Properties

Label 845.2.c.d.506.1
Level 845845
Weight 22
Character 845.506
Analytic conductor 6.7476.747
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 845=5132 845 = 5 \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 845.c (of order 22, degree 11, not minimal)

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: no
Analytic conductor: 6.747358970806.74735897080
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,13)\Q(i, \sqrt{13})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+7x2+9 x^{4} + 7x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 506.1
Root 2.30278i-2.30278i of defining polynomial
Character χ\chi == 845.506
Dual form 845.2.c.d.506.4

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q2.30278iq2+1.00000q33.30278q41.00000iq52.30278iq6+1.00000iq7+3.00000iq82.00000q92.30278q101.60555iq113.30278q12+2.30278q141.00000iq15+0.302776q167.60555q17+4.60555iq185.60555iq19+3.30278iq20+1.00000iq213.69722q22+3.00000q23+3.00000iq241.00000q255.00000q273.30278iq286.21110q292.30278q304.00000iq31+5.30278iq321.60555iq33+17.5139iq34+1.00000q35+6.60555q363.60555iq3712.9083q38+3.00000q40+3.00000iq41+2.30278q42+10.2111q43+5.30278iq44+2.00000iq456.90833iq469.21110iq47+0.302776q48+6.00000q49+2.30278iq507.60555q513.21110q53+11.5139iq541.60555q553.00000q565.60555iq57+14.3028iq58+10.8167iq59+3.30278iq601.00000q619.21110q622.00000iq63+12.8167q643.69722q667.00000iq67+25.1194q68+3.00000q692.30278iq70+4.81665iq716.00000iq72+0.788897iq738.30278q741.00000q75+18.5139iq76+1.60555q77+5.21110q790.302776iq80+1.00000q81+6.90833q829.21110iq833.30278iq84+7.60555iq8523.5139iq866.21110q87+4.81665q88+6.21110iq89+4.60555q909.90833q924.00000iq9321.2111q945.60555q95+5.30278iq968.39445iq9713.8167iq98+3.21110iq99+O(q100)q-2.30278i q^{2} +1.00000 q^{3} -3.30278 q^{4} -1.00000i q^{5} -2.30278i q^{6} +1.00000i q^{7} +3.00000i q^{8} -2.00000 q^{9} -2.30278 q^{10} -1.60555i q^{11} -3.30278 q^{12} +2.30278 q^{14} -1.00000i q^{15} +0.302776 q^{16} -7.60555 q^{17} +4.60555i q^{18} -5.60555i q^{19} +3.30278i q^{20} +1.00000i q^{21} -3.69722 q^{22} +3.00000 q^{23} +3.00000i q^{24} -1.00000 q^{25} -5.00000 q^{27} -3.30278i q^{28} -6.21110 q^{29} -2.30278 q^{30} -4.00000i q^{31} +5.30278i q^{32} -1.60555i q^{33} +17.5139i q^{34} +1.00000 q^{35} +6.60555 q^{36} -3.60555i q^{37} -12.9083 q^{38} +3.00000 q^{40} +3.00000i q^{41} +2.30278 q^{42} +10.2111 q^{43} +5.30278i q^{44} +2.00000i q^{45} -6.90833i q^{46} -9.21110i q^{47} +0.302776 q^{48} +6.00000 q^{49} +2.30278i q^{50} -7.60555 q^{51} -3.21110 q^{53} +11.5139i q^{54} -1.60555 q^{55} -3.00000 q^{56} -5.60555i q^{57} +14.3028i q^{58} +10.8167i q^{59} +3.30278i q^{60} -1.00000 q^{61} -9.21110 q^{62} -2.00000i q^{63} +12.8167 q^{64} -3.69722 q^{66} -7.00000i q^{67} +25.1194 q^{68} +3.00000 q^{69} -2.30278i q^{70} +4.81665i q^{71} -6.00000i q^{72} +0.788897i q^{73} -8.30278 q^{74} -1.00000 q^{75} +18.5139i q^{76} +1.60555 q^{77} +5.21110 q^{79} -0.302776i q^{80} +1.00000 q^{81} +6.90833 q^{82} -9.21110i q^{83} -3.30278i q^{84} +7.60555i q^{85} -23.5139i q^{86} -6.21110 q^{87} +4.81665 q^{88} +6.21110i q^{89} +4.60555 q^{90} -9.90833 q^{92} -4.00000i q^{93} -21.2111 q^{94} -5.60555 q^{95} +5.30278i q^{96} -8.39445i q^{97} -13.8167i q^{98} +3.21110i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q36q48q92q106q12+2q146q1616q1722q22+12q234q2520q27+4q292q30+4q35+12q3630q38+12q40+8q95+O(q100) 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9} - 2 q^{10} - 6 q^{12} + 2 q^{14} - 6 q^{16} - 16 q^{17} - 22 q^{22} + 12 q^{23} - 4 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 12 q^{36} - 30 q^{38} + 12 q^{40}+ \cdots - 8 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/845Z)×\left(\mathbb{Z}/845\mathbb{Z}\right)^\times.

nn 171171 677677
χ(n)\chi(n) 1-1 11

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 − 2.30278i − 1.62831i −0.580649 0.814154i 0.697201π-0.697201\pi
0.580649 0.814154i 0.302799π-0.302799\pi
33 1.00000 0.577350 0.288675 0.957427i 0.406785π-0.406785\pi
0.288675 + 0.957427i 0.406785π0.406785\pi
44 −3.30278 −1.65139
55 − 1.00000i − 0.447214i
66 − 2.30278i − 0.940104i
77 1.00000i 0.377964i 0.981981 + 0.188982i 0.0605189π0.0605189\pi
−0.981981 + 0.188982i 0.939481π0.939481\pi
88 3.00000i 1.06066i
99 −2.00000 −0.666667
1010 −2.30278 −0.728202
1111 − 1.60555i − 0.484092i −0.970265 0.242046i 0.922182π-0.922182\pi
0.970265 0.242046i 0.0778185π-0.0778185\pi
1212 −3.30278 −0.953429
1313 0 0
1414 2.30278 0.615443
1515 − 1.00000i − 0.258199i
1616 0.302776 0.0756939
1717 −7.60555 −1.84462 −0.922309 0.386454i 0.873700π-0.873700\pi
−0.922309 + 0.386454i 0.873700π0.873700\pi
1818 4.60555i 1.08554i
1919 − 5.60555i − 1.28600i −0.765865 0.643001i 0.777689π-0.777689\pi
0.765865 0.643001i 0.222311π-0.222311\pi
2020 3.30278i 0.738523i
2121 1.00000i 0.218218i
2222 −3.69722 −0.788251
2323 3.00000 0.625543 0.312772 0.949828i 0.398743π-0.398743\pi
0.312772 + 0.949828i 0.398743π0.398743\pi
2424 3.00000i 0.612372i
2525 −1.00000 −0.200000
2626 0 0
2727 −5.00000 −0.962250
2828 − 3.30278i − 0.624166i
2929 −6.21110 −1.15337 −0.576686 0.816966i 0.695655π-0.695655\pi
−0.576686 + 0.816966i 0.695655π0.695655\pi
3030 −2.30278 −0.420427
3131 − 4.00000i − 0.718421i −0.933257 0.359211i 0.883046π-0.883046\pi
0.933257 0.359211i 0.116954π-0.116954\pi
3232 5.30278i 0.937407i
3333 − 1.60555i − 0.279491i
3434 17.5139i 3.00361i
3535 1.00000 0.169031
3636 6.60555 1.10093
3737 − 3.60555i − 0.592749i −0.955072 0.296374i 0.904222π-0.904222\pi
0.955072 0.296374i 0.0957776π-0.0957776\pi
3838 −12.9083 −2.09401
3939 0 0
4040 3.00000 0.474342
4141 3.00000i 0.468521i 0.972174 + 0.234261i 0.0752669π0.0752669\pi
−0.972174 + 0.234261i 0.924733π0.924733\pi
4242 2.30278 0.355326
4343 10.2111 1.55718 0.778589 0.627534i 0.215936π-0.215936\pi
0.778589 + 0.627534i 0.215936π0.215936\pi
4444 5.30278i 0.799424i
4545 2.00000i 0.298142i
4646 − 6.90833i − 1.01858i
4747 − 9.21110i − 1.34358i −0.740743 0.671789i 0.765526π-0.765526\pi
0.740743 0.671789i 0.234474π-0.234474\pi
4848 0.302776 0.0437019
4949 6.00000 0.857143
5050 2.30278i 0.325662i
5151 −7.60555 −1.06499
5252 0 0
5353 −3.21110 −0.441079 −0.220539 0.975378i 0.570782π-0.570782\pi
−0.220539 + 0.975378i 0.570782π0.570782\pi
5454 11.5139i 1.56684i
5555 −1.60555 −0.216492
5656 −3.00000 −0.400892
5757 − 5.60555i − 0.742473i
5858 14.3028i 1.87805i
5959 10.8167i 1.40821i 0.710097 + 0.704104i 0.248651π0.248651\pi
−0.710097 + 0.704104i 0.751349π0.751349\pi
6060 3.30278i 0.426387i
6161 −1.00000 −0.128037 −0.0640184 0.997949i 0.520392π-0.520392\pi
−0.0640184 + 0.997949i 0.520392π0.520392\pi
6262 −9.21110 −1.16981
6363 − 2.00000i − 0.251976i
6464 12.8167 1.60208
6565 0 0
6666 −3.69722 −0.455097
6767 − 7.00000i − 0.855186i −0.903971 0.427593i 0.859362π-0.859362\pi
0.903971 0.427593i 0.140638π-0.140638\pi
6868 25.1194 3.04618
6969 3.00000 0.361158
7070 − 2.30278i − 0.275234i
7171 4.81665i 0.571632i 0.958285 + 0.285816i 0.0922646π0.0922646\pi
−0.958285 + 0.285816i 0.907735π0.907735\pi
7272 − 6.00000i − 0.707107i
7373 0.788897i 0.0923335i 0.998934 + 0.0461667i 0.0147006π0.0147006\pi
−0.998934 + 0.0461667i 0.985299π0.985299\pi
7474 −8.30278 −0.965178
7575 −1.00000 −0.115470
7676 18.5139i 2.12369i
7777 1.60555 0.182970
7878 0 0
7979 5.21110 0.586295 0.293147 0.956067i 0.405297π-0.405297\pi
0.293147 + 0.956067i 0.405297π0.405297\pi
8080 − 0.302776i − 0.0338513i
8181 1.00000 0.111111
8282 6.90833 0.762897
8383 − 9.21110i − 1.01105i −0.862812 0.505525i 0.831299π-0.831299\pi
0.862812 0.505525i 0.168701π-0.168701\pi
8484 − 3.30278i − 0.360362i
8585 7.60555i 0.824938i
8686 − 23.5139i − 2.53557i
8787 −6.21110 −0.665900
8888 4.81665 0.513457
8989 6.21110i 0.658376i 0.944264 + 0.329188i 0.106775π0.106775\pi
−0.944264 + 0.329188i 0.893225π0.893225\pi
9090 4.60555 0.485468
9191 0 0
9292 −9.90833 −1.03301
9393 − 4.00000i − 0.414781i
9494 −21.2111 −2.18776
9595 −5.60555 −0.575117
9696 5.30278i 0.541212i
9797 − 8.39445i − 0.852327i −0.904646 0.426164i 0.859865π-0.859865\pi
0.904646 0.426164i 0.140135π-0.140135\pi
9898 − 13.8167i − 1.39569i
9999 3.21110i 0.322728i
100100 3.30278 0.330278
101101 9.00000 0.895533 0.447767 0.894150i 0.352219π-0.352219\pi
0.447767 + 0.894150i 0.352219π0.352219\pi
102102 17.5139i 1.73413i
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 1.00000 0.0975900
106106 7.39445i 0.718212i
107107 6.21110 0.600450 0.300225 0.953868i 0.402938π-0.402938\pi
0.300225 + 0.953868i 0.402938π0.402938\pi
108108 16.5139 1.58905
109109 − 19.2111i − 1.84009i −0.391813 0.920045i 0.628152π-0.628152\pi
0.391813 0.920045i 0.371848π-0.371848\pi
110110 3.69722i 0.352517i
111111 − 3.60555i − 0.342224i
112112 0.302776i 0.0286096i
113113 −1.60555 −0.151038 −0.0755188 0.997144i 0.524061π-0.524061\pi
−0.0755188 + 0.997144i 0.524061π0.524061\pi
114114 −12.9083 −1.20898
115115 − 3.00000i − 0.279751i
116116 20.5139 1.90467
117117 0 0
118118 24.9083 2.29300
119119 − 7.60555i − 0.697200i
120120 3.00000 0.273861
121121 8.42221 0.765655
122122 2.30278i 0.208484i
123123 3.00000i 0.270501i
124124 13.2111i 1.18639i
125125 1.00000i 0.0894427i
126126 −4.60555 −0.410295
127127 4.21110 0.373675 0.186837 0.982391i 0.440176π-0.440176\pi
0.186837 + 0.982391i 0.440176π0.440176\pi
128128 − 18.9083i − 1.67128i
129129 10.2111 0.899037
130130 0 0
131131 −21.2111 −1.85322 −0.926611 0.376021i 0.877292π-0.877292\pi
−0.926611 + 0.376021i 0.877292π0.877292\pi
132132 5.30278i 0.461547i
133133 5.60555 0.486063
134134 −16.1194 −1.39251
135135 5.00000i 0.430331i
136136 − 22.8167i − 1.95651i
137137 1.60555i 0.137172i 0.997645 + 0.0685858i 0.0218487π0.0218487\pi
−0.997645 + 0.0685858i 0.978151π0.978151\pi
138138 − 6.90833i − 0.588076i
139139 6.39445 0.542370 0.271185 0.962527i 0.412584π-0.412584\pi
0.271185 + 0.962527i 0.412584π0.412584\pi
140140 −3.30278 −0.279135
141141 − 9.21110i − 0.775715i
142142 11.0917 0.930793
143143 0 0
144144 −0.605551 −0.0504626
145145 6.21110i 0.515804i
146146 1.81665 0.150347
147147 6.00000 0.494872
148148 11.9083i 0.978858i
149149 3.00000i 0.245770i 0.992421 + 0.122885i 0.0392146π0.0392146\pi
−0.992421 + 0.122885i 0.960785π0.960785\pi
150150 2.30278i 0.188021i
151151 1.21110i 0.0985581i 0.998785 + 0.0492791i 0.0156924π0.0156924\pi
−0.998785 + 0.0492791i 0.984308π0.984308\pi
152152 16.8167 1.36401
153153 15.2111 1.22974
154154 − 3.69722i − 0.297931i
155155 −4.00000 −0.321288
156156 0 0
157157 11.2111 0.894743 0.447372 0.894348i 0.352360π-0.352360\pi
0.447372 + 0.894348i 0.352360π0.352360\pi
158158 − 12.0000i − 0.954669i
159159 −3.21110 −0.254657
160160 5.30278 0.419221
161161 3.00000i 0.236433i
162162 − 2.30278i − 0.180923i
163163 3.78890i 0.296769i 0.988930 + 0.148385i 0.0474074π0.0474074\pi
−0.988930 + 0.148385i 0.952593π0.952593\pi
164164 − 9.90833i − 0.773710i
165165 −1.60555 −0.124992
166166 −21.2111 −1.64630
167167 − 9.00000i − 0.696441i −0.937413 0.348220i 0.886786π-0.886786\pi
0.937413 0.348220i 0.113214π-0.113214\pi
168168 −3.00000 −0.231455
169169 0 0
170170 17.5139 1.34325
171171 11.2111i 0.857334i
172172 −33.7250 −2.57151
173173 −4.81665 −0.366203 −0.183102 0.983094i 0.558614π-0.558614\pi
−0.183102 + 0.983094i 0.558614π0.558614\pi
174174 14.3028i 1.08429i
175175 − 1.00000i − 0.0755929i
176176 − 0.486122i − 0.0366428i
177177 10.8167i 0.813029i
178178 14.3028 1.07204
179179 −22.8167 −1.70540 −0.852698 0.522404i 0.825035π-0.825035\pi
−0.852698 + 0.522404i 0.825035π0.825035\pi
180180 − 6.60555i − 0.492349i
181181 −17.6333 −1.31067 −0.655337 0.755337i 0.727473π-0.727473\pi
−0.655337 + 0.755337i 0.727473π0.727473\pi
182182 0 0
183183 −1.00000 −0.0739221
184184 9.00000i 0.663489i
185185 −3.60555 −0.265085
186186 −9.21110 −0.675391
187187 12.2111i 0.892964i
188188 30.4222i 2.21877i
189189 − 5.00000i − 0.363696i
190190 12.9083i 0.936468i
191191 16.8167 1.21681 0.608405 0.793627i 0.291810π-0.291810\pi
0.608405 + 0.793627i 0.291810π0.291810\pi
192192 12.8167 0.924962
193193 − 15.6056i − 1.12331i −0.827371 0.561656i 0.810164π-0.810164\pi
0.827371 0.561656i 0.189836π-0.189836\pi
194194 −19.3305 −1.38785
195195 0 0
196196 −19.8167 −1.41548
197197 1.18335i 0.0843099i 0.999111 + 0.0421550i 0.0134223π0.0134223\pi
−0.999111 + 0.0421550i 0.986578π0.986578\pi
198198 7.39445 0.525501
199199 −12.8167 −0.908549 −0.454274 0.890862i 0.650101π-0.650101\pi
−0.454274 + 0.890862i 0.650101π0.650101\pi
200200 − 3.00000i − 0.212132i
201201 − 7.00000i − 0.493742i
202202 − 20.7250i − 1.45820i
203203 − 6.21110i − 0.435934i
204204 25.1194 1.75871
205205 3.00000 0.209529
206206 − 9.21110i − 0.641768i
207207 −6.00000 −0.417029
208208 0 0
209209 −9.00000 −0.622543
210210 − 2.30278i − 0.158907i
211211 −23.6056 −1.62507 −0.812537 0.582910i 0.801914π-0.801914\pi
−0.812537 + 0.582910i 0.801914π0.801914\pi
212212 10.6056 0.728392
213213 4.81665i 0.330032i
214214 − 14.3028i − 0.977718i
215215 − 10.2111i − 0.696391i
216216 − 15.0000i − 1.02062i
217217 4.00000 0.271538
218218 −44.2389 −2.99623
219219 0.788897i 0.0533087i
220220 5.30278 0.357513
221221 0 0
222222 −8.30278 −0.557246
223223 − 4.21110i − 0.281996i −0.990010 0.140998i 0.954969π-0.954969\pi
0.990010 0.140998i 0.0450312π-0.0450312\pi
224224 −5.30278 −0.354307
225225 2.00000 0.133333
226226 3.69722i 0.245936i
227227 − 27.4222i − 1.82008i −0.414525 0.910038i 0.636052π-0.636052\pi
0.414525 0.910038i 0.363948π-0.363948\pi
228228 18.5139i 1.22611i
229229 − 14.0000i − 0.925146i −0.886581 0.462573i 0.846926π-0.846926\pi
0.886581 0.462573i 0.153074π-0.153074\pi
230230 −6.90833 −0.455522
231231 1.60555 0.105638
232232 − 18.6333i − 1.22334i
233233 −15.2111 −0.996512 −0.498256 0.867030i 0.666026π-0.666026\pi
−0.498256 + 0.867030i 0.666026π0.666026\pi
234234 0 0
235235 −9.21110 −0.600866
236236 − 35.7250i − 2.32550i
237237 5.21110 0.338497
238238 −17.5139 −1.13526
239239 0 0 1.00000 00
−1.00000 π\pi
240240 − 0.302776i − 0.0195441i
241241 − 1.78890i − 0.115233i −0.998339 0.0576165i 0.981650π-0.981650\pi
0.998339 0.0576165i 0.0183501π-0.0183501\pi
242242 − 19.3944i − 1.24672i
243243 16.0000 1.02640
244244 3.30278 0.211439
245245 − 6.00000i − 0.383326i
246246 6.90833 0.440459
247247 0 0
248248 12.0000 0.762001
249249 − 9.21110i − 0.583730i
250250 2.30278 0.145640
251251 7.18335 0.453409 0.226704 0.973964i 0.427205π-0.427205\pi
0.226704 + 0.973964i 0.427205π0.427205\pi
252252 6.60555i 0.416111i
253253 − 4.81665i − 0.302820i
254254 − 9.69722i − 0.608458i
255255 7.60555i 0.476278i
256256 −17.9083 −1.11927
257257 16.3944 1.02266 0.511329 0.859385i 0.329153π-0.329153\pi
0.511329 + 0.859385i 0.329153π0.329153\pi
258258 − 23.5139i − 1.46391i
259259 3.60555 0.224038
260260 0 0
261261 12.4222 0.768915
262262 48.8444i 3.01762i
263263 11.7889 0.726935 0.363467 0.931607i 0.381593π-0.381593\pi
0.363467 + 0.931607i 0.381593π0.381593\pi
264264 4.81665 0.296445
265265 3.21110i 0.197256i
266266 − 12.9083i − 0.791460i
267267 6.21110i 0.380113i
268268 23.1194i 1.41224i
269269 −9.00000 −0.548740 −0.274370 0.961624i 0.588469π-0.588469\pi
−0.274370 + 0.961624i 0.588469π0.588469\pi
270270 11.5139 0.700712
271271 20.8167i 1.26452i 0.774756 + 0.632261i 0.217873π0.217873\pi
−0.774756 + 0.632261i 0.782127π0.782127\pi
272272 −2.30278 −0.139626
273273 0 0
274274 3.69722 0.223357
275275 1.60555i 0.0968184i
276276 −9.90833 −0.596411
277277 −27.6056 −1.65866 −0.829328 0.558761i 0.811277π-0.811277\pi
−0.829328 + 0.558761i 0.811277π0.811277\pi
278278 − 14.7250i − 0.883146i
279279 8.00000i 0.478947i
280280 3.00000i 0.179284i
281281 6.00000i 0.357930i 0.983855 + 0.178965i 0.0572749π0.0572749\pi
−0.983855 + 0.178965i 0.942725π0.942725\pi
282282 −21.2111 −1.26310
283283 −5.00000 −0.297219 −0.148610 0.988896i 0.547480π-0.547480\pi
−0.148610 + 0.988896i 0.547480π0.547480\pi
284284 − 15.9083i − 0.943986i
285285 −5.60555 −0.332044
286286 0 0
287287 −3.00000 −0.177084
288288 − 10.6056i − 0.624938i
289289 40.8444 2.40261
290290 14.3028 0.839888
291291 − 8.39445i − 0.492091i
292292 − 2.60555i − 0.152478i
293293 − 10.3944i − 0.607250i −0.952792 0.303625i 0.901803π-0.901803\pi
0.952792 0.303625i 0.0981970π-0.0981970\pi
294294 − 13.8167i − 0.805804i
295295 10.8167 0.629770
296296 10.8167 0.628705
297297 8.02776i 0.465818i
298298 6.90833 0.400189
299299 0 0
300300 3.30278 0.190686
301301 10.2111i 0.588558i
302302 2.78890 0.160483
303303 9.00000 0.517036
304304 − 1.69722i − 0.0973425i
305305 1.00000i 0.0572598i
306306 − 35.0278i − 2.00240i
307307 16.0000i 0.913168i 0.889680 + 0.456584i 0.150927π0.150927\pi
−0.889680 + 0.456584i 0.849073π0.849073\pi
308308 −5.30278 −0.302154
309309 4.00000 0.227552
310310 9.21110i 0.523155i
311311 9.21110 0.522314 0.261157 0.965296i 0.415896π-0.415896\pi
0.261157 + 0.965296i 0.415896π0.415896\pi
312312 0 0
313313 14.0000 0.791327 0.395663 0.918396i 0.370515π-0.370515\pi
0.395663 + 0.918396i 0.370515π0.370515\pi
314314 − 25.8167i − 1.45692i
315315 −2.00000 −0.112687
316316 −17.2111 −0.968200
317317 6.00000i 0.336994i 0.985702 + 0.168497i 0.0538913π0.0538913\pi
−0.985702 + 0.168497i 0.946109π0.946109\pi
318318 7.39445i 0.414660i
319319 9.97224i 0.558338i
320320 − 12.8167i − 0.716473i
321321 6.21110 0.346670
322322 6.90833 0.384986
323323 42.6333i 2.37218i
324324 −3.30278 −0.183488
325325 0 0
326326 8.72498 0.483232
327327 − 19.2111i − 1.06238i
328328 −9.00000 −0.496942
329329 9.21110 0.507825
330330 3.69722i 0.203526i
331331 10.0278i 0.551175i 0.961276 + 0.275588i 0.0888724π0.0888724\pi
−0.961276 + 0.275588i 0.911128π0.911128\pi
332332 30.4222i 1.66964i
333333 7.21110i 0.395166i
334334 −20.7250 −1.13402
335335 −7.00000 −0.382451
336336 0.302776i 0.0165178i
337337 25.6333 1.39634 0.698168 0.715934i 0.253999π-0.253999\pi
0.698168 + 0.715934i 0.253999π0.253999\pi
338338 0 0
339339 −1.60555 −0.0872016
340340 − 25.1194i − 1.36229i
341341 −6.42221 −0.347782
342342 25.8167 1.39600
343343 13.0000i 0.701934i
344344 30.6333i 1.65164i
345345 − 3.00000i − 0.161515i
346346 11.0917i 0.596292i
347347 5.78890 0.310764 0.155382 0.987854i 0.450339π-0.450339\pi
0.155382 + 0.987854i 0.450339π0.450339\pi
348348 20.5139 1.09966
349349 3.78890i 0.202815i 0.994845 + 0.101408i 0.0323346π0.0323346\pi
−0.994845 + 0.101408i 0.967665π0.967665\pi
350350 −2.30278 −0.123089
351351 0 0
352352 8.51388 0.453791
353353 16.8167i 0.895060i 0.894269 + 0.447530i 0.147696π0.147696\pi
−0.894269 + 0.447530i 0.852304π0.852304\pi
354354 24.9083 1.32386
355355 4.81665 0.255641
356356 − 20.5139i − 1.08723i
357357 − 7.60555i − 0.402528i
358358 52.5416i 2.77691i
359359 − 18.4222i − 0.972287i −0.873879 0.486143i 0.838403π-0.838403\pi
0.873879 0.486143i 0.161597π-0.161597\pi
360360 −6.00000 −0.316228
361361 −12.4222 −0.653800
362362 40.6056i 2.13418i
363363 8.42221 0.442051
364364 0 0
365365 0.788897 0.0412928
366366 2.30278i 0.120368i
367367 11.4222 0.596234 0.298117 0.954529i 0.403641π-0.403641\pi
0.298117 + 0.954529i 0.403641π0.403641\pi
368368 0.908327 0.0473498
369369 − 6.00000i − 0.312348i
370370 8.30278i 0.431641i
371371 − 3.21110i − 0.166712i
372372 13.2111i 0.684964i
373373 −20.3944 −1.05598 −0.527992 0.849249i 0.677055π-0.677055\pi
−0.527992 + 0.849249i 0.677055π0.677055\pi
374374 28.1194 1.45402
375375 1.00000i 0.0516398i
376376 27.6333 1.42508
377377 0 0
378378 −11.5139 −0.592210
379379 9.60555i 0.493404i 0.969091 + 0.246702i 0.0793469π0.0793469\pi
−0.969091 + 0.246702i 0.920653π0.920653\pi
380380 18.5139 0.949742
381381 4.21110 0.215741
382382 − 38.7250i − 1.98134i
383383 − 24.6333i − 1.25870i −0.777121 0.629352i 0.783321π-0.783321\pi
0.777121 0.629352i 0.216679π-0.216679\pi
384384 − 18.9083i − 0.964912i
385385 − 1.60555i − 0.0818265i
386386 −35.9361 −1.82910
387387 −20.4222 −1.03812
388388 27.7250i 1.40752i
389389 15.2111 0.771234 0.385617 0.922659i 0.373989π-0.373989\pi
0.385617 + 0.922659i 0.373989π0.373989\pi
390390 0 0
391391 −22.8167 −1.15389
392392 18.0000i 0.909137i
393393 −21.2111 −1.06996
394394 2.72498 0.137283
395395 − 5.21110i − 0.262199i
396396 − 10.6056i − 0.532949i
397397 − 22.0278i − 1.10554i −0.833333 0.552771i 0.813571π-0.813571\pi
0.833333 0.552771i 0.186429π-0.186429\pi
398398 29.5139i 1.47940i
399399 5.60555 0.280629
400400 −0.302776 −0.0151388
401401 − 12.2111i − 0.609793i −0.952385 0.304897i 0.901378π-0.901378\pi
0.952385 0.304897i 0.0986219π-0.0986219\pi
402402 −16.1194 −0.803964
403403 0 0
404404 −29.7250 −1.47887
405405 − 1.00000i − 0.0496904i
406406 −14.3028 −0.709835
407407 −5.78890 −0.286945
408408 − 22.8167i − 1.12959i
409409 8.21110i 0.406013i 0.979177 + 0.203006i 0.0650712π0.0650712\pi
−0.979177 + 0.203006i 0.934929π0.934929\pi
410410 − 6.90833i − 0.341178i
411411 1.60555i 0.0791960i
412412 −13.2111 −0.650864
413413 −10.8167 −0.532253
414414 13.8167i 0.679051i
415415 −9.21110 −0.452155
416416 0 0
417417 6.39445 0.313138
418418 20.7250i 1.01369i
419419 17.2389 0.842173 0.421087 0.907020i 0.361649π-0.361649\pi
0.421087 + 0.907020i 0.361649π0.361649\pi
420420 −3.30278 −0.161159
421421 32.4222i 1.58016i 0.613003 + 0.790081i 0.289961π0.289961\pi
−0.613003 + 0.790081i 0.710039π0.710039\pi
422422 54.3583i 2.64612i
423423 18.4222i 0.895718i
424424 − 9.63331i − 0.467835i
425425 7.60555 0.368923
426426 11.0917 0.537393
427427 − 1.00000i − 0.0483934i
428428 −20.5139 −0.991576
429429 0 0
430430 −23.5139 −1.13394
431431 29.2389i 1.40839i 0.710008 + 0.704193i 0.248691π0.248691\pi
−0.710008 + 0.704193i 0.751309π0.751309\pi
432432 −1.51388 −0.0728365
433433 −3.60555 −0.173272 −0.0866359 0.996240i 0.527612π-0.527612\pi
−0.0866359 + 0.996240i 0.527612π0.527612\pi
434434 − 9.21110i − 0.442147i
435435 6.21110i 0.297800i
436436 63.4500i 3.03870i
437437 − 16.8167i − 0.804450i
438438 1.81665 0.0868031
439439 27.2389 1.30004 0.650020 0.759917i 0.274761π-0.274761\pi
0.650020 + 0.759917i 0.274761π0.274761\pi
440440 − 4.81665i − 0.229625i
441441 −12.0000 −0.571429
442442 0 0
443443 6.42221 0.305128 0.152564 0.988294i 0.451247π-0.451247\pi
0.152564 + 0.988294i 0.451247π0.451247\pi
444444 11.9083i 0.565144i
445445 6.21110 0.294434
446446 −9.69722 −0.459177
447447 3.00000i 0.141895i
448448 12.8167i 0.605530i
449449 − 30.6333i − 1.44568i −0.691018 0.722838i 0.742837π-0.742837\pi
0.691018 0.722838i 0.257163π-0.257163\pi
450450 − 4.60555i − 0.217108i
451451 4.81665 0.226807
452452 5.30278 0.249422
453453 1.21110i 0.0569026i
454454 −63.1472 −2.96364
455455 0 0
456456 16.8167 0.787512
457457 − 26.8167i − 1.25443i −0.778846 0.627215i 0.784195π-0.784195\pi
0.778846 0.627215i 0.215805π-0.215805\pi
458458 −32.2389 −1.50642
459459 38.0278 1.77498
460460 9.90833i 0.461978i
461461 36.2111i 1.68652i 0.537507 + 0.843260i 0.319366π0.319366\pi
−0.537507 + 0.843260i 0.680634π0.680634\pi
462462 − 3.69722i − 0.172010i
463463 34.4222i 1.59974i 0.600176 + 0.799868i 0.295097π0.295097\pi
−0.600176 + 0.799868i 0.704903π0.704903\pi
464464 −1.88057 −0.0873033
465465 −4.00000 −0.185496
466466 35.0278i 1.62263i
467467 −2.78890 −0.129055 −0.0645274 0.997916i 0.520554π-0.520554\pi
−0.0645274 + 0.997916i 0.520554π0.520554\pi
468468 0 0
469469 7.00000 0.323230
470470 21.2111i 0.978395i
471471 11.2111 0.516580
472472 −32.4500 −1.49363
473473 − 16.3944i − 0.753818i
474474 − 12.0000i − 0.551178i
475475 5.60555i 0.257200i
476476 25.1194i 1.15135i
477477 6.42221 0.294053
478478 0 0
479479 28.8167i 1.31667i 0.752727 + 0.658333i 0.228738π0.228738\pi
−0.752727 + 0.658333i 0.771262π0.771262\pi
480480 5.30278 0.242037
481481 0 0
482482 −4.11943 −0.187635
483483 3.00000i 0.136505i
484484 −27.8167 −1.26439
485485 −8.39445 −0.381172
486486 − 36.8444i − 1.67130i
487487 − 1.00000i − 0.0453143i −0.999743 0.0226572i 0.992787π-0.992787\pi
0.999743 0.0226572i 0.00721262π-0.00721262\pi
488488 − 3.00000i − 0.135804i
489489 3.78890i 0.171340i
490490 −13.8167 −0.624173
491491 16.8167 0.758925 0.379462 0.925207i 0.376109π-0.376109\pi
0.379462 + 0.925207i 0.376109π0.376109\pi
492492 − 9.90833i − 0.446702i
493493 47.2389 2.12753
494494 0 0
495495 3.21110 0.144328
496496 − 1.21110i − 0.0543801i
497497 −4.81665 −0.216056
498498 −21.2111 −0.950492
499499 2.42221i 0.108433i 0.998529 + 0.0542164i 0.0172661π0.0172661\pi
−0.998529 + 0.0542164i 0.982734π0.982734\pi
500500 − 3.30278i − 0.147705i
501501 − 9.00000i − 0.402090i
502502 − 16.5416i − 0.738289i
503503 3.00000 0.133763 0.0668817 0.997761i 0.478695π-0.478695\pi
0.0668817 + 0.997761i 0.478695π0.478695\pi
504504 6.00000 0.267261
505505 − 9.00000i − 0.400495i
506506 −11.0917 −0.493085
507507 0 0
508508 −13.9083 −0.617082
509509 3.00000i 0.132973i 0.997787 + 0.0664863i 0.0211789π0.0211789\pi
−0.997787 + 0.0664863i 0.978821π0.978821\pi
510510 17.5139 0.775528
511511 −0.788897 −0.0348988
512512 3.42221i 0.151242i
513513 28.0278i 1.23746i
514514 − 37.7527i − 1.66520i
515515 − 4.00000i − 0.176261i
516516 −33.7250 −1.48466
517517 −14.7889 −0.650415
518518 − 8.30278i − 0.364803i
519519 −4.81665 −0.211428
520520 0 0
521521 18.0000 0.788594 0.394297 0.918983i 0.370988π-0.370988\pi
0.394297 + 0.918983i 0.370988π0.370988\pi
522522 − 28.6056i − 1.25203i
523523 −1.42221 −0.0621887 −0.0310943 0.999516i 0.509899π-0.509899\pi
−0.0310943 + 0.999516i 0.509899π0.509899\pi
524524 70.0555 3.06039
525525 − 1.00000i − 0.0436436i
526526 − 27.1472i − 1.18367i
527527 30.4222i 1.32521i
528528 − 0.486122i − 0.0211557i
529529 −14.0000 −0.608696
530530 7.39445 0.321194
531531 − 21.6333i − 0.938806i
532532 −18.5139 −0.802678
533533 0 0
534534 14.3028 0.618942
535535 − 6.21110i − 0.268529i
536536 21.0000 0.907062
537537 −22.8167 −0.984611
538538 20.7250i 0.893517i
539539 − 9.63331i − 0.414936i
540540 − 16.5139i − 0.710644i
541541 25.6333i 1.10206i 0.834485 + 0.551031i 0.185765π0.185765\pi
−0.834485 + 0.551031i 0.814235π0.814235\pi
542542 47.9361 2.05903
543543 −17.6333 −0.756718
544544 − 40.3305i − 1.72916i
545545 −19.2111 −0.822913
546546 0 0
547547 32.8444 1.40433 0.702163 0.712016i 0.252218π-0.252218\pi
0.702163 + 0.712016i 0.252218π0.252218\pi
548548 − 5.30278i − 0.226523i
549549 2.00000 0.0853579
550550 3.69722 0.157650
551551 34.8167i 1.48324i
552552 9.00000i 0.383065i
553553 5.21110i 0.221599i
554554 63.5694i 2.70080i
555555 −3.60555 −0.153047
556556 −21.1194 −0.895663
557557 1.60555i 0.0680294i 0.999421 + 0.0340147i 0.0108293π0.0108293\pi
−0.999421 + 0.0340147i 0.989171π0.989171\pi
558558 18.4222 0.779874
559559 0 0
560560 0.302776 0.0127946
561561 12.2111i 0.515553i
562562 13.8167 0.582820
563563 −9.42221 −0.397099 −0.198549 0.980091i 0.563623π-0.563623\pi
−0.198549 + 0.980091i 0.563623π0.563623\pi
564564 30.4222i 1.28101i
565565 1.60555i 0.0675460i
566566 11.5139i 0.483964i
567567 1.00000i 0.0419961i
568568 −14.4500 −0.606307
569569 27.4222 1.14960 0.574799 0.818294i 0.305080π-0.305080\pi
0.574799 + 0.818294i 0.305080π0.305080\pi
570570 12.9083i 0.540670i
571571 −20.8444 −0.872311 −0.436156 0.899871i 0.643660π-0.643660\pi
−0.436156 + 0.899871i 0.643660π0.643660\pi
572572 0 0
573573 16.8167 0.702526
574574 6.90833i 0.288348i
575575 −3.00000 −0.125109
576576 −25.6333 −1.06805
577577 − 13.6333i − 0.567562i −0.958889 0.283781i 0.908411π-0.908411\pi
0.958889 0.283781i 0.0915889π-0.0915889\pi
578578 − 94.0555i − 3.91219i
579579 − 15.6056i − 0.648545i
580580 − 20.5139i − 0.851792i
581581 9.21110 0.382141
582582 −19.3305 −0.801276
583583 5.15559i 0.213523i
584584 −2.36669 −0.0979344
585585 0 0
586586 −23.9361 −0.988790
587587 − 33.4222i − 1.37948i −0.724056 0.689741i 0.757724π-0.757724\pi
0.724056 0.689741i 0.242276π-0.242276\pi
588588 −19.8167 −0.817225
589589 −22.4222 −0.923891
590590 − 24.9083i − 1.02546i
591591 1.18335i 0.0486764i
592592 − 1.09167i − 0.0448675i
593593 − 20.7889i − 0.853698i −0.904323 0.426849i 0.859624π-0.859624\pi
0.904323 0.426849i 0.140376π-0.140376\pi
594594 18.4861 0.758495
595595 −7.60555 −0.311797
596596 − 9.90833i − 0.405861i
597597 −12.8167 −0.524551
598598 0 0
599599 −21.2111 −0.866662 −0.433331 0.901235i 0.642662π-0.642662\pi
−0.433331 + 0.901235i 0.642662π0.642662\pi
600600 − 3.00000i − 0.122474i
601601 13.7889 0.562461 0.281230 0.959640i 0.409257π-0.409257\pi
0.281230 + 0.959640i 0.409257π0.409257\pi
602602 23.5139 0.958354
603603 14.0000i 0.570124i
604604 − 4.00000i − 0.162758i
605605 − 8.42221i − 0.342411i
606606 − 20.7250i − 0.841895i
607607 −34.2111 −1.38859 −0.694293 0.719693i 0.744283π-0.744283\pi
−0.694293 + 0.719693i 0.744283π0.744283\pi
608608 29.7250 1.20551
609609 − 6.21110i − 0.251687i
610610 2.30278 0.0932367
611611 0 0
612612 −50.2389 −2.03079
613613 − 5.60555i − 0.226406i −0.993572 0.113203i 0.963889π-0.963889\pi
0.993572 0.113203i 0.0361111π-0.0361111\pi
614614 36.8444 1.48692
615615 3.00000 0.120972
616616 4.81665i 0.194069i
617617 − 38.4500i − 1.54794i −0.633224 0.773969i 0.718269π-0.718269\pi
0.633224 0.773969i 0.281731π-0.281731\pi
618618 − 9.21110i − 0.370525i
619619 − 14.4222i − 0.579677i −0.957076 0.289839i 0.906398π-0.906398\pi
0.957076 0.289839i 0.0936017π-0.0936017\pi
620620 13.2111 0.530571
621621 −15.0000 −0.601929
622622 − 21.2111i − 0.850488i
623623 −6.21110 −0.248843
624624 0 0
625625 1.00000 0.0400000
626626 − 32.2389i − 1.28852i
627627 −9.00000 −0.359425
628628 −37.0278 −1.47757
629629 27.4222i 1.09339i
630630 4.60555i 0.183490i
631631 36.0278i 1.43424i 0.696949 + 0.717121i 0.254541π0.254541\pi
−0.696949 + 0.717121i 0.745459π0.745459\pi
632632 15.6333i 0.621860i
633633 −23.6056 −0.938236
634634 13.8167 0.548729
635635 − 4.21110i − 0.167113i
636636 10.6056 0.420537
637637 0 0
638638 22.9638 0.909147
639639 − 9.63331i − 0.381088i
640640 −18.9083 −0.747417
641641 −9.42221 −0.372155 −0.186077 0.982535i 0.559578π-0.559578\pi
−0.186077 + 0.982535i 0.559578π0.559578\pi
642642 − 14.3028i − 0.564486i
643643 2.63331i 0.103848i 0.998651 + 0.0519238i 0.0165353π0.0165353\pi
−0.998651 + 0.0519238i 0.983465π0.983465\pi
644644 − 9.90833i − 0.390443i
645645 − 10.2111i − 0.402062i
646646 98.1749 3.86264
647647 −39.4222 −1.54985 −0.774923 0.632055i 0.782212π-0.782212\pi
−0.774923 + 0.632055i 0.782212π0.782212\pi
648648 3.00000i 0.117851i
649649 17.3667 0.681702
650650 0 0
651651 4.00000 0.156772
652652 − 12.5139i − 0.490081i
653653 −7.18335 −0.281106 −0.140553 0.990073i 0.544888π-0.544888\pi
−0.140553 + 0.990073i 0.544888π0.544888\pi
654654 −44.2389 −1.72988
655655 21.2111i 0.828786i
656656 0.908327i 0.0354642i
657657 − 1.57779i − 0.0615556i
658658 − 21.2111i − 0.826895i
659659 −34.8167 −1.35626 −0.678132 0.734940i 0.737210π-0.737210\pi
−0.678132 + 0.734940i 0.737210π0.737210\pi
660660 5.30278 0.206410
661661 4.63331i 0.180215i 0.995932 + 0.0901074i 0.0287210π0.0287210\pi
−0.995932 + 0.0901074i 0.971279π0.971279\pi
662662 23.0917 0.897483
663663 0 0
664664 27.6333 1.07238
665665 − 5.60555i − 0.217374i
666666 16.6056 0.643452
667667 −18.6333 −0.721485
668668 29.7250i 1.15009i
669669 − 4.21110i − 0.162811i
670670 16.1194i 0.622748i
671671 1.60555i 0.0619816i
672672 −5.30278 −0.204559
673673 17.6056 0.678644 0.339322 0.940670i 0.389802π-0.389802\pi
0.339322 + 0.940670i 0.389802π0.389802\pi
674674 − 59.0278i − 2.27366i
675675 5.00000 0.192450
676676 0 0
677677 −9.63331 −0.370238 −0.185119 0.982716i 0.559267π-0.559267\pi
−0.185119 + 0.982716i 0.559267π0.559267\pi
678678 3.69722i 0.141991i
679679 8.39445 0.322149
680680 −22.8167 −0.874979
681681 − 27.4222i − 1.05082i
682682 14.7889i 0.566296i
683683 − 36.2111i − 1.38558i −0.721140 0.692790i 0.756381π-0.756381\pi
0.721140 0.692790i 0.243619π-0.243619\pi
684684 − 37.0278i − 1.41579i
685685 1.60555 0.0613450
686686 29.9361 1.14296
687687 − 14.0000i − 0.534133i
688688 3.09167 0.117869
689689 0 0
690690 −6.90833 −0.262996
691691 − 30.0278i − 1.14231i −0.820842 0.571155i 0.806496π-0.806496\pi
0.820842 0.571155i 0.193504π-0.193504\pi
692692 15.9083 0.604744
693693 −3.21110 −0.121980
694694 − 13.3305i − 0.506020i
695695 − 6.39445i − 0.242555i
696696 − 18.6333i − 0.706294i
697697 − 22.8167i − 0.864242i
698698 8.72498 0.330245
699699 −15.2111 −0.575337
700700 3.30278i 0.124833i
701701 36.4222 1.37565 0.687824 0.725878i 0.258566π-0.258566\pi
0.687824 + 0.725878i 0.258566π0.258566\pi
702702 0 0
703703 −20.2111 −0.762276
704704 − 20.5778i − 0.775555i
705705 −9.21110 −0.346910
706706 38.7250 1.45743
707707 9.00000i 0.338480i
708708 − 35.7250i − 1.34263i
709709 13.8444i 0.519938i 0.965617 + 0.259969i 0.0837123π0.0837123\pi
−0.965617 + 0.259969i 0.916288π0.916288\pi
710710 − 11.0917i − 0.416263i
711711 −10.4222 −0.390863
712712 −18.6333 −0.698313
713713 − 12.0000i − 0.449404i
714714 −17.5139 −0.655440
715715 0 0
716716 75.3583 2.81627
717717 0 0
718718 −42.4222 −1.58318
719719 25.6056 0.954926 0.477463 0.878652i 0.341556π-0.341556\pi
0.477463 + 0.878652i 0.341556π0.341556\pi
720720 0.605551i 0.0225676i
721721 4.00000i 0.148968i
722722 28.6056i 1.06459i
723723 − 1.78890i − 0.0665298i
724724 58.2389 2.16443
725725 6.21110 0.230675
726726 − 19.3944i − 0.719796i
727727 −13.5778 −0.503573 −0.251786 0.967783i 0.581018π-0.581018\pi
−0.251786 + 0.967783i 0.581018π0.581018\pi
728728 0 0
729729 13.0000 0.481481
730730 − 1.81665i − 0.0672374i
731731 −77.6611 −2.87240
732732 3.30278 0.122074
733733 − 46.8444i − 1.73024i −0.501567 0.865119i 0.667243π-0.667243\pi
0.501567 0.865119i 0.332757π-0.332757\pi
734734 − 26.3028i − 0.970853i
735735 − 6.00000i − 0.221313i
736736 15.9083i 0.586389i
737737 −11.2389 −0.413989
738738 −13.8167 −0.508598
739739 35.6056i 1.30977i 0.755728 + 0.654886i 0.227283π0.227283\pi
−0.755728 + 0.654886i 0.772717π0.772717\pi
740740 11.9083 0.437759
741741 0 0
742742 −7.39445 −0.271459
743743 36.6333i 1.34395i 0.740576 + 0.671973i 0.234553π0.234553\pi
−0.740576 + 0.671973i 0.765447π0.765447\pi
744744 12.0000 0.439941
745745 3.00000 0.109911
746746 46.9638i 1.71947i
747747 18.4222i 0.674033i
748748 − 40.3305i − 1.47463i
749749 6.21110i 0.226949i
750750 2.30278 0.0840855
751751 −46.4500 −1.69498 −0.847492 0.530809i 0.821888π-0.821888\pi
−0.847492 + 0.530809i 0.821888π0.821888\pi
752752 − 2.78890i − 0.101701i
753753 7.18335 0.261776
754754 0 0
755755 1.21110 0.0440765
756756 16.5139i 0.600604i
757757 0.816654 0.0296818 0.0148409 0.999890i 0.495276π-0.495276\pi
0.0148409 + 0.999890i 0.495276π0.495276\pi
758758 22.1194 0.803414
759759 − 4.81665i − 0.174833i
760760 − 16.8167i − 0.610004i
761761 − 18.6333i − 0.675457i −0.941244 0.337728i 0.890341π-0.890341\pi
0.941244 0.337728i 0.109659π-0.109659\pi
762762 − 9.69722i − 0.351293i
763763 19.2111 0.695489
764764 −55.5416 −2.00943
765765 − 15.2111i − 0.549959i
766766 −56.7250 −2.04956
767767 0 0
768768 −17.9083 −0.646211
769769 11.0000i 0.396670i 0.980134 + 0.198335i 0.0635534π0.0635534\pi
−0.980134 + 0.198335i 0.936447π0.936447\pi
770770 −3.69722 −0.133239
771771 16.3944 0.590432
772772 51.5416i 1.85502i
773773 22.3944i 0.805472i 0.915316 + 0.402736i 0.131941π0.131941\pi
−0.915316 + 0.402736i 0.868059π0.868059\pi
774774 47.0278i 1.69038i
775775 4.00000i 0.143684i
776776 25.1833 0.904029
777777 3.60555 0.129348
778778 − 35.0278i − 1.25581i
779779 16.8167 0.602519
780780 0 0
781781 7.73338 0.276722
782782 52.5416i 1.87889i
783783 31.0555 1.10983
784784 1.81665 0.0648805
785785 − 11.2111i − 0.400141i
786786 48.8444i 1.74222i
787787 − 14.6333i − 0.521621i −0.965390 0.260811i 0.916010π-0.916010\pi
0.965390 0.260811i 0.0839898π-0.0839898\pi
788788 − 3.90833i − 0.139228i
789789 11.7889 0.419696
790790 −12.0000 −0.426941
791791 − 1.60555i − 0.0570868i
792792 −9.63331 −0.342305
793793 0 0
794794 −50.7250 −1.80016
795795 3.21110i 0.113886i
796796 42.3305 1.50037
797797 14.4500 0.511844 0.255922 0.966697i 0.417621π-0.417621\pi
0.255922 + 0.966697i 0.417621π0.417621\pi
798798 − 12.9083i − 0.456950i
799799 70.0555i 2.47839i
800800 − 5.30278i − 0.187481i
801801 − 12.4222i − 0.438917i
802802 −28.1194 −0.992932
803803 1.26662 0.0446979
804804 23.1194i 0.815359i
805805 3.00000 0.105736
806806 0 0
807807 −9.00000 −0.316815
808808 27.0000i 0.949857i
809809 −55.0555 −1.93565 −0.967824 0.251627i 0.919034π-0.919034\pi
−0.967824 + 0.251627i 0.919034π0.919034\pi
810810 −2.30278 −0.0809113
811811 − 46.4222i − 1.63010i −0.579388 0.815052i 0.696708π-0.696708\pi
0.579388 0.815052i 0.303292π-0.303292\pi
812812 20.5139i 0.719896i
813813 20.8167i 0.730072i
814814 13.3305i 0.467235i
815815 3.78890 0.132719
816816 −2.30278 −0.0806133
817817 − 57.2389i − 2.00253i
818818 18.9083 0.661114
819819 0 0
820820 −9.90833 −0.346014
821821 21.4222i 0.747640i 0.927501 + 0.373820i 0.121952π0.121952\pi
−0.927501 + 0.373820i 0.878048π0.878048\pi
822822 3.69722 0.128956
823823 16.6333 0.579801 0.289900 0.957057i 0.406378π-0.406378\pi
0.289900 + 0.957057i 0.406378π0.406378\pi
824824 12.0000i 0.418040i
825825 1.60555i 0.0558981i
826826 24.9083i 0.866672i
827827 42.4222i 1.47516i 0.675257 + 0.737582i 0.264033π0.264033\pi
−0.675257 + 0.737582i 0.735967π0.735967\pi
828828 19.8167 0.688676
829829 −29.4222 −1.02188 −0.510938 0.859618i 0.670702π-0.670702\pi
−0.510938 + 0.859618i 0.670702π0.670702\pi
830830 21.2111i 0.736248i
831831 −27.6056 −0.957626
832832 0 0
833833 −45.6333 −1.58110
834834 − 14.7250i − 0.509884i
835835 −9.00000 −0.311458
836836 29.7250 1.02806
837837 20.0000i 0.691301i
838838 − 39.6972i − 1.37132i
839839 20.0278i 0.691435i 0.938339 + 0.345717i 0.112364π0.112364\pi
−0.938339 + 0.345717i 0.887636π0.887636\pi
840840 3.00000i 0.103510i
841841 9.57779 0.330269
842842 74.6611 2.57299
843843 6.00000i 0.206651i
844844 77.9638 2.68363
845845 0 0
846846 42.4222 1.45851
847847 8.42221i 0.289390i
848848 −0.972244 −0.0333870
849849 −5.00000 −0.171600
850850 − 17.5139i − 0.600721i
851851 − 10.8167i − 0.370790i
852852 − 15.9083i − 0.545010i
853853 − 47.2111i − 1.61648i −0.588855 0.808239i 0.700421π-0.700421\pi
0.588855 0.808239i 0.299579π-0.299579\pi
854854 −2.30278 −0.0787994
855855 11.2111 0.383412
856856 18.6333i 0.636873i
857857 −6.00000 −0.204956 −0.102478 0.994735i 0.532677π-0.532677\pi
−0.102478 + 0.994735i 0.532677π0.532677\pi
858858 0 0
859859 10.7889 0.368112 0.184056 0.982916i 0.441077π-0.441077\pi
0.184056 + 0.982916i 0.441077π0.441077\pi
860860 33.7250i 1.15001i
861861 −3.00000 −0.102240
862862 67.3305 2.29329
863863 36.0000i 1.22545i 0.790295 + 0.612727i 0.209928π0.209928\pi
−0.790295 + 0.612727i 0.790072π0.790072\pi
864864 − 26.5139i − 0.902020i
865865 4.81665i 0.163771i
866866 8.30278i 0.282140i
867867 40.8444 1.38715
868868 −13.2111 −0.448414
869869 − 8.36669i − 0.283821i
870870 14.3028 0.484910
871871 0 0
872872 57.6333 1.95171
873873 16.7889i 0.568218i
874874 −38.7250 −1.30989
875875 −1.00000 −0.0338062
876876 − 2.60555i − 0.0880334i
877877 − 1.97224i − 0.0665979i −0.999445 0.0332990i 0.989399π-0.989399\pi
0.999445 0.0332990i 0.0106014π-0.0106014\pi
878878 − 62.7250i − 2.11687i
879879 − 10.3944i − 0.350596i
880880 −0.486122 −0.0163872
881881 21.8444 0.735957 0.367978 0.929834i 0.380050π-0.380050\pi
0.367978 + 0.929834i 0.380050π0.380050\pi
882882 27.6333i 0.930462i
883883 −11.6333 −0.391492 −0.195746 0.980655i 0.562713π-0.562713\pi
−0.195746 + 0.980655i 0.562713π0.562713\pi
884884 0 0
885885 10.8167 0.363598
886886 − 14.7889i − 0.496843i
887887 −37.0555 −1.24420 −0.622101 0.782937i 0.713721π-0.713721\pi
−0.622101 + 0.782937i 0.713721π0.713721\pi
888888 10.8167 0.362983
889889 4.21110i 0.141236i
890890 − 14.3028i − 0.479430i
891891 − 1.60555i − 0.0537880i
892892 13.9083i 0.465685i
893893 −51.6333 −1.72784
894894 6.90833 0.231049
895895 22.8167i 0.762677i
896896 18.9083 0.631683
897897 0 0
898898 −70.5416 −2.35400
899899 24.8444i 0.828607i
900900 −6.60555 −0.220185
901901 24.4222 0.813622
902902 − 11.0917i − 0.369312i
903903 10.2111i 0.339804i
904904 − 4.81665i − 0.160200i
905905 17.6333i 0.586151i
906906 2.78890 0.0926549
907907 38.2666 1.27062 0.635311 0.772256i 0.280872π-0.280872\pi
0.635311 + 0.772256i 0.280872π0.280872\pi
908908 90.5694i 3.00565i
909909 −18.0000 −0.597022
910910 0 0
911911 −36.0000 −1.19273 −0.596367 0.802712i 0.703390π-0.703390\pi
−0.596367 + 0.802712i 0.703390π0.703390\pi
912912 − 1.69722i − 0.0562007i
913913 −14.7889 −0.489441
914914 −61.7527 −2.04260
915915 1.00000i 0.0330590i
916916 46.2389i 1.52777i
917917 − 21.2111i − 0.700452i
918918 − 87.5694i − 2.89022i
919919 −38.8167 −1.28044 −0.640222 0.768190i 0.721157π-0.721157\pi
−0.640222 + 0.768190i 0.721157π0.721157\pi
920920 9.00000 0.296721
921921 16.0000i 0.527218i
922922 83.3860 2.74617
923923 0 0
924924 −5.30278 −0.174449
925925 3.60555i 0.118550i
926926 79.2666 2.60486
927927 −8.00000 −0.262754
928928 − 32.9361i − 1.08118i
929929 − 15.4222i − 0.505986i −0.967468 0.252993i 0.918585π-0.918585\pi
0.967468 0.252993i 0.0814150π-0.0814150\pi
930930 9.21110i 0.302044i
931931 − 33.6333i − 1.10229i
932932 50.2389 1.64563
933933 9.21110 0.301558
934934 6.42221i 0.210141i
935935 12.2111 0.399346
936936 0 0
937937 54.4777 1.77971 0.889855 0.456244i 0.150806π-0.150806\pi
0.889855 + 0.456244i 0.150806π0.150806\pi
938938 − 16.1194i − 0.526318i
939939 14.0000 0.456873
940940 30.4222 0.992263
941941 − 9.63331i − 0.314037i −0.987596 0.157018i 0.949812π-0.949812\pi
0.987596 0.157018i 0.0501882π-0.0501882\pi
942942 − 25.8167i − 0.841152i
943943 9.00000i 0.293080i
944944 3.27502i 0.106593i
945945 −5.00000 −0.162650
946946 −37.7527 −1.22745
947947 18.6333i 0.605501i 0.953070 + 0.302751i 0.0979049π0.0979049\pi
−0.953070 + 0.302751i 0.902095π0.902095\pi
948948 −17.2111 −0.558991
949949 0 0
950950 12.9083 0.418801
951951 6.00000i 0.194563i
952952 22.8167 0.739492
953953 14.4500 0.468080 0.234040 0.972227i 0.424805π-0.424805\pi
0.234040 + 0.972227i 0.424805π0.424805\pi
954954 − 14.7889i − 0.478808i
955955 − 16.8167i − 0.544174i
956956 0 0
957957 9.97224i 0.322357i
958958 66.3583 2.14394
959959 −1.60555 −0.0518460
960960 − 12.8167i − 0.413656i
961961 15.0000 0.483871
962962 0 0
963963 −12.4222 −0.400300
964964 5.90833i 0.190294i
965965 −15.6056 −0.502360
966966 6.90833 0.222272
967967 − 44.4777i − 1.43031i −0.698967 0.715153i 0.746357π-0.746357\pi
0.698967 0.715153i 0.253643π-0.253643\pi
968968 25.2666i 0.812100i
969969 42.6333i 1.36958i
970970 19.3305i 0.620666i
971971 −44.0278 −1.41292 −0.706459 0.707754i 0.749709π-0.749709\pi
−0.706459 + 0.707754i 0.749709π0.749709\pi
972972 −52.8444 −1.69499
973973 6.39445i 0.204997i
974974 −2.30278 −0.0737857
975975 0 0
976976 −0.302776 −0.00969161
977977 28.8167i 0.921926i 0.887419 + 0.460963i 0.152496π0.152496\pi
−0.887419 + 0.460963i 0.847504π0.847504\pi
978978 8.72498 0.278994
979979 9.97224 0.318714
980980 19.8167i 0.633020i
981981 38.4222i 1.22673i
982982 − 38.7250i − 1.23576i
983983 − 18.4222i − 0.587577i −0.955870 0.293789i 0.905084π-0.905084\pi
0.955870 0.293789i 0.0949162π-0.0949162\pi
984984 −9.00000 −0.286910
985985 1.18335 0.0377045
986986 − 108.780i − 3.46428i
987987 9.21110 0.293193
988988 0 0
989989 30.6333 0.974083
990990 − 7.39445i − 0.235011i
991991 40.0278 1.27152 0.635762 0.771885i 0.280686π-0.280686\pi
0.635762 + 0.771885i 0.280686π0.280686\pi
992992 21.2111 0.673453
993993 10.0278i 0.318221i
994994 11.0917i 0.351807i
995995 12.8167i 0.406315i
996996 30.4222i 0.963964i
997997 −18.4500 −0.584316 −0.292158 0.956370i 0.594373π-0.594373\pi
−0.292158 + 0.956370i 0.594373π0.594373\pi
998998 5.57779 0.176562
999999 18.0278i 0.570373i
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 845.2.c.d.506.1 4
13.2 odd 12 65.2.e.b.61.2 yes 4
13.3 even 3 845.2.m.d.316.4 8
13.4 even 6 845.2.m.d.361.4 8
13.5 odd 4 845.2.a.c.1.1 2
13.6 odd 12 65.2.e.b.16.2 4
13.7 odd 12 845.2.e.d.146.1 4
13.8 odd 4 845.2.a.f.1.2 2
13.9 even 3 845.2.m.d.361.1 8
13.10 even 6 845.2.m.d.316.1 8
13.11 odd 12 845.2.e.d.191.1 4
13.12 even 2 inner 845.2.c.d.506.4 4
39.2 even 12 585.2.j.d.451.1 4
39.5 even 4 7605.2.a.bg.1.2 2
39.8 even 4 7605.2.a.bb.1.1 2
39.32 even 12 585.2.j.d.406.1 4
52.15 even 12 1040.2.q.o.321.2 4
52.19 even 12 1040.2.q.o.81.2 4
65.2 even 12 325.2.o.b.74.4 8
65.19 odd 12 325.2.e.a.276.1 4
65.28 even 12 325.2.o.b.74.1 8
65.32 even 12 325.2.o.b.224.1 8
65.34 odd 4 4225.2.a.t.1.1 2
65.44 odd 4 4225.2.a.x.1.2 2
65.54 odd 12 325.2.e.a.126.1 4
65.58 even 12 325.2.o.b.224.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.e.b.16.2 4 13.6 odd 12
65.2.e.b.61.2 yes 4 13.2 odd 12
325.2.e.a.126.1 4 65.54 odd 12
325.2.e.a.276.1 4 65.19 odd 12
325.2.o.b.74.1 8 65.28 even 12
325.2.o.b.74.4 8 65.2 even 12
325.2.o.b.224.1 8 65.32 even 12
325.2.o.b.224.4 8 65.58 even 12
585.2.j.d.406.1 4 39.32 even 12
585.2.j.d.451.1 4 39.2 even 12
845.2.a.c.1.1 2 13.5 odd 4
845.2.a.f.1.2 2 13.8 odd 4
845.2.c.d.506.1 4 1.1 even 1 trivial
845.2.c.d.506.4 4 13.12 even 2 inner
845.2.e.d.146.1 4 13.7 odd 12
845.2.e.d.191.1 4 13.11 odd 12
845.2.m.d.316.1 8 13.10 even 6
845.2.m.d.316.4 8 13.3 even 3
845.2.m.d.361.1 8 13.9 even 3
845.2.m.d.361.4 8 13.4 even 6
1040.2.q.o.81.2 4 52.19 even 12
1040.2.q.o.321.2 4 52.15 even 12
4225.2.a.t.1.1 2 65.34 odd 4
4225.2.a.x.1.2 2 65.44 odd 4
7605.2.a.bb.1.1 2 39.8 even 4
7605.2.a.bg.1.2 2 39.5 even 4