Properties

Label 9576.2.a.bi.1.2
Level 95769576
Weight 22
Character 9576.1
Self dual yes
Analytic conductor 76.46576.465
Analytic rank 11
Dimension 22
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9576,2,Mod(1,9576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("9576.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 9576=2332719 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 9576.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: 76.464744975676.4647449756
Analytic rank: 11
Dimension: 22
Coefficient field: Q(ζ10)+\Q(\zeta_{10})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x1 x^{2} - x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 2
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 0.618034-0.618034 of defining polynomial
Character χ\chi == 9576.1

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) == q+1.23607q5+1.00000q72.00000q114.47214q131.23607q171.00000q19+2.00000q233.47214q25+5.70820q291.52786q31+1.23607q35+8.47214q378.47214q41+8.00000q43+1.70820q47+1.00000q49+11.2361q532.47214q558.94427q5913.4164q615.52786q651.52786q67+7.70820q71+12.4721q732.00000q77+4.00000q7913.7082q831.52786q85+4.47214q894.47214q911.23607q9517.4164q97+O(q100)q+1.23607 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.47214 q^{13} -1.23607 q^{17} -1.00000 q^{19} +2.00000 q^{23} -3.47214 q^{25} +5.70820 q^{29} -1.52786 q^{31} +1.23607 q^{35} +8.47214 q^{37} -8.47214 q^{41} +8.00000 q^{43} +1.70820 q^{47} +1.00000 q^{49} +11.2361 q^{53} -2.47214 q^{55} -8.94427 q^{59} -13.4164 q^{61} -5.52786 q^{65} -1.52786 q^{67} +7.70820 q^{71} +12.4721 q^{73} -2.00000 q^{77} +4.00000 q^{79} -13.7082 q^{83} -1.52786 q^{85} +4.47214 q^{89} -4.47214 q^{91} -1.23607 q^{95} -17.4164 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q5+2q74q11+2q172q19+4q23+2q252q2912q312q35+8q378q41+16q4310q47+2q49+18q53+4q5520q65+8q97+O(q100) 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 16 q^{43} - 10 q^{47} + 2 q^{49} + 18 q^{53} + 4 q^{55} - 20 q^{65}+ \cdots - 8 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 1.23607 0.552786 0.276393 0.961045i 0.410861π-0.410861\pi
0.276393 + 0.961045i 0.410861π0.410861\pi
66 0 0
77 1.00000 0.377964
88 0 0
99 0 0
1010 0 0
1111 −2.00000 −0.603023 −0.301511 0.953463i 0.597491π-0.597491\pi
−0.301511 + 0.953463i 0.597491π0.597491\pi
1212 0 0
1313 −4.47214 −1.24035 −0.620174 0.784465i 0.712938π-0.712938\pi
−0.620174 + 0.784465i 0.712938π0.712938\pi
1414 0 0
1515 0 0
1616 0 0
1717 −1.23607 −0.299791 −0.149895 0.988702i 0.547894π-0.547894\pi
−0.149895 + 0.988702i 0.547894π0.547894\pi
1818 0 0
1919 −1.00000 −0.229416
2020 0 0
2121 0 0
2222 0 0
2323 2.00000 0.417029 0.208514 0.978019i 0.433137π-0.433137\pi
0.208514 + 0.978019i 0.433137π0.433137\pi
2424 0 0
2525 −3.47214 −0.694427
2626 0 0
2727 0 0
2828 0 0
2929 5.70820 1.05999 0.529993 0.848002i 0.322194π-0.322194\pi
0.529993 + 0.848002i 0.322194π0.322194\pi
3030 0 0
3131 −1.52786 −0.274412 −0.137206 0.990543i 0.543812π-0.543812\pi
−0.137206 + 0.990543i 0.543812π0.543812\pi
3232 0 0
3333 0 0
3434 0 0
3535 1.23607 0.208934
3636 0 0
3737 8.47214 1.39281 0.696405 0.717649i 0.254782π-0.254782\pi
0.696405 + 0.717649i 0.254782π0.254782\pi
3838 0 0
3939 0 0
4040 0 0
4141 −8.47214 −1.32313 −0.661563 0.749890i 0.730106π-0.730106\pi
−0.661563 + 0.749890i 0.730106π0.730106\pi
4242 0 0
4343 8.00000 1.21999 0.609994 0.792406i 0.291172π-0.291172\pi
0.609994 + 0.792406i 0.291172π0.291172\pi
4444 0 0
4545 0 0
4646 0 0
4747 1.70820 0.249167 0.124584 0.992209i 0.460241π-0.460241\pi
0.124584 + 0.992209i 0.460241π0.460241\pi
4848 0 0
4949 1.00000 0.142857
5050 0 0
5151 0 0
5252 0 0
5353 11.2361 1.54339 0.771696 0.635991i 0.219409π-0.219409\pi
0.771696 + 0.635991i 0.219409π0.219409\pi
5454 0 0
5555 −2.47214 −0.333343
5656 0 0
5757 0 0
5858 0 0
5959 −8.94427 −1.16445 −0.582223 0.813029i 0.697817π-0.697817\pi
−0.582223 + 0.813029i 0.697817π0.697817\pi
6060 0 0
6161 −13.4164 −1.71780 −0.858898 0.512148i 0.828850π-0.828850\pi
−0.858898 + 0.512148i 0.828850π0.828850\pi
6262 0 0
6363 0 0
6464 0 0
6565 −5.52786 −0.685647
6666 0 0
6767 −1.52786 −0.186658 −0.0933292 0.995635i 0.529751π-0.529751\pi
−0.0933292 + 0.995635i 0.529751π0.529751\pi
6868 0 0
6969 0 0
7070 0 0
7171 7.70820 0.914796 0.457398 0.889262i 0.348782π-0.348782\pi
0.457398 + 0.889262i 0.348782π0.348782\pi
7272 0 0
7373 12.4721 1.45975 0.729877 0.683579i 0.239578π-0.239578\pi
0.729877 + 0.683579i 0.239578π0.239578\pi
7474 0 0
7575 0 0
7676 0 0
7777 −2.00000 −0.227921
7878 0 0
7979 4.00000 0.450035 0.225018 0.974355i 0.427756π-0.427756\pi
0.225018 + 0.974355i 0.427756π0.427756\pi
8080 0 0
8181 0 0
8282 0 0
8383 −13.7082 −1.50467 −0.752335 0.658780i 0.771073π-0.771073\pi
−0.752335 + 0.658780i 0.771073π0.771073\pi
8484 0 0
8585 −1.52786 −0.165720
8686 0 0
8787 0 0
8888 0 0
8989 4.47214 0.474045 0.237023 0.971504i 0.423828π-0.423828\pi
0.237023 + 0.971504i 0.423828π0.423828\pi
9090 0 0
9191 −4.47214 −0.468807
9292 0 0
9393 0 0
9494 0 0
9595 −1.23607 −0.126818
9696 0 0
9797 −17.4164 −1.76837 −0.884184 0.467139i 0.845285π-0.845285\pi
−0.884184 + 0.467139i 0.845285π0.845285\pi
9898 0 0
9999 0 0
100100 0 0
101101 −10.7639 −1.07105 −0.535526 0.844519i 0.679886π-0.679886\pi
−0.535526 + 0.844519i 0.679886π0.679886\pi
102102 0 0
103103 4.00000 0.394132 0.197066 0.980390i 0.436859π-0.436859\pi
0.197066 + 0.980390i 0.436859π0.436859\pi
104104 0 0
105105 0 0
106106 0 0
107107 −18.1803 −1.75756 −0.878780 0.477227i 0.841642π-0.841642\pi
−0.878780 + 0.477227i 0.841642π0.841642\pi
108108 0 0
109109 10.0000 0.957826 0.478913 0.877862i 0.341031π-0.341031\pi
0.478913 + 0.877862i 0.341031π0.341031\pi
110110 0 0
111111 0 0
112112 0 0
113113 −7.23607 −0.680712 −0.340356 0.940297i 0.610548π-0.610548\pi
−0.340356 + 0.940297i 0.610548π0.610548\pi
114114 0 0
115115 2.47214 0.230528
116116 0 0
117117 0 0
118118 0 0
119119 −1.23607 −0.113310
120120 0 0
121121 −7.00000 −0.636364
122122 0 0
123123 0 0
124124 0 0
125125 −10.4721 −0.936656
126126 0 0
127127 −15.4164 −1.36798 −0.683992 0.729489i 0.739758π-0.739758\pi
−0.683992 + 0.729489i 0.739758π0.739758\pi
128128 0 0
129129 0 0
130130 0 0
131131 −15.2361 −1.33118 −0.665591 0.746317i 0.731820π-0.731820\pi
−0.665591 + 0.746317i 0.731820π0.731820\pi
132132 0 0
133133 −1.00000 −0.0867110
134134 0 0
135135 0 0
136136 0 0
137137 −4.00000 −0.341743 −0.170872 0.985293i 0.554658π-0.554658\pi
−0.170872 + 0.985293i 0.554658π0.554658\pi
138138 0 0
139139 9.52786 0.808143 0.404071 0.914727i 0.367595π-0.367595\pi
0.404071 + 0.914727i 0.367595π0.367595\pi
140140 0 0
141141 0 0
142142 0 0
143143 8.94427 0.747958
144144 0 0
145145 7.05573 0.585946
146146 0 0
147147 0 0
148148 0 0
149149 13.5279 1.10825 0.554123 0.832435i 0.313054π-0.313054\pi
0.554123 + 0.832435i 0.313054π0.313054\pi
150150 0 0
151151 4.00000 0.325515 0.162758 0.986666i 0.447961π-0.447961\pi
0.162758 + 0.986666i 0.447961π0.447961\pi
152152 0 0
153153 0 0
154154 0 0
155155 −1.88854 −0.151691
156156 0 0
157157 −3.52786 −0.281554 −0.140777 0.990041i 0.544960π-0.544960\pi
−0.140777 + 0.990041i 0.544960π0.544960\pi
158158 0 0
159159 0 0
160160 0 0
161161 2.00000 0.157622
162162 0 0
163163 −16.0000 −1.25322 −0.626608 0.779334i 0.715557π-0.715557\pi
−0.626608 + 0.779334i 0.715557π0.715557\pi
164164 0 0
165165 0 0
166166 0 0
167167 −23.4164 −1.81202 −0.906008 0.423261i 0.860886π-0.860886\pi
−0.906008 + 0.423261i 0.860886π0.860886\pi
168168 0 0
169169 7.00000 0.538462
170170 0 0
171171 0 0
172172 0 0
173173 −20.4721 −1.55647 −0.778234 0.627975i 0.783884π-0.783884\pi
−0.778234 + 0.627975i 0.783884π0.783884\pi
174174 0 0
175175 −3.47214 −0.262469
176176 0 0
177177 0 0
178178 0 0
179179 −7.70820 −0.576138 −0.288069 0.957610i 0.593013π-0.593013\pi
−0.288069 + 0.957610i 0.593013π0.593013\pi
180180 0 0
181181 −3.52786 −0.262224 −0.131112 0.991368i 0.541855π-0.541855\pi
−0.131112 + 0.991368i 0.541855π0.541855\pi
182182 0 0
183183 0 0
184184 0 0
185185 10.4721 0.769927
186186 0 0
187187 2.47214 0.180780
188188 0 0
189189 0 0
190190 0 0
191191 −10.0000 −0.723575 −0.361787 0.932261i 0.617833π-0.617833\pi
−0.361787 + 0.932261i 0.617833π0.617833\pi
192192 0 0
193193 −8.47214 −0.609838 −0.304919 0.952378i 0.598629π-0.598629\pi
−0.304919 + 0.952378i 0.598629π0.598629\pi
194194 0 0
195195 0 0
196196 0 0
197197 −12.0000 −0.854965 −0.427482 0.904024i 0.640599π-0.640599\pi
−0.427482 + 0.904024i 0.640599π0.640599\pi
198198 0 0
199199 −20.9443 −1.48470 −0.742350 0.670012i 0.766289π-0.766289\pi
−0.742350 + 0.670012i 0.766289π0.766289\pi
200200 0 0
201201 0 0
202202 0 0
203203 5.70820 0.400637
204204 0 0
205205 −10.4721 −0.731406
206206 0 0
207207 0 0
208208 0 0
209209 2.00000 0.138343
210210 0 0
211211 −6.47214 −0.445560 −0.222780 0.974869i 0.571513π-0.571513\pi
−0.222780 + 0.974869i 0.571513π0.571513\pi
212212 0 0
213213 0 0
214214 0 0
215215 9.88854 0.674393
216216 0 0
217217 −1.52786 −0.103718
218218 0 0
219219 0 0
220220 0 0
221221 5.52786 0.371844
222222 0 0
223223 −13.5279 −0.905893 −0.452946 0.891538i 0.649627π-0.649627\pi
−0.452946 + 0.891538i 0.649627π0.649627\pi
224224 0 0
225225 0 0
226226 0 0
227227 5.52786 0.366897 0.183449 0.983029i 0.441274π-0.441274\pi
0.183449 + 0.983029i 0.441274π0.441274\pi
228228 0 0
229229 14.9443 0.987545 0.493773 0.869591i 0.335618π-0.335618\pi
0.493773 + 0.869591i 0.335618π0.335618\pi
230230 0 0
231231 0 0
232232 0 0
233233 7.41641 0.485865 0.242933 0.970043i 0.421891π-0.421891\pi
0.242933 + 0.970043i 0.421891π0.421891\pi
234234 0 0
235235 2.11146 0.137736
236236 0 0
237237 0 0
238238 0 0
239239 13.4164 0.867835 0.433918 0.900953i 0.357131π-0.357131\pi
0.433918 + 0.900953i 0.357131π0.357131\pi
240240 0 0
241241 21.4164 1.37955 0.689776 0.724023i 0.257709π-0.257709\pi
0.689776 + 0.724023i 0.257709π0.257709\pi
242242 0 0
243243 0 0
244244 0 0
245245 1.23607 0.0789695
246246 0 0
247247 4.47214 0.284555
248248 0 0
249249 0 0
250250 0 0
251251 12.7639 0.805652 0.402826 0.915277i 0.368028π-0.368028\pi
0.402826 + 0.915277i 0.368028π0.368028\pi
252252 0 0
253253 −4.00000 −0.251478
254254 0 0
255255 0 0
256256 0 0
257257 2.94427 0.183659 0.0918293 0.995775i 0.470729π-0.470729\pi
0.0918293 + 0.995775i 0.470729π0.470729\pi
258258 0 0
259259 8.47214 0.526433
260260 0 0
261261 0 0
262262 0 0
263263 −20.4721 −1.26237 −0.631183 0.775634i 0.717430π-0.717430\pi
−0.631183 + 0.775634i 0.717430π0.717430\pi
264264 0 0
265265 13.8885 0.853166
266266 0 0
267267 0 0
268268 0 0
269269 −2.00000 −0.121942 −0.0609711 0.998140i 0.519420π-0.519420\pi
−0.0609711 + 0.998140i 0.519420π0.519420\pi
270270 0 0
271271 7.41641 0.450515 0.225257 0.974299i 0.427678π-0.427678\pi
0.225257 + 0.974299i 0.427678π0.427678\pi
272272 0 0
273273 0 0
274274 0 0
275275 6.94427 0.418755
276276 0 0
277277 21.4164 1.28679 0.643394 0.765536i 0.277526π-0.277526\pi
0.643394 + 0.765536i 0.277526π0.277526\pi
278278 0 0
279279 0 0
280280 0 0
281281 −11.2361 −0.670288 −0.335144 0.942167i 0.608785π-0.608785\pi
−0.335144 + 0.942167i 0.608785π0.608785\pi
282282 0 0
283283 0.944272 0.0561311 0.0280656 0.999606i 0.491065π-0.491065\pi
0.0280656 + 0.999606i 0.491065π0.491065\pi
284284 0 0
285285 0 0
286286 0 0
287287 −8.47214 −0.500094
288288 0 0
289289 −15.4721 −0.910126
290290 0 0
291291 0 0
292292 0 0
293293 −7.52786 −0.439783 −0.219891 0.975524i 0.570570π-0.570570\pi
−0.219891 + 0.975524i 0.570570π0.570570\pi
294294 0 0
295295 −11.0557 −0.643689
296296 0 0
297297 0 0
298298 0 0
299299 −8.94427 −0.517261
300300 0 0
301301 8.00000 0.461112
302302 0 0
303303 0 0
304304 0 0
305305 −16.5836 −0.949574
306306 0 0
307307 −17.5279 −1.00037 −0.500184 0.865919i 0.666734π-0.666734\pi
−0.500184 + 0.865919i 0.666734π0.666734\pi
308308 0 0
309309 0 0
310310 0 0
311311 −1.34752 −0.0764111 −0.0382055 0.999270i 0.512164π-0.512164\pi
−0.0382055 + 0.999270i 0.512164π0.512164\pi
312312 0 0
313313 5.41641 0.306153 0.153077 0.988214i 0.451082π-0.451082\pi
0.153077 + 0.988214i 0.451082π0.451082\pi
314314 0 0
315315 0 0
316316 0 0
317317 14.6525 0.822965 0.411483 0.911418i 0.365011π-0.365011\pi
0.411483 + 0.911418i 0.365011π0.365011\pi
318318 0 0
319319 −11.4164 −0.639196
320320 0 0
321321 0 0
322322 0 0
323323 1.23607 0.0687767
324324 0 0
325325 15.5279 0.861331
326326 0 0
327327 0 0
328328 0 0
329329 1.70820 0.0941763
330330 0 0
331331 −11.0557 −0.607678 −0.303839 0.952723i 0.598268π-0.598268\pi
−0.303839 + 0.952723i 0.598268π0.598268\pi
332332 0 0
333333 0 0
334334 0 0
335335 −1.88854 −0.103182
336336 0 0
337337 −6.94427 −0.378279 −0.189139 0.981950i 0.560570π-0.560570\pi
−0.189139 + 0.981950i 0.560570π0.560570\pi
338338 0 0
339339 0 0
340340 0 0
341341 3.05573 0.165477
342342 0 0
343343 1.00000 0.0539949
344344 0 0
345345 0 0
346346 0 0
347347 −30.3607 −1.62985 −0.814923 0.579569i 0.803221π-0.803221\pi
−0.814923 + 0.579569i 0.803221π0.803221\pi
348348 0 0
349349 34.9443 1.87052 0.935262 0.353956i 0.115164π-0.115164\pi
0.935262 + 0.353956i 0.115164π0.115164\pi
350350 0 0
351351 0 0
352352 0 0
353353 −30.1803 −1.60634 −0.803169 0.595752i 0.796854π-0.796854\pi
−0.803169 + 0.595752i 0.796854π0.796854\pi
354354 0 0
355355 9.52786 0.505687
356356 0 0
357357 0 0
358358 0 0
359359 −14.9443 −0.788729 −0.394364 0.918954i 0.629035π-0.629035\pi
−0.394364 + 0.918954i 0.629035π0.629035\pi
360360 0 0
361361 1.00000 0.0526316
362362 0 0
363363 0 0
364364 0 0
365365 15.4164 0.806932
366366 0 0
367367 20.9443 1.09328 0.546641 0.837367i 0.315906π-0.315906\pi
0.546641 + 0.837367i 0.315906π0.315906\pi
368368 0 0
369369 0 0
370370 0 0
371371 11.2361 0.583348
372372 0 0
373373 31.8885 1.65113 0.825563 0.564310i 0.190858π-0.190858\pi
0.825563 + 0.564310i 0.190858π0.190858\pi
374374 0 0
375375 0 0
376376 0 0
377377 −25.5279 −1.31475
378378 0 0
379379 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
380380 0 0
381381 0 0
382382 0 0
383383 −1.52786 −0.0780702 −0.0390351 0.999238i 0.512428π-0.512428\pi
−0.0390351 + 0.999238i 0.512428π0.512428\pi
384384 0 0
385385 −2.47214 −0.125992
386386 0 0
387387 0 0
388388 0 0
389389 −24.3607 −1.23514 −0.617568 0.786518i 0.711882π-0.711882\pi
−0.617568 + 0.786518i 0.711882π0.711882\pi
390390 0 0
391391 −2.47214 −0.125021
392392 0 0
393393 0 0
394394 0 0
395395 4.94427 0.248773
396396 0 0
397397 −19.8885 −0.998177 −0.499089 0.866551i 0.666332π-0.666332\pi
−0.499089 + 0.866551i 0.666332π0.666332\pi
398398 0 0
399399 0 0
400400 0 0
401401 −4.76393 −0.237899 −0.118950 0.992900i 0.537953π-0.537953\pi
−0.118950 + 0.992900i 0.537953π0.537953\pi
402402 0 0
403403 6.83282 0.340367
404404 0 0
405405 0 0
406406 0 0
407407 −16.9443 −0.839896
408408 0 0
409409 35.3050 1.74572 0.872859 0.487973i 0.162264π-0.162264\pi
0.872859 + 0.487973i 0.162264π0.162264\pi
410410 0 0
411411 0 0
412412 0 0
413413 −8.94427 −0.440119
414414 0 0
415415 −16.9443 −0.831762
416416 0 0
417417 0 0
418418 0 0
419419 −27.5967 −1.34819 −0.674095 0.738645i 0.735466π-0.735466\pi
−0.674095 + 0.738645i 0.735466π0.735466\pi
420420 0 0
421421 15.5279 0.756782 0.378391 0.925646i 0.376478π-0.376478\pi
0.378391 + 0.925646i 0.376478π0.376478\pi
422422 0 0
423423 0 0
424424 0 0
425425 4.29180 0.208183
426426 0 0
427427 −13.4164 −0.649265
428428 0 0
429429 0 0
430430 0 0
431431 28.6525 1.38014 0.690071 0.723742i 0.257579π-0.257579\pi
0.690071 + 0.723742i 0.257579π0.257579\pi
432432 0 0
433433 10.5836 0.508615 0.254307 0.967123i 0.418152π-0.418152\pi
0.254307 + 0.967123i 0.418152π0.418152\pi
434434 0 0
435435 0 0
436436 0 0
437437 −2.00000 −0.0956730
438438 0 0
439439 12.3607 0.589943 0.294972 0.955506i 0.404690π-0.404690\pi
0.294972 + 0.955506i 0.404690π0.404690\pi
440440 0 0
441441 0 0
442442 0 0
443443 13.4164 0.637433 0.318716 0.947850i 0.396748π-0.396748\pi
0.318716 + 0.947850i 0.396748π0.396748\pi
444444 0 0
445445 5.52786 0.262046
446446 0 0
447447 0 0
448448 0 0
449449 23.2361 1.09658 0.548289 0.836289i 0.315279π-0.315279\pi
0.548289 + 0.836289i 0.315279π0.315279\pi
450450 0 0
451451 16.9443 0.797875
452452 0 0
453453 0 0
454454 0 0
455455 −5.52786 −0.259150
456456 0 0
457457 21.4164 1.00182 0.500909 0.865500i 0.332999π-0.332999\pi
0.500909 + 0.865500i 0.332999π0.332999\pi
458458 0 0
459459 0 0
460460 0 0
461461 28.0689 1.30730 0.653649 0.756798i 0.273237π-0.273237\pi
0.653649 + 0.756798i 0.273237π0.273237\pi
462462 0 0
463463 −41.8885 −1.94673 −0.973363 0.229270i 0.926366π-0.926366\pi
−0.973363 + 0.229270i 0.926366π0.926366\pi
464464 0 0
465465 0 0
466466 0 0
467467 20.7639 0.960840 0.480420 0.877039i 0.340484π-0.340484\pi
0.480420 + 0.877039i 0.340484π0.340484\pi
468468 0 0
469469 −1.52786 −0.0705502
470470 0 0
471471 0 0
472472 0 0
473473 −16.0000 −0.735681
474474 0 0
475475 3.47214 0.159313
476476 0 0
477477 0 0
478478 0 0
479479 −25.7082 −1.17464 −0.587319 0.809356i 0.699817π-0.699817\pi
−0.587319 + 0.809356i 0.699817π0.699817\pi
480480 0 0
481481 −37.8885 −1.72757
482482 0 0
483483 0 0
484484 0 0
485485 −21.5279 −0.977530
486486 0 0
487487 −39.4164 −1.78613 −0.893064 0.449930i 0.851449π-0.851449\pi
−0.893064 + 0.449930i 0.851449π0.851449\pi
488488 0 0
489489 0 0
490490 0 0
491491 31.3050 1.41277 0.706386 0.707826i 0.250324π-0.250324\pi
0.706386 + 0.707826i 0.250324π0.250324\pi
492492 0 0
493493 −7.05573 −0.317774
494494 0 0
495495 0 0
496496 0 0
497497 7.70820 0.345760
498498 0 0
499499 5.88854 0.263607 0.131804 0.991276i 0.457923π-0.457923\pi
0.131804 + 0.991276i 0.457923π0.457923\pi
500500 0 0
501501 0 0
502502 0 0
503503 −2.29180 −0.102186 −0.0510931 0.998694i 0.516271π-0.516271\pi
−0.0510931 + 0.998694i 0.516271π0.516271\pi
504504 0 0
505505 −13.3050 −0.592063
506506 0 0
507507 0 0
508508 0 0
509509 −27.8885 −1.23614 −0.618069 0.786124i 0.712085π-0.712085\pi
−0.618069 + 0.786124i 0.712085π0.712085\pi
510510 0 0
511511 12.4721 0.551735
512512 0 0
513513 0 0
514514 0 0
515515 4.94427 0.217871
516516 0 0
517517 −3.41641 −0.150253
518518 0 0
519519 0 0
520520 0 0
521521 8.11146 0.355369 0.177685 0.984087i 0.443139π-0.443139\pi
0.177685 + 0.984087i 0.443139π0.443139\pi
522522 0 0
523523 41.3050 1.80614 0.903070 0.429494i 0.141308π-0.141308\pi
0.903070 + 0.429494i 0.141308π0.141308\pi
524524 0 0
525525 0 0
526526 0 0
527527 1.88854 0.0822663
528528 0 0
529529 −19.0000 −0.826087
530530 0 0
531531 0 0
532532 0 0
533533 37.8885 1.64114
534534 0 0
535535 −22.4721 −0.971555
536536 0 0
537537 0 0
538538 0 0
539539 −2.00000 −0.0861461
540540 0 0
541541 22.0000 0.945854 0.472927 0.881102i 0.343197π-0.343197\pi
0.472927 + 0.881102i 0.343197π0.343197\pi
542542 0 0
543543 0 0
544544 0 0
545545 12.3607 0.529473
546546 0 0
547547 32.0000 1.36822 0.684111 0.729378i 0.260191π-0.260191\pi
0.684111 + 0.729378i 0.260191π0.260191\pi
548548 0 0
549549 0 0
550550 0 0
551551 −5.70820 −0.243178
552552 0 0
553553 4.00000 0.170097
554554 0 0
555555 0 0
556556 0 0
557557 33.3050 1.41118 0.705588 0.708622i 0.250683π-0.250683\pi
0.705588 + 0.708622i 0.250683π0.250683\pi
558558 0 0
559559 −35.7771 −1.51321
560560 0 0
561561 0 0
562562 0 0
563563 18.4721 0.778508 0.389254 0.921131i 0.372733π-0.372733\pi
0.389254 + 0.921131i 0.372733π0.372733\pi
564564 0 0
565565 −8.94427 −0.376288
566566 0 0
567567 0 0
568568 0 0
569569 −45.1246 −1.89172 −0.945861 0.324572i 0.894780π-0.894780\pi
−0.945861 + 0.324572i 0.894780π0.894780\pi
570570 0 0
571571 −13.8885 −0.581217 −0.290609 0.956842i 0.593858π-0.593858\pi
−0.290609 + 0.956842i 0.593858π0.593858\pi
572572 0 0
573573 0 0
574574 0 0
575575 −6.94427 −0.289596
576576 0 0
577577 −23.5279 −0.979478 −0.489739 0.871869i 0.662908π-0.662908\pi
−0.489739 + 0.871869i 0.662908π0.662908\pi
578578 0 0
579579 0 0
580580 0 0
581581 −13.7082 −0.568712
582582 0 0
583583 −22.4721 −0.930701
584584 0 0
585585 0 0
586586 0 0
587587 −45.1246 −1.86249 −0.931246 0.364391i 0.881277π-0.881277\pi
−0.931246 + 0.364391i 0.881277π0.881277\pi
588588 0 0
589589 1.52786 0.0629545
590590 0 0
591591 0 0
592592 0 0
593593 −46.5410 −1.91121 −0.955605 0.294650i 0.904797π-0.904797\pi
−0.955605 + 0.294650i 0.904797π0.904797\pi
594594 0 0
595595 −1.52786 −0.0626363
596596 0 0
597597 0 0
598598 0 0
599599 −37.5967 −1.53616 −0.768081 0.640353i 0.778788π-0.778788\pi
−0.768081 + 0.640353i 0.778788π0.778788\pi
600600 0 0
601601 27.3050 1.11379 0.556896 0.830582i 0.311992π-0.311992\pi
0.556896 + 0.830582i 0.311992π0.311992\pi
602602 0 0
603603 0 0
604604 0 0
605605 −8.65248 −0.351773
606606 0 0
607607 −11.0557 −0.448738 −0.224369 0.974504i 0.572032π-0.572032\pi
−0.224369 + 0.974504i 0.572032π0.572032\pi
608608 0 0
609609 0 0
610610 0 0
611611 −7.63932 −0.309054
612612 0 0
613613 42.3607 1.71093 0.855466 0.517859i 0.173271π-0.173271\pi
0.855466 + 0.517859i 0.173271π0.173271\pi
614614 0 0
615615 0 0
616616 0 0
617617 −5.88854 −0.237064 −0.118532 0.992950i 0.537819π-0.537819\pi
−0.118532 + 0.992950i 0.537819π0.537819\pi
618618 0 0
619619 35.4164 1.42351 0.711753 0.702430i 0.247902π-0.247902\pi
0.711753 + 0.702430i 0.247902π0.247902\pi
620620 0 0
621621 0 0
622622 0 0
623623 4.47214 0.179172
624624 0 0
625625 4.41641 0.176656
626626 0 0
627627 0 0
628628 0 0
629629 −10.4721 −0.417551
630630 0 0
631631 8.94427 0.356066 0.178033 0.984025i 0.443027π-0.443027\pi
0.178033 + 0.984025i 0.443027π0.443027\pi
632632 0 0
633633 0 0
634634 0 0
635635 −19.0557 −0.756204
636636 0 0
637637 −4.47214 −0.177192
638638 0 0
639639 0 0
640640 0 0
641641 27.2361 1.07576 0.537880 0.843021i 0.319225π-0.319225\pi
0.537880 + 0.843021i 0.319225π0.319225\pi
642642 0 0
643643 −15.0557 −0.593740 −0.296870 0.954918i 0.595943π-0.595943\pi
−0.296870 + 0.954918i 0.595943π0.595943\pi
644644 0 0
645645 0 0
646646 0 0
647647 17.7082 0.696181 0.348091 0.937461i 0.386830π-0.386830\pi
0.348091 + 0.937461i 0.386830π0.386830\pi
648648 0 0
649649 17.8885 0.702187
650650 0 0
651651 0 0
652652 0 0
653653 −48.7214 −1.90661 −0.953307 0.302003i 0.902345π-0.902345\pi
−0.953307 + 0.302003i 0.902345π0.902345\pi
654654 0 0
655655 −18.8328 −0.735859
656656 0 0
657657 0 0
658658 0 0
659659 −30.7639 −1.19839 −0.599196 0.800602i 0.704513π-0.704513\pi
−0.599196 + 0.800602i 0.704513π0.704513\pi
660660 0 0
661661 −36.2492 −1.40993 −0.704966 0.709241i 0.749038π-0.749038\pi
−0.704966 + 0.709241i 0.749038π0.749038\pi
662662 0 0
663663 0 0
664664 0 0
665665 −1.23607 −0.0479327
666666 0 0
667667 11.4164 0.442045
668668 0 0
669669 0 0
670670 0 0
671671 26.8328 1.03587
672672 0 0
673673 −18.3607 −0.707752 −0.353876 0.935292i 0.615137π-0.615137\pi
−0.353876 + 0.935292i 0.615137π0.615137\pi
674674 0 0
675675 0 0
676676 0 0
677677 −30.0000 −1.15299 −0.576497 0.817099i 0.695581π-0.695581\pi
−0.576497 + 0.817099i 0.695581π0.695581\pi
678678 0 0
679679 −17.4164 −0.668380
680680 0 0
681681 0 0
682682 0 0
683683 −18.7639 −0.717982 −0.358991 0.933341i 0.616879π-0.616879\pi
−0.358991 + 0.933341i 0.616879π0.616879\pi
684684 0 0
685685 −4.94427 −0.188911
686686 0 0
687687 0 0
688688 0 0
689689 −50.2492 −1.91434
690690 0 0
691691 −24.3607 −0.926724 −0.463362 0.886169i 0.653357π-0.653357\pi
−0.463362 + 0.886169i 0.653357π0.653357\pi
692692 0 0
693693 0 0
694694 0 0
695695 11.7771 0.446730
696696 0 0
697697 10.4721 0.396660
698698 0 0
699699 0 0
700700 0 0
701701 −27.7771 −1.04913 −0.524563 0.851372i 0.675771π-0.675771\pi
−0.524563 + 0.851372i 0.675771π0.675771\pi
702702 0 0
703703 −8.47214 −0.319533
704704 0 0
705705 0 0
706706 0 0
707707 −10.7639 −0.404819
708708 0 0
709709 35.3050 1.32591 0.662953 0.748661i 0.269303π-0.269303\pi
0.662953 + 0.748661i 0.269303π0.269303\pi
710710 0 0
711711 0 0
712712 0 0
713713 −3.05573 −0.114438
714714 0 0
715715 11.0557 0.413461
716716 0 0
717717 0 0
718718 0 0
719719 −32.7639 −1.22189 −0.610944 0.791674i 0.709210π-0.709210\pi
−0.610944 + 0.791674i 0.709210π0.709210\pi
720720 0 0
721721 4.00000 0.148968
722722 0 0
723723 0 0
724724 0 0
725725 −19.8197 −0.736084
726726 0 0
727727 23.4164 0.868466 0.434233 0.900800i 0.357019π-0.357019\pi
0.434233 + 0.900800i 0.357019π0.357019\pi
728728 0 0
729729 0 0
730730 0 0
731731 −9.88854 −0.365741
732732 0 0
733733 37.4164 1.38201 0.691003 0.722852i 0.257169π-0.257169\pi
0.691003 + 0.722852i 0.257169π0.257169\pi
734734 0 0
735735 0 0
736736 0 0
737737 3.05573 0.112559
738738 0 0
739739 24.0000 0.882854 0.441427 0.897297i 0.354472π-0.354472\pi
0.441427 + 0.897297i 0.354472π0.354472\pi
740740 0 0
741741 0 0
742742 0 0
743743 49.5967 1.81953 0.909764 0.415126i 0.136262π-0.136262\pi
0.909764 + 0.415126i 0.136262π0.136262\pi
744744 0 0
745745 16.7214 0.612623
746746 0 0
747747 0 0
748748 0 0
749749 −18.1803 −0.664295
750750 0 0
751751 −32.9443 −1.20215 −0.601077 0.799191i 0.705261π-0.705261\pi
−0.601077 + 0.799191i 0.705261π0.705261\pi
752752 0 0
753753 0 0
754754 0 0
755755 4.94427 0.179940
756756 0 0
757757 −15.8885 −0.577479 −0.288739 0.957408i 0.593236π-0.593236\pi
−0.288739 + 0.957408i 0.593236π0.593236\pi
758758 0 0
759759 0 0
760760 0 0
761761 25.2361 0.914807 0.457403 0.889259i 0.348780π-0.348780\pi
0.457403 + 0.889259i 0.348780π0.348780\pi
762762 0 0
763763 10.0000 0.362024
764764 0 0
765765 0 0
766766 0 0
767767 40.0000 1.44432
768768 0 0
769769 19.5279 0.704193 0.352096 0.935964i 0.385469π-0.385469\pi
0.352096 + 0.935964i 0.385469π0.385469\pi
770770 0 0
771771 0 0
772772 0 0
773773 44.2492 1.59153 0.795767 0.605603i 0.207068π-0.207068\pi
0.795767 + 0.605603i 0.207068π0.207068\pi
774774 0 0
775775 5.30495 0.190559
776776 0 0
777777 0 0
778778 0 0
779779 8.47214 0.303546
780780 0 0
781781 −15.4164 −0.551642
782782 0 0
783783 0 0
784784 0 0
785785 −4.36068 −0.155639
786786 0 0
787787 −21.5279 −0.767385 −0.383693 0.923461i 0.625348π-0.625348\pi
−0.383693 + 0.923461i 0.625348π0.625348\pi
788788 0 0
789789 0 0
790790 0 0
791791 −7.23607 −0.257285
792792 0 0
793793 60.0000 2.13066
794794 0 0
795795 0 0
796796 0 0
797797 21.4164 0.758608 0.379304 0.925272i 0.376163π-0.376163\pi
0.379304 + 0.925272i 0.376163π0.376163\pi
798798 0 0
799799 −2.11146 −0.0746979
800800 0 0
801801 0 0
802802 0 0
803803 −24.9443 −0.880264
804804 0 0
805805 2.47214 0.0871313
806806 0 0
807807 0 0
808808 0 0
809809 −1.88854 −0.0663977 −0.0331988 0.999449i 0.510569π-0.510569\pi
−0.0331988 + 0.999449i 0.510569π0.510569\pi
810810 0 0
811811 −3.05573 −0.107301 −0.0536506 0.998560i 0.517086π-0.517086\pi
−0.0536506 + 0.998560i 0.517086π0.517086\pi
812812 0 0
813813 0 0
814814 0 0
815815 −19.7771 −0.692761
816816 0 0
817817 −8.00000 −0.279885
818818 0 0
819819 0 0
820820 0 0
821821 −30.2492 −1.05571 −0.527853 0.849336i 0.677003π-0.677003\pi
−0.527853 + 0.849336i 0.677003π0.677003\pi
822822 0 0
823823 −42.8328 −1.49306 −0.746529 0.665353i 0.768281π-0.768281\pi
−0.746529 + 0.665353i 0.768281π0.768281\pi
824824 0 0
825825 0 0
826826 0 0
827827 26.7639 0.930673 0.465337 0.885134i 0.345933π-0.345933\pi
0.465337 + 0.885134i 0.345933π0.345933\pi
828828 0 0
829829 1.41641 0.0491939 0.0245969 0.999697i 0.492170π-0.492170\pi
0.0245969 + 0.999697i 0.492170π0.492170\pi
830830 0 0
831831 0 0
832832 0 0
833833 −1.23607 −0.0428272
834834 0 0
835835 −28.9443 −1.00166
836836 0 0
837837 0 0
838838 0 0
839839 −8.94427 −0.308791 −0.154395 0.988009i 0.549343π-0.549343\pi
−0.154395 + 0.988009i 0.549343π0.549343\pi
840840 0 0
841841 3.58359 0.123572
842842 0 0
843843 0 0
844844 0 0
845845 8.65248 0.297654
846846 0 0
847847 −7.00000 −0.240523
848848 0 0
849849 0 0
850850 0 0
851851 16.9443 0.580842
852852 0 0
853853 −29.7771 −1.01955 −0.509774 0.860308i 0.670271π-0.670271\pi
−0.509774 + 0.860308i 0.670271π0.670271\pi
854854 0 0
855855 0 0
856856 0 0
857857 32.2492 1.10161 0.550806 0.834633i 0.314320π-0.314320\pi
0.550806 + 0.834633i 0.314320π0.314320\pi
858858 0 0
859859 −7.05573 −0.240738 −0.120369 0.992729i 0.538408π-0.538408\pi
−0.120369 + 0.992729i 0.538408π0.538408\pi
860860 0 0
861861 0 0
862862 0 0
863863 47.1246 1.60414 0.802070 0.597230i 0.203732π-0.203732\pi
0.802070 + 0.597230i 0.203732π0.203732\pi
864864 0 0
865865 −25.3050 −0.860394
866866 0 0
867867 0 0
868868 0 0
869869 −8.00000 −0.271381
870870 0 0
871871 6.83282 0.231521
872872 0 0
873873 0 0
874874 0 0
875875 −10.4721 −0.354023
876876 0 0
877877 28.8328 0.973615 0.486808 0.873509i 0.338161π-0.338161\pi
0.486808 + 0.873509i 0.338161π0.338161\pi
878878 0 0
879879 0 0
880880 0 0
881881 −46.1803 −1.55586 −0.777928 0.628354i 0.783729π-0.783729\pi
−0.777928 + 0.628354i 0.783729π0.783729\pi
882882 0 0
883883 5.88854 0.198165 0.0990826 0.995079i 0.468409π-0.468409\pi
0.0990826 + 0.995079i 0.468409π0.468409\pi
884884 0 0
885885 0 0
886886 0 0
887887 −24.5836 −0.825436 −0.412718 0.910859i 0.635421π-0.635421\pi
−0.412718 + 0.910859i 0.635421π0.635421\pi
888888 0 0
889889 −15.4164 −0.517050
890890 0 0
891891 0 0
892892 0 0
893893 −1.70820 −0.0571629
894894 0 0
895895 −9.52786 −0.318481
896896 0 0
897897 0 0
898898 0 0
899899 −8.72136 −0.290874
900900 0 0
901901 −13.8885 −0.462694
902902 0 0
903903 0 0
904904 0 0
905905 −4.36068 −0.144954
906906 0 0
907907 −25.5279 −0.847639 −0.423819 0.905747i 0.639311π-0.639311\pi
−0.423819 + 0.905747i 0.639311π0.639311\pi
908908 0 0
909909 0 0
910910 0 0
911911 −26.5410 −0.879343 −0.439672 0.898159i 0.644905π-0.644905\pi
−0.439672 + 0.898159i 0.644905π0.644905\pi
912912 0 0
913913 27.4164 0.907351
914914 0 0
915915 0 0
916916 0 0
917917 −15.2361 −0.503139
918918 0 0
919919 −35.7771 −1.18018 −0.590089 0.807338i 0.700907π-0.700907\pi
−0.590089 + 0.807338i 0.700907π0.700907\pi
920920 0 0
921921 0 0
922922 0 0
923923 −34.4721 −1.13466
924924 0 0
925925 −29.4164 −0.967206
926926 0 0
927927 0 0
928928 0 0
929929 −6.40325 −0.210084 −0.105042 0.994468i 0.533498π-0.533498\pi
−0.105042 + 0.994468i 0.533498π0.533498\pi
930930 0 0
931931 −1.00000 −0.0327737
932932 0 0
933933 0 0
934934 0 0
935935 3.05573 0.0999330
936936 0 0
937937 16.1115 0.526338 0.263169 0.964750i 0.415232π-0.415232\pi
0.263169 + 0.964750i 0.415232π0.415232\pi
938938 0 0
939939 0 0
940940 0 0
941941 12.8328 0.418338 0.209169 0.977880i 0.432924π-0.432924\pi
0.209169 + 0.977880i 0.432924π0.432924\pi
942942 0 0
943943 −16.9443 −0.551781
944944 0 0
945945 0 0
946946 0 0
947947 −26.0000 −0.844886 −0.422443 0.906389i 0.638827π-0.638827\pi
−0.422443 + 0.906389i 0.638827π0.638827\pi
948948 0 0
949949 −55.7771 −1.81060
950950 0 0
951951 0 0
952952 0 0
953953 13.3475 0.432369 0.216184 0.976353i 0.430639π-0.430639\pi
0.216184 + 0.976353i 0.430639π0.430639\pi
954954 0 0
955955 −12.3607 −0.399982
956956 0 0
957957 0 0
958958 0 0
959959 −4.00000 −0.129167
960960 0 0
961961 −28.6656 −0.924698
962962 0 0
963963 0 0
964964 0 0
965965 −10.4721 −0.337110
966966 0 0
967967 18.8328 0.605623 0.302811 0.953051i 0.402075π-0.402075\pi
0.302811 + 0.953051i 0.402075π0.402075\pi
968968 0 0
969969 0 0
970970 0 0
971971 −46.8328 −1.50294 −0.751468 0.659769i 0.770654π-0.770654\pi
−0.751468 + 0.659769i 0.770654π0.770654\pi
972972 0 0
973973 9.52786 0.305449
974974 0 0
975975 0 0
976976 0 0
977977 50.0689 1.60185 0.800923 0.598768i 0.204343π-0.204343\pi
0.800923 + 0.598768i 0.204343π0.204343\pi
978978 0 0
979979 −8.94427 −0.285860
980980 0 0
981981 0 0
982982 0 0
983983 48.7214 1.55397 0.776985 0.629519i 0.216748π-0.216748\pi
0.776985 + 0.629519i 0.216748π0.216748\pi
984984 0 0
985985 −14.8328 −0.472613
986986 0 0
987987 0 0
988988 0 0
989989 16.0000 0.508770
990990 0 0
991991 52.3607 1.66329 0.831646 0.555306i 0.187399π-0.187399\pi
0.831646 + 0.555306i 0.187399π0.187399\pi
992992 0 0
993993 0 0
994994 0 0
995995 −25.8885 −0.820722
996996 0 0
997997 −16.4721 −0.521678 −0.260839 0.965382i 0.583999π-0.583999\pi
−0.260839 + 0.965382i 0.583999π0.583999\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 9576.2.a.bi.1.2 2
3.2 odd 2 9576.2.a.bv.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bi.1.2 2 1.1 even 1 trivial
9576.2.a.bv.1.1 yes 2 3.2 odd 2