Properties

Label 32.13.d.a.15.1
Level 3232
Weight 1313
Character 32.15
Self dual yes
Analytic conductor 29.24829.248
Analytic rank 00
Dimension 11
CM discriminant -8
Inner twists 22

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,13,Mod(15,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: N N == 32=25 32 = 2^{5}
Weight: k k == 13 13
Character orbit: [χ][\chi] == 32.d (of order 22, degree 11, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 29.247802152829.2478021528
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

Embedding invariants

Embedding label 15.1
Character χ\chi == 32.15

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q658.000q398477.0q91.92312e6q114.52961e7q17+8.79314e7q19+2.44141e8q25+4.14486e8q27+1.26541e9q33+8.62826e9q41+7.03062e9q43+1.38413e10q49+2.98048e10q515.78589e10q578.63831e9q591.75046e11q67+4.91395e10q731.60645e11q752.20397e11q81+1.92940e11q838.66326e11q89+1.65649e12q97+1.89383e11q99+O(q100)q-658.000 q^{3} -98477.0 q^{9} -1.92312e6 q^{11} -4.52961e7 q^{17} +8.79314e7 q^{19} +2.44141e8 q^{25} +4.14486e8 q^{27} +1.26541e9 q^{33} +8.62826e9 q^{41} +7.03062e9 q^{43} +1.38413e10 q^{49} +2.98048e10 q^{51} -5.78589e10 q^{57} -8.63831e9 q^{59} -1.75046e11 q^{67} +4.91395e10 q^{73} -1.60645e11 q^{75} -2.20397e11 q^{81} +1.92940e11 q^{83} -8.66326e11 q^{89} +1.65649e12 q^{97} +1.89383e11 q^{99} +O(q^{100})

Character values

We give the values of χ\chi on generators for (Z/32Z)×\left(\mathbb{Z}/32\mathbb{Z}\right)^\times.

nn 55 3131
χ(n)\chi(n) 1-1 1-1

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0
33 −658.000 −0.902606 −0.451303 0.892371i 0.649041π-0.649041\pi
−0.451303 + 0.892371i 0.649041π0.649041\pi
44 0 0
55 0 0 1.00000 00
−1.00000 π\pi
66 0 0
77 0 0 1.00000 00
−1.00000 π\pi
88 0 0
99 −98477.0 −0.185302
1010 0 0
1111 −1.92312e6 −1.08555 −0.542776 0.839877i 0.682627π-0.682627\pi
−0.542776 + 0.839877i 0.682627π0.682627\pi
1212 0 0
1313 0 0 1.00000 00
−1.00000 π\pi
1414 0 0
1515 0 0
1616 0 0
1717 −4.52961e7 −1.87658 −0.938290 0.345850i 0.887590π-0.887590\pi
−0.938290 + 0.345850i 0.887590π0.887590\pi
1818 0 0
1919 8.79314e7 1.86906 0.934529 0.355888i 0.115822π-0.115822\pi
0.934529 + 0.355888i 0.115822π0.115822\pi
2020 0 0
2121 0 0
2222 0 0
2323 0 0 1.00000 00
−1.00000 π\pi
2424 0 0
2525 2.44141e8 1.00000
2626 0 0
2727 4.14486e8 1.06986
2828 0 0
2929 0 0 1.00000 00
−1.00000 π\pi
3030 0 0
3131 0 0 1.00000 00
−1.00000 π\pi
3232 0 0
3333 1.26541e9 0.979826
3434 0 0
3535 0 0
3636 0 0
3737 0 0 1.00000 00
−1.00000 π\pi
3838 0 0
3939 0 0
4040 0 0
4141 8.62826e9 1.81644 0.908218 0.418498i 0.137443π-0.137443\pi
0.908218 + 0.418498i 0.137443π0.137443\pi
4242 0 0
4343 7.03062e9 1.11220 0.556100 0.831115i 0.312297π-0.312297\pi
0.556100 + 0.831115i 0.312297π0.312297\pi
4444 0 0
4545 0 0
4646 0 0
4747 0 0 1.00000 00
−1.00000 π\pi
4848 0 0
4949 1.38413e10 1.00000
5050 0 0
5151 2.98048e10 1.69381
5252 0 0
5353 0 0 1.00000 00
−1.00000 π\pi
5454 0 0
5555 0 0
5656 0 0
5757 −5.78589e10 −1.68702
5858 0 0
5959 −8.63831e9 −0.204794 −0.102397 0.994744i 0.532651π-0.532651\pi
−0.102397 + 0.994744i 0.532651π0.532651\pi
6060 0 0
6161 0 0 1.00000 00
−1.00000 π\pi
6262 0 0
6363 0 0
6464 0 0
6565 0 0
6666 0 0
6767 −1.75046e11 −1.93510 −0.967549 0.252684i 0.918687π-0.918687\pi
−0.967549 + 0.252684i 0.918687π0.918687\pi
6868 0 0
6969 0 0
7070 0 0
7171 0 0 1.00000 00
−1.00000 π\pi
7272 0 0
7373 4.91395e10 0.324708 0.162354 0.986733i 0.448091π-0.448091\pi
0.162354 + 0.986733i 0.448091π0.448091\pi
7474 0 0
7575 −1.60645e11 −0.902606
7676 0 0
7777 0 0
7878 0 0
7979 0 0 1.00000 00
−1.00000 π\pi
8080 0 0
8181 −2.20397e11 −0.780361
8282 0 0
8383 1.92940e11 0.590139 0.295069 0.955476i 0.404657π-0.404657\pi
0.295069 + 0.955476i 0.404657π0.404657\pi
8484 0 0
8585 0 0
8686 0 0
8787 0 0
8888 0 0
8989 −8.66326e11 −1.74318 −0.871589 0.490238i 0.836910π-0.836910\pi
−0.871589 + 0.490238i 0.836910π0.836910\pi
9090 0 0
9191 0 0
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 1.65649e12 1.98865 0.994324 0.106394i 0.0339306π-0.0339306\pi
0.994324 + 0.106394i 0.0339306π0.0339306\pi
9898 0 0
9999 1.89383e11 0.201155
100100 0 0
101101 0 0 1.00000 00
−1.00000 π\pi
102102 0 0
103103 0 0 1.00000 00
−1.00000 π\pi
104104 0 0
105105 0 0
106106 0 0
107107 2.77677e12 1.85028 0.925140 0.379626i 0.123947π-0.123947\pi
0.925140 + 0.379626i 0.123947π0.123947\pi
108108 0 0
109109 0 0 1.00000 00
−1.00000 π\pi
110110 0 0
111111 0 0
112112 0 0
113113 3.74847e12 1.80046 0.900230 0.435416i 0.143399π-0.143399\pi
0.900230 + 0.435416i 0.143399π0.143399\pi
114114 0 0
115115 0 0
116116 0 0
117117 0 0
118118 0 0
119119 0 0
120120 0 0
121121 5.59970e11 0.178424
122122 0 0
123123 −5.67739e12 −1.63953
124124 0 0
125125 0 0
126126 0 0
127127 0 0 1.00000 00
−1.00000 π\pi
128128 0 0
129129 −4.62615e12 −1.00388
130130 0 0
131131 1.38397e12 0.273841 0.136920 0.990582i 0.456280π-0.456280\pi
0.136920 + 0.990582i 0.456280π0.456280\pi
132132 0 0
133133 0 0
134134 0 0
135135 0 0
136136 0 0
137137 −1.32173e13 −1.99903 −0.999514 0.0311883i 0.990071π-0.990071\pi
−0.999514 + 0.0311883i 0.990071π0.990071\pi
138138 0 0
139139 4.19430e12 0.581528 0.290764 0.956795i 0.406091π-0.406091\pi
0.290764 + 0.956795i 0.406091π0.406091\pi
140140 0 0
141141 0 0
142142 0 0
143143 0 0
144144 0 0
145145 0 0
146146 0 0
147147 −9.10757e12 −0.902606
148148 0 0
149149 0 0 1.00000 00
−1.00000 π\pi
150150 0 0
151151 0 0 1.00000 00
−1.00000 π\pi
152152 0 0
153153 4.46062e12 0.347734
154154 0 0
155155 0 0
156156 0 0
157157 0 0 1.00000 00
−1.00000 π\pi
158158 0 0
159159 0 0
160160 0 0
161161 0 0
162162 0 0
163163 −2.20968e13 −1.17816 −0.589080 0.808075i 0.700510π-0.700510\pi
−0.589080 + 0.808075i 0.700510π0.700510\pi
164164 0 0
165165 0 0
166166 0 0
167167 0 0 1.00000 00
−1.00000 π\pi
168168 0 0
169169 2.32981e13 1.00000
170170 0 0
171171 −8.65922e12 −0.346340
172172 0 0
173173 0 0 1.00000 00
−1.00000 π\pi
174174 0 0
175175 0 0
176176 0 0
177177 5.68401e12 0.184848
178178 0 0
179179 5.53626e13 1.68305 0.841527 0.540215i 0.181657π-0.181657\pi
0.841527 + 0.540215i 0.181657π0.181657\pi
180180 0 0
181181 0 0 1.00000 00
−1.00000 π\pi
182182 0 0
183183 0 0
184184 0 0
185185 0 0
186186 0 0
187187 8.71099e13 2.03712
188188 0 0
189189 0 0
190190 0 0
191191 0 0 1.00000 00
−1.00000 π\pi
192192 0 0
193193 −3.16470e12 −0.0612335 −0.0306167 0.999531i 0.509747π-0.509747\pi
−0.0306167 + 0.999531i 0.509747π0.509747\pi
194194 0 0
195195 0 0
196196 0 0
197197 0 0 1.00000 00
−1.00000 π\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 0 0
201201 1.15180e14 1.74663
202202 0 0
203203 0 0
204204 0 0
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 −1.69103e14 −2.02896
210210 0 0
211211 −1.71035e14 −1.93816 −0.969079 0.246750i 0.920637π-0.920637\pi
−0.969079 + 0.246750i 0.920637π0.920637\pi
212212 0 0
213213 0 0
214214 0 0
215215 0 0
216216 0 0
217217 0 0
218218 0 0
219219 −3.23338e13 −0.293084
220220 0 0
221221 0 0
222222 0 0
223223 0 0 1.00000 00
−1.00000 π\pi
224224 0 0
225225 −2.40422e13 −0.185302
226226 0 0
227227 −1.17238e14 −0.856870 −0.428435 0.903573i 0.640935π-0.640935\pi
−0.428435 + 0.903573i 0.640935π0.640935\pi
228228 0 0
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 0 0
232232 0 0
233233 −1.97639e14 −1.23520 −0.617599 0.786494i 0.711894π-0.711894\pi
−0.617599 + 0.786494i 0.711894π0.711894\pi
234234 0 0
235235 0 0
236236 0 0
237237 0 0
238238 0 0
239239 0 0 1.00000 00
−1.00000 π\pi
240240 0 0
241241 3.10482e14 1.58465 0.792326 0.610097i 0.208870π-0.208870\pi
0.792326 + 0.610097i 0.208870π0.208870\pi
242242 0 0
243243 −7.52536e13 −0.365502
244244 0 0
245245 0 0
246246 0 0
247247 0 0
248248 0 0
249249 −1.26955e14 −0.532663
250250 0 0
251251 3.28002e14 1.31170 0.655849 0.754892i 0.272311π-0.272311\pi
0.655849 + 0.754892i 0.272311π0.272311\pi
252252 0 0
253253 0 0
254254 0 0
255255 0 0
256256 0 0
257257 −2.16326e14 −0.750775 −0.375388 0.926868i 0.622490π-0.622490\pi
−0.375388 + 0.926868i 0.622490π0.622490\pi
258258 0 0
259259 0 0
260260 0 0
261261 0 0
262262 0 0
263263 0 0 1.00000 00
−1.00000 π\pi
264264 0 0
265265 0 0
266266 0 0
267267 5.70043e14 1.57340
268268 0 0
269269 0 0 1.00000 00
−1.00000 π\pi
270270 0 0
271271 0 0 1.00000 00
−1.00000 π\pi
272272 0 0
273273 0 0
274274 0 0
275275 −4.69512e14 −1.08555
276276 0 0
277277 0 0 1.00000 00
−1.00000 π\pi
278278 0 0
279279 0 0
280280 0 0
281281 8.55524e14 1.73778 0.868889 0.495007i 0.164834π-0.164834\pi
0.868889 + 0.495007i 0.164834π0.164834\pi
282282 0 0
283283 6.60679e14 1.28609 0.643046 0.765828i 0.277671π-0.277671\pi
0.643046 + 0.765828i 0.277671π0.277671\pi
284284 0 0
285285 0 0
286286 0 0
287287 0 0
288288 0 0
289289 1.46911e15 2.52155
290290 0 0
291291 −1.08997e15 −1.79497
292292 0 0
293293 0 0 1.00000 00
−1.00000 π\pi
294294 0 0
295295 0 0
296296 0 0
297297 −7.97107e14 −1.16139
298298 0 0
299299 0 0
300300 0 0
301301 0 0
302302 0 0
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 1.63875e15 1.95741 0.978706 0.205269i 0.0658069π-0.0658069\pi
0.978706 + 0.205269i 0.0658069π0.0658069\pi
308308 0 0
309309 0 0
310310 0 0
311311 0 0 1.00000 00
−1.00000 π\pi
312312 0 0
313313 −1.79845e15 −1.91264 −0.956319 0.292326i 0.905571π-0.905571\pi
−0.956319 + 0.292326i 0.905571π0.905571\pi
314314 0 0
315315 0 0
316316 0 0
317317 0 0 1.00000 00
−1.00000 π\pi
318318 0 0
319319 0 0
320320 0 0
321321 −1.82712e15 −1.67007
322322 0 0
323323 −3.98295e15 −3.50743
324324 0 0
325325 0 0
326326 0 0
327327 0 0
328328 0 0
329329 0 0
330330 0 0
331331 2.60911e15 1.98392 0.991959 0.126557i 0.0403928π-0.0403928\pi
0.991959 + 0.126557i 0.0403928π0.0403928\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 −5.20794e13 −0.0355538 −0.0177769 0.999842i 0.505659π-0.505659\pi
−0.0177769 + 0.999842i 0.505659π0.505659\pi
338338 0 0
339339 −2.46649e15 −1.62511
340340 0 0
341341 0 0
342342 0 0
343343 0 0
344344 0 0
345345 0 0
346346 0 0
347347 1.26334e15 0.723675 0.361837 0.932241i 0.382150π-0.382150\pi
0.361837 + 0.932241i 0.382150π0.382150\pi
348348 0 0
349349 0 0 1.00000 00
−1.00000 π\pi
350350 0 0
351351 0 0
352352 0 0
353353 3.84078e14 0.198505 0.0992523 0.995062i 0.468355π-0.468355\pi
0.0992523 + 0.995062i 0.468355π0.468355\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 0 0
359359 0 0 1.00000 00
−1.00000 π\pi
360360 0 0
361361 5.51862e15 2.49337
362362 0 0
363363 −3.68460e14 −0.161046
364364 0 0
365365 0 0
366366 0 0
367367 0 0 1.00000 00
−1.00000 π\pi
368368 0 0
369369 −8.49685e14 −0.336589
370370 0 0
371371 0 0
372372 0 0
373373 0 0 1.00000 00
−1.00000 π\pi
374374 0 0
375375 0 0
376376 0 0
377377 0 0
378378 0 0
379379 5.18672e15 1.75008 0.875039 0.484052i 0.160835π-0.160835\pi
0.875039 + 0.484052i 0.160835π0.160835\pi
380380 0 0
381381 0 0
382382 0 0
383383 0 0 1.00000 00
−1.00000 π\pi
384384 0 0
385385 0 0
386386 0 0
387387 −6.92354e14 −0.206093
388388 0 0
389389 0 0 1.00000 00
−1.00000 π\pi
390390 0 0
391391 0 0
392392 0 0
393393 −9.10650e14 −0.247170
394394 0 0
395395 0 0
396396 0 0
397397 0 0 1.00000 00
−1.00000 π\pi
398398 0 0
399399 0 0
400400 0 0
401401 −1.92614e15 −0.463257 −0.231629 0.972804i 0.574405π-0.574405\pi
−0.231629 + 0.972804i 0.574405π0.574405\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 0 0
408408 0 0
409409 7.63062e15 1.63012 0.815061 0.579375i 0.196703π-0.196703\pi
0.815061 + 0.579375i 0.196703π0.196703\pi
410410 0 0
411411 8.69697e15 1.80433
412412 0 0
413413 0 0
414414 0 0
415415 0 0
416416 0 0
417417 −2.75985e15 −0.524891
418418 0 0
419419 −7.38079e15 −1.36401 −0.682007 0.731346i 0.738892π-0.738892\pi
−0.682007 + 0.731346i 0.738892π0.738892\pi
420420 0 0
421421 0 0 1.00000 00
−1.00000 π\pi
422422 0 0
423423 0 0
424424 0 0
425425 −1.10586e16 −1.87658
426426 0 0
427427 0 0
428428 0 0
429429 0 0
430430 0 0
431431 0 0 1.00000 00
−1.00000 π\pi
432432 0 0
433433 4.24415e15 0.643966 0.321983 0.946745i 0.395651π-0.395651\pi
0.321983 + 0.946745i 0.395651π0.395651\pi
434434 0 0
435435 0 0
436436 0 0
437437 0 0
438438 0 0
439439 0 0 1.00000 00
−1.00000 π\pi
440440 0 0
441441 −1.36305e15 −0.185302
442442 0 0
443443 −1.04567e16 −1.38348 −0.691738 0.722149i 0.743155π-0.743155\pi
−0.691738 + 0.722149i 0.743155π0.743155\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 0 0
449449 −5.86206e14 −0.0715438 −0.0357719 0.999360i 0.511389π-0.511389\pi
−0.0357719 + 0.999360i 0.511389π0.511389\pi
450450 0 0
451451 −1.65932e16 −1.97184
452452 0 0
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 1.78261e14 0.0195686 0.00978428 0.999952i 0.496886π-0.496886\pi
0.00978428 + 0.999952i 0.496886π0.496886\pi
458458 0 0
459459 −1.87746e16 −2.00768
460460 0 0
461461 0 0 1.00000 00
−1.00000 π\pi
462462 0 0
463463 0 0 1.00000 00
−1.00000 π\pi
464464 0 0
465465 0 0
466466 0 0
467467 2.02528e16 1.95246 0.976232 0.216730i 0.0695392π-0.0695392\pi
0.976232 + 0.216730i 0.0695392π0.0695392\pi
468468 0 0
469469 0 0
470470 0 0
471471 0 0
472472 0 0
473473 −1.35207e16 −1.20735
474474 0 0
475475 2.14676e16 1.86906
476476 0 0
477477 0 0
478478 0 0
479479 0 0 1.00000 00
−1.00000 π\pi
480480 0 0
481481 0 0
482482 0 0
483483 0 0
484484 0 0
485485 0 0
486486 0 0
487487 0 0 1.00000 00
−1.00000 π\pi
488488 0 0
489489 1.45397e16 1.06341
490490 0 0
491491 2.03908e16 1.45527 0.727637 0.685963i 0.240619π-0.240619\pi
0.727637 + 0.685963i 0.240619π0.240619\pi
492492 0 0
493493 0 0
494494 0 0
495495 0 0
496496 0 0
497497 0 0
498498 0 0
499499 2.39468e16 1.55112 0.775559 0.631275i 0.217468π-0.217468\pi
0.775559 + 0.631275i 0.217468π0.217468\pi
500500 0 0
501501 0 0
502502 0 0
503503 0 0 1.00000 00
−1.00000 π\pi
504504 0 0
505505 0 0
506506 0 0
507507 −1.53301e16 −0.902606
508508 0 0
509509 0 0 1.00000 00
−1.00000 π\pi
510510 0 0
511511 0 0
512512 0 0
513513 3.64464e16 1.99963
514514 0 0
515515 0 0
516516 0 0
517517 0 0
518518 0 0
519519 0 0
520520 0 0
521521 −4.38740e14 −0.0219372 −0.0109686 0.999940i 0.503491π-0.503491\pi
−0.0109686 + 0.999940i 0.503491π0.503491\pi
522522 0 0
523523 −2.85281e16 −1.39400 −0.697001 0.717070i 0.745483π-0.745483\pi
−0.697001 + 0.717070i 0.745483π0.745483\pi
524524 0 0
525525 0 0
526526 0 0
527527 0 0
528528 0 0
529529 2.19146e16 1.00000
530530 0 0
531531 8.50675e14 0.0379487
532532 0 0
533533 0 0
534534 0 0
535535 0 0
536536 0 0
537537 −3.64286e16 −1.51914
538538 0 0
539539 −2.66185e16 −1.08555
540540 0 0
541541 0 0 1.00000 00
−1.00000 π\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 3.10476e16 1.15906 0.579528 0.814952i 0.303237π-0.303237\pi
0.579528 + 0.814952i 0.303237π0.303237\pi
548548 0 0
549549 0 0
550550 0 0
551551 0 0
552552 0 0
553553 0 0
554554 0 0
555555 0 0
556556 0 0
557557 0 0 1.00000 00
−1.00000 π\pi
558558 0 0
559559 0 0
560560 0 0
561561 −5.73183e16 −1.83872
562562 0 0
563563 2.23353e16 0.701361 0.350681 0.936495i 0.385950π-0.385950\pi
0.350681 + 0.936495i 0.385950π0.385950\pi
564564 0 0
565565 0 0
566566 0 0
567567 0 0
568568 0 0
569569 6.36836e16 1.87652 0.938262 0.345925i 0.112435π-0.112435\pi
0.938262 + 0.345925i 0.112435π0.112435\pi
570570 0 0
571571 −5.93581e16 −1.71263 −0.856314 0.516455i 0.827251π-0.827251\pi
−0.856314 + 0.516455i 0.827251π0.827251\pi
572572 0 0
573573 0 0
574574 0 0
575575 0 0
576576 0 0
577577 −7.38009e16 −1.99989 −0.999946 0.0103984i 0.996690π-0.996690\pi
−0.999946 + 0.0103984i 0.996690π0.996690\pi
578578 0 0
579579 2.08237e15 0.0552697
580580 0 0
581581 0 0
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 −6.62782e15 −0.162010 −0.0810051 0.996714i 0.525813π-0.525813\pi
−0.0810051 + 0.996714i 0.525813π0.525813\pi
588588 0 0
589589 0 0
590590 0 0
591591 0 0
592592 0 0
593593 −1.46818e16 −0.337638 −0.168819 0.985647i 0.553995π-0.553995\pi
−0.168819 + 0.985647i 0.553995π0.553995\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 0 0
599599 0 0 1.00000 00
−1.00000 π\pi
600600 0 0
601601 −4.27830e16 −0.907872 −0.453936 0.891034i 0.649980π-0.649980\pi
−0.453936 + 0.891034i 0.649980π0.649980\pi
602602 0 0
603603 1.72380e16 0.358577
604604 0 0
605605 0 0
606606 0 0
607607 0 0 1.00000 00
−1.00000 π\pi
608608 0 0
609609 0 0
610610 0 0
611611 0 0
612612 0 0
613613 0 0 1.00000 00
−1.00000 π\pi
614614 0 0
615615 0 0
616616 0 0
617617 8.12519e15 0.147273 0.0736364 0.997285i 0.476540π-0.476540\pi
0.0736364 + 0.997285i 0.476540π0.476540\pi
618618 0 0
619619 −1.06992e17 −1.90198 −0.950989 0.309225i 0.899930π-0.899930\pi
−0.950989 + 0.309225i 0.899930π0.899930\pi
620620 0 0
621621 0 0
622622 0 0
623623 0 0
624624 0 0
625625 5.96046e16 1.00000
626626 0 0
627627 1.11270e17 1.83135
628628 0 0
629629 0 0
630630 0 0
631631 0 0 1.00000 00
−1.00000 π\pi
632632 0 0
633633 1.12541e17 1.74939
634634 0 0
635635 0 0
636636 0 0
637637 0 0
638638 0 0
639639 0 0
640640 0 0
641641 9.37398e16 1.35137 0.675687 0.737188i 0.263847π-0.263847\pi
0.675687 + 0.737188i 0.263847π0.263847\pi
642642 0 0
643643 6.10304e16 0.863536 0.431768 0.901985i 0.357890π-0.357890\pi
0.431768 + 0.901985i 0.357890π0.357890\pi
644644 0 0
645645 0 0
646646 0 0
647647 0 0 1.00000 00
−1.00000 π\pi
648648 0 0
649649 1.66125e16 0.222314
650650 0 0
651651 0 0
652652 0 0
653653 0 0 1.00000 00
−1.00000 π\pi
654654 0 0
655655 0 0
656656 0 0
657657 −4.83911e15 −0.0601691
658658 0 0
659659 6.58930e16 0.804501 0.402251 0.915530i 0.368228π-0.368228\pi
0.402251 + 0.915530i 0.368228π0.368228\pi
660660 0 0
661661 0 0 1.00000 00
−1.00000 π\pi
662662 0 0
663663 0 0
664664 0 0
665665 0 0
666666 0 0
667667 0 0
668668 0 0
669669 0 0
670670 0 0
671671 0 0
672672 0 0
673673 −1.27565e17 −1.37290 −0.686451 0.727176i 0.740833π-0.740833\pi
−0.686451 + 0.727176i 0.740833π0.740833\pi
674674 0 0
675675 1.01193e17 1.06986
676676 0 0
677677 0 0 1.00000 00
−1.00000 π\pi
678678 0 0
679679 0 0
680680 0 0
681681 7.71429e16 0.773416
682682 0 0
683683 −4.09526e16 −0.403419 −0.201710 0.979445i 0.564650π-0.564650\pi
−0.201710 + 0.979445i 0.564650π0.564650\pi
684684 0 0
685685 0 0
686686 0 0
687687 0 0
688688 0 0
689689 0 0
690690 0 0
691691 −2.12572e17 −1.95271 −0.976354 0.216178i 0.930641π-0.930641\pi
−0.976354 + 0.216178i 0.930641π0.930641\pi
692692 0 0
693693 0 0
694694 0 0
695695 0 0
696696 0 0
697697 −3.90826e17 −3.40869
698698 0 0
699699 1.30046e17 1.11490
700700 0 0
701701 0 0 1.00000 00
−1.00000 π\pi
702702 0 0
703703 0 0
704704 0 0
705705 0 0
706706 0 0
707707 0 0
708708 0 0
709709 0 0 1.00000 00
−1.00000 π\pi
710710 0 0
711711 0 0
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 0 0
717717 0 0
718718 0 0
719719 0 0 1.00000 00
−1.00000 π\pi
720720 0 0
721721 0 0
722722 0 0
723723 −2.04297e17 −1.43032
724724 0 0
725725 0 0
726726 0 0
727727 0 0 1.00000 00
−1.00000 π\pi
728728 0 0
729729 1.66645e17 1.11027
730730 0 0
731731 −3.18459e17 −2.08713
732732 0 0
733733 0 0 1.00000 00
−1.00000 π\pi
734734 0 0
735735 0 0
736736 0 0
737737 3.36634e17 2.10065
738738 0 0
739739 8.15585e16 0.500729 0.250364 0.968152i 0.419450π-0.419450\pi
0.250364 + 0.968152i 0.419450π0.419450\pi
740740 0 0
741741 0 0
742742 0 0
743743 0 0 1.00000 00
−1.00000 π\pi
744744 0 0
745745 0 0
746746 0 0
747747 −1.90002e16 −0.109354
748748 0 0
749749 0 0
750750 0 0
751751 0 0 1.00000 00
−1.00000 π\pi
752752 0 0
753753 −2.15825e17 −1.18395
754754 0 0
755755 0 0
756756 0 0
757757 0 0 1.00000 00
−1.00000 π\pi
758758 0 0
759759 0 0
760760 0 0
761761 −3.06091e17 −1.57595 −0.787975 0.615707i 0.788871π-0.788871\pi
−0.787975 + 0.615707i 0.788871π0.788871\pi
762762 0 0
763763 0 0
764764 0 0
765765 0 0
766766 0 0
767767 0 0
768768 0 0
769769 7.49631e16 0.362485 0.181242 0.983438i 0.441988π-0.441988\pi
0.181242 + 0.983438i 0.441988π0.441988\pi
770770 0 0
771771 1.42342e17 0.677654
772772 0 0
773773 0 0 1.00000 00
−1.00000 π\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 7.58695e17 3.39502
780780 0 0
781781 0 0
782782 0 0
783783 0 0
784784 0 0
785785 0 0
786786 0 0
787787 −3.83356e17 −1.61345 −0.806723 0.590930i 0.798761π-0.798761\pi
−0.806723 + 0.590930i 0.798761π0.798761\pi
788788 0 0
789789 0 0
790790 0 0
791791 0 0
792792 0 0
793793 0 0
794794 0 0
795795 0 0
796796 0 0
797797 0 0 1.00000 00
−1.00000 π\pi
798798 0 0
799799 0 0
800800 0 0
801801 8.53132e16 0.323014
802802 0 0
803803 −9.45012e16 −0.352488
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 1.67164e17 0.596283 0.298142 0.954522i 0.403633π-0.403633\pi
0.298142 + 0.954522i 0.403633π0.403633\pi
810810 0 0
811811 4.49628e17 1.58026 0.790130 0.612939i 0.210013π-0.210013\pi
0.790130 + 0.612939i 0.210013π0.210013\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 6.18212e17 2.07876
818818 0 0
819819 0 0
820820 0 0
821821 0 0 1.00000 00
−1.00000 π\pi
822822 0 0
823823 0 0 1.00000 00
−1.00000 π\pi
824824 0 0
825825 3.08939e17 0.979826
826826 0 0
827827 3.04079e17 0.950502 0.475251 0.879850i 0.342357π-0.342357\pi
0.475251 + 0.879850i 0.342357π0.342357\pi
828828 0 0
829829 0 0 1.00000 00
−1.00000 π\pi
830830 0 0
831831 0 0
832832 0 0
833833 −6.26956e17 −1.87658
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 0 0 1.00000 00
−1.00000 π\pi
840840 0 0
841841 3.53815e17 1.00000
842842 0 0
843843 −5.62935e17 −1.56853
844844 0 0
845845 0 0
846846 0 0
847847 0 0
848848 0 0
849849 −4.34727e17 −1.16083
850850 0 0
851851 0 0
852852 0 0
853853 0 0 1.00000 00
−1.00000 π\pi
854854 0 0
855855 0 0
856856 0 0
857857 3.89760e16 0.0983813 0.0491907 0.998789i 0.484336π-0.484336\pi
0.0491907 + 0.998789i 0.484336π0.484336\pi
858858 0 0
859859 1.37849e17 0.343119 0.171560 0.985174i 0.445119π-0.445119\pi
0.171560 + 0.985174i 0.445119π0.445119\pi
860860 0 0
861861 0 0
862862 0 0
863863 0 0 1.00000 00
−1.00000 π\pi
864864 0 0
865865 0 0
866866 0 0
867867 −9.66675e17 −2.27597
868868 0 0
869869 0 0
870870 0 0
871871 0 0
872872 0 0
873873 −1.63126e17 −0.368500
874874 0 0
875875 0 0
876876 0 0
877877 0 0 1.00000 00
−1.00000 π\pi
878878 0 0
879879 0 0
880880 0 0
881881 −7.94685e17 −1.69957 −0.849786 0.527128i 0.823269π-0.823269\pi
−0.849786 + 0.527128i 0.823269π0.823269\pi
882882 0 0
883883 −8.71688e17 −1.83906 −0.919532 0.393015i 0.871432π-0.871432\pi
−0.919532 + 0.393015i 0.871432π0.871432\pi
884884 0 0
885885 0 0
886886 0 0
887887 0 0 1.00000 00
−1.00000 π\pi
888888 0 0
889889 0 0
890890 0 0
891891 4.23851e17 0.847123
892892 0 0
893893 0 0
894894 0 0
895895 0 0
896896 0 0
897897 0 0
898898 0 0
899899 0 0
900900 0 0
901901 0 0
902902 0 0
903903 0 0
904904 0 0
905905 0 0
906906 0 0
907907 4.64915e17 0.835084 0.417542 0.908658i 0.362892π-0.362892\pi
0.417542 + 0.908658i 0.362892π0.362892\pi
908908 0 0
909909 0 0
910910 0 0
911911 0 0 1.00000 00
−1.00000 π\pi
912912 0 0
913913 −3.71048e17 −0.640627
914914 0 0
915915 0 0
916916 0 0
917917 0 0
918918 0 0
919919 0 0 1.00000 00
−1.00000 π\pi
920920 0 0
921921 −1.07830e18 −1.76677
922922 0 0
923923 0 0
924924 0 0
925925 0 0
926926 0 0
927927 0 0
928928 0 0
929929 1.13239e18 1.76157 0.880786 0.473515i 0.157015π-0.157015\pi
0.880786 + 0.473515i 0.157015π0.157015\pi
930930 0 0
931931 1.21708e18 1.86906
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 9.59909e17 1.41838 0.709190 0.705018i 0.249061π-0.249061\pi
0.709190 + 0.705018i 0.249061π0.249061\pi
938938 0 0
939939 1.18338e18 1.72636
940940 0 0
941941 0 0 1.00000 00
−1.00000 π\pi
942942 0 0
943943 0 0
944944 0 0
945945 0 0
946946 0 0
947947 −1.42092e18 −1.97001 −0.985005 0.172528i 0.944807π-0.944807\pi
−0.985005 + 0.172528i 0.944807π0.944807\pi
948948 0 0
949949 0 0
950950 0 0
951951 0 0
952952 0 0
953953 −1.35078e18 −1.80313 −0.901565 0.432645i 0.857581π-0.857581\pi
−0.901565 + 0.432645i 0.857581π0.857581\pi
954954 0 0
955955 0 0
956956 0 0
957957 0 0
958958 0 0
959959 0 0
960960 0 0
961961 7.87663e17 1.00000
962962 0 0
963963 −2.73448e17 −0.342860
964964 0 0
965965 0 0
966966 0 0
967967 0 0 1.00000 00
−1.00000 π\pi
968968 0 0
969969 2.62078e18 3.16583
970970 0 0
971971 −1.67618e18 −1.99989 −0.999943 0.0107080i 0.996591π-0.996591\pi
−0.999943 + 0.0107080i 0.996591π0.996591\pi
972972 0 0
973973 0 0
974974 0 0
975975 0 0
976976 0 0
977977 7.04930e17 0.810548 0.405274 0.914195i 0.367176π-0.367176\pi
0.405274 + 0.914195i 0.367176π0.367176\pi
978978 0 0
979979 1.66605e18 1.89231
980980 0 0
981981 0 0
982982 0 0
983983 0 0 1.00000 00
−1.00000 π\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 0 0
990990 0 0
991991 0 0 1.00000 00
−1.00000 π\pi
992992 0 0
993993 −1.71679e18 −1.79070
994994 0 0
995995 0 0
996996 0 0
997997 0 0 1.00000 00
−1.00000 π\pi
998998 0 0
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 32.13.d.a.15.1 1
4.3 odd 2 8.13.d.a.3.1 1
8.3 odd 2 CM 32.13.d.a.15.1 1
8.5 even 2 8.13.d.a.3.1 1
12.11 even 2 72.13.b.a.19.1 1
24.5 odd 2 72.13.b.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.13.d.a.3.1 1 4.3 odd 2
8.13.d.a.3.1 1 8.5 even 2
32.13.d.a.15.1 1 1.1 even 1 trivial
32.13.d.a.15.1 1 8.3 odd 2 CM
72.13.b.a.19.1 1 12.11 even 2
72.13.b.a.19.1 1 24.5 odd 2