Properties

Label 169.8.b.d.168.10
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 168.10
Root 8.41902i8.41902i of defining polynomial
Character χ\chi == 169.168
Dual form 169.8.b.d.168.5

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+8.41902iq21.14749q3+57.1202q4+399.024iq59.66073iq6944.231iq7+1558.53iq82185.68q93359.39q10+5673.16iq1165.5447q12+7949.50q14457.876iq155809.90q1612214.2q1718401.3iq1848622.1iq19+22792.3iq20+1083.49iq2147762.4q223416.52q231788.40iq2481095.5q25+5017.60q2753934.6iq28126824.q29+3854.87q30+170057.iq31+150578.iq326509.89iq33102832.iq34+376771.q35124847.q36+149165.iq37+409350.q38621892.q40645325.iq419121.96q42129943.q43+324052.iq44872141.iq4528763.7iq46+338496.iq47+6666.80q4868029.2q49682745.iq50+14015.7q51226916.q53+42243.3iq542.26373e6q55+1.47161e6q56+55793.3iq571.06773e6iq58+1.60203e6iq5926154.0iq602.00107e6q611.43171e6q62+2.06379e6iq632.01139e6q64+54806.9q661.38038e6iq67697677.q68+3920.41q69+3.17204e6iq705.13680e6iq713.40645e6iq725.76900e6iq731.25582e6q74+93056.2q752.77730e6iq76+5.35678e6q774.45623e6q792.31829e6iq80+4.77433e6q81+5.43300e6q82811479.iq83+61889.4iq844.87377e6iq851.09399e6iq86+145529.q878.84179e6q885.15613e6iq89+7.34257e6q90195152.q92195138.iq932.84980e6q94+1.94014e7q95172787.iq963.83876e6iq97572739.iq981.23997e7iq99+O(q100)q+8.41902i q^{2} -1.14749 q^{3} +57.1202 q^{4} +399.024i q^{5} -9.66073i q^{6} -944.231i q^{7} +1558.53i q^{8} -2185.68 q^{9} -3359.39 q^{10} +5673.16i q^{11} -65.5447 q^{12} +7949.50 q^{14} -457.876i q^{15} -5809.90 q^{16} -12214.2 q^{17} -18401.3i q^{18} -48622.1i q^{19} +22792.3i q^{20} +1083.49i q^{21} -47762.4 q^{22} -3416.52 q^{23} -1788.40i q^{24} -81095.5 q^{25} +5017.60 q^{27} -53934.6i q^{28} -126824. q^{29} +3854.87 q^{30} +170057. i q^{31} +150578. i q^{32} -6509.89i q^{33} -102832. i q^{34} +376771. q^{35} -124847. q^{36} +149165. i q^{37} +409350. q^{38} -621892. q^{40} -645325. i q^{41} -9121.96 q^{42} -129943. q^{43} +324052. i q^{44} -872141. i q^{45} -28763.7i q^{46} +338496. i q^{47} +6666.80 q^{48} -68029.2 q^{49} -682745. i q^{50} +14015.7 q^{51} -226916. q^{53} +42243.3i q^{54} -2.26373e6 q^{55} +1.47161e6 q^{56} +55793.3i q^{57} -1.06773e6i q^{58} +1.60203e6i q^{59} -26154.0i q^{60} -2.00107e6 q^{61} -1.43171e6 q^{62} +2.06379e6i q^{63} -2.01139e6 q^{64} +54806.9 q^{66} -1.38038e6i q^{67} -697677. q^{68} +3920.41 q^{69} +3.17204e6i q^{70} -5.13680e6i q^{71} -3.40645e6i q^{72} -5.76900e6i q^{73} -1.25582e6 q^{74} +93056.2 q^{75} -2.77730e6i q^{76} +5.35678e6 q^{77} -4.45623e6 q^{79} -2.31829e6i q^{80} +4.77433e6 q^{81} +5.43300e6 q^{82} -811479. i q^{83} +61889.4i q^{84} -4.87377e6i q^{85} -1.09399e6i q^{86} +145529. q^{87} -8.84179e6 q^{88} -5.15613e6i q^{89} +7.34257e6 q^{90} -195152. q^{92} -195138. i q^{93} -2.84980e6 q^{94} +1.94014e7 q^{95} -172787. i q^{96} -3.83876e6i q^{97} -572739. i q^{98} -1.23997e7i 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 8.41902i 0.744143i 0.928204 + 0.372071i 0.121352π0.121352\pi
−0.928204 + 0.372071i 0.878648π0.878648\pi
33 −1.14749 −0.0245371 −0.0122686 0.999925i 0.503905π-0.503905\pi
−0.0122686 + 0.999925i 0.503905π0.503905\pi
44 57.1202 0.446251
55 399.024i 1.42759i 0.700353 + 0.713797i 0.253026π0.253026\pi
−0.700353 + 0.713797i 0.746974π0.746974\pi
66 − 9.66073i − 0.0182591i
77 − 944.231i − 1.04048i −0.854019 0.520242i 0.825842π-0.825842\pi
0.854019 0.520242i 0.174158π-0.174158\pi
88 1558.53i 1.07622i
99 −2185.68 −0.999398
1010 −3359.39 −1.06233
1111 5673.16i 1.28514i 0.766227 + 0.642571i 0.222132π0.222132\pi
−0.766227 + 0.642571i 0.777868π0.777868\pi
1212 −65.5447 −0.0109497
1313 0 0
1414 7949.50 0.774268
1515 − 457.876i − 0.0350291i
1616 −5809.90 −0.354608
1717 −12214.2 −0.602968 −0.301484 0.953471i 0.597482π-0.597482\pi
−0.301484 + 0.953471i 0.597482π0.597482\pi
1818 − 18401.3i − 0.743695i
1919 − 48622.1i − 1.62628i −0.582067 0.813141i 0.697756π-0.697756\pi
0.582067 0.813141i 0.302244π-0.302244\pi
2020 22792.3i 0.637065i
2121 1083.49i 0.0255305i
2222 −47762.4 −0.956329
2323 −3416.52 −0.0585512 −0.0292756 0.999571i 0.509320π-0.509320\pi
−0.0292756 + 0.999571i 0.509320π0.509320\pi
2424 − 1788.40i − 0.0264073i
2525 −81095.5 −1.03802
2626 0 0
2727 5017.60 0.0490595
2828 − 53934.6i − 0.464317i
2929 −126824. −0.965626 −0.482813 0.875724i 0.660385π-0.660385\pi
−0.482813 + 0.875724i 0.660385π0.660385\pi
3030 3854.87 0.0260666
3131 170057.i 1.02525i 0.858614 + 0.512623i 0.171326π0.171326\pi
−0.858614 + 0.512623i 0.828674π0.828674\pi
3232 150578.i 0.812338i
3333 − 6509.89i − 0.0315337i
3434 − 102832.i − 0.448694i
3535 376771. 1.48539
3636 −124847. −0.445983
3737 149165.i 0.484129i 0.970260 + 0.242065i 0.0778246π0.0778246\pi
−0.970260 + 0.242065i 0.922175π0.922175\pi
3838 409350. 1.21019
3939 0 0
4040 −621892. −1.53640
4141 − 645325.i − 1.46230i −0.682219 0.731148i 0.738985π-0.738985\pi
0.682219 0.731148i 0.261015π-0.261015\pi
4242 −9121.96 −0.0189983
4343 −129943. −0.249238 −0.124619 0.992205i 0.539771π-0.539771\pi
−0.124619 + 0.992205i 0.539771π0.539771\pi
4444 324052.i 0.573496i
4545 − 872141.i − 1.42673i
4646 − 28763.7i − 0.0435705i
4747 338496.i 0.475566i 0.971318 + 0.237783i 0.0764207π0.0764207\pi
−0.971318 + 0.237783i 0.923579π0.923579\pi
4848 6666.80 0.00870108
4949 −68029.2 −0.0826056
5050 − 682745.i − 0.772437i
5151 14015.7 0.0147951
5252 0 0
5353 −226916. −0.209362 −0.104681 0.994506i 0.533382π-0.533382\pi
−0.104681 + 0.994506i 0.533382π0.533382\pi
5454 42243.3i 0.0365073i
5555 −2.26373e6 −1.83466
5656 1.47161e6 1.11979
5757 55793.3i 0.0399043i
5858 − 1.06773e6i − 0.718564i
5959 1.60203e6i 1.01552i 0.861498 + 0.507762i 0.169527π0.169527\pi
−0.861498 + 0.507762i 0.830473π0.830473\pi
6060 − 26154.0i − 0.0156318i
6161 −2.00107e6 −1.12877 −0.564387 0.825510i 0.690887π-0.690887\pi
−0.564387 + 0.825510i 0.690887π0.690887\pi
6262 −1.43171e6 −0.762930
6363 2.06379e6i 1.03986i
6464 −2.01139e6 −0.959104
6565 0 0
6666 54806.9 0.0234656
6767 − 1.38038e6i − 0.560706i −0.959897 0.280353i 0.909548π-0.909548\pi
0.959897 0.280353i 0.0904516π-0.0904516\pi
6868 −697677. −0.269075
6969 3920.41 0.00143668
7070 3.17204e6i 1.10534i
7171 − 5.13680e6i − 1.70329i −0.524120 0.851644i 0.675606π-0.675606\pi
0.524120 0.851644i 0.324394π-0.324394\pi
7272 − 3.40645e6i − 1.07557i
7373 − 5.76900e6i − 1.73568i −0.496839 0.867842i 0.665506π-0.665506\pi
0.496839 0.867842i 0.334494π-0.334494\pi
7474 −1.25582e6 −0.360261
7575 93056.2 0.0254701
7676 − 2.77730e6i − 0.725731i
7777 5.35678e6 1.33717
7878 0 0
7979 −4.45623e6 −1.01689 −0.508444 0.861095i 0.669779π-0.669779\pi
−0.508444 + 0.861095i 0.669779π0.669779\pi
8080 − 2.31829e6i − 0.506237i
8181 4.77433e6 0.998194
8282 5.43300e6 1.08816
8383 − 811479.i − 0.155777i −0.996962 0.0778886i 0.975182π-0.975182\pi
0.996962 0.0778886i 0.0248179π-0.0248179\pi
8484 61889.4i 0.0113930i
8585 − 4.87377e6i − 0.860793i
8686 − 1.09399e6i − 0.185468i
8787 145529. 0.0236937
8888 −8.84179e6 −1.38309
8989 − 5.15613e6i − 0.775281i −0.921811 0.387640i 0.873290π-0.873290\pi
0.921811 0.387640i 0.126710π-0.126710\pi
9090 7.34257e6 1.06169
9191 0 0
9292 −195152. −0.0261286
9393 − 195138.i − 0.0251566i
9494 −2.84980e6 −0.353889
9595 1.94014e7 2.32167
9696 − 172787.i − 0.0199325i
9797 − 3.83876e6i − 0.427061i −0.976936 0.213530i 0.931504π-0.931504\pi
0.976936 0.213530i 0.0684963π-0.0684963\pi
9898 − 572739.i − 0.0614703i
9999 − 1.23997e7i − 1.28437i
100100 −4.63219e6 −0.463219
101101 9.61610e6 0.928697 0.464348 0.885653i 0.346289π-0.346289\pi
0.464348 + 0.885653i 0.346289π0.346289\pi
102102 117998.i 0.0110097i
103103 −2.86658e6 −0.258484 −0.129242 0.991613i 0.541254π-0.541254\pi
−0.129242 + 0.991613i 0.541254π0.541254\pi
104104 0 0
105105 −432341. −0.0364472
106106 − 1.91041e6i − 0.155796i
107107 −198466. −0.0156618 −0.00783091 0.999969i 0.502493π-0.502493\pi
−0.00783091 + 0.999969i 0.502493π0.502493\pi
108108 286606. 0.0218929
109109 1.82103e7i 1.34687i 0.739249 + 0.673433i 0.235181π0.235181\pi
−0.739249 + 0.673433i 0.764819π0.764819\pi
110110 − 1.90584e7i − 1.36525i
111111 − 171165.i − 0.0118791i
112112 5.48589e6i 0.368964i
113113 −8.14016e6 −0.530711 −0.265356 0.964151i 0.585489π-0.585489\pi
−0.265356 + 0.964151i 0.585489π0.585489\pi
114114 −469725. −0.0296945
115115 − 1.36327e6i − 0.0835873i
116116 −7.24421e6 −0.430912
117117 0 0
118118 −1.34876e7 −0.755694
119119 1.15330e7i 0.627378i
120120 713613. 0.0376989
121121 −1.26976e7 −0.651588
122122 − 1.68470e7i − 0.839969i
123123 740504.i 0.0358806i
124124 9.71368e6i 0.457517i
125125 − 1.18532e6i − 0.0542813i
126126 −1.73751e7 −0.773802
127127 −3.41396e7 −1.47892 −0.739460 0.673200i 0.764919π-0.764919\pi
−0.739460 + 0.673200i 0.764919π0.764919\pi
128128 2.34010e6i 0.0986277i
129129 149108. 0.00611558
130130 0 0
131131 6.55064e6 0.254586 0.127293 0.991865i 0.459371π-0.459371\pi
0.127293 + 0.991865i 0.459371π0.459371\pi
132132 − 371846.i − 0.0140720i
133133 −4.59105e7 −1.69212
134134 1.16214e7 0.417246
135135 2.00215e6i 0.0700370i
136136 − 1.90362e7i − 0.648924i
137137 − 1.53188e7i − 0.508984i −0.967075 0.254492i 0.918092π-0.918092\pi
0.967075 0.254492i 0.0819082π-0.0819082\pi
138138 33006.0i 0.00106910i
139139 1.12089e7 0.354007 0.177004 0.984210i 0.443360π-0.443360\pi
0.177004 + 0.984210i 0.443360π0.443360\pi
140140 2.15212e7 0.662856
141141 − 388420.i − 0.0116690i
142142 4.32468e7 1.26749
143143 0 0
144144 1.26986e7 0.354395
145145 − 5.06059e7i − 1.37852i
146146 4.85693e7 1.29160
147147 78062.8 0.00202690
148148 8.52034e6i 0.216043i
149149 3.85311e7i 0.954243i 0.878837 + 0.477121i 0.158320π0.158320\pi
−0.878837 + 0.477121i 0.841680π0.841680\pi
150150 783442.i 0.0189534i
151151 4.08938e7i 0.966581i 0.875460 + 0.483290i 0.160558π0.160558\pi
−0.875460 + 0.483290i 0.839442π0.839442\pi
152152 7.57790e7 1.75023
153153 2.66964e7 0.602605
154154 4.50988e7i 0.995044i
155155 −6.78568e7 −1.46363
156156 0 0
157157 −2.52905e7 −0.521565 −0.260783 0.965398i 0.583981π-0.583981\pi
−0.260783 + 0.965398i 0.583981π0.583981\pi
158158 − 3.75171e7i − 0.756710i
159159 260383. 0.00513716
160160 −6.00844e7 −1.15969
161161 3.22598e6i 0.0609216i
162162 4.01952e7i 0.742799i
163163 6.83917e7i 1.23693i 0.785810 + 0.618467i 0.212246π0.212246\pi
−0.785810 + 0.618467i 0.787754π0.787754\pi
164164 − 3.68611e7i − 0.652551i
165165 2.59761e6 0.0450173
166166 6.83186e6 0.115921
167167 − 2.92588e7i − 0.486126i −0.970010 0.243063i 0.921848π-0.921848\pi
0.970010 0.243063i 0.0781522π-0.0781522\pi
168168 −1.68866e6 −0.0274764
169169 0 0
170170 4.10323e7 0.640553
171171 1.06272e8i 1.62530i
172172 −7.42237e6 −0.111223
173173 −9.01124e7 −1.32319 −0.661597 0.749860i 0.730121π-0.730121\pi
−0.661597 + 0.749860i 0.730121π0.730121\pi
174174 1.22521e6i 0.0176315i
175175 7.65729e7i 1.08005i
176176 − 3.29605e7i − 0.455722i
177177 − 1.83832e6i − 0.0249180i
178178 4.34096e7 0.576920
179179 −1.69729e7 −0.221193 −0.110596 0.993865i 0.535276π-0.535276\pi
−0.110596 + 0.993865i 0.535276π0.535276\pi
180180 − 4.98168e7i − 0.636682i
181181 −4.68911e7 −0.587781 −0.293890 0.955839i 0.594950π-0.594950\pi
−0.293890 + 0.955839i 0.594950π0.594950\pi
182182 0 0
183183 2.29620e6 0.0276969
184184 − 5.32474e6i − 0.0630139i
185185 −5.95206e7 −0.691140
186186 1.64287e6 0.0187201
187187 − 6.92932e7i − 0.774898i
188188 1.93349e7i 0.212222i
189189 − 4.73778e6i − 0.0510456i
190190 1.63341e8i 1.72765i
191191 −1.32211e8 −1.37294 −0.686470 0.727158i 0.740841π-0.740841\pi
−0.686470 + 0.727158i 0.740841π0.740841\pi
192192 2.30804e6 0.0235337
193193 1.37264e8i 1.37438i 0.726480 + 0.687188i 0.241155π0.241155\pi
−0.726480 + 0.687188i 0.758845π0.758845\pi
194194 3.23186e7 0.317794
195195 0 0
196196 −3.88584e6 −0.0368628
197197 1.66568e8i 1.55225i 0.630581 + 0.776123i 0.282817π0.282817\pi
−0.630581 + 0.776123i 0.717183π0.717183\pi
198198 1.04394e8 0.955753
199199 −5.79904e7 −0.521639 −0.260819 0.965388i 0.583993π-0.583993\pi
−0.260819 + 0.965388i 0.583993π0.583993\pi
200200 − 1.26390e8i − 1.11714i
201201 1.58396e6i 0.0137581i
202202 8.09581e7i 0.691083i
203203 1.19751e8i 1.00472i
204204 800577. 0.00660233
205205 2.57501e8 2.08756
206206 − 2.41338e7i − 0.192349i
207207 7.46742e6 0.0585160
208208 0 0
209209 2.75841e8 2.09000
210210 − 3.63988e6i − 0.0271219i
211211 1.21082e8 0.887345 0.443673 0.896189i 0.353675π-0.353675\pi
0.443673 + 0.896189i 0.353675π0.353675\pi
212212 −1.29615e7 −0.0934283
213213 5.89442e6i 0.0417938i
214214 − 1.67089e6i − 0.0116546i
215215 − 5.18505e7i − 0.355810i
216216 7.82009e6i 0.0527987i
217217 1.60573e8 1.06675
218218 −1.53313e8 −1.00226
219219 6.61987e6i 0.0425887i
220220 −1.29305e8 −0.818719
221221 0 0
222222 1.44104e6 0.00883979
223223 − 4.62753e7i − 0.279436i −0.990191 0.139718i 0.955380π-0.955380\pi
0.990191 0.139718i 0.0446196π-0.0446196\pi
224224 1.42181e8 0.845224
225225 1.77249e8 1.03740
226226 − 6.85321e7i − 0.394925i
227227 3.36801e8i 1.91110i 0.294834 + 0.955548i 0.404736π0.404736\pi
−0.294834 + 0.955548i 0.595264π0.595264\pi
228228 3.18692e6i 0.0178074i
229229 − 2.88661e8i − 1.58841i −0.607648 0.794206i 0.707887π-0.707887\pi
0.607648 0.794206i 0.292113π-0.292113\pi
230230 1.14774e7 0.0622009
231231 −6.14684e6 −0.0328103
232232 − 1.97659e8i − 1.03922i
233233 9.81949e7 0.508561 0.254281 0.967130i 0.418161π-0.418161\pi
0.254281 + 0.967130i 0.418161π0.418161\pi
234234 0 0
235235 −1.35068e8 −0.678914
236236 9.15085e7i 0.453179i
237237 5.11348e6 0.0249515
238238 −9.70968e7 −0.466859
239239 1.13856e8i 0.539465i 0.962935 + 0.269733i 0.0869353π0.0869353\pi
−0.962935 + 0.269733i 0.913065π0.913065\pi
240240 2.66022e6i 0.0124216i
241241 − 7.42915e7i − 0.341884i −0.985281 0.170942i 0.945319π-0.945319\pi
0.985281 0.170942i 0.0546811π-0.0546811\pi
242242 − 1.06901e8i − 0.484874i
243243 −1.64520e7 −0.0735523
244244 −1.14301e8 −0.503717
245245 − 2.71453e7i − 0.117927i
246246 −6.23431e6 −0.0267003
247247 0 0
248248 −2.65039e8 −1.10339
249249 931163.i 0.00382233i
250250 9.97923e6 0.0403931
251251 −1.42394e8 −0.568375 −0.284188 0.958769i 0.591724π-0.591724\pi
−0.284188 + 0.958769i 0.591724π0.591724\pi
252252 1.17884e8i 0.464038i
253253 − 1.93824e7i − 0.0752466i
254254 − 2.87421e8i − 1.10053i
255255 5.59259e6i 0.0211214i
256256 −2.77159e8 −1.03250
257257 2.92823e8 1.07607 0.538033 0.842924i 0.319168π-0.319168\pi
0.538033 + 0.842924i 0.319168π0.319168\pi
258258 1.25534e6i 0.00455086i
259259 1.40846e8 0.503728
260260 0 0
261261 2.77197e8 0.965044
262262 5.51500e7i 0.189448i
263263 −3.05065e8 −1.03406 −0.517032 0.855966i 0.672963π-0.672963\pi
−0.517032 + 0.855966i 0.672963π0.672963\pi
264264 1.01459e7 0.0339371
265265 − 9.05449e7i − 0.298884i
266266 − 3.86521e8i − 1.25918i
267267 5.91660e6i 0.0190232i
268268 − 7.88473e7i − 0.250216i
269269 2.31855e8 0.726246 0.363123 0.931741i 0.381710π-0.381710\pi
0.363123 + 0.931741i 0.381710π0.381710\pi
270270 −1.68561e7 −0.0521176
271271 4.92648e8i 1.50364i 0.659368 + 0.751821i 0.270824π0.270824\pi
−0.659368 + 0.751821i 0.729176π0.729176\pi
272272 7.09634e7 0.213817
273273 0 0
274274 1.28970e8 0.378757
275275 − 4.60068e8i − 1.33401i
276276 223935. 0.000641120 0
277277 4.33329e8 1.22501 0.612503 0.790468i 0.290163π-0.290163\pi
0.612503 + 0.790468i 0.290163π0.290163\pi
278278 9.43681e7i 0.263432i
279279 − 3.71690e8i − 1.02463i
280280 5.87209e8i 1.59860i
281281 2.88578e8i 0.775874i 0.921686 + 0.387937i 0.126812π0.126812\pi
−0.921686 + 0.387937i 0.873188π0.873188\pi
282282 3.27011e6 0.00868342
283283 −5.98945e8 −1.57085 −0.785424 0.618958i 0.787555π-0.787555\pi
−0.785424 + 0.618958i 0.787555π0.787555\pi
284284 − 2.93415e8i − 0.760095i
285285 −2.22629e7 −0.0569671
286286 0 0
287287 −6.09336e8 −1.52149
288288 − 3.29116e8i − 0.811849i
289289 −2.61152e8 −0.636430
290290 4.26052e8 1.02582
291291 4.40493e6i 0.0104789i
292292 − 3.29527e8i − 0.774552i
293293 − 3.12931e7i − 0.0726794i −0.999339 0.0363397i 0.988430π-0.988430\pi
0.999339 0.0363397i 0.0115698π-0.0115698\pi
294294 657212.i 0.00150831i
295295 −6.39251e8 −1.44975
296296 −2.32478e8 −0.521029
297297 2.84657e7i 0.0630484i
298298 −3.24394e8 −0.710093
299299 0 0
300300 5.31539e6 0.0113661
301301 1.22696e8i 0.259328i
302302 −3.44286e8 −0.719274
303303 −1.10344e7 −0.0227876
304304 2.82490e8i 0.576693i
305305 − 7.98474e8i − 1.61143i
306306 2.24757e8i 0.448424i
307307 2.40062e8i 0.473520i 0.971568 + 0.236760i 0.0760856π0.0760856\pi
−0.971568 + 0.236760i 0.923914π0.923914\pi
308308 3.05980e8 0.596713
309309 3.28937e6 0.00634247
310310 − 5.71288e8i − 1.08915i
311311 1.13878e8 0.214674 0.107337 0.994223i 0.465768π-0.465768\pi
0.107337 + 0.994223i 0.465768π0.465768\pi
312312 0 0
313313 −2.86637e8 −0.528356 −0.264178 0.964474i 0.585101π-0.585101\pi
−0.264178 + 0.964474i 0.585101π0.585101\pi
314314 − 2.12921e8i − 0.388119i
315315 −8.23503e8 −1.48449
316316 −2.54541e8 −0.453788
317317 9.47644e8i 1.67085i 0.549603 + 0.835426i 0.314779π0.314779\pi
−0.549603 + 0.835426i 0.685221π0.685221\pi
318318 2.19217e6i 0.00382278i
319319 − 7.19494e8i − 1.24097i
320320 − 8.02593e8i − 1.36921i
321321 227737. 0.000384296 0
322322 −2.71596e7 −0.0453344
323323 5.93880e8i 0.980596i
324324 2.72711e8 0.445445
325325 0 0
326326 −5.75791e8 −0.920456
327327 − 2.08961e7i − 0.0330482i
328328 1.00576e9 1.57375
329329 3.19618e8 0.494818
330330 2.18693e7i 0.0334993i
331331 6.84847e8i 1.03800i 0.854776 + 0.518998i 0.173695π0.173695\pi
−0.854776 + 0.518998i 0.826305π0.826305\pi
332332 − 4.63518e7i − 0.0695158i
333333 − 3.26028e8i − 0.483838i
334334 2.46330e8 0.361747
335335 5.50804e8 0.800460
336336 − 6.29500e6i − 0.00905333i
337337 1.12788e9 1.60531 0.802654 0.596445i 0.203421π-0.203421\pi
0.802654 + 0.596445i 0.203421π0.203421\pi
338338 0 0
339339 9.34074e6 0.0130221
340340 − 2.78390e8i − 0.384130i
341341 −9.64760e8 −1.31759
342342 −8.94710e8 −1.20946
343343 − 7.13380e8i − 0.954534i
344344 − 2.02520e8i − 0.268234i
345345 1.56434e6i 0.00205099i
346346 − 7.58658e8i − 0.984645i
347347 −3.84792e8 −0.494393 −0.247197 0.968965i 0.579509π-0.579509\pi
−0.247197 + 0.968965i 0.579509π0.579509\pi
348348 8.31265e6 0.0105733
349349 − 2.00731e8i − 0.252770i −0.991981 0.126385i 0.959663π-0.959663\pi
0.991981 0.126385i 0.0403375π-0.0403375\pi
350350 −6.44669e8 −0.803708
351351 0 0
352352 −8.54254e8 −1.04397
353353 − 3.55684e8i − 0.430381i −0.976572 0.215191i 0.930963π-0.930963\pi
0.976572 0.215191i 0.0690373π-0.0690373\pi
354354 1.54768e7 0.0185426
355355 2.04971e9 2.43160
356356 − 2.94519e8i − 0.345970i
357357 − 1.32340e7i − 0.0153941i
358358 − 1.42895e8i − 0.164599i
359359 1.16954e8i 0.133409i 0.997773 + 0.0667044i 0.0212485π0.0212485\pi
−0.997773 + 0.0667044i 0.978752π0.978752\pi
360360 1.35926e9 1.53548
361361 −1.47023e9 −1.64479
362362 − 3.94777e8i − 0.437393i
363363 1.45703e7 0.0159881
364364 0 0
365365 2.30197e9 2.47785
366366 1.93318e7i 0.0206104i
367367 1.69538e9 1.79034 0.895169 0.445726i 0.147055π-0.147055\pi
0.895169 + 0.445726i 0.147055π0.147055\pi
368368 1.98496e7 0.0207628
369369 1.41048e9i 1.46142i
370370 − 5.01104e8i − 0.514307i
371371 2.14261e8i 0.217838i
372372 − 1.11463e7i − 0.0112262i
373373 −1.30728e9 −1.30433 −0.652164 0.758078i 0.726138π-0.726138\pi
−0.652164 + 0.758078i 0.726138π0.726138\pi
374374 5.83380e8 0.576635
375375 1.36014e6i 0.00133191i
376376 −5.27555e8 −0.511812
377377 0 0
378378 3.98874e7 0.0379852
379379 8.39591e8i 0.792192i 0.918209 + 0.396096i 0.129635π0.129635\pi
−0.918209 + 0.396096i 0.870365π0.870365\pi
380380 1.10821e9 1.03605
381381 3.91748e7 0.0362885
382382 − 1.11309e9i − 1.02166i
383383 6.31720e8i 0.574551i 0.957848 + 0.287276i 0.0927496π0.0927496\pi
−0.957848 + 0.287276i 0.907250π0.907250\pi
384384 − 2.68523e6i − 0.00242004i
385385 2.13748e9i 1.90893i
386386 −1.15563e9 −1.02273
387387 2.84014e8 0.249088
388388 − 2.19271e8i − 0.190577i
389389 −4.49938e8 −0.387551 −0.193776 0.981046i 0.562073π-0.562073\pi
−0.193776 + 0.981046i 0.562073π0.562073\pi
390390 0 0
391391 4.17300e7 0.0353045
392392 − 1.06026e8i − 0.0889016i
393393 −7.51679e6 −0.00624681
394394 −1.40234e9 −1.15509
395395 − 1.77815e9i − 1.45170i
396396 − 7.08275e8i − 0.573151i
397397 − 1.42468e9i − 1.14275i −0.820690 0.571374i 0.806411π-0.806411\pi
0.820690 0.571374i 0.193589π-0.193589\pi
398398 − 4.88222e8i − 0.388174i
399399 5.26817e7 0.0415198
400400 4.71157e8 0.368092
401401 8.70028e8i 0.673795i 0.941541 + 0.336898i 0.109378π0.109378\pi
−0.941541 + 0.336898i 0.890622π0.890622\pi
402402 −1.33354e7 −0.0102380
403403 0 0
404404 5.49273e8 0.414432
405405 1.90508e9i 1.42502i
406406 −1.00819e9 −0.747653
407407 −8.46238e8 −0.622174
408408 2.18438e7i 0.0159228i
409409 2.23157e9i 1.61280i 0.591373 + 0.806398i 0.298586π0.298586\pi
−0.591373 + 0.806398i 0.701414π0.701414\pi
410410 2.16790e9i 1.55345i
411411 1.75782e7i 0.0124890i
412412 −1.63740e8 −0.115349
413413 1.51269e9 1.05663
414414 6.28683e7i 0.0435442i
415415 3.23800e8 0.222387
416416 0 0
417417 −1.28621e7 −0.00868632
418418 2.32231e9i 1.55526i
419419 −8.31533e8 −0.552243 −0.276122 0.961123i 0.589049π-0.589049\pi
−0.276122 + 0.961123i 0.589049π0.589049\pi
420420 −2.46954e7 −0.0162646
421421 − 9.53842e8i − 0.623002i −0.950246 0.311501i 0.899168π-0.899168\pi
0.950246 0.311501i 0.100832π-0.100832\pi
422422 1.01940e9i 0.660312i
423423 − 7.39844e8i − 0.475279i
424424 − 3.53655e8i − 0.225320i
425425 9.90518e8 0.625894
426426 −4.96252e7 −0.0311006
427427 1.88947e9i 1.17447i
428428 −1.13364e7 −0.00698911
429429 0 0
430430 4.36530e8 0.264773
431431 − 1.01803e9i − 0.612479i −0.951954 0.306240i 0.900929π-0.900929\pi
0.951954 0.306240i 0.0990709π-0.0990709\pi
432432 −2.91518e7 −0.0173969
433433 −1.81766e9 −1.07598 −0.537990 0.842951i 0.680816π-0.680816\pi
−0.537990 + 0.842951i 0.680816π0.680816\pi
434434 1.35187e9i 0.793816i
435435 5.80697e7i 0.0338250i
436436 1.04018e9i 0.601040i
437437 1.66118e8i 0.0952208i
438438 −5.57328e7 −0.0316921
439439 7.03870e8 0.397069 0.198535 0.980094i 0.436382π-0.436382\pi
0.198535 + 0.980094i 0.436382π0.436382\pi
440440 − 3.52809e9i − 1.97449i
441441 1.48690e8 0.0825558
442442 0 0
443443 9.00901e7 0.0492339 0.0246169 0.999697i 0.492163π-0.492163\pi
0.0246169 + 0.999697i 0.492163π0.492163\pi
444444 − 9.77699e6i − 0.00530109i
445445 2.05742e9 1.10679
446446 3.89592e8 0.207940
447447 − 4.42139e7i − 0.0234144i
448448 1.89921e9i 0.997932i
449449 − 1.03765e9i − 0.540992i −0.962721 0.270496i 0.912812π-0.912812\pi
0.962721 0.270496i 0.0871876π-0.0871876\pi
450450 1.49226e9i 0.771972i
451451 3.66104e9 1.87926
452452 −4.64967e8 −0.236831
453453 − 4.69252e7i − 0.0237171i
454454 −2.83553e9 −1.42213
455455 0 0
456456 −8.69555e7 −0.0429457
457457 − 1.66801e9i − 0.817507i −0.912645 0.408754i 0.865964π-0.865964\pi
0.912645 0.408754i 0.134036π-0.134036\pi
458458 2.43024e9 1.18201
459459 −6.12860e7 −0.0295813
460460 − 7.78704e7i − 0.0373010i
461461 2.90964e9i 1.38320i 0.722280 + 0.691601i 0.243094π0.243094\pi
−0.722280 + 0.691601i 0.756906π0.756906\pi
462462 − 5.17503e7i − 0.0244155i
463463 1.89500e9i 0.887310i 0.896198 + 0.443655i 0.146318π0.146318\pi
−0.896198 + 0.443655i 0.853682π0.853682\pi
464464 7.36836e8 0.342419
465465 7.78650e7 0.0359134
466466 8.26704e8i 0.378442i
467467 9.95155e8 0.452149 0.226075 0.974110i 0.427411π-0.427411\pi
0.226075 + 0.974110i 0.427411π0.427411\pi
468468 0 0
469469 −1.30339e9 −0.583405
470470 − 1.13714e9i − 0.505209i
471471 2.90206e7 0.0127977
472472 −2.49682e9 −1.09292
473473 − 7.37188e8i − 0.320305i
474474 4.30504e7i 0.0185675i
475475 3.94303e9i 1.68812i
476476 6.58769e8i 0.279968i
477477 4.95966e8 0.209236
478478 −9.58556e8 −0.401439
479479 1.39127e9i 0.578411i 0.957267 + 0.289206i 0.0933912π0.0933912\pi
−0.957267 + 0.289206i 0.906609π0.906609\pi
480480 6.89461e7 0.0284555
481481 0 0
482482 6.25461e8 0.254411
483483 − 3.70178e6i − 0.00149484i
484484 −7.25289e8 −0.290772
485485 1.53176e9 0.609669
486486 − 1.38510e8i − 0.0547335i
487487 − 2.89212e9i − 1.13466i −0.823491 0.567329i 0.807977π-0.807977\pi
0.823491 0.567329i 0.192023π-0.192023\pi
488488 − 3.11872e9i − 1.21481i
489489 − 7.84787e7i − 0.0303508i
490490 2.28537e8 0.0877547
491491 3.80132e9 1.44927 0.724635 0.689133i 0.242009π-0.242009\pi
0.724635 + 0.689133i 0.242009π0.242009\pi
492492 4.22977e7i 0.0160117i
493493 1.54906e9 0.582241
494494 0 0
495495 4.94780e9 1.83355
496496 − 9.88014e8i − 0.363561i
497497 −4.85032e9 −1.77224
498498 −7.83948e6 −0.00284436
499499 4.75421e8i 0.171288i 0.996326 + 0.0856439i 0.0272947π0.0272947\pi
−0.996326 + 0.0856439i 0.972705π0.972705\pi
500500 − 6.77057e7i − 0.0242231i
501501 3.35742e7i 0.0119281i
502502 − 1.19882e9i − 0.422952i
503503 −1.70972e9 −0.599013 −0.299506 0.954094i 0.596822π-0.596822\pi
−0.299506 + 0.954094i 0.596822π0.596822\pi
504504 −3.21648e9 −1.11911
505505 3.83706e9i 1.32580i
506506 1.63181e8 0.0559942
507507 0 0
508508 −1.95006e9 −0.659970
509509 7.59754e8i 0.255365i 0.991815 + 0.127682i 0.0407538π0.0407538\pi
−0.991815 + 0.127682i 0.959246π0.959246\pi
510510 −4.70841e7 −0.0157173
511511 −5.44727e9 −1.80595
512512 − 2.03387e9i − 0.669698i
513513 − 2.43966e8i − 0.0797846i
514514 2.46528e9i 0.800747i
515515 − 1.14384e9i − 0.369011i
516516 8.51709e6 0.00272908
517517 −1.92034e9 −0.611169
518518 1.18579e9i 0.374846i
519519 1.03403e8 0.0324674
520520 0 0
521521 −4.09809e9 −1.26955 −0.634774 0.772698i 0.718907π-0.718907\pi
−0.634774 + 0.772698i 0.718907π0.718907\pi
522522 2.33373e9i 0.718131i
523523 −3.85404e9 −1.17804 −0.589021 0.808118i 0.700486π-0.700486\pi
−0.589021 + 0.808118i 0.700486π0.700486\pi
524524 3.74174e8 0.113609
525525 − 8.78666e7i − 0.0265012i
526526 − 2.56835e9i − 0.769491i
527527 − 2.07711e9i − 0.618190i
528528 3.78218e7i 0.0111821i
529529 −3.39315e9 −0.996572
530530 7.62299e8 0.222413
531531 − 3.50154e9i − 1.01491i
532532 −2.62241e9 −0.755111
533533 0 0
534534 −4.98120e7 −0.0141560
535535 − 7.91927e7i − 0.0223587i
536536 2.15136e9 0.603442
537537 1.94762e7 0.00542743
538538 1.95199e9i 0.540431i
539539 − 3.85941e8i − 0.106160i
540540 1.14363e8i 0.0312541i
541541 1.31488e8i 0.0357022i 0.999841 + 0.0178511i 0.00568248π0.00568248\pi
−0.999841 + 0.0178511i 0.994318π0.994318\pi
542542 −4.14761e9 −1.11892
543543 5.38070e7 0.0144225
544544 − 1.83919e9i − 0.489814i
545545 −7.26635e9 −1.92278
546546 0 0
547547 2.11326e9 0.552073 0.276037 0.961147i 0.410979π-0.410979\pi
0.276037 + 0.961147i 0.410979π0.410979\pi
548548 − 8.75015e8i − 0.227135i
549549 4.37370e9 1.12809
550550 3.87332e9 0.992691
551551 6.16645e9i 1.57038i
552552 6.11008e6i 0.00154618i
553553 4.20771e9i 1.05805i
554554 3.64820e9i 0.911579i
555555 6.82992e7 0.0169586
556556 6.40255e8 0.157976
557557 − 2.39033e9i − 0.586091i −0.956098 0.293046i 0.905331π-0.905331\pi
0.956098 0.293046i 0.0946688π-0.0946688\pi
558558 3.12927e9 0.762470
559559 0 0
560560 −2.18901e9 −0.526731
561561 7.95131e7i 0.0190138i
562562 −2.42954e9 −0.577361
563563 5.42284e9 1.28070 0.640350 0.768083i 0.278789π-0.278789\pi
0.640350 + 0.768083i 0.278789π0.278789\pi
564564 − 2.21866e7i − 0.00520732i
565565 − 3.24812e9i − 0.757640i
566566 − 5.04253e9i − 1.16894i
567567 − 4.50807e9i − 1.03860i
568568 8.00585e9 1.83311
569569 −1.61095e9 −0.366598 −0.183299 0.983057i 0.558678π-0.558678\pi
−0.183299 + 0.983057i 0.558678π0.558678\pi
570570 − 1.87432e8i − 0.0423917i
571571 5.46762e9 1.22906 0.614528 0.788895i 0.289347π-0.289347\pi
0.614528 + 0.788895i 0.289347π0.289347\pi
572572 0 0
573573 1.51711e8 0.0336880
574574 − 5.13001e9i − 1.13221i
575575 2.77064e8 0.0607775
576576 4.39626e9 0.958527
577577 − 3.08434e9i − 0.668417i −0.942499 0.334209i 0.891531π-0.891531\pi
0.942499 0.334209i 0.108469π-0.108469\pi
578578 − 2.19864e9i − 0.473595i
579579 − 1.57509e8i − 0.0337233i
580580 − 2.89062e9i − 0.615167i
581581 −7.66224e8 −0.162084
582582 −3.70852e7 −0.00779777
583583 − 1.28733e9i − 0.269060i
584584 8.99117e9 1.86797
585585 0 0
586586 2.63457e8 0.0540839
587587 1.47215e9i 0.300413i 0.988655 + 0.150206i 0.0479938π0.0479938\pi
−0.988655 + 0.150206i 0.952006π0.952006\pi
588588 4.45896e6 0.000904509 0
589589 8.26852e9 1.66734
590590 − 5.38186e9i − 1.07882i
591591 − 1.91135e8i − 0.0380877i
592592 − 8.66635e8i − 0.171676i
593593 − 5.98992e9i − 1.17959i −0.807554 0.589793i 0.799209π-0.799209\pi
0.807554 0.589793i 0.200791π-0.200791\pi
594594 −2.39653e8 −0.0469170
595595 −4.60196e9 −0.895640
596596 2.20090e9i 0.425832i
597597 6.65433e7 0.0127995
598598 0 0
599599 −6.44870e9 −1.22597 −0.612983 0.790096i 0.710031π-0.710031\pi
−0.612983 + 0.790096i 0.710031π0.710031\pi
600600 1.45031e8i 0.0274114i
601601 5.19885e9 0.976892 0.488446 0.872594i 0.337564π-0.337564\pi
0.488446 + 0.872594i 0.337564π0.337564\pi
602602 −1.03298e9 −0.192977
603603 3.01706e9i 0.560369i
604604 2.33586e9i 0.431338i
605605 − 5.06665e9i − 0.930202i
606606 − 9.28985e7i − 0.0169572i
607607 −3.63606e9 −0.659890 −0.329945 0.944000i 0.607030π-0.607030\pi
−0.329945 + 0.944000i 0.607030π0.607030\pi
608608 7.32142e9 1.32109
609609 − 1.37413e8i − 0.0246529i
610610 6.72237e9 1.19913
611611 0 0
612612 1.52490e9 0.268913
613613 − 1.64970e9i − 0.289263i −0.989486 0.144632i 0.953800π-0.953800\pi
0.989486 0.144632i 0.0461997π-0.0461997\pi
614614 −2.02108e9 −0.352367
615615 −2.95479e8 −0.0512229
616616 8.34869e9i 1.43908i
617617 6.85487e8i 0.117490i 0.998273 + 0.0587450i 0.0187099π0.0187099\pi
−0.998273 + 0.0587450i 0.981290π0.981290\pi
618618 2.76933e7i 0.00471970i
619619 3.07866e9i 0.521728i 0.965376 + 0.260864i 0.0840074π0.0840074\pi
−0.965376 + 0.260864i 0.915993π0.915993\pi
620620 −3.87599e9 −0.653149
621621 −1.71427e7 −0.00287249
622622 9.58744e8i 0.159748i
623623 −4.86858e9 −0.806667
624624 0 0
625625 −5.86262e9 −0.960531
626626 − 2.41320e9i − 0.393172i
627627 −3.16524e8 −0.0512827
628628 −1.44460e9 −0.232749
629629 − 1.82193e9i − 0.291914i
630630 − 6.93308e9i − 1.10467i
631631 − 8.91118e9i − 1.41199i −0.708215 0.705996i 0.750499π-0.750499\pi
0.708215 0.705996i 0.249501π-0.249501\pi
632632 − 6.94517e9i − 1.09439i
633633 −1.38941e8 −0.0217729
634634 −7.97823e9 −1.24335
635635 − 1.36225e10i − 2.11130i
636636 1.48731e7 0.00229246
637637 0 0
638638 6.05743e9 0.923455
639639 1.12274e10i 1.70226i
640640 −9.33756e8 −0.140800
641641 −1.19624e9 −0.179397 −0.0896984 0.995969i 0.528590π-0.528590\pi
−0.0896984 + 0.995969i 0.528590π0.528590\pi
642642 1.91732e6i 0 0.000285971i
643643 − 7.69066e9i − 1.14084i −0.821353 0.570421i 0.806780π-0.806780\pi
0.821353 0.570421i 0.193220π-0.193220\pi
644644 1.84269e8i 0.0271863i
645645 5.94978e7i 0.00873056i
646646 −4.99989e9 −0.729703
647647 −2.75099e9 −0.399322 −0.199661 0.979865i 0.563984π-0.563984\pi
−0.199661 + 0.979865i 0.563984π0.563984\pi
648648 7.44094e9i 1.07427i
649649 −9.08860e9 −1.30509
650650 0 0
651651 −1.84256e8 −0.0261750
652652 3.90655e9i 0.551984i
653653 −4.79202e9 −0.673477 −0.336738 0.941598i 0.609324π-0.609324\pi
−0.336738 + 0.941598i 0.609324π0.609324\pi
654654 1.75925e8 0.0245926
655655 2.61387e9i 0.363445i
656656 3.74928e9i 0.518542i
657657 1.26092e10i 1.73464i
658658 2.69087e9i 0.368215i
659659 −4.70316e9 −0.640164 −0.320082 0.947390i 0.603710π-0.603710\pi
−0.320082 + 0.947390i 0.603710π0.603710\pi
660660 1.48376e8 0.0200890
661661 − 4.42747e8i − 0.0596280i −0.999555 0.0298140i 0.990508π-0.990508\pi
0.999555 0.0298140i 0.00949150π-0.00949150\pi
662662 −5.76574e9 −0.772417
663663 0 0
664664 1.26471e9 0.167650
665665 − 1.83194e10i − 2.41566i
666666 2.74483e9 0.360044
667667 4.33297e8 0.0565386
668668 − 1.67127e9i − 0.216934i
669669 5.31003e7i 0.00685656i
670670 4.63722e9i 0.595657i
671671 − 1.13524e10i − 1.45063i
672672 −1.63151e8 −0.0207394
673673 −4.22977e9 −0.534890 −0.267445 0.963573i 0.586179π-0.586179\pi
−0.267445 + 0.963573i 0.586179π0.586179\pi
674674 9.49564e9i 1.19458i
675675 −4.06905e8 −0.0509249
676676 0 0
677677 2.02881e9 0.251294 0.125647 0.992075i 0.459899π-0.459899\pi
0.125647 + 0.992075i 0.459899π0.459899\pi
678678 7.86398e7i 0.00969033i
679679 −3.62468e9 −0.444350
680680 7.59591e9 0.926400
681681 − 3.86475e8i − 0.0468929i
682682 − 8.12233e9i − 0.980472i
683683 1.47606e9i 0.177268i 0.996064 + 0.0886340i 0.0282502π0.0282502\pi
−0.996064 + 0.0886340i 0.971750π0.971750\pi
684684 6.07030e9i 0.725294i
685685 6.11259e9 0.726622
686686 6.00595e9 0.710309
687687 3.31235e8i 0.0389751i
688688 7.54957e8 0.0883818
689689 0 0
690690 −1.31702e7 −0.00152623
691691 8.97569e9i 1.03489i 0.855716 + 0.517445i 0.173117π0.173117\pi
−0.855716 + 0.517445i 0.826883π0.826883\pi
692692 −5.14724e9 −0.590477
693693 −1.17082e10 −1.33636
694694 − 3.23957e9i − 0.367899i
695695 4.47263e9i 0.505378i
696696 2.26812e8i 0.0254996i
697697 7.88214e9i 0.881717i
698698 1.68996e9 0.188097
699699 −1.12678e8 −0.0124786
700700 4.37386e9i 0.481972i
701701 5.08142e9 0.557149 0.278575 0.960415i 0.410138π-0.410138\pi
0.278575 + 0.960415i 0.410138π0.410138\pi
702702 0 0
703703 7.25272e9 0.787331
704704 − 1.14109e10i − 1.23258i
705705 1.54989e8 0.0166586
706706 2.99451e9 0.320265
707707 − 9.07982e9i − 0.966294i
708708 − 1.05005e8i − 0.0111197i
709709 3.63793e9i 0.383348i 0.981459 + 0.191674i 0.0613916π0.0613916\pi
−0.981459 + 0.191674i 0.938608π0.938608\pi
710710 1.72565e10i 1.80946i
711711 9.73991e9 1.01628
712712 8.03598e9 0.834371
713713 − 5.81002e8i − 0.0600294i
714714 1.11417e8 0.0114554
715715 0 0
716716 −9.69495e8 −0.0987075
717717 − 1.30649e8i − 0.0132369i
718718 −9.84638e8 −0.0992753
719719 −8.62579e9 −0.865462 −0.432731 0.901523i 0.642450π-0.642450\pi
−0.432731 + 0.901523i 0.642450π0.642450\pi
720720 5.06706e9i 0.505932i
721721 2.70672e9i 0.268949i
722722 − 1.23779e10i − 1.22396i
723723 8.52486e7i 0.00838887i
724724 −2.67843e9 −0.262298
725725 1.02849e10 1.00234
726726 1.22668e8i 0.0118974i
727727 1.23787e10 1.19482 0.597412 0.801934i 0.296196π-0.296196\pi
0.597412 + 0.801934i 0.296196π0.296196\pi
728728 0 0
729729 −1.04226e10 −0.996389
730730 1.93804e10i 1.84388i
731731 1.58715e9 0.150282
732732 1.31159e8 0.0123598
733733 1.32984e10i 1.24720i 0.781745 + 0.623598i 0.214330π0.214330\pi
−0.781745 + 0.623598i 0.785670π0.785670\pi
734734 1.42734e10i 1.33227i
735735 3.11490e7i 0.00289360i
736736 − 5.14452e8i − 0.0475634i
737737 7.83109e9 0.720587
738738 −1.18748e10 −1.08750
739739 − 1.28451e10i − 1.17079i −0.810747 0.585397i 0.800939π-0.800939\pi
0.810747 0.585397i 0.199061π-0.199061\pi
740740 −3.39982e9 −0.308422
741741 0 0
742742 −1.80386e9 −0.162103
743743 − 2.01042e10i − 1.79815i −0.437793 0.899076i 0.644240π-0.644240\pi
0.437793 0.899076i 0.355760π-0.355760\pi
744744 3.04129e8 0.0270740
745745 −1.53748e10 −1.36227
746746 − 1.10060e10i − 0.970606i
747747 1.77364e9i 0.155683i
748748 − 3.95804e9i − 0.345799i
749749 1.87397e8i 0.0162959i
750750 −1.14511e7 −0.000991131 0
751751 −1.99539e8 −0.0171905 −0.00859524 0.999963i 0.502736π-0.502736\pi
−0.00859524 + 0.999963i 0.502736π0.502736\pi
752752 − 1.96663e9i − 0.168640i
753753 1.63396e8 0.0139463
754754 0 0
755755 −1.63176e10 −1.37988
756756 − 2.70623e8i − 0.0227792i
757757 2.34770e10 1.96701 0.983507 0.180868i 0.0578908π-0.0578908\pi
0.983507 + 0.180868i 0.0578908π0.0578908\pi
758758 −7.06853e9 −0.589504
759759 2.22411e7i 0.00184634i
760760 3.02377e10i 2.49862i
761761 1.60366e10i 1.31907i 0.751676 + 0.659533i 0.229246π0.229246\pi
−0.751676 + 0.659533i 0.770754π0.770754\pi
762762 3.29813e8i 0.0270038i
763763 1.71947e10 1.40139
764764 −7.55193e9 −0.612676
765765 1.06525e10i 0.860274i
766766 −5.31846e9 −0.427548
767767 0 0
768768 3.18037e8 0.0253345
769769 1.49614e10i 1.18639i 0.805057 + 0.593197i 0.202135π0.202135\pi
−0.805057 + 0.593197i 0.797865π0.797865\pi
770770 −1.79955e10 −1.42052
771771 −3.36011e8 −0.0264036
772772 7.84053e9i 0.613317i
773773 − 3.25072e9i − 0.253135i −0.991958 0.126567i 0.959604π-0.959604\pi
0.991958 0.126567i 0.0403960π-0.0403960\pi
774774 2.39112e9i 0.185357i
775775 − 1.37909e10i − 1.06423i
776776 5.98282e9 0.459611
777777 −1.61620e8 −0.0123601
778778 − 3.78804e9i − 0.288394i
779779 −3.13771e10 −2.37811
780780 0 0
781781 2.91419e10 2.18897
782782 3.51326e8i 0.0262716i
783783 −6.36353e8 −0.0473731
784784 3.95243e8 0.0292926
785785 − 1.00915e10i − 0.744583i
786786 − 6.32840e7i − 0.00464852i
787787 1.13640e10i 0.831033i 0.909586 + 0.415517i 0.136399π0.136399\pi
−0.909586 + 0.415517i 0.863601π0.863601\pi
788788 9.51441e9i 0.692692i
789789 3.50059e8 0.0253730
790790 1.49702e10 1.08027
791791 7.68619e9i 0.552196i
792792 1.93254e10 1.38226
793793 0 0
794794 1.19944e10 0.850367
795795 1.03899e8i 0.00733377i
796796 −3.31242e9 −0.232782
797797 5.72603e9 0.400636 0.200318 0.979731i 0.435803π-0.435803\pi
0.200318 + 0.979731i 0.435803π0.435803\pi
798798 4.43528e8i 0.0308967i
799799 − 4.13445e9i − 0.286751i
800800 − 1.22112e10i − 0.843226i
801801 1.12697e10i 0.774814i
802802 −7.32478e9 −0.501400
803803 3.27285e10 2.23060
804804 9.04763e7i 0.00613958i
805805 −1.28725e9 −0.0869712
806806 0 0
807807 −2.66051e8 −0.0178200
808808 1.49870e10i 0.999480i
809809 1.88681e10 1.25288 0.626440 0.779470i 0.284512π-0.284512\pi
0.626440 + 0.779470i 0.284512π0.284512\pi
810810 −1.60389e10 −1.06042
811811 − 2.28943e10i − 1.50714i −0.657367 0.753570i 0.728330π-0.728330\pi
0.657367 0.753570i 0.271670π-0.271670\pi
812812 6.84021e9i 0.448356i
813813 − 5.65308e8i − 0.0368951i
814814 − 7.12449e9i − 0.462987i
815815 −2.72900e10 −1.76584
816816 −8.14297e7 −0.00524647
817817 6.31810e9i 0.405331i
818818 −1.87876e10 −1.20015
819819 0 0
820820 1.47085e10 0.931578
821821 3.01467e9i 0.190125i 0.995471 + 0.0950625i 0.0303051π0.0303051\pi
−0.995471 + 0.0950625i 0.969695π0.969695\pi
822822 −1.47991e8 −0.00929361
823823 −9.51460e9 −0.594964 −0.297482 0.954727i 0.596147π-0.596147\pi
−0.297482 + 0.954727i 0.596147π0.596147\pi
824824 − 4.46766e9i − 0.278185i
825825 5.27923e8i 0.0327327i
826826 1.27354e10i 0.786287i
827827 3.09369e10i 1.90199i 0.309210 + 0.950994i 0.399936π0.399936\pi
−0.309210 + 0.950994i 0.600064π0.600064\pi
828828 4.26540e8 0.0261128
829829 −1.75499e10 −1.06988 −0.534938 0.844891i 0.679665π-0.679665\pi
−0.534938 + 0.844891i 0.679665π0.679665\pi
830830 2.72608e9i 0.165487i
831831 −4.97240e8 −0.0300581
832832 0 0
833833 8.30923e8 0.0498085
834834 − 1.08286e8i − 0.00646386i
835835 1.16750e10 0.693991
836836 1.57561e10 0.932666
837837 8.53278e8i 0.0502981i
838838 − 7.00069e9i − 0.410948i
839839 − 1.26056e10i − 0.736878i −0.929652 0.368439i 0.879892π-0.879892\pi
0.929652 0.368439i 0.120108π-0.120108\pi
840840 − 6.73816e8i − 0.0392251i
841841 −1.16552e9 −0.0675671
842842 8.03041e9 0.463602
843843 − 3.31140e8i − 0.0190377i
844844 6.91625e9 0.395979
845845 0 0
846846 6.22876e9 0.353676
847847 1.19895e10i 0.677966i
848848 1.31836e9 0.0742417
849849 6.87282e8 0.0385441
850850 8.33918e9i 0.465755i
851851 − 5.09625e8i − 0.0283464i
852852 3.36690e8i 0.0186506i
853853 − 1.77001e10i − 0.976457i −0.872716 0.488229i 0.837643π-0.837643\pi
0.872716 0.488229i 0.162357π-0.162357\pi
854854 −1.59075e10 −0.873974
855855 −4.24053e10 −2.32027
856856 − 3.09315e8i − 0.0168555i
857857 6.38662e9 0.346608 0.173304 0.984868i 0.444556π-0.444556\pi
0.173304 + 0.984868i 0.444556π0.444556\pi
858858 0 0
859859 9.57498e9 0.515421 0.257710 0.966222i 0.417032π-0.417032\pi
0.257710 + 0.966222i 0.417032π0.417032\pi
860860 − 2.96171e9i − 0.158781i
861861 6.99206e8 0.0373331
862862 8.57084e9 0.455772
863863 − 3.52438e10i − 1.86658i −0.359130 0.933288i 0.616926π-0.616926\pi
0.359130 0.933288i 0.383074π-0.383074\pi
864864 7.55541e8i 0.0398529i
865865 − 3.59571e10i − 1.88898i
866866 − 1.53029e10i − 0.800683i
867867 2.99669e8 0.0156162
868868 9.17195e9 0.476039
869869 − 2.52809e10i − 1.30684i
870870 −4.88890e8 −0.0251706
871871 0 0
872872 −2.83813e10 −1.44952
873873 8.39031e9i 0.426804i
874874 −1.39855e9 −0.0708579
875875 −1.11922e9 −0.0564788
876876 3.78128e8i 0.0190053i
877877 − 6.34498e9i − 0.317638i −0.987308 0.158819i 0.949231π-0.949231\pi
0.987308 0.158819i 0.0507686π-0.0507686\pi
878878 5.92589e9i 0.295476i
879879 3.59084e7i 0.00178335i
880880 1.31521e10 0.650586
881881 2.92173e8 0.0143954 0.00719771 0.999974i 0.497709π-0.497709\pi
0.00719771 + 0.999974i 0.497709π0.497709\pi
882882 1.25183e9i 0.0614333i
883883 3.68103e10 1.79931 0.899656 0.436600i 0.143817π-0.143817\pi
0.899656 + 0.436600i 0.143817π0.143817\pi
884884 0 0
885885 7.33533e8 0.0355728
886886 7.58470e8i 0.0366370i
887887 4.03158e9 0.193973 0.0969867 0.995286i 0.469080π-0.469080\pi
0.0969867 + 0.995286i 0.469080π0.469080\pi
888888 2.66766e8 0.0127846
889889 3.22356e10i 1.53879i
890890 1.73215e10i 0.823607i
891891 2.70856e10i 1.28282i
892892 − 2.64325e9i − 0.124699i
893893 1.64584e10 0.773404
894894 3.72238e8 0.0174237
895895 − 6.77261e9i − 0.315773i
896896 2.20959e9 0.102620
897897 0 0
898898 8.73603e9 0.402575
899899 − 2.15673e10i − 0.990004i
900900 1.01245e10 0.462940
901901 2.77159e9 0.126239
902902 3.08223e10i 1.39844i
903903 − 1.40793e8i − 0.00636316i
904904 − 1.26867e10i − 0.571161i
905905 − 1.87107e10i − 0.839112i
906906 3.95064e8 0.0176489
907907 2.17702e10 0.968807 0.484403 0.874845i 0.339037π-0.339037\pi
0.484403 + 0.874845i 0.339037π0.339037\pi
908908 1.92381e10i 0.852830i
909909 −2.10177e10 −0.928138
910910 0 0
911911 −3.27391e10 −1.43467 −0.717335 0.696728i 0.754638π-0.754638\pi
−0.717335 + 0.696728i 0.754638π0.754638\pi
912912 − 3.24154e8i − 0.0141504i
913913 4.60365e9 0.200196
914914 1.40430e10 0.608342
915915 9.16240e8i 0.0395399i
916916 − 1.64883e10i − 0.708831i
917917 − 6.18532e9i − 0.264892i
918918 − 5.15968e8i − 0.0220127i
919919 1.03618e10 0.440385 0.220193 0.975456i 0.429331π-0.429331\pi
0.220193 + 0.975456i 0.429331π0.429331\pi
920920 2.12470e9 0.0899582
921921 − 2.75468e8i − 0.0116188i
922922 −2.44963e10 −1.02930
923923 0 0
924924 −3.51108e8 −0.0146416
925925 − 1.20966e10i − 0.502537i
926926 −1.59540e10 −0.660286
927927 6.26544e9 0.258329
928928 − 1.90969e10i − 0.784415i
929929 2.15053e10i 0.880018i 0.897993 + 0.440009i 0.145025π0.145025\pi
−0.897993 + 0.440009i 0.854975π0.854975\pi
930930 6.55546e8i 0.0267247i
931931 3.30772e9i 0.134340i
932932 5.60891e9 0.226946
933933 −1.30674e8 −0.00526750
934934 8.37823e9i 0.336464i
935935 2.76497e10 1.10624
936936 0 0
937937 2.31411e8 0.00918957 0.00459479 0.999989i 0.498537π-0.498537\pi
0.00459479 + 0.999989i 0.498537π0.498537\pi
938938 − 1.09733e10i − 0.434137i
939939 3.28912e8 0.0129643
940940 −7.71511e9 −0.302966
941941 2.99031e10i 1.16991i 0.811065 + 0.584956i 0.198888π0.198888\pi
−0.811065 + 0.584956i 0.801112π0.801112\pi
942942 2.44325e8i 0.00952334i
943943 2.20476e9i 0.0856192i
944944 − 9.30767e9i − 0.360113i
945945 1.89049e9 0.0728724
946946 6.20640e9 0.238353
947947 − 3.37889e10i − 1.29285i −0.762976 0.646427i 0.776263π-0.776263\pi
0.762976 0.646427i 0.223737π-0.223737\pi
948948 2.92083e8 0.0111347
949949 0 0
950950 −3.31965e10 −1.25620
951951 − 1.08741e9i − 0.0409979i
952952 −1.79746e10 −0.675195
953953 4.50079e10 1.68447 0.842236 0.539110i 0.181239π-0.181239\pi
0.842236 + 0.539110i 0.181239π0.181239\pi
954954 4.17554e9i 0.155702i
955955 − 5.27555e10i − 1.96000i
956956 6.50348e9i 0.240737i
957957 8.25611e8i 0.0304497i
958958 −1.17131e10 −0.430421
959959 −1.44645e10 −0.529589
960960 9.20966e8i 0.0335965i
961961 −1.40672e9 −0.0511298
962962 0 0
963963 4.33783e8 0.0156524
964964 − 4.24354e9i − 0.152566i
965965 −5.47716e10 −1.96205
966966 3.11653e7 0.00111238
967967 5.77459e9i 0.205366i 0.994714 + 0.102683i 0.0327427π0.0327427\pi
−0.994714 + 0.102683i 0.967257π0.967257\pi
968968 − 1.97896e10i − 0.701250i
969969 − 6.81471e8i − 0.0240610i
970970 1.28959e10i 0.453681i
971971 −5.40196e10 −1.89358 −0.946790 0.321851i 0.895695π-0.895695\pi
−0.946790 + 0.321851i 0.895695π0.895695\pi
972972 −9.39741e8 −0.0328228
973973 − 1.05838e10i − 0.368338i
974974 2.43488e10 0.844347
975975 0 0
976976 1.16260e10 0.400273
977977 − 3.21144e10i − 1.10171i −0.834600 0.550857i 0.814301π-0.814301\pi
0.834600 0.550857i 0.185699π-0.185699\pi
978978 6.60713e8 0.0225854
979979 2.92516e10 0.996345
980980 − 1.55055e9i − 0.0526252i
981981 − 3.98019e10i − 1.34605i
982982 3.20034e10i 1.07846i
983983 − 1.36597e10i − 0.458673i −0.973347 0.229337i 0.926344π-0.926344\pi
0.973347 0.229337i 0.0736556π-0.0736556\pi
984984 −1.15410e9 −0.0386153
985985 −6.64648e10 −2.21598
986986 1.30415e10i 0.433271i
987987 −3.66758e8 −0.0121414
988988 0 0
989989 4.43953e8 0.0145932
990990 4.16556e10i 1.36443i
991991 −3.04984e10 −0.995450 −0.497725 0.867335i 0.665831π-0.665831\pi
−0.497725 + 0.867335i 0.665831π0.665831\pi
992992 −2.56068e10 −0.832847
993993 − 7.85854e8i − 0.0254694i
994994 − 4.08349e10i − 1.31880i
995995 − 2.31396e10i − 0.744688i
996996 5.31882e7i 0.00170572i
997997 3.41604e10 1.09167 0.545833 0.837894i 0.316213π-0.316213\pi
0.545833 + 0.837894i 0.316213π0.316213\pi
998998 −4.00258e9 −0.127463
999999 7.48452e8i 0.0237511i
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.168.10 14
13.3 even 3 13.8.e.a.4.3 14
13.4 even 6 13.8.e.a.10.3 yes 14
13.5 odd 4 169.8.a.g.1.10 14
13.8 odd 4 169.8.a.g.1.5 14
13.12 even 2 inner 169.8.b.d.168.5 14
39.17 odd 6 117.8.q.b.10.5 14
39.29 odd 6 117.8.q.b.82.5 14
52.3 odd 6 208.8.w.a.17.4 14
52.43 odd 6 208.8.w.a.49.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.8.e.a.4.3 14 13.3 even 3
13.8.e.a.10.3 yes 14 13.4 even 6
117.8.q.b.10.5 14 39.17 odd 6
117.8.q.b.82.5 14 39.29 odd 6
169.8.a.g.1.5 14 13.8 odd 4
169.8.a.g.1.10 14 13.5 odd 4
169.8.b.d.168.5 14 13.12 even 2 inner
169.8.b.d.168.10 14 1.1 even 1 trivial
208.8.w.a.17.4 14 52.3 odd 6
208.8.w.a.49.4 14 52.43 odd 6