Properties

Label 169.8.b.d.168.9
Level 169169
Weight 88
Character 169.168
Analytic conductor 52.79352.793
Analytic rank 00
Dimension 1414
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,8,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 169=132 169 = 13^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 169.b (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: 52.793069306852.7930693068
Analytic rank: 00
Dimension: 1414
Coefficient field: Q[x]/(x14+)\mathbb{Q}[x]/(x^{14} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x14+1279x12+629380x10+148562016x8+16872573312x6+790180980480x4++4669637050368 x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 21733136 2^{17}\cdot 3^{3}\cdot 13^{6}
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 168.9
Root 4.51724i4.51724i of defining polynomial
Character χ\chi == 169.168
Dual form 169.8.b.d.168.6

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+4.51724iq243.8390q3+107.595q4+49.0275iq5198.031iq6850.411iq7+1064.24iq8265.146q9221.469q10124.845iq114716.83q12+3841.52q142149.31iq15+8964.67q1611326.0q171197.73iq18+7864.97iq19+5275.09iq20+37281.1iq21+563.954q2286232.1q2346655.1iq24+75721.3q25+107500.q2791499.6iq28+125681.q29+9708.97q3078827.3iq31+176718.iq32+5473.06iq3351162.4iq34+41693.5q3528528.3q36+168074.iq3735528.0q3852176.9q40346835.iq41168408.q42+486181.q4313432.6iq4412999.4iq45389531.iq46+443195.iq47393002.q48+100343.q49+342052.iq50+496521.q511.44734e6q53+485602.iq54+6120.82q55+905040.q56344792.iq57+567730.iq58+2.66402e6iq59231254.iq601.29692e6q61+356082.q62+225483.iq63+349200.q6424723.2q66+4.65420e6iq671.21862e6q68+3.78033e6q69+188340.iq70+4.09074e6iq71282178.iq72+6.09222e6iq73759231.q743.31954e6q75+846228.iq76106169.q771131.00q79+439515.iq804.13279e6q81+1.56674e6q825.26434e6iq83+4.01125e6iq84555286.iq85+2.19620e6iq865.50971e6q87+132864.q881.89261e6iq89+58721.6q909.27810e6q92+3.45570e6iq932.00202e6q94385600.q957.74714e6iq96+1.68363e7iq97+453276.iq98+33102.1iq99+O(q100)q+4.51724i q^{2} -43.8390 q^{3} +107.595 q^{4} +49.0275i q^{5} -198.031i q^{6} -850.411i q^{7} +1064.24i q^{8} -265.146 q^{9} -221.469 q^{10} -124.845i q^{11} -4716.83 q^{12} +3841.52 q^{14} -2149.31i q^{15} +8964.67 q^{16} -11326.0 q^{17} -1197.73i q^{18} +7864.97i q^{19} +5275.09i q^{20} +37281.1i q^{21} +563.954 q^{22} -86232.1 q^{23} -46655.1i q^{24} +75721.3 q^{25} +107500. q^{27} -91499.6i q^{28} +125681. q^{29} +9708.97 q^{30} -78827.3i q^{31} +176718. i q^{32} +5473.06i q^{33} -51162.4i q^{34} +41693.5 q^{35} -28528.3 q^{36} +168074. i q^{37} -35528.0 q^{38} -52176.9 q^{40} -346835. i q^{41} -168408. q^{42} +486181. q^{43} -13432.6i q^{44} -12999.4i q^{45} -389531. i q^{46} +443195. i q^{47} -393002. q^{48} +100343. q^{49} +342052. i q^{50} +496521. q^{51} -1.44734e6 q^{53} +485602. i q^{54} +6120.82 q^{55} +905040. q^{56} -344792. i q^{57} +567730. i q^{58} +2.66402e6i q^{59} -231254. i q^{60} -1.29692e6 q^{61} +356082. q^{62} +225483. i q^{63} +349200. q^{64} -24723.2 q^{66} +4.65420e6i q^{67} -1.21862e6 q^{68} +3.78033e6 q^{69} +188340. i q^{70} +4.09074e6i q^{71} -282178. i q^{72} +6.09222e6i q^{73} -759231. q^{74} -3.31954e6 q^{75} +846228. i q^{76} -106169. q^{77} -1131.00 q^{79} +439515. i q^{80} -4.13279e6 q^{81} +1.56674e6 q^{82} -5.26434e6i q^{83} +4.01125e6i q^{84} -555286. i q^{85} +2.19620e6i q^{86} -5.50971e6 q^{87} +132864. q^{88} -1.89261e6i q^{89} +58721.6 q^{90} -9.27810e6 q^{92} +3.45570e6i q^{93} -2.00202e6 q^{94} -385600. q^{95} -7.74714e6i q^{96} +1.68363e7i q^{97} +453276. i q^{98} +33102.1i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 14q52q3766q4+6982q9+1018q10+38380q1247916q14+1266q16+76806q17+251764q22+137100q23+39380q25432400q27443166q29+315780q30++86840772q95+O(q100) 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/169Z)×\left(\mathbb{Z}/169\mathbb{Z}\right)^\times.

nn 22
χ(n)\chi(n) 1-1

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 4.51724i 0.399272i 0.979870 + 0.199636i 0.0639759π0.0639759\pi
−0.979870 + 0.199636i 0.936024π0.936024\pi
33 −43.8390 −0.937423 −0.468712 0.883351i 0.655282π-0.655282\pi
−0.468712 + 0.883351i 0.655282π0.655282\pi
44 107.595 0.840582
55 49.0275i 0.175406i 0.996147 + 0.0877030i 0.0279526π0.0279526\pi
−0.996147 + 0.0877030i 0.972047π0.972047\pi
66 − 198.031i − 0.374287i
77 − 850.411i − 0.937100i −0.883437 0.468550i 0.844777π-0.844777\pi
0.883437 0.468550i 0.155223π-0.155223\pi
88 1064.24i 0.734892i
99 −265.146 −0.121237
1010 −221.469 −0.0700346
1111 − 124.845i − 0.0282811i −0.999900 0.0141405i 0.995499π-0.995499\pi
0.999900 0.0141405i 0.00450122π-0.00450122\pi
1212 −4716.83 −0.787981
1313 0 0
1414 3841.52 0.374158
1515 − 2149.31i − 0.164430i
1616 8964.67 0.547160
1717 −11326.0 −0.559122 −0.279561 0.960128i 0.590189π-0.590189\pi
−0.279561 + 0.960128i 0.590189π0.590189\pi
1818 − 1197.73i − 0.0484066i
1919 7864.97i 0.263063i 0.991312 + 0.131531i 0.0419894π0.0419894\pi
−0.991312 + 0.131531i 0.958011π0.958011\pi
2020 5275.09i 0.147443i
2121 37281.1i 0.878459i
2222 563.954 0.0112918
2323 −86232.1 −1.47782 −0.738910 0.673804i 0.764659π-0.764659\pi
−0.738910 + 0.673804i 0.764659π0.764659\pi
2424 − 46655.1i − 0.688905i
2525 75721.3 0.969233
2626 0 0
2727 107500. 1.05107
2828 − 91499.6i − 0.787709i
2929 125681. 0.956920 0.478460 0.878109i 0.341195π-0.341195\pi
0.478460 + 0.878109i 0.341195π0.341195\pi
3030 9708.97 0.0656521
3131 − 78827.3i − 0.475237i −0.971359 0.237619i 0.923633π-0.923633\pi
0.971359 0.237619i 0.0763668π-0.0763668\pi
3232 176718.i 0.953358i
3333 5473.06i 0.0265113i
3434 − 51162.4i − 0.223241i
3535 41693.5 0.164373
3636 −28528.3 −0.101910
3737 168074.i 0.545500i 0.962085 + 0.272750i 0.0879331π0.0879331\pi
−0.962085 + 0.272750i 0.912067π0.912067\pi
3838 −35528.0 −0.105034
3939 0 0
4040 −52176.9 −0.128905
4141 − 346835.i − 0.785921i −0.919555 0.392960i 0.871451π-0.871451\pi
0.919555 0.392960i 0.128549π-0.128549\pi
4242 −168408. −0.350744
4343 486181. 0.932520 0.466260 0.884648i 0.345601π-0.345601\pi
0.466260 + 0.884648i 0.345601π0.345601\pi
4444 − 13432.6i − 0.0237726i
4545 − 12999.4i − 0.0212658i
4646 − 389531.i − 0.590052i
4747 443195.i 0.622662i 0.950302 + 0.311331i 0.100775π0.100775\pi
−0.950302 + 0.311331i 0.899225π0.899225\pi
4848 −393002. −0.512921
4949 100343. 0.121844
5050 342052.i 0.386987i
5151 496521. 0.524134
5252 0 0
5353 −1.44734e6 −1.33538 −0.667692 0.744438i 0.732718π-0.732718\pi
−0.667692 + 0.744438i 0.732718π0.732718\pi
5454 485602.i 0.419664i
5555 6120.82 0.00496067
5656 905040. 0.688668
5757 − 344792.i − 0.246601i
5858 567730.i 0.382071i
5959 2.66402e6i 1.68871i 0.535784 + 0.844355i 0.320016π0.320016\pi
−0.535784 + 0.844355i 0.679984π0.679984\pi
6060 − 231254.i − 0.138217i
6161 −1.29692e6 −0.731578 −0.365789 0.930698i 0.619201π-0.619201\pi
−0.365789 + 0.930698i 0.619201π0.619201\pi
6262 356082. 0.189749
6363 225483.i 0.113612i
6464 349200. 0.166511
6565 0 0
6666 −24723.2 −0.0105852
6767 4.65420e6i 1.89053i 0.326309 + 0.945263i 0.394195π0.394195\pi
−0.326309 + 0.945263i 0.605805π0.605805\pi
6868 −1.21862e6 −0.469988
6969 3.78033e6 1.38534
7070 188340.i 0.0656295i
7171 4.09074e6i 1.35643i 0.734864 + 0.678215i 0.237246π0.237246\pi
−0.734864 + 0.678215i 0.762754π0.762754\pi
7272 − 282178.i − 0.0890964i
7373 6.09222e6i 1.83293i 0.400117 + 0.916464i 0.368969π0.368969\pi
−0.400117 + 0.916464i 0.631031π0.631031\pi
7474 −759231. −0.217803
7575 −3.31954e6 −0.908581
7676 846228.i 0.221126i
7777 −106169. −0.0265022
7878 0 0
7979 −1131.00 −0.000258087 0 −0.000129044 1.00000i 0.500041π-0.500041\pi
−0.000129044 1.00000i 0.500041π0.500041\pi
8080 439515.i 0.0959752i
8181 −4.13279e6 −0.864064
8282 1.56674e6 0.313796
8383 − 5.26434e6i − 1.01058i −0.862950 0.505290i 0.831386π-0.831386\pi
0.862950 0.505290i 0.168614π-0.168614\pi
8484 4.01125e6i 0.738417i
8585 − 555286.i − 0.0980733i
8686 2.19620e6i 0.372329i
8787 −5.50971e6 −0.897039
8888 132864. 0.0207835
8989 − 1.89261e6i − 0.284574i −0.989825 0.142287i 0.954554π-0.954554\pi
0.989825 0.142287i 0.0454457π-0.0454457\pi
9090 58721.6 0.00849081
9191 0 0
9292 −9.27810e6 −1.24223
9393 3.45570e6i 0.445499i
9494 −2.00202e6 −0.248611
9595 −385600. −0.0461428
9696 − 7.74714e6i − 0.893700i
9797 1.68363e7i 1.87304i 0.350619 + 0.936518i 0.385971π0.385971\pi
−0.350619 + 0.936518i 0.614029π0.614029\pi
9898 453276.i 0.0486487i
9999 33102.1i 0.00342872i
100100 8.14720e6 0.814720
101101 −1.92121e7 −1.85546 −0.927728 0.373257i 0.878241π-0.878241\pi
−0.927728 + 0.373257i 0.878241π0.878241\pi
102102 2.24291e6i 0.209272i
103103 1.21968e7 1.09980 0.549902 0.835229i 0.314665π-0.314665\pi
0.549902 + 0.835229i 0.314665π0.314665\pi
104104 0 0
105105 −1.82780e6 −0.154087
106106 − 6.53801e6i − 0.533181i
107107 −6.84075e6 −0.539834 −0.269917 0.962884i 0.586996π-0.586996\pi
−0.269917 + 0.962884i 0.586996π0.586996\pi
108108 1.15664e7 0.883514
109109 − 3.26600e6i − 0.241559i −0.992679 0.120779i 0.961461π-0.961461\pi
0.992679 0.120779i 0.0385394π-0.0385394\pi
110110 27649.2i 0.00198065i
111111 − 7.36819e6i − 0.511364i
112112 − 7.62366e6i − 0.512744i
113113 173551. 0.0113150 0.00565748 0.999984i 0.498199π-0.498199\pi
0.00565748 + 0.999984i 0.498199π0.498199\pi
114114 1.55751e6 0.0984610
115115 − 4.22774e6i − 0.259218i
116116 1.35226e7 0.804370
117117 0 0
118118 −1.20340e7 −0.674254
119119 9.63178e6i 0.523953i
120120 2.28738e6 0.120838
121121 1.94716e7 0.999200
122122 − 5.85853e6i − 0.292098i
123123 1.52049e7i 0.736741i
124124 − 8.48138e6i − 0.399476i
125125 7.54269e6i 0.345415i
126126 −1.01856e6 −0.0453619
127127 1.80804e7 0.783239 0.391620 0.920127i 0.371915π-0.371915\pi
0.391620 + 0.920127i 0.371915π0.371915\pi
128128 2.41973e7i 1.01984i
129129 −2.13137e7 −0.874166
130130 0 0
131131 1.46819e7 0.570601 0.285301 0.958438i 0.407907π-0.407907\pi
0.285301 + 0.958438i 0.407907π0.407907\pi
132132 588871.i 0.0222849i
133133 6.68846e6 0.246516
134134 −2.10241e7 −0.754834
135135 5.27043e6i 0.184365i
136136 − 1.20536e7i − 0.410894i
137137 4.91779e7i 1.63399i 0.576648 + 0.816993i 0.304360π0.304360\pi
−0.576648 + 0.816993i 0.695640π0.695640\pi
138138 1.70767e7i 0.553128i
139139 −2.30684e7 −0.728561 −0.364280 0.931289i 0.618685π-0.618685\pi
−0.364280 + 0.931289i 0.618685π0.618685\pi
140140 4.48599e6 0.138169
141141 − 1.94292e7i − 0.583698i
142142 −1.84789e7 −0.541584
143143 0 0
144144 −2.37695e6 −0.0663363
145145 6.16180e6i 0.167849i
146146 −2.75200e7 −0.731836
147147 −4.39895e6 −0.114219
148148 1.80838e7i 0.458537i
149149 − 5.02309e7i − 1.24400i −0.783018 0.621998i 0.786321π-0.786321\pi
0.783018 0.621998i 0.213679π-0.213679\pi
150150 − 1.49952e7i − 0.362771i
151151 2.96842e7i 0.701626i 0.936446 + 0.350813i 0.114095π0.114095\pi
−0.936446 + 0.350813i 0.885905π0.885905\pi
152152 −8.37020e6 −0.193323
153153 3.00305e6 0.0677864
154154 − 479593.i − 0.0105816i
155155 3.86470e6 0.0833595
156156 0 0
157157 3.56376e7 0.734952 0.367476 0.930033i 0.380222π-0.380222\pi
0.367476 + 0.930033i 0.380222π0.380222\pi
158158 − 5108.99i 0 0.000103047i
159159 6.34500e7 1.25182
160160 −8.66404e6 −0.167225
161161 7.33328e7i 1.38487i
162162 − 1.86688e7i − 0.344996i
163163 1.04922e7i 0.189763i 0.995489 + 0.0948814i 0.0302472π0.0302472\pi
−0.995489 + 0.0948814i 0.969753π0.969753\pi
164164 − 3.73175e7i − 0.660631i
165165 −268330. −0.00465025
166166 2.37803e7 0.403496
167167 − 1.94057e7i − 0.322419i −0.986920 0.161210i 0.948460π-0.948460\pi
0.986920 0.161210i 0.0515395π-0.0515395\pi
168168 −3.96760e7 −0.645573
169169 0 0
170170 2.50836e6 0.0391579
171171 − 2.08537e6i − 0.0318930i
172172 5.23104e7 0.783860
173173 −4.57500e7 −0.671785 −0.335892 0.941900i 0.609038π-0.609038\pi
−0.335892 + 0.941900i 0.609038π0.609038\pi
174174 − 2.48887e7i − 0.358162i
175175 − 6.43943e7i − 0.908268i
176176 − 1.11919e6i − 0.0154743i
177177 − 1.16788e8i − 1.58304i
178178 8.54938e6 0.113623
179179 3.92270e7 0.511211 0.255605 0.966781i 0.417725π-0.417725\pi
0.255605 + 0.966781i 0.417725π0.417725\pi
180180 − 1.39867e6i − 0.0178756i
181181 6.74932e7 0.846029 0.423015 0.906123i 0.360972π-0.360972\pi
0.423015 + 0.906123i 0.360972π0.360972\pi
182182 0 0
183183 5.68558e7 0.685798
184184 − 9.17715e7i − 1.08604i
185185 −8.24024e6 −0.0956839
186186 −1.56103e7 −0.177875
187187 1.41399e6i 0.0158126i
188188 4.76853e7i 0.523398i
189189 − 9.14188e7i − 0.984962i
190190 − 1.74185e6i − 0.0184235i
191191 1.71927e8 1.78536 0.892682 0.450688i 0.148821π-0.148821\pi
0.892682 + 0.450688i 0.148821π0.148821\pi
192192 −1.53085e7 −0.156092
193193 2.48491e7i 0.248806i 0.992232 + 0.124403i 0.0397016π0.0397016\pi
−0.992232 + 0.124403i 0.960298π0.960298\pi
194194 −7.60538e7 −0.747850
195195 0 0
196196 1.07964e7 0.102420
197197 − 1.37653e8i − 1.28279i −0.767213 0.641393i 0.778357π-0.778357\pi
0.767213 0.641393i 0.221643π-0.221643\pi
198198 −149530. −0.00136899
199199 −9.81839e7 −0.883190 −0.441595 0.897214i 0.645587π-0.645587\pi
−0.441595 + 0.897214i 0.645587π0.645587\pi
200200 8.05855e7i 0.712282i
201201 − 2.04035e8i − 1.77222i
202202 − 8.67859e7i − 0.740831i
203203 − 1.06880e8i − 0.896730i
204204 5.34229e7 0.440577
205205 1.70044e7 0.137855
206206 5.50959e7i 0.439121i
207207 2.28641e7 0.179167
208208 0 0
209209 981900. 0.00743970
210210 − 8.25662e6i − 0.0615226i
211211 −9.35140e7 −0.685312 −0.342656 0.939461i 0.611326π-0.611326\pi
−0.342656 + 0.939461i 0.611326π0.611326\pi
212212 −1.55726e8 −1.12250
213213 − 1.79334e8i − 1.27155i
214214 − 3.09013e7i − 0.215541i
215215 2.38362e7i 0.163570i
216216 1.14405e8i 0.772426i
217217 −6.70356e7 −0.445345
218218 1.47533e7 0.0964477
219219 − 2.67076e8i − 1.71823i
220220 658566. 0.00416985
221221 0 0
222222 3.32839e7 0.204173
223223 − 6.52040e7i − 0.393738i −0.980430 0.196869i 0.936923π-0.936923\pi
0.980430 0.196869i 0.0630774π-0.0630774\pi
224224 1.50283e8 0.893392
225225 −2.00772e7 −0.117507
226226 783973.i 0.00451774i
227227 1.31044e8i 0.743578i 0.928317 + 0.371789i 0.121255π0.121255\pi
−0.928317 + 0.371789i 0.878745π0.878745\pi
228228 − 3.70977e7i − 0.207289i
229229 − 1.87132e8i − 1.02973i −0.857272 0.514864i 0.827842π-0.827842\pi
0.857272 0.514864i 0.172158π-0.172158\pi
230230 1.90977e7 0.103499
231231 4.65435e6 0.0248438
232232 1.33754e8i 0.703233i
233233 −8.77467e7 −0.454449 −0.227225 0.973842i 0.572965π-0.572965\pi
−0.227225 + 0.973842i 0.572965π0.572965\pi
234234 0 0
235235 −2.17287e7 −0.109219
236236 2.86634e8i 1.41950i
237237 49581.7 0.000241937 0
238238 −4.35091e7 −0.209200
239239 2.99804e8i 1.42051i 0.703943 + 0.710257i 0.251421π0.251421\pi
−0.703943 + 0.710257i 0.748579π0.748579\pi
240240 − 1.92679e7i − 0.0899694i
241241 1.20929e8i 0.556509i 0.960507 + 0.278254i 0.0897558π0.0897558\pi
−0.960507 + 0.278254i 0.910244π0.910244\pi
242242 8.79579e7i 0.398952i
243243 −5.39242e7 −0.241080
244244 −1.39542e8 −0.614951
245245 4.91958e6i 0.0213721i
246246 −6.86841e7 −0.294160
247247 0 0
248248 8.38910e7 0.349248
249249 2.30783e8i 0.947341i
250250 −3.40722e7 −0.137915
251251 3.42122e8 1.36560 0.682800 0.730606i 0.260762π-0.260762\pi
0.682800 + 0.730606i 0.260762π0.260762\pi
252252 2.42608e7i 0.0954998i
253253 1.07656e7i 0.0417943i
254254 8.16735e7i 0.312725i
255255 2.43432e7i 0.0919362i
256256 −6.46077e7 −0.240682
257257 1.33787e8 0.491641 0.245820 0.969315i 0.420943π-0.420943\pi
0.245820 + 0.969315i 0.420943π0.420943\pi
258258 − 9.62790e7i − 0.349030i
259259 1.42932e8 0.511188
260260 0 0
261261 −3.33237e7 −0.116014
262262 6.63217e7i 0.227825i
263263 −4.16302e8 −1.41112 −0.705559 0.708651i 0.749304π-0.749304\pi
−0.705559 + 0.708651i 0.749304π0.749304\pi
264264 −5.82464e6 −0.0194830
265265 − 7.09596e7i − 0.234234i
266266 3.02134e7i 0.0984270i
267267 8.29700e7i 0.266767i
268268 5.00766e8i 1.58914i
269269 −2.98622e8 −0.935381 −0.467690 0.883892i 0.654914π-0.654914\pi
−0.467690 + 0.883892i 0.654914π0.654914\pi
270270 −2.38078e7 −0.0736116
271271 5.32417e7i 0.162502i 0.996694 + 0.0812512i 0.0258916π0.0258916\pi
−0.996694 + 0.0812512i 0.974108π0.974108\pi
272272 −1.01534e8 −0.305929
273273 0 0
274274 −2.22149e8 −0.652404
275275 − 9.45340e6i − 0.0274109i
276276 4.06742e8 1.16449
277277 1.94332e8 0.549369 0.274685 0.961534i 0.411427π-0.411427\pi
0.274685 + 0.961534i 0.411427π0.411427\pi
278278 − 1.04206e8i − 0.290894i
279279 2.09007e7i 0.0576165i
280280 4.43718e7i 0.120796i
281281 − 7.35301e7i − 0.197694i −0.995103 0.0988468i 0.968485π-0.968485\pi
0.995103 0.0988468i 0.0315154π-0.0315154\pi
282282 8.77665e7 0.233054
283283 2.77363e8 0.727437 0.363719 0.931509i 0.381507π-0.381507\pi
0.363719 + 0.931509i 0.381507π0.381507\pi
284284 4.40141e8i 1.14019i
285285 1.69043e7 0.0432554
286286 0 0
287287 −2.94952e8 −0.736486
288288 − 4.68561e7i − 0.115583i
289289 −2.82060e8 −0.687383
290290 −2.78344e7 −0.0670175
291291 − 7.38087e8i − 1.75583i
292292 6.55489e8i 1.54073i
293293 8.92358e7i 0.207254i 0.994616 + 0.103627i 0.0330448π0.0330448\pi
−0.994616 + 0.103627i 0.966955π0.966955\pi
294294 − 1.98711e7i − 0.0456044i
295295 −1.30610e8 −0.296210
296296 −1.78871e8 −0.400883
297297 − 1.34207e7i − 0.0297255i
298298 2.26905e8 0.496693
299299 0 0
300300 −3.57165e8 −0.763737
301301 − 4.13454e8i − 0.873864i
302302 −1.34091e8 −0.280140
303303 8.42240e8 1.73935
304304 7.05069e7i 0.143938i
305305 − 6.35849e7i − 0.128323i
306306 1.35655e7i 0.0270652i
307307 2.03142e8i 0.400696i 0.979725 + 0.200348i 0.0642072π0.0642072\pi
−0.979725 + 0.200348i 0.935793π0.935793\pi
308308 −1.14232e7 −0.0222773
309309 −5.34694e8 −1.03098
310310 1.74578e7i 0.0332831i
311311 7.82046e8 1.47425 0.737125 0.675757i 0.236183π-0.236183\pi
0.737125 + 0.675757i 0.236183π0.236183\pi
312312 0 0
313313 3.84527e8 0.708797 0.354399 0.935094i 0.384686π-0.384686\pi
0.354399 + 0.935094i 0.384686π0.384686\pi
314314 1.60984e8i 0.293446i
315315 −1.10549e7 −0.0199281
316316 −121689. −0.000216944 0
317317 4.19845e8i 0.740256i 0.928981 + 0.370128i 0.120686π0.120686\pi
−0.928981 + 0.370128i 0.879314π0.879314\pi
318318 2.86619e8i 0.499817i
319319 − 1.56906e7i − 0.0270627i
320320 1.71204e7i 0.0292071i
321321 2.99891e8 0.506053
322322 −3.31262e8 −0.552938
323323 − 8.90789e7i − 0.147084i
324324 −4.44666e8 −0.726317
325325 0 0
326326 −4.73960e7 −0.0757669
327327 1.43178e8i 0.226443i
328328 3.69114e8 0.577567
329329 3.76898e8 0.583497
330330 − 1.21211e6i − 0.00185671i
331331 8.18268e8i 1.24022i 0.784516 + 0.620109i 0.212911π0.212911\pi
−0.784516 + 0.620109i 0.787089π0.787089\pi
332332 − 5.66414e8i − 0.849475i
333333 − 4.45641e7i − 0.0661349i
334334 8.76601e7 0.128733
335335 −2.28183e8 −0.331610
336336 3.34213e8i 0.480658i
337337 4.50190e8 0.640753 0.320377 0.947290i 0.396191π-0.396191\pi
0.320377 + 0.947290i 0.396191π0.396191\pi
338338 0 0
339339 −7.60830e6 −0.0106069
340340 − 5.97458e7i − 0.0824386i
341341 −9.84117e6 −0.0134402
342342 9.42011e6 0.0127340
343343 − 7.85684e8i − 1.05128i
344344 5.17412e8i 0.685302i
345345 1.85340e8i 0.242997i
346346 − 2.06664e8i − 0.268225i
347347 −1.12644e9 −1.44729 −0.723647 0.690170i 0.757536π-0.757536\pi
−0.723647 + 0.690170i 0.757536π0.757536\pi
348348 −5.92815e8 −0.754035
349349 − 1.02048e9i − 1.28504i −0.766269 0.642520i 0.777889π-0.777889\pi
0.766269 0.642520i 0.222111π-0.222111\pi
350350 2.90885e8 0.362646
351351 0 0
352352 2.20623e7 0.0269620
353353 − 3.28291e6i − 0.00397235i −0.999998 0.00198618i 0.999368π-0.999368\pi
0.999998 0.00198618i 0.000632220π-0.000632220\pi
354354 5.27559e8 0.632062
355355 −2.00558e8 −0.237926
356356 − 2.03634e8i − 0.239208i
357357 − 4.22247e8i − 0.491166i
358358 1.77198e8i 0.204112i
359359 1.01744e9i 1.16059i 0.814406 + 0.580296i 0.197063π0.197063\pi
−0.814406 + 0.580296i 0.802937π0.802937\pi
360360 1.38345e7 0.0156280
361361 8.32014e8 0.930798
362362 3.04883e8i 0.337795i
363363 −8.53614e8 −0.936674
364364 0 0
365365 −2.98686e8 −0.321507
366366 2.56832e8i 0.273820i
367367 −1.36503e9 −1.44149 −0.720743 0.693202i 0.756199π-0.756199\pi
−0.720743 + 0.693202i 0.756199π0.756199\pi
368368 −7.73043e8 −0.808604
369369 9.19618e7i 0.0952829i
370370 − 3.72232e7i − 0.0382039i
371371 1.23084e9i 1.25139i
372372 3.71815e8i 0.374478i
373373 9.65141e8 0.962963 0.481482 0.876456i 0.340099π-0.340099\pi
0.481482 + 0.876456i 0.340099π0.340099\pi
374374 −6.38736e6 −0.00631351
375375 − 3.30664e8i − 0.323800i
376376 −4.71665e8 −0.457590
377377 0 0
378378 4.12961e8 0.393267
379379 8.15097e8i 0.769081i 0.923108 + 0.384540i 0.125640π0.125640\pi
−0.923108 + 0.384540i 0.874360π0.874360\pi
380380 −4.14884e7 −0.0387868
381381 −7.92625e8 −0.734227
382382 7.76635e8i 0.712845i
383383 − 1.28104e9i − 1.16511i −0.812790 0.582556i 0.802053π-0.802053\pi
0.812790 0.582556i 0.197947π-0.197947\pi
384384 − 1.06079e9i − 0.956023i
385385 − 5.20521e6i − 0.00464864i
386386 −1.12250e8 −0.0993412
387387 −1.28909e8 −0.113056
388388 1.81150e9i 1.57444i
389389 −9.51453e8 −0.819528 −0.409764 0.912192i 0.634389π-0.634389\pi
−0.409764 + 0.912192i 0.634389π0.634389\pi
390390 0 0
391391 9.76667e8 0.826281
392392 1.06789e8i 0.0895419i
393393 −6.43639e8 −0.534895
394394 6.21812e8 0.512180
395395 − 55449.9i 0 4.52701e-5i
396396 3.56160e6i 0.00288212i
397397 − 2.46693e8i − 0.197875i −0.995094 0.0989375i 0.968456π-0.968456\pi
0.995094 0.0989375i 0.0315444π-0.0315444\pi
398398 − 4.43520e8i − 0.352633i
399399 −2.93215e8 −0.231090
400400 6.78817e8 0.530326
401401 − 2.02504e9i − 1.56830i −0.620571 0.784150i 0.713099π-0.713099\pi
0.620571 0.784150i 0.286901π-0.286901\pi
402402 9.21676e8 0.707599
403403 0 0
404404 −2.06712e9 −1.55966
405405 − 2.02620e8i − 0.151562i
406406 4.82804e8 0.358039
407407 2.09831e7 0.0154273
408408 5.28417e8i 0.385182i
409409 − 1.45005e9i − 1.04798i −0.851725 0.523989i 0.824443π-0.824443\pi
0.851725 0.523989i 0.175557π-0.175557\pi
410410 7.68131e7i 0.0550417i
411411 − 2.15591e9i − 1.53174i
412412 1.31231e9 0.924475
413413 2.26551e9 1.58249
414414 1.03283e8i 0.0715363i
415415 2.58097e8 0.177262
416416 0 0
417417 1.01129e9 0.682970
418418 4.43548e6i 0.00297046i
419419 7.01637e8 0.465976 0.232988 0.972480i 0.425150π-0.425150\pi
0.232988 + 0.972480i 0.425150π0.425150\pi
420420 −1.96661e8 −0.129523
421421 6.04587e8i 0.394886i 0.980314 + 0.197443i 0.0632637π0.0632637\pi
−0.980314 + 0.197443i 0.936736π0.936736\pi
422422 − 4.22426e8i − 0.273626i
423423 − 1.17511e8i − 0.0754899i
424424 − 1.54032e9i − 0.981364i
425425 −8.57622e8 −0.541919
426426 8.10094e8 0.507693
427427 1.10292e9i 0.685561i
428428 −7.36027e8 −0.453775
429429 0 0
430430 −1.07674e8 −0.0653087
431431 4.89565e7i 0.0294537i 0.999892 + 0.0147269i 0.00468788π0.00468788\pi
−0.999892 + 0.0147269i 0.995312π0.995312\pi
432432 9.63698e8 0.575106
433433 −1.63499e9 −0.967851 −0.483925 0.875109i 0.660789π-0.660789\pi
−0.483925 + 0.875109i 0.660789π0.660789\pi
434434 − 3.02816e8i − 0.177814i
435435 − 2.70127e8i − 0.157346i
436436 − 3.51403e8i − 0.203050i
437437 − 6.78213e8i − 0.388760i
438438 1.20645e9 0.686041
439439 −1.47539e9 −0.832304 −0.416152 0.909295i 0.636622π-0.636622\pi
−0.416152 + 0.909295i 0.636622π0.636622\pi
440440 6.51401e6i 0.00364556i
441441 −2.66057e7 −0.0147720
442442 0 0
443443 2.19329e8 0.119863 0.0599313 0.998203i 0.480912π-0.480912\pi
0.0599313 + 0.998203i 0.480912π0.480912\pi
444444 − 7.92776e8i − 0.429843i
445445 9.27898e7 0.0499161
446446 2.94542e8 0.157208
447447 2.20207e9i 1.16615i
448448 − 2.96963e8i − 0.156038i
449449 − 1.00005e9i − 0.521387i −0.965422 0.260693i 0.916049π-0.916049\pi
0.965422 0.260693i 0.0839512π-0.0839512\pi
450450 − 9.06936e7i − 0.0469173i
451451 −4.33004e7 −0.0222267
452452 1.86731e7 0.00951115
453453 − 1.30132e9i − 0.657721i
454454 −5.91957e8 −0.296890
455455 0 0
456456 3.66941e8 0.181225
457457 − 3.30790e9i − 1.62123i −0.585578 0.810616i 0.699132π-0.699132\pi
0.585578 0.810616i 0.300868π-0.300868\pi
458458 8.45319e8 0.411142
459459 −1.21754e9 −0.587678
460460 − 4.54882e8i − 0.217894i
461461 − 3.94229e9i − 1.87411i −0.349180 0.937056i 0.613540π-0.613540\pi
0.349180 0.937056i 0.386460π-0.386460\pi
462462 2.10249e7i 0.00991941i
463463 5.62775e8i 0.263513i 0.991282 + 0.131756i 0.0420616π0.0420616\pi
−0.991282 + 0.131756i 0.957938π0.957938\pi
464464 1.12669e9 0.523589
465465 −1.69424e8 −0.0781431
466466 − 3.96373e8i − 0.181449i
467467 1.78505e7 0.00811038 0.00405519 0.999992i 0.498709π-0.498709\pi
0.00405519 + 0.999992i 0.498709π0.498709\pi
468468 0 0
469469 3.95798e9 1.77161
470470 − 9.81540e7i − 0.0436079i
471471 −1.56231e9 −0.688961
472472 −2.83515e9 −1.24102
473473 − 6.06971e7i − 0.0263727i
474474 223973.i 0 9.65987e-5i
475475 5.95546e8i 0.254969i
476476 1.03633e9i 0.440425i
477477 3.83758e8 0.161898
478478 −1.35429e9 −0.567171
479479 − 6.41286e8i − 0.266611i −0.991075 0.133305i 0.957441π-0.957441\pi
0.991075 0.133305i 0.0425591π-0.0425591\pi
480480 3.79822e8 0.156760
481481 0 0
482482 −5.46267e8 −0.222198
483483 − 3.21483e9i − 1.29820i
484484 2.09504e9 0.839910
485485 −8.25442e8 −0.328542
486486 − 2.43589e8i − 0.0962565i
487487 2.14300e9i 0.840759i 0.907348 + 0.420380i 0.138103π0.138103\pi
−0.907348 + 0.420380i 0.861897π0.861897\pi
488488 − 1.38024e9i − 0.537631i
489489 − 4.59968e8i − 0.177888i
490490 −2.22230e7 −0.00853327
491491 1.50315e9 0.573084 0.286542 0.958068i 0.407494π-0.407494\pi
0.286542 + 0.958068i 0.407494π0.407494\pi
492492 1.63596e9i 0.619291i
493493 −1.42346e9 −0.535035
494494 0 0
495495 −1.62291e6 −0.000601418 0
496496 − 7.06661e8i − 0.260031i
497497 3.47881e9 1.27111
498498 −1.04250e9 −0.378247
499499 − 3.29943e9i − 1.18874i −0.804192 0.594369i 0.797402π-0.797402\pi
0.804192 0.594369i 0.202598π-0.202598\pi
500500 8.11552e8i 0.290350i
501501 8.50724e8i 0.302243i
502502 1.54545e9i 0.545245i
503503 −1.85586e9 −0.650215 −0.325107 0.945677i 0.605400π-0.605400\pi
−0.325107 + 0.945677i 0.605400π0.605400\pi
504504 −2.39968e8 −0.0834922
505505 − 9.41922e8i − 0.325458i
506506 −4.86309e7 −0.0166873
507507 0 0
508508 1.94535e9 0.658377
509509 4.17614e9i 1.40366i 0.712342 + 0.701832i 0.247634π0.247634\pi
−0.712342 + 0.701832i 0.752366π0.752366\pi
510510 −1.09964e8 −0.0367075
511511 5.18089e9 1.71764
512512 2.80541e9i 0.923744i
513513 8.45481e8i 0.276499i
514514 6.04348e8i 0.196298i
515515 5.97978e8i 0.192912i
516516 −2.29323e9 −0.734808
517517 5.53305e7 0.0176095
518518 6.45659e8i 0.204103i
519519 2.00563e9 0.629747
520520 0 0
521521 1.46095e9 0.452589 0.226295 0.974059i 0.427339π-0.427339\pi
0.226295 + 0.974059i 0.427339π0.427339\pi
522522 − 1.50531e8i − 0.0463213i
523523 6.21674e8 0.190023 0.0950116 0.995476i 0.469711π-0.469711\pi
0.0950116 + 0.995476i 0.469711π0.469711\pi
524524 1.57969e9 0.479637
525525 2.82298e9i 0.851432i
526526 − 1.88054e9i − 0.563420i
527527 8.92800e8i 0.265715i
528528 4.90642e7i 0.0145059i
529529 4.03115e9 1.18395
530530 3.20542e8 0.0935232
531531 − 7.06354e8i − 0.204735i
532532 7.19642e8 0.207217
533533 0 0
534534 −3.74796e8 −0.106512
535535 − 3.35385e8i − 0.0946902i
536536 −4.95317e9 −1.38933
537537 −1.71967e9 −0.479221
538538 − 1.34895e9i − 0.373471i
539539 − 1.25273e7i − 0.00344587i
540540 5.67069e8i 0.154974i
541541 − 3.06369e9i − 0.831868i −0.909395 0.415934i 0.863455π-0.863455\pi
0.909395 0.415934i 0.136545π-0.136545\pi
542542 −2.40506e8 −0.0648826
543543 −2.95883e9 −0.793087
544544 − 2.00151e9i − 0.533043i
545545 1.60124e8 0.0423709
546546 0 0
547547 −7.27564e9 −1.90071 −0.950354 0.311171i 0.899279π-0.899279\pi
−0.950354 + 0.311171i 0.899279π0.899279\pi
548548 5.29127e9i 1.37350i
549549 3.43875e8 0.0886945
550550 4.27033e7 0.0109444
551551 9.88475e8i 0.251730i
552552 4.02317e9i 1.01808i
553553 961813.i 0 0.000241854i
554554 8.77844e8i 0.219348i
555555 3.61243e8 0.0896963
556556 −2.48203e9 −0.612415
557557 5.92128e9i 1.45185i 0.687773 + 0.725926i 0.258589π0.258589\pi
−0.687773 + 0.725926i 0.741411π0.741411\pi
558558 −9.44137e7 −0.0230046
559559 0 0
560560 3.73769e8 0.0899383
561561 − 6.19880e7i − 0.0148231i
562562 3.32153e8 0.0789335
563563 6.42377e9 1.51709 0.758543 0.651623i 0.225911π-0.225911\pi
0.758543 + 0.651623i 0.225911π0.225911\pi
564564 − 2.09048e9i − 0.490646i
565565 8.50877e6i 0.00198471i
566566 1.25292e9i 0.290445i
567567 3.51457e9i 0.809715i
568568 −4.35352e9 −0.996830
569569 5.59404e9 1.27301 0.636506 0.771272i 0.280379π-0.280379\pi
0.636506 + 0.771272i 0.280379π0.280379\pi
570570 7.63608e7i 0.0172706i
571571 2.51307e9 0.564908 0.282454 0.959281i 0.408852π-0.408852\pi
0.282454 + 0.959281i 0.408852π0.408852\pi
572572 0 0
573573 −7.53709e9 −1.67364
574574 − 1.33237e9i − 0.294058i
575575 −6.52961e9 −1.43235
576576 −9.25889e7 −0.0201874
577577 − 5.64709e9i − 1.22380i −0.790936 0.611899i 0.790406π-0.790406\pi
0.790936 0.611899i 0.209594π-0.209594\pi
578578 − 1.27413e9i − 0.274453i
579579 − 1.08936e9i − 0.233237i
580580 6.62976e8i 0.141091i
581581 −4.47686e9 −0.947014
582582 3.33412e9 0.701052
583583 1.80693e8i 0.0377661i
584584 −6.48357e9 −1.34700
585585 0 0
586586 −4.03100e8 −0.0827506
587587 8.31882e9i 1.69757i 0.528736 + 0.848786i 0.322666π0.322666\pi
−0.528736 + 0.848786i 0.677334π0.677334\pi
588588 −4.73303e8 −0.0960105
589589 6.19974e8 0.125017
590590 − 5.89998e8i − 0.118268i
591591 6.03456e9i 1.20251i
592592 1.50673e9i 0.298476i
593593 − 2.05874e9i − 0.405425i −0.979238 0.202713i 0.935024π-0.935024\pi
0.979238 0.202713i 0.0649757π-0.0649757\pi
594594 6.06248e7 0.0118685
595595 −4.72222e8 −0.0919045
596596 − 5.40457e9i − 1.04568i
597597 4.30428e9 0.827923
598598 0 0
599599 −3.14417e9 −0.597740 −0.298870 0.954294i 0.596610π-0.596610\pi
−0.298870 + 0.954294i 0.596610π0.596610\pi
600600 − 3.53278e9i − 0.667710i
601601 −4.17491e9 −0.784489 −0.392244 0.919861i 0.628301π-0.628301\pi
−0.392244 + 0.919861i 0.628301π0.628301\pi
602602 1.86767e9 0.348909
603603 − 1.23404e9i − 0.229202i
604604 3.19385e9i 0.589774i
605605 9.54642e8i 0.175266i
606606 3.80460e9i 0.694473i
607607 1.55868e9 0.282877 0.141438 0.989947i 0.454827π-0.454827\pi
0.141438 + 0.989947i 0.454827π0.454827\pi
608608 −1.38988e9 −0.250793
609609 4.68552e9i 0.840615i
610610 2.87229e8 0.0512358
611611 0 0
612612 3.23112e8 0.0569801
613613 2.10767e9i 0.369566i 0.982779 + 0.184783i 0.0591582π0.0591582\pi
−0.982779 + 0.184783i 0.940842π0.940842\pi
614614 −9.17641e8 −0.159987
615615 −7.45456e8 −0.129229
616616 − 1.12989e8i − 0.0194763i
617617 3.64167e9i 0.624169i 0.950054 + 0.312084i 0.101027π0.101027\pi
−0.950054 + 0.312084i 0.898973π0.898973\pi
618618 − 2.41535e9i − 0.411642i
619619 3.99887e9i 0.677672i 0.940845 + 0.338836i 0.110033π0.110033\pi
−0.940845 + 0.338836i 0.889967π0.889967\pi
620620 4.15821e8 0.0700705
621621 −9.26991e9 −1.55330
622622 3.53269e9i 0.588626i
623623 −1.60950e9 −0.266675
624624 0 0
625625 5.54593e9 0.908645
626626 1.73700e9i 0.283003i
627627 −4.30455e7 −0.00697415
628628 3.83440e9 0.617788
629629 − 1.90361e9i − 0.305001i
630630 − 4.99375e7i − 0.00795674i
631631 − 5.76033e9i − 0.912735i −0.889791 0.456368i 0.849150π-0.849150\pi
0.889791 0.456368i 0.150850π-0.150850\pi
632632 − 1.20365e6i 0 0.000189666i
633633 4.09956e9 0.642427
634634 −1.89654e9 −0.295563
635635 8.86435e8i 0.137385i
636636 6.82688e9 1.05226
637637 0 0
638638 7.08781e7 0.0108054
639639 − 1.08464e9i − 0.164450i
640640 −1.18633e9 −0.178886
641641 6.04766e9 0.906952 0.453476 0.891268i 0.350184π-0.350184\pi
0.453476 + 0.891268i 0.350184π0.350184\pi
642642 1.35468e9i 0.202053i
643643 9.02692e9i 1.33906i 0.742784 + 0.669532i 0.233505π0.233505\pi
−0.742784 + 0.669532i 0.766495π0.766495\pi
644644 7.89020e9i 1.16409i
645645 − 1.04495e9i − 0.153334i
646646 4.02391e8 0.0587266
647647 −1.99531e9 −0.289631 −0.144816 0.989459i 0.546259π-0.546259\pi
−0.144816 + 0.989459i 0.546259π0.546259\pi
648648 − 4.39827e9i − 0.634994i
649649 3.32589e8 0.0477585
650650 0 0
651651 2.93877e9 0.417477
652652 1.12891e9i 0.159511i
653653 5.54932e9 0.779909 0.389954 0.920834i 0.372491π-0.372491\pi
0.389954 + 0.920834i 0.372491π0.372491\pi
654654 −6.46770e8 −0.0904123
655655 7.19816e8i 0.100087i
656656 − 3.10926e9i − 0.430025i
657657 − 1.61533e9i − 0.222219i
658658 1.70254e9i 0.232974i
659659 −8.54955e7 −0.0116371 −0.00581854 0.999983i 0.501852π-0.501852\pi
−0.00581854 + 0.999983i 0.501852π0.501852\pi
660660 −2.88709e7 −0.00390891
661661 − 1.03382e10i − 1.39233i −0.717882 0.696164i 0.754888π-0.754888\pi
0.717882 0.696164i 0.245112π-0.245112\pi
662662 −3.69632e9 −0.495184
663663 0 0
664664 5.60251e9 0.742667
665665 3.27918e8i 0.0432404i
666666 2.01307e8 0.0264058
667667 −1.08377e10 −1.41416
668668 − 2.08794e9i − 0.271020i
669669 2.85848e9i 0.369099i
670670 − 1.03076e9i − 0.132402i
671671 1.61914e8i 0.0206898i
672672 −6.58825e9 −0.837486
673673 −8.90146e9 −1.12566 −0.562832 0.826571i 0.690288π-0.690288\pi
−0.562832 + 0.826571i 0.690288π0.690288\pi
674674 2.03362e9i 0.255835i
675675 8.14000e9 1.01874
676676 0 0
677677 −8.37714e9 −1.03761 −0.518807 0.854892i 0.673624π-0.673624\pi
−0.518807 + 0.854892i 0.673624π0.673624\pi
678678 − 3.43685e7i − 0.00423504i
679679 1.43178e10 1.75522
680680 5.90957e8 0.0720733
681681 − 5.74483e9i − 0.697047i
682682 − 4.44549e7i − 0.00536630i
683683 1.08780e10i 1.30640i 0.757185 + 0.653200i 0.226574π0.226574\pi
−0.757185 + 0.653200i 0.773426π0.773426\pi
684684 − 2.24374e8i − 0.0268087i
685685 −2.41107e9 −0.286611
686686 3.54912e9 0.419746
687687 8.20365e9i 0.965292i
688688 4.35845e9 0.510238
689689 0 0
690690 −8.37225e8 −0.0970220
691691 − 1.00168e10i − 1.15493i −0.816414 0.577467i 0.804041π-0.804041\pi
0.816414 0.577467i 0.195959π-0.195959\pi
692692 −4.92245e9 −0.564690
693693 2.81504e7 0.00321305
694694 − 5.08843e9i − 0.577864i
695695 − 1.13099e9i − 0.127794i
696696 − 5.86364e9i − 0.659227i
697697 3.92826e9i 0.439425i
698698 4.60977e9 0.513080
699699 3.84673e9 0.426011
700700 − 6.92847e9i − 0.763474i
701701 −1.48395e10 −1.62707 −0.813537 0.581513i 0.802461π-0.802461\pi
−0.813537 + 0.581513i 0.802461π0.802461\pi
702702 0 0
703703 −1.32190e9 −0.143501
704704 − 4.35957e7i − 0.00470912i
705705 9.52565e8 0.102384
706706 1.48297e7 0.00158605
707707 1.63382e10i 1.73875i
708708 − 1.25657e10i − 1.33067i
709709 − 1.35650e10i − 1.42941i −0.699424 0.714707i 0.746560π-0.746560\pi
0.699424 0.714707i 0.253440π-0.253440\pi
710710 − 9.05971e8i − 0.0949971i
711711 299879. 3.12898e−5 0
712712 2.01419e9 0.209132
713713 6.79744e9i 0.702315i
714714 1.90739e9 0.196109
715715 0 0
716716 4.22061e9 0.429715
717717 − 1.31431e10i − 1.33162i
718718 −4.59604e9 −0.463392
719719 −8.17292e9 −0.820023 −0.410011 0.912080i 0.634475π-0.634475\pi
−0.410011 + 0.912080i 0.634475π0.634475\pi
720720 − 1.16536e8i − 0.0116358i
721721 − 1.03723e10i − 1.03063i
722722 3.75841e9i 0.371641i
723723 − 5.30141e9i − 0.521684i
724724 7.26190e9 0.711157
725725 9.51671e9 0.927478
726726 − 3.85598e9i − 0.373987i
727727 −7.62775e8 −0.0736251 −0.0368126 0.999322i 0.511720π-0.511720\pi
−0.0368126 + 0.999322i 0.511720π0.511720\pi
728728 0 0
729729 1.14024e10 1.09006
730730 − 1.34924e9i − 0.128368i
731731 −5.50650e9 −0.521392
732732 6.11738e9 0.576470
733733 8.21437e9i 0.770389i 0.922835 + 0.385195i 0.125866π0.125866\pi
−0.922835 + 0.385195i 0.874134π0.874134\pi
734734 − 6.16617e9i − 0.575545i
735735 − 2.15669e8i − 0.0200347i
736736 − 1.52388e10i − 1.40889i
737737 5.81052e8 0.0534661
738738 −4.15414e8 −0.0380438
739739 8.22305e9i 0.749510i 0.927124 + 0.374755i 0.122273π0.122273\pi
−0.927124 + 0.374755i 0.877727π0.877727\pi
740740 −8.86604e8 −0.0804301
741741 0 0
742742 −5.56000e9 −0.499644
743743 1.37678e10i 1.23142i 0.787975 + 0.615708i 0.211130π0.211130\pi
−0.787975 + 0.615708i 0.788870π0.788870\pi
744744 −3.67769e9 −0.327394
745745 2.46269e9 0.218204
746746 4.35978e9i 0.384484i
747747 1.39582e9i 0.122520i
748748 1.52138e8i 0.0132917i
749749 5.81745e9i 0.505879i
750750 1.49369e9 0.129284
751751 1.23300e10 1.06224 0.531119 0.847297i 0.321772π-0.321772\pi
0.531119 + 0.847297i 0.321772π0.321772\pi
752752 3.97310e9i 0.340696i
753753 −1.49983e10 −1.28014
754754 0 0
755755 −1.45534e9 −0.123069
756756 − 9.83616e9i − 0.827941i
757757 1.84529e10 1.54607 0.773036 0.634362i 0.218737π-0.218737\pi
0.773036 + 0.634362i 0.218737π0.218737\pi
758758 −3.68199e9 −0.307072
759759 − 4.71954e8i − 0.0391790i
760760 − 4.10370e8i − 0.0339100i
761761 1.45618e10i 1.19776i 0.800839 + 0.598880i 0.204387π0.204387\pi
−0.800839 + 0.598880i 0.795613π0.795613\pi
762762 − 3.58048e9i − 0.293156i
763763 −2.77744e9 −0.226365
764764 1.84984e10 1.50074
765765 1.47232e8i 0.0118901i
766766 5.78678e9 0.465196
767767 0 0
768768 2.83233e9 0.225621
769769 4.69706e9i 0.372464i 0.982506 + 0.186232i 0.0596276π0.0596276\pi
−0.982506 + 0.186232i 0.940372π0.940372\pi
770770 2.35132e7 0.00185607
771771 −5.86508e9 −0.460876
772772 2.67363e9i 0.209142i
773773 − 1.42447e10i − 1.10924i −0.832104 0.554619i 0.812864π-0.812864\pi
0.832104 0.554619i 0.187136π-0.187136\pi
774774 − 5.82313e8i − 0.0451402i
775775 − 5.96890e9i − 0.460616i
776776 −1.79179e10 −1.37648
777777 −6.26599e9 −0.479199
778778 − 4.29795e9i − 0.327214i
779779 2.72784e9 0.206747
780780 0 0
781781 5.10707e8 0.0383613
782782 4.41184e9i 0.329911i
783783 1.35106e10 1.00579
784784 8.99546e8 0.0666680
785785 1.74722e9i 0.128915i
786786 − 2.90747e9i − 0.213568i
787787 − 2.03693e10i − 1.48958i −0.667300 0.744789i 0.732550π-0.732550\pi
0.667300 0.744789i 0.267450π-0.267450\pi
788788 − 1.48107e10i − 1.07829i
789789 1.82502e10 1.32282
790790 250481. 1.80751e−5 0
791791 − 1.47590e8i − 0.0106032i
792792 −3.52285e7 −0.00251974
793793 0 0
794794 1.11437e9 0.0790059
795795 3.11079e9i 0.219577i
796796 −1.05640e10 −0.742394
797797 −1.18633e10 −0.830045 −0.415022 0.909811i 0.636226π-0.636226\pi
−0.415022 + 0.909811i 0.636226π0.636226\pi
798798 − 1.32452e9i − 0.0922678i
799799 − 5.01964e9i − 0.348144i
800800 1.33813e10i 0.924026i
801801 5.01818e8i 0.0345011i
802802 9.14762e9 0.626178
803803 7.60581e8 0.0518371
804804 − 2.19531e10i − 1.48970i
805805 −3.59532e9 −0.242914
806806 0 0
807807 1.30913e10 0.876848
808808 − 2.04463e10i − 1.36356i
809809 −8.79991e9 −0.584330 −0.292165 0.956368i 0.594376π-0.594376\pi
−0.292165 + 0.956368i 0.594376π0.594376\pi
810810 9.15285e8 0.0605144
811811 6.04080e9i 0.397669i 0.980033 + 0.198834i 0.0637156π0.0637156\pi
−0.980033 + 0.198834i 0.936284π0.936284\pi
812812 − 1.14997e10i − 0.753775i
813813 − 2.33406e9i − 0.152333i
814814 9.47860e7i 0.00615969i
815815 −5.14407e8 −0.0332855
816816 4.45115e9 0.286785
817817 3.82380e9i 0.245311i
818818 6.55024e9 0.418428
819819 0 0
820820 1.82958e9 0.115879
821821 1.79851e10i 1.13426i 0.823629 + 0.567129i 0.191946π0.191946\pi
−0.823629 + 0.567129i 0.808054π0.808054\pi
822822 9.73877e9 0.611579
823823 −2.38986e10 −1.49442 −0.747210 0.664588i 0.768607π-0.768607\pi
−0.747210 + 0.664588i 0.768607π0.768607\pi
824824 1.29803e10i 0.808237i
825825 4.14427e8i 0.0256956i
826826 1.02339e10i 0.631844i
827827 2.40368e10i 1.47777i 0.673829 + 0.738887i 0.264648π0.264648\pi
−0.673829 + 0.738887i 0.735352π0.735352\pi
828828 2.46005e9 0.150605
829829 −2.77111e10 −1.68933 −0.844663 0.535298i 0.820199π-0.820199\pi
−0.844663 + 0.535298i 0.820199π0.820199\pi
830830 1.16589e9i 0.0707756i
831831 −8.51930e9 −0.514992
832832 0 0
833833 −1.13649e9 −0.0681254
834834 4.56827e9i 0.272691i
835835 9.51411e8 0.0565543
836836 1.05647e8 0.00625368
837837 − 8.47389e9i − 0.499510i
838838 3.16947e9i 0.186051i
839839 − 5.42082e9i − 0.316883i −0.987368 0.158441i 0.949353π-0.949353\pi
0.987368 0.158441i 0.0506469π-0.0506469\pi
840840 − 1.94521e9i − 0.113237i
841841 −1.45424e9 −0.0843045
842842 −2.73107e9 −0.157667
843843 3.22348e9i 0.185323i
844844 −1.00616e10 −0.576061
845845 0 0
846846 5.30828e8 0.0301410
847847 − 1.65589e10i − 0.936350i
848848 −1.29750e10 −0.730669
849849 −1.21593e10 −0.681917
850850 − 3.87409e9i − 0.216373i
851851 − 1.44934e10i − 0.806150i
852852 − 1.92953e10i − 1.06884i
853853 9.15495e8i 0.0505050i 0.999681 + 0.0252525i 0.00803898π0.00803898\pi
−0.999681 + 0.0252525i 0.991961π0.991961\pi
854854 −4.98216e9 −0.273725
855855 1.02240e8 0.00559423
856856 − 7.28019e9i − 0.396720i
857857 −5.57192e9 −0.302393 −0.151197 0.988504i 0.548313π-0.548313\pi
−0.151197 + 0.988504i 0.548313π0.548313\pi
858858 0 0
859859 −3.79832e9 −0.204463 −0.102232 0.994761i 0.532598π-0.532598\pi
−0.102232 + 0.994761i 0.532598π0.532598\pi
860860 2.56464e9i 0.137494i
861861 1.29304e10 0.690400
862862 −2.21149e8 −0.0117600
863863 8.87979e9i 0.470289i 0.971960 + 0.235145i 0.0755563π0.0755563\pi
−0.971960 + 0.235145i 0.924444π0.924444\pi
864864 1.89971e10i 1.00205i
865865 − 2.24301e9i − 0.117835i
866866 − 7.38566e9i − 0.386435i
867867 1.23652e10 0.644369
868868 −7.21266e9 −0.374349
869869 141199.i 0 7.29899e-6i
870870 1.22023e9 0.0628238
871871 0 0
872872 3.47580e9 0.177520
873873 − 4.46408e9i − 0.227082i
874874 3.06365e9 0.155221
875875 6.41439e9 0.323689
876876 − 2.87360e10i − 1.44431i
877877 2.78832e10i 1.39587i 0.716162 + 0.697934i 0.245897π0.245897\pi
−0.716162 + 0.697934i 0.754103π0.754103\pi
878878 − 6.66471e9i − 0.332315i
879879 − 3.91200e9i − 0.194285i
880880 5.48711e7 0.00271428
881881 3.07787e10 1.51647 0.758237 0.651979i 0.226061π-0.226061\pi
0.758237 + 0.651979i 0.226061π0.226061\pi
882882 − 1.20184e8i − 0.00589804i
883883 1.45010e9 0.0708820 0.0354410 0.999372i 0.488716π-0.488716\pi
0.0354410 + 0.999372i 0.488716π0.488716\pi
884884 0 0
885885 5.72581e9 0.277674
886886 9.90763e8i 0.0478577i
887887 −9.25333e9 −0.445211 −0.222605 0.974909i 0.571456π-0.571456\pi
−0.222605 + 0.974909i 0.571456π0.571456\pi
888888 7.84150e9 0.375798
889889 − 1.53758e10i − 0.733973i
890890 4.19154e8i 0.0199301i
891891 5.15957e8i 0.0244367i
892892 − 7.01559e9i − 0.330969i
893893 −3.48572e9 −0.163799
894894 −9.94729e9 −0.465611
895895 1.92320e9i 0.0896694i
896896 2.05777e10 0.955693
897897 0 0
898898 4.51748e9 0.208175
899899 − 9.90706e9i − 0.454764i
900900 −2.16020e9 −0.0987744
901901 1.63927e10 0.746642
902902 − 1.95599e8i − 0.00887448i
903903 1.81254e10i 0.819181i
904904 1.84700e8i 0.00831527i
905905 3.30902e9i 0.148399i
906906 5.87840e9 0.262609
907907 −2.13718e10 −0.951078 −0.475539 0.879695i 0.657747π-0.657747\pi
−0.475539 + 0.879695i 0.657747π0.657747\pi
908908 1.40996e10i 0.625038i
909909 5.09402e9 0.224951
910910 0 0
911911 −2.45761e10 −1.07696 −0.538479 0.842639i 0.681001π-0.681001\pi
−0.538479 + 0.842639i 0.681001π0.681001\pi
912912 − 3.09095e9i − 0.134930i
913913 −6.57225e8 −0.0285803
914914 1.49426e10 0.647312
915915 2.78750e9i 0.120293i
916916 − 2.01343e10i − 0.865572i
917917 − 1.24857e10i − 0.534710i
918918 − 5.49994e9i − 0.234643i
919919 −1.04226e9 −0.0442966 −0.0221483 0.999755i 0.507051π-0.507051\pi
−0.0221483 + 0.999755i 0.507051π0.507051\pi
920920 4.49932e9 0.190498
921921 − 8.90552e9i − 0.375622i
922922 1.78083e10 0.748280
923923 0 0
924924 5.00783e8 0.0208832
925925 1.27268e10i 0.528716i
926926 −2.54219e9 −0.105213
927927 −3.23393e9 −0.133337
928928 2.22100e10i 0.912287i
929929 2.61948e10i 1.07191i 0.844245 + 0.535957i 0.180049π0.180049\pi
−0.844245 + 0.535957i 0.819951π0.819951\pi
930930 − 7.65331e8i − 0.0312003i
931931 7.89198e8i 0.0320525i
932932 −9.44107e9 −0.382002
933933 −3.42841e10 −1.38200
934934 8.06350e7i 0.00323824i
935935 −6.93245e7 −0.00277362
936936 0 0
937937 −1.84464e10 −0.732525 −0.366263 0.930512i 0.619363π-0.619363\pi
−0.366263 + 0.930512i 0.619363π0.619363\pi
938938 1.78792e10i 0.707355i
939939 −1.68573e10 −0.664443
940940 −2.33789e9 −0.0918072
941941 − 2.74685e10i − 1.07466i −0.843373 0.537329i 0.819433π-0.819433\pi
0.843373 0.537329i 0.180567π-0.180567\pi
942942 − 7.05735e9i − 0.275083i
943943 2.99083e10i 1.16145i
944944 2.38821e10i 0.923995i
945945 4.48203e9 0.172768
946946 2.74184e8 0.0105299
947947 1.25081e10i 0.478593i 0.970947 + 0.239297i 0.0769168π0.0769168\pi
−0.970947 + 0.239297i 0.923083π0.923083\pi
948948 5.33472e6 0.000203368 0
949949 0 0
950950 −2.69023e9 −0.101802
951951 − 1.84056e10i − 0.693933i
952952 −1.02505e10 −0.385049
953953 4.44972e10 1.66536 0.832679 0.553755i 0.186806π-0.186806\pi
0.832679 + 0.553755i 0.186806π0.186806\pi
954954 1.73353e9i 0.0646415i
955955 8.42913e9i 0.313163i
956956 3.22573e10i 1.19406i
957957 6.87858e8i 0.0253692i
958958 2.89685e9 0.106450
959959 4.18215e10 1.53121
960960 − 7.50539e8i − 0.0273794i
961961 2.12989e10 0.774150
962962 0 0
963963 1.81380e9 0.0654481
964964 1.30113e10i 0.467791i
965965 −1.21829e9 −0.0436421
966966 1.45222e10 0.518337
967967 1.03373e10i 0.367632i 0.982961 + 0.183816i 0.0588450π0.0588450\pi
−0.982961 + 0.183816i 0.941155π0.941155\pi
968968 2.07224e10i 0.734305i
969969 3.90513e9i 0.137880i
970970 − 3.72872e9i − 0.131177i
971971 −3.56148e10 −1.24843 −0.624214 0.781254i 0.714581π-0.714581\pi
−0.624214 + 0.781254i 0.714581π0.714581\pi
972972 −5.80194e9 −0.202648
973973 1.96176e10i 0.682734i
974974 −9.68047e9 −0.335691
975975 0 0
976976 −1.16265e10 −0.400290
977977 4.78316e10i 1.64091i 0.571713 + 0.820454i 0.306279π0.306279\pi
−0.571713 + 0.820454i 0.693721π0.693721\pi
978978 2.07779e9 0.0710257
979979 −2.36282e8 −0.00804807
980980 5.29320e8i 0.0179650i
981981 8.65966e8i 0.0292860i
982982 6.79011e9i 0.228816i
983983 1.01346e10i 0.340306i 0.985418 + 0.170153i 0.0544262π0.0544262\pi
−0.985418 + 0.170153i 0.945574π0.945574\pi
984984 −1.61816e10 −0.541425
985985 6.74878e9 0.225008
986986 − 6.43013e9i − 0.213624i
987987 −1.65228e10 −0.546983
988988 0 0
989989 −4.19244e10 −1.37810
990990 − 7.33108e6i 0 0.000240129i
991991 −1.22781e10 −0.400749 −0.200375 0.979719i 0.564216π-0.564216\pi
−0.200375 + 0.979719i 0.564216π0.564216\pi
992992 1.39302e10 0.453071
993993 − 3.58720e10i − 1.16261i
994994 1.57146e10i 0.507518i
995995 − 4.81371e9i − 0.154917i
996996 2.48310e10i 0.796318i
997997 −3.28608e10 −1.05013 −0.525067 0.851061i 0.675960π-0.675960\pi
−0.525067 + 0.851061i 0.675960π0.675960\pi
998998 1.49043e10 0.474630
999999 1.80679e10i 0.573360i
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 169.8.b.d.168.9 14
13.5 odd 4 169.8.a.g.1.9 14
13.8 odd 4 169.8.a.g.1.6 14
13.9 even 3 13.8.e.a.10.5 yes 14
13.10 even 6 13.8.e.a.4.5 14
13.12 even 2 inner 169.8.b.d.168.6 14
39.23 odd 6 117.8.q.b.82.3 14
39.35 odd 6 117.8.q.b.10.3 14
52.23 odd 6 208.8.w.a.17.3 14
52.35 odd 6 208.8.w.a.49.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.8.e.a.4.5 14 13.10 even 6
13.8.e.a.10.5 yes 14 13.9 even 3
117.8.q.b.10.3 14 39.35 odd 6
117.8.q.b.82.3 14 39.23 odd 6
169.8.a.g.1.6 14 13.8 odd 4
169.8.a.g.1.9 14 13.5 odd 4
169.8.b.d.168.6 14 13.12 even 2 inner
169.8.b.d.168.9 14 1.1 even 1 trivial
208.8.w.a.17.3 14 52.23 odd 6
208.8.w.a.49.3 14 52.35 odd 6