Properties

Label 8100.2.a.w.1.2
Level 81008100
Weight 22
Character 8100.1
Self dual yes
Analytic conductor 64.67964.679
Analytic rank 11
Dimension 44
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8100,2,Mod(1,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 8100=223452 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 8100.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 64.678825637264.6788256372
Analytic rank: 11
Dimension: 44
Coefficient field: Q(3,7)\Q(\sqrt{3}, \sqrt{7})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x45x2+1 x^{4} - 5x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 3 3
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 0.456850-0.456850 of defining polynomial
Character χ\chi == 8100.1

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) == q2.79129q7+3.10260q112.79129q134.83465q17+3.58258q194.83465q23+3.46410q29+6.58258q31+3.37386q37+6.56670q412.79129q43+8.66025q47+0.791288q49+4.83465q538.29875q59+10.3739q615.00000q677.93725q715.00000q738.66025q774.00000q791.00905q8313.1334q89+7.79129q91+1.16515q97+O(q100)q-2.79129 q^{7} +3.10260 q^{11} -2.79129 q^{13} -4.83465 q^{17} +3.58258 q^{19} -4.83465 q^{23} +3.46410 q^{29} +6.58258 q^{31} +3.37386 q^{37} +6.56670 q^{41} -2.79129 q^{43} +8.66025 q^{47} +0.791288 q^{49} +4.83465 q^{53} -8.29875 q^{59} +10.3739 q^{61} -5.00000 q^{67} -7.93725 q^{71} -5.00000 q^{73} -8.66025 q^{77} -4.00000 q^{79} -1.00905 q^{83} -13.1334 q^{89} +7.79129 q^{91} +1.16515 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q2q72q134q19+8q3114q372q436q49+14q6120q6720q7316q79+22q9132q97+O(q100) 4 q - 2 q^{7} - 2 q^{13} - 4 q^{19} + 8 q^{31} - 14 q^{37} - 2 q^{43} - 6 q^{49} + 14 q^{61} - 20 q^{67} - 20 q^{73} - 16 q^{79} + 22 q^{91} - 32 q^{97}+O(q^{100}) Copy content Toggle raw display

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 0 0
66 0 0
77 −2.79129 −1.05501 −0.527504 0.849553i 0.676872π-0.676872\pi
−0.527504 + 0.849553i 0.676872π0.676872\pi
88 0 0
99 0 0
1010 0 0
1111 3.10260 0.935470 0.467735 0.883869i 0.345070π-0.345070\pi
0.467735 + 0.883869i 0.345070π0.345070\pi
1212 0 0
1313 −2.79129 −0.774164 −0.387082 0.922045i 0.626517π-0.626517\pi
−0.387082 + 0.922045i 0.626517π0.626517\pi
1414 0 0
1515 0 0
1616 0 0
1717 −4.83465 −1.17258 −0.586288 0.810103i 0.699411π-0.699411\pi
−0.586288 + 0.810103i 0.699411π0.699411\pi
1818 0 0
1919 3.58258 0.821899 0.410950 0.911658i 0.365197π-0.365197\pi
0.410950 + 0.911658i 0.365197π0.365197\pi
2020 0 0
2121 0 0
2222 0 0
2323 −4.83465 −1.00809 −0.504047 0.863676i 0.668156π-0.668156\pi
−0.504047 + 0.863676i 0.668156π0.668156\pi
2424 0 0
2525 0 0
2626 0 0
2727 0 0
2828 0 0
2929 3.46410 0.643268 0.321634 0.946864i 0.395768π-0.395768\pi
0.321634 + 0.946864i 0.395768π0.395768\pi
3030 0 0
3131 6.58258 1.18227 0.591133 0.806574i 0.298681π-0.298681\pi
0.591133 + 0.806574i 0.298681π0.298681\pi
3232 0 0
3333 0 0
3434 0 0
3535 0 0
3636 0 0
3737 3.37386 0.554660 0.277330 0.960775i 0.410551π-0.410551\pi
0.277330 + 0.960775i 0.410551π0.410551\pi
3838 0 0
3939 0 0
4040 0 0
4141 6.56670 1.02555 0.512773 0.858524i 0.328618π-0.328618\pi
0.512773 + 0.858524i 0.328618π0.328618\pi
4242 0 0
4343 −2.79129 −0.425667 −0.212834 0.977088i 0.568269π-0.568269\pi
−0.212834 + 0.977088i 0.568269π0.568269\pi
4444 0 0
4545 0 0
4646 0 0
4747 8.66025 1.26323 0.631614 0.775283i 0.282393π-0.282393\pi
0.631614 + 0.775283i 0.282393π0.282393\pi
4848 0 0
4949 0.791288 0.113041
5050 0 0
5151 0 0
5252 0 0
5353 4.83465 0.664091 0.332045 0.943263i 0.392261π-0.392261\pi
0.332045 + 0.943263i 0.392261π0.392261\pi
5454 0 0
5555 0 0
5656 0 0
5757 0 0
5858 0 0
5959 −8.29875 −1.08041 −0.540203 0.841535i 0.681653π-0.681653\pi
−0.540203 + 0.841535i 0.681653π0.681653\pi
6060 0 0
6161 10.3739 1.32824 0.664119 0.747627i 0.268807π-0.268807\pi
0.664119 + 0.747627i 0.268807π0.268807\pi
6262 0 0
6363 0 0
6464 0 0
6565 0 0
6666 0 0
6767 −5.00000 −0.610847 −0.305424 0.952217i 0.598798π-0.598798\pi
−0.305424 + 0.952217i 0.598798π0.598798\pi
6868 0 0
6969 0 0
7070 0 0
7171 −7.93725 −0.941979 −0.470989 0.882139i 0.656103π-0.656103\pi
−0.470989 + 0.882139i 0.656103π0.656103\pi
7272 0 0
7373 −5.00000 −0.585206 −0.292603 0.956234i 0.594521π-0.594521\pi
−0.292603 + 0.956234i 0.594521π0.594521\pi
7474 0 0
7575 0 0
7676 0 0
7777 −8.66025 −0.986928
7878 0 0
7979 −4.00000 −0.450035 −0.225018 0.974355i 0.572244π-0.572244\pi
−0.225018 + 0.974355i 0.572244π0.572244\pi
8080 0 0
8181 0 0
8282 0 0
8383 −1.00905 −0.110758 −0.0553789 0.998465i 0.517637π-0.517637\pi
−0.0553789 + 0.998465i 0.517637π0.517637\pi
8484 0 0
8585 0 0
8686 0 0
8787 0 0
8888 0 0
8989 −13.1334 −1.39214 −0.696069 0.717975i 0.745069π-0.745069\pi
−0.696069 + 0.717975i 0.745069π0.745069\pi
9090 0 0
9191 7.79129 0.816749
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 1.16515 0.118303 0.0591516 0.998249i 0.481160π-0.481160\pi
0.0591516 + 0.998249i 0.481160π0.481160\pi
9898 0 0
9999 0 0
100100 0 0
101101 −15.2270 −1.51514 −0.757569 0.652755i 0.773613π-0.773613\pi
−0.757569 + 0.652755i 0.773613π0.773613\pi
102102 0 0
103103 3.37386 0.332437 0.166218 0.986089i 0.446844π-0.446844\pi
0.166218 + 0.986089i 0.446844π0.446844\pi
104104 0 0
105105 0 0
106106 0 0
107107 −8.66025 −0.837218 −0.418609 0.908166i 0.637482π-0.637482\pi
−0.418609 + 0.908166i 0.637482π0.637482\pi
108108 0 0
109109 −16.9564 −1.62413 −0.812066 0.583565i 0.801657π-0.801657\pi
−0.812066 + 0.583565i 0.801657π0.801657\pi
110110 0 0
111111 0 0
112112 0 0
113113 5.84370 0.549729 0.274865 0.961483i 0.411367π-0.411367\pi
0.274865 + 0.961483i 0.411367π0.411367\pi
114114 0 0
115115 0 0
116116 0 0
117117 0 0
118118 0 0
119119 13.4949 1.23708
120120 0 0
121121 −1.37386 −0.124897
122122 0 0
123123 0 0
124124 0 0
125125 0 0
126126 0 0
127127 −21.7477 −1.92980 −0.964899 0.262620i 0.915413π-0.915413\pi
−0.964899 + 0.262620i 0.915413π0.915413\pi
128128 0 0
129129 0 0
130130 0 0
131131 0.723000 0.0631688 0.0315844 0.999501i 0.489945π-0.489945\pi
0.0315844 + 0.999501i 0.489945π0.489945\pi
132132 0 0
133133 −10.0000 −0.867110
134134 0 0
135135 0 0
136136 0 0
137137 22.1552 1.89284 0.946422 0.322934i 0.104669π-0.104669\pi
0.946422 + 0.322934i 0.104669π0.104669\pi
138138 0 0
139139 −15.3739 −1.30399 −0.651997 0.758222i 0.726069π-0.726069\pi
−0.651997 + 0.758222i 0.726069π0.726069\pi
140140 0 0
141141 0 0
142142 0 0
143143 −8.66025 −0.724207
144144 0 0
145145 0 0
146146 0 0
147147 0 0
148148 0 0
149149 10.6784 0.874805 0.437402 0.899266i 0.355899π-0.355899\pi
0.437402 + 0.899266i 0.355899π0.355899\pi
150150 0 0
151151 2.62614 0.213712 0.106856 0.994275i 0.465922π-0.465922\pi
0.106856 + 0.994275i 0.465922π0.465922\pi
152152 0 0
153153 0 0
154154 0 0
155155 0 0
156156 0 0
157157 −9.41742 −0.751592 −0.375796 0.926702i 0.622631π-0.622631\pi
−0.375796 + 0.926702i 0.622631π0.622631\pi
158158 0 0
159159 0 0
160160 0 0
161161 13.4949 1.06355
162162 0 0
163163 −0.582576 −0.0456309 −0.0228154 0.999740i 0.507263π-0.507263\pi
−0.0228154 + 0.999740i 0.507263π0.507263\pi
164164 0 0
165165 0 0
166166 0 0
167167 −12.4859 −0.966185 −0.483092 0.875569i 0.660486π-0.660486\pi
−0.483092 + 0.875569i 0.660486π0.660486\pi
168168 0 0
169169 −5.20871 −0.400670
170170 0 0
171171 0 0
172172 0 0
173173 19.3386 1.47029 0.735144 0.677911i 0.237115π-0.237115\pi
0.735144 + 0.677911i 0.237115π0.237115\pi
174174 0 0
175175 0 0
176176 0 0
177177 0 0
178178 0 0
179179 14.1425 1.05706 0.528528 0.848916i 0.322744π-0.322744\pi
0.528528 + 0.848916i 0.322744π0.322744\pi
180180 0 0
181181 6.58258 0.489279 0.244639 0.969614i 0.421330π-0.421330\pi
0.244639 + 0.969614i 0.421330π0.421330\pi
182182 0 0
183183 0 0
184184 0 0
185185 0 0
186186 0 0
187187 −15.0000 −1.09691
188188 0 0
189189 0 0
190190 0 0
191191 −20.0616 −1.45161 −0.725804 0.687902i 0.758532π-0.758532\pi
−0.725804 + 0.687902i 0.758532π0.758532\pi
192192 0 0
193193 −21.7477 −1.56544 −0.782718 0.622377i 0.786167π-0.786167\pi
−0.782718 + 0.622377i 0.786167π0.786167\pi
194194 0 0
195195 0 0
196196 0 0
197197 15.5130 1.10526 0.552628 0.833428i 0.313625π-0.313625\pi
0.552628 + 0.833428i 0.313625π0.313625\pi
198198 0 0
199199 −6.20871 −0.440124 −0.220062 0.975486i 0.570626π-0.570626\pi
−0.220062 + 0.975486i 0.570626π0.570626\pi
200200 0 0
201201 0 0
202202 0 0
203203 −9.66930 −0.678652
204204 0 0
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 11.1153 0.768862
210210 0 0
211211 3.41742 0.235265 0.117633 0.993057i 0.462469π-0.462469\pi
0.117633 + 0.993057i 0.462469π0.462469\pi
212212 0 0
213213 0 0
214214 0 0
215215 0 0
216216 0 0
217217 −18.3739 −1.24730
218218 0 0
219219 0 0
220220 0 0
221221 13.4949 0.907766
222222 0 0
223223 16.1652 1.08250 0.541249 0.840862i 0.317952π-0.317952\pi
0.541249 + 0.840862i 0.317952π0.317952\pi
224224 0 0
225225 0 0
226226 0 0
227227 25.9808 1.72440 0.862202 0.506565i 0.169085π-0.169085\pi
0.862202 + 0.506565i 0.169085π0.169085\pi
228228 0 0
229229 −5.41742 −0.357993 −0.178997 0.983850i 0.557285π-0.557285\pi
−0.178997 + 0.983850i 0.557285π0.557285\pi
230230 0 0
231231 0 0
232232 0 0
233233 −10.6784 −0.699562 −0.349781 0.936831i 0.613744π-0.613744\pi
−0.349781 + 0.936831i 0.613744π0.613744\pi
234234 0 0
235235 0 0
236236 0 0
237237 0 0
238238 0 0
239239 −26.9898 −1.74583 −0.872913 0.487876i 0.837772π-0.837772\pi
−0.872913 + 0.487876i 0.837772π0.837772\pi
240240 0 0
241241 −23.9129 −1.54036 −0.770182 0.637824i 0.779835π-0.779835\pi
−0.770182 + 0.637824i 0.779835π0.779835\pi
242242 0 0
243243 0 0
244244 0 0
245245 0 0
246246 0 0
247247 −10.0000 −0.636285
248248 0 0
249249 0 0
250250 0 0
251251 24.1733 1.52580 0.762901 0.646515i 0.223774π-0.223774\pi
0.762901 + 0.646515i 0.223774π0.223774\pi
252252 0 0
253253 −15.0000 −0.943042
254254 0 0
255255 0 0
256256 0 0
257257 −26.9898 −1.68358 −0.841789 0.539806i 0.818497π-0.818497\pi
−0.841789 + 0.539806i 0.818497π0.818497\pi
258258 0 0
259259 −9.41742 −0.585170
260260 0 0
261261 0 0
262262 0 0
263263 26.9898 1.66426 0.832132 0.554578i 0.187120π-0.187120\pi
0.832132 + 0.554578i 0.187120π0.187120\pi
264264 0 0
265265 0 0
266266 0 0
267267 0 0
268268 0 0
269269 −7.28970 −0.444461 −0.222231 0.974994i 0.571334π-0.571334\pi
−0.222231 + 0.974994i 0.571334π0.571334\pi
270270 0 0
271271 0.252273 0.0153245 0.00766224 0.999971i 0.497561π-0.497561\pi
0.00766224 + 0.999971i 0.497561π0.497561\pi
272272 0 0
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 −32.3303 −1.94254 −0.971270 0.237981i 0.923514π-0.923514\pi
−0.971270 + 0.237981i 0.923514π0.923514\pi
278278 0 0
279279 0 0
280280 0 0
281281 −16.3115 −0.973060 −0.486530 0.873664i 0.661738π-0.661738\pi
−0.486530 + 0.873664i 0.661738π0.661738\pi
282282 0 0
283283 24.5390 1.45869 0.729347 0.684144i 0.239824π-0.239824\pi
0.729347 + 0.684144i 0.239824π0.239824\pi
284284 0 0
285285 0 0
286286 0 0
287287 −18.3296 −1.08196
288288 0 0
289289 6.37386 0.374933
290290 0 0
291291 0 0
292292 0 0
293293 −18.3296 −1.07082 −0.535412 0.844591i 0.679844π-0.679844\pi
−0.535412 + 0.844591i 0.679844π0.679844\pi
294294 0 0
295295 0 0
296296 0 0
297297 0 0
298298 0 0
299299 13.4949 0.780431
300300 0 0
301301 7.79129 0.449082
302302 0 0
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 −3.25227 −0.185617 −0.0928085 0.995684i 0.529584π-0.529584\pi
−0.0928085 + 0.995684i 0.529584π0.529584\pi
308308 0 0
309309 0 0
310310 0 0
311311 5.91915 0.335644 0.167822 0.985817i 0.446327π-0.446327\pi
0.167822 + 0.985817i 0.446327π0.446327\pi
312312 0 0
313313 −7.20871 −0.407461 −0.203730 0.979027i 0.565307π-0.565307\pi
−0.203730 + 0.979027i 0.565307π0.565307\pi
314314 0 0
315315 0 0
316316 0 0
317317 16.3115 0.916143 0.458071 0.888915i 0.348540π-0.348540\pi
0.458071 + 0.888915i 0.348540π0.348540\pi
318318 0 0
319319 10.7477 0.601757
320320 0 0
321321 0 0
322322 0 0
323323 −17.3205 −0.963739
324324 0 0
325325 0 0
326326 0 0
327327 0 0
328328 0 0
329329 −24.1733 −1.33272
330330 0 0
331331 30.7477 1.69005 0.845024 0.534728i 0.179586π-0.179586\pi
0.845024 + 0.534728i 0.179586π0.179586\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 −32.3303 −1.76114 −0.880572 0.473913i 0.842841π-0.842841\pi
−0.880572 + 0.473913i 0.842841π0.842841\pi
338338 0 0
339339 0 0
340340 0 0
341341 20.4231 1.10597
342342 0 0
343343 17.3303 0.935748
344344 0 0
345345 0 0
346346 0 0
347347 −3.82560 −0.205369 −0.102685 0.994714i 0.532743π-0.532743\pi
−0.102685 + 0.994714i 0.532743π0.532743\pi
348348 0 0
349349 −25.4955 −1.36474 −0.682370 0.731007i 0.739051π-0.739051\pi
−0.682370 + 0.731007i 0.739051π0.739051\pi
350350 0 0
351351 0 0
352352 0 0
353353 −25.1823 −1.34032 −0.670160 0.742217i 0.733774π-0.733774\pi
−0.670160 + 0.742217i 0.733774π0.733774\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 0 0
359359 −2.37960 −0.125591 −0.0627953 0.998026i 0.520002π-0.520002\pi
−0.0627953 + 0.998026i 0.520002π0.520002\pi
360360 0 0
361361 −6.16515 −0.324482
362362 0 0
363363 0 0
364364 0 0
365365 0 0
366366 0 0
367367 −13.3739 −0.698110 −0.349055 0.937102i 0.613497π-0.613497\pi
−0.349055 + 0.937102i 0.613497π0.613497\pi
368368 0 0
369369 0 0
370370 0 0
371371 −13.4949 −0.700621
372372 0 0
373373 −13.8348 −0.716341 −0.358171 0.933656i 0.616599π-0.616599\pi
−0.358171 + 0.933656i 0.616599π0.616599\pi
374374 0 0
375375 0 0
376376 0 0
377377 −9.66930 −0.497995
378378 0 0
379379 −28.7913 −1.47891 −0.739455 0.673206i 0.764917π-0.764917\pi
−0.739455 + 0.673206i 0.764917π0.764917\pi
380380 0 0
381381 0 0
382382 0 0
383383 5.84370 0.298599 0.149300 0.988792i 0.452298π-0.452298\pi
0.149300 + 0.988792i 0.452298π0.452298\pi
384384 0 0
385385 0 0
386386 0 0
387387 0 0
388388 0 0
389389 16.3115 0.827024 0.413512 0.910499i 0.364302π-0.364302\pi
0.413512 + 0.910499i 0.364302π0.364302\pi
390390 0 0
391391 23.3739 1.18207
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 0 0
397397 −13.8348 −0.694351 −0.347176 0.937800i 0.612859π-0.612859\pi
−0.347176 + 0.937800i 0.612859π0.612859\pi
398398 0 0
399399 0 0
400400 0 0
401401 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
402402 0 0
403403 −18.3739 −0.915267
404404 0 0
405405 0 0
406406 0 0
407407 10.4678 0.518867
408408 0 0
409409 −32.2867 −1.59648 −0.798238 0.602342i 0.794234π-0.794234\pi
−0.798238 + 0.602342i 0.794234π0.794234\pi
410410 0 0
411411 0 0
412412 0 0
413413 23.1642 1.13984
414414 0 0
415415 0 0
416416 0 0
417417 0 0
418418 0 0
419419 −13.4949 −0.659269 −0.329635 0.944109i 0.606926π-0.606926\pi
−0.329635 + 0.944109i 0.606926π0.606926\pi
420420 0 0
421421 29.4955 1.43752 0.718760 0.695258i 0.244710π-0.244710\pi
0.718760 + 0.695258i 0.244710π0.244710\pi
422422 0 0
423423 0 0
424424 0 0
425425 0 0
426426 0 0
427427 −28.9564 −1.40130
428428 0 0
429429 0 0
430430 0 0
431431 −15.5885 −0.750870 −0.375435 0.926849i 0.622507π-0.622507\pi
−0.375435 + 0.926849i 0.622507π0.622507\pi
432432 0 0
433433 5.58258 0.268281 0.134141 0.990962i 0.457173π-0.457173\pi
0.134141 + 0.990962i 0.457173π0.457173\pi
434434 0 0
435435 0 0
436436 0 0
437437 −17.3205 −0.828552
438438 0 0
439439 12.1216 0.578532 0.289266 0.957249i 0.406589π-0.406589\pi
0.289266 + 0.957249i 0.406589π0.406589\pi
440440 0 0
441441 0 0
442442 0 0
443443 31.8245 1.51203 0.756013 0.654557i 0.227145π-0.227145\pi
0.756013 + 0.654557i 0.227145π0.227145\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 0 0
449449 −25.6193 −1.20905 −0.604524 0.796587i 0.706637π-0.706637\pi
−0.604524 + 0.796587i 0.706637π0.706637\pi
450450 0 0
451451 20.3739 0.959368
452452 0 0
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 −3.25227 −0.152135 −0.0760675 0.997103i 0.524236π-0.524236\pi
−0.0760675 + 0.997103i 0.524236π0.524236\pi
458458 0 0
459459 0 0
460460 0 0
461461 −30.0924 −1.40154 −0.700772 0.713386i 0.747161π-0.747161\pi
−0.700772 + 0.713386i 0.747161π0.747161\pi
462462 0 0
463463 −15.1216 −0.702760 −0.351380 0.936233i 0.614287π-0.614287\pi
−0.351380 + 0.936233i 0.614287π0.614287\pi
464464 0 0
465465 0 0
466466 0 0
467467 −36.6591 −1.69638 −0.848191 0.529691i 0.822308π-0.822308\pi
−0.848191 + 0.529691i 0.822308π0.822308\pi
468468 0 0
469469 13.9564 0.644448
470470 0 0
471471 0 0
472472 0 0
473473 −8.66025 −0.398199
474474 0 0
475475 0 0
476476 0 0
477477 0 0
478478 0 0
479479 10.0308 0.458319 0.229160 0.973389i 0.426402π-0.426402\pi
0.229160 + 0.973389i 0.426402π0.426402\pi
480480 0 0
481481 −9.41742 −0.429398
482482 0 0
483483 0 0
484484 0 0
485485 0 0
486486 0 0
487487 22.3303 1.01188 0.505941 0.862568i 0.331145π-0.331145\pi
0.505941 + 0.862568i 0.331145π0.331145\pi
488488 0 0
489489 0 0
490490 0 0
491491 30.7400 1.38728 0.693638 0.720324i 0.256007π-0.256007\pi
0.693638 + 0.720324i 0.256007π0.256007\pi
492492 0 0
493493 −16.7477 −0.754280
494494 0 0
495495 0 0
496496 0 0
497497 22.1552 0.993795
498498 0 0
499499 −19.6261 −0.878587 −0.439293 0.898344i 0.644771π-0.644771\pi
−0.439293 + 0.898344i 0.644771π0.644771\pi
500500 0 0
501501 0 0
502502 0 0
503503 −12.4859 −0.556717 −0.278358 0.960477i 0.589790π-0.589790\pi
−0.278358 + 0.960477i 0.589790π0.589790\pi
504504 0 0
505505 0 0
506506 0 0
507507 0 0
508508 0 0
509509 10.6784 0.473310 0.236655 0.971594i 0.423949π-0.423949\pi
0.236655 + 0.971594i 0.423949π0.423949\pi
510510 0 0
511511 13.9564 0.617397
512512 0 0
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 26.8693 1.18171
518518 0 0
519519 0 0
520520 0 0
521521 −21.3567 −0.935654 −0.467827 0.883820i 0.654963π-0.654963\pi
−0.467827 + 0.883820i 0.654963π0.654963\pi
522522 0 0
523523 −21.7477 −0.950962 −0.475481 0.879726i 0.657726π-0.657726\pi
−0.475481 + 0.879726i 0.657726π0.657726\pi
524524 0 0
525525 0 0
526526 0 0
527527 −31.8245 −1.38630
528528 0 0
529529 0.373864 0.0162549
530530 0 0
531531 0 0
532532 0 0
533533 −18.3296 −0.793941
534534 0 0
535535 0 0
536536 0 0
537537 0 0
538538 0 0
539539 2.45505 0.105747
540540 0 0
541541 11.7913 0.506947 0.253474 0.967342i 0.418427π-0.418427\pi
0.253474 + 0.967342i 0.418427π0.418427\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 11.7477 0.502297 0.251148 0.967949i 0.419192π-0.419192\pi
0.251148 + 0.967949i 0.419192π0.419192\pi
548548 0 0
549549 0 0
550550 0 0
551551 12.4104 0.528701
552552 0 0
553553 11.1652 0.474791
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 7.79129 0.329536
560560 0 0
561561 0 0
562562 0 0
563563 8.66025 0.364986 0.182493 0.983207i 0.441583π-0.441583\pi
0.182493 + 0.983207i 0.441583π0.441583\pi
564564 0 0
565565 0 0
566566 0 0
567567 0 0
568568 0 0
569569 33.1950 1.39161 0.695804 0.718232i 0.255048π-0.255048\pi
0.695804 + 0.718232i 0.255048π0.255048\pi
570570 0 0
571571 1.66970 0.0698747 0.0349373 0.999390i 0.488877π-0.488877\pi
0.0349373 + 0.999390i 0.488877π0.488877\pi
572572 0 0
573573 0 0
574574 0 0
575575 0 0
576576 0 0
577577 −38.4955 −1.60259 −0.801293 0.598272i 0.795854π-0.795854\pi
−0.801293 + 0.598272i 0.795854π0.795854\pi
578578 0 0
579579 0 0
580580 0 0
581581 2.81655 0.116850
582582 0 0
583583 15.0000 0.621237
584584 0 0
585585 0 0
586586 0 0
587587 47.3375 1.95383 0.976913 0.213636i 0.0685307π-0.0685307\pi
0.976913 + 0.213636i 0.0685307π0.0685307\pi
588588 0 0
589589 23.5826 0.971703
590590 0 0
591591 0 0
592592 0 0
593593 30.8154 1.26544 0.632719 0.774382i 0.281939π-0.281939\pi
0.632719 + 0.774382i 0.281939π0.281939\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 0 0
599599 19.9862 0.816612 0.408306 0.912845i 0.366120π-0.366120\pi
0.408306 + 0.912845i 0.366120π0.366120\pi
600600 0 0
601601 −9.70417 −0.395841 −0.197921 0.980218i 0.563419π-0.563419\pi
−0.197921 + 0.980218i 0.563419π0.563419\pi
602602 0 0
603603 0 0
604604 0 0
605605 0 0
606606 0 0
607607 26.7477 1.08566 0.542828 0.839844i 0.317353π-0.317353\pi
0.542828 + 0.839844i 0.317353π0.317353\pi
608608 0 0
609609 0 0
610610 0 0
611611 −24.1733 −0.977945
612612 0 0
613613 −36.7477 −1.48423 −0.742113 0.670274i 0.766176π-0.766176\pi
−0.742113 + 0.670274i 0.766176π0.766176\pi
614614 0 0
615615 0 0
616616 0 0
617617 −18.3296 −0.737920 −0.368960 0.929445i 0.620286π-0.620286\pi
−0.368960 + 0.929445i 0.620286π0.620286\pi
618618 0 0
619619 −20.7477 −0.833922 −0.416961 0.908924i 0.636905π-0.636905\pi
−0.416961 + 0.908924i 0.636905π0.636905\pi
620620 0 0
621621 0 0
622622 0 0
623623 36.6591 1.46872
624624 0 0
625625 0 0
626626 0 0
627627 0 0
628628 0 0
629629 −16.3115 −0.650380
630630 0 0
631631 19.7042 0.784410 0.392205 0.919878i 0.371712π-0.371712\pi
0.392205 + 0.919878i 0.371712π0.371712\pi
632632 0 0
633633 0 0
634634 0 0
635635 0 0
636636 0 0
637637 −2.20871 −0.0875124
638638 0 0
639639 0 0
640640 0 0
641641 27.9234 1.10291 0.551454 0.834205i 0.314073π-0.314073\pi
0.551454 + 0.834205i 0.314073π0.314073\pi
642642 0 0
643643 −46.8693 −1.84835 −0.924173 0.381975i 0.875244π-0.875244\pi
−0.924173 + 0.381975i 0.875244π0.875244\pi
644644 0 0
645645 0 0
646646 0 0
647647 20.3477 0.799949 0.399975 0.916526i 0.369019π-0.369019\pi
0.399975 + 0.916526i 0.369019π0.369019\pi
648648 0 0
649649 −25.7477 −1.01069
650650 0 0
651651 0 0
652652 0 0
653653 −23.1642 −0.906486 −0.453243 0.891387i 0.649733π-0.649733\pi
−0.453243 + 0.891387i 0.649733π0.649733\pi
654654 0 0
655655 0 0
656656 0 0
657657 0 0
658658 0 0
659659 −24.1733 −0.941657 −0.470828 0.882225i 0.656045π-0.656045\pi
−0.470828 + 0.882225i 0.656045π0.656045\pi
660660 0 0
661661 27.7477 1.07926 0.539631 0.841902i 0.318564π-0.318564\pi
0.539631 + 0.841902i 0.318564π0.318564\pi
662662 0 0
663663 0 0
664664 0 0
665665 0 0
666666 0 0
667667 −16.7477 −0.648475
668668 0 0
669669 0 0
670670 0 0
671671 32.1860 1.24253
672672 0 0
673673 −35.0000 −1.34915 −0.674575 0.738206i 0.735673π-0.735673\pi
−0.674575 + 0.738206i 0.735673π0.735673\pi
674674 0 0
675675 0 0
676676 0 0
677677 20.3477 0.782024 0.391012 0.920386i 0.372125π-0.372125\pi
0.391012 + 0.920386i 0.372125π0.372125\pi
678678 0 0
679679 −3.25227 −0.124811
680680 0 0
681681 0 0
682682 0 0
683683 −20.1371 −0.770523 −0.385262 0.922807i 0.625889π-0.625889\pi
−0.385262 + 0.922807i 0.625889π0.625889\pi
684684 0 0
685685 0 0
686686 0 0
687687 0 0
688688 0 0
689689 −13.4949 −0.514115
690690 0 0
691691 −10.4955 −0.399266 −0.199633 0.979871i 0.563975π-0.563975\pi
−0.199633 + 0.979871i 0.563975π0.563975\pi
692692 0 0
693693 0 0
694694 0 0
695695 0 0
696696 0 0
697697 −31.7477 −1.20253
698698 0 0
699699 0 0
700700 0 0
701701 14.5794 0.550657 0.275328 0.961350i 0.411213π-0.411213\pi
0.275328 + 0.961350i 0.411213π0.411213\pi
702702 0 0
703703 12.0871 0.455874
704704 0 0
705705 0 0
706706 0 0
707707 42.5028 1.59848
708708 0 0
709709 28.0780 1.05449 0.527246 0.849713i 0.323225π-0.323225\pi
0.527246 + 0.849713i 0.323225π0.323225\pi
710710 0 0
711711 0 0
712712 0 0
713713 −31.8245 −1.19184
714714 0 0
715715 0 0
716716 0 0
717717 0 0
718718 0 0
719719 9.02175 0.336455 0.168227 0.985748i 0.446196π-0.446196\pi
0.168227 + 0.985748i 0.446196π0.446196\pi
720720 0 0
721721 −9.41742 −0.350723
722722 0 0
723723 0 0
724724 0 0
725725 0 0
726726 0 0
727727 3.37386 0.125130 0.0625648 0.998041i 0.480072π-0.480072\pi
0.0625648 + 0.998041i 0.480072π0.480072\pi
728728 0 0
729729 0 0
730730 0 0
731731 13.4949 0.499127
732732 0 0
733733 5.12159 0.189170 0.0945851 0.995517i 0.469848π-0.469848\pi
0.0945851 + 0.995517i 0.469848π0.469848\pi
734734 0 0
735735 0 0
736736 0 0
737737 −15.5130 −0.571429
738738 0 0
739739 −33.8693 −1.24590 −0.622951 0.782260i 0.714067π-0.714067\pi
−0.622951 + 0.782260i 0.714067π0.714067\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 0 0
748748 0 0
749749 24.1733 0.883272
750750 0 0
751751 18.1216 0.661266 0.330633 0.943759i 0.392738π-0.392738\pi
0.330633 + 0.943759i 0.392738π0.392738\pi
752752 0 0
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 18.3739 0.667809 0.333905 0.942607i 0.391634π-0.391634\pi
0.333905 + 0.942607i 0.391634π0.391634\pi
758758 0 0
759759 0 0
760760 0 0
761761 −34.5656 −1.25300 −0.626500 0.779421i 0.715513π-0.715513\pi
−0.626500 + 0.779421i 0.715513π0.715513\pi
762762 0 0
763763 47.3303 1.71347
764764 0 0
765765 0 0
766766 0 0
767767 23.1642 0.836411
768768 0 0
769769 16.5390 0.596412 0.298206 0.954502i 0.403612π-0.403612\pi
0.298206 + 0.954502i 0.403612π0.403612\pi
770770 0 0
771771 0 0
772772 0 0
773773 6.64215 0.238902 0.119451 0.992840i 0.461887π-0.461887\pi
0.119451 + 0.992840i 0.461887π0.461887\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 23.5257 0.842896
780780 0 0
781781 −24.6261 −0.881192
782782 0 0
783783 0 0
784784 0 0
785785 0 0
786786 0 0
787787 3.37386 0.120265 0.0601326 0.998190i 0.480848π-0.480848\pi
0.0601326 + 0.998190i 0.480848π0.480848\pi
788788 0 0
789789 0 0
790790 0 0
791791 −16.3115 −0.579969
792792 0 0
793793 −28.9564 −1.02827
794794 0 0
795795 0 0
796796 0 0
797797 44.5209 1.57701 0.788506 0.615027i 0.210855π-0.210855\pi
0.788506 + 0.615027i 0.210855π0.210855\pi
798798 0 0
799799 −41.8693 −1.48123
800800 0 0
801801 0 0
802802 0 0
803803 −15.5130 −0.547442
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 −5.19615 −0.182687 −0.0913435 0.995819i 0.529116π-0.529116\pi
−0.0913435 + 0.995819i 0.529116π0.529116\pi
810810 0 0
811811 44.9564 1.57863 0.789317 0.613986i 0.210435π-0.210435\pi
0.789317 + 0.613986i 0.210435π0.210435\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 −10.0000 −0.349856
818818 0 0
819819 0 0
820820 0 0
821821 −56.2838 −1.96432 −0.982159 0.188054i 0.939782π-0.939782\pi
−0.982159 + 0.188054i 0.939782π0.939782\pi
822822 0 0
823823 33.8348 1.17941 0.589704 0.807619i 0.299244π-0.299244\pi
0.589704 + 0.807619i 0.299244π0.299244\pi
824824 0 0
825825 0 0
826826 0 0
827827 7.86180 0.273382 0.136691 0.990614i 0.456353π-0.456353\pi
0.136691 + 0.990614i 0.456353π0.456353\pi
828828 0 0
829829 41.4955 1.44120 0.720598 0.693353i 0.243867π-0.243867\pi
0.720598 + 0.693353i 0.243867π0.243867\pi
830830 0 0
831831 0 0
832832 0 0
833833 −3.82560 −0.132549
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 −13.1334 −0.453416 −0.226708 0.973963i 0.572796π-0.572796\pi
−0.226708 + 0.973963i 0.572796π0.572796\pi
840840 0 0
841841 −17.0000 −0.586207
842842 0 0
843843 0 0
844844 0 0
845845 0 0
846846 0 0
847847 3.83485 0.131767
848848 0 0
849849 0 0
850850 0 0
851851 −16.3115 −0.559150
852852 0 0
853853 −5.00000 −0.171197 −0.0855984 0.996330i 0.527280π-0.527280\pi
−0.0855984 + 0.996330i 0.527280π0.527280\pi
854854 0 0
855855 0 0
856856 0 0
857857 −42.5028 −1.45187 −0.725934 0.687764i 0.758592π-0.758592\pi
−0.725934 + 0.687764i 0.758592π0.758592\pi
858858 0 0
859859 0.252273 0.00860744 0.00430372 0.999991i 0.498630π-0.498630\pi
0.00430372 + 0.999991i 0.498630π0.498630\pi
860860 0 0
861861 0 0
862862 0 0
863863 −30.8154 −1.04897 −0.524484 0.851420i 0.675742π-0.675742\pi
−0.524484 + 0.851420i 0.675742π0.675742\pi
864864 0 0
865865 0 0
866866 0 0
867867 0 0
868868 0 0
869869 −12.4104 −0.420994
870870 0 0
871871 13.9564 0.472896
872872 0 0
873873 0 0
874874 0 0
875875 0 0
876876 0 0
877877 −2.79129 −0.0942551 −0.0471275 0.998889i 0.515007π-0.515007\pi
−0.0471275 + 0.998889i 0.515007π0.515007\pi
878878 0 0
879879 0 0
880880 0 0
881881 36.9452 1.24471 0.622357 0.782733i 0.286175π-0.286175\pi
0.622357 + 0.782733i 0.286175π0.286175\pi
882882 0 0
883883 34.6606 1.16642 0.583211 0.812321i 0.301796π-0.301796\pi
0.583211 + 0.812321i 0.301796π0.301796\pi
884884 0 0
885885 0 0
886886 0 0
887887 −23.1642 −0.777778 −0.388889 0.921285i 0.627141π-0.627141\pi
−0.388889 + 0.921285i 0.627141π0.627141\pi
888888 0 0
889889 60.7042 2.03595
890890 0 0
891891 0 0
892892 0 0
893893 31.0260 1.03825
894894 0 0
895895 0 0
896896 0 0
897897 0 0
898898 0 0
899899 22.8027 0.760513
900900 0 0
901901 −23.3739 −0.778696
902902 0 0
903903 0 0
904904 0 0
905905 0 0
906906 0 0
907907 1.62614 0.0539950 0.0269975 0.999636i 0.491405π-0.491405\pi
0.0269975 + 0.999636i 0.491405π0.491405\pi
908908 0 0
909909 0 0
910910 0 0
911911 −0.286051 −0.00947728 −0.00473864 0.999989i 0.501508π-0.501508\pi
−0.00473864 + 0.999989i 0.501508π0.501508\pi
912912 0 0
913913 −3.13068 −0.103610
914914 0 0
915915 0 0
916916 0 0
917917 −2.01810 −0.0666436
918918 0 0
919919 −13.6261 −0.449485 −0.224742 0.974418i 0.572154π-0.572154\pi
−0.224742 + 0.974418i 0.572154π0.572154\pi
920920 0 0
921921 0 0
922922 0 0
923923 22.1552 0.729246
924924 0 0
925925 0 0
926926 0 0
927927 0 0
928928 0 0
929929 14.1425 0.463999 0.231999 0.972716i 0.425473π-0.425473\pi
0.231999 + 0.972716i 0.425473π0.425473\pi
930930 0 0
931931 2.83485 0.0929084
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 −9.41742 −0.307654 −0.153827 0.988098i 0.549160π-0.549160\pi
−0.153827 + 0.988098i 0.549160π0.549160\pi
938938 0 0
939939 0 0
940940 0 0
941941 −2.81655 −0.0918169 −0.0459085 0.998946i 0.514618π-0.514618\pi
−0.0459085 + 0.998946i 0.514618π0.514618\pi
942942 0 0
943943 −31.7477 −1.03385
944944 0 0
945945 0 0
946946 0 0
947947 −57.0068 −1.85247 −0.926236 0.376945i 0.876975π-0.876975\pi
−0.926236 + 0.376945i 0.876975π0.876975\pi
948948 0 0
949949 13.9564 0.453045
950950 0 0
951951 0 0
952952 0 0
953953 53.9796 1.74857 0.874286 0.485412i 0.161330π-0.161330\pi
0.874286 + 0.485412i 0.161330π0.161330\pi
954954 0 0
955955 0 0
956956 0 0
957957 0 0
958958 0 0
959959 −61.8414 −1.99696
960960 0 0
961961 12.3303 0.397752
962962 0 0
963963 0 0
964964 0 0
965965 0 0
966966 0 0
967967 8.25227 0.265375 0.132688 0.991158i 0.457639π-0.457639\pi
0.132688 + 0.991158i 0.457639π0.457639\pi
968968 0 0
969969 0 0
970970 0 0
971971 −5.55765 −0.178354 −0.0891768 0.996016i 0.528424π-0.528424\pi
−0.0891768 + 0.996016i 0.528424π0.528424\pi
972972 0 0
973973 42.9129 1.37572
974974 0 0
975975 0 0
976976 0 0
977977 −35.8607 −1.14728 −0.573642 0.819106i 0.694470π-0.694470\pi
−0.573642 + 0.819106i 0.694470π0.694470\pi
978978 0 0
979979 −40.7477 −1.30230
980980 0 0
981981 0 0
982982 0 0
983983 −22.3658 −0.713357 −0.356679 0.934227i 0.616091π-0.616091\pi
−0.356679 + 0.934227i 0.616091π0.616091\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 13.4949 0.429113
990990 0 0
991991 −41.1216 −1.30627 −0.653135 0.757241i 0.726547π-0.726547\pi
−0.653135 + 0.757241i 0.726547π0.726547\pi
992992 0 0
993993 0 0
994994 0 0
995995 0 0
996996 0 0
997997 −6.74773 −0.213703 −0.106851 0.994275i 0.534077π-0.534077\pi
−0.106851 + 0.994275i 0.534077π0.534077\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 8100.2.a.w.1.2 yes 4
3.2 odd 2 inner 8100.2.a.w.1.1 4
5.2 odd 4 8100.2.d.r.649.2 8
5.3 odd 4 8100.2.d.r.649.8 8
5.4 even 2 8100.2.a.bb.1.4 yes 4
15.2 even 4 8100.2.d.r.649.1 8
15.8 even 4 8100.2.d.r.649.7 8
15.14 odd 2 8100.2.a.bb.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8100.2.a.w.1.1 4 3.2 odd 2 inner
8100.2.a.w.1.2 yes 4 1.1 even 1 trivial
8100.2.a.bb.1.3 yes 4 15.14 odd 2
8100.2.a.bb.1.4 yes 4 5.4 even 2
8100.2.d.r.649.1 8 15.2 even 4
8100.2.d.r.649.2 8 5.2 odd 4
8100.2.d.r.649.7 8 15.8 even 4
8100.2.d.r.649.8 8 5.3 odd 4