Properties

Label 9576.2.a.ci.1.2
Level 95769576
Weight 22
Character 9576.1
Self dual yes
Analytic conductor 76.46576.465
Analytic rank 00
Dimension 44
CM no
Inner twists 11

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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: 00
Dimension: 44
Coefficient field: 4.4.25857.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x38x2+9x+3 x^{4} - x^{3} - 8x^{2} + 9x + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 1064)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 1.522801.52280 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) == q1.52280q5+1.00000q75.50666q11+1.15829q13+2.98386q17+1.00000q190.841710q232.68109q25+7.95392q29+6.66728q311.52280q35+9.49052q379.95392q411.41010q430.635493q47+1.00000q490.348371q53+8.38553q5514.0961q597.01565q611.76384q653.30044q6714.5756q712.26562q735.50666q776.62120q792.33993q834.54382q851.56839q89+1.15829q911.52280q95+13.4977q97+O(q100)q-1.52280 q^{5} +1.00000 q^{7} -5.50666 q^{11} +1.15829 q^{13} +2.98386 q^{17} +1.00000 q^{19} -0.841710 q^{23} -2.68109 q^{25} +7.95392 q^{29} +6.66728 q^{31} -1.52280 q^{35} +9.49052 q^{37} -9.95392 q^{41} -1.41010 q^{43} -0.635493 q^{47} +1.00000 q^{49} -0.348371 q^{53} +8.38553 q^{55} -14.0961 q^{59} -7.01565 q^{61} -1.76384 q^{65} -3.30044 q^{67} -14.5756 q^{71} -2.26562 q^{73} -5.50666 q^{77} -6.62120 q^{79} -2.33993 q^{83} -4.54382 q^{85} -1.56839 q^{89} +1.15829 q^{91} -1.52280 q^{95} +13.4977 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4qq5+4q72q11+2q133q17+4q196q233q25+17q31q35+3q378q41+7q435q47+4q49+16q53+17q557q59q61++3q97+O(q100) 4 q - q^{5} + 4 q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{17} + 4 q^{19} - 6 q^{23} - 3 q^{25} + 17 q^{31} - q^{35} + 3 q^{37} - 8 q^{41} + 7 q^{43} - 5 q^{47} + 4 q^{49} + 16 q^{53} + 17 q^{55} - 7 q^{59} - q^{61}+ \cdots + 3 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 −1.52280 −0.681016 −0.340508 0.940242i 0.610599π-0.610599\pi
−0.340508 + 0.940242i 0.610599π0.610599\pi
66 0 0
77 1.00000 0.377964
88 0 0
99 0 0
1010 0 0
1111 −5.50666 −1.66032 −0.830160 0.557525i 0.811751π-0.811751\pi
−0.830160 + 0.557525i 0.811751π0.811751\pi
1212 0 0
1313 1.15829 0.321252 0.160626 0.987015i 0.448649π-0.448649\pi
0.160626 + 0.987015i 0.448649π0.448649\pi
1414 0 0
1515 0 0
1616 0 0
1717 2.98386 0.723693 0.361847 0.932238i 0.382146π-0.382146\pi
0.361847 + 0.932238i 0.382146π0.382146\pi
1818 0 0
1919 1.00000 0.229416
2020 0 0
2121 0 0
2222 0 0
2323 −0.841710 −0.175509 −0.0877543 0.996142i 0.527969π-0.527969\pi
−0.0877543 + 0.996142i 0.527969π0.527969\pi
2424 0 0
2525 −2.68109 −0.536218
2626 0 0
2727 0 0
2828 0 0
2929 7.95392 1.47701 0.738503 0.674250i 0.235533π-0.235533\pi
0.738503 + 0.674250i 0.235533π0.235533\pi
3030 0 0
3131 6.66728 1.19748 0.598740 0.800944i 0.295668π-0.295668\pi
0.598740 + 0.800944i 0.295668π0.295668\pi
3232 0 0
3333 0 0
3434 0 0
3535 −1.52280 −0.257400
3636 0 0
3737 9.49052 1.56023 0.780116 0.625635i 0.215160π-0.215160\pi
0.780116 + 0.625635i 0.215160π0.215160\pi
3838 0 0
3939 0 0
4040 0 0
4141 −9.95392 −1.55454 −0.777271 0.629166i 0.783396π-0.783396\pi
−0.777271 + 0.629166i 0.783396π0.783396\pi
4242 0 0
4343 −1.41010 −0.215039 −0.107519 0.994203i 0.534291π-0.534291\pi
−0.107519 + 0.994203i 0.534291π0.534291\pi
4444 0 0
4545 0 0
4646 0 0
4747 −0.635493 −0.0926962 −0.0463481 0.998925i 0.514758π-0.514758\pi
−0.0463481 + 0.998925i 0.514758π0.514758\pi
4848 0 0
4949 1.00000 0.142857
5050 0 0
5151 0 0
5252 0 0
5353 −0.348371 −0.0478524 −0.0239262 0.999714i 0.507617π-0.507617\pi
−0.0239262 + 0.999714i 0.507617π0.507617\pi
5454 0 0
5555 8.38553 1.13070
5656 0 0
5757 0 0
5858 0 0
5959 −14.0961 −1.83515 −0.917576 0.397560i 0.869857π-0.869857\pi
−0.917576 + 0.397560i 0.869857π0.869857\pi
6060 0 0
6161 −7.01565 −0.898262 −0.449131 0.893466i 0.648266π-0.648266\pi
−0.449131 + 0.893466i 0.648266π0.648266\pi
6262 0 0
6363 0 0
6464 0 0
6565 −1.76384 −0.218778
6666 0 0
6767 −3.30044 −0.403213 −0.201607 0.979467i 0.564616π-0.564616\pi
−0.201607 + 0.979467i 0.564616π0.564616\pi
6868 0 0
6969 0 0
7070 0 0
7171 −14.5756 −1.72981 −0.864903 0.501939i 0.832620π-0.832620\pi
−0.864903 + 0.501939i 0.832620π0.832620\pi
7272 0 0
7373 −2.26562 −0.265170 −0.132585 0.991172i 0.542328π-0.542328\pi
−0.132585 + 0.991172i 0.542328π0.542328\pi
7474 0 0
7575 0 0
7676 0 0
7777 −5.50666 −0.627542
7878 0 0
7979 −6.62120 −0.744944 −0.372472 0.928043i 0.621490π-0.621490\pi
−0.372472 + 0.928043i 0.621490π0.621490\pi
8080 0 0
8181 0 0
8282 0 0
8383 −2.33993 −0.256841 −0.128421 0.991720i 0.540991π-0.540991\pi
−0.128421 + 0.991720i 0.540991π0.540991\pi
8484 0 0
8585 −4.54382 −0.492846
8686 0 0
8787 0 0
8888 0 0
8989 −1.56839 −0.166249 −0.0831246 0.996539i 0.526490π-0.526490\pi
−0.0831246 + 0.996539i 0.526490π0.526490\pi
9090 0 0
9191 1.15829 0.121422
9292 0 0
9393 0 0
9494 0 0
9595 −1.52280 −0.156236
9696 0 0
9797 13.4977 1.37049 0.685244 0.728314i 0.259696π-0.259696\pi
0.685244 + 0.728314i 0.259696π0.259696\pi
9898 0 0
9999 0 0
100100 0 0
101101 0.552739 0.0549996 0.0274998 0.999622i 0.491245π-0.491245\pi
0.0274998 + 0.999622i 0.491245π0.491245\pi
102102 0 0
103103 −3.56606 −0.351374 −0.175687 0.984446i 0.556215π-0.556215\pi
−0.175687 + 0.984446i 0.556215π0.556215\pi
104104 0 0
105105 0 0
106106 0 0
107107 16.0205 1.54876 0.774382 0.632718i 0.218061π-0.218061\pi
0.774382 + 0.632718i 0.218061π0.218061\pi
108108 0 0
109109 14.2268 1.36268 0.681338 0.731969i 0.261398π-0.261398\pi
0.681338 + 0.731969i 0.261398π0.261398\pi
110110 0 0
111111 0 0
112112 0 0
113113 −14.1735 −1.33333 −0.666664 0.745359i 0.732278π-0.732278\pi
−0.666664 + 0.745359i 0.732278π0.732278\pi
114114 0 0
115115 1.28175 0.119524
116116 0 0
117117 0 0
118118 0 0
119119 2.98386 0.273530
120120 0 0
121121 19.3233 1.75666
122122 0 0
123123 0 0
124124 0 0
125125 11.6967 1.04619
126126 0 0
127127 13.8850 1.23209 0.616046 0.787710i 0.288734π-0.288734\pi
0.616046 + 0.787710i 0.288734π0.288734\pi
128128 0 0
129129 0 0
130130 0 0
131131 5.30511 0.463509 0.231755 0.972774i 0.425553π-0.425553\pi
0.231755 + 0.972774i 0.425553π0.425553\pi
132132 0 0
133133 1.00000 0.0867110
134134 0 0
135135 0 0
136136 0 0
137137 1.06221 0.0907511 0.0453756 0.998970i 0.485552π-0.485552\pi
0.0453756 + 0.998970i 0.485552π0.485552\pi
138138 0 0
139139 18.8407 1.59805 0.799025 0.601297i 0.205349π-0.205349\pi
0.799025 + 0.601297i 0.205349π0.205349\pi
140140 0 0
141141 0 0
142142 0 0
143143 −6.37831 −0.533381
144144 0 0
145145 −12.1122 −1.00586
146146 0 0
147147 0 0
148148 0 0
149149 23.0661 1.88965 0.944826 0.327574i 0.106231π-0.106231\pi
0.944826 + 0.327574i 0.106231π0.106231\pi
150150 0 0
151151 −1.01614 −0.0826920 −0.0413460 0.999145i 0.513165π-0.513165\pi
−0.0413460 + 0.999145i 0.513165π0.513165\pi
152152 0 0
153153 0 0
154154 0 0
155155 −10.1529 −0.815502
156156 0 0
157157 5.06662 0.404360 0.202180 0.979348i 0.435197π-0.435197\pi
0.202180 + 0.979348i 0.435197π0.435197\pi
158158 0 0
159159 0 0
160160 0 0
161161 −0.841710 −0.0663360
162162 0 0
163163 −12.6794 −0.993126 −0.496563 0.868001i 0.665405π-0.665405\pi
−0.496563 + 0.868001i 0.665405π0.665405\pi
164164 0 0
165165 0 0
166166 0 0
167167 15.3827 1.19035 0.595175 0.803596i 0.297083π-0.297083\pi
0.595175 + 0.803596i 0.297083π0.297083\pi
168168 0 0
169169 −11.6584 −0.896797
170170 0 0
171171 0 0
172172 0 0
173173 15.3299 1.16551 0.582755 0.812648i 0.301975π-0.301975\pi
0.582755 + 0.812648i 0.301975π0.301975\pi
174174 0 0
175175 −2.68109 −0.202671
176176 0 0
177177 0 0
178178 0 0
179179 15.3640 1.14836 0.574181 0.818729i 0.305321π-0.305321\pi
0.574181 + 0.818729i 0.305321π0.305321\pi
180180 0 0
181181 −3.99719 −0.297108 −0.148554 0.988904i 0.547462π-0.547462\pi
−0.148554 + 0.988904i 0.547462π0.547462\pi
182182 0 0
183183 0 0
184184 0 0
185185 −14.4521 −1.06254
186186 0 0
187187 −16.4311 −1.20156
188188 0 0
189189 0 0
190190 0 0
191191 9.79330 0.708618 0.354309 0.935128i 0.384716π-0.384716\pi
0.354309 + 0.935128i 0.384716π0.384716\pi
192192 0 0
193193 −20.1206 −1.44832 −0.724158 0.689634i 0.757771π-0.757771\pi
−0.724158 + 0.689634i 0.757771π0.757771\pi
194194 0 0
195195 0 0
196196 0 0
197197 16.5223 1.17717 0.588583 0.808437i 0.299686π-0.299686\pi
0.588583 + 0.808437i 0.299686π0.299686\pi
198198 0 0
199199 12.8574 0.911434 0.455717 0.890125i 0.349383π-0.349383\pi
0.455717 + 0.890125i 0.349383π0.349383\pi
200200 0 0
201201 0 0
202202 0 0
203203 7.95392 0.558256
204204 0 0
205205 15.1578 1.05867
206206 0 0
207207 0 0
208208 0 0
209209 −5.50666 −0.380904
210210 0 0
211211 −15.1555 −1.04335 −0.521673 0.853146i 0.674692π-0.674692\pi
−0.521673 + 0.853146i 0.674692π0.674692\pi
212212 0 0
213213 0 0
214214 0 0
215215 2.14730 0.146445
216216 0 0
217217 6.66728 0.452605
218218 0 0
219219 0 0
220220 0 0
221221 3.45618 0.232488
222222 0 0
223223 12.8094 0.857783 0.428892 0.903356i 0.358904π-0.358904\pi
0.428892 + 0.903356i 0.358904π0.358904\pi
224224 0 0
225225 0 0
226226 0 0
227227 24.2539 1.60979 0.804893 0.593419i 0.202222π-0.202222\pi
0.804893 + 0.593419i 0.202222π0.202222\pi
228228 0 0
229229 −17.1945 −1.13624 −0.568122 0.822945i 0.692330π-0.692330\pi
−0.568122 + 0.822945i 0.692330π0.692330\pi
230230 0 0
231231 0 0
232232 0 0
233233 28.0056 1.83471 0.917355 0.398071i 0.130320π-0.130320\pi
0.917355 + 0.398071i 0.130320π0.130320\pi
234234 0 0
235235 0.967727 0.0631275
236236 0 0
237237 0 0
238238 0 0
239239 0.460583 0.0297926 0.0148963 0.999889i 0.495258π-0.495258\pi
0.0148963 + 0.999889i 0.495258π0.495258\pi
240240 0 0
241241 −2.16673 −0.139571 −0.0697856 0.997562i 0.522232π-0.522232\pi
−0.0697856 + 0.997562i 0.522232π0.522232\pi
242242 0 0
243243 0 0
244244 0 0
245245 −1.52280 −0.0972880
246246 0 0
247247 1.15829 0.0737002
248248 0 0
249249 0 0
250250 0 0
251251 −28.0705 −1.77180 −0.885898 0.463880i 0.846457π-0.846457\pi
−0.885898 + 0.463880i 0.846457π0.846457\pi
252252 0 0
253253 4.63501 0.291401
254254 0 0
255255 0 0
256256 0 0
257257 1.63124 0.101754 0.0508769 0.998705i 0.483798π-0.483798\pi
0.0508769 + 0.998705i 0.483798π0.483798\pi
258258 0 0
259259 9.49052 0.589712
260260 0 0
261261 0 0
262262 0 0
263263 −3.56791 −0.220007 −0.110003 0.993931i 0.535086π-0.535086\pi
−0.110003 + 0.993931i 0.535086π0.535086\pi
264264 0 0
265265 0.530498 0.0325882
266266 0 0
267267 0 0
268268 0 0
269269 −15.6422 −0.953723 −0.476862 0.878978i 0.658226π-0.658226\pi
−0.476862 + 0.878978i 0.658226π0.658226\pi
270270 0 0
271271 32.1461 1.95273 0.976367 0.216117i 0.0693393π-0.0693393\pi
0.976367 + 0.216117i 0.0693393π0.0693393\pi
272272 0 0
273273 0 0
274274 0 0
275275 14.7638 0.890293
276276 0 0
277277 5.01332 0.301221 0.150611 0.988593i 0.451876π-0.451876\pi
0.150611 + 0.988593i 0.451876π0.451876\pi
278278 0 0
279279 0 0
280280 0 0
281281 16.0481 0.957352 0.478676 0.877992i 0.341117π-0.341117\pi
0.478676 + 0.877992i 0.341117π0.341117\pi
282282 0 0
283283 1.41314 0.0840024 0.0420012 0.999118i 0.486627π-0.486627\pi
0.0420012 + 0.999118i 0.486627π0.486627\pi
284284 0 0
285285 0 0
286286 0 0
287287 −9.95392 −0.587561
288288 0 0
289289 −8.09656 −0.476268
290290 0 0
291291 0 0
292292 0 0
293293 −10.9006 −0.636821 −0.318411 0.947953i 0.603149π-0.603149\pi
−0.318411 + 0.947953i 0.603149π0.603149\pi
294294 0 0
295295 21.4655 1.24977
296296 0 0
297297 0 0
298298 0 0
299299 −0.974944 −0.0563825
300300 0 0
301301 −1.41010 −0.0812769
302302 0 0
303303 0 0
304304 0 0
305305 10.6834 0.611731
306306 0 0
307307 −4.15170 −0.236950 −0.118475 0.992957i 0.537801π-0.537801\pi
−0.118475 + 0.992957i 0.537801π0.537801\pi
308308 0 0
309309 0 0
310310 0 0
311311 −5.00770 −0.283961 −0.141980 0.989869i 0.545347π-0.545347\pi
−0.141980 + 0.989869i 0.545347π0.545347\pi
312312 0 0
313313 19.2556 1.08839 0.544195 0.838959i 0.316835π-0.316835\pi
0.544195 + 0.838959i 0.316835π0.316835\pi
314314 0 0
315315 0 0
316316 0 0
317317 8.95818 0.503141 0.251571 0.967839i 0.419053π-0.419053\pi
0.251571 + 0.967839i 0.419053π0.419053\pi
318318 0 0
319319 −43.7996 −2.45230
320320 0 0
321321 0 0
322322 0 0
323323 2.98386 0.166027
324324 0 0
325325 −3.10548 −0.172261
326326 0 0
327327 0 0
328328 0 0
329329 −0.635493 −0.0350359
330330 0 0
331331 17.9588 0.987105 0.493553 0.869716i 0.335698π-0.335698\pi
0.493553 + 0.869716i 0.335698π0.335698\pi
332332 0 0
333333 0 0
334334 0 0
335335 5.02591 0.274595
336336 0 0
337337 −6.75381 −0.367903 −0.183952 0.982935i 0.558889π-0.558889\pi
−0.183952 + 0.982935i 0.558889π0.558889\pi
338338 0 0
339339 0 0
340340 0 0
341341 −36.7145 −1.98820
342342 0 0
343343 1.00000 0.0539949
344344 0 0
345345 0 0
346346 0 0
347347 26.0033 1.39593 0.697965 0.716132i 0.254089π-0.254089\pi
0.697965 + 0.716132i 0.254089π0.254089\pi
348348 0 0
349349 20.4634 1.09538 0.547690 0.836681i 0.315507π-0.315507\pi
0.547690 + 0.836681i 0.315507π0.315507\pi
350350 0 0
351351 0 0
352352 0 0
353353 15.9696 0.849975 0.424987 0.905199i 0.360279π-0.360279\pi
0.424987 + 0.905199i 0.360279π0.360279\pi
354354 0 0
355355 22.1957 1.17803
356356 0 0
357357 0 0
358358 0 0
359359 −13.6038 −0.717984 −0.358992 0.933341i 0.616879π-0.616879\pi
−0.358992 + 0.933341i 0.616879π0.616879\pi
360360 0 0
361361 1.00000 0.0526316
362362 0 0
363363 0 0
364364 0 0
365365 3.45008 0.180585
366366 0 0
367367 −24.9655 −1.30319 −0.651595 0.758567i 0.725900π-0.725900\pi
−0.651595 + 0.758567i 0.725900π0.725900\pi
368368 0 0
369369 0 0
370370 0 0
371371 −0.348371 −0.0180865
372372 0 0
373373 15.3965 0.797202 0.398601 0.917125i 0.369496π-0.369496\pi
0.398601 + 0.917125i 0.369496π0.369496\pi
374374 0 0
375375 0 0
376376 0 0
377377 9.21295 0.474491
378378 0 0
379379 −0.932900 −0.0479198 −0.0239599 0.999713i 0.507627π-0.507627\pi
−0.0239599 + 0.999713i 0.507627π0.507627\pi
380380 0 0
381381 0 0
382382 0 0
383383 −0.841710 −0.0430094 −0.0215047 0.999769i 0.506846π-0.506846\pi
−0.0215047 + 0.999769i 0.506846π0.506846\pi
384384 0 0
385385 8.38553 0.427366
386386 0 0
387387 0 0
388388 0 0
389389 3.75492 0.190382 0.0951911 0.995459i 0.469654π-0.469654\pi
0.0951911 + 0.995459i 0.469654π0.469654\pi
390390 0 0
391391 −2.51155 −0.127014
392392 0 0
393393 0 0
394394 0 0
395395 10.0828 0.507318
396396 0 0
397397 28.7052 1.44067 0.720336 0.693626i 0.243988π-0.243988\pi
0.720336 + 0.693626i 0.243988π0.243988\pi
398398 0 0
399399 0 0
400400 0 0
401401 −18.9660 −0.947118 −0.473559 0.880762i 0.657031π-0.657031\pi
−0.473559 + 0.880762i 0.657031π0.657031\pi
402402 0 0
403403 7.72265 0.384693
404404 0 0
405405 0 0
406406 0 0
407407 −52.2611 −2.59049
408408 0 0
409409 −3.08716 −0.152650 −0.0763250 0.997083i 0.524319π-0.524319\pi
−0.0763250 + 0.997083i 0.524319π0.524319\pi
410410 0 0
411411 0 0
412412 0 0
413413 −14.0961 −0.693623
414414 0 0
415415 3.56325 0.174913
416416 0 0
417417 0 0
418418 0 0
419419 11.2219 0.548224 0.274112 0.961698i 0.411616π-0.411616\pi
0.274112 + 0.961698i 0.411616π0.411616\pi
420420 0 0
421421 5.58146 0.272024 0.136012 0.990707i 0.456571π-0.456571\pi
0.136012 + 0.990707i 0.456571π0.456571\pi
422422 0 0
423423 0 0
424424 0 0
425425 −8.00000 −0.388057
426426 0 0
427427 −7.01565 −0.339511
428428 0 0
429429 0 0
430430 0 0
431431 −0.555071 −0.0267368 −0.0133684 0.999911i 0.504255π-0.504255\pi
−0.0133684 + 0.999911i 0.504255π0.504255\pi
432432 0 0
433433 −11.8215 −0.568107 −0.284053 0.958808i 0.591679π-0.591679\pi
−0.284053 + 0.958808i 0.591679π0.591679\pi
434434 0 0
435435 0 0
436436 0 0
437437 −0.841710 −0.0402644
438438 0 0
439439 −26.3432 −1.25729 −0.628647 0.777691i 0.716391π-0.716391\pi
−0.628647 + 0.777691i 0.716391π0.716391\pi
440440 0 0
441441 0 0
442442 0 0
443443 16.0246 0.761350 0.380675 0.924709i 0.375692π-0.375692\pi
0.380675 + 0.924709i 0.375692π0.375692\pi
444444 0 0
445445 2.38834 0.113218
446446 0 0
447447 0 0
448448 0 0
449449 33.4589 1.57902 0.789511 0.613736i 0.210334π-0.210334\pi
0.789511 + 0.613736i 0.210334π0.210334\pi
450450 0 0
451451 54.8129 2.58104
452452 0 0
453453 0 0
454454 0 0
455455 −1.76384 −0.0826902
456456 0 0
457457 40.1051 1.87604 0.938020 0.346582i 0.112658π-0.112658\pi
0.938020 + 0.346582i 0.112658π0.112658\pi
458458 0 0
459459 0 0
460460 0 0
461461 −15.9719 −0.743886 −0.371943 0.928256i 0.621308π-0.621308\pi
−0.371943 + 0.928256i 0.621308π0.621308\pi
462462 0 0
463463 16.8094 0.781201 0.390600 0.920560i 0.372267π-0.372267\pi
0.390600 + 0.920560i 0.372267π0.372267\pi
464464 0 0
465465 0 0
466466 0 0
467467 35.5178 1.64357 0.821784 0.569799i 0.192979π-0.192979\pi
0.821784 + 0.569799i 0.192979π0.192979\pi
468468 0 0
469469 −3.30044 −0.152400
470470 0 0
471471 0 0
472472 0 0
473473 7.76495 0.357033
474474 0 0
475475 −2.68109 −0.123017
476476 0 0
477477 0 0
478478 0 0
479479 −30.2336 −1.38141 −0.690703 0.723138i 0.742699π-0.742699\pi
−0.690703 + 0.723138i 0.742699π0.742699\pi
480480 0 0
481481 10.9928 0.501228
482482 0 0
483483 0 0
484484 0 0
485485 −20.5543 −0.933324
486486 0 0
487487 −4.06847 −0.184360 −0.0921799 0.995742i 0.529383π-0.529383\pi
−0.0921799 + 0.995742i 0.529383π0.529383\pi
488488 0 0
489489 0 0
490490 0 0
491491 26.6046 1.20065 0.600324 0.799757i 0.295038π-0.295038\pi
0.600324 + 0.799757i 0.295038π0.295038\pi
492492 0 0
493493 23.7334 1.06890
494494 0 0
495495 0 0
496496 0 0
497497 −14.5756 −0.653805
498498 0 0
499499 −14.9098 −0.667453 −0.333726 0.942670i 0.608306π-0.608306\pi
−0.333726 + 0.942670i 0.608306π0.608306\pi
500500 0 0
501501 0 0
502502 0 0
503503 −17.4054 −0.776070 −0.388035 0.921645i 0.626846π-0.626846\pi
−0.388035 + 0.921645i 0.626846π0.626846\pi
504504 0 0
505505 −0.841710 −0.0374556
506506 0 0
507507 0 0
508508 0 0
509509 35.7414 1.58421 0.792104 0.610386i 0.208986π-0.208986\pi
0.792104 + 0.610386i 0.208986π0.208986\pi
510510 0 0
511511 −2.26562 −0.100225
512512 0 0
513513 0 0
514514 0 0
515515 5.43039 0.239292
516516 0 0
517517 3.49944 0.153905
518518 0 0
519519 0 0
520520 0 0
521521 0.905290 0.0396615 0.0198307 0.999803i 0.493687π-0.493687\pi
0.0198307 + 0.999803i 0.493687π0.493687\pi
522522 0 0
523523 −29.3182 −1.28199 −0.640997 0.767543i 0.721479π-0.721479\pi
−0.640997 + 0.767543i 0.721479π0.721479\pi
524524 0 0
525525 0 0
526526 0 0
527527 19.8943 0.866608
528528 0 0
529529 −22.2915 −0.969197
530530 0 0
531531 0 0
532532 0 0
533533 −11.5295 −0.499399
534534 0 0
535535 −24.3960 −1.05473
536536 0 0
537537 0 0
538538 0 0
539539 −5.50666 −0.237189
540540 0 0
541541 13.6521 0.586950 0.293475 0.955967i 0.405188π-0.405188\pi
0.293475 + 0.955967i 0.405188π0.405188\pi
542542 0 0
543543 0 0
544544 0 0
545545 −21.6645 −0.928004
546546 0 0
547547 21.0489 0.899985 0.449993 0.893032i 0.351427π-0.351427\pi
0.449993 + 0.893032i 0.351427π0.351427\pi
548548 0 0
549549 0 0
550550 0 0
551551 7.95392 0.338848
552552 0 0
553553 −6.62120 −0.281562
554554 0 0
555555 0 0
556556 0 0
557557 43.6750 1.85057 0.925284 0.379275i 0.123827π-0.123827\pi
0.925284 + 0.379275i 0.123827π0.123827\pi
558558 0 0
559559 −1.63331 −0.0690816
560560 0 0
561561 0 0
562562 0 0
563563 14.6201 0.616163 0.308082 0.951360i 0.400313π-0.400313\pi
0.308082 + 0.951360i 0.400313π0.400313\pi
564564 0 0
565565 21.5833 0.908017
566566 0 0
567567 0 0
568568 0 0
569569 1.54311 0.0646907 0.0323453 0.999477i 0.489702π-0.489702\pi
0.0323453 + 0.999477i 0.489702π0.489702\pi
570570 0 0
571571 3.23031 0.135184 0.0675921 0.997713i 0.478468π-0.478468\pi
0.0675921 + 0.997713i 0.478468π0.478468\pi
572572 0 0
573573 0 0
574574 0 0
575575 2.25670 0.0941108
576576 0 0
577577 12.0678 0.502391 0.251195 0.967936i 0.419176π-0.419176\pi
0.251195 + 0.967936i 0.419176π0.419176\pi
578578 0 0
579579 0 0
580580 0 0
581581 −2.33993 −0.0970768
582582 0 0
583583 1.91836 0.0794503
584584 0 0
585585 0 0
586586 0 0
587587 18.9165 0.780768 0.390384 0.920652i 0.372342π-0.372342\pi
0.390384 + 0.920652i 0.372342π0.372342\pi
588588 0 0
589589 6.66728 0.274721
590590 0 0
591591 0 0
592592 0 0
593593 42.9703 1.76458 0.882289 0.470708i 0.156002π-0.156002\pi
0.882289 + 0.470708i 0.156002π0.156002\pi
594594 0 0
595595 −4.54382 −0.186278
596596 0 0
597597 0 0
598598 0 0
599599 −45.3930 −1.85471 −0.927354 0.374186i 0.877922π-0.877922\pi
−0.927354 + 0.374186i 0.877922π0.877922\pi
600600 0 0
601601 −24.9712 −1.01859 −0.509297 0.860591i 0.670095π-0.670095\pi
−0.509297 + 0.860591i 0.670095π0.670095\pi
602602 0 0
603603 0 0
604604 0 0
605605 −29.4255 −1.19632
606606 0 0
607607 −16.8131 −0.682424 −0.341212 0.939986i 0.610837π-0.610837\pi
−0.341212 + 0.939986i 0.610837π0.610837\pi
608608 0 0
609609 0 0
610610 0 0
611611 −0.736085 −0.0297788
612612 0 0
613613 0.894115 0.0361130 0.0180565 0.999837i 0.494252π-0.494252\pi
0.0180565 + 0.999837i 0.494252π0.494252\pi
614614 0 0
615615 0 0
616616 0 0
617617 −13.4396 −0.541056 −0.270528 0.962712i 0.587198π-0.587198\pi
−0.270528 + 0.962712i 0.587198π0.587198\pi
618618 0 0
619619 2.96332 0.119106 0.0595530 0.998225i 0.481032π-0.481032\pi
0.0595530 + 0.998225i 0.481032π0.481032\pi
620620 0 0
621621 0 0
622622 0 0
623623 −1.56839 −0.0628363
624624 0 0
625625 −4.40633 −0.176253
626626 0 0
627627 0 0
628628 0 0
629629 28.3184 1.12913
630630 0 0
631631 −17.6849 −0.704023 −0.352012 0.935996i 0.614502π-0.614502\pi
−0.352012 + 0.935996i 0.614502π0.614502\pi
632632 0 0
633633 0 0
634634 0 0
635635 −21.1440 −0.839074
636636 0 0
637637 1.15829 0.0458931
638638 0 0
639639 0 0
640640 0 0
641641 −13.6002 −0.537174 −0.268587 0.963255i 0.586557π-0.586557\pi
−0.268587 + 0.963255i 0.586557π0.586557\pi
642642 0 0
643643 −13.5492 −0.534328 −0.267164 0.963651i 0.586087π-0.586087\pi
−0.267164 + 0.963651i 0.586087π0.586087\pi
644644 0 0
645645 0 0
646646 0 0
647647 −10.0396 −0.394697 −0.197349 0.980333i 0.563233π-0.563233\pi
−0.197349 + 0.980333i 0.563233π0.563233\pi
648648 0 0
649649 77.6223 3.04694
650650 0 0
651651 0 0
652652 0 0
653653 −32.3408 −1.26559 −0.632797 0.774318i 0.718093π-0.718093\pi
−0.632797 + 0.774318i 0.718093π0.718093\pi
654654 0 0
655655 −8.07860 −0.315657
656656 0 0
657657 0 0
658658 0 0
659659 −30.5883 −1.19155 −0.595776 0.803151i 0.703155π-0.703155\pi
−0.595776 + 0.803151i 0.703155π0.703155\pi
660660 0 0
661661 4.03597 0.156981 0.0784905 0.996915i 0.474990π-0.474990\pi
0.0784905 + 0.996915i 0.474990π0.474990\pi
662662 0 0
663663 0 0
664664 0 0
665665 −1.52280 −0.0590516
666666 0 0
667667 −6.69489 −0.259227
668668 0 0
669669 0 0
670670 0 0
671671 38.6328 1.49140
672672 0 0
673673 −9.29434 −0.358270 −0.179135 0.983824i 0.557330π-0.557330\pi
−0.179135 + 0.983824i 0.557330π0.557330\pi
674674 0 0
675675 0 0
676676 0 0
677677 32.4111 1.24566 0.622829 0.782358i 0.285983π-0.285983\pi
0.622829 + 0.782358i 0.285983π0.285983\pi
678678 0 0
679679 13.4977 0.517996
680680 0 0
681681 0 0
682682 0 0
683683 −29.1840 −1.11669 −0.558347 0.829608i 0.688564π-0.688564\pi
−0.558347 + 0.829608i 0.688564π0.688564\pi
684684 0 0
685685 −1.61754 −0.0618029
686686 0 0
687687 0 0
688688 0 0
689689 −0.403514 −0.0153727
690690 0 0
691691 −43.9489 −1.67190 −0.835948 0.548809i 0.815081π-0.815081\pi
−0.835948 + 0.548809i 0.815081π0.815081\pi
692692 0 0
693693 0 0
694694 0 0
695695 −28.6906 −1.08830
696696 0 0
697697 −29.7011 −1.12501
698698 0 0
699699 0 0
700700 0 0
701701 13.5446 0.511571 0.255785 0.966734i 0.417666π-0.417666\pi
0.255785 + 0.966734i 0.417666π0.417666\pi
702702 0 0
703703 9.49052 0.357942
704704 0 0
705705 0 0
706706 0 0
707707 0.552739 0.0207879
708708 0 0
709709 26.2496 0.985825 0.492913 0.870079i 0.335932π-0.335932\pi
0.492913 + 0.870079i 0.335932π0.335932\pi
710710 0 0
711711 0 0
712712 0 0
713713 −5.61192 −0.210168
714714 0 0
715715 9.71288 0.363241
716716 0 0
717717 0 0
718718 0 0
719719 −21.9069 −0.816989 −0.408494 0.912761i 0.633946π-0.633946\pi
−0.408494 + 0.912761i 0.633946π0.633946\pi
720720 0 0
721721 −3.56606 −0.132807
722722 0 0
723723 0 0
724724 0 0
725725 −21.3252 −0.791997
726726 0 0
727727 29.5100 1.09446 0.547232 0.836981i 0.315682π-0.315682\pi
0.547232 + 0.836981i 0.315682π0.315682\pi
728728 0 0
729729 0 0
730730 0 0
731731 −4.20755 −0.155622
732732 0 0
733733 35.1131 1.29693 0.648465 0.761244i 0.275411π-0.275411\pi
0.648465 + 0.761244i 0.275411π0.275411\pi
734734 0 0
735735 0 0
736736 0 0
737737 18.1744 0.669464
738738 0 0
739739 40.2785 1.48167 0.740833 0.671689i 0.234431π-0.234431\pi
0.740833 + 0.671689i 0.234431π0.234431\pi
740740 0 0
741741 0 0
742742 0 0
743743 7.19082 0.263805 0.131903 0.991263i 0.457891π-0.457891\pi
0.131903 + 0.991263i 0.457891π0.457891\pi
744744 0 0
745745 −35.1251 −1.28688
746746 0 0
747747 0 0
748748 0 0
749749 16.0205 0.585378
750750 0 0
751751 0.902222 0.0329226 0.0164613 0.999865i 0.494760π-0.494760\pi
0.0164613 + 0.999865i 0.494760π0.494760\pi
752752 0 0
753753 0 0
754754 0 0
755755 1.54737 0.0563146
756756 0 0
757757 −49.3719 −1.79445 −0.897227 0.441570i 0.854422π-0.854422\pi
−0.897227 + 0.441570i 0.854422π0.854422\pi
758758 0 0
759759 0 0
760760 0 0
761761 25.7582 0.933734 0.466867 0.884327i 0.345383π-0.345383\pi
0.466867 + 0.884327i 0.345383π0.345383\pi
762762 0 0
763763 14.2268 0.515043
764764 0 0
765765 0 0
766766 0 0
767767 −16.3273 −0.589546
768768 0 0
769769 47.7844 1.72315 0.861574 0.507631i 0.169479π-0.169479\pi
0.861574 + 0.507631i 0.169479π0.169479\pi
770770 0 0
771771 0 0
772772 0 0
773773 26.5911 0.956416 0.478208 0.878247i 0.341287π-0.341287\pi
0.478208 + 0.878247i 0.341287π0.341287\pi
774774 0 0
775775 −17.8756 −0.642109
776776 0 0
777777 0 0
778778 0 0
779779 −9.95392 −0.356636
780780 0 0
781781 80.2629 2.87203
782782 0 0
783783 0 0
784784 0 0
785785 −7.71543 −0.275376
786786 0 0
787787 51.9142 1.85054 0.925270 0.379309i 0.123838π-0.123838\pi
0.925270 + 0.379309i 0.123838π0.123838\pi
788788 0 0
789789 0 0
790790 0 0
791791 −14.1735 −0.503950
792792 0 0
793793 −8.12616 −0.288569
794794 0 0
795795 0 0
796796 0 0
797797 20.2226 0.716320 0.358160 0.933660i 0.383404π-0.383404\pi
0.358160 + 0.933660i 0.383404π0.383404\pi
798798 0 0
799799 −1.89622 −0.0670836
800800 0 0
801801 0 0
802802 0 0
803803 12.4760 0.440268
804804 0 0
805805 1.28175 0.0451759
806806 0 0
807807 0 0
808808 0 0
809809 −16.2186 −0.570215 −0.285107 0.958496i 0.592029π-0.592029\pi
−0.285107 + 0.958496i 0.592029π0.592029\pi
810810 0 0
811811 −15.3107 −0.537632 −0.268816 0.963191i 0.586632π-0.586632\pi
−0.268816 + 0.963191i 0.586632π0.586632\pi
812812 0 0
813813 0 0
814814 0 0
815815 19.3081 0.676335
816816 0 0
817817 −1.41010 −0.0493332
818818 0 0
819819 0 0
820820 0 0
821821 28.8945 1.00843 0.504213 0.863579i 0.331783π-0.331783\pi
0.504213 + 0.863579i 0.331783π0.331783\pi
822822 0 0
823823 −55.3879 −1.93070 −0.965350 0.260960i 0.915961π-0.915961\pi
−0.965350 + 0.260960i 0.915961π0.915961\pi
824824 0 0
825825 0 0
826826 0 0
827827 −14.8094 −0.514975 −0.257487 0.966282i 0.582895π-0.582895\pi
−0.257487 + 0.966282i 0.582895π0.582895\pi
828828 0 0
829829 8.75100 0.303935 0.151967 0.988386i 0.451439π-0.451439\pi
0.151967 + 0.988386i 0.451439π0.451439\pi
830830 0 0
831831 0 0
832832 0 0
833833 2.98386 0.103385
834834 0 0
835835 −23.4248 −0.810647
836836 0 0
837837 0 0
838838 0 0
839839 37.9859 1.31142 0.655709 0.755014i 0.272370π-0.272370\pi
0.655709 + 0.755014i 0.272370π0.272370\pi
840840 0 0
841841 34.2649 1.18155
842842 0 0
843843 0 0
844844 0 0
845845 17.7533 0.610733
846846 0 0
847847 19.3233 0.663957
848848 0 0
849849 0 0
850850 0 0
851851 −7.98827 −0.273834
852852 0 0
853853 −6.37246 −0.218189 −0.109094 0.994031i 0.534795π-0.534795\pi
−0.109094 + 0.994031i 0.534795π0.534795\pi
854854 0 0
855855 0 0
856856 0 0
857857 58.3717 1.99394 0.996970 0.0777879i 0.0247857π-0.0247857\pi
0.996970 + 0.0777879i 0.0247857π0.0247857\pi
858858 0 0
859859 −14.4372 −0.492592 −0.246296 0.969195i 0.579214π-0.579214\pi
−0.246296 + 0.969195i 0.579214π0.579214\pi
860860 0 0
861861 0 0
862862 0 0
863863 −35.4975 −1.20835 −0.604174 0.796852i 0.706497π-0.706497\pi
−0.604174 + 0.796852i 0.706497π0.706497\pi
864864 0 0
865865 −23.3443 −0.793731
866866 0 0
867867 0 0
868868 0 0
869869 36.4607 1.23685
870870 0 0
871871 −3.82287 −0.129533
872872 0 0
873873 0 0
874874 0 0
875875 11.6967 0.395422
876876 0 0
877877 12.3429 0.416789 0.208394 0.978045i 0.433176π-0.433176\pi
0.208394 + 0.978045i 0.433176π0.433176\pi
878878 0 0
879879 0 0
880880 0 0
881881 28.5321 0.961270 0.480635 0.876921i 0.340406π-0.340406\pi
0.480635 + 0.876921i 0.340406π0.340406\pi
882882 0 0
883883 −35.2677 −1.18685 −0.593426 0.804888i 0.702225π-0.702225\pi
−0.593426 + 0.804888i 0.702225π0.702225\pi
884884 0 0
885885 0 0
886886 0 0
887887 −14.2146 −0.477280 −0.238640 0.971108i 0.576702π-0.576702\pi
−0.238640 + 0.971108i 0.576702π0.576702\pi
888888 0 0
889889 13.8850 0.465687
890890 0 0
891891 0 0
892892 0 0
893893 −0.635493 −0.0212660
894894 0 0
895895 −23.3963 −0.782052
896896 0 0
897897 0 0
898898 0 0
899899 53.0310 1.76868
900900 0 0
901901 −1.03949 −0.0346304
902902 0 0
903903 0 0
904904 0 0
905905 6.08690 0.202336
906906 0 0
907907 0.342749 0.0113808 0.00569040 0.999984i 0.498189π-0.498189\pi
0.00569040 + 0.999984i 0.498189π0.498189\pi
908908 0 0
909909 0 0
910910 0 0
911911 −38.8036 −1.28562 −0.642810 0.766026i 0.722232π-0.722232\pi
−0.642810 + 0.766026i 0.722232π0.722232\pi
912912 0 0
913913 12.8852 0.426439
914914 0 0
915915 0 0
916916 0 0
917917 5.30511 0.175190
918918 0 0
919919 5.52158 0.182140 0.0910700 0.995844i 0.470971π-0.470971\pi
0.0910700 + 0.995844i 0.470971π0.470971\pi
920920 0 0
921921 0 0
922922 0 0
923923 −16.8828 −0.555704
924924 0 0
925925 −25.4449 −0.836624
926926 0 0
927927 0 0
928928 0 0
929929 15.7718 0.517455 0.258727 0.965950i 0.416697π-0.416697\pi
0.258727 + 0.965950i 0.416697π0.416697\pi
930930 0 0
931931 1.00000 0.0327737
932932 0 0
933933 0 0
934934 0 0
935935 25.0213 0.818283
936936 0 0
937937 29.4650 0.962579 0.481290 0.876562i 0.340169π-0.340169\pi
0.481290 + 0.876562i 0.340169π0.340169\pi
938938 0 0
939939 0 0
940940 0 0
941941 −1.12235 −0.0365876 −0.0182938 0.999833i 0.505823π-0.505823\pi
−0.0182938 + 0.999833i 0.505823π0.505823\pi
942942 0 0
943943 8.37831 0.272835
944944 0 0
945945 0 0
946946 0 0
947947 21.8952 0.711497 0.355748 0.934582i 0.384226π-0.384226\pi
0.355748 + 0.934582i 0.384226π0.384226\pi
948948 0 0
949949 −2.62424 −0.0851865
950950 0 0
951951 0 0
952952 0 0
953953 1.88912 0.0611947 0.0305974 0.999532i 0.490259π-0.490259\pi
0.0305974 + 0.999532i 0.490259π0.490259\pi
954954 0 0
955955 −14.9132 −0.482580
956956 0 0
957957 0 0
958958 0 0
959959 1.06221 0.0343007
960960 0 0
961961 13.4527 0.433957
962962 0 0
963963 0 0
964964 0 0
965965 30.6397 0.986326
966966 0 0
967967 −19.9874 −0.642752 −0.321376 0.946952i 0.604145π-0.604145\pi
−0.321376 + 0.946952i 0.604145π0.604145\pi
968968 0 0
969969 0 0
970970 0 0
971971 48.0256 1.54121 0.770607 0.637311i 0.219953π-0.219953\pi
0.770607 + 0.637311i 0.219953π0.219953\pi
972972 0 0
973973 18.8407 0.604007
974974 0 0
975975 0 0
976976 0 0
977977 14.9221 0.477401 0.238701 0.971093i 0.423279π-0.423279\pi
0.238701 + 0.971093i 0.423279π0.423279\pi
978978 0 0
979979 8.63661 0.276027
980980 0 0
981981 0 0
982982 0 0
983983 15.0194 0.479045 0.239523 0.970891i 0.423009π-0.423009\pi
0.239523 + 0.970891i 0.423009π0.423009\pi
984984 0 0
985985 −25.1601 −0.801669
986986 0 0
987987 0 0
988988 0 0
989989 1.18690 0.0377411
990990 0 0
991991 40.2089 1.27728 0.638640 0.769506i 0.279497π-0.279497\pi
0.638640 + 0.769506i 0.279497π0.279497\pi
992992 0 0
993993 0 0
994994 0 0
995995 −19.5792 −0.620701
996996 0 0
997997 −25.0968 −0.794824 −0.397412 0.917640i 0.630092π-0.630092\pi
−0.397412 + 0.917640i 0.630092π0.630092\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.ci.1.2 4
3.2 odd 2 1064.2.a.h.1.2 4
12.11 even 2 2128.2.a.t.1.4 4
21.20 even 2 7448.2.a.bj.1.3 4
24.5 odd 2 8512.2.a.bq.1.3 4
24.11 even 2 8512.2.a.bu.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.a.h.1.2 4 3.2 odd 2
2128.2.a.t.1.4 4 12.11 even 2
7448.2.a.bj.1.3 4 21.20 even 2
8512.2.a.bq.1.3 4 24.5 odd 2
8512.2.a.bu.1.1 4 24.11 even 2
9576.2.a.ci.1.2 4 1.1 even 1 trivial