Author: William A. Stein
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8\title{A mod five approach to modularity of icosahedral
9       Galois representations}
10\author{Kevin Buzzard and William A. Stein}
11
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108\begin{document}
109\maketitle
110\begin{abstract}
111We give eight new examples of icosahedral Galois representations that
112satisfy Artin's conjecture on holomorphicity of their $L$-function.
113We give in detail one example of an icosahedral representation of
114conductor ${\bf 1376}=2^5\cdot 43$ that satisfies Artin's conjecture.
115We also briefly explain the computations behind seven additional
116examples of conductors ${\bf 2416}=2^4\cdot 151$, ${\bf 3184}=2^4\cdot 117199$, ${\bf 3556}=2^2\cdot 7\cdot 127$, ${\bf 3756}=2^2\cdot 3\cdot 118313$, ${\bf 4108}=2^2\cdot 13\cdot 79$, ${\bf 4288}=2^6\cdot 67$, and
119${\bf 5373}=3^3\cdot 199$.
120\end{abstract}
121
122\section*{Introduction}
123Consider a continuous irreducible Galois representation
124$$\rho:\galq\ra\GL_n(\C)$$
125with $n > 1$.
126Inspired by his reciprocity law,
127Artin conjectured in~\cite{artin:conjecture} that
128$L(\rho,s)$ has an analytic continuation to the whole complex plane.
129Many of the known cases of this conjecture were obtained by
130proving the apparently stronger assertion that~$\rho$ is \defn{automorphic},
131in the sense that the $L$-function of~$\rho$ is equal to the $L$-function
132of a certain automorphic representation (whose $L$-function is known to have
133analytic continuation). In the special case where $n=2$ and $\rho$ is in
134addition assumed to be odd, the automorphic representation in question
135should be the one associated to a classical weight~$1$
136modular eigenform, and in fact there is conjectured to be a
137bijection between such~$\rho$ and the set of all weight~$1$
138cuspidal newforms, which should
139preserve $L$-functions. It is this bijection
140that we are concerned with in this paper, so assume for the rest
141of the paper that $n=2$ and~$\rho$ is odd.
142
143In this special case, the construction
144of~\cite{deligne-serre} shows how to construct a continuous irreducible
145odd 2-dimensional representation from a weight~$1$ newform, and the problem
146is to go the other way. Say that a representation is \defn{modular}
147if it arises in this way.
148
149If the image of~$\rho$ is solvable,
150then~$\rho$  is known to be modular
151\cite{langlands:basechange, tunnell:artin};
152if the image is not solvable, then $\im(\rho)$ in $\PGL_2(\C)$
153is isomorphic to the
154alternating group~$A_5$, and the modularity of~$\rho$
155is, in general, unknown. We call such a 2-dimensional representation an
156icosahedral representation''.
157The published literature contains only eight examples (up to twist)
158of odd icosahedral Galois representations that are known to satisfy Artin's
159conjecture: one of conductor $800=2^5\cdot 5^2$
160(see \cite{buhler:thesis}), and seven of conductors:
161$2083,\, 2^2\cdot 487,\, 2^2\cdot 751,\, 162 2^2 \cdot 887,\, 2^2\cdot 919,\, 163 2^5\cdot 73,\,\text{ and } 2^5\cdot 193$
164(see \cite{freyetal}).
165
166After the first draft of this paper was written, the
167preprint~\cite{bdsbt} appeared, which contains a general theorem that
168yields infinitely many (up to twist) modular icosahedral representations.
169However, we feel that our work, although much less powerful, is still
170of some worth, because it gives an effective computational approach to
171proving that certain mod~5 representations are modular, without
172computing any spaces of weight~1 forms or using effective versions of
173the Chebotar\"ev density theorem. We also note that the
174main theorem of~\cite{bdsbt} does not apply to any of the examples
175considered in the present paper.  Very recently,
176the preprint~\cite{taylor:artin2}
177appeared, which gives local conditions under which an icosahedral
178representation is modular.  In particular, \cite{taylor:artin2} also
179proves that the first three
180examples in the present paper, of conductors 1376, 2416, 3184,
181are modular; these correspond to the first, third, and fourth
182equations at the end of~\cite{taylor:artin2}.
183However, \cite{taylor:artin2} does not apply to
184our remaining five examples.
185
186
187In this paper we give eight new examples of modular icosahedral
188representations that were computed
189by applying the main theorem  of~\cite{buzzard-taylor} to
190the mod~$5$ reduction of~$\rho$.
191We verify modularity mod~$5$ on a case-by-case basis. Later we shall
192explain our approach more carefully, but let us briefly summarise it here.
193By~\cite{buzzard-taylor},
194the problem is to show that the mod~5 reduction of~$\rho$ is modular.
195We do this by finding a candidate mod~5 modular form at weight~5
196and then, using the table of icosahedral extensions of $\Q$ in~\cite{freyetal}
197and what we know about the 5-adic representation attached to our candidate
198form, we deduce that the mod~5 representation attached to our candidate
199form must be the reduction of~$\rho$. In particular, this paper gives
200a computational methods for checking the modularity of certain mod~5
201representations whose conductors are not too large. We now give
202more details.
203
204In each of our examples it is easy to compute a few Hecke operators
205and be morally convinced that a mod~$5$ representation should be modular;
206it is far more difficult to prove this.
207Effective variants of the Chebotarev density theorem require
208that we check vastly more traces of Frobenius than is practical.
209Instead we use the Local Langlands theorem for $\GL_2$, the
210theory of companion forms, and Table~2 of~\cite{freyetal},
211to provide proofs of modularity
212in certain cases.
213
214More precisely, let~$K$ be an icosahedral extension of~$\Q$ that is not
215totally real, and consider a minimal lift $\rho:\GQ\ra \GL_2(\C)$
216of
217   $$\GQ\ra \Gal(K/\Q)\ncisom{}A_5\subset \PGL_2(\C);$$
218the lift is minimal in the sense  that its conductor is minimal.
219Assume that~$5$ does not ramify in~$K$, and that
220a Frobenius element at~$5$ in $\Gal(K/\Q)$ does not have order~$1$ or~$5$.
221Inspired by the possibility that~$\rho$ is modular,
222we search for a mod~$5$ modular form of weight~$5$ whose existence would
223be forced by modularity of~$\rho$.  Indeed, we find
224a candidate mod~$5$ form~$f$, and then prove that the fixed field
225of the kernel of the projective mod~$5$ representation
226associated to a certain twist of~$f$  must be~$K$.
227This proves that the mod~$5$ reduction of a twist
228of~$\rho$ is modular, and the main theorem
229of \cite{buzzard-taylor} then implies
230that~$\rho$ is modular.
231We carried out this program for icosahedral representations
232of the following conductors:
233${\bf 1376} = 2^5\cdot 43$,
234${\bf 2416}=2^4\cdot 151$,
235${\bf 3184}=2^4\cdot 199$,
236${\bf 3556}=2^2\cdot 7\cdot 127$,
237${\bf 3756}=2^2\cdot 3\cdot 313$,
238${\bf 4108}=2^2\cdot 13\cdot 79$,
239${\bf 4288}=2^6\cdot 67$, and
240${\bf 5373}=3^3\cdot 199$.
241
242
243We choose an icosahedral field~$K$ and representation~$\rho$,
244then proceed as follows:
245\vspace{.5ex}
246\begin{numlist}
247\item Search for a form~$f \in S_5(N,\eps;\Fbar_5)$ whose
248      associated mod~$5$ Galois representation looks like
249      it is the mod~$5$ reduction of~$\rho$.
250\item Twist~$f$ to obtain an eigenform~$g$ with coefficients in~$\F_5$.
251\item Prove that~$\rho_g$ is unramified at~$5$ by finding a companion form.
252\item Prove that the image of $\proj\rho_g$ is~$A_5$ by ruling out all
253      other possibilities.
254\item Prove that the fixed field~$L$ of $\proj\rho_g$ has
255      root field of discriminant at most $2083^2$,
256      so~$L$ is in Table~2 of~\cite{freyetal}; deduce that~$L=K$.
257\item Apply the main theorem of~\cite{buzzard-taylor}
258      to a lift of $\rhobar=\rho_g$
259      to conclude that~$\rho$ is modular.
260\end{numlist}
261
262
263
264\section{Modularity of an icosahedral representation of
265conductor~$1376=2^5\cdot 43$}\label{sec:1376}
266In this section we prove the following theorem.
267\begin{theorem}\label{thm:1376}
268The icosahedral representations whose corresponding
269icosahedral extension
270is the splitting field of $x^5 + 2x^4+6x^3+8x^2+10x+8$
271are modular.
272\end{theorem}
273
274Let~$K$ be the splitting field of $h=x^5 + 2x^4+6x^3+8x^2+10x+8$.
275The Galois group of~$K$ is~$A_5$, so we obtain a homomorphism
276$G_\Q\ra{}A_5\subset \PGL_2(\C)$;
277let $\rho:G_\Q\ra\GL_2(\C)$ be a minimal lift, minimal
278in the sense that the Artin conductor of~$\rho$ is minimal.
279By Table~$A_5$ of~\cite{buhler:thesis}, the conductor of~$\rho$
280is $N=1376=2^5\cdot 43$.  Since
281$h\con (x-1)(x^2-x+1)(x^2-x+2)\pmod{5}$,
282and ${\rm disc}(h)$ is coprime to~$5$,
283any Frobenius element at~$5$ in $\Gal(K/\Q)$ has order~$2$.
284
285We use the notation of Tables 3.1 and 3.2 of~\cite[pg. 46]{buhler:thesis};
286from Table 3.2 we see that the type of~$\rho$ at~$2$
287is~$17$ and the type at~$43$ is~$2$.
288The mod~$N$ Dirichlet character~$\eps=\det(\rho)$
289factors as~$\eps=\eps_2\cdot \eps_{43}$ where~$\eps_2$ is
290a character mod~$2^5$ and~$\eps_{43}$ is a character mod~$43$.
291Corresponding to each type in Buhler's table, there is a character,
292and fortunately Buhler's level $800$ example also was of type~$17$ at~$2$
293(see the first line of~\cite[Table~3.2]{buhler:thesis}).
294By~\cite[pg.~80]{buhler:thesis} $\eps_2$ is the unique
295character of conductor~$4$ and order~$2$.
296A local computation shows that the image
297of~$\eps_{43}$ has order~$3$.
298
299If~$\rho$ is modular,  then there is a weight~$1$
300newform $f_?\in S_1(N,\eps;\Qbar)$ that gives rise to~$\rho$.
301Suppose for the moment that~$\rho$ is modular, so that~$f_?$ exists.
302Choose a prime of~$\overline{\Z}$ lying
303over~$5$, and denote by~$\fbar_?$ the reduction
304of $f_?$ modulo this prime. The Eisenstein series
305$E_4\in M_4(1;\F_5)$  is congruent to~$1$ modulo~$5$, so
306$E_4\cdot{}\fbar_?\in S_5(N,\eps;\Fbar_5)$ has the same $q$-expansion
307as $\fbar_?$.  Using a computer, we can search for a
308form $f\in S_5(N,\eps,\Fbar_5)$ that has the same
309$q$-expansion as the conjectural form $E_4\cdot{}\fbar_?$.
310
311Instead of multiplying $\fbar_?$ by~$E_4$, we could have multiplied
312it by an  Eisenstein series of weight~$1$, level~$5$, and character $\eps'$.
313We used $E_4$ because the dimension of $S_5(N,\eps;\Fbar_5)$
314is~$696$ whereas the dimension of the relevant space
315$S_2(5\cdot 1376, \eps_{43})$ of weight~$2$ cusp forms is~$1040$.
316
317\subsection{Searching for the newform~$f$}
318Using modular symbols (see Section~\ref{sec:modsym}) we
319compute (at least up to semi-simplification) the space
320$S_5(1376,\eps;\F_{25})$. Note that there is injective map
321from the image of~$\eps$ into $\F_{25}^*$.  By computing
322the kernels of various Hecke operators on this space, we find~$f$.
323In the following computations, we represent nonzero elements of~$\F_{25}$
324as powers of a generator~$\alp$ of~$\F_{25}^*$, which satisfies
325$$\alp^2 + 4\alp + 2=0.$$
326Our character $\eps_{43}$ was represented
327as the map sending $(1,3)\in(\Z/2^5\Z)^*\cross(\Z/43\Z)^*$ to
328$2\alp+1$. Note that~3 is a primitive root mod~43, and that $2\alp+1$
329has order~3.
330
331If the least common multiple of the degrees of the factors of
332the polynomial~$h$ modulo an
333unramified prime~$p$ is~$2$, then $\Frob_p\in\Gal(K/\Q)$
334has order~$2$.  The minimal polynomial of $\rho(\Frob_p)\in\GL_2(\C)$
335is then $x^2-1$, so $\rho(\Frob_p)$ has trace~$0$.
336The first three primes $p \nmid 5\cdot 1376$ such
337that $\rho(\Frob_p)$ has  order~$2$ are $p= 19,31,97$.
338We computed the mod~$5$ reduction $\sS_5(1376,\eps;\F_{25})^{+}$
339of the $\Z_5[\zeta_3]$-lattice of
340modular symbols of level~$1376$ and
341character~$\tilde{\eps}$ where
342complex conjugation acts as $+1$.
343Here~$\tilde{\eps}$ denotes the Teichm\"uller lift of~$\eps$.
344
345Let~$V$ be the intersection of the kernels of $T_{19}$, $T_{31}$, and
346$T_{97}$ inside of the space $\sS_5(1376,\eps;\F_{25})^{+}$ of mod~5
347modular symbols.
348The space~$V$ is $8$-dimensional,
349and no doubt all the eigenforms in this space give rise to~$\rho$ or one
350of its twists. One of the eigenvalues of~$T_3$ on this space
351is~$\alp^{16}$, and the kernel $V_1$ of $T_3-\alp^{16}$ is $2$-dimensional
352over $\F_{25}$. The Hecke operator~$T_5$ acted as a diagonalisable matrix on
353$V_1$, with eigenvalues $\alp^{10}$ and $\alp^{22}$, so the corresponding
354two systems of eigenvalues must correspond to mod~$5$ modular eigenforms,
355and furthermore we must have found all mod~$5$ modular eigenforms
356of this level, weight and character,
357such that $a_{19}=a_{31}=0$ and $a_3=\alp^{16}$.
358
359\begin{remark}
360The careful reader might wonder how we know that the
361systems of mod~$5$ eigenvalues really do correspond to mod~$5$ modular
362forms, and not to perhaps some strange mod~$5$ torsion in the space of
363modular symbols. However, we eliminated this possibility by
364computing the dimension of the full space of mod~$5$ modular
365symbols where complex conjugation acts as~$+1$, and checking that it
366equals $696$, the dimension of $S_5(1376,\tilde{\eps},\C)$, which we
367computed using the formula in \cite{cohen-oesterle}.
368\end{remark}
369
370Let~$f$ be the eigenform in~$V_1$ that satisfies
371$a_5=\alp^{22}$; the $q$-expansion of~$f$ begins
372$$f=q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7 373 + \alp^{14}q^9 + 4q^{11}+\cdots.$$
374Further eigenvalues are given in Table~\ref{table:1376}.
375The primes~$p$ in the table such that~$a_p=0$ are
376exactly those
377predicted by considering the splitting behavior of~$h$.
378This is strong evidence that~$\rho$ is modular,
379and also that our modular symbols algorithm have been correctly
380implemented.
381
382\begin{table}
383\caption{\label{table:1376}Eigenvalues of~$f$}
384\begin{center}
385$$\begin{array}{|rl|}\hline 3862&0\\ 3873&\alpha^{16}\\ 3885&\alpha^{22}\\ 3897&\alpha^{14}\\ 39011&4\\ 39113&\alpha^{14}\\ 39217&\alpha^{14}\\ 39319&0\\ 39423&\alpha^{16}\\ 39529&\alpha^{8}\\ 39631&0\\ 39737&\alpha^{10}\\ 39841&1\\ 39943&\alpha^{10}\\ 40047&1\\ 40153&\alpha^{22}\\ 402\hline\end{array} 403\begin{array}{|rl|}\hline 40459&4\\ 40561&\alpha^{14}\\ 40667&\alpha^{4}\\ 40771&\alpha^{20}\\ 40873&\alpha^{2}\\ 40979&\alpha^{20}\\ 41083&\alpha^{4}\\ 41189&\alpha^{10}\\ 41297&0\\ 413101&\alpha^{8}\\ 414103&\alpha^{14}\\ 415107&0\\ 416109&\alpha^{10}\\ 417113&2\\ 418127&0\\ 419131&2\\ 420\hline\end{array} 421\begin{array}{|rl|}\hline 422137&0\\ 423139&\alpha^{22}\\ 424149&\alpha^{4}\\ 425151&1\\ 426157&\alpha^{14}\\ 427163&0\\ 428167&\alpha^{22}\\ 429173&4\\ 430179&\alpha^{2}\\ 431181&\alpha^{14}\\ 432191&\alpha^{10}\\ 433193&4\\ 434197&0\\ 435199&3\\ 436211&0\\ 437223&0\\ 438\hline\end{array} 439\begin{array}{|rl|}\hline 440227&\alpha^{10}\\ 441229&0\\ 442233&\alpha^{14}\\ 443239&0\\ 444241&\alpha^{2}\\ 445251&\alpha^{2}\\ 446257&3\\ 447263&\alpha^{16}\\ 448269&2\\ 449271&\alpha^{8}\\ 450277&0\\ 451281&\alpha^{16}\\ 452283&0\\ 453293&3\\ 454307&\alpha^{4}\\ 455311&\alpha^{22}\\ 456\hline\end{array} 457\begin{array}{|rl|}\hline 458313&0\\ 459317&0\\ 460331&\alpha^{14}\\ 461337&0\\ 462347&\alpha^{16}\\ 463349&\alpha^{4}\\ 464353&0\\ 465359&0\\ 466367&\alpha^{22}\\ 467373&0\\ 468379&3\\ 469383&3\\ 470389&1\\ 471397&\alpha^{16}\\ 472401&0\\ 473409&2\\ 474\hline\end{array} 475\begin{array}{|rl|}\hline 476419&3\\ 477421&\alpha^{20}\\ 478431&4\\ 479433&\alpha^{4}\\ 480439&\alpha^{20}\\ 481443&0\\ 482449&0\\ 483457&0\\ 484461&0\\ 485463&\alpha^{10}\\ 486467&0\\ 487479&0\\ 488487&\alpha^{8}\\ 489491&\alpha^{2}\\ 490499&\alpha^{20}\\ 491503&\alpha^{2}\\ 492\hline\end{array} 493\begin{array}{|rl|}\hline 494509&\alpha^{8}\\ 495521&\alpha^{10}\\ 496523&\alpha^{14}\\ 497541&\alpha^{20}\\ 498547&\alpha^{22}\\ 499557&3\\ 500563&1\\ 501569&\alpha^{16}\\ 502571&\alpha^{22}\\ 503577&\alpha^{14}\\ 504587&\alpha^{20}\\ 505593&0\\ 506599&\alpha^{22}\\ 507601&0\\ 508607&\alpha^{16}\\ 509613&2\\ 510\hline\end{array} 511\comment{\begin{array}{|rl|}\hline 512617&0\\ 513619&\alpha^{20}\\ 514631&\alpha^{20}\\ 515641&4\\ 516643&1\\ 517647&4\\ 518653&1\\ 519659&\alpha^{14}\\ 520661&2\\ 521673&\alpha^{8}\\ 522677&4\\ 523683&0\\ 524691&\alpha^{16}\\ 525701&\alpha^{14}\\ 526709&4\\ 527719&\alpha^{4}\\ 528\hline\end{array} 529} 530$$
531\end{center}
532\end{table}
533
534\subsection{Twisting into $\GL(2,\F_5)$}
535Although there is a representation
536$\rho_f:\GQ\ra\GL(2,\F_{25})$ attached to $f$,
537it is difficult to say anything about its image without further
538work. We use a trick to show that the image of $\rho_f$ is small.
539Firstly, for a character~$\chi:\GQ\to\Fbar_5$, let~$\tilde\chi$
540denote its Teichm\"uller lift to~$\Qbar_5$. By a result of Carayol,
541there is a characteristic 0 eigenform
542$\tilde{f}\in S_5(N,\tilde{\eps};\Qbar_5)$ lifting $f$.
543The twist $\tilde{g}=\tilde{f} \tensor \tilde{\eps}_{43}$ is, by
544\cite[Prop. 3.64]{shimura:intro}, an eigenform in
545$S_5(43N, \tilde{\eps}_2; \Qbar_5)$, and its reduction is
546a form $g\in S_5(43N,\eps_2,\F_{25})$.
547The eigenvalues $a_p(g) = a_p(f) \eps_{43}(p)$, for the
548first few
549$p\nmid 5N$, are given in Table~\ref{table:1376twist}.
550
551\begin{table}
552\caption{\label{table:1376twist}Eigenvalues of~$g=f\tensor\eps_{43}$}
553\begin{center}
554$$555\begin{array}{|rl|}\hline 5562&*\\%0\\ 5573&1\\ 5585&*\\%3\\ 5597&2\\ 56011&4\\ 56113&2\\ 56217&2\\ 56319&0\\ 56423&1\\ 56529&1\\ 56631&0\\ 56737&3\\ 56841&1\\ 56943&*\\%0\\ 57047&1\\ 57153&2\\ 572\hline\end{array} 573\begin{array}{|rl|}\hline 57459&4\\ 57561&2\\ 57667&4\\ 57771&4\\ 57873&3\\ 57979&4\\ 58083&4\\ 58189&3\\ 58297&0\\ 583101&1\\ 584103&2\\ 585107&0\\ 586109&3\\ 587113&2\\ 588127&0\\ 589131&2\\ 590\hline\end{array} 591\begin{array}{|rl|}\hline 592137&0\\ 593139&2\\ 594149&4\\ 595151&1\\ 596157&2\\ 597163&0\\ 598167&2\\ 599173&4\\ 600179&3\\ 601181&2\\ 602191&3\\ 603193&4\\ 604197&0\\ 605199&3\\ 606211&0\\ 607223&0\\ 608\hline\end{array} 609\begin{array}{|rl|}\hline 610227&3\\ 611229&0\\ 612233&2\\ 613239&0\\ 614241&3\\ 615251&3\\ 616257&3\\ 617263&1\\ 618269&2\\ 619271&1\\ 620277&0\\ 621281&1\\ 622283&0\\ 623293&3\\ 624307&4\\ 625311&2\\ 626\hline\end{array} 627\begin{array}{|rl|}\hline 628313&0\\ 629317&0\\ 630331&2\\ 631337&0\\ 632347&1\\ 633349&4\\ 634353&0\\ 635359&0\\ 636367&2\\ 637373&0\\ 638379&3\\ 639383&3\\ 640389&1\\ 641397&1\\ 642401&0\\ 643409&2\\ 644\hline\end{array} 645\begin{array}{|rl|}\hline 646419&3\\ 647421&4\\ 648431&4\\ 649433&4\\ 650439&4\\ 651443&0\\ 652449&0\\ 653457&0\\ 654461&0\\ 655463&3\\ 656467&0\\ 657479&0\\ 658487&1\\ 659491&3\\ 660499&4\\ 661503&3\\ 662\hline\end{array} 663\begin{array}{|rl|}\hline 664509&1\\ 665521&3\\ 666523&2\\ 667541&4\\ 668547&2\\ 669557&3\\ 670563&1\\ 671569&1\\ 672571&2\\ 673577&2\\ 674587&4\\ 675593&0\\ 676599&2\\ 677601&0\\ 678607&1\\ 679613&2\\ 680\hline\end{array} 681\begin{array}{|rl|}\hline 682617&0\\ 683619&4\\ 684631&4\\ 685641&4\\ 686643&1\\ 687647&4\\ 688653&1\\ 689659&2\\ 690661&2\\ 691673&1\\ 692677&4\\ 693683&0\\ 694691&1\\ 695701&2\\ 696709&4\\ 697719&4\\ 698\hline\end{array} 699\comment{ 700\begin{array}{|rl|}\hline 701727&4\\ 702733&0\\ 703739&2\\ 704743&2\\ 705751&4\\ 706757&3\\ 707761&3\\ 708769&0\\ 709773&0\\ 710787&4\\ 711797&1\\ 712809&3\\ 713811&1\\ 714821&2\\ 715823&3\\ 716827&3\\ 717\hline\end{array} 718\begin{array}{|rl|}\hline 719829&2\\ 720839&0\\ 721853&2\\ 722857&0\\ 723859&0\\ 724863&4\\ 725877&1\\ 726881&1\\ 727883&0\\ 728887&2\\ 729907&0\\ 730911&1\\ 731919&1\\ 732929&0\\ 733937&3\\ 734941&4\\ 735\hline\end{array} 736\begin{array}{|rl|}\hline 737947&2\\ 738953&1\\ 739967&4\\ 740971&3\\ 741977&0\\ 742983&0\\ 743991&3\\ 744997&3\\ 745&\\ 746&\\ 747&\\ 748&\\ 749&\\ 750&\\ 751&\\ 752&\\ 753\hline\end{array}} 754$$
755\end{center}
756\end{table}
757
758\begin{proposition}\label{prop:1376-g}
759Let $g=f\tensor \eps_{43}$.  Then $a_p(g)\in \F_5$
760for all~$p\nmid \ell N$.
761\end{proposition}
762\begin{proof}
763Consider an eigenform $\tilde{f}\in S_5(N,\tilde{\eps};\Qbar_5)$
764lifting~$f$ as above.
765Associated to~$\tilde{f}$ there is an automorphic
766representation~$\pi=\tensor_v'\pi_v$ of $\GL(2,\bA)$, where~$\bA$
767is the ad\{e}le ring of~$\Q$.
768Because $43\mid\mid N$, and~$43$ divides the conductor
769of $\eps$, we see that the local component $\pi_{43}$ of $\pi$ at
770$43$ must be ramified principal series. By Carayol's theorem,
771$\rho_{\tilde{f}}|_{D_{43}} \sim 772 \abcd{\Psi_1}{0}{0}{\Psi_2}$
773with, without loss of generality,~$\Psi_2$ unramified.  We have
774$(\Psi_1\cdot \Psi_2)|_{I_{43}}=\tilde{\eps}|_{I_{43}}=\tilde{\eps}_{43}$,
775therefore, $\rho_{\tilde{f}}|_{I_{43}} \sim 776 \abcd{\tilde{\eps}_{43}}{0}{0}{1}$.
777
778Now twist~$\tilde{f}$ by $\tilde{\eps}_{43}^{-1}$; we find that
779$\rho_{\tilde{f}\tensor\tilde{\eps}_{43}^{-1}}|_{I_{43}} \sim 780 \abcd{1}{0}{0}{\tilde{\eps}^{-1}_{43}}$.
781In particular, there is an
782eigenform~$\tilde{f}'\in S_5(N,\tilde{\eps}_2\tilde{\eps}^{-1}_{43},\Qbar_5)$
783whose associated Galois representation is the twist by $\tilde{\eps}^{-1}_{43}$
784of that of $\tilde{f}$ (recall that $N=1376$ and so~$43$ divides~$N$
785exactly once). Let~$f'$ denote the mod~$5$ reduction of~$\tilde{f}'$. Then
786one checks easily that $f'\in S_5(N,\eps_2\eps^{-1}_{43},\F_{25})=S_5(N,\eps^5,\F_{25})$.
787
788For all primes $p\nmid5N$ we have $a_p(f')=\eps_{43}(p)^{-1}a_p(f)$.
789In particular, we have $a_p(f')=0$ for
790$p=19,31$.
791Also, $\eps_{43}(3)=\alp^8$ and $\eps_{43}(5) =\alp^8$, so
792$$a_3(f')=\alp^{16}/\alp^8 = \alp^8 = (\alp^{16})^5$$
793$$a_5(f')=\alp^{22}/\alp^8 = \alp^{14} = (\alp^{22})^5.$$
794Now if $\sigma$ is the non-trivial automorphism of $\F_{25}$,
795then $\sigma(f')$ and $f$ both lie in
796$S_5(1376,\eps;\F_{25})$ and have same~$a_p$ for
797$p=3,5,19,31$, so they are equal because we found~$f$
798by computing the unique eigenform with given~$a_p$ for $p=3,5,19,31$.
799So $g = f\tensor\eps_{43} = \sigma(f)\tensor\eps_{43}^2$.
800Thus for all $p\nmid 5N$, we see that
801$a_p(g) = a_p(f)^5 \eps_{43}^2$ has fifth power
802$a_p(g)^5 = a_p(f)^{25} \eps_{43}^{10} 803 = a_p(f) \eps_{43} = a_p(g)$.
804\end{proof}
805
806\subsection{Proof that~$\rho_g$ is unramified at~$5$}
807
808We begin with a generalisation of~\cite{sturm:cong}.
809Let $M>4$ be an integer, and let $h=\sum_{n\geq1}c_nq^n$ be a
810normalised cuspidal eigenform
811of some weight~$k\geq1$, level~$M$ and character~$\chi$, defined over some
812field of characteristic not dividing~$M$. Even though the base field
813might not have characteristic zero, we may still define the conductor
814of $\chi$ to the the largest divisor $f$ of $M$ such that~$\chi$
815factors through $(\Z/f\Z)^\times$.
816Let~$I$ be a set of primes, with the property that for all~$p$
817in~$I$, one of the following conditions hold:
818
819(i) $p$ divides~$M$ but~$p$ does not divide~$M/\cond(\chi)$, or
820
821(ii) $p$ divides~$M$ exactly once, and $h$ is $p$-new, in the sense
822that there is no eigenform $h'$ of level $M/p$ such
823that the $T_n$-eigenvalues of $h$ and $h'$ agree for all $n$
824prime to $p$.
825
826Let~$C$ denote the orbit of the cusp~$\infty$ in $X_1(M)$
827under the action of the group generated by $w_p$ for $p\in I$, and
828the Diamond operators $\langle d\rangle_M$. The orbit of~$\infty$
829under the Diamond operators has size $\phi(M)/2$, and each
830$w_p$ increases the size of the orbit by a factor of~2. In this
831situation, we have
832
833\begin{lemma} The first~$t$ terms of the $q$-expansion
834of~$h$ at any cusp in~$C$ are determined by~$M$,~$k$, $\chi$, $c_p$
835for~$p$ in~$I$, and $c_n$ for $1\leq n\leq t$.
836
837\end{lemma}
838
839\begin{remark} Our proof is just a translation of Corollary~4.6.18
840of \cite{miyake} into the language of moduli problems (Miyake's argument
841technically is only valid over the complex numbers).
842\end{remark}
843\begin{proof}
844If $J\subseteq I$ is any subset, and $w_J$ denotes the product
845of $w_p$ for $p\in J$, then $h|w_J$ is an eigenform for all the
846Diamond operators, and this observation reduces the proof
847of the lemma to showing that for $p\in I$, if $h|w_p=\sum_n d_nq^n$,
848then $d_j$ for $1\leq j\leq n$ and $d_q$ for all $q\in I$
849are determined by $M$, $k$, $\chi$, $p$, $c_j$ for $1\leq j\leq n$
850and $c_q$ for all $q\in I$.
851
852We first deal with primes $p$ of the form (i).
853Say $M=p^mR$, where $R$ is prime to $p$.
854Thinking of~$h$ as a rule for attaching
855$k$-fold differentials to elliptic curves equipped with points
856of order $p^m$ and $R$, we have by definition that
857$$h(\G_m/q^\Z,\zeta,\zeta_R)=\bigg(\sum c_nq^n\bigg)(dt/t)^k,$$
858where $\zeta=\zeta_{p^m}$ and $\zeta_R$ are fixed $p^m$th and $R$th roots
859of unity in $\G_m$ which correspond to the cusp~$\infty$,
860and $dt/t$ is the canonical differential on the Tate
861curve $\G_m/q^\Z$. We normalise things such that
862$$h(\G_m/q^{p^m\Z},q,\zeta_R)=\bigg(\sum d_nq^n\bigg)(dt/t)^k,$$
863and remark that because $h$ is an action for the diamond operators,
864we do not have to worry too much about whether this corresponds to
865the standard normalisation of the $w_p$-operator.
866
867We recall that the operator $pU_p$ in this setting can be thought
868of as being defined by the rule:
869$$(pU_ph)(E,P,Q)=\sum_C\pi^*h(E/C,\overline{P},\overline{Q}),$$
870where $C$ runs through the subgroups of $E$ of order $p$ which have
871trivial intersection with $\langle P\rangle$, and $\pi$ denotes the canonical
872projection $E\to E/C$.
873We see that
874\begin{align*}
875(pc_p)^m\big(\sum d_nq^n\big)(dt/t)^k&=(p^mU_{p^m}h)(\G_m/q^{p^m\Z},q,\zeta_R)\\
876&=\sum_{c=0}^{p^m-1}\pi^c*h(\G_m/\langle q^{p^m},\zeta q^c\rangle,q,\zeta_R),
877\end{align*}
878where $\pi$ denotes the canonical projection from $\G_m/\langle q^{p^m}\rangle$
879to the appropriate quotient. This last sum can be written as a
880double sum
881\begin{align*}
882&\sum_{c\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle q^{p^m},\zeta q^c\rangle,q,\zeta_R)+\sum_{a=0}^{p^{m-1}-1}\pi^*h(\G_m/\langle q^{p^m},\zeta q^{pa}\rangle,q,\zeta_R)\\
883=&\sum_{b\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle q^{p^m},\zeta^{-b}q\rangle,q,\zeta_R)+p^{m-1}\pi^*U_{p^{m-1}}h(\G_m/\langle q^{p^m},\zeta^{p^{m-1}}\rangle,q,\zeta_R)\\
884=&\sum_{b\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle\zeta^{-b}q\rangle,\zeta^b,\zeta_R)+(pc_p)^{m-1}\pi^*h(\G_m/\langle q^{p^m},\zeta^{p^{m-1}}\rangle,q,\zeta_R)\\
885=&\sum_{b}\chi_p(b)\sum_{n\geq1}c_n(\zeta^{-b}q)^n(dt/t)^k+p^k(pc_p)^{m-1}\pi^*h(\G_m/\langle q^{p^{m+1}}\rangle,q^p,\zeta_R^p),
886\end{align*}
887where we have written $\chi=\chi_R\chi_p,$ for $\chi_R$ a character of
888level~$R$ and $\chi_p$ a character of level~$p^m$. We deduce that
889\begin{align*}
890&(pc_p)^m\big(\sum d_nq^n\big)(dt/t)^k-p^k(pc_p)^{m-1}\chi_R(p)\pi^*h(\G_m/\langle q^{p^{m+1}}\rangle,q^p,\zeta_R)\\
891=&\bigg(\sum_n\big(\sum_b\chi_p(b)\zeta^{-bn}\big)c_nq^n\bigg)(dt/t)^k\\
892=&W(\chi_p)\big(\sum_{p\nmid n}\chi_p(-n)^{-1}c_nq^n\big)(dt/t)^k
893\end{align*}
894where $W(\chi_p)=\sum_{b\in(\Z/p^m\Z)^\times}\chi_p(b)\zeta^b$ can be
895checked to be nonzero because the conductor of $\chi_p$ is $p^m$.
896Hence
897$$(pc_p)^m\sum_n d_nq^n-p^k(pc_p)^{m-1}\chi_R(p)\sum_n d_nq^{np}=W(\chi_p)\chi_p(-1)\sum_{p\nmid n}\chi_p(n)^{-1}c_nq^n.$$
898Equating coefficients of $q$ we deduce that $W(\chi_p)\chi_p(-1)=(pc_p)^md_1$,
899and because $h|w_p$ is an eigenform for $T_n$ for all $n$ prime to $p$,
900with eigenvalues determined by $\chi$ and $c_n$, we deduce that we can
901determine $d_n$ for $n$ prime to $p$ from $c_n$. It remains to establish
902what $d_p$ is, and equating coefficients of $q^p$ in the above equation
903gives us that $(pc_p)^md_p=p^k(pc_p)^{m-1}\chi_R(p)d_1$ and hence
904that $d_p$ is determined by $\chi$ and $c_p$.
905Note that as a consequence we see that $d_p/d_1=p^{k-1}\chi_R(p)/c_p$,
906a classical formula if the base field is the complexes.
907
908Now we deal with primes of the form (ii) (note that we never use
909this case in the rest of the paper). We think of $h$ as a rule associating
910$k$-fold differentials to triples $(E,C,Q)$ where $C$ a cyclic subgroup of
911order~$p$ and $Q$ a point of order~$R=M/p$. Because $h$ is $p$-new, the
912trace of $h$ down to $X_1(M/p)$ must be zero, and hence we see
913that for any elliptic curve $E$ equipped with a point $Q$ of order $R$,
914$$\sum_C\pi^* h(E/C,E[p]/C,\overline{Q})=0.$$ As before, normalise things so
915that
916$$h(\G_m/q^\Z,\mu_p,\zeta_R)=\bigg(\sum_n c_nq^n\bigg)(dt/t)^k$$
917and
918$$h(\G_m/q^{p\Z},\langle q\rangle,\zeta_R)=\bigg(\sum_n d_nq^n\bigg)(dt/t)^k.$$
919The fact that the trace of~$h$ is zero implies that
920$$(pU_p)h(\G_m/q^{p\Z},\langle q\rangle,\zeta_R)+\pi^*h(\G_m/q^\Z,\mu_p,\zeta_R)=0,$$
921and hence that
922$$c_p\sum d_nq^n+p^{k-1}\sum c_nq^n=0$$
923from which we deduce that the $d_n$ can be read off from $c_p$ and the $c_n$.
924\end{proof}
925\begin{remark} The size of~$C$ is
926$\phi(M).2^{|I|-1}$, and the usefulness of this lemma is that
927if $h_1$ and $h_2$ are two normalised eigenforms of the same level,
928weight and character as above, both new at all primes in~$I$,
929and the coefficients of $q^n$ in the
930$q$-expansions of $h_1$ and $h_2$ agree for $n\in I$ and $n\leq t$,
931then $h_1-h_2$ has a zero of order at least $t+1$ at all cusps in~$C$,
932and in particular if
933$\phi(M).2^{|I|-1}(t+1)>k/12[\SL_2(\Z):\Gamma_1(M)]=\deg(\omega^k)$ on
934$X_1(M)$ then $h_1=h_2$. Using the fact that
935$[\Gamma_0(M):\Gamma_1(M)]=\phi(M)/2$,
936we deduce
937\end{remark}
938\begin{corollary}\label{cor:bound}
939Let $h_1$ and $h_2$ be two normalised eigenforms as above.
940If the coefficients of $q^n$ in the $q$-expansions of $h_1$ and $h_2$ agree
941for all primes in $I$ and for all
942$n\leq\frac{k}{12}[\SL_2(\Z):\Gamma_0(M)]/2^{|I|}$ then $h_1=h_2$.
943\end{corollary}
944\begin{remark} One can certainly do better than this corollary
945in many cases. For example, when $n>1$ and
946$p^n$ exactly divides both the level
947of an eigenform and the conductor of its character, then one can compute
948the $q$-expansion of the eigenform at many middle cusps'' too,
949and hence increase the size of $C$ in the result above.
950\end{remark}
951
952We now go back to the explicit situation we are concerned with.
953Although~$g$ is an eigenform of level $59168=2^5\cdot 43^2$,
954we can still consider the corresponding representation
955$\rho_g :\GQ\ra \GL(2,\F_5)$, and then directly analyze
956its ramification.
957\begin{proposition}
958The representation~$\rho_g$ is unramified at~$5$.
959\end{proposition}
960\begin{proof}
961Continuing the modular symbols computations as above,
962we find that~$V_1$ is spanned by the two eigenforms
963\begin{align*}
964f\,\,&=q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7
965 + \alp^{14}q^9 + 4q^{11}+\cdots\\
966f_1&=q + \alp^{16}q^3 + \alp^{10}q^5 + \alp^{14}q^7
967 + \alp^{14}q^9 + 4q^{11}+\cdots.
968\end{align*}
969For $p\neq 5$ and $p\leq 997$, we have $a_p(f_1)=a_p(f)$.
970To check that $a_p(f) = a_p(f_1)$ for all $p\neq 5$,
971it suffices to show that the difference~$f-f_1$ has
972$q$-expansion involving only powers of~$q^5$;
973for this we use the $\theta$-operator
974$q\frac{d}{dq}:S_5(1376,\eps,\F_{25})\ra S_{11}(1376,\eps;\F_{25})$.
975Since~$\theta$ sends normalized eigenforms to normalized eigenforms,
976it suffices to check that the subspace of
977$S_{11}(1376,\eps;\F_{25})$ generated by~$\theta(f)$
978and~$\theta(f_1)$ has dimension~$1$.
979Corollary~\ref{cor:bound} implies that it suffices to verify that the
980coefficients $a_p(\theta(f))$ and $a_p(\theta(f_1))$ are equal for all
981$$p \leq \frac{11}{12}\cdot [\SL_2(\Z):\Gamma_0(1376)]\cdot \frac{1}{2} 982 = 968.$$
983The eigenform~$f$ must be new because we computed it by finding
984the intersections of the kernels of Hecke operators $T_p$ with
985$p\nmid 1376$; if~$f$ were an oldform then the intersection of the
986kernels of these Hecke operators
987would necessarily have dimension greater than~$1$.
988Because it takes less than a second
989to compute each $a_p(\theta(f))$, we were easily able to verify that the
990space generated by $\theta(f)$ and $\theta(f_1)$ has dimension~$1$.
991
992\begin{remark}
993It is possible to avoid appealing to Corollary~\ref{cor:bound} by using
994one of the following two alternative methods:
995\begin{enumerate}
996\item Define~$\theta$ directly on modular symbols and compute it.
997\item Compute the intersection
998  $$\bigcap_{p\geq 2} \ker(T_p - pa_p(f)) 999 \subset S_{11}(1376,\eps;\F_{25}).$$
1000  Since~$\theta(f)$ and~$\theta(f_1)$ both lie
1001  in the intersection, the moment the dimension
1002  of a partial intersection is~$1$, it follows
1003  that $\theta(f-f_1)=0$.
1004\end{enumerate}
1005We successfully carried out both alternatives.
1006For the first, we showed that~$\theta$ on modular symbols is
1007induced by multiplication by
1008$X^5Y - Y^5X$.
1009For the second, we find that after intersecting
1010kernels for $p\leq 11$, the dimension is already~$1$.
1011The first of these two methods took much less
1012time than the second.
1013\end{remark}
1014
1015Next we use that $\theta(f-f_1)=0$ to show that $\rho_g$ is unramified,
1016thus finishing the proof of the proposition.
1017Since~$f$ is ordinary, Deligne's theorem (see~\cite[\S12]{gross:tameness})
1018implies that
1019$$\rho_f|_{D_5}\sim 1020 \mtwo{\alp}{*}{0}{\beta}\qquad\text{over \Fbar_5}$$
1021with both~$\alp, \beta$ unramified,
1022$\alp(\Frob_5)=\eps(5)/a_5=\alp^8/\alp^{22}=\alp^{10}$, and
1023$\beta(\Frob_5)=\alp^{22}$.
1024Since $a_p(f_1)=a_p(f)$, for $p\neq 5$, we have
1025$${\rho_f}|_{D_5}\sim {\rho_{f_1}}|_{D_5} \sim \mtwo{\alp'}{*}{0}{\beta'}$$
1026with
1027$\alp'(\Frob_5)=\alp^8/\alp^{10}=\alp^{22}$ and
1028$\beta'(\Frob_5)=\alp^{10}$;
1029in particular, $\alp'=\beta$.
1030Thus $\rho_f|_{D_5}$ contains $\alp\oplus \beta$, so
1031$\rho_f|_{D_5}\sim\alp\oplus\beta$ and hence there is a choice
1032of basis so that $*=0$.
1033
1034\end{proof}
1035
1036
1037
1038\subsection{The image of $\proj \rho_g$}
1039\begin{proposition}\label{prop:image_is_A5}
1040The image of $\proj \rho_g$ is $A_5$.
1041\end{proposition}
1042\begin{proof}
1043
1044The image~$H$ of $\proj \rho_g$ in $\PGL_2(\F_5)$ is easily checked to
1045lie in $\PSL_2(\F_5)\cong A_5$ because of what we know about the
1046determinant of $\rho_g$. Hence $H$ is a subgroup of $A_5$ that
1047contains an element of order~$2$ (complex conjugation) and an element
1048of order~$3$ (for example, $\rho_g(\Frob_7)$ has characteristic
1049polynomial $x^2-2x-1$).  This proves that~$H$ is isomorphic to
1050either~$S_3$,~$A_4$, or~$A_5$.  Let $L$ be the number field cut out
1051by~$H$.  If~$L$ were an $S_3$-extension, then there would be a
1052quadratic extension contained in it which is unramified outside
1053$2\cdot 5\cdot 43$; it is furthermore unramified at~$5$ by the
1054previous section and unramified at $43$ because $I_{43}$ has
1055order~$3$.  Thus it is one of the three quadratic fields unramified
1056outside~$2$.
1057In particular, the trace of $\Frob_p$
1058would be zero for all primes in a certain congruence class
1059modulo~8.
1060However, there are primes~$p$ congruent to $3$, $5$, and $7$
1061mod $8$ such that $a_p(g)\neq 0$, e.g., $3$, $7$, and $13$.
1062
1063
1064If $H$ were isomorphic to $A_4$, then let~$M$ denote the cyclic
1065extension of degree~3 over~$\Q$ contained in~$L$. Now~$M$ is unramified
1066at~2 and~5, and hence is the subfield of $\Q(\zeta_{43})$ of degree~3.
1067Choose $p\nmid 1376\cdot 5$ that is inert in~$M$, i.e., so that
1068$p$ is not a cube mod $43$.  The order of
1069$\rho_g(\Frob_p)$ in $\GL_2(\F_5)$ must be divisible by~$3$.  However,
1070a quick check using Table~\ref{table:1376twist} shows that this is
1071usually not the case, even for $p=3$.
1072\end{proof}
1073
1074
1075\subsection{Bounding the ramification at~$2$ and~$43$}
1076Let~$L$ be the fixed field of $\ker(\proj(\rho_g))$. We have just
1077shown that $\Gal(L/\Q)$ is isomorphic to $A_5$.
1078By a root field for~$L$, we mean
1079a non-Galois extension of $\Q$ of degree~5 whose Galois closure is~$L$.
1080\begin{proposition}
1081The discriminant of a root
1082field for~$L$ divides $(43\cdot 8)^2=344^2$, and
1083in particular,~$L$ must be mentioned in Table~1
1084of \cite[pg 122]{freyetal}.\end{proposition}
1085\begin{proof}
1086The analysis of the local behavior of~$\rho_f$ at~$43$ given in
1087Proposition~\ref{prop:1376-g}
1088shows that the inertia group at~$43$ in $\Gal(L/\Q)$ has order~$3$. Using
1089Table~3.1 of~\cite{buhler:thesis}, we see that if
1090$\Gal(L/\Q)\isom A_5$
1091then it must be type~$2$'' at 43, and hence the discriminant of a root
1092field of~$L$, that is, of a non-Galois extension of~$\Q$ of degree~$5$
1093whose Galois closure is~$L$, must be $43^2$ at~$43$.
1094
1095At~$2$ the behavior of~$\rho$ is more subtle and we shall not analyze
1096it fully. But we can say that, because~$\rho$ has arisen from
1097a form of level $1376=2^5.43$, we must be either of type~$5$
1098or one of types~$14$--$17$. In particular, the discriminant at~$2$ of a root
1099field for~$L$ will be at most~$2^6$.
1100
1101Finally,~$L$ is unramified at all other primes, because~$\rho$ is.
1102Hence the discriminant of a root field for~$L$, assuming that
1103$\Gal(L/\Q)\cong A_5$, divides $(43.8)^2=344^2$.
1104\end{proof}
1105
1106We know that~$L$ is an icosahedral extension of~$\Q$ with
1107discriminant dividing $43^2\cdot 2^6$.  Table~1 of \cite[pg 122]{freyetal}
1108contains all icosahedral extensions, such that the discriminant
1109of a root field is bounded by $2083^2$.  The table
1110must contain~$L$; there is only one icosahedral extension with
1111discriminant dividing $43^2\cdot 2^6$, so $L=K$.
1112
1113\subsection{Obtaining a classical weight one form}
1114We have shown that a twist of the icosahedral
1115representation $\rho:\GQ\ra\GL(2,\C)$,
1116nobtained by lifting $\GQ\ra \Gal(K/\Q)\ncisom A_5$,
1117has a mod~$5$ reduction $\rho_g:\GQ\ra \GL_2(\F_5)$ that
1118is modular.  Since~$\rho$ ramifies at only finitely many primes,
1119and~$\rho$ is unramified at~$5$ with distinct eigenvalues,
1120\cite{buzzard-taylor} implies that~$\rho$ arises from
1121a classical weight~$1$ newform.
1122
1123
1124
1125\section{More examples}
1126The data necessary to deduce modularity of each of our eight
1127icosahedral examples is summarized in
1128Tables~\ref{table:more1}--\ref{table:more4}.
1129
1130The notation in Table~\ref{table:more1} is as follows.
1131The first column contains the conductor.
1132The second column contains a $5$-tuple $[a_4,a_3,a_2,a_1,a_0]$ such
1133that the $A_5$-extension is the splitting field of the polynomial
1134$h=x^5+a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$.
1135The column labeled $\ord(\Frob_5)$ contains the order of the image
1136of $\Frob_5$ in $A_5$.  The next column, which is labeled $p$ with
1137$a_p=0$'', contains the first few~$p$ such that $a_p$ is easily seen
1138to equal~$0$ by considering the splitting of~$h$ mod~$p$.
1139The $\eps$ column contains the character of the representation, where the
1140notation is as follows. Write $(\Z/N\Z)^*$ as a product of cyclic groups
1141corresponding to the prime divisors of~$N$ in ascending order, and then
1142the tuples give the orders of the images of these cyclic factors; when
1143$8\mid N$, there are two cyclic factors corresponding to the prime~$2$.
1144Finally, the last column records the dimension of $S_5(\Gamma_1(N),\eps)$.
1145
1146The notation in Table~\ref{table:more2} is as follows.  The first column
1147contains the conductor.  The second column contains an eigenform that
1148was found by first intersecting the kernels of the Hecke operators
1149$T_p$ with~$p$ as in Table~\ref{table:more1}, and then locating an
1150eigenform.
1151In each case, a companion form was found, by computing $a_p(f)$ for
1152$p\leq$ bound, where bound is the bound from Corollary~\ref{cor:bound}.
1153
1154Table~\ref{table:more3} shows that the fixed field
1155of the image of each $\proj(\rho_g)$ is icosahedral.
1156The first column contains the
1157conductor~$N$.  The second column contains a twist~$g$ of~$f$ such that
1158$a_p(g)\in\F_5$ for all $p\nmid 5N$.  The third column contains
1159a $\Frob_p$ such that $\proj(\rho_g(\Frob_p))$ has order~$3$,
1160along with the characteristic polynomial of $\rho_g(\Frob_p)$.
1161As in the proof of Proposition~\ref{prop:image_is_A5},
1162the other two boxes give data that allows us to deduce
1163that the fixed field of the image of $\proj(\rho_g)$ is icosahedral.
1164The case $5373$ must be treated separately, because there are
1165three possibilities $M_1$, $M_2$, and $M_3$
1166for the cubic field~$M$ of the analogue of
1167Proposition~\ref{prop:image_is_A5}.
1168For $M_1$ we find a prime~$p$ such that
1169$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,1),(7,1)\}$$
1170with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1171for this, $p=2$ suffices, since the characteristic polynomial
1172of $\rho_g(\Frob_2)$ is $(x+2)^2$
1173and $(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199) = (4,106)$.
1174For $M_2$ we find a prime~$p$ such that
1175$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,92),(7,106)\}$$
1176with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1177again, $p=2$ suffices.
1178For $M_3$ we find a prime~$p$ such that
1179$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,106),(7,92)\}$$
1180with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1181here, $p=13$ suffices, as the characteristic polynomial
1182of $\rho_g(\Frob_p)$ is $(x+4)^2$ and
1183$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199) = (7,106)$.
1184
1185
1186
1187Table~\ref{table:more4} gives upper bounds on the ramification of the
1188fixed field of the image of $\proj(\rho_g)$.  These bounds
1189were deduced using Table~3.1
1190of~\cite{buhler:thesis} by restricting the possible types'' using
1191information about the character $\eps$.  Note that though
1192the bounds are not sharp, e.g., the discriminant of
1193the representation of conductor $2416$ is $2^4\cdot 151^2$, they
1194are all less than $2083^2$, so the corresponding
1195field must appear in Table~2 of~\cite{freyetal}.
1196
1197\begin{table}
1198\caption{\label{table:more1}Data on icosahedral representations mod~$5$}
1199\begin{center}
1200\begin{tabular}{|clclll|}\hline
1201$N$&\hspace{2em}$h$&$\hspace{-1.5em}\ord(\Frob_5)$&$p$ with $a_p=0$&
1202   \hspace{1em}$\eps$ &$\hspace{-1.5em}\dim S_5(N,\eps)$\\\hline
1203{\bf 1376}&$[2,6,8,10,8]$ & \hspace{-1.5em}$2$&$19,31,97$&$[2,1,3]$&$696$\\
1204{\bf 2416}&$[0,-2,2,5,6]$ & \hspace{-1.5em}$2$&$53,97,127$&$[2,1,3]$&$1210$\\
1205{\bf 3184}&$[5,8,-20,-21,-5]$& \hspace{-1.5em}$2$&$31,89,97$&$[2,1,3]$&$1594$\\
1206{\bf 3556}&$[3,9,-6,-4,-40]$&\hspace{-1.5em}$3$&$19,29,89$&$[1,2,3]$&$2042$\\
1207{\bf 3756}&$[0,-3,10,30,-18]$&\hspace{-1.5em}$3$&$17,61,67$&$[1,2,3]$&$2506$\\
1208{\bf 4108}&$[4,3,9,4,5]$& \hspace{-1.5em}$3$&$17,23,31,89$&$[1,3,2]$&$2234$\\
1209{\bf 4288}&$[4,5,8,3,2]$& \hspace{-1.5em}$3$&$19,23,47$&$[1,2,3]$&$2164$\\
1210{\bf 5373}&$[2,1,7,23,-11]$& \hspace{-1.5em}$2$&$7,23,37,79,89$&$[2,3]$&$2394$\\
1211\hline\end{tabular}
1212\end{center}
1213\end{table}
1214
1215\begin{table}
1216\caption{\label{table:more2}The newform $f$ and the companion form bound}
1217\begin{center}
1218\begin{tabular}{|cll|}\hline
1219$N$&\hspace{7em}$f$ &\hspace{-.3em}bound \\\hline
1220{\bf 1376}&
1221    $q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7 1222 + \alp^{14}q^9 + 4q^{11}+ \alp^{14}q^{13} + \cdots$
1223    & $968$\\
1224{\bf 2416}&
1225    $q +3q^3 + \alp^{22}q^5 + \alp^{16}q^7 + \alp^{4} q^{11} 1226 + \alp^2 q^{13} + \alp^{16}q^{15} + \cdots$
1227    & $1672$ \\
1228{\bf 3184}&
1229    $q + \alp^{16}q^3 + 3q^5 + \alp^{22}q^7 + \alp^{14}q^9 + 3q^{11} 1230 + \alp^{22}q^{13} + \cdots$
1231    & $2200$\\
1232{\bf 3556}&
1233    $q + \alp^{16}q^{3} + \alp^{14}q^5 + \alp^{10}q^7 + \alp^{14}q^9 1234 + \alp^{2}q^{11} + \alp^{22}q^{13} + \cdots$
1235    & $1408$ \\
1236{\bf 3756}&
1237      $q + \alp^{14}q^3 + \alp^{14}q^5 + 3q^7 + \alp^4q^9 + \alp^{16}q^{11} + \alp^{10}q^{13} + \cdots$
1238    & $1727$ \\
1239{\bf 4108}&
1240    $q + \alp^{16}q^{3} + \alp^{11}q^{5} + \alp^{20}q^{7} + \alp^{14}q^{9} + \alp^{10}q^{11} + 4q^{13} + \cdots$
1241    & $1540$\\
1242
1243{\bf 4288}&
1244     $q + 3q^3 + \alp^{14}q^{5} + \alp^{20}q^{7} + 3q^9 + \alp^{20}q^{11} + \alp^{16}q^{13} + \cdots$
1245     & $2992$ \\
1246
1247
1248{\bf 5373}&
1249    $q + \alp^{16}q^{2} + \alp^{14}q^{4} + 4q^5 + 3q^8 + \alp^{4}q^{10} + 2q^{11}+\cdots$
1250     & $3300$ \\
1251\hline\end{tabular}
1252\end{center}
1253\end{table}
1254
1255
1256\begin{table}
1257\caption{\label{table:more3}Verification that the image of $\proj(\rho_g)$ is $A_5$}
1258\begin{center}
1259 Find a Frobenius element with projective order $3$.\vspace{1ex}\\
1260\begin{tabular}{|c|l|ll|}\hline
1261$N$ & \hspace{1em} $g$ & proj. order $3$&\hspace{.7em}charpoly \\\hline
1262{\bf 1376} & $f\tensor \eps_{43}$
1263    &  $\quad\Frob_7$ & $x^2-2x-1$
1264 \\
1265{\bf 2416}& $f\tensor \eps_{151}$
1266    &  $\quad\Frob_{19}$ & $x^2+2x-1$
1267 \\
1268{\bf 3184}& $f\tensor \eps_{199}$
1269    & $\quad\Frob_7$ & $x^2+3x+4$
1270 \\
1271{\bf 3556}& $f\tensor \eps_{127}$
1272    &  $\quad\Frob_{13}$ & $x^2+3x+4$
1273  \\
1274{\bf 3756}&$f\tensor\eps_{313}$
1275    & $\quad\Frob_{23}$ & $x^2 + 2x + 4$
1276  \\
1277{\bf 4108} & $f\tensor\eps_{13}$
1278    & $\quad\Frob_{29}$ & $x^2+3x+4$
1279 \\
1280{\bf 4288}& $f\tensor\eps_{67}$
1281    & $\quad\Frob_{11}$ & $x^2+x+1$
1282\\
1283
1284{\bf 5373}& $f\tensor\eps_{199}$
1285    & $\quad\Frob_{11}$ & $x^2+3x+4$
1286\\
1287\hline\end{tabular}\vspace{3ex}
1288
1289Not $S_3$: For all $t\in T$, find unramified $p$ s.t.\ $t\not\equiv \Box \mod p$ and $a_p(g)\neq 0$. \vspace{1ex}\\
1290\begin{tabular}{|c|l|l|}\hline
1291$N$ & $\qquad\quad{}T$ & $\qquad{}p$ \\\hline
1292{\bf 1376}
1293    &  $\{-1,-2\}$ & $3$, $7$\\
1294{\bf 2416}
1295    &  $\{-1, -2\}$ & $3$, $7$ \\
1296{\bf 3184}
1297    & $\{-1, -2\}$ &  $3$, $7$ \\
1298{\bf 3556}
1299    &  $\{-1, -2, -7, -14\}$ & $3$, $13$, $3$, $11$\\
1300{\bf 3756}
1301    & $\{-1,-2,-3,-6\}$ &  $7$, $7$, $11$, $13$\\
1302{\bf 4108}
1303    & $\{-1, -2, -79, -158\}$ &$3$, $7$, $3$, $7$ \\
1304{\bf 4288}
1305    & $\{-1,-2\}$ &  $3$, $7$ \\
1306{\bf 5373}
1307    & $\{-3\}$ & $11$\\
1308\hline\end{tabular}\vspace{3ex}\\
1309\comment{
1310function FindS3(t, aplist, N)
1311   P:=[p : p in [2..97] |IsPrime(p)];
1312   for i in [1..#aplist] do
1313      p := P[i];
1314      if (5*N mod p ne 0) and not IsSquare(GF(p)!t) and aplist[i] ne 0 then
1315         return p;
1316      end if;
1317   end for;
1318end function;
1319}
1320
1321
1322Not $A_4$: Unramified~$p$, not cube mod $\ell$,
1323order of $\rho_g(\Frob_p)$ not divisible by $3$.
1324\vspace{1ex}\\
1325\begin{tabular}{|c|c|cl|}\hline
1326$N$ & $\ell$ & $p$&\hspace{.7em}charpoly$(\rho_g(\Frob_p))$ \\\hline
1327{\bf 1376}
1328    & $43$  & $3$ & $\qquad(x+2)^2$ \\
1329{\bf 2416}
1330    &  $151$ & $7$ & $\qquad(x+2)^2$ \\
1331{\bf 3184}
1332    & $199$ &  $3$ & $\qquad(x+2)^2$ \\
1333{\bf 3556}
1334    &  $127$ & $3$ & $\qquad(x+2)^2$\\
1335{\bf 3756}
1336    & $313$ &  $11$ & $\qquad(x+2)^2$\\
1337{\bf 4108}
1338    & $13$ & $3$ & $\qquad(x+2)^2$\\
1339{\bf 4288}
1340    & $67$ & $7$ & $\qquad(x+3)^2$\\
1341{\bf 5373}
1342    & --- & &(see text)\\
1343\hline\end{tabular}\vspace{3ex}\\
1344\comment{
1345function IsCubeMod(p, ell) // is p a cube in F_ell
1346   R<x>:=PolynomialRing(GF(ell));
1347   return not IsIrreducible(x^3-p);
1348end function;
1349
1350procedure FindA4(ell, aplist, N, e1, e2)
1351   P:=[p : p in [2..97] |IsPrime(p)];
1352   for i in [1..#aplist] do
1353      p := P[i];
1354      if (5*N mod p ne 0) then
1355//      if (5*N mod p ne 0) and not IsCubeMod(p,ell) then
1356         t := GF(5)!(Evaluate(e2,p)*aplist[i]);
1357         d := GF(5)!Evaluate(e1,p);
1358         R<x> := PolynomialRing(GF(5));
1359         f := x^2 - t*x + d;
1360"p =",p;
1361"f =",f;
1362"factor(f) =",Factorization(f);
1363      end if;
1364   end for;
1365end procedure;
1366}
1367
1368\end{center}
1369\end{table}
1370
1371
1372\begin{table}
1373\caption{\label{table:more4}Bounding the discrimant of the fixed field
1374of $\proj(\rho_g)$}
1375\begin{center}
1376\begin{tabular}{|cl|}\hline
1377$N$ & Bound on discriminant\\
1378{\bf 1376}& $\qquad2^6\cdot 43^2$\\
1379{\bf 2416}& $\qquad 2^6\cdot 151^2$\\
1380{\bf 3184}& $\qquad 2^6\cdot 199^2$\\
1381{\bf 3556}& $\qquad 2^2\cdot 7^2 \cdot 127^2$\\
1382{\bf 3756}& $\qquad 2^2\cdot 3^2 \cdot 313^2$\\
1383{\bf 4108}& $\qquad 2^2\cdot 13^2 \cdot 79^2$\\
1384{\bf 4288}& $\qquad 2^6\cdot 67^2$\\
1385{\bf 5373}& $\qquad 3^4\cdot 199^2$\\
1386\hline\end{tabular}
1387\end{center}
1388\end{table}
1389
1390\section{Computing mod~$p$ modular forms}
1391\subsection{Higher weight modular symbols}
1392\label{sec:modsym}
1393The second author developed software that computes the space of
1394weight~$k$ modular symbols $\sS_k(N,\eps)$, for $k\geq 2$ and
1395arbitrary~$\eps$.
1396See~\cite{merel:1585} for the standard facts about higher weight
1397modular symbols, and~\cite{stein:phd} for a description of
1398how to compute with them.
1399
1400Let $K=\Q(\eps)$ be the field generated by the values of~$\eps$.
1401 The cuspidal modular symbols $\sS_k(N,\eps)$ are a
1402finite dimensional vector space over~$K$, which is generated by all
1403linear combinations of  higher weight modular symbols
1404    $$X^i Y^{k-2-i}\{\alp,\beta\}$$
1405that lie in the kernel of an appropriate boundary map.  There is an
1406involution~$*$ that acts on $\sS_k(N,\eps)$, and
1407$\sS_k(N,\eps)^+\tensor_K\C$ is isomorphic, as a module over the Hecke
1408algebra, to the space $S_k(N,\eps;\C)$ of cusp forms.
1409
1410Fix $k=5$.  In each case considered in this paper,
1411there is a prime ideal~$\lambda$
1412of the ring of integers $\mathcal{O}$ of~$K$
1413such that $\mathcal{O}/\lambda\isom \F_{25}$.
1414Let~$\cL$ be the $\mathcal{O}$-module generated by all modular
1415symbols of the form $X^iY^{3-i}\{\alp,\beta\}$,
1416and let
1417 $$\sS_5(N,\eps;\F_{25})=(\cL\tensor_{\mathcal{O}}\F_{25})\cap \sS_5(N,\eps).$$
1418This is the space that we computed.
1419The Hecke algebra acts on $\sS_5(N,\eps;\F_{25})$, so when
1420we find an eigenform we find a maximal ideal of the Hecke algebra.
1421
1422As an extra check on our computation of
1423$\sS_5(N,\eps;\F_{25})$, we computed the dimension
1424of $S_5(N,\eps;\C)$ using both the formula of~\cite{cohen-oesterle}
1425and the Hijikata trace formula (see~\cite{hijikata:trace})
1426applied to the identity Hecke operator.
1427
1428
1429\comment{%it's all in my thesis and it's not that relevant.
1430The Manin symbols are
1431$[i, (c,d)]$ where $0\leq i\leq k-2=3$ and
1432$(c,d)$ vary over points in the projective plane.
1433The Manin symbol $[i,(c',d')]$ corresponds to the
1434modular symbol $(g.X^iY^{3-i})\{g(0),g(\infty)\}$
1435where $g=\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ is a matrix whose lower
1436two entries are congruent to $(c',d')$ modulo $N$,
1437and $g.X^iY^{3-i} := (dX-bY)^i(-cX+aY)^{3-i}$.
1438Let $\sigma=\abcd{0}{-1}{1}{0}$, $\tau={0}{-1}{1}{-1}$
1439and for $\gamma\in\SL_2(\Z)$, let
1440$[i,(c,d)]\gamma = [\gamma.X^iY^{3-i}, (c,d)\gamma]$.
1441Since there are only finitely many
1442Manin symbols, we  can
1443compute $\sS_5(N,\eps)$ as the quotient of the $\F$-vector
1444space generated by Manin symbols modulo
1445the following relations:
1446\begin{align*}
1447    {[i,(c,d)] + [i,(c,d)]\sigma} &= 0\\
1448    {[i,(c,d)] + [i,(c,d)]\tau + [i,(c,d)]\tau^2} &= 0\\
1449    {[i,(n c,n d)]}&=\eps(n)[i,(c,d)]
1451\end{align*}
1452The quotient was computed by using a fast hashing'' function
1453to quotient out by the $2$-term relations.  The quotient
1454by the $3$-term relations was then computed using sparse
1455Gauss elimination.  One important subtlety is that, e.g., $\sigma$
1456and~$\tau$ do not commute so, after modding out by
1457the~$\sigma$ relations, it is important to mod out by~$3$
1458term relations coming both from~$\tau$ and~$\sigma\tau$.
1459
1460The main result of~\cite{merel:1585} gives
1461a way to compute the action of $T_p$ directly
1462on the Manin symbols.
1463Suppose $f\in\sS_5(N,\eps;\F_{25})$ is an eigenvector; to
1464naively compute the action of~$T_p$ on~$f$ requires computing
1465the action of~$T_p$ on each Manin symbol involved in~$f$,
1466and then summing the result. This requires roughly
1467$\dim\sS$ times as long as computing~$T_p$ on a single
1468Manin symbol.
1469In order to quickly compute a large number of
1470Hecke eigenvalues we use the following projection trick.
1471Let $\vphi\in\Hom(\sS_5(N,\eps;\F_{25}),\F_{25})$ be a (left) eigenvector for all
1472Hecke operators~$T_p$ having the same eigenvalues as~$f$.
1473Choose a Manin symbol $x=[i,(c,d)]$ such
1474that $\vphi(x)\neq 0$.  Since~$x$ is of a very simple form,
1475it is easy to compute~$T_p(x)$ quickly.  We have
1476 $\vphi(T_p(x)) = (T_p(\vphi))(x) = a_p \vphi(x)$,
1477so since $\vphi(x)\neq 0$ we divide and find
1478$a_p = \vphi(T_p(x))/\vphi(x)$.
1479In fact, we use a generalization of this trick to
1480quickly compute the action of~$T_p$ on any Hecke stable subspace
1481$V\subset \Hom(\sS(N,\eps;\F_{25}),\F_{25})$.
1482}
1483
1484
1485\subsection{Complexity}
1486We implemented the modular symbols algorithms mentioned above
1487in \magma{} (see \cite{magma}) because of its robust support
1488for linear algebra over small finite fields.
1489
1490The following table gives a flavor of the complexity of the
1491machine computations appearing in this paper.
1492The table indicates how much
1493CPU time on a Sun Ultra E450 was required to compute all data
1494for the given level,
1495including the matrices $T_p$ on the $2$-dimensional spaces,
1496for $p<2000$.   For example, the total time for level $N=1376$
1497was~$6$ minutes and~$58$ seconds.
1498\begin{center}
1499\begin{tabular}{|cr|}\hline
1500\vspace{-2ex}&\\
1501 N & time (minutes)\\
1502\vspace{-2ex}&\\
1503 1376&  6:58\hspace{2.5em}\mbox{}\\
1504 2416&  10:42\hspace{2.5em}\mbox{}\\
1505 3184&  14:16\hspace{2.5em}\mbox{}\\
1506 3556&  19:55\hspace{2.5em}\mbox{}\\
1507 3756&  27:47\hspace{2.5em}\mbox{}\\
1508 4108&  23:11\hspace{2.5em}\mbox{}\\
1509 4288&  15:18\hspace{2.5em}\mbox{}\\
1510 5376&  24:49\hspace{2.5em}\mbox{}\\\hline
1511\end{tabular}
1512\end{center}
1513
1514\subsection{Acknowledgment}
1515Some of the computing equipment was purchased
1516by the second author using a UC Berkeley Vice Chancellor Research Grant.
1518made on the Sun Ultra E450 of the Computational Algebra Group at
1519the University of Sydney.   Allan Steel was very helpful in optimizing our
1520code.
1521
1522\comment{\bibliographystyle{amsplain}
1523\bibliography{biblio}}
1524
1525\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
1526\begin{thebibliography}{10}
1527
1528\bibitem{artin:conjecture}
1529E.~Artin, \emph{{\"U}ber eine neue {A}rt von {L}-reihen}, Abh. Math. Sem. in
1530  Univ. Hamburg \textbf{3} (1923/1924), no. 1, 89--108.
1531
1532\bibitem{buhler:thesis}
1533J.\thinspace{}P. Buhler, \emph{Icosahedral \protect{G}alois representations},
1534  Springer-Verlag, Berlin, 1978, Lecture Notes in Mathematics, Vol. 654.
1535
1536\bibitem{bdsbt}
1537K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor, \emph{On
1538  icosahedral {A}rtin representations}, in preparation.
1539
1540
1541\bibitem{buzzard-taylor}
1542K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one forms}, Ann. of
1543  Math. (2) \textbf{149} (1999), no.~3, 905--919.
1544
1545\bibitem{cohen-oesterle}
1546H.~Cohen and J.~Oesterl{\'e}, \emph{Dimensions des espaces de formes
1547  modulaires},  (1977), 69--78. Lecture Notes in Math., Vol. 627.
1548
1549\bibitem{magma}
1550W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system {I}:
1551  {T}he user language}, J. Symb. Comp. \textbf{24} (1997), no.~3-4, 235--265,
1552  \\\protect{\sf http://www.maths.usyd.edu.au:8000/u/magma/}.
1553
1554\bibitem{deligne-serre}
1555P.~Deligne and J-P. Serre, \emph{Formes modulaires de poids $1$}, Ann. Sci.
1556  \'Ecole Norm. Sup. (4) \textbf{7} (1974), 507--530 (1975).
1557
1558\bibitem{freyetal}
1559G.~Frey (ed.), \emph{On {A}rtin's conjecture for odd \protect{$2$}-dimensional
1560  representations}, Springer-Verlag, Berlin, 1994.
1561
1562\bibitem{gross:tameness}
1563B.\thinspace{}H. Gross, \emph{A tameness criterion for \protect{G}alois
1564  representations associated to modular forms (mod \protect{$p$})}, Duke Math.
1565  J. \textbf{61} (1990), no.~2, 445--517.
1566
1567\bibitem{hijikata:trace}
1568H.~Hijikata, \emph{Explicit formula of the traces of \protect{H}ecke operators
1569  for \protect{$\Gamma_0(N)$}}, J. Math. Soc. Japan \textbf{26} (1974), no.~1,
1570  56--82.
1571
1572\bibitem{langlands:basechange}
1573R.\thinspace{}P. Langlands, \emph{Base change for \protect{${\rm {G}{L}}(2)$}},
1574  Princeton University Press, Princeton, N.J., 1980.
1575
1576\bibitem{merel:1585}
1577L.~Merel, \emph{Universal \protect{F}ourier expansions of modular forms}, On
1578  {A}rtin's conjecture for odd \protect{$2$}-dimensional representations
1579  (Berlin), Springer, 1994, pp.~59--94. Lecture Notes in Math., Vol. 1585.
1580
1581\bibitem{miyake}
1582T.~Miyake, \emph{Modular forms}, Springer-Verlag, Berlin, 1989, Translated from
1583  the Japanese by Yoshitaka Maeda.
1584
1585\bibitem{shimura:intro}
1586G.~Shimura, \emph{Introduction to the arithmetic theory of automorphic
1587  functions}, Princeton University Press, Princeton, NJ, 1994, Reprint of the
1588  1971 original, Kan Memorial Lectures, 1.
1589
1590\bibitem{stein:phd}
1591W.\thinspace{}A. Stein, \emph{Explicit approaches to modular abelian
1592  varieties}, U.\thinspace{}C. Berkeley Ph.D. thesis (2000).
1593
1594\bibitem{sturm:cong}
1595J.~Sturm, \emph{On the congruence of modular forms}, Number theory (New York,
1596  1984--1985), Springer, Berlin, 1987, pp.~275--280.
1597   Lecture Notes in Math., Vol. 1240.
1598
1599\bibitem{taylor:artin2}
1600R.~Taylor, \emph{On icosahedral {A}rtin representations II},
1601  in preparation.
1602
1603\bibitem{tunnell:artin}
1604J.~Tunnell, \emph{Artin's conjecture for representations of octahedral type},
1605  Bull. Amer. Math. Soc. (N.S.) \textbf{5} (1981), no.~2, 173--175.
1606
1607\end{thebibliography}
1608
1609
1610\end{document}
1611
1612
1613
1614
1615
1616
1617***
1618
1619[8458981, 509]
1620
1621
1622// Cohen-Oesterle Dimension computations in MAGMA:
1623
1624> G<a2,b2,c> := DirichletGroup(1376,CyclotomicField(EulerPhi(1376)));
1625> eps:=a2*(c^(Order(c) div 3));
1626> Order(eps);
16276
1628> DimensionCuspForms(eps,5);
1629
1630
1631
1632> G<a2,b2,c> := DirichletGroup(2416,CyclotomicField(EulerPhi(2416)));
1633> eps:=a2*(c^(Order(c) div 3));
1634> Order(eps);
16356
1636> DimensionCuspForms(eps,5);
16371210
1638
1639> G<a2,b2,c> := DirichletGroup(3184,CyclotomicField(EulerPhi(3184)));
1640> eps:=a2*(c^(Order(c) div 3));
1641> Order(eps);
16426
1643> DimensionCuspForms(eps,5);
16441594
1645
1646
1647> G<a,b,c> := DirichletGroup(3556,CyclotomicField(EulerPhi(3556)));
1648> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 3));
1649> Order(eps);
16506
1651> DimensionCuspForms(eps,5);
16522042;
1653
1654
1655> G<a,b,c> := DirichletGroup(3756,CyclotomicField(EulerPhi(3756)));
1656> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 3));
1657> Order(eps);
16586
1659> DimensionCuspForms(eps,5);
1660
1661
1662> G<a,b,c> := DirichletGroup(4108,CyclotomicField(EulerPhi(4108)));
1663> eps:=(b^(Order(b) div 3))*(c^(Order(c) div 2));
1664> DimensionCuspForms(eps,5);
1665
1666> G<a,b,c> := DirichletGroup(4288,CyclotomicField(EulerPhi(4288)));
1667> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 2));
1668> DimensionCuspForms(eps,5);
1669
1670
1671> G<a,b> := DirichletGroup(5373,CyclotomicField(EulerPhi(5373)));
1672> eps:=(a^(Order(a) div 2))*(b^(Order(b) div 3));
1673> DimensionCuspForms(eps,5);
1674
1675
1676///////////////////////////////////////////
1677
1678procedure powfrob(p, e1, e2, aplist)
1679   Primes := [p : p in [2..97] |IsPrime(p)];
1680   n := Index(Primes,p);
1681   a := GF(5)!(Evaluate(e2,p)*aplist[n]);
1682   b := GF(5)!Evaluate(e1,p);
1683   R<x>:=PolynomialRing(GF(5));
1684   Q<y> := quo<R|x^2-a*x+b>;
1685   x^2 - a*x + b;
1686   for i in [1..24] do
1687      f := MinimalPolynomial(y^i);
1688      if Degree(f) le 1 then
1689         printf "rho_g(Frob_%o)^%o satisfies %o.\n", p, i, f;
1690      end if;
1691   end for;
1692end procedure;
1693
1694
1695
1696> // 1376
1697> h := x^5+2*x^4+6*x^3+8*x^2+10*x+8;
1698> N := 2^5*43;
1699> F<alp> := GF(25);
1700> G<a2,b2,c>:=DirichletGroup(N, F);
1701> eps := a2 * (c^(Order(c) div 3));
1702> aplist := [0,alp^16,alp^22,alp^14,4,alp^14,alp^14,0, alp^16,alp^8,0,alp^10,1,alp^10,1,alp^22,4,alp^14,alp^4,alp^20,alp^2,alp^20,alp^4,alp^10];
1703> qEigenform(aplist,eps,5);
1704//q + alp^16*q^3 + alp^22*q^5 + alp^14*q^7 + alp^14*q^9 + 4*q^11 + alp^14*q^13 + alp^14*q^15 + alp^14*q^17 + O(q^20)
1705
1706
1707
1708> // 2416
1709> h := x^5-2*x^3+2*x^2+5*x+6;
1710> N := 2^4*151;
1711> k:=5;
1712> F<alp> := GF(25);
1713> G<a2,b2,c>:=DirichletGroup(N, F);
1714> eps := a2 * (c^(Order(c) div 3));
1715> aplist:=[0,3,alp^22,alp^16, alp^4, alp^2, alp^22,3,alp^22,3,alp^16,alp^22,2,alp^8,alp^8,0,1,alp^8,3,alp^8,2,2,2,alp^20,0,2,alp^16,1];
1716> qEigenform(aplist,eps,k);
1717// q + 3*q^3 + alp^22*q^5 + alp^16*q^7 + alp^4*q^11 + alp^2*q^13 + alp^16*q^15 + alp^22*q^17 + 3*q^19 + O(q^20)
1718
1719> // 3184
1720> h := x^5+5*x^4+8*x^3-20*x^2-21*x-5;
1721> N:=2^4*199;
1722> F<alp> := GF(25);
1723> G<a2,b2,c>:=DirichletGroup(N, F);
1724> eps := a2 * (c^(Order(c) div 3));
1725> aplist:=[0,alp^16,3,alp^22,3,alp^22,3,alp^16];
1726> qEigenform(aplist,eps,5);
1727// q + alp^16*q^3 + 3*q^5 + alp^22*q^7 + alp^14*q^9 + 3*q^11 + alp^22*q^13 + alp^10*q^15 + 3*q^17 + alp^16*q^19 + O(q^20)
1728
1729> // 3556
1730> h := x^5+3*x^4+9*x^3-6*x^2-4*x-40;
1731> N := 2^2*7*127;
1732> F<alp> := GF(25);
1733> G<a,b,c>:=DirichletGroup(N, F);
1734> eps := b^(Order(b) div 2) * (c^(Order(c) div 3));
1735> aplist := [0,alp^16,alp^14,alp^10,alp^2,alp^22,alp^14,0, alp^10,0,alp^16,alp^20];
1736
1737
1738> // 3756
1739> h := x^5-3*x^3+10*x^2+30*x-18;
1740> N := 2^2*3*313;
1741> F<alp> := GF(25);
1742> G<a,b,c>:=DirichletGroup(N, F);
1743> eps := b^(Order(b) div 2) * (c^(Order(c) div 3));
1744> aplist:=[0,alp^14,alp^14,3,alp^16,alp^10,0,3,3,alp^2,alp^22,alp^22,alp^20,alp^16,alp^4,4,alp^8,0];
1745> qEigenform(aplist,eps,5);
1746// q + alp^14*q^3 + alp^14*q^5 + 3*q^7 + alp^4*q^9 + alp^16*q^11 + alp^10*q^13 + alp^4*q^15 + 3*q^19 + alp^8*q^21 + 3*q^23 + alp^4*q^25 + 3*q^27 + alp^2*q^29 + alp^22*q^31 + 2*q^33 + alp^8*q^35 + alp^22*q^37 + q^39 + alp^20*q^41 + alp^16*q^43 + 3*q^45 + alp^4*q^47 + 3*q^49 + 4*q^53 + 2*q^55 + alp^8*q^57 + alp^8*q^59 + O(q^62)
1747
1748
1749// 4108
1750> h := x^5+4*x^4+3*x^3+9*x^2+4*x+5;
1751> N := 2^2*13*79;
1752> F<alp> := GF(25);
1753> G<a,b,c>:=DirichletGroup(N, F);
1754> eps := b^(Order(b) div 3) * (c^(Order(c) div 2));
1755> aplist := [0,alp^16,alp^11, alp^20,alp^10,4,0,alp^14,0, alp^22, 0, alp^22,alp^10,alp^2,3,4,alp^14,alp^2,alp^10];
1756> qEigenform(aplist,eps,5);
1757//q + alp^16*q^3 + alp^11*q^5  + alp^20*q^7 + alp^14*q^9 + alp^10*q^11
1758//       + 4*q^13 + alp^3*q^15 + alp^14*q^19     + 4*q^21 + O(q^24)
1759
1760
1761// 4288
1762> h := x^5+4*x^4+5*x^3+8*x^2+3*x+2;
1763> N := 2^6*67;
1764> F<alp> := GF(25);
1765> G<a2,b2,c>:=DirichletGroup(N, F);
1766> eps := a2*b2^(Order(b2) div 2)*(c^(Order(c) div 3));
1767> aplist := [0,3,alp^14, alp^20, alp^20, alp^16, alp^16,0,0, alp^14,alp^20,alp^4,alp^8,2,0,4,1,alp^16,alp^4,alp^20,0,alp^20,0,2];
1768> qEigenform(aplist,eps,5);
1769//q + 3*q^3 + alp^14*q^5 + alp^20*q^7 + 3*q^9 + alp^20*q^11 + alp^16*q^13 + alp^8*q^15 + alp^16*q^17 + alp^14*q^21 + O(q^24)
1770
1771
1772// 5373
1773> h := x^5+2*x^4+x^3+7*x^2+23*x-11;
1774> N := 3^3*199;
1775> F<alp> := GF(25);
1776> G<a,b>:=DirichletGroup(N, F);
1777> eps := a^(Order(a) div 2) * (b^(Order(b) div 3));
1778> aplist := [alp^16, 0, 4, 0, 2, alp^22, 1, alp^16, 0];
1779> qEigenform(aplist,eps,5);
1780//q + alp^16*q^2 + alp^14*q^4 + 4*q^5 + 3*q^8 + alp^4*q^10 + 2*q^11 + alp^22*q^13 + q^17 + alp^16*q^19 + alp^2*q^20 + alp^22*q^22 + O(q^24)
1781
1782
1783
`