Properties

Label 288.8.a.b.1.1
Level 288288
Weight 88
Character 288.1
Self dual yes
Analytic conductor 89.96789.967
Analytic rank 11
Dimension 11
CM discriminant -4
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,8,Mod(1,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 288=2532 288 = 2^{5} \cdot 3^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 288.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,58,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 89.966887339489.9668873394
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 32)
Fricke sign: 1-1
Sato-Tate group: N(U(1))N(\mathrm{U}(1))

Embedding invariants

Embedding label 1.1
Character χ\chi == 288.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+58.0000q58898.00q13+40094.0q1774761.0q25233230.q29+563974.q379530.00q41823543.q49+798602.q533.50533e6q61516084.q65+3.91742e6q73+2.32545e6q859.24617e6q891.75674e7q97+O(q100)q+58.0000 q^{5} -8898.00 q^{13} +40094.0 q^{17} -74761.0 q^{25} -233230. q^{29} +563974. q^{37} -9530.00 q^{41} -823543. q^{49} +798602. q^{53} -3.50533e6 q^{61} -516084. q^{65} +3.91742e6 q^{73} +2.32545e6 q^{85} -9.24617e6 q^{89} -1.75674e7 q^{97} +O(q^{100})

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 0 0
44 0 0
55 58.0000 0.207507 0.103754 0.994603i 0.466915π-0.466915\pi
0.103754 + 0.994603i 0.466915π0.466915\pi
66 0 0
77 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
88 0 0
99 0 0
1010 0 0
1111 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
1212 0 0
1313 −8898.00 −1.12329 −0.561643 0.827379i 0.689831π-0.689831\pi
−0.561643 + 0.827379i 0.689831π0.689831\pi
1414 0 0
1515 0 0
1616 0 0
1717 40094.0 1.97928 0.989642 0.143557i 0.0458539π-0.0458539\pi
0.989642 + 0.143557i 0.0458539π0.0458539\pi
1818 0 0
1919 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
2020 0 0
2121 0 0
2222 0 0
2323 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
2424 0 0
2525 −74761.0 −0.956941
2626 0 0
2727 0 0
2828 0 0
2929 −233230. −1.77579 −0.887895 0.460047i 0.847833π-0.847833\pi
−0.887895 + 0.460047i 0.847833π0.847833\pi
3030 0 0
3131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3232 0 0
3333 0 0
3434 0 0
3535 0 0
3636 0 0
3737 563974. 1.83043 0.915215 0.402966i 0.132021π-0.132021\pi
0.915215 + 0.402966i 0.132021π0.132021\pi
3838 0 0
3939 0 0
4040 0 0
4141 −9530.00 −0.0215948 −0.0107974 0.999942i 0.503437π-0.503437\pi
−0.0107974 + 0.999942i 0.503437π0.503437\pi
4242 0 0
4343 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4444 0 0
4545 0 0
4646 0 0
4747 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4848 0 0
4949 −823543. −1.00000
5050 0 0
5151 0 0
5252 0 0
5353 798602. 0.736826 0.368413 0.929662i 0.379901π-0.379901\pi
0.368413 + 0.929662i 0.379901π0.379901\pi
5454 0 0
5555 0 0
5656 0 0
5757 0 0
5858 0 0
5959 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6060 0 0
6161 −3.50533e6 −1.97731 −0.988654 0.150208i 0.952006π-0.952006\pi
−0.988654 + 0.150208i 0.952006π0.952006\pi
6262 0 0
6363 0 0
6464 0 0
6565 −516084. −0.233090
6666 0 0
6767 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6868 0 0
6969 0 0
7070 0 0
7171 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
7272 0 0
7373 3.91742e6 1.17861 0.589305 0.807911i 0.299402π-0.299402\pi
0.589305 + 0.807911i 0.299402π0.299402\pi
7474 0 0
7575 0 0
7676 0 0
7777 0 0
7878 0 0
7979 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
8080 0 0
8181 0 0
8282 0 0
8383 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
8484 0 0
8585 2.32545e6 0.410716
8686 0 0
8787 0 0
8888 0 0
8989 −9.24617e6 −1.39026 −0.695131 0.718883i 0.744654π-0.744654\pi
−0.695131 + 0.718883i 0.744654π0.744654\pi
9090 0 0
9191 0 0
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 −1.75674e7 −1.95437 −0.977185 0.212392i 0.931875π-0.931875\pi
−0.977185 + 0.212392i 0.931875π0.931875\pi
9898 0 0
9999 0 0
100100 0 0
101101 −1.33042e7 −1.28488 −0.642442 0.766334i 0.722079π-0.722079\pi
−0.642442 + 0.766334i 0.722079π0.722079\pi
102102 0 0
103103 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
104104 0 0
105105 0 0
106106 0 0
107107 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
108108 0 0
109109 −2.41162e7 −1.78368 −0.891839 0.452352i 0.850585π-0.850585\pi
−0.891839 + 0.452352i 0.850585π0.850585\pi
110110 0 0
111111 0 0
112112 0 0
113113 2.91247e7 1.89883 0.949417 0.314017i 0.101675π-0.101675\pi
0.949417 + 0.314017i 0.101675π0.101675\pi
114114 0 0
115115 0 0
116116 0 0
117117 0 0
118118 0 0
119119 0 0
120120 0 0
121121 −1.94872e7 −1.00000
122122 0 0
123123 0 0
124124 0 0
125125 −8.86739e6 −0.406079
126126 0 0
127127 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
128128 0 0
129129 0 0
130130 0 0
131131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
132132 0 0
133133 0 0
134134 0 0
135135 0 0
136136 0 0
137137 −4.60312e7 −1.52943 −0.764716 0.644367i 0.777121π-0.777121\pi
−0.764716 + 0.644367i 0.777121π0.777121\pi
138138 0 0
139139 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
140140 0 0
141141 0 0
142142 0 0
143143 0 0
144144 0 0
145145 −1.35273e7 −0.368489
146146 0 0
147147 0 0
148148 0 0
149149 −7.31577e7 −1.81179 −0.905895 0.423502i 0.860801π-0.860801\pi
−0.905895 + 0.423502i 0.860801π0.860801\pi
150150 0 0
151151 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
152152 0 0
153153 0 0
154154 0 0
155155 0 0
156156 0 0
157157 −9.09715e7 −1.87610 −0.938051 0.346497i 0.887371π-0.887371\pi
−0.938051 + 0.346497i 0.887371π0.887371\pi
158158 0 0
159159 0 0
160160 0 0
161161 0 0
162162 0 0
163163 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
164164 0 0
165165 0 0
166166 0 0
167167 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
168168 0 0
169169 1.64259e7 0.261773
170170 0 0
171171 0 0
172172 0 0
173173 6.55688e7 0.962800 0.481400 0.876501i 0.340129π-0.340129\pi
0.481400 + 0.876501i 0.340129π0.340129\pi
174174 0 0
175175 0 0
176176 0 0
177177 0 0
178178 0 0
179179 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
180180 0 0
181181 −1.45860e8 −1.82836 −0.914180 0.405309i 0.867164π-0.867164\pi
−0.914180 + 0.405309i 0.867164π0.867164\pi
182182 0 0
183183 0 0
184184 0 0
185185 3.27105e7 0.379827
186186 0 0
187187 0 0
188188 0 0
189189 0 0
190190 0 0
191191 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
192192 0 0
193193 −1.05243e8 −1.05377 −0.526883 0.849938i 0.676639π-0.676639\pi
−0.526883 + 0.849938i 0.676639π0.676639\pi
194194 0 0
195195 0 0
196196 0 0
197197 −1.02733e8 −0.957363 −0.478681 0.877989i 0.658885π-0.658885\pi
−0.478681 + 0.877989i 0.658885π0.658885\pi
198198 0 0
199199 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
200200 0 0
201201 0 0
202202 0 0
203203 0 0
204204 0 0
205205 −552740. −0.00448108
206206 0 0
207207 0 0
208208 0 0
209209 0 0
210210 0 0
211211 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
212212 0 0
213213 0 0
214214 0 0
215215 0 0
216216 0 0
217217 0 0
218218 0 0
219219 0 0
220220 0 0
221221 −3.56756e8 −2.22330
222222 0 0
223223 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
224224 0 0
225225 0 0
226226 0 0
227227 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
228228 0 0
229229 2.17910e8 1.19909 0.599547 0.800339i 0.295347π-0.295347\pi
0.599547 + 0.800339i 0.295347π0.295347\pi
230230 0 0
231231 0 0
232232 0 0
233233 2.90325e8 1.50362 0.751811 0.659379i 0.229181π-0.229181\pi
0.751811 + 0.659379i 0.229181π0.229181\pi
234234 0 0
235235 0 0
236236 0 0
237237 0 0
238238 0 0
239239 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
240240 0 0
241241 1.08961e8 0.501433 0.250717 0.968061i 0.419334π-0.419334\pi
0.250717 + 0.968061i 0.419334π0.419334\pi
242242 0 0
243243 0 0
244244 0 0
245245 −4.77655e7 −0.207507
246246 0 0
247247 0 0
248248 0 0
249249 0 0
250250 0 0
251251 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
252252 0 0
253253 0 0
254254 0 0
255255 0 0
256256 0 0
257257 2.30304e8 0.846323 0.423161 0.906054i 0.360920π-0.360920\pi
0.423161 + 0.906054i 0.360920π0.360920\pi
258258 0 0
259259 0 0
260260 0 0
261261 0 0
262262 0 0
263263 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
264264 0 0
265265 4.63189e7 0.152897
266266 0 0
267267 0 0
268268 0 0
269269 −7.80336e7 −0.244427 −0.122213 0.992504i 0.538999π-0.538999\pi
−0.122213 + 0.992504i 0.538999π0.538999\pi
270270 0 0
271271 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
272272 0 0
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 −5.35122e8 −1.51277 −0.756386 0.654126i 0.773037π-0.773037\pi
−0.756386 + 0.654126i 0.773037π0.773037\pi
278278 0 0
279279 0 0
280280 0 0
281281 6.34409e8 1.70568 0.852839 0.522174i 0.174879π-0.174879\pi
0.852839 + 0.522174i 0.174879π0.174879\pi
282282 0 0
283283 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
284284 0 0
285285 0 0
286286 0 0
287287 0 0
288288 0 0
289289 1.19719e9 2.91757
290290 0 0
291291 0 0
292292 0 0
293293 5.87629e8 1.36479 0.682396 0.730983i 0.260938π-0.260938\pi
0.682396 + 0.730983i 0.260938π0.260938\pi
294294 0 0
295295 0 0
296296 0 0
297297 0 0
298298 0 0
299299 0 0
300300 0 0
301301 0 0
302302 0 0
303303 0 0
304304 0 0
305305 −2.03309e8 −0.410306
306306 0 0
307307 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
308308 0 0
309309 0 0
310310 0 0
311311 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
312312 0 0
313313 5.25457e8 0.968573 0.484286 0.874910i 0.339079π-0.339079\pi
0.484286 + 0.874910i 0.339079π0.339079\pi
314314 0 0
315315 0 0
316316 0 0
317317 −1.13287e9 −1.99744 −0.998720 0.0505827i 0.983892π-0.983892\pi
−0.998720 + 0.0505827i 0.983892π0.983892\pi
318318 0 0
319319 0 0
320320 0 0
321321 0 0
322322 0 0
323323 0 0
324324 0 0
325325 6.65223e8 1.07492
326326 0 0
327327 0 0
328328 0 0
329329 0 0
330330 0 0
331331 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 6.05044e8 0.861157 0.430578 0.902553i 0.358310π-0.358310\pi
0.430578 + 0.902553i 0.358310π0.358310\pi
338338 0 0
339339 0 0
340340 0 0
341341 0 0
342342 0 0
343343 0 0
344344 0 0
345345 0 0
346346 0 0
347347 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
348348 0 0
349349 1.50468e9 1.89477 0.947385 0.320097i 0.103715π-0.103715\pi
0.947385 + 0.320097i 0.103715π0.103715\pi
350350 0 0
351351 0 0
352352 0 0
353353 1.64963e9 1.99607 0.998035 0.0626535i 0.0199563π-0.0199563\pi
0.998035 + 0.0626535i 0.0199563π0.0199563\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 0 0
359359 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
360360 0 0
361361 −8.93872e8 −1.00000
362362 0 0
363363 0 0
364364 0 0
365365 2.27210e8 0.244570
366366 0 0
367367 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
368368 0 0
369369 0 0
370370 0 0
371371 0 0
372372 0 0
373373 −1.03979e9 −1.03744 −0.518720 0.854944i 0.673591π-0.673591\pi
−0.518720 + 0.854944i 0.673591π0.673591\pi
374374 0 0
375375 0 0
376376 0 0
377377 2.07528e9 1.99472
378378 0 0
379379 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
380380 0 0
381381 0 0
382382 0 0
383383 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
384384 0 0
385385 0 0
386386 0 0
387387 0 0
388388 0 0
389389 −1.94162e9 −1.67240 −0.836202 0.548422i 0.815229π-0.815229\pi
−0.836202 + 0.548422i 0.815229π0.815229\pi
390390 0 0
391391 0 0
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 0 0
397397 −1.34628e9 −1.07986 −0.539932 0.841709i 0.681550π-0.681550\pi
−0.539932 + 0.841709i 0.681550π0.681550\pi
398398 0 0
399399 0 0
400400 0 0
401401 8.84817e8 0.685248 0.342624 0.939473i 0.388684π-0.388684\pi
0.342624 + 0.939473i 0.388684π0.388684\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 0 0
408408 0 0
409409 2.38968e9 1.72706 0.863531 0.504296i 0.168248π-0.168248\pi
0.863531 + 0.504296i 0.168248π0.168248\pi
410410 0 0
411411 0 0
412412 0 0
413413 0 0
414414 0 0
415415 0 0
416416 0 0
417417 0 0
418418 0 0
419419 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
420420 0 0
421421 1.58512e9 1.03532 0.517661 0.855586i 0.326803π-0.326803\pi
0.517661 + 0.855586i 0.326803π0.326803\pi
422422 0 0
423423 0 0
424424 0 0
425425 −2.99747e9 −1.89406
426426 0 0
427427 0 0
428428 0 0
429429 0 0
430430 0 0
431431 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
432432 0 0
433433 −1.34596e9 −0.796757 −0.398378 0.917221i 0.630427π-0.630427\pi
−0.398378 + 0.917221i 0.630427π0.630427\pi
434434 0 0
435435 0 0
436436 0 0
437437 0 0
438438 0 0
439439 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
440440 0 0
441441 0 0
442442 0 0
443443 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
444444 0 0
445445 −5.36278e8 −0.288489
446446 0 0
447447 0 0
448448 0 0
449449 −2.71111e9 −1.41346 −0.706732 0.707481i 0.749832π-0.749832\pi
−0.706732 + 0.707481i 0.749832π0.749832\pi
450450 0 0
451451 0 0
452452 0 0
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 1.02241e9 0.501095 0.250548 0.968104i 0.419389π-0.419389\pi
0.250548 + 0.968104i 0.419389π0.419389\pi
458458 0 0
459459 0 0
460460 0 0
461461 4.07657e9 1.93795 0.968974 0.247164i 0.0794988π-0.0794988\pi
0.968974 + 0.247164i 0.0794988π0.0794988\pi
462462 0 0
463463 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
464464 0 0
465465 0 0
466466 0 0
467467 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
468468 0 0
469469 0 0
470470 0 0
471471 0 0
472472 0 0
473473 0 0
474474 0 0
475475 0 0
476476 0 0
477477 0 0
478478 0 0
479479 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
480480 0 0
481481 −5.01824e9 −2.05610
482482 0 0
483483 0 0
484484 0 0
485485 −1.01891e9 −0.405545
486486 0 0
487487 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
488488 0 0
489489 0 0
490490 0 0
491491 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
492492 0 0
493493 −9.35112e9 −3.51479
494494 0 0
495495 0 0
496496 0 0
497497 0 0
498498 0 0
499499 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
500500 0 0
501501 0 0
502502 0 0
503503 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
504504 0 0
505505 −7.71643e8 −0.266623
506506 0 0
507507 0 0
508508 0 0
509509 −5.95022e9 −1.99996 −0.999979 0.00646007i 0.997944π-0.997944\pi
−0.999979 + 0.00646007i 0.997944π0.997944\pi
510510 0 0
511511 0 0
512512 0 0
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 0 0
518518 0 0
519519 0 0
520520 0 0
521521 2.38452e9 0.738701 0.369350 0.929290i 0.379580π-0.379580\pi
0.369350 + 0.929290i 0.379580π0.379580\pi
522522 0 0
523523 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
524524 0 0
525525 0 0
526526 0 0
527527 0 0
528528 0 0
529529 −3.40483e9 −1.00000
530530 0 0
531531 0 0
532532 0 0
533533 8.47979e7 0.0242572
534534 0 0
535535 0 0
536536 0 0
537537 0 0
538538 0 0
539539 0 0
540540 0 0
541541 7.36235e9 1.99906 0.999530 0.0306534i 0.00975880π-0.00975880\pi
0.999530 + 0.0306534i 0.00975880π0.00975880\pi
542542 0 0
543543 0 0
544544 0 0
545545 −1.39874e9 −0.370126
546546 0 0
547547 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\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 2.15369e9 0.528067 0.264034 0.964513i 0.414947π-0.414947\pi
0.264034 + 0.964513i 0.414947π0.414947\pi
558558 0 0
559559 0 0
560560 0 0
561561 0 0
562562 0 0
563563 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
564564 0 0
565565 1.68923e9 0.394022
566566 0 0
567567 0 0
568568 0 0
569569 −6.84617e9 −1.55796 −0.778978 0.627052i 0.784262π-0.784262\pi
−0.778978 + 0.627052i 0.784262π0.784262\pi
570570 0 0
571571 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
572572 0 0
573573 0 0
574574 0 0
575575 0 0
576576 0 0
577577 −2.65224e9 −0.574775 −0.287387 0.957814i 0.592787π-0.592787\pi
−0.287387 + 0.957814i 0.592787π0.592787\pi
578578 0 0
579579 0 0
580580 0 0
581581 0 0
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
588588 0 0
589589 0 0
590590 0 0
591591 0 0
592592 0 0
593593 −7.08713e9 −1.39566 −0.697829 0.716265i 0.745850π-0.745850\pi
−0.697829 + 0.716265i 0.745850π0.745850\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 0 0
599599 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
600600 0 0
601601 −1.05496e10 −1.98233 −0.991165 0.132637i 0.957656π-0.957656\pi
−0.991165 + 0.132637i 0.957656π0.957656\pi
602602 0 0
603603 0 0
604604 0 0
605605 −1.13026e9 −0.207507
606606 0 0
607607 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
608608 0 0
609609 0 0
610610 0 0
611611 0 0
612612 0 0
613613 −9.50665e9 −1.66692 −0.833462 0.552577i 0.813645π-0.813645\pi
−0.833462 + 0.552577i 0.813645π0.813645\pi
614614 0 0
615615 0 0
616616 0 0
617617 2.16779e9 0.371551 0.185776 0.982592i 0.440520π-0.440520\pi
0.185776 + 0.982592i 0.440520π0.440520\pi
618618 0 0
619619 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
620620 0 0
621621 0 0
622622 0 0
623623 0 0
624624 0 0
625625 5.32639e9 0.872676
626626 0 0
627627 0 0
628628 0 0
629629 2.26120e10 3.62294
630630 0 0
631631 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
632632 0 0
633633 0 0
634634 0 0
635635 0 0
636636 0 0
637637 7.32789e9 1.12329
638638 0 0
639639 0 0
640640 0 0
641641 −5.92310e9 −0.888272 −0.444136 0.895959i 0.646489π-0.646489\pi
−0.444136 + 0.895959i 0.646489π0.646489\pi
642642 0 0
643643 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
644644 0 0
645645 0 0
646646 0 0
647647 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
648648 0 0
649649 0 0
650650 0 0
651651 0 0
652652 0 0
653653 7.96894e9 1.11997 0.559983 0.828504i 0.310808π-0.310808\pi
0.559983 + 0.828504i 0.310808π0.310808\pi
654654 0 0
655655 0 0
656656 0 0
657657 0 0
658658 0 0
659659 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
660660 0 0
661661 1.15742e9 0.155879 0.0779394 0.996958i 0.475166π-0.475166\pi
0.0779394 + 0.996958i 0.475166π0.475166\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 1.54182e10 1.94976 0.974879 0.222734i 0.0714982π-0.0714982\pi
0.974879 + 0.222734i 0.0714982π0.0714982\pi
674674 0 0
675675 0 0
676676 0 0
677677 −4.29286e9 −0.531724 −0.265862 0.964011i 0.585657π-0.585657\pi
−0.265862 + 0.964011i 0.585657π0.585657\pi
678678 0 0
679679 0 0
680680 0 0
681681 0 0
682682 0 0
683683 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
684684 0 0
685685 −2.66981e9 −0.317368
686686 0 0
687687 0 0
688688 0 0
689689 −7.10596e9 −0.827667
690690 0 0
691691 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
692692 0 0
693693 0 0
694694 0 0
695695 0 0
696696 0 0
697697 −3.82096e8 −0.0427423
698698 0 0
699699 0 0
700700 0 0
701701 −1.77141e10 −1.94226 −0.971129 0.238556i 0.923326π-0.923326\pi
−0.971129 + 0.238556i 0.923326π0.923326\pi
702702 0 0
703703 0 0
704704 0 0
705705 0 0
706706 0 0
707707 0 0
708708 0 0
709709 1.64383e10 1.73219 0.866096 0.499877i 0.166621π-0.166621\pi
0.866096 + 0.499877i 0.166621π0.166621\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 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 0 0
721721 0 0
722722 0 0
723723 0 0
724724 0 0
725725 1.74365e10 1.69933
726726 0 0
727727 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
728728 0 0
729729 0 0
730730 0 0
731731 0 0
732732 0 0
733733 1.85321e10 1.73804 0.869022 0.494773i 0.164749π-0.164749\pi
0.869022 + 0.494773i 0.164749π0.164749\pi
734734 0 0
735735 0 0
736736 0 0
737737 0 0
738738 0 0
739739 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
740740 0 0
741741 0 0
742742 0 0
743743 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
744744 0 0
745745 −4.24315e9 −0.375959
746746 0 0
747747 0 0
748748 0 0
749749 0 0
750750 0 0
751751 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
752752 0 0
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 1.72709e10 1.44703 0.723516 0.690307i 0.242525π-0.242525\pi
0.723516 + 0.690307i 0.242525π0.242525\pi
758758 0 0
759759 0 0
760760 0 0
761761 1.99861e10 1.64392 0.821961 0.569543i 0.192880π-0.192880\pi
0.821961 + 0.569543i 0.192880π0.192880\pi
762762 0 0
763763 0 0
764764 0 0
765765 0 0
766766 0 0
767767 0 0
768768 0 0
769769 −2.52206e10 −1.99992 −0.999960 0.00895271i 0.997150π-0.997150\pi
−0.999960 + 0.00895271i 0.997150π0.997150\pi
770770 0 0
771771 0 0
772772 0 0
773773 2.55364e10 1.98853 0.994263 0.106964i 0.0341129π-0.0341129\pi
0.994263 + 0.106964i 0.0341129π0.0341129\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 0 0
780780 0 0
781781 0 0
782782 0 0
783783 0 0
784784 0 0
785785 −5.27635e9 −0.389305
786786 0 0
787787 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
788788 0 0
789789 0 0
790790 0 0
791791 0 0
792792 0 0
793793 3.11904e10 2.22108
794794 0 0
795795 0 0
796796 0 0
797797 9.53016e9 0.666801 0.333400 0.942785i 0.391804π-0.391804\pi
0.333400 + 0.942785i 0.391804π0.391804\pi
798798 0 0
799799 0 0
800800 0 0
801801 0 0
802802 0 0
803803 0 0
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 2.84568e10 1.88959 0.944793 0.327668i 0.106263π-0.106263\pi
0.944793 + 0.327668i 0.106263π0.106263\pi
810810 0 0
811811 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 0 0
818818 0 0
819819 0 0
820820 0 0
821821 −2.88014e10 −1.81641 −0.908203 0.418530i 0.862546π-0.862546\pi
−0.908203 + 0.418530i 0.862546π0.862546\pi
822822 0 0
823823 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
824824 0 0
825825 0 0
826826 0 0
827827 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
828828 0 0
829829 −2.59446e10 −1.58163 −0.790817 0.612052i 0.790344π-0.790344\pi
−0.790817 + 0.612052i 0.790344π0.790344\pi
830830 0 0
831831 0 0
832832 0 0
833833 −3.30191e10 −1.97928
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
840840 0 0
841841 3.71464e10 2.15343
842842 0 0
843843 0 0
844844 0 0
845845 9.52701e8 0.0543198
846846 0 0
847847 0 0
848848 0 0
849849 0 0
850850 0 0
851851 0 0
852852 0 0
853853 −2.32103e9 −0.128044 −0.0640220 0.997948i 0.520393π-0.520393\pi
−0.0640220 + 0.997948i 0.520393π0.520393\pi
854854 0 0
855855 0 0
856856 0 0
857857 −2.11516e10 −1.14792 −0.573960 0.818884i 0.694593π-0.694593\pi
−0.573960 + 0.818884i 0.694593π0.694593\pi
858858 0 0
859859 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
860860 0 0
861861 0 0
862862 0 0
863863 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
864864 0 0
865865 3.80299e9 0.199788
866866 0 0
867867 0 0
868868 0 0
869869 0 0
870870 0 0
871871 0 0
872872 0 0
873873 0 0
874874 0 0
875875 0 0
876876 0 0
877877 −5.68052e9 −0.284374 −0.142187 0.989840i 0.545413π-0.545413\pi
−0.142187 + 0.989840i 0.545413π0.545413\pi
878878 0 0
879879 0 0
880880 0 0
881881 2.69850e10 1.32956 0.664778 0.747041i 0.268526π-0.268526\pi
0.664778 + 0.747041i 0.268526π0.268526\pi
882882 0 0
883883 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
884884 0 0
885885 0 0
886886 0 0
887887 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
888888 0 0
889889 0 0
890890 0 0
891891 0 0
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 3.20191e10 1.45839
902902 0 0
903903 0 0
904904 0 0
905905 −8.45989e9 −0.379398
906906 0 0
907907 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
908908 0 0
909909 0 0
910910 0 0
911911 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
912912 0 0
913913 0 0
914914 0 0
915915 0 0
916916 0 0
917917 0 0
918918 0 0
919919 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
920920 0 0
921921 0 0
922922 0 0
923923 0 0
924924 0 0
925925 −4.21633e10 −1.75161
926926 0 0
927927 0 0
928928 0 0
929929 1.43415e10 0.586866 0.293433 0.955980i 0.405202π-0.405202\pi
0.293433 + 0.955980i 0.405202π0.405202\pi
930930 0 0
931931 0 0
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 −5.03482e10 −1.99938 −0.999690 0.0249138i 0.992069π-0.992069\pi
−0.999690 + 0.0249138i 0.992069π0.992069\pi
938938 0 0
939939 0 0
940940 0 0
941941 −3.49808e10 −1.36857 −0.684283 0.729217i 0.739885π-0.739885\pi
−0.684283 + 0.729217i 0.739885π0.739885\pi
942942 0 0
943943 0 0
944944 0 0
945945 0 0
946946 0 0
947947 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
948948 0 0
949949 −3.48572e10 −1.32392
950950 0 0
951951 0 0
952952 0 0
953953 5.27653e9 0.197480 0.0987400 0.995113i 0.468519π-0.468519\pi
0.0987400 + 0.995113i 0.468519π0.468519\pi
954954 0 0
955955 0 0
956956 0 0
957957 0 0
958958 0 0
959959 0 0
960960 0 0
961961 −2.75126e10 −1.00000
962962 0 0
963963 0 0
964964 0 0
965965 −6.10411e9 −0.218664
966966 0 0
967967 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
968968 0 0
969969 0 0
970970 0 0
971971 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
972972 0 0
973973 0 0
974974 0 0
975975 0 0
976976 0 0
977977 3.63185e10 1.24594 0.622970 0.782246i 0.285926π-0.285926\pi
0.622970 + 0.782246i 0.285926π0.285926\pi
978978 0 0
979979 0 0
980980 0 0
981981 0 0
982982 0 0
983983 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
984984 0 0
985985 −5.95849e9 −0.198660
986986 0 0
987987 0 0
988988 0 0
989989 0 0
990990 0 0
991991 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
992992 0 0
993993 0 0
994994 0 0
995995 0 0
996996 0 0
997997 1.44203e10 0.460832 0.230416 0.973092i 0.425991π-0.425991\pi
0.230416 + 0.973092i 0.425991π0.425991\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 288.8.a.b.1.1 1
3.2 odd 2 32.8.a.a.1.1 1
4.3 odd 2 CM 288.8.a.b.1.1 1
8.3 odd 2 576.8.a.n.1.1 1
8.5 even 2 576.8.a.n.1.1 1
12.11 even 2 32.8.a.a.1.1 1
24.5 odd 2 64.8.a.d.1.1 1
24.11 even 2 64.8.a.d.1.1 1
48.5 odd 4 256.8.b.d.129.1 2
48.11 even 4 256.8.b.d.129.1 2
48.29 odd 4 256.8.b.d.129.2 2
48.35 even 4 256.8.b.d.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.a.1.1 1 3.2 odd 2
32.8.a.a.1.1 1 12.11 even 2
64.8.a.d.1.1 1 24.5 odd 2
64.8.a.d.1.1 1 24.11 even 2
256.8.b.d.129.1 2 48.5 odd 4
256.8.b.d.129.1 2 48.11 even 4
256.8.b.d.129.2 2 48.29 odd 4
256.8.b.d.129.2 2 48.35 even 4
288.8.a.b.1.1 1 1.1 even 1 trivial
288.8.a.b.1.1 1 4.3 odd 2 CM
576.8.a.n.1.1 1 8.3 odd 2
576.8.a.n.1.1 1 8.5 even 2