Sharedwww / tables / dischecke.texOpen in CoCalc
Author: William A. Stein
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% dischecke.tex
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\documentclass[10pt]{article}
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\textwidth=1.2\textwidth
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\hoffset=-.5in
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\textheight=1.2\textheight
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\voffset=-.55in
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\title{Discriminants of Hecke Algebras}
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\include{macros}
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\begin{document}
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\maketitle
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\begin{abstract}
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For $N<250$ we compute the discriminants of the Hecke algebras associated to
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weight $2$ cusp forms and newforms of level $N$ and determine
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the primes $p$ so that the cusp $\infty$ is a Weierstrass point
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on $X_0(N)/\Fp$.
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\end{abstract}
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\section{Introduction}
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Let $S_2(N)$ be the space of
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weight $2$ cusp forms for $\Gamma_0(N)$. The
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Hecke algebra $\T\subset\End(S_2(N))$ is a finite commutative
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$\Z$-algebra. Its discriminant $\Delta=\disc(\T)$ is important in
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studying congruences between modular forms. Let $\Delta^{\new}$ be
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the discriminant of the new Hecke algebra
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$\T^{\new}\subset\End(S_2^{\new}(N))$.
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By ``multiplicity one'' $\T^{\new}$ is a subring
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of a product of fields and $\tilde{\T}^{\new}$ is the product
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of the rings of integers of these fields.
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Let $W$ be the sub $\Z$-module of $\T$ generated by the Hecke
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operator $T_1,T_2,\ldots T_g$ where $g$ is the genus of $X_0(N)$.
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When $N$ is prime $W$ has finite index
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in $\T$, but when $\infty$ is a Weierstrass point on $X_0(N)$ the index
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will not be finite. If $p\nmid N$, the cusp $\infty$ is a Weierstrass point on
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$X_0(N)/\Fp$ iff $p|[\T:W]$.
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%\section{Algorithms}
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\section{Tables}
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The notation is as above.
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\newpage
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\begin{center}
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\begin{tabular}{|l||c|c|c|c|}\hline
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$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
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& $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
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44
11 & $1$ & $1$ & $1$ & $1$\\ \hline
45
46
14 & $1$ & $1$ & $1$ & $1$\\ \hline
47
48
15 & $1$ & $1$ & $1$ & $1$\\ \hline
49
50
17 & $1$ & $1$ & $1$ & $1$\\ \hline
51
52
19 & $1$ & $1$ & $1$ & $1$\\ \hline
53
54
20 & $1$ & $1$ & $1$ & $1$\\ \hline
55
56
21 & $1$ & $1$ & $1$ & $1$\\ \hline
57
58
23 & $5$ & $1$ & $1$ & $1$\\ \hline
59
60
24 & $1$ & $1$ & $1$ & $1$\\ \hline
61
62
26 & $2^2$ & $1$ & $2$ & $1$\\ \hline
63
64
27 & $1$ & $1$ & $1$ & $1$\\ \hline
65
66
29 & $2^3$ & $1$ & $1$ & $1$\\ \hline
67
68
30 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline
69
70
31 & $5$ & $1$ & $1$ & $1$\\ \hline
71
72
32 & $1$ & $1$ & $1$ & $1$\\ \hline
73
74
33 & $1$ & $-3^2.11$ & $1$ & $1$\\ \hline
75
76
34 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline
77
78
35 & $2^2.17$ & $1$ & $2$ & $1$\\ \hline
79
80
36 & $1$ & $1$ & $1$ & $1$\\ \hline
81
82
37 & $2^2$ & $1$ & $2$ & $1$\\ \hline
83
84
38 & $2^2$ & $-2^3.3^2$ & $2$ & $1$\\ \hline
85
86
39 & $2^5$ & $1$ & $2$ & $1$\\ \hline
87
88
40 & $1$ & $0$ & $1$ & $?$\\ \hline
89
90
41 & $2^2.37$ & $1$ & $1$ & $1$\\ \hline
91
92
42 & $1$ & $2^9.7$ & $1$ & $1$\\ \hline
93
94
43 & $2^5$ & $1$ & $2$ & $2$\\ \hline
95
96
44 & $1$ & $-2^6$ & $1$ & $1$\\ \hline
97
98
45 & $1$ & $2^2$ & $1$ & $1$\\ \hline
99
100
46 & $1$ & $2^2.5^4.41$ & $1$ & $1$\\ \hline
101
102
47 & $19.103$ & $1$ & $1$ & $1$\\ \hline
103
104
48 & $1$ & $0$ & $1$ & $?$\\ \hline
105
106
49 & $1$ & $1$ & $1$ & $1$\\ \hline
107
108
50 & $2^2$ & $1$ & $2$ & $1$\\ \hline
109
110
51 & $2^2.17$ & $-2^6.3$ & $2$ & $3$\\ \hline
111
112
52 & $1$ & $2^4.3^2$ & $1$ & $3$\\ \hline
113
114
53 & $2^4.37$ & $1$ & $2$ & $3$\\ \hline
115
116
54 & $2^2$ & $-2^3.3^4$ & $2$ & $\infty$\\ \hline
117
118
55 & $2^5$ & $-7^2.19$ & $2$ & $1$\\ \hline
119
120
56 & $2^2$ & $0$ & $2$ & $?$\\ \hline
121
122
57 & $2^2.3^2$ & $-2^7$ & $2.3$ & $1$\\ \hline
123
124
58 & $2^2$ & $2^{10}.17$ & $2$ & $2$\\ \hline
125
126
59 & $2^3.31.557$ & $1$ & $1$ & $1$\\ \hline
127
128
61 & $2^4.37$ & $1$ & $2$ & $2$\\ \hline
129
130
62 & $2^4.3$ & $2^2.5^2.11^2.41$ & $2$ & $3$\\ \hline
131
132
63 & $2^4.3$ & $2^4$ & $2$ & $2$\\ \hline
133
134
64 & $1$ & $0$ & $1$ & $\infty$\\ \hline
135
136
65 & $2^{11}.3$ & $1$ & $2^3$ & $3$\\ \hline
137
138
66 & $2^4$ & $-2^8.3^4.5^2.7.11^2$ & $2^2$ & $2$\\ \hline
139
140
67 & $2^4.5^4$ & $1$ & $2^2.5$ & $2^3$\\ \hline
141
142
68 & $2^2.3$ & $-2^8.3^2.7$ & $1$ & $2^2$\\ \hline
143
144
69 & $2^4.5$ & $5^2.7^2.11^2$ & $2^2$ & $2$\\ \hline
145
146
70 & $1$ & $2^{18}.3^2.5.17^2$ & $1$ & $3$\\ \hline
147
148
149
71 & $3^4.257^2$ & $1$ & $3^2$ & $1$\\ \hline
150
151
\end{tabular}
152
153
\begin{tabular}{|l||c|c|c|c|}\hline
154
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$
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& $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline
156
157
72 & $1$ & $0$ & $1$ & $\infty$\\ \hline
158
159
73 & $2^4.3^2.5.13$ & $1$ & $2^2.3$ & $2$\\ \hline
160
161
74 & $2^4.5.13$ & $2^9.3^2.5^2$ & $2^2$ & $11$\\ \hline
162
163
75 & $2^2.3^2$ & $2^2.3^2$ & $2.3$ & $1$\\ \hline
164
165
76 & $1$ & $-2^{11}.3^6$ & $1$ & $1$\\ \hline
166
167
77 & $2^6.5^3$ & $-2^7.3^3$ & $2^3.5$ & $5$\\ \hline
168
169
78 & $1$ & $-2^{22}.3.5^2.7^3.11.17$ & $1$ & $2^2$\\ \hline
170
171
79 & $2^2.83.983$ & $1$ & $2$ & $2^3$\\ \hline
172
173
80 & $2^2$ & $0$ & $2$ & $?$\\ \hline
174
175
81 & $2^2.3$ & $0$ & $1$ & $?$\\ \hline
176
177
82 & $2^5$ & $-2^{10}.23.37^2$ & $2$ & $2$\\ \hline
178
179
83 & $2^4.197.11497$ & $1$ & $2$ & $2^2$\\ \hline
180
181
84 & $2^2$ & $-2^{22}.3^4.7$ & $2$ & $3$\\ \hline
182
183
85 & $2^{11}.3$ & $-2^{10}$ & $2^3$ & $7$\\ \hline
184
185
86 & $2^4.3.5.7$ & $-2^{14}.3^2.5^2.7^2$ & $2^2$ & $41$\\ \hline
186
187
87 & $2^4.5.229$ & $2^6.23^2.73$ & $2^2$ & $2^3$\\ \hline
188
189
88 & $2^6.17$ & $0$ & $2^3$ & $?$\\ \hline
190
191
89 & $2^6.5^3.6689$ & $1$ & $2.5$ & $3^2$\\ \hline
192
193
90 & $2^4$ & $-2^{22}.7^3$ & $2^2$ & $1$\\ \hline
194
195
91 & $2^{15}.79$ & $1$ & $2^5$ & $2$\\ \hline
196
197
92 & $2^2$ & $2^{12}.3^2.5^7.41$ & $2$ & $1$\\ \hline
198
199
93 & $2^4.5.229$ & $2^{12}.5^2$ & $2^2$ & $2^3$\\ \hline
200
201
94 & $2^5$ & $2^4.19^2.47^2.103^2.457$ & $2$ & $7$\\ \hline
202
203
95 & $2^{12}.37.709$ & $-3^4.11$ & $2^3$ & $11$\\ \hline
204
205
96 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
206
207
97 & $2^6.7^2.2777$ & $1$ & $2^3$ & $2.3$\\ \hline
208
209
98 & $2^5$ & $-2^{12}.7$ & $2$ & $\infty$\\ \hline
210
211
99 & $2^4.3^4$ & $-2^6.3^8.11$ & $2^2.3^2$ & $2$\\ \hline
212
213
100 & $1$ & $2^8.3^4$ & $1$ & $\infty$\\ \hline
214
215
101 & $2^8.17568767$ & $1$ & $2$ & $3^3$\\ \hline
216
217
102 & $2^4$ & $2^{41}.3^6.7^2.17^2$ & $2^2$ & $3.5^2$\\ \hline
218
219
103 & $2^4.5.17.411721$ & $1$ & $2^2$ & $2^2.7$\\ \hline
220
221
104 & $2^2.17$ & $0$ & $2$ & $?$\\ \hline
222
223
105 & $2^4.5$ & $-2^{30}.5^2.7.11.13.17^2$ & $2^2$ & $2.3$\\ \hline
224
225
106 & $2^4.3^2$ & $2^{18}.5^2.7.37^2.151$ & $2^2.3$ & $2.13$\\ \hline
226
227
107 & $2^6.5.7.1667.19079$ & $1$ & $2^2$ & $2^2.5$\\ \hline
228
229
108 & $1$ & $0$ & $1$ & $\infty$\\ \hline
230
231
109 & $2^{10}.7^2.7537$ & $1$ & $2^5$ & $2.5$\\ \hline
232
233
110 & $2^8.3.11$ & $-2^{22}.5^2.7^7.17.19^2$ & $2^4$ & $2.3$\\ \hline
234
235
111 & $2^{12}.37.389$ & $2^4.3.5^2.7^2.11$ & $2^3$ & $2^2.3$\\ \hline
236
237
112 & $2^6$ & $0$ & $2^3$ & $?$\\ \hline
238
239
113 & $2^{10}.3^4.7^2.11^2.107$ & $1$ & $2^4.3.11$ & $2^2.17$\\ \hline
240
241
114 & $2^4$ & $-2^{36}.3^{12}.5^6.7.11^2$ & $2^2$ & $3^5$\\ \hline
242
243
115 & $2^6.5.17^2.53$ & $2^{12}.5^4.11$ & $2^3$ & $2^2$\\ \hline
244
245
116 & $2^4.5^2$ & $2^{29}.3^4.17$ & $2^2.5$ & $2^2.7$\\ \hline
246
247
117 & $2^{11}.3$ & $2^{24}$ & $2^3$ & $2^4$\\ \hline
248
249
118 & $2^4.3^2$ & $-2^{18}.7.19^2.31^2.557^2$ & $2^2.3$ & $223$\\ \hline
250
251
119 & $2^8.71.131.311.1459$ & $-2^6.3^3$ & $2^4$ & $2^4$\\ \hline
252
253
120 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
254
255
121 & $2^4.3^2$ & $2^6.3^2$ & $2^2.3$ & $\infty$\\ \hline
256
257
122 & $2^6.13.229$ & $2^{18}.7.13^2.37^2.151$ & $2^3$ & $2.83$\\ \hline
258
259
123 & $2^{15}.79$ & $-2^8.5^2.23^2.37^2.191$ & $2^5$ & $13$\\ \hline
260
261
124 & $2^2$ & $2^{20}.3^6.5^3.11^4.41$ & $2$ & $3^2$\\ \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
268
125 & $2^8.5^{12}.11$ & $0$ & $2^2.5^4$ & $\infty$\\ \hline
269
270
126 & $2^2$ & $2^{49}.3^8.5^2.7^3$ & $2$ & $2$\\ \hline
271
272
127 & $2^6.3^4.7.86235899$ & $1$ & $2^3$ & $2^2.29$\\ \hline
273
274
128 & $2^{16}$ & $0$ & $2^8$ & $\infty$\\ \hline
275
276
129 & $2^{12}.5^2.71$ & $-2^{23}.3^2.5^2.7^2$ & $2^3.5$ & $17$\\ \hline
277
278
130 & $2^4$ & $-2^{38}.3^6.5^4.7^3.11.17.19$ & $2^2$ & $5.7$\\ \hline
279
280
131 & $2^9.5.46141.75619573$ & $1$ & $2$ & $2.5^2$\\ \hline
281
282
132 & $2^2$ & $-2^{32}.3^{12}.5^6.7.11^4$ & $2$ & $2^3$\\ \hline
283
284
133 & $2^{16}.5^2.13.229$ & $-3^5.7^2$ & $2^8$ & $2.43$\\ \hline
285
286
134 & $2^6.3^4.11.43$ & $-2^{10}.3^2.5^{12}.19^2.41$ & $2^3$ & $29.31$\\ \hline
287
288
135 & $2^6.3^4.13^2$ & $0$ & $2^3.3^2$ & $\infty$\\ \hline
289
290
136 & $2^8.5$ & $0$ & $2^4$ & $?$\\ \hline
291
292
137 & $2^{10}.5^2.29.401.895241$ & $1$ & $2^4$ & $2^5.5$\\ \hline
293
294
138 & $2^8.5$ & $2^{32}.3^3.5^{10}.7^5.11^6.41^2$ & $2^4$ & $2^4.7^2$\\ \hline
295
296
139 & $2^8.3^2.7^2.997.2151701$ & $1$ & $2^4.3$ & $2^3.41$\\ \hline
297
298
140 & $2^2$ & $2^{43}.3^{11}.5^4.17^3$ & $2$ & $2^2.3$\\ \hline
299
300
141 & $2^{14}.3^2.17$ & $3^2.7^2.19^2.43^2.103^2.3529$ & $2^7.3$ & $2^2.5.13$\\ \hline
301
302
142 & $2^8.3^2$ & $2^6.3^{22}.7.103.257^4$ & $2^4.3$ & $3.67$\\ \hline
303
304
143 & $2^{10}.5.7.19.103.5560463$ & $-2^6.3^6$ & $2^5$ & $2.3^2$\\ \hline
305
306
144 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
307
308
145 & $2^{21}.37^2$ & $2^6.5^4.7^2.19^2$ & $2^7$ & $2.7$\\ \hline
309
310
146 & $2^{12}.101.389$ & $-2^{12}.3^{10}.5^2.7.13^2.17.19^2$ & $2^3$ & $97$\\ \hline
311
312
147 & $2^{14}.3^2.7^4$ & $-2^{14}.3^3$ & $2^4.3.7^2$ & $\infty$\\ \hline
313
314
148 & $2^2.17$ & $2^{27}.3^{10}.5^6.13^2$ & $2$ & $3^2.11^2$\\ \hline
315
316
149 & $2^{12}.7^2.234893.1252037$ & $1$ & $2^3$ & $107$\\ \hline
317
318
150 & $2^4$ & $-2^{34}.3^{12}.5^8.7^3.11^2$ & $2^2$ & $\infty$\\ \hline
319
320
151 & $2^6.7^2.11.67^2.257.439867$ & $1$ & $2^3.67$ & $2.11^2$\\ \hline
321
322
152 & $2^8.31^2$ & $0$ & $2^4$ & $?$\\ \hline
323
324
153 & $2^{12}.3^2.17$ & $-2^{30}.3^5.17^2$ & $2^6.3$ & $3^3$\\ \hline
325
326
154 & $2^8.5$ & $2^{56}.3^{14}.5^8.7.11$ & $2^4$ & $2.23$\\ \hline
327
328
155 & $2^{24}.29.73.5077$ & $5^4.7^4.19^2$ & $2^{10}$ & $2.73$\\ \hline
329
330
156 & $2^2$ & $2^{57}.3^{11}.5^4.7^5.11^2.17$ & $2$ & $2^2.3$\\ \hline
331
332
157 & $2^{13}.61.397.48795779$ & $1$ & $2^5$ & $2.13$\\ \hline
333
334
158 & $2^{13}.3^3.5^2$ & $2^{22}.7.17.53^2.83^2.271.983^2$ & $2^5.3.5$ & $2^2.541$\\ \hline
335
336
159 & $2^8.19.103.1054013$ & $2^8.3.7^2.37^2.107^2.227$ & $2^4$ & $2^4.3.5$\\ \hline
337
338
160 & $2^{13}$ & $0$ & $2^5$ & $\infty$\\ \hline
339
340
161 & $2^{16}.5^3.37.536777$ & $2^{12}.5^2.19^2.29$ & $2^6.5$ & $2.97$\\ \hline
341
342
162 & $2^4.3^4$ & $0$ & $2^2.3^2$ & $\infty$\\ \hline
343
344
163 & $2^{15}.3^2.65657.82536739$ & $1$ & $2^6.3$ & $2^2.347$\\ \hline
345
346
164 & $2^4.1613$ & $-2^{40}.3^6.23.37^3$ & $1$ & $2^3.5$\\ \hline
347
348
165 & $2^{17}.3.37$ & $-2^{50}.3^5.5^2.7^5.11^2.19^2$ & $2^5$ & $2^2.3$\\ \hline
349
350
166 & $2^6.5.229$ & $-2^{26}.7.71.131^2.197^2.11497^2$ & $2^3$ & $2.5.479$\\ \hline
351
352
167 & $2^4.5.8269.5103536431379173$ & $1$ & $2^2$ & $2.3^2$\\ \hline
353
354
168 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline
355
356
169 & $2^8.3.7^4.13^4$ & $1$ & $2^3.13^2$ & $\infty$\\ \hline
357
358
170 & $2^{12}.5^2.17$ & $2^{68}.3^8.5^3.7^5.17$ & $2^6.5$ & $839$\\ \hline
359
360
171 & $2^{16}.3^7.11^2$ & $-2^{33}.3^8$ & $2^7.3^2$ & $2.3^3$\\ \hline
361
362
172 & $2^5$ & $-2^{39}.3^{10}.5^6.7^6$ & $2$ & $41^2$\\ \hline
363
364
173 & $2^{14}.5^2.7.29.5608385124289$ & $1$ & $2^4$ & $79$\\ \hline
365
366
174 & $2^8.5^2$ & $-2^{38}.3^3.5^4.7^4.11^3.13^2.17^2\ldots$ & $2^4.5$ & $2^6.5.11^2$\\
367
& & $\ldots 23^4.41.47.73^2.229^2$ & &\\ \hline
368
369
175 & $2^{12}.3^4.5^6.17$ & $2^{18}.3^4.17^2$ & $2^6.3^2.5^2$ & $2^2$\\ \hline
370
371
176 & $2^{14}.17$ & $0$ & $2^7$ & $?$\\ \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
378
177 & $2^{16}.5^3.229$ & $-2^6.31^4.229^2.557^2.20627$ & $2^8$ & $2^4.149$\\ \hline
379
380
178 & $2^{16}.71$ & $-2^{30}.5^6.7^4.199.6689^2$ & $2^5$ & $2^2.487$\\ \hline
381
382
179 & $2^{12}.3^4.7^2.313.137707.536747147$ & $1$ & $2^3.3^2$ & $2.59$\\ \hline
383
384
180 & $1$ & $-2^{73}.3^{16}.5.7^3$ & $1$ & $\infty$\\ \hline
385
386
181 & $2^{16}.5^2.7.61.397.595051637$ & $1$ & $2^5$ & $2^4.3^3$\\ \hline
387
388
182 & $2^8.3^2$ & $2^{51}.3^{20}.5^4.7^6.11^2.23.79^2$ & $2^4.3$ & $2^2.3^2.5.37$\\ \hline
389
390
183 & $2^{22}.37.127.5623$ & $2^{27}.3^2.19^2.23.37^2$ & $2^5$ & $2^2.5^4$\\ \hline
391
392
184 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline
393
394
185 & $2^{26}.3^2.23029.60869$ & $2^{24}.3^2.5$ & $2^9.3$ & $193$\\ \hline
395
396
186 & $2^8.17$ & $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
399
187 & $2^{28}.3.5^2.17.37.8461$ & $2^{24}.3^4.11$ & $2^{11}.5$ & $2.3^2.11$\\ \hline
400
401
188 & $2^4.5.13$ & $2^{34}.3^6.19^3.47^4.103^3.457$ & $2^2$ & $7^2$\\ \hline
402
403
189 & $2^{16}.3^9.7$ & $0$ & $2^6.3^4$ & $\infty$\\ \hline
404
405
190 & $2^8.17$ & $-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
408
191 & $2^4.3^3.5.382146223.319500117632677$ & $1$ & $2^2$ & $2^2.223$\\ \hline
409
410
192 & $2^{12}$ & $0$ & $2^6$ & $\infty$\\ \hline
411
412
193 & $2^{14}.5.11^2.17.103.401.4153.680059$ & $1$ & $2^7.11$ & $2^7$\\ \hline
413
414
194 & $2^{20}.7^2.137.223$ & $-2^{14}.7^5.67^2.71^2.433.2777^2$ & $2^4.7$ & $3.22283$\\ \hline
415
416
195 & $2^{14}.3^2.79$ & $-2^{77}.3^9.5^2.7^6.11^2.13$ & $2^6.3$ & $2.3^2.5.7$\\ \hline
417
418
196 & $2^5.7^4$ & $-2^{38}.3^8.7$ & $2.7^2$ & $\infty$\\ \hline
419
420
197 & $2^{18}.5^2.61.397.35217676193989$ & $1$ & $2^6.5$ & $7.383$\\ \hline
421
422
198 & $2^{10}$ & $-2^{78}.3^{24}.5^6.7^5.11^2$ & $2^5$ & $2.7^2$\\ \hline
423
424
199 & $2^8.3.5^3.29.31.71^2.347.947.37316093$ & $1$ & $2^4.71$ & $2^4.241$\\ \hline
425
426
200 & $2^8.3^2.5^4$ & $0$ & $2^4.3.5^2$ & $\infty$\\ \hline
427
428
201 & $2^{18}.3^2.37.269.953$ & $-2^{15}.3^2.5^{10}.19^2.29^2.61.109$ & $2^7.3$ & $3.239$\\ \hline
429
430
202 & $2^8.3^4.10273$ & $2^{33}.3^4.17^4.17568767^2$ & $2^4$ & $2^3.2381$\\ \hline
431
432
203 & $2^{35}.3^7.17.29.37.7547$ & $2^{22}.5^2.7^2$ & $2^{14}.3^3$ & $2^3.193$\\ \hline
433
434
204 & $2^2$ & $2^{102}.3^{23}.7^2.11^2.13.17^3$ & $2$ & $2^2.3.5^4$\\ \hline
435
436
205 & $2^{30}.3^2.5.13.229^2$ & $-2^{14}.13^2.31^2.37^2.239$ & $2^{15}.3$ & $7^2.23$\\ \hline
437
438
206 & $2^{16}.3^2.5^2.13.29.359$ & $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
441
207 & $2^{20}.3^4.5^2.11^2$ & $2^{28}.3^8.5^5.7^2.11^4$ & $2^7.3^2.11$ & $2^6$\\ \hline
442
443
208 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline
444
445
209 & $2^{32}.3^4.15427.2002061$ & $2^{10}.3^4.5^5.7.19$ & $2^9$ & $3.11^2$\\ \hline
446
447
210 & $2^8$ & $2^{176}.3^{14}.5^8.7^7.11^2.13^2.17^4$ & $2^4$ & $2^3.3$\\ \hline
448
449
211 & $2^{14}.3.5.7^4.41^2.43.229.52184516509$ & $1$ & $2^6.7.41$ & $2^5.5.23$\\ \hline
450
451
212 & $2^4.3^3.7^3$ & $2^{56}.3^{12}.5^4.7.37^3.151$ & $2.7$ & $2^2.13^3$\\ \hline
452
453
213 & $2^{20}.3^2.5^4.13.89$ & $3^{14}.5^2.19^2.37.61^2.229.257^4$ & $2^{10}.3$ & $2^3.3.17.53$\\ \hline
454
455
214 & $2^{16}.3^6$ & $-2^{24}.5^4.7^4.11^2.41.109^2.521\ldots$ & $2^6.3^2$ & $3.67.4903$\\
456
& & $\ldots 1667^2.19079^2$ && \\ \hline
457
458
215 & $2^{14}.3.7^4.101.107.321821.1933097$ & $-2^{22}.5^2.31^2.41$ & $2^7.7^2$ & $2^2.3^6$\\ \hline
459
460
216 & $2^8.3^4$ & $0$ & $2^4.3^2$ & $\infty$\\ \hline
461
462
217 & $2^{31}.3^8.11.31.557.619$ & $5^2.19^2.31^2.281$ & $2^{14}$ & $2^4.5.43$\\ \hline
463
464
218 & $2^{19}.3^4.5.11^2.23$ & $2^{34}.3^4.7^7.41^2.167.601.7537^2$ & $2^7.11$ & $2.7.21227$\\ \hline
465
466
219 & $2^{24}.3^2.29.73.1189637$ & $-2^{16}.3^8.5^4.13^2.17^2.23.29^2.61$ & $2^{10}.3$ & $3^2.41.71$\\ \hline
467
468
220 & $2^2$ & $2^{77}.3^{16}.5^4.7^{11}.11^4.17.19^3$ & $2$ & $2.3.5$\\ \hline
469
470
221 & $2^{40}.3^3.5^6.7.37.109^2.229$ & $-2^{12}.3^3$ & $2^{17}.3.5^2$ & $5.107$\\ \hline
471
472
222 & $2^8.3^2$ & $-2^{60}.3^{14}.5^{10}.7^4.11^4.13^5.19^2\ldots$ & $2^4.3$ & $2^4.3^5.11^2$\\
473
& & $\ldots23^2.31.37^2.109.389^2.409$ && \\ \hline
474
475
223 & $2^{18}.7^2.19.103.3995922697473293141$ & $1$ & $2^6.7$ & $2.3^2.5.641$\\ \hline
476
477
224 & $2^{18}.5^2$ & $0$ & $2^9$ & $\infty$\\ \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
484
225 & $2^{12}.3^2.5^5$ & $2^{38}.3^{10}$ & $2^6.3.5^2$ & $\infty$\\ \hline
485
486
226 & $2^{23}.3.5^3$ & $-2^{34}.3^{12}.7^9.11^4.23.41^2\ldots$ & $2^7$ & $2^2.190129$\\
487
& & $\ldots 107^2.167$ & & \\\hline
488
489
227 & $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
492
228 & $2^4.3.11$ & $-2^{88}.3^{36}.5^{12}.7.11^4$ & $2^2$ & $2^2.3^9$\\ \hline
493
494
229 & $2^{24}.107.17467.39555937\times\ldots$ & $1$ & $2^9$ & $2^2.11.193$\\
495
& $\ldots 53625889$ & & & \\ \hline
496
497
230 & $2^{16}.3^2.5.7.13.367$ & $2^{52}.3^2.5^{18}.11^4.17^4.19^2\ldots$ & $2^8$ & $2^2.5^2.53$\\
498
& & $\ldots41^2.43^2.47.53^2$ & & \\\hline
499
500
231 & $2^{20}.3^4.5^3.7.31.229$ & $-2^{59}.3^{16}.5^{12}.7^3.11^3$ & $2^{10}.5$ & $2^2.29$\\ \hline
501
502
232 & $2^{20}.71$ & $0$ & $2^7$ & $?$\\ \hline
503
504
233 & $2^{22}.3^7.53.139.653.4127\times \ldots$ & $1$ & $2^7.3^3$ & $2^2.3.11.197$\\
505
& $\ldots24989.8388019$ & & & \\\hline
506
507
234 & $2^6.3^2.5^2$ & $-2^{118}.3^{13}.5^6.7^9.11.17^3$ & $2^3.3.5$ & $2^5.3^2$\\ \hline
508
509
235 & $2^{18}.3^2.73.1453.3851.1916279$ & $2^{24}.3^8.19^3.103^2$ & $2^9.3$ & $2^2.3.17.31$\\ \hline
510
511
236 & $2^2.3.7^2.107$ & $-2^{55}.3^{12}.7.19^4.31^3.557^3$ & $2.7$ & $223^2$\\ \hline
512
513
237 & $2^{18}.19.103.1705391537$ & $2^{28}.3^2.5^2.11^2.31^2.83^2.983^2$ & $2^6$ & $2^6.5.97$\\ \hline
514
515
238 & $2^{16}.5$ & $-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
518
239 & $2^6.7^2.2833.51817.97423\times \ldots$ & $1$ & $2^3$ & $2^2.3.11^2$\\
519
& $\ldots 1174779433.8920940047$ & & & \\\hline
520
521
240 & $2^{10}$ & $0$ & $2^5$ & $\infty$\\ \hline
522
523
241 & $2^{23}.97.1489.20857\times \ldots$ & $1$ & $2^7$ & $2^7.151$\\
524
& $\ldots 651474368435017$ & & &\\\hline
525
526
242 & $2^{18}.3^2.5^2.11^8$ & $2^{41}.3^{12}.5^4.7^2$ & $2^7.11^4$ & $\infty$\\ \hline
527
528
243 & $2^{13}.3^{40}$ & $0$ & $2^4.3^{15}$ & $\infty$\\ \hline
529
530
244 & $2^4.5077$ & $2^{60}.3^{10}.7.13^6.37^3.151.229^2$ & $2$ & $2^2.3.5.83^2$\\ \hline
531
532
245 & $2^{36}.3^4.7^8.17$ & $-2^{28}.3^8.5.17^2$ & $2^{12}.3^2.7^4$ & $2^2$\\ \hline
533
534
246 & $2^{12}.3^2.5^2$ & $2^{96}.3^8.5^{10}.7^6.11^2.23^7.37^4\ldots$ & $2^6.3.5$ & $7.73^2$\\
535
& & $\ldots 79^2.191^2$ & & \\\hline
536
537
247 & $2^{30}.3^4.5^2.11.31^2.619\times\ldots$ & $-2^6.3^4.5^2$ & $2^{15}.31$ & $2^2.29.241$\\
538
& $\ldots 57713.2655049$ & & & \\\hline
539
540
248 & $2^{22}.3.11.79$ & $0$ & $2^{10}$ & $?$\\ \hline
541
542
249 & $2^{33}.389.23029$ & $-2^8.3^2.5^2.11.31^2.197^2.2711^2\ldots$ & $2^{11}$ & $2.13.41.89$\\
543
& & $\ldots 11497^2.295201$ & & \\ \hline
544
545
250 & $2^8.5^{12}$ & $0$ & $2^4.5^4$ & $\infty$\\ \hline
546
547
251 & $2^{14}.5^2.29.373\times\ldots $ & $1$ & $2^4$ & $2^2.3.797$\\
548
& $8768135668531.2006012696666681$ & & & \\ \hline
549
550
252 & $2^2$ & $2^{136}.3^{30}.5^2.7^3$ & $2$ & $\infty$\\ \hline
551
552
253 & $2^{28}.3^4.13^2.2711.3187.170701$ & $-2^{12}.3^2.5^7.7.13.19^2.23^2$ & $2^{14}$ & $2.4507$\\ \hline
553
554
254 & $2^{28}.3^4.17.569$ & $2^{16}.3^{18}.7^2.17^2.41.71\times\ldots$ & $2^{11}$ & $7.189713$\\
555
& & $\ldots 383^2.1231.86235899^2$ & & \\ \hline
556
557
255 & $2^{23}.5.13.229.1721$ & $-2^{99}.3^{14}.5^2.11^3.13^3.17^4$ & $2^{10}$ & $2^3.3^3.379$\\ \hline
558
559
256 & $2^{31}$ & $0$ & $2^{14}$ & $\infty$\\ \hline
560
561
257 & $2^{29}.29.479.71711.409177\times\ldots$ & $1$ & $2^{11}$ & $2.29.3251$\\
562
& $\ldots 654233.32354821$ & & & \\ \hline
563
564
565
258 & $2^{12}.3^2.5^2$ & $-2^{104}.3^{14}.5^{16}.7^{17}\times\ldots$ & $2^6.3.5$ & $7^3.199^2$\\
566
& & $\ldots 11^2.17.19^2.37.71^2.109$ & & \\\hline
567
568
259 & $2^{43}.3^9.5^3.7^2.17^3.29.37.167$ & $2^4.3^{12}.7^4$ & $2^{20}.3.17$ & $2^2.3.29.193$\\ \hline
569
570
260 & $2^6.3.47$ & $-2^{117}.3^{26}.5^6.7^5.11^2.17.19^2$ & $2^2$ & $\infty$\\ \hline
571
572
261 & $2^{19}.3^4.5^3.23^2.229$ & $2^{47}.3^8.5^2.23^4.73.229^2$ & $2^8.3^2.23$ & $2^{10}.5$\\ \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
581
262 & $2^{19}.3^3.5.11^2.13$ & $-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
584
263 & $2^{10}.11.61.397.15631853\times\ldots$& $1$ & $2^5$ & $2^3.19.223$\\
585
& $\ldots 34867513.97092067.252746489$ & & & \\\hline
586
587
588
264 & $2^6.3^2$ & $0$ & $2^3.3$ & $\infty$\\ \hline
589
590
265 & $2^{37}.3^2.5^4.7^3.11^2.13.17^2.113$ & $2^{26}.3^4.5^3.31^2.37^2.239$ & $2^{13}.5.7.11.17$ & $2.3^3.103$\\ \hline
591
592
266 & $2^{16}.5.7.13.29.67$ & $2^{51}.3^{31}.5^6.7^6.11^6.13^4.17.19^3.41.47.229^2$ & $2^8$ & $3.216679$\\ \hline
593
594
267 & $2^{26}.3^4.5^2.7^4.13^2.97.241$ & $-2^{17}.3^4.5^8.11.13^2.17^2.37.113^2.911.6689^2$ & $2^{13}.5.7$ & $3^2.17.41.271$\\ \hline
595
596
268 & $2^4.3^3.5.7$ & $-2^{46}.3^{20}.5^{20}.11^2.19^4.41.43^2$ & $2^2.3$ & $29^2.31^2$\\ \hline
597
598
269 & $2^{22}.3^2.43.151.27767.65657.5550873754172978311$ & $1$ & $2^6.3$ & $11^2.19^2$\\ \hline
599
600
270 & $2^8$ & $0$ & $2^4$ & $\infty$\\ \hline
601
602
271 & $2^{12}.3^2.1367.6091.592661.1132673.14171513.172450541$ & $1$ & $2^6.3$ & $2.3.17239$\\ \hline
603
604
272 & $2^{28}.3^3.5$ & $0$ & $2^{13}.3$ & $?$\\ \hline
605
606
273 & $2^{25}.79.4357$ & $-2^{111}.3^9.5^4.7^6.11.79^2$ & $2^9$ & $2^2.3.313$\\ \hline
607
608
274 & $2^{20}.3^2.19.37.1321$ & $-2^{26}.5^4.7^2.11^2.29^2.109^2.149^2.401^2.769.47143.895241^2$ & $2^7.3$ & $2.11.304937$\\ \hline
609
610
275 & $2^{31}.3^8.5^{12}.7^2.11^2.13^2$ & $-2^{28}.3^6.5^4.7^4.19$ & $2^{12}.3^3.5^5.7$ & $2^2$\\ \hline
611
612
276 & $2^{10}.5$ & $-2^{108}.3^{19}.5^{21}.7^9.11^{11}.41^2$ & $2^2$ & $2^4.7^4$\\ \hline
613
614
277 & $2^{22}.5^2.19.29.37.137^2.92767.1530091.25531570859$ & $1$ & $2^{10}.5.137$ & $2.3^2.8311$\\ \hline
615
616
278 & $2^{22}.3^4.7.17^2.103.107$ & $-2^{38}.3^4.7^7.41^2.167.271^3.997^2.2151701^2$ & $2^8.17$ & $6084451$\\ \hline
617
618
279 & $2^{28}.3^5.5^2.229.1373^2$ & $2^{60}.3^8.5^5.229^2$ & $2^{11}.3^2$ & $2^9.5^3$\\ \hline
619
620
280 & $2^{18}.3.11.17$ & $0$ & $2^9$ & $\infty$\\ \hline
621
622
281 & $2^{22}.3.5.181.857.8388019.2647382149.1778899342669$ & $1$ & $2^7$ & $2^2.79309$\\ \hline
623
624
282 & $2^{23}.3^2.37$ & $2^{96}.3^{13}.5^4.7^6.17^2.19^4.23^2.43^4.47^4.103^4.457^2.3529^2$ & $2^8$ & $2^7.5^6$\\ \hline
625
626
283 & $2^{40}.349.1297.413713.73199099.5832488839$ & $1$ & $2^9$ & $2^7.2687$\\ \hline
627
628
284 & $2^6.3^5.107$ & $2^{52}.3^{52}.7.103.257^6$ & $2^3$ & $3^2.67^2$\\ \hline
629
630
285 & $2^{34}.3^3.7$ & $-2^{80}.3^{23}.5^5.7^2.11^3.19^3.23.37^2.47^2.73.709^2$ & $2^{12}.3$ & $2.3.10771$\\ \hline
631
632
286 & $2^{16}.3^2.31^2$ & $-2^{70}.3^{22}.5^9.7^4.13^2.19^4.103^2.139^2.257.5560463^2$ & $2^8.3$ & $120671$\\ \hline
633
634
287 & $2^{28}.3.5^2.7^2.61^2.97^2.103.257.211039.1798619$ & $-2^8.13^2.19^2.37^4.1847$ & $2^{14}.61.97$ & $2^2.3.823$\\ \hline
635
636
288 & $2^{20}.3^2$ & $0$ & $2^{10}.3$ & $\infty$\\ \hline
637
638
289 & $2^{23}.3^{16}.13^2.17^8$ & $2^6.3^4$ & $2^6.3^4.17^4$ & $\infty$\\ \hline
639
640
290 & $2^{20}.3^3.7.13^2.23.67$ & $-2^{84}.3^6.5^{10}.7^{11}.11.13^2.17^3.19^5.23^2.31.37^4.151$ & $2^{10}$ & $2^3.5.97.421$\\ \hline
641
642
291 & $2^{32}.3^4.5^4.11^2.13.19.457.16657$ & $-2^{22}.7^4.11^2.13^2.23^2.139^2.491.1933.2777^2$ & $2^{15}.3^2.5.11$ & $73.167$\\ \hline
643
644
292 & $2^7.5.41^2$ & $-2^{64}.3^{28}.5^3.7.13^3.17.19^4.101^2.389^2$ & $2^2$ & $2.3^2.97^2$\\ \hline
645
646
293 & $2^{26}.3^2.29.233.2351^2.69763.42711913589792108923$ & $1$ & $2^{10}$ & $67759$\\ \hline
647
648
294 & $2^{12}.3^2.7^4$ & $2^{147}.3^{16}.5^6.7^{21}.17^2$ & $2^6.3.7^2$ & $\infty$\\ \hline
649
650
295 & $2^{36}.3^4.7^4.43.37199.8055869$ & $-2^6.7^2.31^2.107^2.557^2.947^2.271499$ & $2^{15}$ & $2.223.241$\\ \hline
651
652
296 & $2^{18}.11.53.83.229$ & $0$ & $2^9$ & $?$\\ \hline
653
654
297 & $2^{30}.3^{24}.47^2$ & $0$ & $2^{11}.3^{10}$ & $\infty$\\ \hline
655
656
298 & $2^{21}.3.5^2.7.13^2.103.107$ & $2^{36}.3^6.7^4.13^2.23^2.29^2.41.239.1847.234893^2.1252037^2$ & $2^8.5$ & $2^3.2972969$\\ \hline
657
658
299 & $2^{50}.3.5^9.7.17^3.197.5936311524617$ & $5^2.11^2.17^4.43^2$ & $2^{19}.5^3.17$ & $2^3.3769$\\ \hline
659
660
300 & $2^4.5^4$ & $2^{113}.3^{42}.5^{16}.7^3.11^4$ & $2^2.5^2$ & $\infty$\\ \hline
661
662
301 & $2^{38}.19.31.37.103.9739.81509.32366197$ & $-2^{30}.7^3.17^2.113$ & $2^{19}$ & $2^2.3.13.17.61$\\ \hline
663
664
302 & $2^{33}.3^4.389.1613$ & $2^{18}.3^8.5^4.7^7.11^2.41^2.67^4.167.257^2.431^2.16937.439867^2$ & $2^{11}.3^2$ & $17.877.4457$\\ \hline
665
666
303 & $2^{35}.89.3083683.3678833$ & $2^{39}.3^2.7^2.59.1103^2.4021.17568767^2$ & $2^{15}$ & $2^4.61.199$\\ \hline
667
668
304 & $2^{26}.3^4.31^2$ & $0$ & $2^{13}.3^2$ & $?$\\ \hline
669
670
305 & $2^{52}.3^4.31.43.977.2777.82219057$ & $2^8.3^6.11.37^4.139^2.2371$ & $2^{19}$ & $2^2.3^4.223$\\ \hline
671
672
306 & $2^{24}.3^4$ & $2^{183}.3^{30}.5^4.7^4.17^6$ & $2^9.3$ & $3^5.5^4$\\ \hline
673
674
307 & $2^{22}.3^2.5^5.11^2.13^3.107^2.457.3697.21577.974513.568380457$ & $1$ & $2^{11}.3.5^2.11.13.107$ & $2^3.3^2.16529$\\ \hline
675
676
308 & $2^{12}.3.127$ & $-2^{140}.3^{38}.5^{17}.7.11^2$ & $2^3$ & $2^9.23$\\ \hline
677
678
309 & $2^{24}.5.37.81509.109363884517$ & $2^{32}.3^4.5^8.11^2.17^2.89^2.397.411721^2$ & $2^{10}$ & $2^6.3.11^2.421$\\ \hline
679
680
310 & $2^{23}.3^2.37$ & $2^{119}.3^{10}.5^{12}.7^9.11^4.17.19^4.23^2.29^2.41^2.73^2.5077^2$ & $2^8$ & $2.5^2.7.73.991$\\ \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
687
311 & $2^8.5^2.29.3013091897.2106873009119126062143259000543887593$ & $1$ & $2^4$ & $2.610843$\\ \hline
688
689
312 & $2^{18}$ & $0$ & $2^9$ & $\infty$\\ \hline
690
691
313 & $2^{24}.5.41^2.8619587.9614923.130838023.2164322751511$ & $1$ & $2^{11}.41$ & $2^5.19.241$\\ \hline
692
693
314 & $2^{14}.3.47.53.337.907.176130259$ & $2^{30}.5^2.7.19^2.61^2.113^2.397^2.709^2.743.1489.48795779^2$ & $2^7$ & $2.13^2.127.461$\\ \hline
694
695
315 & $2^{24}.3^4.5^3.17$ & $-2^{158}.3^{16}.5^6.7^3.11.13.17^3$ & $2^9.3^2.5$ & $2^3.3$\\ \hline
696
697
316 & $2^6.3^4.13^2$ & $2^{92}.3^{18}.5^4.7.17.53^4.83^3.271.983^3$ & $2^3.3^2$ & $2^4.541^2$\\ \hline
698
699
317 & $2^{26}.7.367.3217.660603043.14989400036918065702697531$ & $1$ & $2^{11}$ & $5.167.14627$\\ \hline
700
701
318 & $2^{16}.5^2.17.41$ & $-2^{84}.3^{16}.5^8.7^7.11^4.13^2.17^2.19^2.37^4.43^2.71.103^2.107^4.151^2.227^2.263.1054013^2$ & $2^8.5$ & $2^4.3^2.5^4$\\ \hline
702
703
319 & $2^{34}.3^4.23^2.127.1063.1433.2459.2777.6709.65551$ & $-2^{12}.3^4.5^2.7.17^2.29.167^2.239$ & $2^{17}.23$ & $2.13.59.197$\\ \hline
704
705
320 & $2^{39}$ & $0$ & $2^{18}$ & $\infty$\\ \hline
706
707
321 & $2^{32}.5^2.21803.24443.826957$ & $-2^{12}.5^4.7^2.29^2.41^2.61.1667^2.14051^2.19079^2.1229279$ & $2^{12}$ & $2^7.11.73.109$\\ \hline
708
709
322 & $2^{28}.3^3.5.79$ & $-2^{100}.3^5.5^{18}.7^3.11^4.19^4.23^2.29^2.37^2.41^3.151.536777^2$ & $2^{12}.3$ & $2^3.13.41.53.67$\\ \hline
710
711
323 & $2^{48}.7^2.17^3.19.73.103.353.1453.1627.1697.3851.17299$ & $2^{12}.3^5.5^3.7^2.59$ & $2^{23}.7.17$ & $2^2.767603$\\ \hline
712
713
324 & $2^4.3^8$ & $0$ & $2^2.3^4$ & $\infty$\\ \hline
714
715
325 & $2^{49}.3^9.5^{12}.7^4.37^2$ & $2^{50}.3^{10}$ & $2^{17}.3^4.5^6.7^2$ & $2^2.3.17$\\ \hline
716
717
326 & $2^{18}.17^2.617.28921.482689$ & $-2^{35}.3^{16}.5^2.13^2.47^2.61^2.263.14831.65657^2.82536739^2$ & $2^9.17$ & $311.102305897$\\ \hline
718
719
327 & $2^{37}.5.37.139.1023203.1033895651$ & $2^{40}.3.7^4.13^2.107^2.139^2.491.577.7537^2$ & $2^{11}$ & $2^6.3.281.463$\\ \hline
720
721
328 & $2^{28}.3^3.37.197$ & $0$ & $2^{11}.3$ & $?$\\ \hline
722
723
329 & $2^{46}.5^3.7^4.11.13^2.17.37^3.1447.1609.51607$ & $3^6.19^4.31^2.83^2.103^2.18617$ & $2^{20}.5.13.37$ & $2^4.11.1801$\\ \hline
724
725
330 & $2^8$ & $2^{264}.3^{26}.5^{21}.7^{22}.11^{10}.17^3.19^4.23.37^2$ & $2^4$ & $2.3.5.1153$\\ \hline
726
727
331 & $2^{32}.3^2.53^2.229.1399.21911.205493.6363601.584461573862449$ & $1$ & $2^{12}.3.53$ & $2^2.1120529$\\ \hline
728
729
332 & $2^9.7^3.29^2$ & $-2^{82}.3^{12}.5^2.7.71.131^4.197^3.229^2.11497^3$ & $2^2.29$ & $2^2.5^2.479^2$\\ \hline
730
731
333 & $2^{42}.3^7.5^2.7^2.37.389$ & $2^{72}.3^9.5^4.7^4.11.37^2.389^2$ & $2^{13}.3^2.5.7$ & $2^{10}.3^5$\\ \hline
732
733
334 & $2^{27}.5^2.7^3.67.733$ & $2^{18}.3^{10}.5^4.7^4.11^4.41.113^2.8269^2.1951993.5103536431379173^2$ & $2^{12}.7$ & $113.3715823$\\ \hline
734
735
335 & $2^{31}.3.5.29.71.83.179^2.887.26393.6262079.23057641$ & $-2^{22}.3^{12}.5^8.11^2.41.59^2$ & $2^{13}.179$ & $2^2.3.331.431$\\ \hline
736
737
336 & $2^{16}.3^2$ & $0$ & $2^8.3$ & $\infty$\\ \hline
738
739
337 & $2^{28}.113.593.2791.2963615537.747945736667.4122851467451$ & $1$ & $2^{13}$ & $2^4.3^3.27299$\\ \hline
740
741
338 & $2^{16}.3^2.7^6.13^8$ & $2^{36}.3^8.5^2.7^{14}.13^8.41^4.167^2$ & $2^8.3.7.13^4$ & $\infty$\\ \hline
742
743
339 & $2^{41}.7^2.13^2.17.71^2.8297.470621$ & $-2^{29}.3^{18}.7^6.11^8.13^5.107^2.167.647$ & $2^{14}.7.13.71$ & $2^2.5^2.127.11801$\\ \hline
744
745
340 & $2^6.101$ & $-2^{198}.3^{37}.5^8.7^7.17^3$ & $2^2$ & $\infty$\\ \hline
746
747
341 & $2^{44}.3.5^3.89.151.1121599.344460847.14444130109$ & $-2^{14}.3^3.5^{10}.7^4.13^2$ & $2^{16}$ & $31.9043$\\ \hline
748
749
342 & $2^{12}.3^4.5^2$ & $-2^{184}.3^{58}.5^{16}.7^3.11^6$ & $2^6.3^2.5$ & $2.3^9.5^2$\\ \hline
750
751
343 & $2^{27}.7^{48}.13^2.29^3.41^2$ & $0$ & $3.33521863.4906673923$ & $\infty$\\ \hline
752
753
344 & $2^{26}.3.229.1999567$ & $0$ & $2^{11}$ & $?$\\ \hline
754
755
345 & $2^{46}.3^2.5^4.79$ & $-2^{100}.3^{13}.5^{19}.7^4.11^{13}.13^2.17^4.23.53^2$ & $2^{18}.5^2$ & $2^2.7^2.15991$\\ \hline
756
757
346 & $2^{18}.7^2.229.2777.2075621$ & $2^{40}.5^4.7^6.29^2.31^2.67^2.79^2.311.1279.1289.5608385124289^2$ & $2^9.7$ & $2^3.18947.256957$\\ \hline
758
759
347 & $2^{27}.5.7^2.19^2.331.349.479.617.1797330450291217.918291275915301361$ & $1$ & $2^{10}.7.19$ & $2^6.3.7.19709$\\ \hline
760
761
348 & $2^8$ & $2^{129}.3^{27}.5^{17}.7^{10}.11^7.13^4.17^2.23^6.41.47.73^3.229^3$ & $2^4$ & $2^8.5.11^4$\\ \hline
762
763
349 & $2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031$ & $1$ & $2^{11}$ & $2^3.3^2.239.1531$\\ \hline
764
765
350 & $2^{24}.3^4.5^6$ & $2^{138}.3^{34}.5^{29}.11^4.17^6.41^2$ & $2^9.3.5^3$ & $2^3.3$\\ \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
772
351 & $2^{32}.3^{17}.5^4.13^2.19.37^2$ & $0$ & $2^{12}.3^8$ & $\infty$\\ \hline
773
774
352 & $2^{42}.17^2$ & $0$ & $2^{21}$ & $\infty$\\ \hline
775
776
353 & $2^{34}.3^2.5.127^2.229.114641.551801.12611821.7779730837.24314514437$ & $1$ & $2^{15}.3.127$ & $2^2.3.47.26927$\\ \hline
777
778
354 & $2^{28}.5^2.11.79$ & $-2^{98}.3^{14}.5^{12}.7^4.11^6.19^6.29^2.31^8.41^2.127.229^6.557^4.20627^2$ & $2^{12}.5$ & $2^{15}.3^4.5$\\ \hline
779
780
355 & $2^{51}.5^2.19.29.103.62581037.933591331$ & $3^{10}.5.7^2.19.61^2.103^2.257^4.359^2.3671$ & $2^{23}$ & $2^6.3^2.5^2.13.37$\\ \hline
781
782
356 & $2^9.3.4079.31547$ & $-2^{114}.3^{14}.5^9.7^6.71^2.199.6689^3$ & $2$ & $2^5.11.487^2$\\ \hline
783
784
357 & $2^{49}.3.79.113$ & $-2^{99}.3^{16}.7^4.17^4.47^2.71^2.131^2.223^2.311^2.397.1459^2.1753$ & $2^{18}$ & $2^2.17.86341$\\ \hline
785
786
358 & $2^{22}.3^3.5^3.7.11^4.17^3$ & $-2^{46}.3^8.5^2.7^4.13^2.17^4.29^2.79^2.239.313^2.3257.137707^2.536747147^2$ & $2^{11}.3.11^2$ & $251.4099597$\\ \hline
787
788
359 & $2^{24}.3^4.2777.16512254293.64542630435970307.2171776478013633068927$ & $1$ & $2^{11}$ & $2^6.5.123059$\\ \hline
789
790
360 & $2^{12}.3^2$ & $0$ & $2^6.3$ & $\infty$\\ \hline
791
792
361 & $2^{28}.3^{10}.5^{11}.11^4.19^{12}$ & $2^6.3^2.5^2$ & $2^{12}.3.5^2.11^2.19^6$ & $\infty$\\ \hline
793
794
362 & $2^{26}.5^4.17^3.6359.1134769$ & $2^{40}.3^2.5^4.7^4.17^2.23^2.29^2.47.61^2.71.97.397^2.797^2.5297.595051637^2$ & $2^{10}.5.17$ & $2^2.2609601041$\\ \hline
795
796
363 & $2^{36}.3^9.5^7.11^{18}$ & $2^{38}.3^{22}.5^8.11^4$ & $2^{15}.3^3.5.11^8$ & $\infty$\\ \hline
797
798
364 & $2^{15}.3^2$ & $2^{125}.3^{63}.5^{10}.7^{12}.11^4.23.79^3$ & $2^5$ & $2^7.3^2.5.17.37$\\ \hline
799
800
365 & $2^{51}.3^3.5^4.7^3.47.313^2.353783.9377894173$ & $-2^{20}.3^{12}.5^4.11^2.13^4.19^2.29.269$ & $2^{19}.3.5^2$ & $3^2.37.71473$\\ \hline
801
802
366 & $2^{16}.3^2.13^2.17$ & $-2^{162}.3^{11}.5^4.7^6.13^{10}.17^3.19^6.23^3.31^2.37^6.127^2.137.151^3.229^2.5623^2$ & $2^8.3.13$ & $2^4.5^7.1277^2$\\ \hline
803
804
367 & $2^{22}.7.81421.251387.418175501.15354151381.13144405392643360366681$ & $1$ & $2^{11}$ & $2^5.3197809$\\ \hline
805
806
368 & $2^{40}.3^4.5^3.17$ & $0$ & $2^{20}.3^2.5$ & $?$\\ \hline
807
808
369 & $2^{41}.3^6.5^2.23^2.37^3.79$ & $-2^{86}.3^{12}.5^4.23^4.37^3.79^2.191$ & $2^{15}.3^3.5.23$ & $7.13^2$\\ \hline
809
810
370 & $2^{28}.3^2.11.223$ & $-2^{148}.3^{28}.5^{10}.7.11^2.13^4.19^2.23.31^2.61.193.269.23029^2.60869^2$ & $2^{12}$ & $2^4.72709379$\\ \hline
811
812
371 & $2^{62}.5^3.7.157.229.64627370911.76723322773093$ & $2^{30}.3^3.5^2.37^2.191^2.643$ & $2^{25}.5$ & $2^5.3.5.76103$\\ \hline
813
814
372 & $2^{10}.5^2.17$ & $-2^{181}.3^{31}.5^{13}.7^4.11^{12}.13^2.17^2.19^4.41^2.127.229^3$ & $2^5.5$ & $2^2.3^5.5^8$\\ \hline
815
816
373 & $2^{32}.7.11^3.23.199.673.2143.1542194372227.72819251148518000363297$ & $1$ & $2^{13}.11$ & $2^4.3.1520821$\\ \hline
817
818
374 & $2^{28}.5^2.157.257.11117.17417$ & $2^{142}.3^{20}.5^{18}.7^2.11^2.13^2.17^2.23.37^4.8461^2$ & $2^{14}$ & $11.3259.109111$\\ \hline
819
820
375 & $2^{24}.5^{24}.101^2$ & $0$ & $2^{12}.5^8$ & $\infty$\\ \hline
821
822
376 & $2^{33}.5^5.41^2.61$ & $0$ & $2^{15}$ & $?$\\ \hline
823
824
377 & $2^{70}.3^3.5^2.7^2.829.991.36497.202817.400031.54126547$ & $2^6.7^2.23^3.31^2.79$ & $2^{30}.3.7$ & $181.118057$\\ \hline
825
826
378 & $2^{16}.3^8$ & $0$ & $2^8.3^4$ & $\infty$\\ \hline
827
828
379 & $2^{28}.59.317.421.278329.5698591.2117788336277.2851210737989187265253$ & $1$ & $2^{13}$ & $2^5.3^3.7.229.809$\\ \hline
829
830
380 & $2^{15}.3$ & $2^{158}.3^{47}.5^9.7^2.11^7.13^4.17^2.19.37^3.113.151.709^3$ & $2^5$ & $2^2.3^4.11.53$\\ \hline
831
832
381 & $2^{32}.11^2.15427.81509.98361184089911$ & $2^{36}.3^{10}.5^2.7^2.19^2.71.109^2.647^2.2003.86235899^2$ & $2^{12}.11$ & $2^6.3.7^3.13.43.113$\\ \hline
833
834
382 & $2^{28}.7^2.11.13^2.619.44171$ & $2^{20}.3^6.5^4.11^4.41.967^2.1511.6277^2.63703.382146223^2.319500117632677^2$ & $2^{13}$ & $5.59.9181.50459$\\ \hline
835
836
383 & $2^{16}.5.11^2.13.72893.3151861.16141144314299.178236551484825400362837637090811$ & $1$ & $2^8.11$ & $2^7.2480227$\\ \hline
837
838
384 & $2^{40}$ & $0$ & $2^{20}$ & $\infty$\\ \hline
839
840
385 & $2^{61}.3.37^3.2837$ & $-2^{114}.3^{18}.5^{17}.7^8.11.17^2.19^5.389$ & $2^{24}$ & $2^4.3.5.29.71$\\ \hline
841
842
386 & $2^{32}.5^2.14653.22961.9659021$ & $-2^{30}.3^2.5^4.7.11^4.17^2.19^2.29^2.31^2.103^2.229^2.271^2.337.401^2.4153^2.244553.680059^2$ & $2^{12}$ & $3^2.13^2.74891419$\\ \hline
843
844
387 & $2^{51}.3^9.5^8.7^2.13^2.71$ & $-2^{106}.3^{16}.5^6.7^4.71^2$ & $2^{20}.3^4.5^3.7$ & $2.7^3.17^2$\\ \hline
845
846
388 & $2^8.7^2.179.1297$ & $-2^{90}.3^{16}.7^{11}.67^4.71^4.137^2.223^2.433.2777^3$ & $2^3$ & $2^3.3^3.22283^2$\\ \hline
847
848
389 & $2^{53}.3^4.5^6.31^2.37.389.3881.215517113148241.477439237737571441$ & $1$ & $2^{17}.3^2.5.31$ & $7^2.67.173.863$\\ \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