Properties

Label 5220.2.a.y.1.5
Level 52205220
Weight 22
Character 5220.1
Self dual yes
Analytic conductor 41.68241.682
Analytic rank 00
Dimension 77
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] = [5220,2,Mod(1,5220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5220.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 5220=2232529 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29
Weight: k k == 2 2
Character orbit: [χ][\chi] == 5220.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: 41.681909855141.6819098551
Analytic rank: 00
Dimension: 77
Coefficient field: Q[x]/(x7)\mathbb{Q}[x]/(x^{7} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x73x622x5+38x4+81x375x242x+18 x^{7} - 3x^{6} - 22x^{5} + 38x^{4} + 81x^{3} - 75x^{2} - 42x + 18 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.5
Root 3.73648-3.73648 of defining polynomial
Character χ\chi == 5220.1

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) == q1.00000q5+2.23366q70.862773q11+6.19586q13+6.72420q17+3.79733q190.114019q23+1.00000q251.00000q29+1.48490q312.23366q35+5.84818q373.84818q412.66310q4311.5650q472.01078q49+6.85596q53+0.862773q55+0.814916q590.312424q616.19586q656.15806q67+2.31242q71+10.9170q731.92714q776.57956q798.62530q836.72420q858.93763q89+13.8394q913.79733q95+6.87902q97+O(q100)q-1.00000 q^{5} +2.23366 q^{7} -0.862773 q^{11} +6.19586 q^{13} +6.72420 q^{17} +3.79733 q^{19} -0.114019 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.48490 q^{31} -2.23366 q^{35} +5.84818 q^{37} -3.84818 q^{41} -2.66310 q^{43} -11.5650 q^{47} -2.01078 q^{49} +6.85596 q^{53} +0.862773 q^{55} +0.814916 q^{59} -0.312424 q^{61} -6.19586 q^{65} -6.15806 q^{67} +2.31242 q^{71} +10.9170 q^{73} -1.92714 q^{77} -6.57956 q^{79} -8.62530 q^{83} -6.72420 q^{85} -8.93763 q^{89} +13.8394 q^{91} -3.79733 q^{95} +6.87902 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 7q7q53q11+2q138q17+6q19q23+7q257q292q31+15q37q41+9q434q47+29q4917q53+3q55+4q59+6q61++15q97+O(q100) 7 q - 7 q^{5} - 3 q^{11} + 2 q^{13} - 8 q^{17} + 6 q^{19} - q^{23} + 7 q^{25} - 7 q^{29} - 2 q^{31} + 15 q^{37} - q^{41} + 9 q^{43} - 4 q^{47} + 29 q^{49} - 17 q^{53} + 3 q^{55} + 4 q^{59} + 6 q^{61}+ \cdots + 15 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 −1.00000 −0.447214
66 0 0
77 2.23366 0.844243 0.422122 0.906539i 0.361286π-0.361286\pi
0.422122 + 0.906539i 0.361286π0.361286\pi
88 0 0
99 0 0
1010 0 0
1111 −0.862773 −0.260136 −0.130068 0.991505i 0.541520π-0.541520\pi
−0.130068 + 0.991505i 0.541520π0.541520\pi
1212 0 0
1313 6.19586 1.71842 0.859211 0.511621i 0.170955π-0.170955\pi
0.859211 + 0.511621i 0.170955π0.170955\pi
1414 0 0
1515 0 0
1616 0 0
1717 6.72420 1.63086 0.815430 0.578856i 0.196501π-0.196501\pi
0.815430 + 0.578856i 0.196501π0.196501\pi
1818 0 0
1919 3.79733 0.871167 0.435583 0.900148i 0.356542π-0.356542\pi
0.435583 + 0.900148i 0.356542π0.356542\pi
2020 0 0
2121 0 0
2222 0 0
2323 −0.114019 −0.0237745 −0.0118873 0.999929i 0.503784π-0.503784\pi
−0.0118873 + 0.999929i 0.503784π0.503784\pi
2424 0 0
2525 1.00000 0.200000
2626 0 0
2727 0 0
2828 0 0
2929 −1.00000 −0.185695
3030 0 0
3131 1.48490 0.266696 0.133348 0.991069i 0.457427π-0.457427\pi
0.133348 + 0.991069i 0.457427π0.457427\pi
3232 0 0
3333 0 0
3434 0 0
3535 −2.23366 −0.377557
3636 0 0
3737 5.84818 0.961436 0.480718 0.876875i 0.340376π-0.340376\pi
0.480718 + 0.876875i 0.340376π0.340376\pi
3838 0 0
3939 0 0
4040 0 0
4141 −3.84818 −0.600985 −0.300493 0.953784i 0.597151π-0.597151\pi
−0.300493 + 0.953784i 0.597151π0.597151\pi
4242 0 0
4343 −2.66310 −0.406119 −0.203059 0.979166i 0.565088π-0.565088\pi
−0.203059 + 0.979166i 0.565088π0.565088\pi
4444 0 0
4545 0 0
4646 0 0
4747 −11.5650 −1.68692 −0.843462 0.537188i 0.819486π-0.819486\pi
−0.843462 + 0.537188i 0.819486π0.819486\pi
4848 0 0
4949 −2.01078 −0.287254
5050 0 0
5151 0 0
5252 0 0
5353 6.85596 0.941738 0.470869 0.882203i 0.343940π-0.343940\pi
0.470869 + 0.882203i 0.343940π0.343940\pi
5454 0 0
5555 0.862773 0.116336
5656 0 0
5757 0 0
5858 0 0
5959 0.814916 0.106093 0.0530465 0.998592i 0.483107π-0.483107\pi
0.0530465 + 0.998592i 0.483107π0.483107\pi
6060 0 0
6161 −0.312424 −0.0400017 −0.0200009 0.999800i 0.506367π-0.506367\pi
−0.0200009 + 0.999800i 0.506367π0.506367\pi
6262 0 0
6363 0 0
6464 0 0
6565 −6.19586 −0.768502
6666 0 0
6767 −6.15806 −0.752327 −0.376164 0.926553i 0.622757π-0.622757\pi
−0.376164 + 0.926553i 0.622757π0.622757\pi
6868 0 0
6969 0 0
7070 0 0
7171 2.31242 0.274434 0.137217 0.990541i 0.456184π-0.456184\pi
0.137217 + 0.990541i 0.456184π0.456184\pi
7272 0 0
7373 10.9170 1.27774 0.638868 0.769316i 0.279403π-0.279403\pi
0.638868 + 0.769316i 0.279403π0.279403\pi
7474 0 0
7575 0 0
7676 0 0
7777 −1.92714 −0.219618
7878 0 0
7979 −6.57956 −0.740258 −0.370129 0.928980i 0.620687π-0.620687\pi
−0.370129 + 0.928980i 0.620687π0.620687\pi
8080 0 0
8181 0 0
8282 0 0
8383 −8.62530 −0.946750 −0.473375 0.880861i 0.656964π-0.656964\pi
−0.473375 + 0.880861i 0.656964π0.656964\pi
8484 0 0
8585 −6.72420 −0.729342
8686 0 0
8787 0 0
8888 0 0
8989 −8.93763 −0.947387 −0.473693 0.880690i 0.657080π-0.657080\pi
−0.473693 + 0.880690i 0.657080π0.657080\pi
9090 0 0
9191 13.8394 1.45077
9292 0 0
9393 0 0
9494 0 0
9595 −3.79733 −0.389598
9696 0 0
9797 6.87902 0.698459 0.349229 0.937037i 0.386443π-0.386443\pi
0.349229 + 0.937037i 0.386443π0.386443\pi
9898 0 0
9999 0 0
100100 0 0
101101 −6.54654 −0.651405 −0.325702 0.945472i 0.605601π-0.605601\pi
−0.325702 + 0.945472i 0.605601π0.605601\pi
102102 0 0
103103 16.1111 1.58747 0.793735 0.608264i 0.208134π-0.208134\pi
0.793735 + 0.608264i 0.208134π0.208134\pi
104104 0 0
105105 0 0
106106 0 0
107107 16.9837 1.64188 0.820939 0.571016i 0.193451π-0.193451\pi
0.820939 + 0.571016i 0.193451π0.193451\pi
108108 0 0
109109 −13.8805 −1.32951 −0.664754 0.747062i 0.731464π-0.731464\pi
−0.664754 + 0.747062i 0.731464π0.731464\pi
110110 0 0
111111 0 0
112112 0 0
113113 11.6770 1.09848 0.549238 0.835666i 0.314918π-0.314918\pi
0.549238 + 0.835666i 0.314918π0.314918\pi
114114 0 0
115115 0.114019 0.0106323
116116 0 0
117117 0 0
118118 0 0
119119 15.0196 1.37684
120120 0 0
121121 −10.2556 −0.932329
122122 0 0
123123 0 0
124124 0 0
125125 −1.00000 −0.0894427
126126 0 0
127127 −9.09773 −0.807293 −0.403647 0.914915i 0.632257π-0.632257\pi
−0.403647 + 0.914915i 0.632257π0.632257\pi
128128 0 0
129129 0 0
130130 0 0
131131 1.28904 0.112624 0.0563121 0.998413i 0.482066π-0.482066\pi
0.0563121 + 0.998413i 0.482066π0.482066\pi
132132 0 0
133133 8.48193 0.735476
134134 0 0
135135 0 0
136136 0 0
137137 16.3598 1.39771 0.698857 0.715261i 0.253692π-0.253692\pi
0.698857 + 0.715261i 0.253692π0.253692\pi
138138 0 0
139139 −3.73162 −0.316512 −0.158256 0.987398i 0.550587π-0.550587\pi
−0.158256 + 0.987398i 0.550587π0.550587\pi
140140 0 0
141141 0 0
142142 0 0
143143 −5.34562 −0.447023
144144 0 0
145145 1.00000 0.0830455
146146 0 0
147147 0 0
148148 0 0
149149 −1.28223 −0.105044 −0.0525222 0.998620i 0.516726π-0.516726\pi
−0.0525222 + 0.998620i 0.516726π0.516726\pi
150150 0 0
151151 20.7941 1.69220 0.846100 0.533024i 0.178945π-0.178945\pi
0.846100 + 0.533024i 0.178945π0.178945\pi
152152 0 0
153153 0 0
154154 0 0
155155 −1.48490 −0.119270
156156 0 0
157157 23.7487 1.89535 0.947677 0.319231i 0.103425π-0.103425\pi
0.947677 + 0.319231i 0.103425π0.103425\pi
158158 0 0
159159 0 0
160160 0 0
161161 −0.254678 −0.0200715
162162 0 0
163163 −13.5073 −1.05797 −0.528987 0.848630i 0.677428π-0.677428\pi
−0.528987 + 0.848630i 0.677428π0.677428\pi
164164 0 0
165165 0 0
166166 0 0
167167 18.6135 1.44036 0.720180 0.693788i 0.244059π-0.244059\pi
0.720180 + 0.693788i 0.244059π0.244059\pi
168168 0 0
169169 25.3887 1.95298
170170 0 0
171171 0 0
172172 0 0
173173 −1.88336 −0.143189 −0.0715947 0.997434i 0.522809π-0.522809\pi
−0.0715947 + 0.997434i 0.522809π0.522809\pi
174174 0 0
175175 2.23366 0.168849
176176 0 0
177177 0 0
178178 0 0
179179 4.11447 0.307530 0.153765 0.988107i 0.450860π-0.450860\pi
0.153765 + 0.988107i 0.450860π0.450860\pi
180180 0 0
181181 15.1334 1.12486 0.562429 0.826846i 0.309867π-0.309867\pi
0.562429 + 0.826846i 0.309867π0.309867\pi
182182 0 0
183183 0 0
184184 0 0
185185 −5.84818 −0.429967
186186 0 0
187187 −5.80146 −0.424245
188188 0 0
189189 0 0
190190 0 0
191191 −18.2790 −1.32262 −0.661311 0.750112i 0.730000π-0.730000\pi
−0.661311 + 0.750112i 0.730000π0.730000\pi
192192 0 0
193193 −8.34771 −0.600881 −0.300441 0.953801i 0.597134π-0.597134\pi
−0.300441 + 0.953801i 0.597134π0.597134\pi
194194 0 0
195195 0 0
196196 0 0
197197 −20.8912 −1.48844 −0.744220 0.667935i 0.767179π-0.767179\pi
−0.744220 + 0.667935i 0.767179π0.767179\pi
198198 0 0
199199 1.29348 0.0916926 0.0458463 0.998949i 0.485402π-0.485402\pi
0.0458463 + 0.998949i 0.485402π0.485402\pi
200200 0 0
201201 0 0
202202 0 0
203203 −2.23366 −0.156772
204204 0 0
205205 3.84818 0.268769
206206 0 0
207207 0 0
208208 0 0
209209 −3.27623 −0.226622
210210 0 0
211211 −17.2837 −1.18986 −0.594930 0.803778i 0.702820π-0.702820\pi
−0.594930 + 0.803778i 0.702820π0.702820\pi
212212 0 0
213213 0 0
214214 0 0
215215 2.66310 0.181622
216216 0 0
217217 3.31676 0.225157
218218 0 0
219219 0 0
220220 0 0
221221 41.6622 2.80250
222222 0 0
223223 19.9516 1.33606 0.668028 0.744136i 0.267139π-0.267139\pi
0.668028 + 0.744136i 0.267139π0.267139\pi
224224 0 0
225225 0 0
226226 0 0
227227 16.9351 1.12402 0.562011 0.827130i 0.310028π-0.310028\pi
0.562011 + 0.827130i 0.310028π0.310028\pi
228228 0 0
229229 26.4182 1.74576 0.872882 0.487931i 0.162248π-0.162248\pi
0.872882 + 0.487931i 0.162248π0.162248\pi
230230 0 0
231231 0 0
232232 0 0
233233 −0.481568 −0.0315485 −0.0157743 0.999876i 0.505021π-0.505021\pi
−0.0157743 + 0.999876i 0.505021π0.505021\pi
234234 0 0
235235 11.5650 0.754416
236236 0 0
237237 0 0
238238 0 0
239239 −10.4889 −0.678468 −0.339234 0.940702i 0.610168π-0.610168\pi
−0.339234 + 0.940702i 0.610168π0.610168\pi
240240 0 0
241241 −6.18410 −0.398353 −0.199177 0.979964i 0.563827π-0.563827\pi
−0.199177 + 0.979964i 0.563827π0.563827\pi
242242 0 0
243243 0 0
244244 0 0
245245 2.01078 0.128464
246246 0 0
247247 23.5277 1.49703
248248 0 0
249249 0 0
250250 0 0
251251 25.1896 1.58995 0.794975 0.606642i 0.207484π-0.207484\pi
0.794975 + 0.606642i 0.207484π0.207484\pi
252252 0 0
253253 0.0983720 0.00618460
254254 0 0
255255 0 0
256256 0 0
257257 −18.2072 −1.13574 −0.567868 0.823120i 0.692231π-0.692231\pi
−0.567868 + 0.823120i 0.692231π0.692231\pi
258258 0 0
259259 13.0628 0.811685
260260 0 0
261261 0 0
262262 0 0
263263 −25.7119 −1.58546 −0.792732 0.609570i 0.791342π-0.791342\pi
−0.792732 + 0.609570i 0.791342π0.791342\pi
264264 0 0
265265 −6.85596 −0.421158
266266 0 0
267267 0 0
268268 0 0
269269 −13.7347 −0.837419 −0.418710 0.908120i 0.637517π-0.637517\pi
−0.418710 + 0.908120i 0.637517π0.637517\pi
270270 0 0
271271 8.26714 0.502193 0.251096 0.967962i 0.419209π-0.419209\pi
0.251096 + 0.967962i 0.419209π0.419209\pi
272272 0 0
273273 0 0
274274 0 0
275275 −0.862773 −0.0520271
276276 0 0
277277 −19.5738 −1.17607 −0.588037 0.808834i 0.700099π-0.700099\pi
−0.588037 + 0.808834i 0.700099π0.700099\pi
278278 0 0
279279 0 0
280280 0 0
281281 −8.15135 −0.486269 −0.243134 0.969993i 0.578176π-0.578176\pi
−0.243134 + 0.969993i 0.578176π0.578176\pi
282282 0 0
283283 −4.92175 −0.292567 −0.146284 0.989243i 0.546731π-0.546731\pi
−0.146284 + 0.989243i 0.546731π0.546731\pi
284284 0 0
285285 0 0
286286 0 0
287287 −8.59552 −0.507378
288288 0 0
289289 28.2149 1.65970
290290 0 0
291291 0 0
292292 0 0
293293 −6.04097 −0.352917 −0.176459 0.984308i 0.556464π-0.556464\pi
−0.176459 + 0.984308i 0.556464π0.556464\pi
294294 0 0
295295 −0.814916 −0.0474462
296296 0 0
297297 0 0
298298 0 0
299299 −0.706443 −0.0408546
300300 0 0
301301 −5.94845 −0.342863
302302 0 0
303303 0 0
304304 0 0
305305 0.312424 0.0178893
306306 0 0
307307 −0.589954 −0.0336704 −0.0168352 0.999858i 0.505359π-0.505359\pi
−0.0168352 + 0.999858i 0.505359π0.505359\pi
308308 0 0
309309 0 0
310310 0 0
311311 6.08054 0.344796 0.172398 0.985027i 0.444849π-0.444849\pi
0.172398 + 0.985027i 0.444849π0.444849\pi
312312 0 0
313313 −16.2276 −0.917241 −0.458620 0.888632i 0.651656π-0.651656\pi
−0.458620 + 0.888632i 0.651656π0.651656\pi
314314 0 0
315315 0 0
316316 0 0
317317 6.96348 0.391108 0.195554 0.980693i 0.437349π-0.437349\pi
0.195554 + 0.980693i 0.437349π0.437349\pi
318318 0 0
319319 0.862773 0.0483060
320320 0 0
321321 0 0
322322 0 0
323323 25.5340 1.42075
324324 0 0
325325 6.19586 0.343684
326326 0 0
327327 0 0
328328 0 0
329329 −25.8322 −1.42417
330330 0 0
331331 −3.21645 −0.176792 −0.0883960 0.996085i 0.528174π-0.528174\pi
−0.0883960 + 0.996085i 0.528174π0.528174\pi
332332 0 0
333333 0 0
334334 0 0
335335 6.15806 0.336451
336336 0 0
337337 13.2124 0.719724 0.359862 0.933006i 0.382824π-0.382824\pi
0.359862 + 0.933006i 0.382824π0.382824\pi
338338 0 0
339339 0 0
340340 0 0
341341 −1.28113 −0.0693773
342342 0 0
343343 −20.1270 −1.08675
344344 0 0
345345 0 0
346346 0 0
347347 16.9781 0.911434 0.455717 0.890125i 0.349383π-0.349383\pi
0.455717 + 0.890125i 0.349383π0.349383\pi
348348 0 0
349349 −0.578713 −0.0309778 −0.0154889 0.999880i 0.504930π-0.504930\pi
−0.0154889 + 0.999880i 0.504930π0.504930\pi
350350 0 0
351351 0 0
352352 0 0
353353 −8.09715 −0.430968 −0.215484 0.976507i 0.569133π-0.569133\pi
−0.215484 + 0.976507i 0.569133π0.569133\pi
354354 0 0
355355 −2.31242 −0.122731
356356 0 0
357357 0 0
358358 0 0
359359 −13.9175 −0.734538 −0.367269 0.930115i 0.619707π-0.619707\pi
−0.367269 + 0.930115i 0.619707π0.619707\pi
360360 0 0
361361 −4.58031 −0.241069
362362 0 0
363363 0 0
364364 0 0
365365 −10.9170 −0.571421
366366 0 0
367367 1.84218 0.0961612 0.0480806 0.998843i 0.484690π-0.484690\pi
0.0480806 + 0.998843i 0.484690π0.484690\pi
368368 0 0
369369 0 0
370370 0 0
371371 15.3139 0.795056
372372 0 0
373373 30.9107 1.60050 0.800249 0.599668i 0.204701π-0.204701\pi
0.800249 + 0.599668i 0.204701π0.204701\pi
374374 0 0
375375 0 0
376376 0 0
377377 −6.19586 −0.319103
378378 0 0
379379 14.1422 0.726438 0.363219 0.931704i 0.381678π-0.381678\pi
0.363219 + 0.931704i 0.381678π0.381678\pi
380380 0 0
381381 0 0
382382 0 0
383383 −35.4290 −1.81034 −0.905170 0.425050i 0.860256π-0.860256\pi
−0.905170 + 0.425050i 0.860256π0.860256\pi
384384 0 0
385385 1.92714 0.0982160
386386 0 0
387387 0 0
388388 0 0
389389 −6.78581 −0.344054 −0.172027 0.985092i 0.555032π-0.555032\pi
−0.172027 + 0.985092i 0.555032π0.555032\pi
390390 0 0
391391 −0.766684 −0.0387729
392392 0 0
393393 0 0
394394 0 0
395395 6.57956 0.331054
396396 0 0
397397 6.07559 0.304925 0.152463 0.988309i 0.451280π-0.451280\pi
0.152463 + 0.988309i 0.451280π0.451280\pi
398398 0 0
399399 0 0
400400 0 0
401401 34.6086 1.72827 0.864135 0.503260i 0.167866π-0.167866\pi
0.864135 + 0.503260i 0.167866π0.167866\pi
402402 0 0
403403 9.20025 0.458297
404404 0 0
405405 0 0
406406 0 0
407407 −5.04565 −0.250104
408408 0 0
409409 20.1817 0.997922 0.498961 0.866624i 0.333715π-0.333715\pi
0.498961 + 0.866624i 0.333715π0.333715\pi
410410 0 0
411411 0 0
412412 0 0
413413 1.82024 0.0895683
414414 0 0
415415 8.62530 0.423399
416416 0 0
417417 0 0
418418 0 0
419419 19.9108 0.972705 0.486353 0.873763i 0.338327π-0.338327\pi
0.486353 + 0.873763i 0.338327π0.338327\pi
420420 0 0
421421 −3.38394 −0.164923 −0.0824617 0.996594i 0.526278π-0.526278\pi
−0.0824617 + 0.996594i 0.526278π0.526278\pi
422422 0 0
423423 0 0
424424 0 0
425425 6.72420 0.326172
426426 0 0
427427 −0.697847 −0.0337712
428428 0 0
429429 0 0
430430 0 0
431431 40.3943 1.94573 0.972863 0.231382i 0.0743246π-0.0743246\pi
0.972863 + 0.231382i 0.0743246π0.0743246\pi
432432 0 0
433433 3.44005 0.165318 0.0826590 0.996578i 0.473659π-0.473659\pi
0.0826590 + 0.996578i 0.473659π0.473659\pi
434434 0 0
435435 0 0
436436 0 0
437437 −0.432966 −0.0207115
438438 0 0
439439 −25.2149 −1.20344 −0.601721 0.798706i 0.705518π-0.705518\pi
−0.601721 + 0.798706i 0.705518π0.705518\pi
440440 0 0
441441 0 0
442442 0 0
443443 −25.0925 −1.19218 −0.596091 0.802917i 0.703280π-0.703280\pi
−0.596091 + 0.802917i 0.703280π0.703280\pi
444444 0 0
445445 8.93763 0.423684
446446 0 0
447447 0 0
448448 0 0
449449 −23.9624 −1.13086 −0.565429 0.824797i 0.691289π-0.691289\pi
−0.565429 + 0.824797i 0.691289π0.691289\pi
450450 0 0
451451 3.32011 0.156338
452452 0 0
453453 0 0
454454 0 0
455455 −13.8394 −0.648802
456456 0 0
457457 11.0245 0.515705 0.257852 0.966184i 0.416985π-0.416985\pi
0.257852 + 0.966184i 0.416985π0.416985\pi
458458 0 0
459459 0 0
460460 0 0
461461 39.9276 1.85962 0.929808 0.368046i 0.119973π-0.119973\pi
0.929808 + 0.368046i 0.119973π0.119973\pi
462462 0 0
463463 −17.9300 −0.833279 −0.416639 0.909072i 0.636792π-0.636792\pi
−0.416639 + 0.909072i 0.636792π0.636792\pi
464464 0 0
465465 0 0
466466 0 0
467467 −2.15485 −0.0997146 −0.0498573 0.998756i 0.515877π-0.515877\pi
−0.0498573 + 0.998756i 0.515877π0.515877\pi
468468 0 0
469469 −13.7550 −0.635147
470470 0 0
471471 0 0
472472 0 0
473473 2.29765 0.105646
474474 0 0
475475 3.79733 0.174233
476476 0 0
477477 0 0
478478 0 0
479479 27.1212 1.23920 0.619600 0.784918i 0.287295π-0.287295\pi
0.619600 + 0.784918i 0.287295π0.287295\pi
480480 0 0
481481 36.2345 1.65215
482482 0 0
483483 0 0
484484 0 0
485485 −6.87902 −0.312360
486486 0 0
487487 −29.9751 −1.35830 −0.679150 0.733999i 0.737651π-0.737651\pi
−0.679150 + 0.733999i 0.737651π0.737651\pi
488488 0 0
489489 0 0
490490 0 0
491491 2.62107 0.118287 0.0591437 0.998249i 0.481163π-0.481163\pi
0.0591437 + 0.998249i 0.481163π0.481163\pi
492492 0 0
493493 −6.72420 −0.302843
494494 0 0
495495 0 0
496496 0 0
497497 5.16516 0.231689
498498 0 0
499499 5.69586 0.254982 0.127491 0.991840i 0.459308π-0.459308\pi
0.127491 + 0.991840i 0.459308π0.459308\pi
500500 0 0
501501 0 0
502502 0 0
503503 −14.9136 −0.664966 −0.332483 0.943109i 0.607886π-0.607886\pi
−0.332483 + 0.943109i 0.607886π0.607886\pi
504504 0 0
505505 6.54654 0.291317
506506 0 0
507507 0 0
508508 0 0
509509 28.6905 1.27168 0.635842 0.771819i 0.280653π-0.280653\pi
0.635842 + 0.771819i 0.280653π0.280653\pi
510510 0 0
511511 24.3848 1.07872
512512 0 0
513513 0 0
514514 0 0
515515 −16.1111 −0.709938
516516 0 0
517517 9.97794 0.438829
518518 0 0
519519 0 0
520520 0 0
521521 15.5920 0.683098 0.341549 0.939864i 0.389048π-0.389048\pi
0.341549 + 0.939864i 0.389048π0.389048\pi
522522 0 0
523523 8.48834 0.371169 0.185584 0.982628i 0.440582π-0.440582\pi
0.185584 + 0.982628i 0.440582π0.440582\pi
524524 0 0
525525 0 0
526526 0 0
527527 9.98479 0.434944
528528 0 0
529529 −22.9870 −0.999435
530530 0 0
531531 0 0
532532 0 0
533533 −23.8428 −1.03275
534534 0 0
535535 −16.9837 −0.734270
536536 0 0
537537 0 0
538538 0 0
539539 1.73484 0.0747249
540540 0 0
541541 23.9357 1.02908 0.514538 0.857468i 0.327963π-0.327963\pi
0.514538 + 0.857468i 0.327963π0.327963\pi
542542 0 0
543543 0 0
544544 0 0
545545 13.8805 0.594574
546546 0 0
547547 24.2718 1.03779 0.518893 0.854839i 0.326344π-0.326344\pi
0.518893 + 0.854839i 0.326344π0.326344\pi
548548 0 0
549549 0 0
550550 0 0
551551 −3.79733 −0.161772
552552 0 0
553553 −14.6965 −0.624958
554554 0 0
555555 0 0
556556 0 0
557557 −24.0359 −1.01843 −0.509216 0.860639i 0.670065π-0.670065\pi
−0.509216 + 0.860639i 0.670065π0.670065\pi
558558 0 0
559559 −16.5002 −0.697884
560560 0 0
561561 0 0
562562 0 0
563563 −12.7209 −0.536123 −0.268061 0.963402i 0.586383π-0.586383\pi
−0.268061 + 0.963402i 0.586383π0.586383\pi
564564 0 0
565565 −11.6770 −0.491253
566566 0 0
567567 0 0
568568 0 0
569569 −9.11139 −0.381970 −0.190985 0.981593i 0.561168π-0.561168\pi
−0.190985 + 0.981593i 0.561168π0.561168\pi
570570 0 0
571571 10.7613 0.450347 0.225173 0.974319i 0.427705π-0.427705\pi
0.225173 + 0.974319i 0.427705π0.427705\pi
572572 0 0
573573 0 0
574574 0 0
575575 −0.114019 −0.00475490
576576 0 0
577577 −1.34669 −0.0560636 −0.0280318 0.999607i 0.508924π-0.508924\pi
−0.0280318 + 0.999607i 0.508924π0.508924\pi
578578 0 0
579579 0 0
580580 0 0
581581 −19.2660 −0.799287
582582 0 0
583583 −5.91513 −0.244980
584584 0 0
585585 0 0
586586 0 0
587587 −11.3469 −0.468335 −0.234168 0.972196i 0.575236π-0.575236\pi
−0.234168 + 0.972196i 0.575236π0.575236\pi
588588 0 0
589589 5.63866 0.232337
590590 0 0
591591 0 0
592592 0 0
593593 −22.5818 −0.927323 −0.463661 0.886012i 0.653465π-0.653465\pi
−0.463661 + 0.886012i 0.653465π0.653465\pi
594594 0 0
595595 −15.0196 −0.615742
596596 0 0
597597 0 0
598598 0 0
599599 −11.9871 −0.489782 −0.244891 0.969551i 0.578752π-0.578752\pi
−0.244891 + 0.969551i 0.578752π0.578752\pi
600600 0 0
601601 −21.3867 −0.872383 −0.436192 0.899854i 0.643673π-0.643673\pi
−0.436192 + 0.899854i 0.643673π0.643673\pi
602602 0 0
603603 0 0
604604 0 0
605605 10.2556 0.416950
606606 0 0
607607 3.91836 0.159042 0.0795208 0.996833i 0.474661π-0.474661\pi
0.0795208 + 0.996833i 0.474661π0.474661\pi
608608 0 0
609609 0 0
610610 0 0
611611 −71.6549 −2.89885
612612 0 0
613613 31.5282 1.27341 0.636706 0.771106i 0.280296π-0.280296\pi
0.636706 + 0.771106i 0.280296π0.280296\pi
614614 0 0
615615 0 0
616616 0 0
617617 −21.5381 −0.867092 −0.433546 0.901131i 0.642738π-0.642738\pi
−0.433546 + 0.901131i 0.642738π0.642738\pi
618618 0 0
619619 −1.44852 −0.0582209 −0.0291105 0.999576i 0.509267π-0.509267\pi
−0.0291105 + 0.999576i 0.509267π0.509267\pi
620620 0 0
621621 0 0
622622 0 0
623623 −19.9636 −0.799825
624624 0 0
625625 1.00000 0.0400000
626626 0 0
627627 0 0
628628 0 0
629629 39.3244 1.56797
630630 0 0
631631 −39.9229 −1.58931 −0.794654 0.607063i 0.792348π-0.792348\pi
−0.794654 + 0.607063i 0.792348π0.792348\pi
632632 0 0
633633 0 0
634634 0 0
635635 9.09773 0.361032
636636 0 0
637637 −12.4585 −0.493623
638638 0 0
639639 0 0
640640 0 0
641641 27.5532 1.08829 0.544143 0.838992i 0.316855π-0.316855\pi
0.544143 + 0.838992i 0.316855π0.316855\pi
642642 0 0
643643 −28.8043 −1.13593 −0.567965 0.823053i 0.692269π-0.692269\pi
−0.567965 + 0.823053i 0.692269π0.692269\pi
644644 0 0
645645 0 0
646646 0 0
647647 21.6477 0.851058 0.425529 0.904945i 0.360088π-0.360088\pi
0.425529 + 0.904945i 0.360088π0.360088\pi
648648 0 0
649649 −0.703087 −0.0275986
650650 0 0
651651 0 0
652652 0 0
653653 −19.9102 −0.779147 −0.389574 0.920995i 0.627378π-0.627378\pi
−0.389574 + 0.920995i 0.627378π0.627378\pi
654654 0 0
655655 −1.28904 −0.0503671
656656 0 0
657657 0 0
658658 0 0
659659 28.6588 1.11639 0.558195 0.829710i 0.311494π-0.311494\pi
0.558195 + 0.829710i 0.311494π0.311494\pi
660660 0 0
661661 11.3667 0.442113 0.221057 0.975261i 0.429049π-0.429049\pi
0.221057 + 0.975261i 0.429049π0.429049\pi
662662 0 0
663663 0 0
664664 0 0
665665 −8.48193 −0.328915
666666 0 0
667667 0.114019 0.00441481
668668 0 0
669669 0 0
670670 0 0
671671 0.269550 0.0104059
672672 0 0
673673 4.10543 0.158253 0.0791264 0.996865i 0.474787π-0.474787\pi
0.0791264 + 0.996865i 0.474787π0.474787\pi
674674 0 0
675675 0 0
676676 0 0
677677 −6.35675 −0.244310 −0.122155 0.992511i 0.538980π-0.538980\pi
−0.122155 + 0.992511i 0.538980π0.538980\pi
678678 0 0
679679 15.3654 0.589669
680680 0 0
681681 0 0
682682 0 0
683683 −25.1306 −0.961595 −0.480798 0.876832i 0.659653π-0.659653\pi
−0.480798 + 0.876832i 0.659653π0.659653\pi
684684 0 0
685685 −16.3598 −0.625077
686686 0 0
687687 0 0
688688 0 0
689689 42.4786 1.61830
690690 0 0
691691 −39.9628 −1.52026 −0.760129 0.649772i 0.774864π-0.774864\pi
−0.760129 + 0.649772i 0.774864π0.774864\pi
692692 0 0
693693 0 0
694694 0 0
695695 3.73162 0.141548
696696 0 0
697697 −25.8760 −0.980122
698698 0 0
699699 0 0
700700 0 0
701701 12.3476 0.466362 0.233181 0.972433i 0.425086π-0.425086\pi
0.233181 + 0.972433i 0.425086π0.425086\pi
702702 0 0
703703 22.2075 0.837570
704704 0 0
705705 0 0
706706 0 0
707707 −14.6227 −0.549944
708708 0 0
709709 4.89993 0.184021 0.0920103 0.995758i 0.470671π-0.470671\pi
0.0920103 + 0.995758i 0.470671π0.470671\pi
710710 0 0
711711 0 0
712712 0 0
713713 −0.169306 −0.00634058
714714 0 0
715715 5.34562 0.199915
716716 0 0
717717 0 0
718718 0 0
719719 −26.5380 −0.989699 −0.494850 0.868979i 0.664777π-0.664777\pi
−0.494850 + 0.868979i 0.664777π0.664777\pi
720720 0 0
721721 35.9866 1.34021
722722 0 0
723723 0 0
724724 0 0
725725 −1.00000 −0.0371391
726726 0 0
727727 −33.7317 −1.25104 −0.625520 0.780208i 0.715113π-0.715113\pi
−0.625520 + 0.780208i 0.715113π0.715113\pi
728728 0 0
729729 0 0
730730 0 0
731731 −17.9072 −0.662323
732732 0 0
733733 52.0372 1.92204 0.961019 0.276482i 0.0891686π-0.0891686\pi
0.961019 + 0.276482i 0.0891686π0.0891686\pi
734734 0 0
735735 0 0
736736 0 0
737737 5.31301 0.195707
738738 0 0
739739 −23.4998 −0.864456 −0.432228 0.901764i 0.642272π-0.642272\pi
−0.432228 + 0.901764i 0.642272π0.642272\pi
740740 0 0
741741 0 0
742742 0 0
743743 −42.6872 −1.56604 −0.783021 0.621995i 0.786322π-0.786322\pi
−0.783021 + 0.621995i 0.786322π0.786322\pi
744744 0 0
745745 1.28223 0.0469773
746746 0 0
747747 0 0
748748 0 0
749749 37.9358 1.38614
750750 0 0
751751 −39.1265 −1.42775 −0.713873 0.700276i 0.753060π-0.753060\pi
−0.713873 + 0.700276i 0.753060π0.753060\pi
752752 0 0
753753 0 0
754754 0 0
755755 −20.7941 −0.756775
756756 0 0
757757 36.9259 1.34209 0.671047 0.741415i 0.265845π-0.265845\pi
0.671047 + 0.741415i 0.265845π0.265845\pi
758758 0 0
759759 0 0
760760 0 0
761761 50.8039 1.84164 0.920820 0.389988i 0.127521π-0.127521\pi
0.920820 + 0.389988i 0.127521π0.127521\pi
762762 0 0
763763 −31.0042 −1.12243
764764 0 0
765765 0 0
766766 0 0
767767 5.04910 0.182313
768768 0 0
769769 −5.85269 −0.211053 −0.105527 0.994416i 0.533653π-0.533653\pi
−0.105527 + 0.994416i 0.533653π0.533653\pi
770770 0 0
771771 0 0
772772 0 0
773773 11.3796 0.409297 0.204649 0.978835i 0.434395π-0.434395\pi
0.204649 + 0.978835i 0.434395π0.434395\pi
774774 0 0
775775 1.48490 0.0533393
776776 0 0
777777 0 0
778778 0 0
779779 −14.6128 −0.523558
780780 0 0
781781 −1.99510 −0.0713901
782782 0 0
783783 0 0
784784 0 0
785785 −23.7487 −0.847628
786786 0 0
787787 24.5905 0.876557 0.438279 0.898839i 0.355588π-0.355588\pi
0.438279 + 0.898839i 0.355588π0.355588\pi
788788 0 0
789789 0 0
790790 0 0
791791 26.0823 0.927380
792792 0 0
793793 −1.93573 −0.0687399
794794 0 0
795795 0 0
796796 0 0
797797 −3.93342 −0.139329 −0.0696644 0.997570i 0.522193π-0.522193\pi
−0.0696644 + 0.997570i 0.522193π0.522193\pi
798798 0 0
799799 −77.7652 −2.75114
800800 0 0
801801 0 0
802802 0 0
803803 −9.41888 −0.332385
804804 0 0
805805 0.254678 0.00897623
806806 0 0
807807 0 0
808808 0 0
809809 −48.9615 −1.72139 −0.860697 0.509117i 0.829972π-0.829972\pi
−0.860697 + 0.509117i 0.829972π0.829972\pi
810810 0 0
811811 −19.6362 −0.689522 −0.344761 0.938691i 0.612040π-0.612040\pi
−0.344761 + 0.938691i 0.612040π0.612040\pi
812812 0 0
813813 0 0
814814 0 0
815815 13.5073 0.473140
816816 0 0
817817 −10.1127 −0.353797
818818 0 0
819819 0 0
820820 0 0
821821 16.6248 0.580211 0.290106 0.956995i 0.406310π-0.406310\pi
0.290106 + 0.956995i 0.406310π0.406310\pi
822822 0 0
823823 −28.6690 −0.999337 −0.499669 0.866217i 0.666545π-0.666545\pi
−0.499669 + 0.866217i 0.666545π0.666545\pi
824824 0 0
825825 0 0
826826 0 0
827827 46.7188 1.62457 0.812286 0.583260i 0.198223π-0.198223\pi
0.812286 + 0.583260i 0.198223π0.198223\pi
828828 0 0
829829 9.89829 0.343782 0.171891 0.985116i 0.445012π-0.445012\pi
0.171891 + 0.985116i 0.445012π0.445012\pi
830830 0 0
831831 0 0
832832 0 0
833833 −13.5209 −0.468470
834834 0 0
835835 −18.6135 −0.644148
836836 0 0
837837 0 0
838838 0 0
839839 29.0236 1.00201 0.501004 0.865445i 0.332964π-0.332964\pi
0.501004 + 0.865445i 0.332964π0.332964\pi
840840 0 0
841841 1.00000 0.0344828
842842 0 0
843843 0 0
844844 0 0
845845 −25.3887 −0.873397
846846 0 0
847847 −22.9075 −0.787113
848848 0 0
849849 0 0
850850 0 0
851851 −0.666801 −0.0228577
852852 0 0
853853 −36.6959 −1.25644 −0.628222 0.778034i 0.716217π-0.716217\pi
−0.628222 + 0.778034i 0.716217π0.716217\pi
854854 0 0
855855 0 0
856856 0 0
857857 0.902417 0.0308260 0.0154130 0.999881i 0.495094π-0.495094\pi
0.0154130 + 0.999881i 0.495094π0.495094\pi
858858 0 0
859859 −43.6590 −1.48963 −0.744813 0.667273i 0.767461π-0.767461\pi
−0.744813 + 0.667273i 0.767461π0.767461\pi
860860 0 0
861861 0 0
862862 0 0
863863 18.8040 0.640094 0.320047 0.947402i 0.396301π-0.396301\pi
0.320047 + 0.947402i 0.396301π0.396301\pi
864864 0 0
865865 1.88336 0.0640362
866866 0 0
867867 0 0
868868 0 0
869869 5.67666 0.192568
870870 0 0
871871 −38.1545 −1.29282
872872 0 0
873873 0 0
874874 0 0
875875 −2.23366 −0.0755114
876876 0 0
877877 −8.49872 −0.286981 −0.143491 0.989652i 0.545833π-0.545833\pi
−0.143491 + 0.989652i 0.545833π0.545833\pi
878878 0 0
879879 0 0
880880 0 0
881881 10.0746 0.339421 0.169710 0.985494i 0.445717π-0.445717\pi
0.169710 + 0.985494i 0.445717π0.445717\pi
882882 0 0
883883 −17.4369 −0.586799 −0.293399 0.955990i 0.594787π-0.594787\pi
−0.293399 + 0.955990i 0.594787π0.594787\pi
884884 0 0
885885 0 0
886886 0 0
887887 6.96047 0.233710 0.116855 0.993149i 0.462719π-0.462719\pi
0.116855 + 0.993149i 0.462719π0.462719\pi
888888 0 0
889889 −20.3212 −0.681552
890890 0 0
891891 0 0
892892 0 0
893893 −43.9160 −1.46959
894894 0 0
895895 −4.11447 −0.137532
896896 0 0
897897 0 0
898898 0 0
899899 −1.48490 −0.0495243
900900 0 0
901901 46.1009 1.53584
902902 0 0
903903 0 0
904904 0 0
905905 −15.1334 −0.503052
906906 0 0
907907 −23.0944 −0.766836 −0.383418 0.923575i 0.625253π-0.625253\pi
−0.383418 + 0.923575i 0.625253π0.625253\pi
908908 0 0
909909 0 0
910910 0 0
911911 8.39120 0.278013 0.139006 0.990291i 0.455609π-0.455609\pi
0.139006 + 0.990291i 0.455609π0.455609\pi
912912 0 0
913913 7.44167 0.246283
914914 0 0
915915 0 0
916916 0 0
917917 2.87928 0.0950822
918918 0 0
919919 −23.0180 −0.759293 −0.379647 0.925132i 0.623954π-0.623954\pi
−0.379647 + 0.925132i 0.623954π0.623954\pi
920920 0 0
921921 0 0
922922 0 0
923923 14.3275 0.471594
924924 0 0
925925 5.84818 0.192287
926926 0 0
927927 0 0
928928 0 0
929929 53.4134 1.75244 0.876219 0.481913i 0.160058π-0.160058\pi
0.876219 + 0.481913i 0.160058π0.160058\pi
930930 0 0
931931 −7.63557 −0.250246
932932 0 0
933933 0 0
934934 0 0
935935 5.80146 0.189728
936936 0 0
937937 −9.18564 −0.300082 −0.150041 0.988680i 0.547941π-0.547941\pi
−0.150041 + 0.988680i 0.547941π0.547941\pi
938938 0 0
939939 0 0
940940 0 0
941941 −4.03797 −0.131634 −0.0658170 0.997832i 0.520965π-0.520965\pi
−0.0658170 + 0.997832i 0.520965π0.520965\pi
942942 0 0
943943 0.438764 0.0142881
944944 0 0
945945 0 0
946946 0 0
947947 50.6471 1.64581 0.822905 0.568179i 0.192352π-0.192352\pi
0.822905 + 0.568179i 0.192352π0.192352\pi
948948 0 0
949949 67.6401 2.19569
950950 0 0
951951 0 0
952952 0 0
953953 −3.24076 −0.104978 −0.0524892 0.998621i 0.516716π-0.516716\pi
−0.0524892 + 0.998621i 0.516716π0.516716\pi
954954 0 0
955955 18.2790 0.591495
956956 0 0
957957 0 0
958958 0 0
959959 36.5422 1.18001
960960 0 0
961961 −28.7951 −0.928873
962962 0 0
963963 0 0
964964 0 0
965965 8.34771 0.268722
966966 0 0
967967 −51.3561 −1.65150 −0.825750 0.564037i 0.809248π-0.809248\pi
−0.825750 + 0.564037i 0.809248π0.809248\pi
968968 0 0
969969 0 0
970970 0 0
971971 −33.7059 −1.08167 −0.540836 0.841128i 0.681892π-0.681892\pi
−0.540836 + 0.841128i 0.681892π0.681892\pi
972972 0 0
973973 −8.33516 −0.267213
974974 0 0
975975 0 0
976976 0 0
977977 51.0353 1.63277 0.816383 0.577511i 0.195976π-0.195976\pi
0.816383 + 0.577511i 0.195976π0.195976\pi
978978 0 0
979979 7.71114 0.246449
980980 0 0
981981 0 0
982982 0 0
983983 −47.8553 −1.52635 −0.763174 0.646193i 0.776360π-0.776360\pi
−0.763174 + 0.646193i 0.776360π0.776360\pi
984984 0 0
985985 20.8912 0.665650
986986 0 0
987987 0 0
988988 0 0
989989 0.303643 0.00965527
990990 0 0
991991 21.6652 0.688216 0.344108 0.938930i 0.388181π-0.388181\pi
0.344108 + 0.938930i 0.388181π0.388181\pi
992992 0 0
993993 0 0
994994 0 0
995995 −1.29348 −0.0410062
996996 0 0
997997 11.7861 0.373270 0.186635 0.982429i 0.440242π-0.440242\pi
0.186635 + 0.982429i 0.440242π0.440242\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 5220.2.a.y.1.5 7
3.2 odd 2 5220.2.a.z.1.5 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5220.2.a.y.1.5 7 1.1 even 1 trivial
5220.2.a.z.1.5 yes 7 3.2 odd 2