Properties

Label 2800.2.k.p.2351.8
Level 28002800
Weight 22
Character 2800.2351
Analytic conductor 22.35822.358
Analytic rank 00
Dimension 88
CM discriminant -35
Inner twists 88

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2800,2,Mod(2351,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2800=24527 2800 = 2^{4} \cdot 5^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2800.k (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: 22.358112566022.3581125660
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.31116960000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+x68x4+9x2+81 x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 26 2^{6}
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

Embedding invariants

Embedding label 2351.8
Root 1.62968+0.586627i1.62968 + 0.586627i of defining polynomial
Character χ\chi == 2800.2351
Dual form 2800.2.k.p.2351.7

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) == q+3.25937q32.64575q7+7.62348q9+0.359964iq11+5.64539iq13+7.99190iq178.62348q21+15.0696q27+0.623475q29+1.17325iq33+18.4004iq395.71383q47+7.00000q49+26.0485iq5120.1698q6311.8322iq7113.4164iq730.952374iq77+8.00809iq79+26.2470q81+15.8745q83+2.03214q8714.9363iq91+12.6849iq97+2.74417iq99+O(q100)q+3.25937 q^{3} -2.64575 q^{7} +7.62348 q^{9} +0.359964i q^{11} +5.64539i q^{13} +7.99190i q^{17} -8.62348 q^{21} +15.0696 q^{27} +0.623475 q^{29} +1.17325i q^{33} +18.4004i q^{39} -5.71383 q^{47} +7.00000 q^{49} +26.0485i q^{51} -20.1698 q^{63} -11.8322i q^{71} -13.4164i q^{73} -0.952374i q^{77} +8.00809i q^{79} +26.2470 q^{81} +15.8745 q^{83} +2.03214 q^{87} -14.9363i q^{91} +12.6849i q^{97} +2.74417i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+20q928q2136q29+56q49+128q81+O(q100) 8 q + 20 q^{9} - 28 q^{21} - 36 q^{29} + 56 q^{49} + 128 q^{81}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2800Z)×\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times.

nn 351351 801801 21012101 25772577
χ(n)\chi(n) 1-1 1-1 11 11

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 3.25937 1.88180 0.940898 0.338689i 0.109984π-0.109984\pi
0.940898 + 0.338689i 0.109984π0.109984\pi
44 0 0
55 0 0
66 0 0
77 −2.64575 −1.00000
88 0 0
99 7.62348 2.54116
1010 0 0
1111 0.359964i 0.108533i 0.998526 + 0.0542666i 0.0172821π0.0172821\pi
−0.998526 + 0.0542666i 0.982718π0.982718\pi
1212 0 0
1313 5.64539i 1.56575i 0.622179 + 0.782875i 0.286247π0.286247\pi
−0.622179 + 0.782875i 0.713753π0.713753\pi
1414 0 0
1515 0 0
1616 0 0
1717 7.99190i 1.93832i 0.246433 + 0.969160i 0.420742π0.420742\pi
−0.246433 + 0.969160i 0.579258π0.579258\pi
1818 0 0
1919 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
2020 0 0
2121 −8.62348 −1.88180
2222 0 0
2323 0 0 1.00000 00
−1.00000 π\pi
2424 0 0
2525 0 0
2626 0 0
2727 15.0696 2.90015
2828 0 0
2929 0.623475 0.115776 0.0578882 0.998323i 0.481563π-0.481563\pi
0.0578882 + 0.998323i 0.481563π0.481563\pi
3030 0 0
3131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3232 0 0
3333 1.17325i 0.204237i
3434 0 0
3535 0 0
3636 0 0
3737 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3838 0 0
3939 18.4004i 2.94642i
4040 0 0
4141 0 0 1.00000 00
−1.00000 π\pi
4242 0 0
4343 0 0 1.00000 00
−1.00000 π\pi
4444 0 0
4545 0 0
4646 0 0
4747 −5.71383 −0.833448 −0.416724 0.909033i 0.636822π-0.636822\pi
−0.416724 + 0.909033i 0.636822π0.636822\pi
4848 0 0
4949 7.00000 1.00000
5050 0 0
5151 26.0485i 3.64752i
5252 0 0
5353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\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 0 0 1.00000 00
−1.00000 π\pi
6262 0 0
6363 −20.1698 −2.54116
6464 0 0
6565 0 0
6666 0 0
6767 0 0 1.00000 00
−1.00000 π\pi
6868 0 0
6969 0 0
7070 0 0
7171 − 11.8322i − 1.40422i −0.712069 0.702109i 0.752242π-0.752242\pi
0.712069 0.702109i 0.247758π-0.247758\pi
7272 0 0
7373 − 13.4164i − 1.57027i −0.619324 0.785136i 0.712593π-0.712593\pi
0.619324 0.785136i 0.287407π-0.287407\pi
7474 0 0
7575 0 0
7676 0 0
7777 − 0.952374i − 0.108533i
7878 0 0
7979 8.00809i 0.900981i 0.892781 + 0.450490i 0.148751π0.148751\pi
−0.892781 + 0.450490i 0.851249π0.851249\pi
8080 0 0
8181 26.2470 2.91633
8282 0 0
8383 15.8745 1.74245 0.871227 0.490881i 0.163325π-0.163325\pi
0.871227 + 0.490881i 0.163325π0.163325\pi
8484 0 0
8585 0 0
8686 0 0
8787 2.03214 0.217868
8888 0 0
8989 0 0 1.00000 00
−1.00000 π\pi
9090 0 0
9191 − 14.9363i − 1.56575i
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 12.6849i 1.28796i 0.765043 + 0.643979i 0.222718π0.222718\pi
−0.765043 + 0.643979i 0.777282π0.777282\pi
9898 0 0
9999 2.74417i 0.275800i
100100 0 0
101101 0 0 1.00000 00
−1.00000 π\pi
102102 0 0
103103 18.7513 1.84762 0.923810 0.382851i 0.125058π-0.125058\pi
0.923810 + 0.382851i 0.125058π0.125058\pi
104104 0 0
105105 0 0
106106 0 0
107107 0 0 1.00000 00
−1.00000 π\pi
108108 0 0
109109 −9.87043 −0.945415 −0.472708 0.881219i 0.656723π-0.656723\pi
−0.472708 + 0.881219i 0.656723π0.656723\pi
110110 0 0
111111 0 0
112112 0 0
113113 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
114114 0 0
115115 0 0
116116 0 0
117117 43.0375i 3.97882i
118118 0 0
119119 − 21.1446i − 1.93832i
120120 0 0
121121 10.8704 0.988221
122122 0 0
123123 0 0
124124 0 0
125125 0 0
126126 0 0
127127 0 0 1.00000 00
−1.00000 π\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 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
138138 0 0
139139 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
140140 0 0
141141 −18.6235 −1.56838
142142 0 0
143143 −2.03214 −0.169936
144144 0 0
145145 0 0
146146 0 0
147147 22.8156 1.88180
148148 0 0
149149 −6.00000 −0.491539 −0.245770 0.969328i 0.579041π-0.579041\pi
−0.245770 + 0.969328i 0.579041π0.579041\pi
150150 0 0
151151 5.84831i 0.475929i 0.971274 + 0.237964i 0.0764802π0.0764802\pi
−0.971274 + 0.237964i 0.923520π0.923520\pi
152152 0 0
153153 60.9260i 4.92558i
154154 0 0
155155 0 0
156156 0 0
157157 − 13.4164i − 1.07075i −0.844616 0.535373i 0.820171π-0.820171\pi
0.844616 0.535373i 0.179829π-0.179829\pi
158158 0 0
159159 0 0
160160 0 0
161161 0 0
162162 0 0
163163 0 0 1.00000 00
−1.00000 π\pi
164164 0 0
165165 0 0
166166 0 0
167167 25.2700 1.95545 0.977727 0.209881i 0.0673075π-0.0673075\pi
0.977727 + 0.209881i 0.0673075π0.0673075\pi
168168 0 0
169169 −18.8704 −1.45157
170170 0 0
171171 0 0
172172 0 0
173173 15.0314i 1.14282i 0.820666 + 0.571409i 0.193603π0.193603\pi
−0.820666 + 0.571409i 0.806397π0.806397\pi
174174 0 0
175175 0 0
176176 0 0
177177 0 0
178178 0 0
179179 − 11.8322i − 0.884377i −0.896922 0.442189i 0.854202π-0.854202\pi
0.896922 0.442189i 0.145798π-0.145798\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 −2.87679 −0.210372
188188 0 0
189189 −39.8704 −2.90015
190190 0 0
191191 − 20.4246i − 1.47788i −0.673774 0.738938i 0.735328π-0.735328\pi
0.673774 0.738938i 0.264672π-0.264672\pi
192192 0 0
193193 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
194194 0 0
195195 0 0
196196 0 0
197197 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\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 −1.64956 −0.115776
204204 0 0
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 0 0
210210 0 0
211211 − 28.7927i − 1.98217i −0.133226 0.991086i 0.542533π-0.542533\pi
0.133226 0.991086i 0.457467π-0.457467\pi
212212 0 0
213213 − 38.5654i − 2.64245i
214214 0 0
215215 0 0
216216 0 0
217217 0 0
218218 0 0
219219 − 43.7290i − 2.95493i
220220 0 0
221221 −45.1174 −3.03492
222222 0 0
223223 −20.3611 −1.36348 −0.681740 0.731594i 0.738777π-0.738777\pi
−0.681740 + 0.731594i 0.738777π0.738777\pi
224224 0 0
225225 0 0
226226 0 0
227227 −21.2058 −1.40748 −0.703738 0.710460i 0.748487π-0.748487\pi
−0.703738 + 0.710460i 0.748487π0.748487\pi
228228 0 0
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 − 3.10414i − 0.204237i
232232 0 0
233233 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
234234 0 0
235235 0 0
236236 0 0
237237 26.1013i 1.69546i
238238 0 0
239239 − 22.5844i − 1.46087i −0.682985 0.730433i 0.739318π-0.739318\pi
0.682985 0.730433i 0.260682π-0.260682\pi
240240 0 0
241241 0 0 1.00000 00
−1.00000 π\pi
242242 0 0
243243 40.3396 2.58779
244244 0 0
245245 0 0
246246 0 0
247247 0 0
248248 0 0
249249 51.7409 3.27894
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 − 4.47214i − 0.278964i −0.990225 0.139482i 0.955456π-0.955456\pi
0.990225 0.139482i 0.0445438π-0.0445438\pi
258258 0 0
259259 0 0
260260 0 0
261261 4.75305 0.294206
262262 0 0
263263 0 0 1.00000 00
−1.00000 π\pi
264264 0 0
265265 0 0
266266 0 0
267267 0 0
268268 0 0
269269 0 0 1.00000 00
−1.00000 π\pi
270270 0 0
271271 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
272272 0 0
273273 − 48.6829i − 2.94642i
274274 0 0
275275 0 0
276276 0 0
277277 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
278278 0 0
279279 0 0
280280 0 0
281281 11.3765 0.678667 0.339333 0.940666i 0.389799π-0.389799\pi
0.339333 + 0.940666i 0.389799π0.389799\pi
282282 0 0
283283 −27.7245 −1.64805 −0.824025 0.566553i 0.808277π-0.808277\pi
−0.824025 + 0.566553i 0.808277π0.808277\pi
284284 0 0
285285 0 0
286286 0 0
287287 0 0
288288 0 0
289289 −46.8704 −2.75708
290290 0 0
291291 41.3448i 2.42367i
292292 0 0
293293 32.9200i 1.92320i 0.274446 + 0.961602i 0.411505π0.411505\pi
−0.274446 + 0.961602i 0.588495π0.588495\pi
294294 0 0
295295 0 0
296296 0 0
297297 5.42451i 0.314762i
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 −11.3879 −0.649942 −0.324971 0.945724i 0.605355π-0.605355\pi
−0.324971 + 0.945724i 0.605355π0.605355\pi
308308 0 0
309309 61.1174 3.47685
310310 0 0
311311 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
312312 0 0
313313 − 35.2665i − 1.99338i −0.0813030 0.996689i 0.525908π-0.525908\pi
0.0813030 0.996689i 0.474092π-0.474092\pi
314314 0 0
315315 0 0
316316 0 0
317317 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
318318 0 0
319319 0.224428i 0.0125656i
320320 0 0
321321 0 0
322322 0 0
323323 0 0
324324 0 0
325325 0 0
326326 0 0
327327 −32.1713 −1.77908
328328 0 0
329329 15.1174 0.833448
330330 0 0
331331 − 35.4965i − 1.95106i −0.219860 0.975531i 0.570560π-0.570560\pi
0.219860 0.975531i 0.429440π-0.429440\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
338338 0 0
339339 0 0
340340 0 0
341341 0 0
342342 0 0
343343 −18.5203 −1.00000
344344 0 0
345345 0 0
346346 0 0
347347 0 0 1.00000 00
−1.00000 π\pi
348348 0 0
349349 0 0 1.00000 00
−1.00000 π\pi
350350 0 0
351351 85.0738i 4.54090i
352352 0 0
353353 6.08715i 0.323986i 0.986792 + 0.161993i 0.0517922π0.0517922\pi
−0.986792 + 0.161993i 0.948208π0.948208\pi
354354 0 0
355355 0 0
356356 0 0
357357 − 68.9179i − 3.64752i
358358 0 0
359359 11.8322i 0.624477i 0.950004 + 0.312239i 0.101079π0.101079\pi
−0.950004 + 0.312239i 0.898921π0.898921\pi
360360 0 0
361361 −19.0000 −1.00000
362362 0 0
363363 35.4307 1.85963
364364 0 0
365365 0 0
366366 0 0
367367 38.3075 1.99964 0.999818 0.0190919i 0.00607750π-0.00607750\pi
0.999818 + 0.0190919i 0.00607750π0.00607750\pi
368368 0 0
369369 0 0
370370 0 0
371371 0 0
372372 0 0
373373 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
374374 0 0
375375 0 0
376376 0 0
377377 3.51976i 0.181277i
378378 0 0
379379 35.4965i 1.82333i 0.410932 + 0.911666i 0.365203π0.365203\pi
−0.410932 + 0.911666i 0.634797π0.634797\pi
380380 0 0
381381 0 0
382382 0 0
383383 −15.8745 −0.811149 −0.405575 0.914062i 0.632929π-0.632929\pi
−0.405575 + 0.914062i 0.632929π0.632929\pi
384384 0 0
385385 0 0
386386 0 0
387387 0 0
388388 0 0
389389 24.6235 1.24846 0.624230 0.781241i 0.285413π-0.285413\pi
0.624230 + 0.781241i 0.285413π0.285413\pi
390390 0 0
391391 0 0
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 0 0
397397 − 19.7244i − 0.989941i −0.868910 0.494971i 0.835179π-0.835179\pi
0.868910 0.494971i 0.164821π-0.164821\pi
398398 0 0
399399 0 0
400400 0 0
401401 −39.1174 −1.95343 −0.976714 0.214544i 0.931173π-0.931173\pi
−0.976714 + 0.214544i 0.931173π0.931173\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 0 0
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 33.8704 1.65074 0.825372 0.564590i 0.190966π-0.190966\pi
0.825372 + 0.564590i 0.190966π0.190966\pi
422422 0 0
423423 −43.5593 −2.11792
424424 0 0
425425 0 0
426426 0 0
427427 0 0
428428 0 0
429429 −6.62348 −0.319784
430430 0 0
431431 − 23.3044i − 1.12253i −0.827636 0.561266i 0.810315π-0.810315\pi
0.827636 0.561266i 0.189685π-0.189685\pi
432432 0 0
433433 40.2492i 1.93425i 0.254293 + 0.967127i 0.418157π0.418157\pi
−0.254293 + 0.967127i 0.581843π0.581843\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 53.3643 2.54116
442442 0 0
443443 0 0 1.00000 00
−1.00000 π\pi
444444 0 0
445445 0 0
446446 0 0
447447 −19.5562 −0.924977
448448 0 0
449449 −40.3643 −1.90491 −0.952455 0.304679i 0.901451π-0.901451\pi
−0.952455 + 0.304679i 0.901451π0.901451\pi
450450 0 0
451451 0 0
452452 0 0
453453 19.0618i 0.895601i
454454 0 0
455455 0 0
456456 0 0
457457 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
458458 0 0
459459 120.435i 5.62141i
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 40.7620 1.88624 0.943119 0.332454i 0.107877π-0.107877\pi
0.943119 + 0.332454i 0.107877π0.107877\pi
468468 0 0
469469 0 0
470470 0 0
471471 − 43.7290i − 2.01493i
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 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 0 0
490490 0 0
491491 43.3690i 1.95722i 0.205731 + 0.978609i 0.434043π0.434043\pi
−0.205731 + 0.978609i 0.565957π0.565957\pi
492492 0 0
493493 4.98275i 0.224412i
494494 0 0
495495 0 0
496496 0 0
497497 31.3050i 1.40422i
498498 0 0
499499 − 26.6329i − 1.19225i −0.802890 0.596127i 0.796706π-0.796706\pi
0.802890 0.596127i 0.203294π-0.203294\pi
500500 0 0
501501 82.3643 3.67977
502502 0 0
503503 44.8262 1.99870 0.999352 0.0360049i 0.0114632π-0.0114632\pi
0.999352 + 0.0360049i 0.0114632π0.0114632\pi
504504 0 0
505505 0 0
506506 0 0
507507 −61.5057 −2.73156
508508 0 0
509509 0 0 1.00000 00
−1.00000 π\pi
510510 0 0
511511 35.4965i 1.57027i
512512 0 0
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 − 2.05677i − 0.0904567i
518518 0 0
519519 48.9929i 2.15055i
520520 0 0
521521 0 0 1.00000 00
−1.00000 π\pi
522522 0 0
523523 37.0405 1.61967 0.809834 0.586659i 0.199557π-0.199557\pi
0.809834 + 0.586659i 0.199557π0.199557\pi
524524 0 0
525525 0 0
526526 0 0
527527 0 0
528528 0 0
529529 23.0000 1.00000
530530 0 0
531531 0 0
532532 0 0
533533 0 0
534534 0 0
535535 0 0
536536 0 0
537537 − 38.5654i − 1.66422i
538538 0 0
539539 2.51975i 0.108533i
540540 0 0
541541 −6.12957 −0.263531 −0.131765 0.991281i 0.542065π-0.542065\pi
−0.131765 + 0.991281i 0.542065π0.542065\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 0 0 1.00000 00
−1.00000 π\pi
548548 0 0
549549 0 0
550550 0 0
551551 0 0
552552 0 0
553553 − 21.1874i − 0.900981i
554554 0 0
555555 0 0
556556 0 0
557557 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
558558 0 0
559559 0 0
560560 0 0
561561 −9.37652 −0.395877
562562 0 0
563563 15.8745 0.669031 0.334515 0.942390i 0.391427π-0.391427\pi
0.334515 + 0.942390i 0.391427π0.391427\pi
564564 0 0
565565 0 0
566566 0 0
567567 −69.4429 −2.91633
568568 0 0
569569 6.00000 0.251533 0.125767 0.992060i 0.459861π-0.459861\pi
0.125767 + 0.992060i 0.459861π0.459861\pi
570570 0 0
571571 − 35.4965i − 1.48548i −0.669579 0.742741i 0.733526π-0.733526\pi
0.669579 0.742741i 0.266474π-0.266474\pi
572572 0 0
573573 − 66.5714i − 2.78106i
574574 0 0
575575 0 0
576576 0 0
577577 − 46.5573i − 1.93820i −0.246661 0.969102i 0.579333π-0.579333\pi
0.246661 0.969102i 0.420667π-0.420667\pi
578578 0 0
579579 0 0
580580 0 0
581581 −42.0000 −1.74245
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 47.6235 1.96563 0.982817 0.184585i 0.0590940π-0.0590940\pi
0.982817 + 0.184585i 0.0590940π0.0590940\pi
588588 0 0
589589 0 0
590590 0 0
591591 0 0
592592 0 0
593593 41.8642i 1.71916i 0.511003 + 0.859579i 0.329274π0.329274\pi
−0.511003 + 0.859579i 0.670726π0.670726\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 0 0
599599 18.9848i 0.775698i 0.921723 + 0.387849i 0.126782π0.126782\pi
−0.921723 + 0.387849i 0.873218π0.873218\pi
600600 0 0
601601 0 0 1.00000 00
−1.00000 π\pi
602602 0 0
603603 0 0
604604 0 0
605605 0 0
606606 0 0
607607 −39.9173 −1.62019 −0.810097 0.586296i 0.800586π-0.800586\pi
−0.810097 + 0.586296i 0.800586π0.800586\pi
608608 0 0
609609 −5.37652 −0.217868
610610 0 0
611611 − 32.2568i − 1.30497i
612612 0 0
613613 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
614614 0 0
615615 0 0
616616 0 0
617617 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\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 0 0
626626 0 0
627627 0 0
628628 0 0
629629 0 0
630630 0 0
631631 − 49.5773i − 1.97364i −0.161817 0.986821i 0.551735π-0.551735\pi
0.161817 0.986821i 0.448265π-0.448265\pi
632632 0 0
633633 − 93.8460i − 3.73004i
634634 0 0
635635 0 0
636636 0 0
637637 39.5177i 1.56575i
638638 0 0
639639 − 90.2022i − 3.56834i
640640 0 0
641641 18.0000 0.710957 0.355479 0.934684i 0.384318π-0.384318\pi
0.355479 + 0.934684i 0.384318π0.384318\pi
642642 0 0
643643 −30.9441 −1.22032 −0.610158 0.792279i 0.708894π-0.708894\pi
−0.610158 + 0.792279i 0.708894π0.708894\pi
644644 0 0
645645 0 0
646646 0 0
647647 −47.6235 −1.87227 −0.936137 0.351636i 0.885626π-0.885626\pi
−0.936137 + 0.351636i 0.885626π0.885626\pi
648648 0 0
649649 0 0
650650 0 0
651651 0 0
652652 0 0
653653 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
654654 0 0
655655 0 0
656656 0 0
657657 − 102.280i − 3.99031i
658658 0 0
659659 41.2093i 1.60528i 0.596461 + 0.802642i 0.296573π0.296573\pi
−0.596461 + 0.802642i 0.703427π0.703427\pi
660660 0 0
661661 0 0 1.00000 00
−1.00000 π\pi
662662 0 0
663663 −147.054 −5.71111
664664 0 0
665665 0 0
666666 0 0
667667 0 0
668668 0 0
669669 −66.3643 −2.56579
670670 0 0
671671 0 0
672672 0 0
673673 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
674674 0 0
675675 0 0
676676 0 0
677677 − 18.8409i − 0.724115i −0.932156 0.362058i 0.882074π-0.882074\pi
0.932156 0.362058i 0.117926π-0.117926\pi
678678 0 0
679679 − 33.5611i − 1.28796i
680680 0 0
681681 −69.1174 −2.64858
682682 0 0
683683 0 0 1.00000 00
−1.00000 π\pi
684684 0 0
685685 0 0
686686 0 0
687687 0 0
688688 0 0
689689 0 0
690690 0 0
691691 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
692692 0 0
693693 − 7.26040i − 0.275800i
694694 0 0
695695 0 0
696696 0 0
697697 0 0
698698 0 0
699699 0 0
700700 0 0
701701 52.3643 1.97777 0.988887 0.148671i 0.0474996π-0.0474996\pi
0.988887 + 0.148671i 0.0474996π0.0474996\pi
702702 0 0
703703 0 0
704704 0 0
705705 0 0
706706 0 0
707707 0 0
708708 0 0
709709 45.6113 1.71297 0.856484 0.516174i 0.172644π-0.172644\pi
0.856484 + 0.516174i 0.172644π0.172644\pi
710710 0 0
711711 61.0495i 2.28954i
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 0 0
717717 − 73.6109i − 2.74905i
718718 0 0
719719 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 0 0
721721 −49.6113 −1.84762
722722 0 0
723723 0 0
724724 0 0
725725 0 0
726726 0 0
727727 −5.29150 −0.196251 −0.0981255 0.995174i 0.531285π-0.531285\pi
−0.0981255 + 0.995174i 0.531285π0.531285\pi
728728 0 0
729729 52.7409 1.95336
730730 0 0
731731 0 0
732732 0 0
733733 − 53.5968i − 1.97964i −0.142318 0.989821i 0.545455π-0.545455\pi
0.142318 0.989821i 0.454545π-0.454545\pi
734734 0 0
735735 0 0
736736 0 0
737737 0 0
738738 0 0
739739 − 17.0961i − 0.628889i −0.949276 0.314445i 0.898182π-0.898182\pi
0.949276 0.314445i 0.101818π-0.101818\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 121.019 4.42785
748748 0 0
749749 0 0
750750 0 0
751751 37.8807i 1.38229i 0.722718 + 0.691143i 0.242893π0.242893\pi
−0.722718 + 0.691143i 0.757107π0.757107\pi
752752 0 0
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
758758 0 0
759759 0 0
760760 0 0
761761 0 0 1.00000 00
−1.00000 π\pi
762762 0 0
763763 26.1147 0.945415
764764 0 0
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 − 14.5763i − 0.524954i
772772 0 0
773773 46.9990i 1.69044i 0.534421 + 0.845218i 0.320530π0.320530\pi
−0.534421 + 0.845218i 0.679470π0.679470\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 0 0
780780 0 0
781781 4.25915 0.152404
782782 0 0
783783 9.39553 0.335769
784784 0 0
785785 0 0
786786 0 0
787787 6.55849 0.233785 0.116892 0.993145i 0.462707π-0.462707\pi
0.116892 + 0.993145i 0.462707π0.462707\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 34.8247i 1.23355i 0.787138 + 0.616777i 0.211562π0.211562\pi
−0.787138 + 0.616777i 0.788438π0.788438\pi
798798 0 0
799799 − 45.6644i − 1.61549i
800800 0 0
801801 0 0
802802 0 0
803803 4.82942 0.170427
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 −16.3643 −0.575339 −0.287670 0.957730i 0.592880π-0.592880\pi
−0.287670 + 0.957730i 0.592880π0.592880\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 − 113.866i − 3.97882i
820820 0 0
821821 23.3765 0.815846 0.407923 0.913016i 0.366253π-0.366253\pi
0.407923 + 0.913016i 0.366253π0.366253\pi
822822 0 0
823823 0 0 1.00000 00
−1.00000 π\pi
824824 0 0
825825 0 0
826826 0 0
827827 0 0 1.00000 00
−1.00000 π\pi
828828 0 0
829829 0 0 1.00000 00
−1.00000 π\pi
830830 0 0
831831 0 0
832832 0 0
833833 55.9433i 1.93832i
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 −28.6113 −0.986596
842842 0 0
843843 37.0803 1.27711
844844 0 0
845845 0 0
846846 0 0
847847 −28.7604 −0.988221
848848 0 0
849849 −90.3643 −3.10130
850850 0 0
851851 0 0
852852 0 0
853853 − 40.2492i − 1.37811i −0.724710 0.689054i 0.758026π-0.758026\pi
0.724710 0.689054i 0.241974π-0.241974\pi
854854 0 0
855855 0 0
856856 0 0
857857 49.1935i 1.68042i 0.542263 + 0.840209i 0.317568π0.317568\pi
−0.542263 + 0.840209i 0.682432π0.682432\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 1.00000 00
−1.00000 π\pi
864864 0 0
865865 0 0
866866 0 0
867867 −152.768 −5.18827
868868 0 0
869869 −2.88262 −0.0977863
870870 0 0
871871 0 0
872872 0 0
873873 96.7031i 3.27290i
874874 0 0
875875 0 0
876876 0 0
877877 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
878878 0 0
879879 107.298i 3.61908i
880880 0 0
881881 0 0 1.00000 00
−1.00000 π\pi
882882 0 0
883883 0 0 1.00000 00
−1.00000 π\pi
884884 0 0
885885 0 0
886886 0 0
887887 −47.6235 −1.59904 −0.799521 0.600639i 0.794913π-0.794913\pi
−0.799521 + 0.600639i 0.794913π0.794913\pi
888888 0 0
889889 0 0
890890 0 0
891891 9.44795i 0.316518i
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 0 0 1.00000 00
−1.00000 π\pi
908908 0 0
909909 0 0
910910 0 0
911911 − 59.1608i − 1.96008i −0.198789 0.980042i 0.563701π-0.563701\pi
0.198789 0.980042i 0.436299π-0.436299\pi
912912 0 0
913913 5.71425i 0.189114i
914914 0 0
915915 0 0
916916 0 0
917917 0 0
918918 0 0
919919 − 51.7371i − 1.70665i −0.521380 0.853325i 0.674583π-0.674583\pi
0.521380 0.853325i 0.325417π-0.325417\pi
920920 0 0
921921 −37.1174 −1.22306
922922 0 0
923923 66.7972 2.19866
924924 0 0
925925 0 0
926926 0 0
927927 142.950 4.69510
928928 0 0
929929 0 0 1.00000 00
−1.00000 π\pi
930930 0 0
931931 0 0
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 49.3455i 1.61205i 0.591883 + 0.806024i 0.298385π0.298385\pi
−0.591883 + 0.806024i 0.701615π0.701615\pi
938938 0 0
939939 − 114.946i − 3.75113i
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 0 0 1.00000 00
−1.00000 π\pi
948948 0 0
949949 75.7409 2.45865
950950 0 0
951951 0 0
952952 0 0
953953 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
954954 0 0
955955 0 0
956956 0 0
957957 0.731495i 0.0236459i
958958 0 0
959959 0 0
960960 0 0
961961 −31.0000 −1.00000
962962 0 0
963963 0 0
964964 0 0
965965 0 0
966966 0 0
967967 0 0 1.00000 00
−1.00000 π\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 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
978978 0 0
979979 0 0
980980 0 0
981981 −75.2470 −2.40245
982982 0 0
983983 52.9548 1.68900 0.844498 0.535559i 0.179899π-0.179899\pi
0.844498 + 0.535559i 0.179899π0.179899\pi
984984 0 0
985985 0 0
986986 0 0
987987 49.2731 1.56838
988988 0 0
989989 0 0
990990 0 0
991991 35.4965i 1.12758i 0.825917 + 0.563791i 0.190658π0.190658\pi
−0.825917 + 0.563791i 0.809342π0.809342\pi
992992 0 0
993993 − 115.696i − 3.67150i
994994 0 0
995995 0 0
996996 0 0
997997 − 14.1479i − 0.448069i −0.974581 0.224034i 0.928077π-0.928077\pi
0.974581 0.224034i 0.0719228π-0.0719228\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 2800.2.k.p.2351.8 8
4.3 odd 2 inner 2800.2.k.p.2351.1 8
5.2 odd 4 560.2.e.c.559.1 8
5.3 odd 4 560.2.e.c.559.8 yes 8
5.4 even 2 inner 2800.2.k.p.2351.2 8
7.6 odd 2 inner 2800.2.k.p.2351.2 8
20.3 even 4 560.2.e.c.559.2 yes 8
20.7 even 4 560.2.e.c.559.7 yes 8
20.19 odd 2 inner 2800.2.k.p.2351.7 8
28.27 even 2 inner 2800.2.k.p.2351.7 8
35.13 even 4 560.2.e.c.559.1 8
35.27 even 4 560.2.e.c.559.8 yes 8
35.34 odd 2 CM 2800.2.k.p.2351.8 8
40.3 even 4 2240.2.e.d.2239.7 8
40.13 odd 4 2240.2.e.d.2239.1 8
40.27 even 4 2240.2.e.d.2239.2 8
40.37 odd 4 2240.2.e.d.2239.8 8
140.27 odd 4 560.2.e.c.559.2 yes 8
140.83 odd 4 560.2.e.c.559.7 yes 8
140.139 even 2 inner 2800.2.k.p.2351.1 8
280.13 even 4 2240.2.e.d.2239.8 8
280.27 odd 4 2240.2.e.d.2239.7 8
280.83 odd 4 2240.2.e.d.2239.2 8
280.237 even 4 2240.2.e.d.2239.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.e.c.559.1 8 5.2 odd 4
560.2.e.c.559.1 8 35.13 even 4
560.2.e.c.559.2 yes 8 20.3 even 4
560.2.e.c.559.2 yes 8 140.27 odd 4
560.2.e.c.559.7 yes 8 20.7 even 4
560.2.e.c.559.7 yes 8 140.83 odd 4
560.2.e.c.559.8 yes 8 5.3 odd 4
560.2.e.c.559.8 yes 8 35.27 even 4
2240.2.e.d.2239.1 8 40.13 odd 4
2240.2.e.d.2239.1 8 280.237 even 4
2240.2.e.d.2239.2 8 40.27 even 4
2240.2.e.d.2239.2 8 280.83 odd 4
2240.2.e.d.2239.7 8 40.3 even 4
2240.2.e.d.2239.7 8 280.27 odd 4
2240.2.e.d.2239.8 8 40.37 odd 4
2240.2.e.d.2239.8 8 280.13 even 4
2800.2.k.p.2351.1 8 4.3 odd 2 inner
2800.2.k.p.2351.1 8 140.139 even 2 inner
2800.2.k.p.2351.2 8 5.4 even 2 inner
2800.2.k.p.2351.2 8 7.6 odd 2 inner
2800.2.k.p.2351.7 8 20.19 odd 2 inner
2800.2.k.p.2351.7 8 28.27 even 2 inner
2800.2.k.p.2351.8 8 1.1 even 1 trivial
2800.2.k.p.2351.8 8 35.34 odd 2 CM