Properties

Label 63.2.c.a.62.2
Level 6363
Weight 22
Character 63.62
Analytic conductor 0.5030.503
Analytic rank 00
Dimension 44
CM discriminant -7
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,2,Mod(62,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.62");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 63=327 63 = 3^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 63.c (of order 22, degree 11, 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: no
Analytic conductor: 0.5030575327340.503057532734
Analytic rank: 00
Dimension: 44
Coefficient field: Q(2,7)\Q(\sqrt{-2}, \sqrt{7})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+8x2+9 x^{4} + 8x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 3 3
Twist minimal: yes
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

Embedding invariants

Embedding label 62.2
Root 1.16372i-1.16372i of defining polynomial
Character χ\chi == 63.62
Dual form 63.2.c.a.62.3

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) == q1.16372iq2+0.645751q42.64575q73.07892iq8+6.57008iq11+3.07892iq142.29150q16+7.64575q221.91520iq235.00000q251.70850q288.89753iq293.49117iq32+10.5830q375.29150q43+4.24264iq442.22876q46+7.00000q49+5.81861iq500.412247iq53+8.14605iq5610.3542q588.64575q644.00000q67+15.0554iq7112.3157iq7417.3828iq77+8.00000q79+6.15784iq86+20.2288q881.23674iq928.14605iq98+O(q100)q-1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} -3.07892i q^{8} +6.57008i q^{11} +3.07892i q^{14} -2.29150 q^{16} +7.64575 q^{22} -1.91520i q^{23} -5.00000 q^{25} -1.70850 q^{28} -8.89753i q^{29} -3.49117i q^{32} +10.5830 q^{37} -5.29150 q^{43} +4.24264i q^{44} -2.22876 q^{46} +7.00000 q^{49} +5.81861i q^{50} -0.412247i q^{53} +8.14605i q^{56} -10.3542 q^{58} -8.64575 q^{64} -4.00000 q^{67} +15.0554i q^{71} -12.3157i q^{74} -17.3828i q^{77} +8.00000 q^{79} +6.15784i q^{86} +20.2288 q^{88} -1.23674i q^{92} -8.14605i q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q8q4+12q16+20q2220q2528q28+44q46+28q4952q5824q6416q67+32q79+28q88+O(q100) 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/63Z)×\left(\mathbb{Z}/63\mathbb{Z}\right)^\times.

nn 1010 2929
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 − 1.16372i − 0.822876i −0.911438 0.411438i 0.865027π-0.865027\pi
0.911438 0.411438i 0.134973π-0.134973\pi
33 0 0
44 0.645751 0.322876
55 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
66 0 0
77 −2.64575 −1.00000
88 − 3.07892i − 1.08856i
99 0 0
1010 0 0
1111 6.57008i 1.98096i 0.137675 + 0.990478i 0.456037π0.456037\pi
−0.137675 + 0.990478i 0.543963π0.543963\pi
1212 0 0
1313 0 0 1.00000 00
−1.00000 π\pi
1414 3.07892i 0.822876i
1515 0 0
1616 −2.29150 −0.572876
1717 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
1818 0 0
1919 0 0 1.00000 00
−1.00000 π\pi
2020 0 0
2121 0 0
2222 7.64575 1.63008
2323 − 1.91520i − 0.399346i −0.979863 0.199673i 0.936012π-0.936012\pi
0.979863 0.199673i 0.0639880π-0.0639880\pi
2424 0 0
2525 −5.00000 −1.00000
2626 0 0
2727 0 0
2828 −1.70850 −0.322876
2929 − 8.89753i − 1.65223i −0.563502 0.826115i 0.690546π-0.690546\pi
0.563502 0.826115i 0.309454π-0.309454\pi
3030 0 0
3131 0 0 1.00000 00
−1.00000 π\pi
3232 − 3.49117i − 0.617157i
3333 0 0
3434 0 0
3535 0 0
3636 0 0
3737 10.5830 1.73984 0.869918 0.493197i 0.164172π-0.164172\pi
0.869918 + 0.493197i 0.164172π0.164172\pi
3838 0 0
3939 0 0
4040 0 0
4141 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4242 0 0
4343 −5.29150 −0.806947 −0.403473 0.914991i 0.632197π-0.632197\pi
−0.403473 + 0.914991i 0.632197π0.632197\pi
4444 4.24264i 0.639602i
4545 0 0
4646 −2.22876 −0.328612
4747 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4848 0 0
4949 7.00000 1.00000
5050 5.81861i 0.822876i
5151 0 0
5252 0 0
5353 − 0.412247i − 0.0566265i −0.999599 0.0283132i 0.990986π-0.990986\pi
0.999599 0.0283132i 0.00901359π-0.00901359\pi
5454 0 0
5555 0 0
5656 8.14605i 1.08856i
5757 0 0
5858 −10.3542 −1.35958
5959 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6060 0 0
6161 0 0 1.00000 00
−1.00000 π\pi
6262 0 0
6363 0 0
6464 −8.64575 −1.08072
6565 0 0
6666 0 0
6767 −4.00000 −0.488678 −0.244339 0.969690i 0.578571π-0.578571\pi
−0.244339 + 0.969690i 0.578571π0.578571\pi
6868 0 0
6969 0 0
7070 0 0
7171 15.0554i 1.78674i 0.449319 + 0.893372i 0.351667π0.351667\pi
−0.449319 + 0.893372i 0.648333π0.648333\pi
7272 0 0
7373 0 0 1.00000 00
−1.00000 π\pi
7474 − 12.3157i − 1.43167i
7575 0 0
7676 0 0
7777 − 17.3828i − 1.98096i
7878 0 0
7979 8.00000 0.900070 0.450035 0.893011i 0.351411π-0.351411\pi
0.450035 + 0.893011i 0.351411π0.351411\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 0 0
8686 6.15784i 0.664017i
8787 0 0
8888 20.2288 2.15639
8989 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
9090 0 0
9191 0 0
9292 − 1.23674i − 0.128939i
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 0 0 1.00000 00
−1.00000 π\pi
9898 − 8.14605i − 0.822876i
9999 0 0
100100 −3.22876 −0.322876
101101 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
102102 0 0
103103 0 0 1.00000 00
−1.00000 π\pi
104104 0 0
105105 0 0
106106 −0.479741 −0.0465965
107107 − 10.4005i − 1.00545i −0.864446 0.502726i 0.832330π-0.832330\pi
0.864446 0.502726i 0.167670π-0.167670\pi
108108 0 0
109109 10.5830 1.01367 0.506834 0.862044i 0.330816π-0.330816\pi
0.506834 + 0.862044i 0.330816π0.330816\pi
110110 0 0
111111 0 0
112112 6.06275 0.572876
113113 13.5524i 1.27490i 0.770490 + 0.637452i 0.220012π0.220012\pi
−0.770490 + 0.637452i 0.779988π0.779988\pi
114114 0 0
115115 0 0
116116 − 5.74559i − 0.533465i
117117 0 0
118118 0 0
119119 0 0
120120 0 0
121121 −32.1660 −2.92418
122122 0 0
123123 0 0
124124 0 0
125125 0 0
126126 0 0
127127 −16.0000 −1.41977 −0.709885 0.704317i 0.751253π-0.751253\pi
−0.709885 + 0.704317i 0.751253π0.751253\pi
128128 3.07892i 0.272141i
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 4.65489i 0.402121i
135135 0 0
136136 0 0
137137 22.0377i 1.88281i 0.337282 + 0.941404i 0.390493π0.390493\pi
−0.337282 + 0.941404i 0.609507π0.609507\pi
138138 0 0
139139 0 0 1.00000 00
−1.00000 π\pi
140140 0 0
141141 0 0
142142 17.5203 1.47027
143143 0 0
144144 0 0
145145 0 0
146146 0 0
147147 0 0
148148 6.83399 0.561750
149149 8.07303i 0.661369i 0.943741 + 0.330684i 0.107280π0.107280\pi
−0.943741 + 0.330684i 0.892720π0.892720\pi
150150 0 0
151151 −5.29150 −0.430616 −0.215308 0.976546i 0.569076π-0.569076\pi
−0.215308 + 0.976546i 0.569076π0.569076\pi
152152 0 0
153153 0 0
154154 −20.2288 −1.63008
155155 0 0
156156 0 0
157157 0 0 1.00000 00
−1.00000 π\pi
158158 − 9.30978i − 0.740646i
159159 0 0
160160 0 0
161161 5.06713i 0.399346i
162162 0 0
163163 20.0000 1.56652 0.783260 0.621694i 0.213555π-0.213555\pi
0.783260 + 0.621694i 0.213555π0.213555\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 13.0000 1.00000
170170 0 0
171171 0 0
172172 −3.41699 −0.260543
173173 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
174174 0 0
175175 13.2288 1.00000
176176 − 15.0554i − 1.13484i
177177 0 0
178178 0 0
179179 − 15.8799i − 1.18692i −0.804865 0.593458i 0.797762π-0.797762\pi
0.804865 0.593458i 0.202238π-0.202238\pi
180180 0 0
181181 0 0 1.00000 00
−1.00000 π\pi
182182 0 0
183183 0 0
184184 −5.89674 −0.434713
185185 0 0
186186 0 0
187187 0 0
188188 0 0
189189 0 0
190190 0 0
191191 − 24.3651i − 1.76300i −0.472184 0.881500i 0.656534π-0.656534\pi
0.472184 0.881500i 0.343466π-0.343466\pi
192192 0 0
193193 −21.1660 −1.52356 −0.761781 0.647834i 0.775675π-0.775675\pi
−0.761781 + 0.647834i 0.775675π0.775675\pi
194194 0 0
195195 0 0
196196 4.52026 0.322876
197197 − 25.8681i − 1.84303i −0.388348 0.921513i 0.626954π-0.626954\pi
0.388348 0.921513i 0.373046π-0.373046\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 15.3946i 1.08856i
201201 0 0
202202 0 0
203203 23.5406i 1.65223i
204204 0 0
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 0 0
210210 0 0
211211 26.4575 1.82141 0.910705 0.413057i 0.135539π-0.135539\pi
0.910705 + 0.413057i 0.135539π0.135539\pi
212212 − 0.266209i − 0.0182833i
213213 0 0
214214 −12.1033 −0.827362
215215 0 0
216216 0 0
217217 0 0
218218 − 12.3157i − 0.834123i
219219 0 0
220220 0 0
221221 0 0
222222 0 0
223223 0 0 1.00000 00
−1.00000 π\pi
224224 9.23676i 0.617157i
225225 0 0
226226 15.7712 1.04909
227227 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
228228 0 0
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 0 0
232232 −27.3948 −1.79855
233233 30.5230i 1.99963i 0.0193169 + 0.999813i 0.493851π0.493851\pi
−0.0193169 + 0.999813i 0.506149π0.506149\pi
234234 0 0
235235 0 0
236236 0 0
237237 0 0
238238 0 0
239239 − 7.39458i − 0.478316i −0.970981 0.239158i 0.923129π-0.923129\pi
0.970981 0.239158i 0.0768713π-0.0768713\pi
240240 0 0
241241 0 0 1.00000 00
−1.00000 π\pi
242242 37.4323i 2.40624i
243243 0 0
244244 0 0
245245 0 0
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 12.5830 0.791087
254254 18.6196i 1.16829i
255255 0 0
256256 −13.7085 −0.856781
257257 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
258258 0 0
259259 −28.0000 −1.73984
260260 0 0
261261 0 0
262262 0 0
263263 − 18.8858i − 1.16455i −0.812993 0.582273i 0.802164π-0.802164\pi
0.812993 0.582273i 0.197836π-0.197836\pi
264264 0 0
265265 0 0
266266 0 0
267267 0 0
268268 −2.58301 −0.157782
269269 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
270270 0 0
271271 0 0 1.00000 00
−1.00000 π\pi
272272 0 0
273273 0 0
274274 25.6458 1.54932
275275 − 32.8504i − 1.98096i
276276 0 0
277277 −10.0000 −0.600842 −0.300421 0.953807i 0.597127π-0.597127\pi
−0.300421 + 0.953807i 0.597127π0.597127\pi
278278 0 0
279279 0 0
280280 0 0
281281 − 3.41815i − 0.203910i −0.994789 0.101955i 0.967490π-0.967490\pi
0.994789 0.101955i 0.0325097π-0.0325097\pi
282282 0 0
283283 0 0 1.00000 00
−1.00000 π\pi
284284 9.72202i 0.576896i
285285 0 0
286286 0 0
287287 0 0
288288 0 0
289289 −17.0000 −1.00000
290290 0 0
291291 0 0
292292 0 0
293293 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
294294 0 0
295295 0 0
296296 − 32.5842i − 1.89392i
297297 0 0
298298 9.39477 0.544224
299299 0 0
300300 0 0
301301 14.0000 0.806947
302302 6.15784i 0.354344i
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 0 0 1.00000 00
−1.00000 π\pi
308308 − 11.2250i − 0.639602i
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 0 0 1.00000 00
−1.00000 π\pi
314314 0 0
315315 0 0
316316 5.16601 0.290611
317317 16.5583i 0.930008i 0.885309 + 0.465004i 0.153947π0.153947\pi
−0.885309 + 0.465004i 0.846053π0.846053\pi
318318 0 0
319319 58.4575 3.27299
320320 0 0
321321 0 0
322322 5.89674 0.328612
323323 0 0
324324 0 0
325325 0 0
326326 − 23.2744i − 1.28905i
327327 0 0
328328 0 0
329329 0 0
330330 0 0
331331 −5.29150 −0.290847 −0.145424 0.989369i 0.546455π-0.546455\pi
−0.145424 + 0.989369i 0.546455π0.546455\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 −21.1660 −1.15299 −0.576493 0.817102i 0.695579π-0.695579\pi
−0.576493 + 0.817102i 0.695579π0.695579\pi
338338 − 15.1284i − 0.822876i
339339 0 0
340340 0 0
341341 0 0
342342 0 0
343343 −18.5203 −1.00000
344344 16.2921i 0.878412i
345345 0 0
346346 0 0
347347 29.0200i 1.55788i 0.627100 + 0.778938i 0.284242π0.284242\pi
−0.627100 + 0.778938i 0.715758π0.715758\pi
348348 0 0
349349 0 0 1.00000 00
−1.00000 π\pi
350350 − 15.3946i − 0.822876i
351351 0 0
352352 22.9373 1.22256
353353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 −18.4797 −0.976685
359359 20.5347i 1.08378i 0.840449 + 0.541891i 0.182292π0.182292\pi
−0.840449 + 0.541891i 0.817708π0.817708\pi
360360 0 0
361361 19.0000 1.00000
362362 0 0
363363 0 0
364364 0 0
365365 0 0
366366 0 0
367367 0 0 1.00000 00
−1.00000 π\pi
368368 4.38868i 0.228776i
369369 0 0
370370 0 0
371371 1.09070i 0.0566265i
372372 0 0
373373 −22.0000 −1.13912 −0.569558 0.821951i 0.692886π-0.692886\pi
−0.569558 + 0.821951i 0.692886π0.692886\pi
374374 0 0
375375 0 0
376376 0 0
377377 0 0
378378 0 0
379379 −37.0405 −1.90264 −0.951322 0.308199i 0.900274π-0.900274\pi
−0.951322 + 0.308199i 0.900274π0.900274\pi
380380 0 0
381381 0 0
382382 −28.3542 −1.45073
383383 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
384384 0 0
385385 0 0
386386 24.6314i 1.25370i
387387 0 0
388388 0 0
389389 − 34.3534i − 1.74179i −0.491473 0.870893i 0.663542π-0.663542\pi
0.491473 0.870893i 0.336458π-0.336458\pi
390390 0 0
391391 0 0
392392 − 21.5524i − 1.08856i
393393 0 0
394394 −30.1033 −1.51658
395395 0 0
396396 0 0
397397 0 0 1.00000 00
−1.00000 π\pi
398398 0 0
399399 0 0
400400 11.4575 0.572876
401401 39.0083i 1.94798i 0.226592 + 0.973990i 0.427242π0.427242\pi
−0.226592 + 0.973990i 0.572758π0.572758\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 27.3948 1.35958
407407 69.5312i 3.44654i
408408 0 0
409409 0 0 1.00000 00
−1.00000 π\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 26.0000 1.26716 0.633581 0.773676i 0.281584π-0.281584\pi
0.633581 + 0.773676i 0.281584π0.281584\pi
422422 − 30.7892i − 1.49879i
423423 0 0
424424 −1.26927 −0.0616414
425425 0 0
426426 0 0
427427 0 0
428428 − 6.71612i − 0.324636i
429429 0 0
430430 0 0
431431 − 41.3357i − 1.99107i −0.0943889 0.995535i 0.530090π-0.530090\pi
0.0943889 0.995535i 0.469910π-0.469910\pi
432432 0 0
433433 0 0 1.00000 00
−1.00000 π\pi
434434 0 0
435435 0 0
436436 6.83399 0.327289
437437 0 0
438438 0 0
439439 0 0 1.00000 00
−1.00000 π\pi
440440 0 0
441441 0 0
442442 0 0
443443 12.0495i 0.572487i 0.958157 + 0.286244i 0.0924067π0.0924067\pi
−0.958157 + 0.286244i 0.907593π0.907593\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 22.8745 1.08072
449449 − 31.3475i − 1.47938i −0.672948 0.739689i 0.734972π-0.734972\pi
0.672948 0.739689i 0.265028π-0.265028\pi
450450 0 0
451451 0 0
452452 8.75149i 0.411635i
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 42.3320 1.98021 0.990104 0.140334i 0.0448177π-0.0448177\pi
0.990104 + 0.140334i 0.0448177π0.0448177\pi
458458 0 0
459459 0 0
460460 0 0
461461 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
462462 0 0
463463 −40.0000 −1.85896 −0.929479 0.368875i 0.879743π-0.879743\pi
−0.929479 + 0.368875i 0.879743π0.879743\pi
464464 20.3887i 0.946522i
465465 0 0
466466 35.5203 1.64544
467467 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
468468 0 0
469469 10.5830 0.488678
470470 0 0
471471 0 0
472472 0 0
473473 − 34.7656i − 1.59852i
474474 0 0
475475 0 0
476476 0 0
477477 0 0
478478 −8.60523 −0.393594
479479 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
480480 0 0
481481 0 0
482482 0 0
483483 0 0
484484 −20.7712 −0.944147
485485 0 0
486486 0 0
487487 −37.0405 −1.67847 −0.839233 0.543772i 0.816996π-0.816996\pi
−0.839233 + 0.543772i 0.816996π0.816996\pi
488488 0 0
489489 0 0
490490 0 0
491491 − 27.3710i − 1.23524i −0.786478 0.617619i 0.788097π-0.788097\pi
0.786478 0.617619i 0.211903π-0.211903\pi
492492 0 0
493493 0 0
494494 0 0
495495 0 0
496496 0 0
497497 − 39.8328i − 1.78674i
498498 0 0
499499 26.4575 1.18440 0.592200 0.805791i 0.298259π-0.298259\pi
0.592200 + 0.805791i 0.298259π0.298259\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 0 0
506506 − 14.6431i − 0.650966i
507507 0 0
508508 −10.3320 −0.458409
509509 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
510510 0 0
511511 0 0
512512 22.1107i 0.977165i
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 0 0
518518 32.5842i 1.43167i
519519 0 0
520520 0 0
521521 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
522522 0 0
523523 0 0 1.00000 00
−1.00000 π\pi
524524 0 0
525525 0 0
526526 −21.9778 −0.958276
527527 0 0
528528 0 0
529529 19.3320 0.840523
530530 0 0
531531 0 0
532532 0 0
533533 0 0
534534 0 0
535535 0 0
536536 12.3157i 0.531956i
537537 0 0
538538 0 0
539539 45.9906i 1.98096i
540540 0 0
541541 −34.0000 −1.46177 −0.730887 0.682498i 0.760893π-0.760893\pi
−0.730887 + 0.682498i 0.760893π0.760893\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 44.0000 1.88130 0.940652 0.339372i 0.110215π-0.110215\pi
0.940652 + 0.339372i 0.110215π0.110215\pi
548548 14.2309i 0.607913i
549549 0 0
550550 −38.2288 −1.63008
551551 0 0
552552 0 0
553553 −21.1660 −0.900070
554554 11.6372i 0.494418i
555555 0 0
556556 0 0
557557 25.0436i 1.06113i 0.847644 + 0.530566i 0.178020π0.178020\pi
−0.847644 + 0.530566i 0.821980π0.821980\pi
558558 0 0
559559 0 0
560560 0 0
561561 0 0
562562 −3.97777 −0.167792
563563 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
564564 0 0
565565 0 0
566566 0 0
567567 0 0
568568 46.3542 1.94498
569569 − 14.3769i − 0.602711i −0.953512 0.301356i 0.902561π-0.902561\pi
0.953512 0.301356i 0.0974392π-0.0974392\pi
570570 0 0
571571 −4.00000 −0.167395 −0.0836974 0.996491i 0.526673π-0.526673\pi
−0.0836974 + 0.996491i 0.526673π0.526673\pi
572572 0 0
573573 0 0
574574 0 0
575575 9.57598i 0.399346i
576576 0 0
577577 0 0 1.00000 00
−1.00000 π\pi
578578 19.7833i 0.822876i
579579 0 0
580580 0 0
581581 0 0
582582 0 0
583583 2.70850 0.112174
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 −24.2510 −0.996709
593593 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
594594 0 0
595595 0 0
596596 5.21317i 0.213540i
597597 0 0
598598 0 0
599599 3.56418i 0.145629i 0.997346 + 0.0728143i 0.0231980π0.0231980\pi
−0.997346 + 0.0728143i 0.976802π0.976802\pi
600600 0 0
601601 0 0 1.00000 00
−1.00000 π\pi
602602 − 16.2921i − 0.664017i
603603 0 0
604604 −3.41699 −0.139036
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 38.0000 1.53481 0.767403 0.641165i 0.221549π-0.221549\pi
0.767403 + 0.641165i 0.221549π0.221549\pi
614614 0 0
615615 0 0
616616 −53.5203 −2.15639
617617 − 48.3180i − 1.94521i −0.232462 0.972605i 0.574678π-0.574678\pi
0.232462 0.972605i 0.425322π-0.425322\pi
618618 0 0
619619 0 0 1.00000 00
−1.00000 π\pi
620620 0 0
621621 0 0
622622 0 0
623623 0 0
624624 0 0
625625 25.0000 1.00000
626626 0 0
627627 0 0
628628 0 0
629629 0 0
630630 0 0
631631 −16.0000 −0.636950 −0.318475 0.947931i 0.603171π-0.603171\pi
−0.318475 + 0.947931i 0.603171π0.603171\pi
632632 − 24.6314i − 0.979783i
633633 0 0
634634 19.2693 0.765281
635635 0 0
636636 0 0
637637 0 0
638638 − 68.0283i − 2.69327i
639639 0 0
640640 0 0
641641 47.4935i 1.87588i 0.346795 + 0.937941i 0.387270π0.387270\pi
−0.346795 + 0.937941i 0.612730π0.612730\pi
642642 0 0
643643 0 0 1.00000 00
−1.00000 π\pi
644644 3.27211i 0.128939i
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 12.9150 0.505791
653653 − 42.8387i − 1.67641i −0.545358 0.838203i 0.683606π-0.683606\pi
0.545358 0.838203i 0.316394π-0.316394\pi
654654 0 0
655655 0 0
656656 0 0
657657 0 0
658658 0 0
659659 − 49.8210i − 1.94075i −0.241604 0.970375i 0.577673π-0.577673\pi
0.241604 0.970375i 0.422327π-0.422327\pi
660660 0 0
661661 0 0 1.00000 00
−1.00000 π\pi
662662 6.15784i 0.239331i
663663 0 0
664664 0 0
665665 0 0
666666 0 0
667667 −17.0405 −0.659812
668668 0 0
669669 0 0
670670 0 0
671671 0 0
672672 0 0
673673 42.3320 1.63178 0.815890 0.578208i 0.196248π-0.196248\pi
0.815890 + 0.578208i 0.196248π0.196248\pi
674674 24.6314i 0.948764i
675675 0 0
676676 8.39477 0.322876
677677 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
678678 0 0
679679 0 0
680680 0 0
681681 0 0
682682 0 0
683683 40.5112i 1.55012i 0.631889 + 0.775059i 0.282280π0.282280\pi
−0.631889 + 0.775059i 0.717720π0.717720\pi
684684 0 0
685685 0 0
686686 21.5524i 0.822876i
687687 0 0
688688 12.1255 0.462280
689689 0 0
690690 0 0
691691 0 0 1.00000 00
−1.00000 π\pi
692692 0 0
693693 0 0
694694 33.7712 1.28194
695695 0 0
696696 0 0
697697 0 0
698698 0 0
699699 0 0
700700 8.54249 0.322876
701701 36.0024i 1.35979i 0.733309 + 0.679895i 0.237975π0.237975\pi
−0.733309 + 0.679895i 0.762025π0.762025\pi
702702 0 0
703703 0 0
704704 − 56.8033i − 2.14086i
705705 0 0
706706 0 0
707707 0 0
708708 0 0
709709 −52.9150 −1.98727 −0.993633 0.112667i 0.964061π-0.964061\pi
−0.993633 + 0.112667i 0.964061π0.964061\pi
710710 0 0
711711 0 0
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 − 10.2544i − 0.383226i
717717 0 0
718718 23.8967 0.891818
719719 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 0 0
721721 0 0
722722 − 22.1107i − 0.822876i
723723 0 0
724724 0 0
725725 44.4876i 1.65223i
726726 0 0
727727 0 0 1.00000 00
−1.00000 π\pi
728728 0 0
729729 0 0
730730 0 0
731731 0 0
732732 0 0
733733 0 0 1.00000 00
−1.00000 π\pi
734734 0 0
735735 0 0
736736 −6.68627 −0.246459
737737 − 26.2803i − 0.968049i
738738 0 0
739739 −52.0000 −1.91285 −0.956425 0.291977i 0.905687π-0.905687\pi
−0.956425 + 0.291977i 0.905687π0.905687\pi
740740 0 0
741741 0 0
742742 1.26927 0.0465965
743743 54.4759i 1.99853i 0.0383863 + 0.999263i 0.487778π0.487778\pi
−0.0383863 + 0.999263i 0.512222π0.512222\pi
744744 0 0
745745 0 0
746746 25.6019i 0.937352i
747747 0 0
748748 0 0
749749 27.5171i 1.00545i
750750 0 0
751751 26.4575 0.965448 0.482724 0.875772i 0.339647π-0.339647\pi
0.482724 + 0.875772i 0.339647π0.339647\pi
752752 0 0
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 10.5830 0.384646 0.192323 0.981332i 0.438398π-0.438398\pi
0.192323 + 0.981332i 0.438398π0.438398\pi
758758 43.1049i 1.56564i
759759 0 0
760760 0 0
761761 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
762762 0 0
763763 −28.0000 −1.01367
764764 − 15.7338i − 0.569230i
765765 0 0
766766 0 0
767767 0 0
768768 0 0
769769 0 0 1.00000 00
−1.00000 π\pi
770770 0 0
771771 0 0
772772 −13.6680 −0.491921
773773 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 −39.9778 −1.43327
779779 0 0
780780 0 0
781781 −98.9150 −3.53946
782782 0 0
783783 0 0
784784 −16.0405 −0.572876
785785 0 0
786786 0 0
787787 0 0 1.00000 00
−1.00000 π\pi
788788 − 16.7044i − 0.595068i
789789 0 0
790790 0 0
791791 − 35.8563i − 1.27490i
792792 0 0
793793 0 0
794794 0 0
795795 0 0
796796 0 0
797797 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
798798 0 0
799799 0 0
800800 17.4558i 0.617157i
801801 0 0
802802 45.3948 1.60294
803803 0 0
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 − 56.8033i − 1.99710i −0.0538482 0.998549i 0.517149π-0.517149\pi
0.0538482 0.998549i 0.482851π-0.482851\pi
810810 0 0
811811 0 0 1.00000 00
−1.00000 π\pi
812812 15.2014i 0.533465i
813813 0 0
814814 80.9150 2.83607
815815 0 0
816816 0 0
817817 0 0
818818 0 0
819819 0 0
820820 0 0
821821 52.9729i 1.84877i 0.381464 + 0.924384i 0.375420π0.375420\pi
−0.381464 + 0.924384i 0.624580π0.624580\pi
822822 0 0
823823 32.0000 1.11545 0.557725 0.830026i 0.311674π-0.311674\pi
0.557725 + 0.830026i 0.311674π0.311674\pi
824824 0 0
825825 0 0
826826 0 0
827827 − 4.92110i − 0.171123i −0.996333 0.0855616i 0.972732π-0.972732\pi
0.996333 0.0855616i 0.0272685π-0.0272685\pi
828828 0 0
829829 0 0 1.00000 00
−1.00000 π\pi
830830 0 0
831831 0 0
832832 0 0
833833 0 0
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 −50.1660 −1.72986
842842 − 30.2568i − 1.04272i
843843 0 0
844844 17.0850 0.588089
845845 0 0
846846 0 0
847847 85.1033 2.92418
848848 0.944665i 0.0324399i
849849 0 0
850850 0 0
851851 − 20.2685i − 0.694797i
852852 0 0
853853 0 0 1.00000 00
−1.00000 π\pi
854854 0 0
855855 0 0
856856 −32.0222 −1.09450
857857 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
858858 0 0
859859 0 0 1.00000 00
−1.00000 π\pi
860860 0 0
861861 0 0
862862 −48.1033 −1.63840
863863 − 46.8151i − 1.59360i −0.604240 0.796802i 0.706523π-0.706523\pi
0.604240 0.796802i 0.293477π-0.293477\pi
864864 0 0
865865 0 0
866866 0 0
867867 0 0
868868 0 0
869869 52.5607i 1.78300i
870870 0 0
871871 0 0
872872 − 32.5842i − 1.10344i
873873 0 0
874874 0 0
875875 0 0
876876 0 0
877877 50.0000 1.68838 0.844190 0.536044i 0.180082π-0.180082\pi
0.844190 + 0.536044i 0.180082π0.180082\pi
878878 0 0
879879 0 0
880880 0 0
881881 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
882882 0 0
883883 58.2065 1.95881 0.979403 0.201916i 0.0647168π-0.0647168\pi
0.979403 + 0.201916i 0.0647168π0.0647168\pi
884884 0 0
885885 0 0
886886 14.0222 0.471086
887887 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
888888 0 0
889889 42.3320 1.41977
890890 0 0
891891 0 0
892892 0 0
893893 0 0
894894 0 0
895895 0 0
896896 − 8.14605i − 0.272141i
897897 0 0
898898 −36.4797 −1.21734
899899 0 0
900900 0 0
901901 0 0
902902 0 0
903903 0 0
904904 41.7268 1.38781
905905 0 0
906906 0 0
907907 −5.29150 −0.175701 −0.0878507 0.996134i 0.528000π-0.528000\pi
−0.0878507 + 0.996134i 0.528000π0.528000\pi
908908 0 0
909909 0 0
910910 0 0
911911 − 29.8445i − 0.988793i −0.869236 0.494397i 0.835389π-0.835389\pi
0.869236 0.494397i 0.164611π-0.164611\pi
912912 0 0
913913 0 0
914914 − 49.2627i − 1.62947i
915915 0 0
916916 0 0
917917 0 0
918918 0 0
919919 −37.0405 −1.22185 −0.610927 0.791687i 0.709203π-0.709203\pi
−0.610927 + 0.791687i 0.709203π0.709203\pi
920920 0 0
921921 0 0
922922 0 0
923923 0 0
924924 0 0
925925 −52.9150 −1.73984
926926 46.5489i 1.52969i
927927 0 0
928928 −31.0627 −1.01968
929929 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
930930 0 0
931931 0 0
932932 19.7103i 0.645631i
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 0 0 1.00000 00
−1.00000 π\pi
938938 − 12.3157i − 0.402121i
939939 0 0
940940 0 0
941941 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
942942 0 0
943943 0 0
944944 0 0
945945 0 0
946946 −40.4575 −1.31539
947947 − 55.3004i − 1.79702i −0.438953 0.898510i 0.644650π-0.644650\pi
0.438953 0.898510i 0.355350π-0.355350\pi
948948 0 0
949949 0 0
950950 0 0
951951 0 0
952952 0 0
953953 55.9788i 1.81333i 0.421849 + 0.906666i 0.361381π0.361381\pi
−0.421849 + 0.906666i 0.638619π0.638619\pi
954954 0 0
955955 0 0
956956 − 4.77506i − 0.154436i
957957 0 0
958958 0 0
959959 − 58.3063i − 1.88281i
960960 0 0
961961 31.0000 1.00000
962962 0 0
963963 0 0
964964 0 0
965965 0 0
966966 0 0
967967 −40.0000 −1.28631 −0.643157 0.765735i 0.722376π-0.722376\pi
−0.643157 + 0.765735i 0.722376π0.722376\pi
968968 99.0365i 3.18315i
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 43.1049i 1.38117i
975975 0 0
976976 0 0
977977 2.59365i 0.0829783i 0.999139 + 0.0414892i 0.0132102π0.0132102\pi
−0.999139 + 0.0414892i 0.986790π0.986790\pi
978978 0 0
979979 0 0
980980 0 0
981981 0 0
982982 −31.8523 −1.01645
983983 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 10.1343i 0.322251i
990990 0 0
991991 58.2065 1.84899 0.924496 0.381193i 0.124487π-0.124487\pi
0.924496 + 0.381193i 0.124487π0.124487\pi
992992 0 0
993993 0 0
994994 −46.3542 −1.47027
995995 0 0
996996 0 0
997997 0 0 1.00000 00
−1.00000 π\pi
998998 − 30.7892i − 0.974615i
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 63.2.c.a.62.2 4
3.2 odd 2 inner 63.2.c.a.62.3 yes 4
4.3 odd 2 1008.2.k.a.881.3 4
5.2 odd 4 1575.2.g.d.1574.5 8
5.3 odd 4 1575.2.g.d.1574.4 8
5.4 even 2 1575.2.b.a.251.3 4
7.2 even 3 441.2.p.b.80.2 8
7.3 odd 6 441.2.p.b.215.3 8
7.4 even 3 441.2.p.b.215.3 8
7.5 odd 6 441.2.p.b.80.2 8
7.6 odd 2 CM 63.2.c.a.62.2 4
8.3 odd 2 4032.2.k.b.3905.4 4
8.5 even 2 4032.2.k.c.3905.1 4
9.2 odd 6 567.2.o.f.377.3 8
9.4 even 3 567.2.o.f.188.3 8
9.5 odd 6 567.2.o.f.188.2 8
9.7 even 3 567.2.o.f.377.2 8
12.11 even 2 1008.2.k.a.881.4 4
15.2 even 4 1575.2.g.d.1574.3 8
15.8 even 4 1575.2.g.d.1574.6 8
15.14 odd 2 1575.2.b.a.251.2 4
21.2 odd 6 441.2.p.b.80.3 8
21.5 even 6 441.2.p.b.80.3 8
21.11 odd 6 441.2.p.b.215.2 8
21.17 even 6 441.2.p.b.215.2 8
21.20 even 2 inner 63.2.c.a.62.3 yes 4
24.5 odd 2 4032.2.k.c.3905.2 4
24.11 even 2 4032.2.k.b.3905.3 4
28.27 even 2 1008.2.k.a.881.3 4
35.13 even 4 1575.2.g.d.1574.4 8
35.27 even 4 1575.2.g.d.1574.5 8
35.34 odd 2 1575.2.b.a.251.3 4
56.13 odd 2 4032.2.k.c.3905.1 4
56.27 even 2 4032.2.k.b.3905.4 4
63.13 odd 6 567.2.o.f.188.3 8
63.20 even 6 567.2.o.f.377.3 8
63.34 odd 6 567.2.o.f.377.2 8
63.41 even 6 567.2.o.f.188.2 8
84.83 odd 2 1008.2.k.a.881.4 4
105.62 odd 4 1575.2.g.d.1574.3 8
105.83 odd 4 1575.2.g.d.1574.6 8
105.104 even 2 1575.2.b.a.251.2 4
168.83 odd 2 4032.2.k.b.3905.3 4
168.125 even 2 4032.2.k.c.3905.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.2 4 1.1 even 1 trivial
63.2.c.a.62.2 4 7.6 odd 2 CM
63.2.c.a.62.3 yes 4 3.2 odd 2 inner
63.2.c.a.62.3 yes 4 21.20 even 2 inner
441.2.p.b.80.2 8 7.2 even 3
441.2.p.b.80.2 8 7.5 odd 6
441.2.p.b.80.3 8 21.2 odd 6
441.2.p.b.80.3 8 21.5 even 6
441.2.p.b.215.2 8 21.11 odd 6
441.2.p.b.215.2 8 21.17 even 6
441.2.p.b.215.3 8 7.3 odd 6
441.2.p.b.215.3 8 7.4 even 3
567.2.o.f.188.2 8 9.5 odd 6
567.2.o.f.188.2 8 63.41 even 6
567.2.o.f.188.3 8 9.4 even 3
567.2.o.f.188.3 8 63.13 odd 6
567.2.o.f.377.2 8 9.7 even 3
567.2.o.f.377.2 8 63.34 odd 6
567.2.o.f.377.3 8 9.2 odd 6
567.2.o.f.377.3 8 63.20 even 6
1008.2.k.a.881.3 4 4.3 odd 2
1008.2.k.a.881.3 4 28.27 even 2
1008.2.k.a.881.4 4 12.11 even 2
1008.2.k.a.881.4 4 84.83 odd 2
1575.2.b.a.251.2 4 15.14 odd 2
1575.2.b.a.251.2 4 105.104 even 2
1575.2.b.a.251.3 4 5.4 even 2
1575.2.b.a.251.3 4 35.34 odd 2
1575.2.g.d.1574.3 8 15.2 even 4
1575.2.g.d.1574.3 8 105.62 odd 4
1575.2.g.d.1574.4 8 5.3 odd 4
1575.2.g.d.1574.4 8 35.13 even 4
1575.2.g.d.1574.5 8 5.2 odd 4
1575.2.g.d.1574.5 8 35.27 even 4
1575.2.g.d.1574.6 8 15.8 even 4
1575.2.g.d.1574.6 8 105.83 odd 4
4032.2.k.b.3905.3 4 24.11 even 2
4032.2.k.b.3905.3 4 168.83 odd 2
4032.2.k.b.3905.4 4 8.3 odd 2
4032.2.k.b.3905.4 4 56.27 even 2
4032.2.k.c.3905.1 4 8.5 even 2
4032.2.k.c.3905.1 4 56.13 odd 2
4032.2.k.c.3905.2 4 24.5 odd 2
4032.2.k.c.3905.2 4 168.125 even 2