Sharedwww / tables / dischecke.texOpen in CoCalc
Author: William A. Stein
1% dischecke.tex
2\documentclass[10pt]{article}
3\textwidth=1.2\textwidth
4\hoffset=-.5in
5\textheight=1.2\textheight
6\voffset=-.55in
7\title{Discriminants of Hecke Algebras}
8\include{macros}
9\begin{document}
10\maketitle
11\begin{abstract}
12For $N<250$ we compute the discriminants of the Hecke algebras associated to
13weight $2$ cusp forms and newforms of level $N$ and determine
14the primes $p$ so that the cusp $\infty$ is a Weierstrass point
15on $X_0(N)/\Fp$.
16\end{abstract}
17\section{Introduction}
18Let $S_2(N)$ be the space of
19weight $2$ cusp forms for $\Gamma_0(N)$. The
20Hecke algebra $\T\subset\End(S_2(N))$ is a finite commutative
21$\Z$-algebra. Its discriminant $\Delta=\disc(\T)$ is important in
22studying congruences between modular forms.  Let $\Delta^{\new}$ be
23the discriminant of the new Hecke algebra
24$\T^{\new}\subset\End(S_2^{\new}(N))$.
25 By multiplicity one'' $\T^{\new}$ is a subring
26of a product of fields and $\tilde{\T}^{\new}$ is the product
27of the rings of integers of these fields.
28
29Let $W$ be the sub $\Z$-module of $\T$ generated by the Hecke
30operator $T_1,T_2,\ldots T_g$ where $g$ is the genus of $X_0(N)$.
31When $N$ is prime $W$ has finite index
32in $\T$, but when $\infty$ is a Weierstrass point on $X_0(N)$ the index
33will not be finite.  If $p\nmid N$, the cusp $\infty$ is a Weierstrass point on
34$X_0(N)/\Fp$ iff $p|[\T:W]$.
35%\section{Algorithms}
36\section{Tables}
37The notation is as above.
38\newpage
39\begin{center}
40\begin{tabular}{|l||c|c|c|c|}\hline
41$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
42    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
43
4411 & $1$ & $1$ & $1$ & $1$\\ \hline
45
4614 & $1$ & $1$ & $1$ & $1$\\ \hline
47
4815 & $1$ & $1$ & $1$ & $1$\\ \hline
49
5017 & $1$ & $1$ & $1$ & $1$\\ \hline
51
5219 & $1$ & $1$ & $1$ & $1$\\ \hline
53
5420 & $1$ & $1$ & $1$ & $1$\\ \hline
55
5621 & $1$ & $1$ & $1$ & $1$\\ \hline
57
5823 & $5$ & $1$ & $1$ & $1$\\ \hline
59
6024 & $1$ & $1$ & $1$ & $1$\\ \hline
61
6226 & $2^2$ & $1$ & $2$ & $1$\\ \hline
63
6427 & $1$ & $1$ & $1$ & $1$\\ \hline
65
6629 & $2^3$ & $1$ & $1$ & $1$\\ \hline
67
6830 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline
69
7031 & $5$ & $1$ & $1$ & $1$\\ \hline
71
7232 & $1$ & $1$ & $1$ & $1$\\ \hline
73
7433 & $1$ & $-3^2.11$ & $1$ & $1$\\ \hline
75
7634 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline
77
7835 & $2^2.17$ & $1$ & $2$ & $1$\\ \hline
79
8036 & $1$ & $1$ & $1$ & $1$\\ \hline
81
8237 & $2^2$ & $1$ & $2$ & $1$\\ \hline
83
8438 & $2^2$ & $-2^3.3^2$ & $2$ & $1$\\ \hline
85
8639 & $2^5$ & $1$ & $2$ & $1$\\ \hline
87
8840 & $1$ & $0$ & $1$ & $?$\\ \hline
89
9041 & $2^2.37$ & $1$ & $1$ & $1$\\ \hline
91
9242 & $1$ & $2^9.7$ & $1$ & $1$\\ \hline
93
9443 & $2^5$ & $1$ & $2$ & $2$\\ \hline
95
9644 & $1$ & $-2^6$ & $1$ & $1$\\ \hline
97
9845 & $1$ & $2^2$ & $1$ & $1$\\ \hline
99
10046 & $1$ & $2^2.5^4.41$ & $1$ & $1$\\ \hline
101
10247 & $19.103$ & $1$ & $1$ & $1$\\ \hline
103
10448 & $1$ & $0$ & $1$ & $?$\\ \hline
105
10649 & $1$ & $1$ & $1$ & $1$\\ \hline
107
10850 & $2^2$ & $1$ & $2$ & $1$\\ \hline
109
11051 & $2^2.17$ & $-2^6.3$ & $2$ & $3$\\ \hline
111
11252 & $1$ & $2^4.3^2$ & $1$ & $3$\\ \hline
113
11453 & $2^4.37$ & $1$ & $2$ & $3$\\ \hline
115
11654 & $2^2$ & $-2^3.3^4$ & $2$ & $\infty$\\ \hline
117
11855 & $2^5$ & $-7^2.19$ & $2$ & $1$\\ \hline
119
12056 & $2^2$ & $0$ & $2$ & $?$\\ \hline
121
12257 & $2^2.3^2$ & $-2^7$ & $2.3$ & $1$\\ \hline
123
12458 & $2^2$ & $2^{10}.17$ & $2$ & $2$\\ \hline
125
12659 & $2^3.31.557$ & $1$ & $1$ & $1$\\ \hline
127
12861 & $2^4.37$ & $1$ & $2$ & $2$\\ \hline
129
13062 & $2^4.3$ & $2^2.5^2.11^2.41$ & $2$ & $3$\\ \hline
131
13263 & $2^4.3$ & $2^4$ & $2$ & $2$\\ \hline
133
13464 & $1$ & $0$ & $1$ & $\infty$\\ \hline
135
13665 & $2^{11}.3$ & $1$ & $2^3$ & $3$\\ \hline
137
13866 & $2^4$ & $-2^8.3^4.5^2.7.11^2$ & $2^2$ & $2$\\ \hline
139
14067 & $2^4.5^4$ & $1$ & $2^2.5$ & $2^3$\\ \hline
141
14268 & $2^2.3$ & $-2^8.3^2.7$ & $1$ & $2^2$\\ \hline
143
14469 & $2^4.5$ & $5^2.7^2.11^2$ & $2^2$ & $2$\\ \hline
145
14670 & $1$ & $2^{18}.3^2.5.17^2$ & $1$ & $3$\\ \hline
147
148
14971 & $3^4.257^2$ & $1$ & $3^2$ & $1$\\ \hline
150
151\end{tabular}
152
153\begin{tabular}{|l||c|c|c|c|}\hline
154$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
155    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
156
15772 & $1$ & $0$ & $1$ & $\infty$\\ \hline
158
15973 & $2^4.3^2.5.13$ & $1$ & $2^2.3$ & $2$\\ \hline
160
16174 & $2^4.5.13$ & $2^9.3^2.5^2$ & $2^2$ & $11$\\ \hline
162
16375 & $2^2.3^2$ & $2^2.3^2$ & $2.3$ & $1$\\ \hline
164
16576 & $1$ & $-2^{11}.3^6$ & $1$ & $1$\\ \hline
166
16777 & $2^6.5^3$ & $-2^7.3^3$ & $2^3.5$ & $5$\\ \hline
168
16978 & $1$ & $-2^{22}.3.5^2.7^3.11.17$ & $1$ & $2^2$\\ \hline
170
17179 & $2^2.83.983$ & $1$ & $2$ & $2^3$\\ \hline
172
17380 & $2^2$ & $0$ & $2$ & $?$\\ \hline
174
17581 & $2^2.3$ & $0$ & $1$ & $?$\\ \hline
176
17782 & $2^5$ & $-2^{10}.23.37^2$ & $2$ & $2$\\ \hline
178
17983 & $2^4.197.11497$ & $1$ & $2$ & $2^2$\\ \hline
180
18184 & $2^2$ & $-2^{22}.3^4.7$ & $2$ & $3$\\ \hline
182
18385 & $2^{11}.3$ & $-2^{10}$ & $2^3$ & $7$\\ \hline
184
18586 & $2^4.3.5.7$ & $-2^{14}.3^2.5^2.7^2$ & $2^2$ & $41$\\ \hline
186
18787 & $2^4.5.229$ & $2^6.23^2.73$ & $2^2$ & $2^3$\\ \hline
188
18988 & $2^6.17$ & $0$ & $2^3$ & $?$\\ \hline
190
19189 & $2^6.5^3.6689$ & $1$ & $2.5$ & $3^2$\\ \hline
192
19390 & $2^4$ & $-2^{22}.7^3$ & $2^2$ & $1$\\ \hline
194
19591 & $2^{15}.79$ & $1$ & $2^5$ & $2$\\ \hline
196
19792 & $2^2$ & $2^{12}.3^2.5^7.41$ & $2$ & $1$\\ \hline
198
19993 & $2^4.5.229$ & $2^{12}.5^2$ & $2^2$ & $2^3$\\ \hline
200
20194 & $2^5$ & $2^4.19^2.47^2.103^2.457$ & $2$ & $7$\\ \hline
202
20395 & $2^{12}.37.709$ & $-3^4.11$ & $2^3$ & $11$\\ \hline
204
20596 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
206
20797 & $2^6.7^2.2777$ & $1$ & $2^3$ & $2.3$\\ \hline
208
20998 & $2^5$ & $-2^{12}.7$ & $2$ & $\infty$\\ \hline
210
21199 & $2^4.3^4$ & $-2^6.3^8.11$ & $2^2.3^2$ & $2$\\ \hline
212
213100 & $1$ & $2^8.3^4$ & $1$ & $\infty$\\ \hline
214
215101 & $2^8.17568767$ & $1$ & $2$ & $3^3$\\ \hline
216
217102 & $2^4$ & $2^{41}.3^6.7^2.17^2$ & $2^2$ & $3.5^2$\\ \hline
218
219103 & $2^4.5.17.411721$ & $1$ & $2^2$ & $2^2.7$\\ \hline
220
221104 & $2^2.17$ & $0$ & $2$ & $?$\\ \hline
222
223105 & $2^4.5$ & $-2^{30}.5^2.7.11.13.17^2$ & $2^2$ & $2.3$\\ \hline
224
225106 & $2^4.3^2$ & $2^{18}.5^2.7.37^2.151$ & $2^2.3$ & $2.13$\\ \hline
226
227107 & $2^6.5.7.1667.19079$ & $1$ & $2^2$ & $2^2.5$\\ \hline
228
229108 & $1$ & $0$ & $1$ & $\infty$\\ \hline
230
231109 & $2^{10}.7^2.7537$ & $1$ & $2^5$ & $2.5$\\ \hline
232
233110 & $2^8.3.11$ & $-2^{22}.5^2.7^7.17.19^2$ & $2^4$ & $2.3$\\ \hline
234
235111 & $2^{12}.37.389$ & $2^4.3.5^2.7^2.11$ & $2^3$ & $2^2.3$\\ \hline
236
237112 & $2^6$ & $0$ & $2^3$ & $?$\\ \hline
238
239113 & $2^{10}.3^4.7^2.11^2.107$ & $1$ & $2^4.3.11$ & $2^2.17$\\ \hline
240
241114 & $2^4$ & $-2^{36}.3^{12}.5^6.7.11^2$ & $2^2$ & $3^5$\\ \hline
242
243115 & $2^6.5.17^2.53$ & $2^{12}.5^4.11$ & $2^3$ & $2^2$\\ \hline
244
245116 & $2^4.5^2$ & $2^{29}.3^4.17$ & $2^2.5$ & $2^2.7$\\ \hline
246
247117 & $2^{11}.3$ & $2^{24}$ & $2^3$ & $2^4$\\ \hline
248
249118 & $2^4.3^2$ & $-2^{18}.7.19^2.31^2.557^2$ & $2^2.3$ & $223$\\ \hline
250
251119 & $2^8.71.131.311.1459$ & $-2^6.3^3$ & $2^4$ & $2^4$\\ \hline
252
253120 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
254
255121 & $2^4.3^2$ & $2^6.3^2$ & $2^2.3$ & $\infty$\\ \hline
256
257122 & $2^6.13.229$ & $2^{18}.7.13^2.37^2.151$ & $2^3$ & $2.83$\\ \hline
258
259123 & $2^{15}.79$ & $-2^8.5^2.23^2.37^2.191$ & $2^5$ & $13$\\ \hline
260
261124 & $2^2$ & $2^{20}.3^6.5^3.11^4.41$ & $2$ & $3^2$\\ \hline
262\end{tabular}
263
264\begin{tabular}{|l||c|c|c|c|}\hline
265$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
266    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
267
268125 & $2^8.5^{12}.11$ & $0$ & $2^2.5^4$ & $\infty$\\ \hline
269
270126 & $2^2$ & $2^{49}.3^8.5^2.7^3$ & $2$ & $2$\\ \hline
271
272127 & $2^6.3^4.7.86235899$ & $1$ & $2^3$ & $2^2.29$\\ \hline
273
274128 & $2^{16}$ & $0$ & $2^8$ & $\infty$\\ \hline
275
276129 & $2^{12}.5^2.71$ & $-2^{23}.3^2.5^2.7^2$ & $2^3.5$ & $17$\\ \hline
277
278130 & $2^4$ & $-2^{38}.3^6.5^4.7^3.11.17.19$ & $2^2$ & $5.7$\\ \hline
279
280131 & $2^9.5.46141.75619573$ & $1$ & $2$ & $2.5^2$\\ \hline
281
282132 & $2^2$ & $-2^{32}.3^{12}.5^6.7.11^4$ & $2$ & $2^3$\\ \hline
283
284133 & $2^{16}.5^2.13.229$ & $-3^5.7^2$ & $2^8$ & $2.43$\\ \hline
285
286134 & $2^6.3^4.11.43$ & $-2^{10}.3^2.5^{12}.19^2.41$ & $2^3$ & $29.31$\\ \hline
287
288135 & $2^6.3^4.13^2$ & $0$ & $2^3.3^2$ & $\infty$\\ \hline
289
290136 & $2^8.5$ & $0$ & $2^4$ & $?$\\ \hline
291
292137 & $2^{10}.5^2.29.401.895241$ & $1$ & $2^4$ & $2^5.5$\\ \hline
293
294138 & $2^8.5$ & $2^{32}.3^3.5^{10}.7^5.11^6.41^2$ & $2^4$ & $2^4.7^2$\\ \hline
295
296139 & $2^8.3^2.7^2.997.2151701$ & $1$ & $2^4.3$ & $2^3.41$\\ \hline
297
298140 & $2^2$ & $2^{43}.3^{11}.5^4.17^3$ & $2$ & $2^2.3$\\ \hline
299
300141 & $2^{14}.3^2.17$ & $3^2.7^2.19^2.43^2.103^2.3529$ & $2^7.3$ & $2^2.5.13$\\ \hline
301
302142 & $2^8.3^2$ & $2^6.3^{22}.7.103.257^4$ & $2^4.3$ & $3.67$\\ \hline
303
304143 & $2^{10}.5.7.19.103.5560463$ & $-2^6.3^6$ & $2^5$ & $2.3^2$\\ \hline
305
306144 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
307
308145 & $2^{21}.37^2$ & $2^6.5^4.7^2.19^2$ & $2^7$ & $2.7$\\ \hline
309
310146 & $2^{12}.101.389$ & $-2^{12}.3^{10}.5^2.7.13^2.17.19^2$ & $2^3$ & $97$\\ \hline
311
312147 & $2^{14}.3^2.7^4$ & $-2^{14}.3^3$ & $2^4.3.7^2$ & $\infty$\\ \hline
313
314148 & $2^2.17$ & $2^{27}.3^{10}.5^6.13^2$ & $2$ & $3^2.11^2$\\ \hline
315
316149 & $2^{12}.7^2.234893.1252037$ & $1$ & $2^3$ & $107$\\ \hline
317
318150 & $2^4$ & $-2^{34}.3^{12}.5^8.7^3.11^2$ & $2^2$ & $\infty$\\ \hline
319
320151 & $2^6.7^2.11.67^2.257.439867$ & $1$ & $2^3.67$ & $2.11^2$\\ \hline
321
322152 & $2^8.31^2$ & $0$ & $2^4$ & $?$\\ \hline
323
324153 & $2^{12}.3^2.17$ & $-2^{30}.3^5.17^2$ & $2^6.3$ & $3^3$\\ \hline
325
326154 & $2^8.5$ & $2^{56}.3^{14}.5^8.7.11$ & $2^4$ & $2.23$\\ \hline
327
328155 & $2^{24}.29.73.5077$ & $5^4.7^4.19^2$ & $2^{10}$ & $2.73$\\ \hline
329
330156 & $2^2$ & $2^{57}.3^{11}.5^4.7^5.11^2.17$ & $2$ & $2^2.3$\\ \hline
331
332157 & $2^{13}.61.397.48795779$ & $1$ & $2^5$ & $2.13$\\ \hline
333
334158 & $2^{13}.3^3.5^2$ & $2^{22}.7.17.53^2.83^2.271.983^2$ & $2^5.3.5$ & $2^2.541$\\ \hline
335
336159 & $2^8.19.103.1054013$ & $2^8.3.7^2.37^2.107^2.227$ & $2^4$ & $2^4.3.5$\\ \hline
337
338160 & $2^{13}$ & $0$ & $2^5$ & $\infty$\\ \hline
339
340161 & $2^{16}.5^3.37.536777$ & $2^{12}.5^2.19^2.29$ & $2^6.5$ & $2.97$\\ \hline
341
342162 & $2^4.3^4$ & $0$ & $2^2.3^2$ & $\infty$\\ \hline
343
344163 & $2^{15}.3^2.65657.82536739$ & $1$ & $2^6.3$ & $2^2.347$\\ \hline
345
346164 & $2^4.1613$ & $-2^{40}.3^6.23.37^3$ & $1$ & $2^3.5$\\ \hline
347
348165 & $2^{17}.3.37$ & $-2^{50}.3^5.5^2.7^5.11^2.19^2$ & $2^5$ & $2^2.3$\\ \hline
349
350166 & $2^6.5.229$ & $-2^{26}.7.71.131^2.197^2.11497^2$ & $2^3$ & $2.5.479$\\ \hline
351
352167 & $2^4.5.8269.5103536431379173$ & $1$ & $2^2$ & $2.3^2$\\ \hline
353
354168 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
355
356169 & $2^8.3.7^4.13^4$ & $1$ & $2^3.13^2$ & $\infty$\\ \hline
357
358170 & $2^{12}.5^2.17$ & $2^{68}.3^8.5^3.7^5.17$ & $2^6.5$ & $839$\\ \hline
359
360171 & $2^{16}.3^7.11^2$ & $-2^{33}.3^8$ & $2^7.3^2$ & $2.3^3$\\ \hline
361
362172 & $2^5$ & $-2^{39}.3^{10}.5^6.7^6$ & $2$ & $41^2$\\ \hline
363
364173 & $2^{14}.5^2.7.29.5608385124289$ & $1$ & $2^4$ & $79$\\ \hline
365
366174 & $2^8.5^2$ & $-2^{38}.3^3.5^4.7^4.11^3.13^2.17^2\ldots$ & $2^4.5$ & $2^6.5.11^2$\\
367    &  & $\ldots 23^4.41.47.73^2.229^2$ & &\\ \hline
368
369175 & $2^{12}.3^4.5^6.17$ & $2^{18}.3^4.17^2$ & $2^6.3^2.5^2$ & $2^2$\\ \hline
370
371176 & $2^{14}.17$ & $0$ & $2^7$ & $?$\\ \hline
372\end{tabular}
373
374\begin{tabular}{|l||c|c|c|c|}\hline
375$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
376    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
377
378177 & $2^{16}.5^3.229$ & $-2^6.31^4.229^2.557^2.20627$ & $2^8$ & $2^4.149$\\ \hline
379
380178 & $2^{16}.71$ & $-2^{30}.5^6.7^4.199.6689^2$ & $2^5$ & $2^2.487$\\ \hline
381
382179 & $2^{12}.3^4.7^2.313.137707.536747147$ & $1$ & $2^3.3^2$ & $2.59$\\ \hline
383
384180 & $1$ & $-2^{73}.3^{16}.5.7^3$ & $1$ & $\infty$\\ \hline
385
386181 & $2^{16}.5^2.7.61.397.595051637$ & $1$ & $2^5$ & $2^4.3^3$\\ \hline
387
388182 & $2^8.3^2$ & $2^{51}.3^{20}.5^4.7^6.11^2.23.79^2$ & $2^4.3$ & $2^2.3^2.5.37$\\ \hline
389
390183 & $2^{22}.37.127.5623$ & $2^{27}.3^2.19^2.23.37^2$ & $2^5$ & $2^2.5^4$\\ \hline
391
392184 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline
393
394185 & $2^{26}.3^2.23029.60869$ & $2^{24}.3^2.5$ & $2^9.3$ & $193$\\ \hline
395
396186 & $2^8.17$ & $2^{62}.3^5.5^8.7^2.11^6.13.19^2\ldots$ & $2^4$ & $2^2.5^5$\\
397    &           & $\ldots 41^2.127.229^2$ && \\ \hline
398
399187 & $2^{28}.3.5^2.17.37.8461$ & $2^{24}.3^4.11$ & $2^{11}.5$ & $2.3^2.11$\\ \hline
400
401188 & $2^4.5.13$ & $2^{34}.3^6.19^3.47^4.103^3.457$ & $2^2$ & $7^2$\\ \hline
402
403189 & $2^{16}.3^9.7$ & $0$ & $2^6.3^4$ & $\infty$\\ \hline
404
405190 & $2^8.17$ & $-2^{56}.3^{16}.5^3.11^4.13^2.37^2\ldots$ & $2^4$ & $11.53$\\
406    &           & $\ldots 113.151.709^2$ && \\ \hline
407
408191 & $2^4.3^3.5.382146223.319500117632677$ & $1$ & $2^2$ & $2^2.223$\\ \hline
409
410192 & $2^{12}$ & $0$ & $2^6$ & $\infty$\\ \hline
411
412193 & $2^{14}.5.11^2.17.103.401.4153.680059$ & $1$ & $2^7.11$ & $2^7$\\ \hline
413
414194 & $2^{20}.7^2.137.223$ & $-2^{14}.7^5.67^2.71^2.433.2777^2$ & $2^4.7$ & $3.22283$\\ \hline
415
416195 & $2^{14}.3^2.79$ & $-2^{77}.3^9.5^2.7^6.11^2.13$ & $2^6.3$ & $2.3^2.5.7$\\ \hline
417
418196 & $2^5.7^4$ & $-2^{38}.3^8.7$ & $2.7^2$ & $\infty$\\ \hline
419
420197 & $2^{18}.5^2.61.397.35217676193989$ & $1$ & $2^6.5$ & $7.383$\\ \hline
421
422198 & $2^{10}$ & $-2^{78}.3^{24}.5^6.7^5.11^2$ & $2^5$ & $2.7^2$\\ \hline
423
424199 & $2^8.3.5^3.29.31.71^2.347.947.37316093$ & $1$ & $2^4.71$ & $2^4.241$\\ \hline
425
426200 & $2^8.3^2.5^4$ & $0$ & $2^4.3.5^2$ & $\infty$\\ \hline
427
428201 & $2^{18}.3^2.37.269.953$ & $-2^{15}.3^2.5^{10}.19^2.29^2.61.109$ & $2^7.3$ & $3.239$\\ \hline
429
430202 & $2^8.3^4.10273$ & $2^{33}.3^4.17^4.17568767^2$ & $2^4$ & $2^3.2381$\\ \hline
431
432203 & $2^{35}.3^7.17.29.37.7547$ & $2^{22}.5^2.7^2$ & $2^{14}.3^3$ & $2^3.193$\\ \hline
433
434204 & $2^2$ & $2^{102}.3^{23}.7^2.11^2.13.17^3$ & $2$ & $2^2.3.5^4$\\ \hline
435
436205 & $2^{30}.3^2.5.13.229^2$ & $-2^{14}.13^2.31^2.37^2.239$ & $2^{15}.3$ & $7^2.23$\\ \hline
437
438206 & $2^{16}.3^2.5^2.13.29.359$ & $2^{12}.3^2.5^2.17^4.19^2.67^2\ldots$ & $2^6.3.5$ & $17.2843$\\
439    &           & $\ldots 1801.411721^2$ && \\ \hline
440
441207 & $2^{20}.3^4.5^2.11^2$ & $2^{28}.3^8.5^5.7^2.11^4$ & $2^7.3^2.11$ & $2^6$\\ \hline
442
443208 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline
444
445209 & $2^{32}.3^4.15427.2002061$ & $2^{10}.3^4.5^5.7.19$ & $2^9$ & $3.11^2$\\ \hline
446
447210 & $2^8$ & $2^{176}.3^{14}.5^8.7^7.11^2.13^2.17^4$ & $2^4$ & $2^3.3$\\ \hline
448
449211 & $2^{14}.3.5.7^4.41^2.43.229.52184516509$ & $1$ & $2^6.7.41$ & $2^5.5.23$\\ \hline
450
451212 & $2^4.3^3.7^3$ & $2^{56}.3^{12}.5^4.7.37^3.151$ & $2.7$ & $2^2.13^3$\\ \hline
452
453213 & $2^{20}.3^2.5^4.13.89$ & $3^{14}.5^2.19^2.37.61^2.229.257^4$ & $2^{10}.3$ & $2^3.3.17.53$\\ \hline
454
455214 & $2^{16}.3^6$ & $-2^{24}.5^4.7^4.11^2.41.109^2.521\ldots$  & $2^6.3^2$ & $3.67.4903$\\
456    &           & $\ldots 1667^2.19079^2$ && \\ \hline
457
458215 & $2^{14}.3.7^4.101.107.321821.1933097$ & $-2^{22}.5^2.31^2.41$ & $2^7.7^2$ & $2^2.3^6$\\ \hline
459
460216 & $2^8.3^4$ & $0$ & $2^4.3^2$ & $\infty$\\ \hline
461
462217 & $2^{31}.3^8.11.31.557.619$ & $5^2.19^2.31^2.281$ & $2^{14}$ & $2^4.5.43$\\ \hline
463
464218 & $2^{19}.3^4.5.11^2.23$ & $2^{34}.3^4.7^7.41^2.167.601.7537^2$ & $2^7.11$ & $2.7.21227$\\ \hline
465
466219 & $2^{24}.3^2.29.73.1189637$ & $-2^{16}.3^8.5^4.13^2.17^2.23.29^2.61$ & $2^{10}.3$ & $3^2.41.71$\\ \hline
467
468220 & $2^2$ & $2^{77}.3^{16}.5^4.7^{11}.11^4.17.19^3$ & $2$ & $2.3.5$\\ \hline
469
470221 & $2^{40}.3^3.5^6.7.37.109^2.229$ & $-2^{12}.3^3$ & $2^{17}.3.5^2$ & $5.107$\\ \hline
471
472222 & $2^8.3^2$ & $-2^{60}.3^{14}.5^{10}.7^4.11^4.13^5.19^2\ldots$ & $2^4.3$ & $2^4.3^5.11^2$\\
473    &           & $\ldots23^2.31.37^2.109.389^2.409$ && \\ \hline
474
475223 & $2^{18}.7^2.19.103.3995922697473293141$ & $1$ & $2^6.7$ & $2.3^2.5.641$\\ \hline
476
477224 & $2^{18}.5^2$ & $0$ & $2^9$ & $\infty$\\ \hline
478\end{tabular}
479
480\begin{tabular}{|l||c|c|c|c|}\hline
481$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
482    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
483
484225 & $2^{12}.3^2.5^5$ & $2^{38}.3^{10}$ & $2^6.3.5^2$ & $\infty$\\ \hline
485
486226 & $2^{23}.3.5^3$ & $-2^{34}.3^{12}.7^9.11^4.23.41^2\ldots$  & $2^7$ & $2^2.190129$\\
487    & & $\ldots 107^2.167$ & & \\\hline
488
489227 & $2^{19}.5^3.7^4.13^2.29.31^2\ldots$ & $1$ & $2^7.5.7.13.31$ & $2^2.29^2$\\
490    & $\ldots 13591.57139.273349$ & & & \\\hline
491
492228 & $2^4.3.11$ & $-2^{88}.3^{36}.5^{12}.7.11^4$ & $2^2$ & $2^2.3^9$\\ \hline
493
494229 & $2^{24}.107.17467.39555937\times\ldots$ & $1$ & $2^9$ & $2^2.11.193$\\
495    & $\ldots 53625889$ & & & \\ \hline
496
497230 & $2^{16}.3^2.5.7.13.367$ & $2^{52}.3^2.5^{18}.11^4.17^4.19^2\ldots$ & $2^8$ & $2^2.5^2.53$\\
498   & & $\ldots41^2.43^2.47.53^2$ & & \\\hline
499
500231 & $2^{20}.3^4.5^3.7.31.229$ & $-2^{59}.3^{16}.5^{12}.7^3.11^3$ & $2^{10}.5$ & $2^2.29$\\ \hline
501
502232 & $2^{20}.71$ & $0$ & $2^7$ & $?$\\ \hline
503
504233 & $2^{22}.3^7.53.139.653.4127\times \ldots$ & $1$ & $2^7.3^3$ & $2^2.3.11.197$\\
505    & $\ldots24989.8388019$ & & & \\\hline
506
507234 & $2^6.3^2.5^2$ & $-2^{118}.3^{13}.5^6.7^9.11.17^3$ & $2^3.3.5$ & $2^5.3^2$\\ \hline
508
509235 & $2^{18}.3^2.73.1453.3851.1916279$ & $2^{24}.3^8.19^3.103^2$ & $2^9.3$ & $2^2.3.17.31$\\ \hline
510
511236 & $2^2.3.7^2.107$ & $-2^{55}.3^{12}.7.19^4.31^3.557^3$ & $2.7$ & $223^2$\\ \hline
512
513237 & $2^{18}.19.103.1705391537$ & $2^{28}.3^2.5^2.11^2.31^2.83^2.983^2$ & $2^6$ & $2^6.5.97$\\ \hline
514
515238 & $2^{16}.5$ & $-2^{73}.3^{11}.5^2.7^4.61^2.71^2.131^2\ldots$ & $2^8$ & $2^2.13.331$\\
516    & &    $\ldots 311^2.337.479.1459^2$ & & \\\hline
517
518239 & $2^6.7^2.2833.51817.97423\times \ldots$ & $1$ & $2^3$ & $2^2.3.11^2$\\
519    & $\ldots 1174779433.8920940047$ & & & \\\hline
520
521240 & $2^{10}$ & $0$ & $2^5$ & $\infty$\\ \hline
522
523241 & $2^{23}.97.1489.20857\times \ldots$ & $1$ & $2^7$ & $2^7.151$\\
524    &      $\ldots 651474368435017$  & & &\\\hline
525
526242 & $2^{18}.3^2.5^2.11^8$ & $2^{41}.3^{12}.5^4.7^2$ & $2^7.11^4$ & $\infty$\\ \hline
527
528243 & $2^{13}.3^{40}$ & $0$ & $2^4.3^{15}$ & $\infty$\\ \hline
529
530244 & $2^4.5077$ & $2^{60}.3^{10}.7.13^6.37^3.151.229^2$ & $2$ & $2^2.3.5.83^2$\\ \hline
531
532245 & $2^{36}.3^4.7^8.17$ & $-2^{28}.3^8.5.17^2$ & $2^{12}.3^2.7^4$ & $2^2$\\ \hline
533
534246 & $2^{12}.3^2.5^2$ & $2^{96}.3^8.5^{10}.7^6.11^2.23^7.37^4\ldots$ & $2^6.3.5$ & $7.73^2$\\
535   & & $\ldots 79^2.191^2$ & & \\\hline
536
537247 & $2^{30}.3^4.5^2.11.31^2.619\times\ldots$ & $-2^6.3^4.5^2$ & $2^{15}.31$ & $2^2.29.241$\\
538   & $\ldots 57713.2655049$ & & & \\\hline
539
540248 & $2^{22}.3.11.79$ & $0$ & $2^{10}$ & $?$\\ \hline
541
542249 & $2^{33}.389.23029$ & $-2^8.3^2.5^2.11.31^2.197^2.2711^2\ldots$ & $2^{11}$ & $2.13.41.89$\\
543    & & $\ldots 11497^2.295201$ & & \\ \hline
544
545250 & $2^8.5^{12}$ & $0$ & $2^4.5^4$ & $\infty$\\ \hline
546
547251 & $2^{14}.5^2.29.373\times\ldots$ & $1$ & $2^4$ & $2^2.3.797$\\
548    & $8768135668531.2006012696666681$ & & & \\ \hline
549
550252 & $2^2$ & $2^{136}.3^{30}.5^2.7^3$ & $2$ & $\infty$\\ \hline
551
552253 & $2^{28}.3^4.13^2.2711.3187.170701$ & $-2^{12}.3^2.5^7.7.13.19^2.23^2$ & $2^{14}$ & $2.4507$\\ \hline
553
554254 & $2^{28}.3^4.17.569$ &  $2^{16}.3^{18}.7^2.17^2.41.71\times\ldots$ & $2^{11}$ & $7.189713$\\
555   & & $\ldots 383^2.1231.86235899^2$ & & \\ \hline
556
557255 & $2^{23}.5.13.229.1721$ & $-2^{99}.3^{14}.5^2.11^3.13^3.17^4$ & $2^{10}$ & $2^3.3^3.379$\\ \hline
558
559256 & $2^{31}$ & $0$ & $2^{14}$ & $\infty$\\ \hline
560
561257 & $2^{29}.29.479.71711.409177\times\ldots$ & $1$ & $2^{11}$ & $2.29.3251$\\
562    & $\ldots 654233.32354821$ & & & \\ \hline
563
564
565258 & $2^{12}.3^2.5^2$ & $-2^{104}.3^{14}.5^{16}.7^{17}\times\ldots$  & $2^6.3.5$ & $7^3.199^2$\\
566   & & $\ldots 11^2.17.19^2.37.71^2.109$ & & \\\hline
567
568259 & $2^{43}.3^9.5^3.7^2.17^3.29.37.167$ & $2^4.3^{12}.7^4$ & $2^{20}.3.17$ & $2^2.3.29.193$\\ \hline
569
570260 & $2^6.3.47$ & $-2^{117}.3^{26}.5^6.7^5.11^2.17.19^2$ & $2^2$ & $\infty$\\ \hline
571
572261 & $2^{19}.3^4.5^3.23^2.229$ & $2^{47}.3^8.5^2.23^4.73.229^2$ & $2^8.3^2.23$ & $2^{10}.5$\\ \hline
573
574\end{tabular}
575
576
577\begin{tabular}{|l||c|c|c|c|}\hline
578$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
579    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
580
581262 & $2^{19}.3^3.5.11^2.13$ & $-2^{43}.3^4.5^2.11^2.71\times\ldots$ & $2^7.3.11$ & $29.482689$\\
582    & & $\ldots 313^2.46141^2.75619573^2$ & & \\\hline
583
584263 & $2^{10}.11.61.397.15631853\times\ldots$& $1$ & $2^5$ & $2^3.19.223$\\
585    & $\ldots 34867513.97092067.252746489$ & & & \\\hline
586
587
588264 & $2^6.3^2$ & $0$ & $2^3.3$ & $\infty$\\ \hline
589
590265 & $2^{37}.3^2.5^4.7^3.11^2.13.17^2.113$ & $2^{26}.3^4.5^3.31^2.37^2.239$ & $2^{13}.5.7.11.17$ & $2.3^3.103$\\ \hline
591
592266 & $2^{16}.5.7.13.29.67$ & $2^{51}.3^{31}.5^6.7^6.11^6.13^4.17.19^3.41.47.229^2$ & $2^8$ & $3.216679$\\ \hline
593
594267 & $2^{26}.3^4.5^2.7^4.13^2.97.241$ & $-2^{17}.3^4.5^8.11.13^2.17^2.37.113^2.911.6689^2$ & $2^{13}.5.7$ & $3^2.17.41.271$\\ \hline
595
596268 & $2^4.3^3.5.7$ & $-2^{46}.3^{20}.5^{20}.11^2.19^4.41.43^2$ & $2^2.3$ & $29^2.31^2$\\ \hline
597
598269 & $2^{22}.3^2.43.151.27767.65657.5550873754172978311$ & $1$ & $2^6.3$ & $11^2.19^2$\\ \hline
599
600270 & $2^8$ & $0$ & $2^4$ & $\infty$\\ \hline
601
602271 & $2^{12}.3^2.1367.6091.592661.1132673.14171513.172450541$ & $1$ & $2^6.3$ & $2.3.17239$\\ \hline
603
604272 & $2^{28}.3^3.5$ & $0$ & $2^{13}.3$ & $?$\\ \hline
605
606273 & $2^{25}.79.4357$ & $-2^{111}.3^9.5^4.7^6.11.79^2$ & $2^9$ & $2^2.3.313$\\ \hline
607
608274 & $2^{20}.3^2.19.37.1321$ & $-2^{26}.5^4.7^2.11^2.29^2.109^2.149^2.401^2.769.47143.895241^2$ & $2^7.3$ & $2.11.304937$\\ \hline
609
610275 & $2^{31}.3^8.5^{12}.7^2.11^2.13^2$ & $-2^{28}.3^6.5^4.7^4.19$ & $2^{12}.3^3.5^5.7$ & $2^2$\\ \hline
611
612276 & $2^{10}.5$ & $-2^{108}.3^{19}.5^{21}.7^9.11^{11}.41^2$ & $2^2$ & $2^4.7^4$\\ \hline
613
614277 & $2^{22}.5^2.19.29.37.137^2.92767.1530091.25531570859$ & $1$ & $2^{10}.5.137$ & $2.3^2.8311$\\ \hline
615
616278 & $2^{22}.3^4.7.17^2.103.107$ & $-2^{38}.3^4.7^7.41^2.167.271^3.997^2.2151701^2$ & $2^8.17$ & $6084451$\\ \hline
617
618279 & $2^{28}.3^5.5^2.229.1373^2$ & $2^{60}.3^8.5^5.229^2$ & $2^{11}.3^2$ & $2^9.5^3$\\ \hline
619
620280 & $2^{18}.3.11.17$ & $0$ & $2^9$ & $\infty$\\ \hline
621
622281 & $2^{22}.3.5.181.857.8388019.2647382149.1778899342669$ & $1$ & $2^7$ & $2^2.79309$\\ \hline
623
624282 & $2^{23}.3^2.37$ & $2^{96}.3^{13}.5^4.7^6.17^2.19^4.23^2.43^4.47^4.103^4.457^2.3529^2$ & $2^8$ & $2^7.5^6$\\ \hline
625
626283 & $2^{40}.349.1297.413713.73199099.5832488839$ & $1$ & $2^9$ & $2^7.2687$\\ \hline
627
628284 & $2^6.3^5.107$ & $2^{52}.3^{52}.7.103.257^6$ & $2^3$ & $3^2.67^2$\\ \hline
629
630285 & $2^{34}.3^3.7$ & $-2^{80}.3^{23}.5^5.7^2.11^3.19^3.23.37^2.47^2.73.709^2$ & $2^{12}.3$ & $2.3.10771$\\ \hline
631
632286 & $2^{16}.3^2.31^2$ & $-2^{70}.3^{22}.5^9.7^4.13^2.19^4.103^2.139^2.257.5560463^2$ & $2^8.3$ & $120671$\\ \hline
633
634287 & $2^{28}.3.5^2.7^2.61^2.97^2.103.257.211039.1798619$ & $-2^8.13^2.19^2.37^4.1847$ & $2^{14}.61.97$ & $2^2.3.823$\\ \hline
635
636288 & $2^{20}.3^2$ & $0$ & $2^{10}.3$ & $\infty$\\ \hline
637
638289 & $2^{23}.3^{16}.13^2.17^8$ & $2^6.3^4$ & $2^6.3^4.17^4$ & $\infty$\\ \hline
639
640290 & $2^{20}.3^3.7.13^2.23.67$ & $-2^{84}.3^6.5^{10}.7^{11}.11.13^2.17^3.19^5.23^2.31.37^4.151$ & $2^{10}$ & $2^3.5.97.421$\\ \hline
641
642291 & $2^{32}.3^4.5^4.11^2.13.19.457.16657$ & $-2^{22}.7^4.11^2.13^2.23^2.139^2.491.1933.2777^2$ & $2^{15}.3^2.5.11$ & $73.167$\\ \hline
643
644292 & $2^7.5.41^2$ & $-2^{64}.3^{28}.5^3.7.13^3.17.19^4.101^2.389^2$ & $2^2$ & $2.3^2.97^2$\\ \hline
645
646293 & $2^{26}.3^2.29.233.2351^2.69763.42711913589792108923$ & $1$ & $2^{10}$ & $67759$\\ \hline
647
648294 & $2^{12}.3^2.7^4$ & $2^{147}.3^{16}.5^6.7^{21}.17^2$ & $2^6.3.7^2$ & $\infty$\\ \hline
649
650295 & $2^{36}.3^4.7^4.43.37199.8055869$ & $-2^6.7^2.31^2.107^2.557^2.947^2.271499$ & $2^{15}$ & $2.223.241$\\ \hline
651
652296 & $2^{18}.11.53.83.229$ & $0$ & $2^9$ & $?$\\ \hline
653
654297 & $2^{30}.3^{24}.47^2$ & $0$ & $2^{11}.3^{10}$ & $\infty$\\ \hline
655
656298 & $2^{21}.3.5^2.7.13^2.103.107$ & $2^{36}.3^6.7^4.13^2.23^2.29^2.41.239.1847.234893^2.1252037^2$ & $2^8.5$ & $2^3.2972969$\\ \hline
657
658299 & $2^{50}.3.5^9.7.17^3.197.5936311524617$ & $5^2.11^2.17^4.43^2$ & $2^{19}.5^3.17$ & $2^3.3769$\\ \hline
659
660300 & $2^4.5^4$ & $2^{113}.3^{42}.5^{16}.7^3.11^4$ & $2^2.5^2$ & $\infty$\\ \hline
661
662301 & $2^{38}.19.31.37.103.9739.81509.32366197$ & $-2^{30}.7^3.17^2.113$ & $2^{19}$ & $2^2.3.13.17.61$\\ \hline
663
664302 & $2^{33}.3^4.389.1613$ & $2^{18}.3^8.5^4.7^7.11^2.41^2.67^4.167.257^2.431^2.16937.439867^2$ & $2^{11}.3^2$ & $17.877.4457$\\ \hline
665
666303 & $2^{35}.89.3083683.3678833$ & $2^{39}.3^2.7^2.59.1103^2.4021.17568767^2$ & $2^{15}$ & $2^4.61.199$\\ \hline
667
668304 & $2^{26}.3^4.31^2$ & $0$ & $2^{13}.3^2$ & $?$\\ \hline
669
670305 & $2^{52}.3^4.31.43.977.2777.82219057$ & $2^8.3^6.11.37^4.139^2.2371$ & $2^{19}$ & $2^2.3^4.223$\\ \hline
671
672306 & $2^{24}.3^4$ & $2^{183}.3^{30}.5^4.7^4.17^6$ & $2^9.3$ & $3^5.5^4$\\ \hline
673
674307 & $2^{22}.3^2.5^5.11^2.13^3.107^2.457.3697.21577.974513.568380457$ & $1$ & $2^{11}.3.5^2.11.13.107$ & $2^3.3^2.16529$\\ \hline
675
676308 & $2^{12}.3.127$ & $-2^{140}.3^{38}.5^{17}.7.11^2$ & $2^3$ & $2^9.23$\\ \hline
677
678309 & $2^{24}.5.37.81509.109363884517$ & $2^{32}.3^4.5^8.11^2.17^2.89^2.397.411721^2$ & $2^{10}$ & $2^6.3.11^2.421$\\ \hline
679
680310 & $2^{23}.3^2.37$ & $2^{119}.3^{10}.5^{12}.7^9.11^4.17.19^4.23^2.29^2.41^2.73^2.5077^2$ & $2^8$ & $2.5^2.7.73.991$\\ \hline
681\end{tabular}
682
683\begin{tabular}{|l||c|c|c|c|}\hline
684$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
685    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
686
687311 & $2^8.5^2.29.3013091897.2106873009119126062143259000543887593$ & $1$ & $2^4$ & $2.610843$\\ \hline
688
689312 & $2^{18}$ & $0$ & $2^9$ & $\infty$\\ \hline
690
691313 & $2^{24}.5.41^2.8619587.9614923.130838023.2164322751511$ & $1$ & $2^{11}.41$ & $2^5.19.241$\\ \hline
692
693314 & $2^{14}.3.47.53.337.907.176130259$ & $2^{30}.5^2.7.19^2.61^2.113^2.397^2.709^2.743.1489.48795779^2$ & $2^7$ & $2.13^2.127.461$\\ \hline
694
695315 & $2^{24}.3^4.5^3.17$ & $-2^{158}.3^{16}.5^6.7^3.11.13.17^3$ & $2^9.3^2.5$ & $2^3.3$\\ \hline
696
697316 & $2^6.3^4.13^2$ & $2^{92}.3^{18}.5^4.7.17.53^4.83^3.271.983^3$ & $2^3.3^2$ & $2^4.541^2$\\ \hline
698
699317 & $2^{26}.7.367.3217.660603043.14989400036918065702697531$ & $1$ & $2^{11}$ & $5.167.14627$\\ \hline
700
701318 & $2^{16}.5^2.17.41$ & $-2^{84}.3^{16}.5^8.7^7.11^4.13^2.17^2.19^2.37^4.43^2.71.103^2.107^4.151^2.227^2.263.1054013^2$ & $2^8.5$ & $2^4.3^2.5^4$\\ \hline
702
703319 & $2^{34}.3^4.23^2.127.1063.1433.2459.2777.6709.65551$ & $-2^{12}.3^4.5^2.7.17^2.29.167^2.239$ & $2^{17}.23$ & $2.13.59.197$\\ \hline
704
705320 & $2^{39}$ & $0$ & $2^{18}$ & $\infty$\\ \hline
706
707321 & $2^{32}.5^2.21803.24443.826957$ & $-2^{12}.5^4.7^2.29^2.41^2.61.1667^2.14051^2.19079^2.1229279$ & $2^{12}$ & $2^7.11.73.109$\\ \hline
708
709322 & $2^{28}.3^3.5.79$ & $-2^{100}.3^5.5^{18}.7^3.11^4.19^4.23^2.29^2.37^2.41^3.151.536777^2$ & $2^{12}.3$ & $2^3.13.41.53.67$\\ \hline
710
711323 & $2^{48}.7^2.17^3.19.73.103.353.1453.1627.1697.3851.17299$ & $2^{12}.3^5.5^3.7^2.59$ & $2^{23}.7.17$ & $2^2.767603$\\ \hline
712
713324 & $2^4.3^8$ & $0$ & $2^2.3^4$ & $\infty$\\ \hline
714
715325 & $2^{49}.3^9.5^{12}.7^4.37^2$ & $2^{50}.3^{10}$ & $2^{17}.3^4.5^6.7^2$ & $2^2.3.17$\\ \hline
716
717326 & $2^{18}.17^2.617.28921.482689$ & $-2^{35}.3^{16}.5^2.13^2.47^2.61^2.263.14831.65657^2.82536739^2$ & $2^9.17$ & $311.102305897$\\ \hline
718
719327 & $2^{37}.5.37.139.1023203.1033895651$ & $2^{40}.3.7^4.13^2.107^2.139^2.491.577.7537^2$ & $2^{11}$ & $2^6.3.281.463$\\ \hline
720
721328 & $2^{28}.3^3.37.197$ & $0$ & $2^{11}.3$ & $?$\\ \hline
722
723329 & $2^{46}.5^3.7^4.11.13^2.17.37^3.1447.1609.51607$ & $3^6.19^4.31^2.83^2.103^2.18617$ & $2^{20}.5.13.37$ & $2^4.11.1801$\\ \hline
724
725330 & $2^8$ & $2^{264}.3^{26}.5^{21}.7^{22}.11^{10}.17^3.19^4.23.37^2$ & $2^4$ & $2.3.5.1153$\\ \hline
726
727331 & $2^{32}.3^2.53^2.229.1399.21911.205493.6363601.584461573862449$ & $1$ & $2^{12}.3.53$ & $2^2.1120529$\\ \hline
728
729332 & $2^9.7^3.29^2$ & $-2^{82}.3^{12}.5^2.7.71.131^4.197^3.229^2.11497^3$ & $2^2.29$ & $2^2.5^2.479^2$\\ \hline
730
731333 & $2^{42}.3^7.5^2.7^2.37.389$ & $2^{72}.3^9.5^4.7^4.11.37^2.389^2$ & $2^{13}.3^2.5.7$ & $2^{10}.3^5$\\ \hline
732
733334 & $2^{27}.5^2.7^3.67.733$ & $2^{18}.3^{10}.5^4.7^4.11^4.41.113^2.8269^2.1951993.5103536431379173^2$ & $2^{12}.7$ & $113.3715823$\\ \hline
734
735335 & $2^{31}.3.5.29.71.83.179^2.887.26393.6262079.23057641$ & $-2^{22}.3^{12}.5^8.11^2.41.59^2$ & $2^{13}.179$ & $2^2.3.331.431$\\ \hline
736
737336 & $2^{16}.3^2$ & $0$ & $2^8.3$ & $\infty$\\ \hline
738
739337 & $2^{28}.113.593.2791.2963615537.747945736667.4122851467451$ & $1$ & $2^{13}$ & $2^4.3^3.27299$\\ \hline
740
741338 & $2^{16}.3^2.7^6.13^8$ & $2^{36}.3^8.5^2.7^{14}.13^8.41^4.167^2$ & $2^8.3.7.13^4$ & $\infty$\\ \hline
742
743339 & $2^{41}.7^2.13^2.17.71^2.8297.470621$ & $-2^{29}.3^{18}.7^6.11^8.13^5.107^2.167.647$ & $2^{14}.7.13.71$ & $2^2.5^2.127.11801$\\ \hline
744
745340 & $2^6.101$ & $-2^{198}.3^{37}.5^8.7^7.17^3$ & $2^2$ & $\infty$\\ \hline
746
747341 & $2^{44}.3.5^3.89.151.1121599.344460847.14444130109$ & $-2^{14}.3^3.5^{10}.7^4.13^2$ & $2^{16}$ & $31.9043$\\ \hline
748
749342 & $2^{12}.3^4.5^2$ & $-2^{184}.3^{58}.5^{16}.7^3.11^6$ & $2^6.3^2.5$ & $2.3^9.5^2$\\ \hline
750
751343 & $2^{27}.7^{48}.13^2.29^3.41^2$ & $0$ & $3.33521863.4906673923$ & $\infty$\\ \hline
752
753344 & $2^{26}.3.229.1999567$ & $0$ & $2^{11}$ & $?$\\ \hline
754
755345 & $2^{46}.3^2.5^4.79$ & $-2^{100}.3^{13}.5^{19}.7^4.11^{13}.13^2.17^4.23.53^2$ & $2^{18}.5^2$ & $2^2.7^2.15991$\\ \hline
756
757346 & $2^{18}.7^2.229.2777.2075621$ & $2^{40}.5^4.7^6.29^2.31^2.67^2.79^2.311.1279.1289.5608385124289^2$ & $2^9.7$ & $2^3.18947.256957$\\ \hline
758
759347 & $2^{27}.5.7^2.19^2.331.349.479.617.1797330450291217.918291275915301361$ & $1$ & $2^{10}.7.19$ & $2^6.3.7.19709$\\ \hline
760
761348 & $2^8$ & $2^{129}.3^{27}.5^{17}.7^{10}.11^7.13^4.17^2.23^6.41.47.73^3.229^3$ & $2^4$ & $2^8.5.11^4$\\ \hline
762
763349 & $2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031$ & $1$ & $2^{11}$ & $2^3.3^2.239.1531$\\ \hline
764
765350 & $2^{24}.3^4.5^6$ & $2^{138}.3^{34}.5^{29}.11^4.17^6.41^2$ & $2^9.3.5^3$ & $2^3.3$\\ \hline
766\end{tabular}
767
768\begin{tabular}{|l||c|c|c|c|}\hline
769$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
770    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
771
772351 & $2^{32}.3^{17}.5^4.13^2.19.37^2$ & $0$ & $2^{12}.3^8$ & $\infty$\\ \hline
773
774352 & $2^{42}.17^2$ & $0$ & $2^{21}$ & $\infty$\\ \hline
775
776353 & $2^{34}.3^2.5.127^2.229.114641.551801.12611821.7779730837.24314514437$ & $1$ & $2^{15}.3.127$ & $2^2.3.47.26927$\\ \hline
777
778354 & $2^{28}.5^2.11.79$ & $-2^{98}.3^{14}.5^{12}.7^4.11^6.19^6.29^2.31^8.41^2.127.229^6.557^4.20627^2$ & $2^{12}.5$ & $2^{15}.3^4.5$\\ \hline
779
780355 & $2^{51}.5^2.19.29.103.62581037.933591331$ & $3^{10}.5.7^2.19.61^2.103^2.257^4.359^2.3671$ & $2^{23}$ & $2^6.3^2.5^2.13.37$\\ \hline
781
782356 & $2^9.3.4079.31547$ & $-2^{114}.3^{14}.5^9.7^6.71^2.199.6689^3$ & $2$ & $2^5.11.487^2$\\ \hline
783
784357 & $2^{49}.3.79.113$ & $-2^{99}.3^{16}.7^4.17^4.47^2.71^2.131^2.223^2.311^2.397.1459^2.1753$ & $2^{18}$ & $2^2.17.86341$\\ \hline
785
786358 & $2^{22}.3^3.5^3.7.11^4.17^3$ & $-2^{46}.3^8.5^2.7^4.13^2.17^4.29^2.79^2.239.313^2.3257.137707^2.536747147^2$ & $2^{11}.3.11^2$ & $251.4099597$\\ \hline
787
788359 & $2^{24}.3^4.2777.16512254293.64542630435970307.2171776478013633068927$ & $1$ & $2^{11}$ & $2^6.5.123059$\\ \hline
789
790360 & $2^{12}.3^2$ & $0$ & $2^6.3$ & $\infty$\\ \hline
791
792361 & $2^{28}.3^{10}.5^{11}.11^4.19^{12}$ & $2^6.3^2.5^2$ & $2^{12}.3.5^2.11^2.19^6$ & $\infty$\\ \hline
793
794362 & $2^{26}.5^4.17^3.6359.1134769$ & $2^{40}.3^2.5^4.7^4.17^2.23^2.29^2.47.61^2.71.97.397^2.797^2.5297.595051637^2$ & $2^{10}.5.17$ & $2^2.2609601041$\\ \hline
795
796363 & $2^{36}.3^9.5^7.11^{18}$ & $2^{38}.3^{22}.5^8.11^4$ & $2^{15}.3^3.5.11^8$ & $\infty$\\ \hline
797
798364 & $2^{15}.3^2$ & $2^{125}.3^{63}.5^{10}.7^{12}.11^4.23.79^3$ & $2^5$ & $2^7.3^2.5.17.37$\\ \hline
799
800365 & $2^{51}.3^3.5^4.7^3.47.313^2.353783.9377894173$ & $-2^{20}.3^{12}.5^4.11^2.13^4.19^2.29.269$ & $2^{19}.3.5^2$ & $3^2.37.71473$\\ \hline
801
802366 & $2^{16}.3^2.13^2.17$ & $-2^{162}.3^{11}.5^4.7^6.13^{10}.17^3.19^6.23^3.31^2.37^6.127^2.137.151^3.229^2.5623^2$ & $2^8.3.13$ & $2^4.5^7.1277^2$\\ \hline
803
804367 & $2^{22}.7.81421.251387.418175501.15354151381.13144405392643360366681$ & $1$ & $2^{11}$ & $2^5.3197809$\\ \hline
805
806368 & $2^{40}.3^4.5^3.17$ & $0$ & $2^{20}.3^2.5$ & $?$\\ \hline
807
808369 & $2^{41}.3^6.5^2.23^2.37^3.79$ & $-2^{86}.3^{12}.5^4.23^4.37^3.79^2.191$ & $2^{15}.3^3.5.23$ & $7.13^2$\\ \hline
809
810370 & $2^{28}.3^2.11.223$ & $-2^{148}.3^{28}.5^{10}.7.11^2.13^4.19^2.23.31^2.61.193.269.23029^2.60869^2$ & $2^{12}$ & $2^4.72709379$\\ \hline
811
812371 & $2^{62}.5^3.7.157.229.64627370911.76723322773093$ & $2^{30}.3^3.5^2.37^2.191^2.643$ & $2^{25}.5$ & $2^5.3.5.76103$\\ \hline
813
814372 & $2^{10}.5^2.17$ & $-2^{181}.3^{31}.5^{13}.7^4.11^{12}.13^2.17^2.19^4.41^2.127.229^3$ & $2^5.5$ & $2^2.3^5.5^8$\\ \hline
815
816373 & $2^{32}.7.11^3.23.199.673.2143.1542194372227.72819251148518000363297$ & $1$ & $2^{13}.11$ & $2^4.3.1520821$\\ \hline
817
818374 & $2^{28}.5^2.157.257.11117.17417$ & $2^{142}.3^{20}.5^{18}.7^2.11^2.13^2.17^2.23.37^4.8461^2$ & $2^{14}$ & $11.3259.109111$\\ \hline
819
820375 & $2^{24}.5^{24}.101^2$ & $0$ & $2^{12}.5^8$ & $\infty$\\ \hline
821
822376 & $2^{33}.5^5.41^2.61$ & $0$ & $2^{15}$ & $?$\\ \hline
823
824377 & $2^{70}.3^3.5^2.7^2.829.991.36497.202817.400031.54126547$ & $2^6.7^2.23^3.31^2.79$ & $2^{30}.3.7$ & $181.118057$\\ \hline
825
826378 & $2^{16}.3^8$ & $0$ & $2^8.3^4$ & $\infty$\\ \hline
827
828379 & $2^{28}.59.317.421.278329.5698591.2117788336277.2851210737989187265253$ & $1$ & $2^{13}$ & $2^5.3^3.7.229.809$\\ \hline
829
830380 & $2^{15}.3$ & $2^{158}.3^{47}.5^9.7^2.11^7.13^4.17^2.19.37^3.113.151.709^3$ & $2^5$ & $2^2.3^4.11.53$\\ \hline
831
832381 & $2^{32}.11^2.15427.81509.98361184089911$ & $2^{36}.3^{10}.5^2.7^2.19^2.71.109^2.647^2.2003.86235899^2$ & $2^{12}.11$ & $2^6.3.7^3.13.43.113$\\ \hline
833
834382 & $2^{28}.7^2.11.13^2.619.44171$ & $2^{20}.3^6.5^4.11^4.41.967^2.1511.6277^2.63703.382146223^2.319500117632677^2$ & $2^{13}$ & $5.59.9181.50459$\\ \hline
835
836383 & $2^{16}.5.11^2.13.72893.3151861.16141144314299.178236551484825400362837637090811$ & $1$ & $2^8.11$ & $2^7.2480227$\\ \hline
837
838384 & $2^{40}$ & $0$ & $2^{20}$ & $\infty$\\ \hline
839
840385 & $2^{61}.3.37^3.2837$ & $-2^{114}.3^{18}.5^{17}.7^8.11.17^2.19^5.389$ & $2^{24}$ & $2^4.3.5.29.71$\\ \hline
841
842386 & $2^{32}.5^2.14653.22961.9659021$ & $-2^{30}.3^2.5^4.7.11^4.17^2.19^2.29^2.31^2.103^2.229^2.271^2.337.401^2.4153^2.244553.680059^2$ & $2^{12}$ & $3^2.13^2.74891419$\\ \hline
843
844387 & $2^{51}.3^9.5^8.7^2.13^2.71$ & $-2^{106}.3^{16}.5^6.7^4.71^2$ & $2^{20}.3^4.5^3.7$ & $2.7^3.17^2$\\ \hline
845
846388 & $2^8.7^2.179.1297$ & $-2^{90}.3^{16}.7^{11}.67^4.71^4.137^2.223^2.433.2777^3$ & $2^3$ & $2^3.3^3.22283^2$\\ \hline
847
848389 & $2^{53}.3^4.5^6.31^2.37.389.3881.215517113148241.477439237737571441$ & $1$ & $2^{17}.3^2.5.31$ & $7^2.67.173.863$\\ \hline
849
850
851
852\end{tabular}
853\end{center}
854
855%\begin{thebibliography}{HHHHHHH}
856%\bibitem[S]{stein} W. Stein, {\em Generating the Hecke algebra
857%as a $\Z$-module}.  preprint, 1998.
858%\end{thebibliography} \normalsize\vspace*{1 cm}
859
860\end{document}
861
862
863
864\\ table.gp -- make the discriminant table.
865
866\\discnew=discnormal=discsk=disct1tg=vector(1000,x,0);
867
868\rdiscnew
869\rdiscnormal
870\rdiscsk
871\rdisct1tg
872
873{pfac(n,
874  i, f)=
875  if(n<0, print1("-"); n=-n);
876  if(n==0 || n==1,print1(n); return;);
877  f=factor(n);
878  for(i=1,matsize(f)[1],
879    if(i>1,print1("."));
880    print1(f[i,1]);
881    if(f[i,2]>1 && f[i,2]<10, print1("^",f[i,2]));
882    if(f[i,2]>=10, print1("^{",f[i,2],"}"));
883  );
884}
885
886{pfac2(n,
887  i, f)=
888  if(n<0, print1("-"); n=-n);
889  if(n==0 || n==1,print1(n); return;);
890  f=factor(n);
891  for(i=1,matsize(f)[1],
892    if(i>1,print1("*"));
893    print1(f[i,1]);
894    if(f[i,2]>1, print1("^",f[i,2]));
895  );
896}
897
898{dsk(N)=
899  if(type(discsk[N])=="t_VEC",return);
900  print1("\ndiscsk[",N,"] = ");
901  pfac2(discsk[N]);
902}
903
904{dsknew(N)=
905  if(type(discnew[N])=="t_VEC",return);
906  print1("\ndiscnew[",N,"] = ");
907  pfac2(discnew[N]);
908}
909
910{entry(N)=
911  if(type(discnew[N])=="t_VEC",return);
912  print1("\n",N," & $"); 913 pfac(discnew[N]); 914 print1("$ & $"); 915 pfac(discsk[N]/discnew[N]); 916 print1("$ & $"); 917 pfac(floor(sqrt(discnew[N]/discnorm[N]))); 918 if(!issquare(discnew[N] % discnorm[N]), 919 print("ERROR ERROR ERROR!!! in [Ttildenew:Tnew] at level ",N);return;); 920 print1("$ & $"); 921 if(type(disct1tg[N])=="t_VEC", 922 print1("\\infty"), 923 \\ else 924 if(disct1tg[N]==0, 925 print1("?"), 926 \\ else 927 pfac(floor(sqrt(disct1tg[N]/discsk[N]))); 928 if(!issquare(disct1tg[N] % discsk[N]), 929 print("ERROR ERROR ERROR!!! in [T:W] at level ",N);return;) 930 )); 931 print1("$\\\\ \\hline\n");
932}
933
934{entry2(N)=
935  if(type(discnew[N])=="t_VEC",return);
936  if(type(disct1tg[N])=="t_VEC",
937    return);
938  \\ else
939  if(disct1tg[N]!=discsk[N],return);
940  if(disct1tg[N]==0,
941     print(N,"?"),
942  \\ else
943     if(disct1tg[N]==discsk[N] && !isprime(N),
944       print(N));
945  );
946}
947
948