052). <-><-> l.227 $\disc(T_q)$ modulo~$p$ for several primes~$q$, and in most cases Not a real error. [169] ! Preview: Snippet 170 started. <-><-> l.227 ...isc(T_q)$ modulo~$p$ for several primes~$ q$, and in most cases Not a real error. ! Preview: Snippet 170 ended.(308974+139537x346102). <-><-> l.227 ...c(T_q)$ modulo~$p$ for several primes~$q$ , and in most cases Not a real error. [170] ! Preview: Snippet 171 started. <-><-> l.228 found a~$ q$ such that this discriminant is nonzero. The following Not a real error. ! Preview: Snippet 171 ended.(308974+139537x346102). <-><-> l.228 found a~$q$ such that this discriminant is nonzero. The following Not a real error. [171] ! Preview: Snippet 172 started. <-><-> l.229 ...summarizes how often we used each prime~$ q$ (note that there Not a real error. ! Preview: Snippet 172 ended.(308974+139537x346102). <-><-> l.229 ...mmarizes how often we used each prime~$q$ (note that there Not a real error. [172] ! Preview: Snippet 173 started. <-><-> l.230 are $ 6057$ primes up to $60000$): Not a real error. ! Preview: Snippet 173 ended.(462465+0x1435240). <-><-> l.230 are $6057$ primes up to $60000$): Not a real error. [173] ! Preview: Snippet 174 started. <-><-> l.230 are $6057$ primes up to $ 60000$): Not a real error. ! Preview: Snippet 174 ended.(462465+0x1794050). <-><-> l.230 are $6057$ primes up to $60000$ ): Not a real error. [174] ! Preview: Snippet 175 started. <-><-> l.233 $ q$ & number of $p< 60000$ where~$q$ smallest Not a real error. ! Preview: Snippet 175 ended.(308974+139537x346102). <-><-> l.233 $q$ & number of $p< 60000$ where~$q$ smallest Not a real error. [175] ! Preview: Snippet 176 started. <-><-> l.233 $q$ & number of $ p< 60000$ where~$q$ smallest Not a real error. ! Preview: Snippet 176 ended.(462465+139537x3111922). <-><-> l.233 $q$ & number of $p< 60000$ where~$q$ smallest Not a real error. [176] ! Preview: Snippet 177 started. <-><-> l.233 $q$ & number of $p< 60000$ where~$ q$ smallest Not a real error. ! Preview: Snippet 177 ended.(308974+139537x346102). <-><-> l.233 $q$ & number of $p< 60000$ where~$q$ smallest Not a real error. [177] ! Preview: Snippet 178 started. <-><-> l.234 s.t. $ \disc(T_q)\neq 0$ mod~$p$\\\hline Not a real error. ! Preview: Snippet 178 ended.(538214+209587x3793445). <-><-> l.234 s.t. $\disc(T_q)\neq 0$ mod~$p$\\\hline Not a real error. [178] ! Preview: Snippet 179 started. <-><-> l.234 s.t. $\disc(T_q)\neq 0$ mod~$ p$\\\hline Not a real error. ! Preview: Snippet 179 ended.(308974+139537x361052). <-><-> l.234 s.t. $\disc(T_q)\neq 0$ mod~$p$ \\\hline Not a real error. [179] ! Preview: Snippet 180 started. <-><-> l.248 $ $ Not a real error. ! Preview: Snippet 180 ended.(1030827+0x28311480). <-><-> l.250 $$ Not a real error. [180] ! Preview: Snippet 181 started. <-><-> l.251 (The last two are still being processed. $ 51437$ has the property Not a real error. ! Preview: Snippet 181 ended.(462465+0x1794050). <-><-> l.251 ...t two are still being processed. $51437$ has the property Not a real error. [181] ! Preview: Snippet 182 started. <-><-> l.252 that $ \disc(T_q)=0$ for $q=2,3,\ldots,17$.) We determined the Not a real error. ! Preview: Snippet 182 ended.(538214+209587x3793445). <-><-> l.252 that $\disc(T_q)=0$ for $q=2,3,\ldots,17$.) We determined the Not a real error. [182] ! Preview: Snippet 183 started. <-><-> l.252 that $\disc(T_q)=0$ for $ q=2,3,\ldots,17$.) We determined the Not a real error. ! Preview: Snippet 183 ended.(462465+139537x4651802). <-><-> l.252 that $\disc(T_q)=0$ for $q=2,3,\ldots,17$ .) We determined the Not a real error. [183] ! Preview: Snippet 184 started. <-><-> l.253 ...emaining 6 levels using Hecke operators $ T_n$ Not a real error. ! Preview: Snippet 184 ended.(490372+107642x790127). <-><-> l.253 ...ning 6 levels using Hecke operators $T_n$ Not a real error. [184] ! Preview: Snippet 185 started. <-><-> l.254 with~$ n$ composite. Not a real error. ! Preview: Snippet 185 ended.(308974+0x430740). <-><-> l.254 with~$n$ composite. Not a real error. [185] ! Preview: Snippet 186 started. <-><-> l.257 $ p$ & How we rule level~$p$ out, if possible\\\hline Not a real error. ! Preview: Snippet 186 ended.(308974+139537x361052). <-><-> l.257 $p$ & How we rule level~$p$ out, if possible\\\hline Not a real error. [186] ! Preview: Snippet 187 started. <-><-> l.257 $p$ & How we rule level~$ p$ out, if possible\\\hline Not a real error. ! Preview: Snippet 187 ended.(308974+139537x361052). <-><-> l.257 $p$ & How we rule level~$p$ out, if possible\\\hline Not a real error. [187] ! Preview: Snippet 188 started. <-><-> l.258 389& $ p$ does divide discriminant\\ Not a real error. ! Preview: Snippet 188 ended.(308974+139537x361052). <-><-> l.258 389& $p$ does divide discriminant\\ Not a real error. [188] ! Preview: Snippet 189 started. <-><-> l.259 487& using charpoly($ T_{12}$)\\ Not a real error. ! Preview: Snippet 189 ended.(490372+107642x1009191). <-><-> l.259 487& using charpoly($T_{12}$ )\\ Not a real error. [189] ! Preview: Snippet 190 started. <-><-> l.260 2341& using charpoly($ T_6$)\\ Not a real error. ! Preview: Snippet 190 ended.(490372+107642x730659). <-><-> l.260 2341& using charpoly($T_6$ )\\ Not a real error. [190] ! Preview: Snippet 191 started. <-><-> l.261 7057& using charpoly($ T_{18}$)\\ Not a real error. ! Preview: Snippet 191 ended.(490372+107642x1009191). <-><-> l.261 7057& using charpoly($T_{18}$ )\\ Not a real error. [191] ! Preview: Snippet 192 started. <-><-> l.262 15641& using charpoly($ T_6$)\\ Not a real error. ! Preview: Snippet 192 ended.(490372+107642x730659). <-><-> l.262 15641& using charpoly($T_6$ )\\ Not a real error. [192] ! Preview: Snippet 193 started. <-><-> l.263 28279& using charpoly($ T_{34}$)\\\hline Not a real error. ! Preview: Snippet 193 ended.(490372+107642x1009191). <-><-> l.263 28279& using charpoly($T_{34}$ )\\\hline Not a real error. [193] ! Preview: Snippet 194 started. <-><-> l.267 Computing $ T_n$ with~$n$ composite is very time consuming when~$p$ is Not a real error. ! Preview: Snippet 194 ended.(490372+107642x790127). <-><-> l.267 Computing $T_n$ with~$n$ composite is very time consuming when~$p$ is Not a real error. [194] ! Preview: Snippet 195 started. <-><-> l.267 Computing $T_n$ with~$ n$ composite is very time consuming when~$p$ is Not a real error. ! Preview: Snippet 195 ended.(308974+0x430740). <-><-> l.267 Computing $T_n$ with~$n$ composite is very time consuming when~$p$ is Not a real error. [195] ! Preview: Snippet 196 started. <-><-> l.267 ...$ composite is very time consuming when~$ p$ is Not a real error. ! Preview: Snippet 196 ended.(308974+139537x361052). <-><-> l.267 ...composite is very time consuming when~$p$ is Not a real error. [196] ! Preview: Snippet 197 started. <-><-> l.268 ... so it is important to choose the right $ T_n$ quickly. For Not a real error. ! Preview: Snippet 197 ended.(490372+107642x790127). <-><-> l.268 ...it is important to choose the right $T_n$ quickly. For Not a real error. [197] ! Preview: Snippet 198 started. <-><-> l.269 $ p=28279$, here is the trick I used to quickly find an~$n$ such that Not a real error. ! Preview: Snippet 198 ended.(462465+139537x3111922). <-><-> l.269 $p=28279$ , here is the trick I used to quickly find an~$n$ such that Not a real error. [198] ! Preview: Snippet 199 started. <-><-> l.269 ... is the trick I used to quickly find an~$ n$ such that Not a real error. ! Preview: Snippet 199 ended.(308974+0x430740). <-><-> l.269 ...s the trick I used to quickly find an~$n$ such that Not a real error. [199] ! Preview: Snippet 200 started. <-><-> l.270 $ \disc(T_n)$ is not divisible by~$p$. This trick might be used to Not a real error. ! Preview: Snippet 200 ended.(538214+179404x2548297). <-><-> l.270 $\disc(T_n)$ is not divisible by~$p$. This trick might be used to Not a real error. [200] ! Preview: Snippet 201 started. <-><-> l.270 $\disc(T_n)$ is not divisible by~$ p$. This trick might be used to Not a real error. ! Preview: Snippet 201 ended.(308974+139537x361052). <-><-> l.270 $\disc(T_n)$ is not divisible by~$p$ . This trick might be used to Not a real error. [201] ! Preview: Snippet 202 started. <-><-> l.272 efficiently discover which $ T_n$ to compute. Though computing $T_n$ Not a real error. ! Preview: Snippet 202 ended.(490372+107642x790127). <-><-> l.272 efficiently discover which $T_n$ to compute. Though computing $T_n$ Not a real error. [202] ! Preview: Snippet 203 started. <-><-> l.272 ...ich $T_n$ to compute. Though computing $ T_n$ Not a real error. ! Preview: Snippet 203 ended.(490372+107642x790127). <-><-> l.272 ...$T_n$ to compute. Though computing $T_n$ Not a real error. [203] ! Preview: Snippet 204 started. <-><-> l.274 ...e is an algorithm that quickly computes $ T_n$ on subspaces of Not a real error. ! Preview: Snippet 204 ended.(490372+107642x790127). <-><-> l.274 ... an algorithm that quickly computes $T_n$ on subspaces of Not a real error. [204] LaTeX Font Info: Try loading font information for OMS+cmr on input line 275. (/usr/share/texmf/tex/latex/base/omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 275. ! Preview: Snippet 205 started. <-><-> l.276 Let~$ M$ be the space of mod~$p$ modular symbols of level $p=28279$, Not a real error. ! Preview: Snippet 205 ended.(490372+0x774430). <-><-> l.276 Let~$M$ be the space of mod~$p$ modular symbols of level $p=28279$, Not a real error. [205] ! Preview: Snippet 206 started. <-><-> l.276 Let~$M$ be the space of mod~$ p$ modular symbols of level $p=28279$, Not a real error. ! Preview: Snippet 206 ended.(308974+139537x361052). <-><-> l.276 Let~$M$ be the space of mod~$p$ modular symbols of level $p=28279$, Not a real error. [206] ! Preview: Snippet 207 started. <-><-> l.276 ...ace of mod~$p$ modular symbols of level $ p=28279$, Not a real error. ! Preview: Snippet 207 ended.(462465+139537x3111922). <-><-> l.276 ...od~$p$ modular symbols of level $p=28279$ , Not a real error. [207] ! Preview: Snippet 208 started. <-><-> l.277 and let $ f=\gcd(\charpoly(T_2),\deriv(\charpoly(T_2)))$. Let~$V$ be Not a real error. ! Preview: Snippet 208 ended.(538214+179404x13467496). <-><-> l.277 ...d(\charpoly(T_2),\deriv(\charpoly(T_2)))$ . Let~$V$ be Not a real error. [208] ! Preview: Snippet 209 started. <-><-> l.277 ...oly(T_2),\deriv(\charpoly(T_2)))$. Let~$ V$ be Not a real error. ! Preview: Snippet 209 ended.(490372+0x578082). <-><-> l.277 ...y(T_2),\deriv(\charpoly(T_2)))$. Let~$V$ be Not a real error. [209] ! Preview: Snippet 210 started. <-><-> l.278 the kernel of $ f(T_2)$ (this takes 7 minutes to compute). If $V=0$, Not a real error. ! Preview: Snippet 210 ended.(538214+179404x1717389). <-><-> l.278 the kernel of $f(T_2)$ (this takes 7 minutes to compute). If $V=0$, Not a real error. [210] ! Preview: Snippet 211 started. <-><-> l.278 ... (this takes 7 minutes to compute). If $ V=0$, Not a real error. ! Preview: Snippet 211 ended.(490372+0x1893712). <-><-> l.278 ...is takes 7 minutes to compute). If $V=0$ , Not a real error. [211] ! Preview: Snippet 212 started. <-><-> l.279 we would be done, since then $ \disc(T_2)\neq 0\in\F_p$. In fact,~$V$ Not a real error. ! Preview: Snippet 212 ended.(538214+209587x5433270). <-><-> l.279 ...one, since then $\disc(T_2)\neq 0\in\F_p$ . In fact,~$V$ Not a real error. [212] ! Preview: Snippet 213 started. <-><-> l.279 ...en $\disc(T_2)\neq 0\in\F_p$. In fact,~$ V$ Not a real error. ! Preview: Snippet 213 ended.(490372+0x578082). <-><-> l.279 ... $\disc(T_2)\neq 0\in\F_p$. In fact,~$V$ Not a real error. [213] ! Preview: Snippet 214 started. <-><-> l.280 has dimension~$ 7$. We find the first few integers~$n$ so that the Not a real error. ! Preview: Snippet 214 ended.(462465+0x358810). <-><-> l.280 has dimension~$7$ . We find the first few integers~$n$ so that the Not a real error. [214] ! Preview: Snippet 215 started. <-><-> l.280 ...on~$7$. We find the first few integers~$ n$ so that the Not a real error. ! Preview: Snippet 215 ended.(308974+0x430740). <-><-> l.280 ...~$7$. We find the first few integers~$n$ so that the Not a real error. [215] ! Preview: Snippet 216 started. <-><-> l.281 charpoly of $ T_n$ on $V_1$ has distinct roots, and they are $n=34$, Not a real error. ! Preview: Snippet 216 ended.(490372+107642x790127). <-><-> l.281 charpoly of $T_n$ on $V_1$ has distinct roots, and they are $n=34$, Not a real error. [216] ! Preview: Snippet 217 started. <-><-> l.281 charpoly of $T_n$ on $ V_1$ has distinct roots, and they are $n=34$, Not a real error. ! Preview: Snippet 217 ended.(490372+107642x729911). <-><-> l.281 charpoly of $T_n$ on $V_1$ has distinct roots, and they are $n=34$, Not a real error. [217] ! Preview: Snippet 218 started. <-><-> l.281 ... $V_1$ has distinct roots, and they are $ n=34$, Not a real error. ! Preview: Snippet 218 ended.(462465+0x2105180). <-><-> l.281 ...$ has distinct roots, and they are $n=34$ , Not a real error. [218] ! Preview: Snippet 219 started. <-><-> l.282 $ 47$, $53$, and $89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. ! Preview: Snippet 219 ended.(462465+0x717620). <-><-> l.282 $47$ , $53$, and $89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. [219] ! Preview: Snippet 220 started. <-><-> l.282 $47$, $ 53$, and $89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. ! Preview: Snippet 220 ended.(462465+0x717620). <-><-> l.282 $47$, $53$ , and $89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. [220] ! Preview: Snippet 221 started. <-><-> l.282 $47$, $53$, and $ 89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. ! Preview: Snippet 221 ended.(462465+0x717620). <-><-> l.282 $47$, $53$, and $89$ . I then computed $\charpoly(T_{34})$ directly on Not a real error. [221] ! Preview: Snippet 222 started. <-><-> l.282 $47$, $53$, and $89$. I then computed $ \charpoly(T_{34})$ directly on Not a real error. ! Preview: Snippet 222 ended.(538214+179404x4270376). <-><-> l.282 ...89$. I then computed $\charpoly(T_{34})$ directly on Not a real error. [222] ! Preview: Snippet 223 started. <-><-> l.283 ...found that it has distinct roots modulo~$ p$. Not a real error. ! Preview: Snippet 223 ended.(308974+139537x361052). <-><-> l.283 ...und that it has distinct roots modulo~$p$ . Not a real error. [223] ! Preview: Snippet 224 started. <-><-> l.286 \subsection {Higher Weight Data} Not a real error. ! Preview: Snippet 224 ended.(546132+152916x28311480). <-><-> l.286 \subsection{Higher Weight Data} Not a real error. [224 ] ! Preview: Snippet 225 started. <-><-> l.288 \item The following are the valuations $ d=d_4(\Gamma_0(p))$ at~$p$ of Not a real error. ! Preview: Snippet 225 ended.(538214+179404x4252307). <-><-> l.288 ...g are the valuations $d=d_4(\Gamma_0(p))$ at~$p$ of Not a real error. [225] ! Preview: Snippet 226 started. <-><-> l.288 ... the valuations $d=d_4(\Gamma_0(p))$ at~$ p$ of Not a real error. ! Preview: Snippet 226 ended.(308974+139537x361052). <-><-> l.288 ...he valuations $d=d_4(\Gamma_0(p))$ at~$p$ of Not a real error. [226] ! Preview: Snippet 227 started. <-><-> l.290 $ S_4(\Gamma_0(p))$ for $p<500$. Not a real error. ! Preview: Snippet 227 ended.(538214+179404x2988505). <-><-> l.290 $S_4(\Gamma_0(p))$ for $p<500$. Not a real error. [227] ! Preview: Snippet 228 started. <-><-> l.290 $S_4(\Gamma_0(p))$ for $ p<500$. Not a real error. ! Preview: Snippet 228 ended.(462465+139537x2394302). <-><-> l.290 $S_4(\Gamma_0(p))$ for $p<500$ . Not a real error. [228] ! Preview: Snippet 229 started. <-><-> l.294 $ p$ &2& 3& 5& 7& 11& 13& 17& 19& 23& 29& 31& 37& 41& 43& 47& 53& 59\\ Not a real error. ! Preview: Snippet 229 ended.(308974+139537x361052). <-><-> l.294 $p$ &2& 3& 5& 7& 11& 13& 17& 19& 23& 29& 31& 37& 41& 43& 47& 53& 59\\ Not a real error. [229] ! Preview: Snippet 230 started. <-><-> l.295 $ d$ &0& 0& 0& 0& 0& 2& 2& 2& 2& 4& 4& 6& 6& 6& 6& 8& 8\\\hline Not a real error. ! Preview: Snippet 230 ended.(498346+0x373511). <-><-> l.295 $d$ &0& 0& 0& 0& 0& 2& 2& 2& 2& 4& 4& 6& 6& 6& 6& 8& 8\\\hline Not a real error. [230] ! Preview: Snippet 231 started. <-><-> l.296 $ p$&61& 67& 71& 73& 79& 83& 89& 97& 101& 103& 107& 109& 113& 127& 131&... Not a real error. ! Preview: Snippet 231 ended.(308974+139537x361052). <-><-> l.296 $p$ &61& 67& 71& 73& 79& 83& 89& 97& 101& 103& 107& 109& 113& 127& 131&... Not a real error. [231] ! Preview: Snippet 232 started. <-><-> l.297 $ d$ & 10& 10& 10& 12& 12& 12& 14& 16& 16& 16& 16& 18& 18& 20& 20& 22&2... Not a real error. ! Preview: Snippet 232 ended.(498346+0x373511). <-><-> l.297 $d$ & 10& 10& 10& 12& 12& 12& 14& 16& 16& 16& 16& 18& 18& 20& 20& 22&2... Not a real error. [232] ! Preview: Snippet 233 started. <-><-> l.298 $ p$ & 149& 151& 157& 163& 167& 173& 179& 181& 191& 193& 197& 199& Not a real error. ! Preview: Snippet 233 ended.(308974+139537x361052). <-><-> l.298 $p$ & 149& 151& 157& 163& 167& 173& 179& 181& 191& 193& 197& 199& Not a real error. [233] ! Preview: Snippet 234 started. <-><-> l.300 $ d$ & 24& 24& 26& 26& 26&28& 28& 30& 30& 32& 32& 32& 34& 36& 36& 38& 3... Not a real error. ! Preview: Snippet 234 ended.(498346+0x373511). <-><-> l.300 $d$ & 24& 24& 26& 26& 26&28& 28& 30& 30& 32& 32& 32& 34& 36& 36& 38& 3... Not a real error. [234] ! Preview: Snippet 235 started. <-><-> l.301 $ p$ & 239& 241& 251& 257& 263& 269& 271& 277& Not a real error. ! Preview: Snippet 235 ended.(308974+139537x361052). <-><-> l.301 $p$ & 239& 241& 251& 257& 263& 269& 271& 277& Not a real error. [235] ! Preview: Snippet 236 started. <-><-> l.303 $ d$ & 38& 40& 40& 42& 42&44& 44& 46& 46& 46& 48& 50& 50& 52& 52& 54& 5... Not a real error. ! Preview: Snippet 236 ended.(498346+0x373511). <-><-> l.303 $d$ & 38& 40& 40& 42& 42&44& 44& 46& 46& 46& 48& 50& 50& 52& 52& 54& 5... Not a real error. [236] ! Preview: Snippet 237 started. <-><-> l.304 $ p$ & 347& 349& 353& 359& 367& 373& 379& 383& 389&397& 401& 409& 419& ... Not a real error. ! Preview: Snippet 237 ended.(308974+139537x361052). <-><-> l.304 $p$ & 347& 349& 353& 359& 367& 373& 379& 383& 389&397& 401& 409& 419& ... Not a real error. [237] ! Preview: Snippet 238 started. <-><-> l.305 $ d$ & 56& 58& 58& 58& 60&62& 62& 62& 65 &66& 66& 68& 68& 70& 70& 72& ... Not a real error. ! Preview: Snippet 238 ended.(498346+0x373511). <-><-> l.305 $d$ & 56& 58& 58& 58& 60&62& 62& 62& 65 &66& 66& 68& 68& 70& 70& 72& ... Not a real error. [238] ! Preview: Snippet 239 started. <-><-> l.306 $ p$ & 443& 449& 457& 461& 463& 467& 479& 487& 491& 499 &&&&&&&\\ Not a real error. ! Preview: Snippet 239 ended.(308974+139537x361052). <-><-> l.306 $p$ & 443& 449& 457& 461& 463& 467& 479& 487& 491& 499 &&&&&&&\\ Not a real error. [239] ! Preview: Snippet 240 started. <-><-> l.307 $ d$ & 72& 74& 76& 76& 76& 76& 78& 80& 80& 82 &&&&&&&\\\hline Not a real error. ! Preview: Snippet 240 ended.(498346+0x373511). <-><-> l.307 $d$ & 72& 74& 76& 76& 76& 76& 78& 80& 80& 82 &&&&&&&\\\hline Not a real error. [240] ! Preview: Snippet 241 started. <-><-> l.322 \section {The Discriminant is Divisible by~$p$} Not a real error. ! Preview: Snippet 241 ended.(655359+183500x28311480). <-><-> l.322 ...ion{The Discriminant is Divisible by~$p$} Not a real error. [241 ] ! Preview: Snippet 242 started. <-><-> l.323 In this section we prove that for $ k\geq 4$ the discriminant of the Not a real error. ! Preview: Snippet 242 ended.(498346+97575x1711816). <-><-> l.323 In this section we prove that for $k\geq 4$ the discriminant of the Not a real error. [242] ! Preview: Snippet 243 started. <-><-> l.324 Hecke algebra associated to $ S_k(\Gamma_0(p))$ is almost always Not a real error. ! Preview: Snippet 243 ended.(538214+179404x3013991). <-><-> l.324 ... algebra associated to $S_k(\Gamma_0(p))$ is almost always Not a real error. [243] ! Preview: Snippet 244 started. <-><-> l.325 divisible by~$ p$. Not a real error. ! Preview: Snippet 244 ended.(308974+139537x361052). <-><-> l.325 divisible by~$p$ . Not a real error. [244] ! Preview: Snippet 245 started. <-><-> l.328 Suppose~$ p$ is a prime and $k\geq 4$ is an even integer. Then Not a real error. ! Preview: Snippet 245 ended.(308974+139537x361052). <-><-> l.328 Suppose~$p$ is a prime and $k\geq 4$ is an even integer. Then Not a real error. [245] ! Preview: Snippet 246 started. <-><-> l.328 Suppose~$p$ is a prime and $ k\geq 4$ is an even integer. Then Not a real error. ! Preview: Snippet 246 ended.(498346+97575x1711816). <-><-> l.328 Suppose~$p$ is a prime and $k\geq 4$ is an even integer. Then Not a real error. [246] ! Preview: Snippet 247 started. <-><-> l.329 $ d_k(\Gamma_0(p))>0$ unless Not a real error. ! Preview: Snippet 247 ended.(538214+179404x4263092). <-><-> l.329 $d_k(\Gamma_0(p))>0$ unless Not a real error. [247] ! Preview: Snippet 248 started. <-><-> l.330 \begin{align*} Not a real error. ! Preview: Snippet 248 ended.(3067085+0x19717160). <-><-> l.334 \end{align*} Not a real error. [248] ! Preview: Snippet 249 started. <-><-> l.335 in which case $ d_k(\Gamma_0(p))=0$. Not a real error. ! Preview: Snippet 249 ended.(538214+179404x4263092). <-><-> l.335 in which case $d_k(\Gamma_0(p))=0$ . Not a real error. [249] ! Preview: Snippet 250 started. <-><-> l.340 Let~$ p$ be a prime. For~$N$ and~$k$ integers, let Not a real error. ! Preview: Snippet 250 ended.(308974+139537x361052). <-><-> l.340 Let~$p$ be a prime. For~$N$ and~$k$ integers, let Not a real error. [250] ! Preview: Snippet 251 started. <-><-> l.340 Let~$p$ be a prime. For~$ N$ and~$k$ integers, let Not a real error. ! Preview: Snippet 251 ended.(490372+0x654826). <-><-> l.340 Let~$p$ be a prime. For~$N$ and~$k$ integers, let Not a real error. [251] ! Preview: Snippet 252 started. <-><-> l.340 Let~$p$ be a prime. For~$N$ and~$ k$ integers, let Not a real error. ! Preview: Snippet 252 ended.(498346+0x396186). <-><-> l.340 Let~$p$ be a prime. For~$N$ and~$k$ integers, let Not a real error. [252] ! Preview: Snippet 253 started. <-><-> l.341 $ $ Not a real error. ! Preview: Snippet 253 ended.(1100877+0x28311480). <-><-> l.343 $$ Not a real error. [253] ! Preview: Snippet 254 started. <-><-> l.345 $ $ Not a real error. ! Preview: Snippet 254 ended.(1100877+0x28311480). <-><-> l.348 $$ Not a real error. [254] ! Preview: Snippet 255 started. <-><-> l.350 Let $ \wp$ denote the maximal ideal of $\Zpbar$. In Not a real error. ! Preview: Snippet 255 ended.(308974+139537x456735). <-><-> l.350 Let $\wp$ denote the maximal ideal of $\Zpbar$. In Not a real error. [255] ! Preview: Snippet 256 started. <-><-> l.350 Let $\wp$ denote the maximal ideal of $ \Zpbar$. In Not a real error. ! Preview: Snippet 256 ended.(637879+209587x791595). <-><-> l.350 ...\wp$ denote the maximal ideal of $\Zpbar$ . In Not a real error. [256] ! Preview: Snippet 257 started. <-><-> l.352 $ $ Not a real error. ! Preview: Snippet 257 ended.(1168399+0x28311480). <-><-> l.354 $$ Not a real error. [257] ! Preview: Snippet 258 started. <-><-> l.356 $ $ Not a real error. ! Preview: Snippet 258 ended.(1070694+0x28311480). <-><-> l.358 $$ Not a real error. [258] ! Preview: Snippet 259 started. <-><-> l.359 where~$ G$ is an Eisenstein series of weight~$p-1$ such that Not a real error. ! Preview: Snippet 259 ended.(490372+0x564227). <-><-> l.359 where~$G$ is an Eisenstein series of weight~$p-1$ such that Not a real error. [259] ! Preview: Snippet 260 started. <-><-> l.359 ...e~$G$ is an Eisenstein series of weight~$ p-1$ such that Not a real error. ! Preview: Snippet 260 ended.(462465+139537x1596948). <-><-> l.359 ...$ is an Eisenstein series of weight~$p-1$ such that Not a real error. [260] ! Preview: Snippet 261 started. <-><-> l.360 $ G\con 1\pmod{\wp}$. Serre shows (see Lemme~9 with $m=0$) that Not a real error. ! Preview: Snippet 261 ended.(538214+179404x4828319). <-><-> l.360 $G\con 1\pmod{\wp}$ . Serre shows (see Lemme~9 with $m=0$) that Not a real error. [261] ! Preview: Snippet 262 started. <-><-> l.360 ...d{\wp}$. Serre shows (see Lemme~9 with $ m=0$) that Not a real error. ! Preview: Snippet 262 ended.(462465+0x1945709). <-><-> l.360 ...p}$. Serre shows (see Lemme~9 with $m=0$ ) that Not a real error. [262] ! Preview: Snippet 263 started. <-><-> l.361 $ t(f) \con f \pmod{\wp}$. Not a real error. ! Preview: Snippet 263 ended.(538214+179404x5579733). <-><-> l.361 $t(f) \con f \pmod{\wp}$ . Not a real error. [263] ! Preview: Snippet 264 started. <-><-> l.363 By ??, there is a basis $ f_1,\ldots,f_n$ of Hecke eigenforms for Not a real error. ! Preview: Snippet 264 ended.(498346+139537x2979440). <-><-> l.363 By ??, there is a basis $f_1,\ldots,f_n$ of Hecke eigenforms for Not a real error. [264] ! Preview: Snippet 265 started. <-><-> l.364 $ S_k(\Gamma_0(p),\Qpbar)$, and we may assume these $f_i$ are Not a real error. ! Preview: Snippet 265 ended.(637879+254316x4204262). <-><-> l.364 $S_k(\Gamma_0(p),\Qpbar)$ , and we may assume these $f_i$ are Not a real error. [265] ! Preview: Snippet 266 started. <-><-> l.364 ..._0(p),\Qpbar)$, and we may assume these $ f_i$ are Not a real error. ! Preview: Snippet 266 ended.(498346+139537x573762). <-><-> l.364 ...),\Qpbar)$, and we may assume these $f_i$ are Not a real error. [266] ! Preview: Snippet 267 started. <-><-> l.365 ...so that the leading coefficient of each~$ q$-expansion Not a real error. ! Preview: Snippet 267 ended.(308974+139537x346102). <-><-> l.365 ... that the leading coefficient of each~$q$ -expansion Not a real error. [267] ! Preview: Snippet 268 started. <-><-> l.366 is~$ 1$. If Not a real error. ! Preview: Snippet 268 ended.(462465+0x358810). <-><-> l.366 is~$1$ . If Not a real error. [268] ! Preview: Snippet 269 started. <-><-> l.367 $ $\dim S_{k+(p-1)}(\Gamma_0(p),\Zpbar) < \dim S_k(\Gamma_0(p), Not a real error. ! Preview: Snippet 269 ended.(1168399+0x28311480). <-><-> l.368 \Zpbar)$$ Not a real error. [269] ! Preview: Snippet 270 started. <-><-> l.369 then the set of $ q$-expansions Not a real error. ! Preview: Snippet 270 ended.(308974+139537x346102). <-><-> l.369 then the set of $q$ -expansions Not a real error. [270] ! Preview: Snippet 271 started. <-><-> l.370 $ $ Not a real error. ! Preview: Snippet 271 ended.(1070694+0x28311480). <-><-> l.372 $$ Not a real error. [271] ! Preview: Snippet 272 started. <-><-> l.374 $ d_k(\Gamma_0(p))>0$. Not a real error. ! Preview: Snippet 272 ended.(538214+179404x4263092). <-><-> l.374 $d_k(\Gamma_0(p))>0$ . Not a real error. [272] ! Preview: Snippet 273 started. <-><-> l.377 \begin{align*} Not a real error. ! Preview: Snippet 273 ended.(3500386+0x27970317). <-><-> l.384 \end{align*} Not a real error. [273] ! Preview: Snippet 274 started. <-><-> l.386 $ $ Not a real error. ! Preview: Snippet 274 ended.(1581475+0x28311480). <-><-> l.388 $$ Not a real error. [274] ! Preview: Snippet 275 started. <-><-> l.390 $ $ Not a real error. ! Preview: Snippet 275 ended.(2218414+0x28311480). <-><-> l.395 $$ Not a real error. [275] ! Preview: Snippet 276 started. <-><-> l.397 $ $ Not a real error. ! Preview: Snippet 276 ended.(1110145+0x28311480). <-><-> l.399 $$ Not a real error. [276] ! Preview: Snippet 277 started. <-><-> l.400 then $ (k-2)p\leq 36$. This reduces the assertion of the theorem Not a real error. ! Preview: Snippet 277 ended.(538214+179404x4225724). <-><-> l.400 then $(k-2)p\leq 36$ . This reduces the assertion of the theorem Not a real error. [277] ! Preview: Snippet 278 started. <-><-> l.408 Suppose $ p>2$ is a prime and $k\geq 3$ is an integer. If Not a real error. ! Preview: Snippet 278 ended.(462465+139537x1676682). <-><-> l.408 Suppose $p>2$ is a prime and $k\geq 3$ is an integer. If Not a real error. [278] ! Preview: Snippet 279 started. <-><-> l.408 Suppose $p>2$ is a prime and $ k\geq 3$ is an integer. If Not a real error. ! Preview: Snippet 279 ended.(498346+97575x1711816). <-><-> l.408 Suppose $p>2$ is a prime and $k\geq 3$ is an integer. If Not a real error. [279] ! Preview: Snippet 280 started. <-><-> l.409 \begin{align*} Not a real error. ! Preview: Snippet 280 ended.(4154983+0x21810215). <-><-> l.414 \end{align*} Not a real error. [280] ! Preview: Snippet 281 started. <-><-> l.415 then $ d_k(\Gamma_1(p))>0$. Not a real error. ! Preview: Snippet 281 ended.(538214+179404x4263092). <-><-> l.415 then $d_k(\Gamma_1(p))>0$ . Not a real error. [281] ! Preview: Snippet 282 started. <-><-> l.419 \section {The Conjecture} Not a real error. ! Preview: Snippet 282 ended.(655359+183500x28311480). <-><-> l.419 \section{The Conjecture} Not a real error. [282 ] ! Preview: Snippet 283 started. <-><-> l.422 Let~$ k=2m$ be an even integer and~$p$ a prime. Let $\T$ be the Hecke Not a real error. ! Preview: Snippet 283 ended.(498346+0x2341895). <-><-> l.422 Let~$k=2m$ be an even integer and~$p$ a prime. Let $\T$ be the Hecke Not a real error. [283] ! Preview: Snippet 284 started. <-><-> l.422 Let~$k=2m$ be an even integer and~$ p$ a prime. Let $\T$ be the Hecke Not a real error. ! Preview: Snippet 284 ended.(308974+139537x361052). <-><-> l.422 Let~$k=2m$ be an even integer and~$p$ a prime. Let $\T$ be the Hecke Not a real error. [284] ! Preview: Snippet 285 started. <-><-> l.422 ...e an even integer and~$p$ a prime. Let $ \T$ be the Hecke Not a real error. ! Preview: Snippet 285 ended.(494359+0x478414). <-><-> l.422 ...n even integer and~$p$ a prime. Let $\T$ be the Hecke Not a real error. [285] ! Preview: Snippet 286 started. <-><-> l.423 algebra associated to $ S_k(\Gamma_0(p))$ and let $\tT$ be the Not a real error. ! Preview: Snippet 286 ended.(538214+179404x3013991). <-><-> l.423 algebra associated to $S_k(\Gamma_0(p))$ and let $\tT$ be the Not a real error. [286] ! Preview: Snippet 287 started. <-><-> l.423 ...ssociated to $S_k(\Gamma_0(p))$ and let $ \tT$ be the Not a real error. ! Preview: Snippet 287 ended.(664653+0x478414). <-><-> l.423 ...iated to $S_k(\Gamma_0(p))$ and let $\tT$ be the Not a real error. [287] ! Preview: Snippet 288 started. <-><-> l.424 normalization of $ \tT$ in $\T\tensor\Q$. Not a real error. ! Preview: Snippet 288 ended.(664653+0x478414). <-><-> l.424 normalization of $\tT$ in $\T\tensor\Q$. Not a real error. [288] ! Preview: Snippet 289 started. <-><-> l.424 normalization of $\tT$ in $ \T\tensor\Q$. Not a real error. ! Preview: Snippet 289 ended.(494359+119603x1913650). <-><-> l.424 normalization of $\tT$ in $\T\tensor\Q$ . Not a real error. [289] ! Preview: Snippet 290 started. <-><-> l.426 $ $ Not a real error. ! Preview: Snippet 290 ended.(1787839+0x28311480). <-><-> l.429 $$ Not a real error. [290] ! Preview: Snippet 291 started. <-><-> l.431 $ $ Not a real error. ! Preview: Snippet 291 ended.(5663007+0x28311480). <-><-> l.439 $$ Not a real error. [291] ! Preview: Snippet 292 started. <-><-> l.440 In particular, when $ k=2$ we conjecture that $[\tT:\T]$ is not Not a real error. ! Preview: Snippet 292 ended.(498346+0x1711816). <-><-> l.440 In particular, when $k=2$ we conjecture that $[\tT:\T]$ is not Not a real error. [292] ! Preview: Snippet 293 started. <-><-> l.440 ...rticular, when $k=2$ we conjecture that $ [\tT:\T]$ is not Not a real error. ! Preview: Snippet 293 ended.(664653+179404x1953515). <-><-> l.440 ... when $k=2$ we conjecture that $[\tT:\T]$ is not Not a real error. [293] ! Preview: Snippet 294 started. <-><-> l.441 divisible by~$ p$. Not a real error. ! Preview: Snippet 294 ended.(308974+139537x361052). <-><-> l.441 divisible by~$p$ . Not a real error. [294] ! Preview: Snippet 295 started. <-><-> l.443 Here $ \binom{x}{y}$ is the binomial coefficient ``$x$ choose $y$'', Not a real error. ! Preview: Snippet 295 ended.(609979+349409x971395). <-><-> l.443 Here $\binom{x}{y}$ is the binomial coefficient ``$x$ choose $y$'', Not a real error. [295] ! Preview: Snippet 296 started. <-><-> l.443 ...om{x}{y}$ is the binomial coefficient ``$ x$ choose $y$'', Not a real error. ! Preview: Snippet 296 ended.(308974+0x410139). <-><-> l.443 ...{x}{y}$ is the binomial coefficient ``$x$ choose $y$'', Not a real error. [296] ! Preview: Snippet 297 started. <-><-> l.443 ...s the binomial coefficient ``$x$ choose $ y$'', Not a real error. ! Preview: Snippet 297 ended.(308974+139537x377582). <-><-> l.443 ...the binomial coefficient ``$x$ choose $y$ '', Not a real error. [297] LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 187 strings out of 19155 2856 string characters out of 180967 80378 words of memory out of 350001 4856 multiletter control sequences out of 10000+15000 15247 words of font info for 58 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 10000 23i,14n,22p,173b,386s stack positions out of 3000i,100n,1500p,50000b,4000s Output written on _region_.dvi (297 pages, 25172 bytes).