Properties

Label 968.2.g.a.483.8
Level 968968
Weight 22
Character 968.483
Analytic conductor 7.7307.730
Analytic rank 00
Dimension 88
CM discriminant -8
Inner twists 44

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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] = [968,2,Mod(483,968)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(968, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("968.483"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 968=23112 968 = 2^{3} \cdot 11^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 968.g (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 7.729518915667.72951891566
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.64000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x82x6+4x48x2+16 x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 24112 2^{4}\cdot 11^{2}
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

Embedding invariants

Embedding label 483.8
Root 1.34500+0.437016i1.34500 + 0.437016i of defining polynomial
Character χ\chi == 968.483
Dual form 968.2.g.a.483.4

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+1.41421iq2+3.30803q32.00000q4+4.67826iq62.82843iq8+7.94305q96.61606q12+4.00000q161.04978iq17+11.2332iq18+8.04031iq199.35652iq24+5.00000q25+16.3517q27+5.65685iq32+1.48461q3415.8861q3611.3707q382.21021iq416.88847iq43+13.2321q487.00000q49+7.07107iq503.47270iq51+23.1249iq54+26.5976iq573.45844q598.00000q6416.3138q67+2.09956iq6822.4663iq727.14631iq73+16.5401q7516.0806iq76+30.2629q81+3.12571q8216.2450iq83+9.74177q86+0.182318q89+18.7130iq9613.0498q979.89949iq98+O(q100)q+1.41421i q^{2} +3.30803 q^{3} -2.00000 q^{4} +4.67826i q^{6} -2.82843i q^{8} +7.94305 q^{9} -6.61606 q^{12} +4.00000 q^{16} -1.04978i q^{17} +11.2332i q^{18} +8.04031i q^{19} -9.35652i q^{24} +5.00000 q^{25} +16.3517 q^{27} +5.65685i q^{32} +1.48461 q^{34} -15.8861 q^{36} -11.3707 q^{38} -2.21021i q^{41} -6.88847i q^{43} +13.2321 q^{48} -7.00000 q^{49} +7.07107i q^{50} -3.47270i q^{51} +23.1249i q^{54} +26.5976i q^{57} -3.45844 q^{59} -8.00000 q^{64} -16.3138 q^{67} +2.09956i q^{68} -22.4663i q^{72} -7.14631i q^{73} +16.5401 q^{75} -16.0806i q^{76} +30.2629 q^{81} +3.12571 q^{82} -16.2450i q^{83} +9.74177 q^{86} +0.182318 q^{89} +18.7130i q^{96} -13.0498 q^{97} -9.89949i q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q4q316q4+28q9+8q12+32q16+40q25+8q27+16q3456q3624q3816q4856q49+12q5964q6428q6720q75+112q81+20q97+O(q100) 8 q - 4 q^{3} - 16 q^{4} + 28 q^{9} + 8 q^{12} + 32 q^{16} + 40 q^{25} + 8 q^{27} + 16 q^{34} - 56 q^{36} - 24 q^{38} - 16 q^{48} - 56 q^{49} + 12 q^{59} - 64 q^{64} - 28 q^{67} - 20 q^{75} + 112 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/968Z)×\left(\mathbb{Z}/968\mathbb{Z}\right)^\times.

nn 485485 727727 849849
χ(n)\chi(n) 1-1 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 1.41421i 1.00000i
33 3.30803 1.90989 0.954945 0.296781i 0.0959133π-0.0959133\pi
0.954945 + 0.296781i 0.0959133π0.0959133\pi
44 −2.00000 −1.00000
55 0 0 1.00000 00
−1.00000 π\pi
66 4.67826i 1.90989i
77 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
88 − 2.82843i − 1.00000i
99 7.94305 2.64768
1010 0 0
1111 0 0
1212 −6.61606 −1.90989
1313 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
1414 0 0
1515 0 0
1616 4.00000 1.00000
1717 − 1.04978i − 0.254609i −0.991864 0.127304i 0.959367π-0.959367\pi
0.991864 0.127304i 0.0406325π-0.0406325\pi
1818 11.2332i 2.64768i
1919 8.04031i 1.84457i 0.386507 + 0.922287i 0.373682π0.373682\pi
−0.386507 + 0.922287i 0.626318π0.626318\pi
2020 0 0
2121 0 0
2222 0 0
2323 0 0 1.00000 00
−1.00000 π\pi
2424 − 9.35652i − 1.90989i
2525 5.00000 1.00000
2626 0 0
2727 16.3517 3.14690
2828 0 0
2929 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3030 0 0
3131 0 0 1.00000 00
−1.00000 π\pi
3232 5.65685i 1.00000i
3333 0 0
3434 1.48461 0.254609
3535 0 0
3636 −15.8861 −2.64768
3737 0 0 1.00000 00
−1.00000 π\pi
3838 −11.3707 −1.84457
3939 0 0
4040 0 0
4141 − 2.21021i − 0.345177i −0.984994 0.172588i 0.944787π-0.944787\pi
0.984994 0.172588i 0.0552131π-0.0552131\pi
4242 0 0
4343 − 6.88847i − 1.05048i −0.850954 0.525241i 0.823975π-0.823975\pi
0.850954 0.525241i 0.176025π-0.176025\pi
4444 0 0
4545 0 0
4646 0 0
4747 0 0 1.00000 00
−1.00000 π\pi
4848 13.2321 1.90989
4949 −7.00000 −1.00000
5050 7.07107i 1.00000i
5151 − 3.47270i − 0.486275i
5252 0 0
5353 0 0 1.00000 00
−1.00000 π\pi
5454 23.1249i 3.14690i
5555 0 0
5656 0 0
5757 26.5976i 3.52293i
5858 0 0
5959 −3.45844 −0.450250 −0.225125 0.974330i 0.572279π-0.572279\pi
−0.225125 + 0.974330i 0.572279π0.572279\pi
6060 0 0
6161 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6262 0 0
6363 0 0
6464 −8.00000 −1.00000
6565 0 0
6666 0 0
6767 −16.3138 −1.99304 −0.996522 0.0833352i 0.973443π-0.973443\pi
−0.996522 + 0.0833352i 0.973443π0.973443\pi
6868 2.09956i 0.254609i
6969 0 0
7070 0 0
7171 0 0 1.00000 00
−1.00000 π\pi
7272 − 22.4663i − 2.64768i
7373 − 7.14631i − 0.836412i −0.908352 0.418206i 0.862659π-0.862659\pi
0.908352 0.418206i 0.137341π-0.137341\pi
7474 0 0
7575 16.5401 1.90989
7676 − 16.0806i − 1.84457i
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 30.2629 3.36254
8282 3.12571 0.345177
8383 − 16.2450i − 1.78312i −0.452904 0.891559i 0.649612π-0.649612\pi
0.452904 0.891559i 0.350388π-0.350388\pi
8484 0 0
8585 0 0
8686 9.74177 1.05048
8787 0 0
8888 0 0
8989 0.182318 0.0193256 0.00966282 0.999953i 0.496924π-0.496924\pi
0.00966282 + 0.999953i 0.496924π0.496924\pi
9090 0 0
9191 0 0
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 18.7130i 1.90989i
9797 −13.0498 −1.32501 −0.662503 0.749059i 0.730506π-0.730506\pi
−0.662503 + 0.749059i 0.730506π0.730506\pi
9898 − 9.89949i − 1.00000i
9999 0 0
100100 −10.0000 −1.00000
101101 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
102102 4.91114 0.486275
103103 0 0 1.00000 00
−1.00000 π\pi
104104 0 0
105105 0 0
106106 0 0
107107 − 11.8246i − 1.14312i −0.820559 0.571562i 0.806338π-0.806338\pi
0.820559 0.571562i 0.193662π-0.193662\pi
108108 −32.7035 −3.14690
109109 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
110110 0 0
111111 0 0
112112 0 0
113113 7.91227 0.744324 0.372162 0.928168i 0.378617π-0.378617\pi
0.372162 + 0.928168i 0.378617π0.378617\pi
114114 −37.6146 −3.52293
115115 0 0
116116 0 0
117117 0 0
118118 − 4.89097i − 0.450250i
119119 0 0
120120 0 0
121121 0 0
122122 0 0
123123 − 7.31144i − 0.659250i
124124 0 0
125125 0 0
126126 0 0
127127 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
128128 − 11.3137i − 1.00000i
129129 − 22.7872i − 2.00631i
130130 0 0
131131 22.0214i 1.92402i 0.273022 + 0.962008i 0.411977π0.411977\pi
−0.273022 + 0.962008i 0.588023π0.588023\pi
132132 0 0
133133 0 0
134134 − 23.0711i − 1.99304i
135135 0 0
136136 −2.96923 −0.254609
137137 −19.6659 −1.68017 −0.840083 0.542457i 0.817494π-0.817494\pi
−0.840083 + 0.542457i 0.817494π0.817494\pi
138138 0 0
139139 8.48528i 0.719712i 0.933008 + 0.359856i 0.117174π0.117174\pi
−0.933008 + 0.359856i 0.882826π0.882826\pi
140140 0 0
141141 0 0
142142 0 0
143143 0 0
144144 31.7722 2.64768
145145 0 0
146146 10.1064 0.836412
147147 −23.1562 −1.90989
148148 0 0
149149 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
150150 23.3913i 1.90989i
151151 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
152152 22.7414 1.84457
153153 − 8.33845i − 0.674124i
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 42.7982i 3.36254i
163163 −13.3445 −1.04522 −0.522612 0.852570i 0.675042π-0.675042\pi
−0.522612 + 0.852570i 0.675042π0.675042\pi
164164 4.42042i 0.345177i
165165 0 0
166166 22.9739 1.78312
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 63.8646i 4.88384i
172172 13.7769i 1.05048i
173173 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
174174 0 0
175175 0 0
176176 0 0
177177 −11.4406 −0.859929
178178 0.257836i 0.0193256i
179179 26.1999 1.95827 0.979135 0.203212i 0.0651381π-0.0651381\pi
0.979135 + 0.203212i 0.0651381π0.0651381\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 0 0
188188 0 0
189189 0 0
190190 0 0
191191 0 0 1.00000 00
−1.00000 π\pi
192192 −26.4642 −1.90989
193193 − 16.9706i − 1.22157i −0.791797 0.610784i 0.790854π-0.790854\pi
0.791797 0.610784i 0.209146π-0.209146\pi
194194 − 18.4552i − 1.32501i
195195 0 0
196196 14.0000 1.00000
197197 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 − 14.1421i − 1.00000i
201201 −53.9664 −3.80649
202202 0 0
203203 0 0
204204 6.94540i 0.486275i
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 0 0
210210 0 0
211211 21.1811i 1.45817i 0.684425 + 0.729083i 0.260053π0.260053\pi
−0.684425 + 0.729083i 0.739947π0.739947\pi
212212 0 0
213213 0 0
214214 16.7225 1.14312
215215 0 0
216216 − 46.2497i − 3.14690i
217217 0 0
218218 0 0
219219 − 23.6402i − 1.59746i
220220 0 0
221221 0 0
222222 0 0
223223 0 0 1.00000 00
−1.00000 π\pi
224224 0 0
225225 39.7152 2.64768
226226 11.1896i 0.744324i
227227 19.9218i 1.32226i 0.750273 + 0.661128i 0.229922π0.229922\pi
−0.750273 + 0.661128i 0.770078π0.770078\pi
228228 − 53.1951i − 3.52293i
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 0 0
232232 0 0
233233 − 30.2798i − 1.98369i −0.127438 0.991847i 0.540675π-0.540675\pi
0.127438 0.991847i 0.459325π-0.459325\pi
234234 0 0
235235 0 0
236236 6.91687 0.450250
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 − 29.0119i − 1.86882i −0.356199 0.934410i 0.615928π-0.615928\pi
0.356199 0.934410i 0.384072π-0.384072\pi
242242 0 0
243243 51.0552 3.27520
244244 0 0
245245 0 0
246246 10.3399 0.659250
247247 0 0
248248 0 0
249249 − 53.7389i − 3.40556i
250250 0 0
251251 6.00000 0.378717 0.189358 0.981908i 0.439359π-0.439359\pi
0.189358 + 0.981908i 0.439359π0.439359\pi
252252 0 0
253253 0 0
254254 0 0
255255 0 0
256256 16.0000 1.00000
257257 20.0305 1.24947 0.624734 0.780838i 0.285208π-0.285208\pi
0.624734 + 0.780838i 0.285208π0.285208\pi
258258 32.2260 2.00631
259259 0 0
260260 0 0
261261 0 0
262262 −31.1429 −1.92402
263263 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
264264 0 0
265265 0 0
266266 0 0
267267 0.603112 0.0369099
268268 32.6275 1.99304
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 − 4.19912i − 0.254609i
273273 0 0
274274 − 27.8117i − 1.68017i
275275 0 0
276276 0 0
277277 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
278278 −12.0000 −0.719712
279279 0 0
280280 0 0
281281 − 25.8593i − 1.54264i −0.636448 0.771319i 0.719597π-0.719597\pi
0.636448 0.771319i 0.280403π-0.280403\pi
282282 0 0
283283 25.4558i 1.51319i 0.653882 + 0.756596i 0.273139π0.273139\pi
−0.653882 + 0.756596i 0.726861π0.726861\pi
284284 0 0
285285 0 0
286286 0 0
287287 0 0
288288 44.9327i 2.64768i
289289 15.8980 0.935174
290290 0 0
291291 −43.1691 −2.53062
292292 14.2926i 0.836412i
293293 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
294294 − 32.7478i − 1.90989i
295295 0 0
296296 0 0
297297 0 0
298298 0 0
299299 0 0
300300 −33.0803 −1.90989
301301 0 0
302302 0 0
303303 0 0
304304 32.1612i 1.84457i
305305 0 0
306306 11.7924 0.674124
307307 34.9580i 1.99516i 0.0695319 + 0.997580i 0.477849π0.477849\pi
−0.0695319 + 0.997580i 0.522151π0.522151\pi
308308 0 0
309309 0 0
310310 0 0
311311 0 0 1.00000 00
−1.00000 π\pi
312312 0 0
313313 −11.8599 −0.670362 −0.335181 0.942154i 0.608798π-0.608798\pi
−0.335181 + 0.942154i 0.608798π0.608798\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 − 39.1160i − 2.18324i
322322 0 0
323323 8.44055 0.469645
324324 −60.5258 −3.36254
325325 0 0
326326 − 18.8720i − 1.04522i
327327 0 0
328328 −6.25142 −0.345177
329329 0 0
330330 0 0
331331 16.1755 0.889086 0.444543 0.895757i 0.353366π-0.353366\pi
0.444543 + 0.895757i 0.353366π0.353366\pi
332332 32.4900i 1.78312i
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 19.2300i 1.04752i 0.851865 + 0.523761i 0.175472π0.175472\pi
−0.851865 + 0.523761i 0.824528π0.824528\pi
338338 − 18.3848i − 1.00000i
339339 26.1740 1.42158
340340 0 0
341341 0 0
342342 −90.3181 −4.88384
343343 0 0
344344 −19.4835 −1.05048
345345 0 0
346346 0 0
347347 − 26.2205i − 1.40759i −0.710404 0.703795i 0.751488π-0.751488\pi
0.710404 0.703795i 0.248512π-0.248512\pi
348348 0 0
349349 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
350350 0 0
351351 0 0
352352 0 0
353353 37.5706 1.99968 0.999840 0.0178943i 0.00569624π-0.00569624\pi
0.999840 + 0.0178943i 0.00569624π0.00569624\pi
354354 − 16.1795i − 0.859929i
355355 0 0
356356 −0.364636 −0.0193256
357357 0 0
358358 37.0522i 1.95827i
359359 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
360360 0 0
361361 −45.6465 −2.40245
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 0 0
369369 − 17.5558i − 0.913919i
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 0 0
378378 0 0
379379 −3.67266 −0.188652 −0.0943260 0.995541i 0.530070π-0.530070\pi
−0.0943260 + 0.995541i 0.530070π0.530070\pi
380380 0 0
381381 0 0
382382 0 0
383383 0 0 1.00000 00
−1.00000 π\pi
384384 − 37.4261i − 1.90989i
385385 0 0
386386 24.0000 1.22157
387387 − 54.7155i − 2.78134i
388388 26.0996 1.32501
389389 0 0 1.00000 00
−1.00000 π\pi
390390 0 0
391391 0 0
392392 19.7990i 1.00000i
393393 72.8473i 3.67466i
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 20.0000 1.00000
401401 39.5140 1.97324 0.986618 0.163049i 0.0521329π-0.0521329\pi
0.986618 + 0.163049i 0.0521329π0.0521329\pi
402402 − 76.3200i − 3.80649i
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 0 0
408408 −9.82228 −0.486275
409409 33.9411i 1.67828i 0.543915 + 0.839140i 0.316941π0.316941\pi
−0.543915 + 0.839140i 0.683059π0.683059\pi
410410 0 0
411411 −65.0552 −3.20894
412412 0 0
413413 0 0
414414 0 0
415415 0 0
416416 0 0
417417 28.0695i 1.37457i
418418 0 0
419419 −29.4076 −1.43666 −0.718328 0.695705i 0.755092π-0.755092\pi
−0.718328 + 0.695705i 0.755092π0.755092\pi
420420 0 0
421421 0 0 1.00000 00
−1.00000 π\pi
422422 −29.9546 −1.45817
423423 0 0
424424 0 0
425425 − 5.24890i − 0.254609i
426426 0 0
427427 0 0
428428 23.6491i 1.14312i
429429 0 0
430430 0 0
431431 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
432432 65.4070 3.14690
433433 −20.7676 −0.998027 −0.499014 0.866594i 0.666304π-0.666304\pi
−0.499014 + 0.866594i 0.666304π0.666304\pi
434434 0 0
435435 0 0
436436 0 0
437437 0 0
438438 33.4323 1.59746
439439 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
440440 0 0
441441 −55.6013 −2.64768
442442 0 0
443443 −10.2887 −0.488832 −0.244416 0.969670i 0.578596π-0.578596\pi
−0.244416 + 0.969670i 0.578596π0.578596\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 0 0
449449 −30.6537 −1.44664 −0.723319 0.690514i 0.757384π-0.757384\pi
−0.723319 + 0.690514i 0.757384π0.757384\pi
450450 56.1658i 2.64768i
451451 0 0
452452 −15.8245 −0.744324
453453 0 0
454454 −28.1737 −1.32226
455455 0 0
456456 75.2293 3.52293
457457 35.2159i 1.64733i 0.567078 + 0.823664i 0.308074π0.308074\pi
−0.567078 + 0.823664i 0.691926π0.691926\pi
458458 0 0
459459 − 17.1657i − 0.801228i
460460 0 0
461461 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
462462 0 0
463463 0 0 1.00000 00
−1.00000 π\pi
464464 0 0
465465 0 0
466466 42.8220 1.98369
467467 30.0000 1.38823 0.694117 0.719862i 0.255795π-0.255795\pi
0.694117 + 0.719862i 0.255795π0.255795\pi
468468 0 0
469469 0 0
470470 0 0
471471 0 0
472472 9.78194i 0.450250i
473473 0 0
474474 0 0
475475 40.2015i 1.84457i
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 41.0290 1.86882
483483 0 0
484484 0 0
485485 0 0
486486 72.2030i 3.27520i
487487 0 0 1.00000 00
−1.00000 π\pi
488488 0 0
489489 −44.1441 −1.99627
490490 0 0
491491 44.3145i 1.99989i 0.0106338 + 0.999943i 0.496615π0.496615\pi
−0.0106338 + 0.999943i 0.503385π0.503385\pi
492492 14.6229i 0.659250i
493493 0 0
494494 0 0
495495 0 0
496496 0 0
497497 0 0
498498 75.9982 3.40556
499499 36.0237 1.61264 0.806321 0.591479i 0.201456π-0.201456\pi
0.806321 + 0.591479i 0.201456π0.201456\pi
500500 0 0
501501 0 0
502502 8.48528i 0.378717i
503503 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
504504 0 0
505505 0 0
506506 0 0
507507 −43.0044 −1.90989
508508 0 0
509509 0 0 1.00000 00
−1.00000 π\pi
510510 0 0
511511 0 0
512512 22.6274i 1.00000i
513513 131.473i 5.80468i
514514 28.3274i 1.24947i
515515 0 0
516516 45.5745i 2.00631i
517517 0 0
518518 0 0
519519 0 0
520520 0 0
521521 −21.7460 −0.952711 −0.476355 0.879253i 0.658042π-0.658042\pi
−0.476355 + 0.879253i 0.658042π0.658042\pi
522522 0 0
523523 − 1.74163i − 0.0761561i −0.999275 0.0380781i 0.987876π-0.987876\pi
0.999275 0.0380781i 0.0121236π-0.0121236\pi
524524 − 44.0427i − 1.92402i
525525 0 0
526526 0 0
527527 0 0
528528 0 0
529529 23.0000 1.00000
530530 0 0
531531 −27.4705 −1.19212
532532 0 0
533533 0 0
534534 0.852930i 0.0369099i
535535 0 0
536536 46.1423i 1.99304i
537537 86.6699 3.74008
538538 0 0
539539 0 0
540540 0 0
541541 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
542542 0 0
543543 0 0
544544 5.93845 0.254609
545545 0 0
546546 0 0
547547 33.9029i 1.44958i 0.688969 + 0.724791i 0.258064π0.258064\pi
−0.688969 + 0.724791i 0.741936π0.741936\pi
548548 39.3317 1.68017
549549 0 0
550550 0 0
551551 0 0
552552 0 0
553553 0 0
554554 0 0
555555 0 0
556556 − 16.9706i − 0.719712i
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 0 0
562562 36.5706 1.54264
563563 − 39.8941i − 1.68134i −0.541551 0.840668i 0.682163π-0.682163\pi
0.541551 0.840668i 0.317837π-0.317837\pi
564564 0 0
565565 0 0
566566 −36.0000 −1.51319
567567 0 0
568568 0 0
569569 − 42.9929i − 1.80236i −0.433447 0.901179i 0.642703π-0.642703\pi
0.433447 0.901179i 0.357297π-0.357297\pi
570570 0 0
571571 42.4264i 1.77549i 0.460336 + 0.887745i 0.347729π0.347729\pi
−0.460336 + 0.887745i 0.652271π0.652271\pi
572572 0 0
573573 0 0
574574 0 0
575575 0 0
576576 −63.5444 −2.64768
577577 −47.4567 −1.97565 −0.987824 0.155579i 0.950276π-0.950276\pi
−0.987824 + 0.155579i 0.950276π0.950276\pi
578578 22.4831i 0.935174i
579579 − 56.1391i − 2.33306i
580580 0 0
581581 0 0
582582 − 61.0503i − 2.53062i
583583 0 0
584584 −20.2128 −0.836412
585585 0 0
586586 0 0
587587 33.1167 1.36687 0.683437 0.730010i 0.260484π-0.260484\pi
0.683437 + 0.730010i 0.260484π0.260484\pi
588588 46.3124 1.90989
589589 0 0
590590 0 0
591591 0 0
592592 0 0
593593 47.1921i 1.93795i 0.247167 + 0.968973i 0.420500π0.420500\pi
−0.247167 + 0.968973i 0.579500π0.579500\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 − 46.7826i − 1.90989i
601601 − 48.9928i − 1.99846i −0.0392649 0.999229i 0.512502π-0.512502\pi
0.0392649 0.999229i 0.487498π-0.487498\pi
602602 0 0
603603 −129.581 −5.27695
604604 0 0
605605 0 0
606606 0 0
607607 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
608608 −45.4828 −1.84457
609609 0 0
610610 0 0
611611 0 0
612612 16.6769i 0.674124i
613613 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
614614 −49.4381 −1.99516
615615 0 0
616616 0 0
617617 0.995401 0.0400733 0.0200367 0.999799i 0.493622π-0.493622\pi
0.0200367 + 0.999799i 0.493622π0.493622\pi
618618 0 0
619619 −45.9721 −1.84777 −0.923887 0.382667i 0.875006π-0.875006\pi
−0.923887 + 0.382667i 0.875006π0.875006\pi
620620 0 0
621621 0 0
622622 0 0
623623 0 0
624624 0 0
625625 25.0000 1.00000
626626 − 16.7725i − 0.670362i
627627 0 0
628628 0 0
629629 0 0
630630 0 0
631631 0 0 1.00000 00
−1.00000 π\pi
632632 0 0
633633 70.0676i 2.78494i
634634 0 0
635635 0 0
636636 0 0
637637 0 0
638638 0 0
639639 0 0
640640 0 0
641641 −39.8787 −1.57511 −0.787556 0.616243i 0.788654π-0.788654\pi
−0.787556 + 0.616243i 0.788654π0.788654\pi
642642 55.3184 2.18324
643643 23.5208 0.927571 0.463786 0.885948i 0.346491π-0.346491\pi
0.463786 + 0.885948i 0.346491π0.346491\pi
644644 0 0
645645 0 0
646646 11.9367i 0.469645i
647647 0 0 1.00000 00
−1.00000 π\pi
648648 − 85.5964i − 3.36254i
649649 0 0
650650 0 0
651651 0 0
652652 26.6891 1.04522
653653 0 0 1.00000 00
−1.00000 π\pi
654654 0 0
655655 0 0
656656 − 8.84084i − 0.345177i
657657 − 56.7635i − 2.21455i
658658 0 0
659659 − 28.3200i − 1.10319i −0.834111 0.551596i 0.814019π-0.814019\pi
0.834111 0.551596i 0.185981π-0.185981\pi
660660 0 0
661661 0 0 1.00000 00
−1.00000 π\pi
662662 22.8756i 0.889086i
663663 0 0
664664 −45.9478 −1.78312
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 − 35.3106i − 1.36112i −0.732691 0.680561i 0.761736π-0.761736\pi
0.732691 0.680561i 0.238264π-0.238264\pi
674674 −27.1953 −1.04752
675675 81.7587 3.14690
676676 26.0000 1.00000
677677 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
678678 37.0157i 1.42158i
679679 0 0
680680 0 0
681681 65.9019i 2.52537i
682682 0 0
683683 −42.0000 −1.60709 −0.803543 0.595247i 0.797054π-0.797054\pi
−0.803543 + 0.595247i 0.797054π0.797054\pi
684684 − 127.729i − 4.88384i
685685 0 0
686686 0 0
687687 0 0
688688 − 27.5539i − 1.05048i
689689 0 0
690690 0 0
691691 22.2522 0.846514 0.423257 0.906010i 0.360887π-0.360887\pi
0.423257 + 0.906010i 0.360887π0.360887\pi
692692 0 0
693693 0 0
694694 37.0814 1.40759
695695 0 0
696696 0 0
697697 −2.32023 −0.0878852
698698 0 0
699699 − 100.166i − 3.78864i
700700 0 0
701701 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
702702 0 0
703703 0 0
704704 0 0
705705 0 0
706706 53.1328i 1.99968i
707707 0 0
708708 22.8812 0.859929
709709 0 0 1.00000 00
−1.00000 π\pi
710710 0 0
711711 0 0
712712 − 0.515673i − 0.0193256i
713713 0 0
714714 0 0
715715 0 0
716716 −52.3997 −1.95827
717717 0 0
718718 0 0
719719 0 0 1.00000 00
−1.00000 π\pi
720720 0 0
721721 0 0
722722 − 64.5540i − 2.40245i
723723 − 95.9721i − 3.56924i
724724 0 0
725725 0 0
726726 0 0
727727 0 0 1.00000 00
−1.00000 π\pi
728728 0 0
729729 78.1035 2.89272
730730 0 0
731731 −7.23137 −0.267462
732732 0 0
733733 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
734734 0 0
735735 0 0
736736 0 0
737737 0 0
738738 24.8277 0.913919
739739 14.3390i 0.527468i 0.964595 + 0.263734i 0.0849541π0.0849541\pi
−0.964595 + 0.263734i 0.915046π0.915046\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 0 0
746746 0 0
747747 − 129.035i − 4.72113i
748748 0 0
749749 0 0
750750 0 0
751751 0 0 1.00000 00
−1.00000 π\pi
752752 0 0
753753 19.8482 0.723307
754754 0 0
755755 0 0
756756 0 0
757757 0 0 1.00000 00
−1.00000 π\pi
758758 − 5.19393i − 0.188652i
759759 0 0
760760 0 0
761761 40.8934i 1.48238i 0.671293 + 0.741192i 0.265739π0.265739\pi
−0.671293 + 0.741192i 0.734261π0.734261\pi
762762 0 0
763763 0 0
764764 0 0
765765 0 0
766766 0 0
767767 0 0
768768 52.9284 1.90989
769769 − 50.9117i − 1.83592i −0.396670 0.917961i 0.629834π-0.629834\pi
0.396670 0.917961i 0.370166π-0.370166\pi
770770 0 0
771771 66.2614 2.38635
772772 33.9411i 1.22157i
773773 0 0 1.00000 00
−1.00000 π\pi
774774 77.3793 2.78134
775775 0 0
776776 36.9104i 1.32501i
777777 0 0
778778 0 0
779779 17.7708 0.636704
780780 0 0
781781 0 0
782782 0 0
783783 0 0
784784 −28.0000 −1.00000
785785 0 0
786786 −103.022 −3.67466
787787 49.9835i 1.78172i 0.454280 + 0.890859i 0.349897π0.349897\pi
−0.454280 + 0.890859i 0.650103π0.650103\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 28.2843i 1.00000i
801801 1.44816 0.0511682
802802 55.8813i 1.97324i
803803 0 0
804804 107.933 3.80649
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 49.2916i 1.73300i 0.499176 + 0.866501i 0.333636π0.333636\pi
−0.499176 + 0.866501i 0.666364π0.666364\pi
810810 0 0
811811 49.2506i 1.72942i 0.502268 + 0.864712i 0.332499π0.332499\pi
−0.502268 + 0.864712i 0.667501π0.667501\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 − 13.8908i − 0.486275i
817817 55.3854 1.93769
818818 −48.0000 −1.67828
819819 0 0
820820 0 0
821821 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
822822 − 92.0019i − 3.20894i
823823 0 0 1.00000 00
−1.00000 π\pi
824824 0 0
825825 0 0
826826 0 0
827827 15.7227i 0.546731i 0.961910 + 0.273366i 0.0881369π0.0881369\pi
−0.961910 + 0.273366i 0.911863π0.911863\pi
828828 0 0
829829 0 0 1.00000 00
−1.00000 π\pi
830830 0 0
831831 0 0
832832 0 0
833833 7.34846i 0.254609i
834834 −39.6963 −1.37457
835835 0 0
836836 0 0
837837 0 0
838838 − 41.5887i − 1.43666i
839839 0 0 1.00000 00
−1.00000 π\pi
840840 0 0
841841 −29.0000 −1.00000
842842 0 0
843843 − 85.5434i − 2.94627i
844844 − 42.3622i − 1.45817i
845845 0 0
846846 0 0
847847 0 0
848848 0 0
849849 84.2086i 2.89003i
850850 7.42306 0.254609
851851 0 0
852852 0 0
853853 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
854854 0 0
855855 0 0
856856 −33.4449 −1.14312
857857 − 58.3493i − 1.99317i −0.0825467 0.996587i 0.526305π-0.526305\pi
0.0825467 0.996587i 0.473695π-0.473695\pi
858858 0 0
859859 −51.9105 −1.77116 −0.885582 0.464483i 0.846240π-0.846240\pi
−0.885582 + 0.464483i 0.846240π0.846240\pi
860860 0 0
861861 0 0
862862 0 0
863863 0 0 1.00000 00
−1.00000 π\pi
864864 92.4994i 3.14690i
865865 0 0
866866 − 29.3698i − 0.998027i
867867 52.5909 1.78608
868868 0 0
869869 0 0
870870 0 0
871871 0 0
872872 0 0
873873 −103.655 −3.50820
874874 0 0
875875 0 0
876876 47.2804i 1.59746i
877877 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
878878 0 0
879879 0 0
880880 0 0
881881 59.3622 1.99996 0.999981 0.00609171i 0.00193906π-0.00193906\pi
0.999981 + 0.00609171i 0.00193906π0.00193906\pi
882882 − 78.6322i − 2.64768i
883883 −55.8718 −1.88024 −0.940119 0.340848i 0.889286π-0.889286\pi
−0.940119 + 0.340848i 0.889286π0.889286\pi
884884 0 0
885885 0 0
886886 − 14.5504i − 0.488832i
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 − 43.3509i − 1.44664i
899899 0 0
900900 −79.4305 −2.64768
901901 0 0
902902 0 0
903903 0 0
904904 − 22.3793i − 0.744324i
905905 0 0
906906 0 0
907907 43.0028 1.42789 0.713943 0.700204i 0.246908π-0.246908\pi
0.713943 + 0.700204i 0.246908π0.246908\pi
908908 − 39.8436i − 1.32226i
909909 0 0
910910 0 0
911911 0 0 1.00000 00
−1.00000 π\pi
912912 106.390i 3.52293i
913913 0 0
914914 −49.8027 −1.64733
915915 0 0
916916 0 0
917917 0 0
918918 24.2760 0.801228
919919 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
920920 0 0
921921 115.642i 3.81054i
922922 0 0
923923 0 0
924924 0 0
925925 0 0
926926 0 0
927927 0 0
928928 0 0
929929 60.3120 1.97877 0.989386 0.145310i 0.0464179π-0.0464179\pi
0.989386 + 0.145310i 0.0464179π0.0464179\pi
930930 0 0
931931 − 56.2822i − 1.84457i
932932 60.5595i 1.98369i
933933 0 0
934934 42.4264i 1.38823i
935935 0 0
936936 0 0
937937 − 61.1731i − 1.99844i −0.0395055 0.999219i 0.512578π-0.512578\pi
0.0395055 0.999219i 0.487422π-0.487422\pi
938938 0 0
939939 −39.2330 −1.28032
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 −13.8337 −0.450250
945945 0 0
946946 0 0
947947 55.8582 1.81515 0.907573 0.419894i 0.137933π-0.137933\pi
0.907573 + 0.419894i 0.137933π0.137933\pi
948948 0 0
949949 0 0
950950 −56.8536 −1.84457
951951 0 0
952952 0 0
953953 53.9289i 1.74693i 0.486889 + 0.873464i 0.338132π0.338132\pi
−0.486889 + 0.873464i 0.661868π0.661868\pi
954954 0 0
955955 0 0
956956 0 0
957957 0 0
958958 0 0
959959 0 0
960960 0 0
961961 31.0000 1.00000
962962 0 0
963963 − 93.9231i − 3.02663i
964964 58.0238i 1.86882i
965965 0 0
966966 0 0
967967 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
968968 0 0
969969 27.9216 0.896970
970970 0 0
971971 54.0000 1.73294 0.866471 0.499227i 0.166383π-0.166383\pi
0.866471 + 0.499227i 0.166383π0.166383\pi
972972 −102.110 −3.27520
973973 0 0
974974 0 0
975975 0 0
976976 0 0
977977 6.00000 0.191957 0.0959785 0.995383i 0.469402π-0.469402\pi
0.0959785 + 0.995383i 0.469402π0.469402\pi
978978 − 62.4292i − 1.99627i
979979 0 0
980980 0 0
981981 0 0
982982 −62.6702 −1.99989
983983 0 0 1.00000 00
−1.00000 π\pi
984984 −20.6799 −0.659250
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 53.5090 1.69806
994994 0 0
995995 0 0
996996 107.478i 3.40556i
997997 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
998998 50.9452i 1.61264i
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 968.2.g.a.483.8 8
4.3 odd 2 3872.2.g.b.1935.1 8
8.3 odd 2 CM 968.2.g.a.483.8 8
8.5 even 2 3872.2.g.b.1935.1 8
11.2 odd 10 968.2.k.d.403.1 8
11.3 even 5 88.2.k.a.35.2 8
11.4 even 5 968.2.k.b.699.1 8
11.5 even 5 968.2.k.d.723.1 8
11.6 odd 10 968.2.k.c.723.2 8
11.7 odd 10 88.2.k.a.83.2 yes 8
11.8 odd 10 968.2.k.b.475.1 8
11.9 even 5 968.2.k.c.403.2 8
11.10 odd 2 inner 968.2.g.a.483.4 8
33.14 odd 10 792.2.bp.a.739.1 8
33.29 even 10 792.2.bp.a.523.1 8
44.3 odd 10 352.2.s.a.79.2 8
44.7 even 10 352.2.s.a.303.2 8
44.43 even 2 3872.2.g.b.1935.2 8
88.3 odd 10 88.2.k.a.35.2 8
88.19 even 10 968.2.k.b.475.1 8
88.21 odd 2 3872.2.g.b.1935.2 8
88.27 odd 10 968.2.k.d.723.1 8
88.29 odd 10 352.2.s.a.303.2 8
88.35 even 10 968.2.k.d.403.1 8
88.43 even 2 inner 968.2.g.a.483.4 8
88.51 even 10 88.2.k.a.83.2 yes 8
88.59 odd 10 968.2.k.b.699.1 8
88.69 even 10 352.2.s.a.79.2 8
88.75 odd 10 968.2.k.c.403.2 8
88.83 even 10 968.2.k.c.723.2 8
264.179 even 10 792.2.bp.a.739.1 8
264.227 odd 10 792.2.bp.a.523.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.k.a.35.2 8 11.3 even 5
88.2.k.a.35.2 8 88.3 odd 10
88.2.k.a.83.2 yes 8 11.7 odd 10
88.2.k.a.83.2 yes 8 88.51 even 10
352.2.s.a.79.2 8 44.3 odd 10
352.2.s.a.79.2 8 88.69 even 10
352.2.s.a.303.2 8 44.7 even 10
352.2.s.a.303.2 8 88.29 odd 10
792.2.bp.a.523.1 8 33.29 even 10
792.2.bp.a.523.1 8 264.227 odd 10
792.2.bp.a.739.1 8 33.14 odd 10
792.2.bp.a.739.1 8 264.179 even 10
968.2.g.a.483.4 8 11.10 odd 2 inner
968.2.g.a.483.4 8 88.43 even 2 inner
968.2.g.a.483.8 8 1.1 even 1 trivial
968.2.g.a.483.8 8 8.3 odd 2 CM
968.2.k.b.475.1 8 11.8 odd 10
968.2.k.b.475.1 8 88.19 even 10
968.2.k.b.699.1 8 11.4 even 5
968.2.k.b.699.1 8 88.59 odd 10
968.2.k.c.403.2 8 11.9 even 5
968.2.k.c.403.2 8 88.75 odd 10
968.2.k.c.723.2 8 11.6 odd 10
968.2.k.c.723.2 8 88.83 even 10
968.2.k.d.403.1 8 11.2 odd 10
968.2.k.d.403.1 8 88.35 even 10
968.2.k.d.723.1 8 11.5 even 5
968.2.k.d.723.1 8 88.27 odd 10
3872.2.g.b.1935.1 8 4.3 odd 2
3872.2.g.b.1935.1 8 8.5 even 2
3872.2.g.b.1935.2 8 44.43 even 2
3872.2.g.b.1935.2 8 88.21 odd 2