Properties

Label 10.32.a.a.1.2
Level $10$
Weight $32$
Character 10.1
Self dual yes
Analytic conductor $60.877$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,32,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.8771328190\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 17573188320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-132563.\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32768.0 q^{2} +2.09091e7 q^{3} +1.07374e9 q^{4} +3.05176e10 q^{5} -6.85150e11 q^{6} +1.72878e13 q^{7} -3.51844e13 q^{8} -1.80482e14 q^{9} -1.00000e15 q^{10} -1.40404e16 q^{11} +2.24510e16 q^{12} +1.12100e17 q^{13} -5.66487e17 q^{14} +6.38096e17 q^{15} +1.15292e18 q^{16} -1.87926e19 q^{17} +5.91403e18 q^{18} -1.17369e20 q^{19} +3.27680e19 q^{20} +3.61473e20 q^{21} +4.60074e20 q^{22} -2.47133e21 q^{23} -7.35674e20 q^{24} +9.31323e20 q^{25} -3.67328e21 q^{26} -1.66887e22 q^{27} +1.85627e22 q^{28} +6.20136e22 q^{29} -2.09091e22 q^{30} +1.42328e23 q^{31} -3.77789e22 q^{32} -2.93572e23 q^{33} +6.15797e23 q^{34} +5.27582e23 q^{35} -1.93791e23 q^{36} +1.53344e24 q^{37} +3.84596e24 q^{38} +2.34391e24 q^{39} -1.07374e24 q^{40} -6.64565e24 q^{41} -1.18448e25 q^{42} +1.23317e24 q^{43} -1.50757e25 q^{44} -5.50787e24 q^{45} +8.09804e25 q^{46} -1.65205e26 q^{47} +2.41066e25 q^{48} +1.41093e26 q^{49} -3.05176e25 q^{50} -3.92938e26 q^{51} +1.20366e26 q^{52} +2.10693e25 q^{53} +5.46856e26 q^{54} -4.28478e26 q^{55} -6.08261e26 q^{56} -2.45409e27 q^{57} -2.03206e27 q^{58} +2.11570e27 q^{59} +6.85150e26 q^{60} -2.54907e27 q^{61} -4.66380e27 q^{62} -3.12014e27 q^{63} +1.23794e27 q^{64} +3.42101e27 q^{65} +9.61975e27 q^{66} +2.58183e28 q^{67} -2.01784e28 q^{68} -5.16733e28 q^{69} -1.72878e28 q^{70} -6.76939e28 q^{71} +6.35014e27 q^{72} -5.88515e28 q^{73} -5.02476e28 q^{74} +1.94731e28 q^{75} -1.26024e29 q^{76} -2.42727e29 q^{77} -7.68051e28 q^{78} -2.13368e29 q^{79} +3.51844e28 q^{80} -2.37468e29 q^{81} +2.17765e29 q^{82} -1.06637e29 q^{83} +3.88129e29 q^{84} -5.73506e29 q^{85} -4.04086e28 q^{86} +1.29665e30 q^{87} +4.94001e29 q^{88} -5.07280e29 q^{89} +1.80482e29 q^{90} +1.93796e30 q^{91} -2.65357e30 q^{92} +2.97595e30 q^{93} +5.41345e30 q^{94} -3.58183e30 q^{95} -7.89924e29 q^{96} +2.99146e30 q^{97} -4.62334e30 q^{98} +2.53403e30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 65536 q^{2} - 5904756 q^{3} + 2147483648 q^{4} + 61035156250 q^{5} + 193487044608 q^{6} + 18016565093212 q^{7} - 70368744177664 q^{8} - 79171117705926 q^{9} - 20\!\cdots\!00 q^{10} - 17\!\cdots\!76 q^{11}+ \cdots + 21\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32768.0 −0.707107
\(3\) 2.09091e7 0.841311 0.420655 0.907221i \(-0.361800\pi\)
0.420655 + 0.907221i \(0.361800\pi\)
\(4\) 1.07374e9 0.500000
\(5\) 3.05176e10 0.447214
\(6\) −6.85150e11 −0.594896
\(7\) 1.72878e13 1.37632 0.688162 0.725557i \(-0.258418\pi\)
0.688162 + 0.725557i \(0.258418\pi\)
\(8\) −3.51844e13 −0.353553
\(9\) −1.80482e14 −0.292196
\(10\) −1.00000e15 −0.316228
\(11\) −1.40404e16 −1.01342 −0.506712 0.862115i \(-0.669139\pi\)
−0.506712 + 0.862115i \(0.669139\pi\)
\(12\) 2.24510e16 0.420655
\(13\) 1.12100e17 0.607410 0.303705 0.952766i \(-0.401776\pi\)
0.303705 + 0.952766i \(0.401776\pi\)
\(14\) −5.66487e17 −0.973208
\(15\) 6.38096e17 0.376246
\(16\) 1.15292e18 0.250000
\(17\) −1.87926e19 −1.59232 −0.796158 0.605088i \(-0.793138\pi\)
−0.796158 + 0.605088i \(0.793138\pi\)
\(18\) 5.91403e18 0.206614
\(19\) −1.17369e20 −1.77367 −0.886837 0.462083i \(-0.847102\pi\)
−0.886837 + 0.462083i \(0.847102\pi\)
\(20\) 3.27680e19 0.223607
\(21\) 3.61473e20 1.15792
\(22\) 4.60074e20 0.716600
\(23\) −2.47133e21 −1.93263 −0.966315 0.257363i \(-0.917146\pi\)
−0.966315 + 0.257363i \(0.917146\pi\)
\(24\) −7.35674e20 −0.297448
\(25\) 9.31323e20 0.200000
\(26\) −3.67328e21 −0.429504
\(27\) −1.66887e22 −1.08714
\(28\) 1.85627e22 0.688162
\(29\) 6.20136e22 1.33450 0.667249 0.744835i \(-0.267472\pi\)
0.667249 + 0.744835i \(0.267472\pi\)
\(30\) −2.09091e22 −0.266046
\(31\) 1.42328e23 1.08939 0.544696 0.838634i \(-0.316645\pi\)
0.544696 + 0.838634i \(0.316645\pi\)
\(32\) −3.77789e22 −0.176777
\(33\) −2.93572e23 −0.852605
\(34\) 6.15797e23 1.12594
\(35\) 5.27582e23 0.615511
\(36\) −1.93791e23 −0.146098
\(37\) 1.53344e24 0.756030 0.378015 0.925799i \(-0.376607\pi\)
0.378015 + 0.925799i \(0.376607\pi\)
\(38\) 3.84596e24 1.25418
\(39\) 2.34391e24 0.511021
\(40\) −1.07374e24 −0.158114
\(41\) −6.64565e24 −0.667403 −0.333701 0.942679i \(-0.608298\pi\)
−0.333701 + 0.942679i \(0.608298\pi\)
\(42\) −1.18448e25 −0.818770
\(43\) 1.23317e24 0.0591920 0.0295960 0.999562i \(-0.490578\pi\)
0.0295960 + 0.999562i \(0.490578\pi\)
\(44\) −1.50757e25 −0.506712
\(45\) −5.50787e24 −0.130674
\(46\) 8.09804e25 1.36658
\(47\) −1.65205e26 −1.99760 −0.998798 0.0490256i \(-0.984388\pi\)
−0.998798 + 0.0490256i \(0.984388\pi\)
\(48\) 2.41066e25 0.210328
\(49\) 1.41093e26 0.894267
\(50\) −3.05176e25 −0.141421
\(51\) −3.92938e26 −1.33963
\(52\) 1.20366e26 0.303705
\(53\) 2.10693e25 0.0395708 0.0197854 0.999804i \(-0.493702\pi\)
0.0197854 + 0.999804i \(0.493702\pi\)
\(54\) 5.46856e26 0.768723
\(55\) −4.28478e26 −0.453217
\(56\) −6.08261e26 −0.486604
\(57\) −2.45409e27 −1.49221
\(58\) −2.03206e27 −0.943633
\(59\) 2.11570e27 0.753786 0.376893 0.926257i \(-0.376992\pi\)
0.376893 + 0.926257i \(0.376992\pi\)
\(60\) 6.85150e26 0.188123
\(61\) −2.54907e27 −0.541712 −0.270856 0.962620i \(-0.587307\pi\)
−0.270856 + 0.962620i \(0.587307\pi\)
\(62\) −4.66380e27 −0.770316
\(63\) −3.12014e27 −0.402157
\(64\) 1.23794e27 0.125000
\(65\) 3.42101e27 0.271642
\(66\) 9.61975e27 0.602883
\(67\) 2.58183e28 1.28165 0.640823 0.767688i \(-0.278593\pi\)
0.640823 + 0.767688i \(0.278593\pi\)
\(68\) −2.01784e28 −0.796158
\(69\) −5.16733e28 −1.62594
\(70\) −1.72878e28 −0.435232
\(71\) −6.76939e28 −1.36787 −0.683936 0.729542i \(-0.739733\pi\)
−0.683936 + 0.729542i \(0.739733\pi\)
\(72\) 6.35014e27 0.103307
\(73\) −5.88515e28 −0.773131 −0.386566 0.922262i \(-0.626339\pi\)
−0.386566 + 0.922262i \(0.626339\pi\)
\(74\) −5.02476e28 −0.534594
\(75\) 1.94731e28 0.168262
\(76\) −1.26024e29 −0.886837
\(77\) −2.42727e29 −1.39480
\(78\) −7.68051e28 −0.361346
\(79\) −2.13368e29 −0.823965 −0.411982 0.911192i \(-0.635163\pi\)
−0.411982 + 0.911192i \(0.635163\pi\)
\(80\) 3.51844e28 0.111803
\(81\) −2.37468e29 −0.622425
\(82\) 2.17765e29 0.471925
\(83\) −1.06637e29 −0.191513 −0.0957564 0.995405i \(-0.530527\pi\)
−0.0957564 + 0.995405i \(0.530527\pi\)
\(84\) 3.88129e29 0.578958
\(85\) −5.73506e29 −0.712106
\(86\) −4.04086e28 −0.0418550
\(87\) 1.29665e30 1.12273
\(88\) 4.94001e29 0.358300
\(89\) −5.07280e29 −0.308818 −0.154409 0.988007i \(-0.549347\pi\)
−0.154409 + 0.988007i \(0.549347\pi\)
\(90\) 1.80482e29 0.0924006
\(91\) 1.93796e30 0.835993
\(92\) −2.65357e30 −0.966315
\(93\) 2.97595e30 0.916517
\(94\) 5.41345e30 1.41251
\(95\) −3.58183e30 −0.793211
\(96\) −7.89924e29 −0.148724
\(97\) 2.99146e30 0.479646 0.239823 0.970817i \(-0.422911\pi\)
0.239823 + 0.970817i \(0.422911\pi\)
\(98\) −4.62334e30 −0.632342
\(99\) 2.53403e30 0.296119
\(100\) 1.00000e30 0.100000
\(101\) 5.56134e30 0.476648 0.238324 0.971186i \(-0.423402\pi\)
0.238324 + 0.971186i \(0.423402\pi\)
\(102\) 1.28758e31 0.947264
\(103\) −9.44476e30 −0.597330 −0.298665 0.954358i \(-0.596541\pi\)
−0.298665 + 0.954358i \(0.596541\pi\)
\(104\) −3.94416e30 −0.214752
\(105\) 1.10313e31 0.517836
\(106\) −6.90400e29 −0.0279808
\(107\) 3.40170e31 1.19192 0.595961 0.803014i \(-0.296771\pi\)
0.595961 + 0.803014i \(0.296771\pi\)
\(108\) −1.79194e31 −0.543569
\(109\) 1.12629e31 0.296169 0.148085 0.988975i \(-0.452689\pi\)
0.148085 + 0.988975i \(0.452689\pi\)
\(110\) 1.40404e31 0.320473
\(111\) 3.20628e31 0.636056
\(112\) 1.99315e31 0.344081
\(113\) −1.71220e31 −0.257537 −0.128769 0.991675i \(-0.541102\pi\)
−0.128769 + 0.991675i \(0.541102\pi\)
\(114\) 8.04156e31 1.05515
\(115\) −7.54189e31 −0.864298
\(116\) 6.65866e31 0.667249
\(117\) −2.02320e31 −0.177483
\(118\) −6.93273e31 −0.533007
\(119\) −3.24884e32 −2.19154
\(120\) −2.24510e31 −0.133023
\(121\) 5.18822e30 0.0270299
\(122\) 8.35278e31 0.383048
\(123\) −1.38955e32 −0.561493
\(124\) 1.52824e32 0.544696
\(125\) 2.84217e31 0.0894427
\(126\) 1.02241e32 0.284368
\(127\) −7.80052e32 −1.91940 −0.959701 0.281022i \(-0.909327\pi\)
−0.959701 + 0.281022i \(0.909327\pi\)
\(128\) −4.05648e31 −0.0883883
\(129\) 2.57846e31 0.0497988
\(130\) −1.12100e32 −0.192080
\(131\) 7.18592e32 1.09340 0.546698 0.837330i \(-0.315885\pi\)
0.546698 + 0.837330i \(0.315885\pi\)
\(132\) −3.15220e32 −0.426303
\(133\) −2.02906e33 −2.44115
\(134\) −8.46014e32 −0.906261
\(135\) −5.09300e32 −0.486183
\(136\) 6.61207e32 0.562969
\(137\) 1.47730e33 1.12279 0.561395 0.827548i \(-0.310265\pi\)
0.561395 + 0.827548i \(0.310265\pi\)
\(138\) 1.69323e33 1.14971
\(139\) 5.43580e32 0.330016 0.165008 0.986292i \(-0.447235\pi\)
0.165008 + 0.986292i \(0.447235\pi\)
\(140\) 5.66487e32 0.307755
\(141\) −3.45430e33 −1.68060
\(142\) 2.21819e33 0.967231
\(143\) −1.57392e33 −0.615565
\(144\) −2.08081e32 −0.0730491
\(145\) 1.89251e33 0.596806
\(146\) 1.92845e33 0.546686
\(147\) 2.95014e33 0.752356
\(148\) 1.64651e33 0.378015
\(149\) −3.63270e33 −0.751350 −0.375675 0.926752i \(-0.622589\pi\)
−0.375675 + 0.926752i \(0.622589\pi\)
\(150\) −6.38096e32 −0.118979
\(151\) 6.05619e33 1.01872 0.509362 0.860552i \(-0.329881\pi\)
0.509362 + 0.860552i \(0.329881\pi\)
\(152\) 4.12957e33 0.627088
\(153\) 3.39173e33 0.465269
\(154\) 7.95368e33 0.986273
\(155\) 4.34351e33 0.487191
\(156\) 2.51675e33 0.255510
\(157\) 2.21456e33 0.203631 0.101815 0.994803i \(-0.467535\pi\)
0.101815 + 0.994803i \(0.467535\pi\)
\(158\) 6.99163e33 0.582631
\(159\) 4.40541e32 0.0332913
\(160\) −1.15292e33 −0.0790569
\(161\) −4.27238e34 −2.65992
\(162\) 7.78135e33 0.440121
\(163\) −1.25964e34 −0.647649 −0.323824 0.946117i \(-0.604969\pi\)
−0.323824 + 0.946117i \(0.604969\pi\)
\(164\) −7.13572e33 −0.333701
\(165\) −8.95909e33 −0.381297
\(166\) 3.49429e33 0.135420
\(167\) −1.61993e34 −0.571992 −0.285996 0.958231i \(-0.592324\pi\)
−0.285996 + 0.958231i \(0.592324\pi\)
\(168\) −1.27182e34 −0.409385
\(169\) −2.14936e34 −0.631052
\(170\) 1.87926e34 0.503535
\(171\) 2.11830e34 0.518261
\(172\) 1.32411e33 0.0295960
\(173\) 8.33177e34 1.70225 0.851124 0.524965i \(-0.175922\pi\)
0.851124 + 0.524965i \(0.175922\pi\)
\(174\) −4.24886e34 −0.793888
\(175\) 1.61005e34 0.275265
\(176\) −1.61874e34 −0.253356
\(177\) 4.42375e34 0.634168
\(178\) 1.66226e34 0.218367
\(179\) −1.19965e35 −1.44488 −0.722442 0.691432i \(-0.756980\pi\)
−0.722442 + 0.691432i \(0.756980\pi\)
\(180\) −5.91403e33 −0.0653371
\(181\) −9.22700e34 −0.935497 −0.467749 0.883861i \(-0.654935\pi\)
−0.467749 + 0.883861i \(0.654935\pi\)
\(182\) −6.35030e34 −0.591137
\(183\) −5.32987e34 −0.455748
\(184\) 8.69521e34 0.683288
\(185\) 4.67968e34 0.338107
\(186\) −9.75160e34 −0.648075
\(187\) 2.63855e35 1.61369
\(188\) −1.77388e35 −0.998798
\(189\) −2.88512e35 −1.49625
\(190\) 1.17369e35 0.560885
\(191\) 9.20859e34 0.405673 0.202837 0.979213i \(-0.434984\pi\)
0.202837 + 0.979213i \(0.434984\pi\)
\(192\) 2.58842e34 0.105164
\(193\) −4.07145e35 −1.52620 −0.763098 0.646283i \(-0.776323\pi\)
−0.763098 + 0.646283i \(0.776323\pi\)
\(194\) −9.80241e34 −0.339161
\(195\) 7.15303e34 0.228536
\(196\) 1.51498e35 0.447133
\(197\) 1.97536e35 0.538791 0.269395 0.963030i \(-0.413176\pi\)
0.269395 + 0.963030i \(0.413176\pi\)
\(198\) −8.30351e34 −0.209388
\(199\) 4.46274e35 1.04083 0.520413 0.853914i \(-0.325778\pi\)
0.520413 + 0.853914i \(0.325778\pi\)
\(200\) −3.27680e34 −0.0707107
\(201\) 5.39838e35 1.07826
\(202\) −1.82234e35 −0.337041
\(203\) 1.07208e36 1.83670
\(204\) −4.21914e35 −0.669816
\(205\) −2.02809e35 −0.298472
\(206\) 3.09486e35 0.422376
\(207\) 4.46030e35 0.564707
\(208\) 1.29242e35 0.151853
\(209\) 1.64791e36 1.79748
\(210\) −3.61473e35 −0.366165
\(211\) 3.17406e35 0.298701 0.149351 0.988784i \(-0.452282\pi\)
0.149351 + 0.988784i \(0.452282\pi\)
\(212\) 2.26230e34 0.0197854
\(213\) −1.41542e36 −1.15080
\(214\) −1.11467e36 −0.842815
\(215\) 3.76334e34 0.0264714
\(216\) 5.87182e35 0.384362
\(217\) 2.46054e36 1.49936
\(218\) −3.69063e35 −0.209423
\(219\) −1.23053e36 −0.650444
\(220\) −4.60074e35 −0.226609
\(221\) −2.10665e36 −0.967190
\(222\) −1.05063e36 −0.449760
\(223\) 1.18359e36 0.472582 0.236291 0.971682i \(-0.424068\pi\)
0.236291 + 0.971682i \(0.424068\pi\)
\(224\) −6.53115e35 −0.243302
\(225\) −1.68087e35 −0.0584393
\(226\) 5.61055e35 0.182106
\(227\) 1.73010e34 0.00524408 0.00262204 0.999997i \(-0.499165\pi\)
0.00262204 + 0.999997i \(0.499165\pi\)
\(228\) −2.63506e36 −0.746105
\(229\) −1.57133e36 −0.415737 −0.207868 0.978157i \(-0.566653\pi\)
−0.207868 + 0.978157i \(0.566653\pi\)
\(230\) 2.47133e36 0.611151
\(231\) −5.07521e36 −1.17346
\(232\) −2.18191e36 −0.471816
\(233\) −6.37806e36 −1.29024 −0.645122 0.764080i \(-0.723193\pi\)
−0.645122 + 0.764080i \(0.723193\pi\)
\(234\) 6.62961e35 0.125500
\(235\) −5.04167e36 −0.893352
\(236\) 2.27172e36 0.376893
\(237\) −4.46133e36 −0.693210
\(238\) 1.06458e37 1.54965
\(239\) 3.04721e36 0.415656 0.207828 0.978165i \(-0.433361\pi\)
0.207828 + 0.978165i \(0.433361\pi\)
\(240\) 7.35674e35 0.0940614
\(241\) 1.21299e36 0.145409 0.0727047 0.997354i \(-0.476837\pi\)
0.0727047 + 0.997354i \(0.476837\pi\)
\(242\) −1.70007e35 −0.0191130
\(243\) 5.34294e36 0.563486
\(244\) −2.73704e36 −0.270856
\(245\) 4.30583e36 0.399928
\(246\) 4.55327e36 0.397036
\(247\) −1.31571e37 −1.07735
\(248\) −5.00772e36 −0.385158
\(249\) −2.22969e36 −0.161122
\(250\) −9.31323e35 −0.0632456
\(251\) −5.23269e36 −0.334027 −0.167014 0.985955i \(-0.553412\pi\)
−0.167014 + 0.985955i \(0.553412\pi\)
\(252\) −3.35022e36 −0.201078
\(253\) 3.46983e37 1.95858
\(254\) 2.55608e37 1.35722
\(255\) −1.19915e37 −0.599102
\(256\) 1.32923e36 0.0625000
\(257\) −2.76137e36 −0.122225 −0.0611126 0.998131i \(-0.519465\pi\)
−0.0611126 + 0.998131i \(0.519465\pi\)
\(258\) −8.44908e35 −0.0352131
\(259\) 2.65098e37 1.04054
\(260\) 3.67328e36 0.135821
\(261\) −1.11923e37 −0.389936
\(262\) −2.35468e37 −0.773147
\(263\) 2.48833e37 0.770183 0.385091 0.922878i \(-0.374170\pi\)
0.385091 + 0.922878i \(0.374170\pi\)
\(264\) 1.03291e37 0.301441
\(265\) 6.42985e35 0.0176966
\(266\) 6.64882e37 1.72615
\(267\) −1.06068e37 −0.259812
\(268\) 2.77222e37 0.640823
\(269\) −8.65600e37 −1.88867 −0.944336 0.328982i \(-0.893295\pi\)
−0.944336 + 0.328982i \(0.893295\pi\)
\(270\) 1.66887e37 0.343783
\(271\) −7.04664e36 −0.137075 −0.0685375 0.997649i \(-0.521833\pi\)
−0.0685375 + 0.997649i \(0.521833\pi\)
\(272\) −2.16664e37 −0.398079
\(273\) 4.05210e37 0.703330
\(274\) −4.84080e37 −0.793933
\(275\) −1.30761e37 −0.202685
\(276\) −5.54838e37 −0.812971
\(277\) 8.82814e37 1.22302 0.611509 0.791238i \(-0.290563\pi\)
0.611509 + 0.791238i \(0.290563\pi\)
\(278\) −1.78120e37 −0.233356
\(279\) −2.56876e37 −0.318316
\(280\) −1.85627e37 −0.217616
\(281\) −2.56349e36 −0.0284370 −0.0142185 0.999899i \(-0.504526\pi\)
−0.0142185 + 0.999899i \(0.504526\pi\)
\(282\) 1.13191e38 1.18836
\(283\) −1.16177e38 −1.15460 −0.577298 0.816533i \(-0.695893\pi\)
−0.577298 + 0.816533i \(0.695893\pi\)
\(284\) −7.26858e37 −0.683936
\(285\) −7.48929e37 −0.667337
\(286\) 5.15742e37 0.435270
\(287\) −1.14889e38 −0.918562
\(288\) 6.81841e36 0.0516535
\(289\) 2.13874e38 1.53547
\(290\) −6.20136e37 −0.422005
\(291\) 6.25488e37 0.403532
\(292\) −6.31913e37 −0.386566
\(293\) −1.14934e38 −0.666805 −0.333402 0.942785i \(-0.608197\pi\)
−0.333402 + 0.942785i \(0.608197\pi\)
\(294\) −9.66701e37 −0.531996
\(295\) 6.45661e37 0.337103
\(296\) −5.39530e37 −0.267297
\(297\) 2.34316e38 1.10173
\(298\) 1.19036e38 0.531285
\(299\) −2.77035e38 −1.17390
\(300\) 2.09091e37 0.0841311
\(301\) 2.13189e37 0.0814673
\(302\) −1.98449e38 −0.720347
\(303\) 1.16283e38 0.401009
\(304\) −1.35318e38 −0.443418
\(305\) −7.77913e37 −0.242261
\(306\) −1.11140e38 −0.328995
\(307\) 1.02249e38 0.287749 0.143874 0.989596i \(-0.454044\pi\)
0.143874 + 0.989596i \(0.454044\pi\)
\(308\) −2.60626e38 −0.697400
\(309\) −1.97482e38 −0.502540
\(310\) −1.42328e38 −0.344496
\(311\) 5.49215e38 1.26461 0.632305 0.774720i \(-0.282109\pi\)
0.632305 + 0.774720i \(0.282109\pi\)
\(312\) −8.24688e37 −0.180673
\(313\) 3.18159e38 0.663294 0.331647 0.943403i \(-0.392396\pi\)
0.331647 + 0.943403i \(0.392396\pi\)
\(314\) −7.25667e37 −0.143989
\(315\) −9.52191e37 −0.179850
\(316\) −2.29102e38 −0.411982
\(317\) 4.84708e38 0.829968 0.414984 0.909829i \(-0.363787\pi\)
0.414984 + 0.909829i \(0.363787\pi\)
\(318\) −1.44357e37 −0.0235405
\(319\) −8.70693e38 −1.35241
\(320\) 3.77789e37 0.0559017
\(321\) 7.11266e38 1.00278
\(322\) 1.39997e39 1.88085
\(323\) 2.20568e39 2.82425
\(324\) −2.54979e38 −0.311212
\(325\) 1.04401e38 0.121482
\(326\) 4.12760e38 0.457957
\(327\) 2.35497e38 0.249170
\(328\) 2.33823e38 0.235962
\(329\) −2.85604e39 −2.74934
\(330\) 2.93572e38 0.269617
\(331\) 1.17621e39 1.03074 0.515371 0.856967i \(-0.327654\pi\)
0.515371 + 0.856967i \(0.327654\pi\)
\(332\) −1.14501e38 −0.0957564
\(333\) −2.76758e38 −0.220909
\(334\) 5.30817e38 0.404459
\(335\) 7.87912e38 0.573170
\(336\) 4.16750e38 0.289479
\(337\) 3.32696e38 0.220690 0.110345 0.993893i \(-0.464804\pi\)
0.110345 + 0.993893i \(0.464804\pi\)
\(338\) 7.04303e38 0.446221
\(339\) −3.58007e38 −0.216669
\(340\) −6.15797e38 −0.356053
\(341\) −1.99834e39 −1.10402
\(342\) −6.94126e38 −0.366466
\(343\) −2.88397e38 −0.145523
\(344\) −4.33884e37 −0.0209275
\(345\) −1.57694e39 −0.727143
\(346\) −2.73015e39 −1.20367
\(347\) −2.52708e39 −1.06540 −0.532700 0.846304i \(-0.678823\pi\)
−0.532700 + 0.846304i \(0.678823\pi\)
\(348\) 1.39227e39 0.561364
\(349\) −2.89784e39 −1.11758 −0.558792 0.829308i \(-0.688735\pi\)
−0.558792 + 0.829308i \(0.688735\pi\)
\(350\) −5.27582e38 −0.194642
\(351\) −1.87080e39 −0.660339
\(352\) 5.30430e38 0.179150
\(353\) 4.87022e39 1.57413 0.787065 0.616871i \(-0.211600\pi\)
0.787065 + 0.616871i \(0.211600\pi\)
\(354\) −1.44957e39 −0.448425
\(355\) −2.06585e39 −0.611731
\(356\) −5.44688e38 −0.154409
\(357\) −6.79303e39 −1.84377
\(358\) 3.93101e39 1.02169
\(359\) −4.94989e39 −1.23206 −0.616030 0.787723i \(-0.711260\pi\)
−0.616030 + 0.787723i \(0.711260\pi\)
\(360\) 1.93791e38 0.0462003
\(361\) 9.39669e39 2.14592
\(362\) 3.02350e39 0.661497
\(363\) 1.08481e38 0.0227406
\(364\) 2.08087e39 0.417997
\(365\) −1.79601e39 −0.345755
\(366\) 1.74649e39 0.322263
\(367\) 2.60673e39 0.461076 0.230538 0.973063i \(-0.425951\pi\)
0.230538 + 0.973063i \(0.425951\pi\)
\(368\) −2.84925e39 −0.483157
\(369\) 1.19942e39 0.195013
\(370\) −1.53344e39 −0.239078
\(371\) 3.64243e38 0.0544622
\(372\) 3.19541e39 0.458258
\(373\) 5.14959e39 0.708412 0.354206 0.935167i \(-0.384751\pi\)
0.354206 + 0.935167i \(0.384751\pi\)
\(374\) −8.64601e39 −1.14105
\(375\) 5.94273e38 0.0752491
\(376\) 5.81265e39 0.706257
\(377\) 6.95170e39 0.810588
\(378\) 9.45395e39 1.05801
\(379\) −1.10887e40 −1.19117 −0.595584 0.803293i \(-0.703079\pi\)
−0.595584 + 0.803293i \(0.703079\pi\)
\(380\) −3.84596e39 −0.396605
\(381\) −1.63102e40 −1.61481
\(382\) −3.01747e39 −0.286854
\(383\) 1.77078e39 0.161653 0.0808266 0.996728i \(-0.474244\pi\)
0.0808266 + 0.996728i \(0.474244\pi\)
\(384\) −8.48175e38 −0.0743621
\(385\) −7.40744e39 −0.623774
\(386\) 1.33413e40 1.07918
\(387\) −2.22565e38 −0.0172957
\(388\) 3.21205e39 0.239823
\(389\) 1.78976e40 1.28403 0.642015 0.766692i \(-0.278099\pi\)
0.642015 + 0.766692i \(0.278099\pi\)
\(390\) −2.34391e39 −0.161599
\(391\) 4.64427e40 3.07736
\(392\) −4.96428e39 −0.316171
\(393\) 1.50251e40 0.919885
\(394\) −6.47287e39 −0.380983
\(395\) −6.51146e39 −0.368488
\(396\) 2.72089e39 0.148060
\(397\) 1.42952e40 0.748063 0.374032 0.927416i \(-0.377975\pi\)
0.374032 + 0.927416i \(0.377975\pi\)
\(398\) −1.46235e40 −0.735976
\(399\) −4.24258e40 −2.05376
\(400\) 1.07374e39 0.0500000
\(401\) 1.59146e38 0.00712948 0.00356474 0.999994i \(-0.498865\pi\)
0.00356474 + 0.999994i \(0.498865\pi\)
\(402\) −1.76894e40 −0.762447
\(403\) 1.59549e40 0.661708
\(404\) 5.97144e39 0.238324
\(405\) −7.24694e39 −0.278357
\(406\) −3.51299e40 −1.29874
\(407\) −2.15300e40 −0.766180
\(408\) 1.38253e40 0.473632
\(409\) 6.33912e39 0.209083 0.104541 0.994521i \(-0.466663\pi\)
0.104541 + 0.994521i \(0.466663\pi\)
\(410\) 6.64565e39 0.211051
\(411\) 3.08890e40 0.944616
\(412\) −1.01412e40 −0.298665
\(413\) 3.65759e40 1.03745
\(414\) −1.46155e40 −0.399308
\(415\) −3.25431e39 −0.0856472
\(416\) −4.23501e39 −0.107376
\(417\) 1.13658e40 0.277646
\(418\) −5.39986e40 −1.27101
\(419\) −9.89389e39 −0.224414 −0.112207 0.993685i \(-0.535792\pi\)
−0.112207 + 0.993685i \(0.535792\pi\)
\(420\) 1.18448e40 0.258918
\(421\) −4.00753e40 −0.844315 −0.422157 0.906523i \(-0.638727\pi\)
−0.422157 + 0.906523i \(0.638727\pi\)
\(422\) −1.04008e40 −0.211214
\(423\) 2.98166e40 0.583690
\(424\) −7.41311e38 −0.0139904
\(425\) −1.75020e40 −0.318463
\(426\) 4.63805e40 0.813742
\(427\) −4.40678e40 −0.745571
\(428\) 3.65255e40 0.595961
\(429\) −3.29093e40 −0.517881
\(430\) −1.23317e39 −0.0187181
\(431\) −1.32321e41 −1.93745 −0.968725 0.248136i \(-0.920182\pi\)
−0.968725 + 0.248136i \(0.920182\pi\)
\(432\) −1.92408e40 −0.271785
\(433\) −7.69902e40 −1.04924 −0.524618 0.851338i \(-0.675792\pi\)
−0.524618 + 0.851338i \(0.675792\pi\)
\(434\) −8.06270e40 −1.06020
\(435\) 3.95706e40 0.502099
\(436\) 1.20934e40 0.148085
\(437\) 2.90058e41 3.42785
\(438\) 4.03221e40 0.459933
\(439\) −6.36959e40 −0.701313 −0.350656 0.936504i \(-0.614041\pi\)
−0.350656 + 0.936504i \(0.614041\pi\)
\(440\) 1.50757e40 0.160237
\(441\) −2.54648e40 −0.261301
\(442\) 6.90306e40 0.683906
\(443\) 6.37157e39 0.0609521 0.0304761 0.999535i \(-0.490298\pi\)
0.0304761 + 0.999535i \(0.490298\pi\)
\(444\) 3.44272e40 0.318028
\(445\) −1.54810e40 −0.138108
\(446\) −3.87840e40 −0.334166
\(447\) −7.59565e40 −0.632119
\(448\) 2.14013e40 0.172040
\(449\) −1.42862e41 −1.10943 −0.554715 0.832040i \(-0.687173\pi\)
−0.554715 + 0.832040i \(0.687173\pi\)
\(450\) 5.50787e39 0.0413228
\(451\) 9.33074e40 0.676362
\(452\) −1.83847e40 −0.128769
\(453\) 1.26630e41 0.857064
\(454\) −5.66918e38 −0.00370812
\(455\) 5.91418e40 0.373868
\(456\) 8.63456e40 0.527576
\(457\) −5.32708e40 −0.314621 −0.157311 0.987549i \(-0.550282\pi\)
−0.157311 + 0.987549i \(0.550282\pi\)
\(458\) 5.14895e40 0.293970
\(459\) 3.13625e41 1.73107
\(460\) −8.09804e40 −0.432149
\(461\) −1.55667e41 −0.803214 −0.401607 0.915812i \(-0.631548\pi\)
−0.401607 + 0.915812i \(0.631548\pi\)
\(462\) 1.66305e41 0.829762
\(463\) −2.90324e41 −1.40080 −0.700401 0.713749i \(-0.746996\pi\)
−0.700401 + 0.713749i \(0.746996\pi\)
\(464\) 7.14968e40 0.333625
\(465\) 9.08189e40 0.409879
\(466\) 2.08996e41 0.912340
\(467\) −1.49482e41 −0.631216 −0.315608 0.948890i \(-0.602209\pi\)
−0.315608 + 0.948890i \(0.602209\pi\)
\(468\) −2.17239e40 −0.0887416
\(469\) 4.46342e41 1.76396
\(470\) 1.65205e41 0.631695
\(471\) 4.63045e40 0.171317
\(472\) −7.44396e40 −0.266504
\(473\) −1.73142e40 −0.0599866
\(474\) 1.46189e41 0.490174
\(475\) −1.09309e41 −0.354735
\(476\) −3.48841e41 −1.09577
\(477\) −3.80263e39 −0.0115624
\(478\) −9.98509e40 −0.293913
\(479\) 9.00830e40 0.256710 0.128355 0.991728i \(-0.459030\pi\)
0.128355 + 0.991728i \(0.459030\pi\)
\(480\) −2.41066e40 −0.0665114
\(481\) 1.71898e41 0.459221
\(482\) −3.97472e40 −0.102820
\(483\) −8.93318e41 −2.23782
\(484\) 5.57081e39 0.0135150
\(485\) 9.12921e40 0.214504
\(486\) −1.75077e41 −0.398445
\(487\) 2.08879e41 0.460463 0.230232 0.973136i \(-0.426052\pi\)
0.230232 + 0.973136i \(0.426052\pi\)
\(488\) 8.96873e40 0.191524
\(489\) −2.63380e41 −0.544874
\(490\) −1.41093e41 −0.282792
\(491\) 5.18801e41 1.00748 0.503742 0.863854i \(-0.331956\pi\)
0.503742 + 0.863854i \(0.331956\pi\)
\(492\) −1.49202e41 −0.280747
\(493\) −1.16540e42 −2.12494
\(494\) 4.31130e41 0.761800
\(495\) 7.73325e40 0.132428
\(496\) 1.64093e41 0.272348
\(497\) −1.17028e42 −1.88263
\(498\) 7.30624e40 0.113930
\(499\) −3.17894e41 −0.480534 −0.240267 0.970707i \(-0.577235\pi\)
−0.240267 + 0.970707i \(0.577235\pi\)
\(500\) 3.05176e40 0.0447214
\(501\) −3.38712e41 −0.481223
\(502\) 1.71465e41 0.236193
\(503\) −2.67214e39 −0.00356908 −0.00178454 0.999998i \(-0.500568\pi\)
−0.00178454 + 0.999998i \(0.500568\pi\)
\(504\) 1.09780e41 0.142184
\(505\) 1.69719e41 0.213164
\(506\) −1.13699e42 −1.38492
\(507\) −4.49413e41 −0.530911
\(508\) −8.37575e41 −0.959701
\(509\) 1.25901e42 1.39928 0.699638 0.714498i \(-0.253345\pi\)
0.699638 + 0.714498i \(0.253345\pi\)
\(510\) 3.92938e41 0.423629
\(511\) −1.01741e42 −1.06408
\(512\) −4.35561e40 −0.0441942
\(513\) 1.95874e42 1.92823
\(514\) 9.04845e40 0.0864263
\(515\) −2.88231e41 −0.267134
\(516\) 2.76860e40 0.0248994
\(517\) 2.31954e42 2.02441
\(518\) −8.68672e41 −0.735774
\(519\) 1.74210e42 1.43212
\(520\) −1.20366e41 −0.0960400
\(521\) 1.93165e42 1.49604 0.748020 0.663676i \(-0.231005\pi\)
0.748020 + 0.663676i \(0.231005\pi\)
\(522\) 3.66750e41 0.275726
\(523\) 5.98151e41 0.436551 0.218275 0.975887i \(-0.429957\pi\)
0.218275 + 0.975887i \(0.429957\pi\)
\(524\) 7.71582e41 0.546698
\(525\) 3.36648e41 0.231583
\(526\) −8.15376e41 −0.544602
\(527\) −2.67472e42 −1.73466
\(528\) −3.38465e41 −0.213151
\(529\) 4.47228e42 2.73506
\(530\) −2.10693e40 −0.0125134
\(531\) −3.81846e41 −0.220254
\(532\) −2.17869e42 −1.22057
\(533\) −7.44976e41 −0.405387
\(534\) 3.47563e41 0.183715
\(535\) 1.03812e42 0.533043
\(536\) −9.08401e41 −0.453131
\(537\) −2.50836e42 −1.21560
\(538\) 2.83640e42 1.33549
\(539\) −1.98100e42 −0.906272
\(540\) −5.46856e41 −0.243092
\(541\) 9.64612e41 0.416673 0.208336 0.978057i \(-0.433195\pi\)
0.208336 + 0.978057i \(0.433195\pi\)
\(542\) 2.30904e41 0.0969266
\(543\) −1.92928e42 −0.787044
\(544\) 7.09966e41 0.281484
\(545\) 3.43717e41 0.132451
\(546\) −1.32779e42 −0.497330
\(547\) 4.34799e42 1.58302 0.791508 0.611159i \(-0.209297\pi\)
0.791508 + 0.611159i \(0.209297\pi\)
\(548\) 1.58623e42 0.561395
\(549\) 4.60060e41 0.158286
\(550\) 4.28478e41 0.143320
\(551\) −7.27849e42 −2.36696
\(552\) 1.81809e42 0.574857
\(553\) −3.68866e42 −1.13404
\(554\) −2.89280e42 −0.864804
\(555\) 9.78479e41 0.284453
\(556\) 5.83665e41 0.165008
\(557\) 4.75748e42 1.30804 0.654021 0.756476i \(-0.273081\pi\)
0.654021 + 0.756476i \(0.273081\pi\)
\(558\) 8.41732e41 0.225084
\(559\) 1.38238e41 0.0359538
\(560\) 6.08261e41 0.153878
\(561\) 5.51698e42 1.35762
\(562\) 8.40004e40 0.0201080
\(563\) −7.57947e42 −1.76506 −0.882529 0.470259i \(-0.844161\pi\)
−0.882529 + 0.470259i \(0.844161\pi\)
\(564\) −3.70903e42 −0.840299
\(565\) −5.22523e41 −0.115174
\(566\) 3.80689e42 0.816423
\(567\) −4.10530e42 −0.856658
\(568\) 2.38177e42 0.483615
\(569\) −1.13132e41 −0.0223536 −0.0111768 0.999938i \(-0.503558\pi\)
−0.0111768 + 0.999938i \(0.503558\pi\)
\(570\) 2.45409e42 0.471878
\(571\) −3.31794e41 −0.0620882 −0.0310441 0.999518i \(-0.509883\pi\)
−0.0310441 + 0.999518i \(0.509883\pi\)
\(572\) −1.68998e42 −0.307782
\(573\) 1.92544e42 0.341297
\(574\) 3.76468e42 0.649522
\(575\) −2.30160e42 −0.386526
\(576\) −2.23426e41 −0.0365245
\(577\) −1.34557e40 −0.00214131 −0.00107066 0.999999i \(-0.500341\pi\)
−0.00107066 + 0.999999i \(0.500341\pi\)
\(578\) −7.00823e42 −1.08574
\(579\) −8.51304e42 −1.28401
\(580\) 2.03206e42 0.298403
\(581\) −1.84352e42 −0.263584
\(582\) −2.04960e42 −0.285340
\(583\) −2.95821e41 −0.0401020
\(584\) 2.07065e42 0.273343
\(585\) −6.17430e41 −0.0793729
\(586\) 3.76614e42 0.471502
\(587\) 1.52213e42 0.185593 0.0927965 0.995685i \(-0.470419\pi\)
0.0927965 + 0.995685i \(0.470419\pi\)
\(588\) 3.16769e42 0.376178
\(589\) −1.67049e43 −1.93223
\(590\) −2.11570e42 −0.238368
\(591\) 4.13031e42 0.453290
\(592\) 1.76793e42 0.189008
\(593\) 1.84601e42 0.192259 0.0961296 0.995369i \(-0.469354\pi\)
0.0961296 + 0.995369i \(0.469354\pi\)
\(594\) −7.67806e42 −0.779043
\(595\) −9.91466e42 −0.980088
\(596\) −3.90058e42 −0.375675
\(597\) 9.33119e42 0.875659
\(598\) 9.07788e42 0.830072
\(599\) −9.39650e41 −0.0837241 −0.0418620 0.999123i \(-0.513329\pi\)
−0.0418620 + 0.999123i \(0.513329\pi\)
\(600\) −6.85150e41 −0.0594896
\(601\) 1.58484e43 1.34101 0.670505 0.741905i \(-0.266077\pi\)
0.670505 + 0.741905i \(0.266077\pi\)
\(602\) −6.98576e41 −0.0576061
\(603\) −4.65974e42 −0.374492
\(604\) 6.50278e42 0.509362
\(605\) 1.58332e41 0.0120882
\(606\) −3.81035e42 −0.283556
\(607\) 1.95801e43 1.42033 0.710167 0.704034i \(-0.248620\pi\)
0.710167 + 0.704034i \(0.248620\pi\)
\(608\) 4.43409e42 0.313544
\(609\) 2.24163e43 1.54524
\(610\) 2.54907e42 0.171304
\(611\) −1.85195e43 −1.21336
\(612\) 3.64184e42 0.232635
\(613\) 1.98433e43 1.23588 0.617939 0.786226i \(-0.287968\pi\)
0.617939 + 0.786226i \(0.287968\pi\)
\(614\) −3.35049e42 −0.203469
\(615\) −4.24056e42 −0.251107
\(616\) 8.54020e42 0.493136
\(617\) −2.01865e43 −1.13669 −0.568344 0.822791i \(-0.692416\pi\)
−0.568344 + 0.822791i \(0.692416\pi\)
\(618\) 6.47108e42 0.355349
\(619\) −1.28098e43 −0.686020 −0.343010 0.939332i \(-0.611447\pi\)
−0.343010 + 0.939332i \(0.611447\pi\)
\(620\) 4.66380e42 0.243595
\(621\) 4.12433e43 2.10104
\(622\) −1.79967e43 −0.894214
\(623\) −8.76977e42 −0.425034
\(624\) 2.70234e42 0.127755
\(625\) 8.67362e41 0.0400000
\(626\) −1.04254e43 −0.469020
\(627\) 3.44563e43 1.51224
\(628\) 2.37787e42 0.101815
\(629\) −2.88173e43 −1.20384
\(630\) 3.12014e42 0.127173
\(631\) −9.11677e42 −0.362565 −0.181283 0.983431i \(-0.558025\pi\)
−0.181283 + 0.983431i \(0.558025\pi\)
\(632\) 7.50720e42 0.291315
\(633\) 6.63668e42 0.251300
\(634\) −1.58829e43 −0.586876
\(635\) −2.38053e43 −0.858383
\(636\) 4.73028e41 0.0166457
\(637\) 1.58165e43 0.543187
\(638\) 2.85309e43 0.956301
\(639\) 1.22175e43 0.399687
\(640\) −1.23794e42 −0.0395285
\(641\) −7.56603e42 −0.235813 −0.117907 0.993025i \(-0.537618\pi\)
−0.117907 + 0.993025i \(0.537618\pi\)
\(642\) −2.33068e43 −0.709070
\(643\) 5.52599e43 1.64112 0.820559 0.571561i \(-0.193662\pi\)
0.820559 + 0.571561i \(0.193662\pi\)
\(644\) −4.58744e43 −1.32996
\(645\) 7.86882e41 0.0222707
\(646\) −7.22757e43 −1.99705
\(647\) −5.65084e43 −1.52439 −0.762197 0.647346i \(-0.775879\pi\)
−0.762197 + 0.647346i \(0.775879\pi\)
\(648\) 8.35516e42 0.220060
\(649\) −2.97052e43 −0.763905
\(650\) −3.42101e42 −0.0859008
\(651\) 5.14477e43 1.26142
\(652\) −1.35253e43 −0.323824
\(653\) −9.93457e42 −0.232271 −0.116135 0.993233i \(-0.537051\pi\)
−0.116135 + 0.993233i \(0.537051\pi\)
\(654\) −7.71678e42 −0.176190
\(655\) 2.19297e43 0.488981
\(656\) −7.66192e42 −0.166851
\(657\) 1.06216e43 0.225906
\(658\) 9.35868e43 1.94408
\(659\) −2.37067e43 −0.481003 −0.240501 0.970649i \(-0.577312\pi\)
−0.240501 + 0.970649i \(0.577312\pi\)
\(660\) −9.61975e42 −0.190648
\(661\) 3.30974e43 0.640725 0.320363 0.947295i \(-0.396195\pi\)
0.320363 + 0.947295i \(0.396195\pi\)
\(662\) −3.85420e43 −0.728845
\(663\) −4.40482e43 −0.813707
\(664\) 3.75196e42 0.0677100
\(665\) −6.19220e43 −1.09171
\(666\) 9.06879e42 0.156206
\(667\) −1.53256e44 −2.57909
\(668\) −1.73938e43 −0.285996
\(669\) 2.47479e43 0.397588
\(670\) −2.58183e43 −0.405292
\(671\) 3.57898e43 0.548985
\(672\) −1.36561e43 −0.204693
\(673\) 3.83414e43 0.561609 0.280805 0.959765i \(-0.409399\pi\)
0.280805 + 0.959765i \(0.409399\pi\)
\(674\) −1.09018e43 −0.156052
\(675\) −1.55426e43 −0.217428
\(676\) −2.30786e43 −0.315526
\(677\) 1.18717e44 1.58631 0.793153 0.609022i \(-0.208438\pi\)
0.793153 + 0.609022i \(0.208438\pi\)
\(678\) 1.17312e43 0.153208
\(679\) 5.17158e43 0.660149
\(680\) 2.01784e43 0.251767
\(681\) 3.61748e41 0.00441190
\(682\) 6.54815e43 0.780658
\(683\) −1.22953e44 −1.43291 −0.716453 0.697635i \(-0.754236\pi\)
−0.716453 + 0.697635i \(0.754236\pi\)
\(684\) 2.27451e43 0.259130
\(685\) 4.50835e43 0.502127
\(686\) 9.45020e42 0.102900
\(687\) −3.28552e43 −0.349764
\(688\) 1.42175e42 0.0147980
\(689\) 2.36187e42 0.0240357
\(690\) 5.16733e43 0.514168
\(691\) 1.32263e44 1.28686 0.643428 0.765507i \(-0.277512\pi\)
0.643428 + 0.765507i \(0.277512\pi\)
\(692\) 8.94617e43 0.851124
\(693\) 4.38079e43 0.407556
\(694\) 8.28075e43 0.753352
\(695\) 1.65888e43 0.147587
\(696\) −4.56218e43 −0.396944
\(697\) 1.24889e44 1.06272
\(698\) 9.49564e43 0.790252
\(699\) −1.33360e44 −1.08550
\(700\) 1.72878e43 0.137632
\(701\) 1.62662e44 1.26665 0.633325 0.773886i \(-0.281690\pi\)
0.633325 + 0.773886i \(0.281690\pi\)
\(702\) 6.13024e43 0.466930
\(703\) −1.79978e44 −1.34095
\(704\) −1.73811e43 −0.126678
\(705\) −1.05417e44 −0.751586
\(706\) −1.59587e44 −1.11308
\(707\) 9.61434e43 0.656022
\(708\) 4.74996e43 0.317084
\(709\) −2.46555e43 −0.161027 −0.0805133 0.996754i \(-0.525656\pi\)
−0.0805133 + 0.996754i \(0.525656\pi\)
\(710\) 6.76939e43 0.432559
\(711\) 3.85090e43 0.240759
\(712\) 1.78483e43 0.109184
\(713\) −3.51739e44 −2.10539
\(714\) 2.22594e44 1.30374
\(715\) −4.80322e43 −0.275289
\(716\) −1.28811e44 −0.722442
\(717\) 6.37144e43 0.349696
\(718\) 1.62198e44 0.871198
\(719\) −1.33187e44 −0.700108 −0.350054 0.936729i \(-0.613837\pi\)
−0.350054 + 0.936729i \(0.613837\pi\)
\(720\) −6.35014e42 −0.0326685
\(721\) −1.63279e44 −0.822119
\(722\) −3.07911e44 −1.51739
\(723\) 2.53625e43 0.122335
\(724\) −9.90742e43 −0.467749
\(725\) 5.77547e43 0.266900
\(726\) −3.55471e42 −0.0160800
\(727\) 3.69856e44 1.63775 0.818877 0.573969i \(-0.194597\pi\)
0.818877 + 0.573969i \(0.194597\pi\)
\(728\) −6.81858e43 −0.295568
\(729\) 2.58394e44 1.09649
\(730\) 5.88515e43 0.244486
\(731\) −2.31746e43 −0.0942523
\(732\) −5.72291e43 −0.227874
\(733\) −2.48074e44 −0.967095 −0.483547 0.875318i \(-0.660652\pi\)
−0.483547 + 0.875318i \(0.660652\pi\)
\(734\) −8.54174e43 −0.326030
\(735\) 9.00310e43 0.336464
\(736\) 9.33641e43 0.341644
\(737\) −3.62498e44 −1.29885
\(738\) −3.93026e43 −0.137895
\(739\) 7.67405e43 0.263655 0.131827 0.991273i \(-0.457916\pi\)
0.131827 + 0.991273i \(0.457916\pi\)
\(740\) 5.02476e43 0.169053
\(741\) −2.75103e44 −0.906384
\(742\) −1.19355e43 −0.0385106
\(743\) −5.47245e44 −1.72924 −0.864620 0.502427i \(-0.832441\pi\)
−0.864620 + 0.502427i \(0.832441\pi\)
\(744\) −1.04707e44 −0.324038
\(745\) −1.10861e44 −0.336014
\(746\) −1.68742e44 −0.500923
\(747\) 1.92461e43 0.0559594
\(748\) 2.83313e44 0.806847
\(749\) 5.88080e44 1.64047
\(750\) −1.94731e43 −0.0532092
\(751\) 4.53013e44 1.21253 0.606263 0.795264i \(-0.292668\pi\)
0.606263 + 0.795264i \(0.292668\pi\)
\(752\) −1.90469e44 −0.499399
\(753\) −1.09411e44 −0.281021
\(754\) −2.27793e44 −0.573173
\(755\) 1.84820e44 0.455587
\(756\) −3.09787e44 −0.748127
\(757\) −3.88934e44 −0.920215 −0.460108 0.887863i \(-0.652189\pi\)
−0.460108 + 0.887863i \(0.652189\pi\)
\(758\) 3.63354e44 0.842282
\(759\) 7.25511e44 1.64777
\(760\) 1.26024e44 0.280442
\(761\) −5.46740e44 −1.19212 −0.596058 0.802941i \(-0.703267\pi\)
−0.596058 + 0.802941i \(0.703267\pi\)
\(762\) 5.34453e44 1.14185
\(763\) 1.94711e44 0.407624
\(764\) 9.88765e43 0.202837
\(765\) 1.03507e44 0.208075
\(766\) −5.80250e43 −0.114306
\(767\) 2.37169e44 0.457858
\(768\) 2.77930e43 0.0525819
\(769\) −8.57160e44 −1.58929 −0.794646 0.607073i \(-0.792343\pi\)
−0.794646 + 0.607073i \(0.792343\pi\)
\(770\) 2.42727e44 0.441075
\(771\) −5.77378e43 −0.102829
\(772\) −4.37168e44 −0.763098
\(773\) −2.48088e44 −0.424447 −0.212224 0.977221i \(-0.568071\pi\)
−0.212224 + 0.977221i \(0.568071\pi\)
\(774\) 7.29302e42 0.0122299
\(775\) 1.32553e44 0.217878
\(776\) −1.05253e44 −0.169581
\(777\) 5.54296e44 0.875419
\(778\) −5.86467e44 −0.907946
\(779\) 7.79996e44 1.18375
\(780\) 7.68051e43 0.114268
\(781\) 9.50447e44 1.38623
\(782\) −1.52184e45 −2.17602
\(783\) −1.03493e45 −1.45078
\(784\) 1.62669e44 0.223567
\(785\) 6.75830e43 0.0910664
\(786\) −4.92343e44 −0.650457
\(787\) 7.42427e44 0.961713 0.480857 0.876799i \(-0.340326\pi\)
0.480857 + 0.876799i \(0.340326\pi\)
\(788\) 2.12103e44 0.269395
\(789\) 5.20288e44 0.647963
\(790\) 2.13368e44 0.260560
\(791\) −2.96003e44 −0.354454
\(792\) −8.91583e43 −0.104694
\(793\) −2.85749e44 −0.329042
\(794\) −4.68426e44 −0.528961
\(795\) 1.34443e43 0.0148883
\(796\) 4.79183e44 0.520413
\(797\) 6.95745e44 0.741047 0.370524 0.928823i \(-0.379178\pi\)
0.370524 + 0.928823i \(0.379178\pi\)
\(798\) 1.39021e45 1.45223
\(799\) 3.10465e45 3.18080
\(800\) −3.51844e43 −0.0353553
\(801\) 9.15549e43 0.0902355
\(802\) −5.21489e42 −0.00504130
\(803\) 8.26296e44 0.783510
\(804\) 5.79647e44 0.539132
\(805\) −1.30383e45 −1.18955
\(806\) −5.22811e44 −0.467898
\(807\) −1.80989e45 −1.58896
\(808\) −1.95672e44 −0.168521
\(809\) −5.39805e44 −0.456073 −0.228037 0.973653i \(-0.573231\pi\)
−0.228037 + 0.973653i \(0.573231\pi\)
\(810\) 2.37468e44 0.196828
\(811\) 6.12229e44 0.497841 0.248920 0.968524i \(-0.419924\pi\)
0.248920 + 0.968524i \(0.419924\pi\)
\(812\) 1.15114e45 0.918351
\(813\) −1.47339e44 −0.115323
\(814\) 7.05495e44 0.541771
\(815\) −3.84412e44 −0.289637
\(816\) −4.53026e44 −0.334908
\(817\) −1.44737e44 −0.104987
\(818\) −2.07720e44 −0.147844
\(819\) −3.49766e44 −0.244274
\(820\) −2.17765e44 −0.149236
\(821\) 2.89962e44 0.194995 0.0974973 0.995236i \(-0.468916\pi\)
0.0974973 + 0.995236i \(0.468916\pi\)
\(822\) −1.01217e45 −0.667944
\(823\) −9.30283e44 −0.602445 −0.301223 0.953554i \(-0.597395\pi\)
−0.301223 + 0.953554i \(0.597395\pi\)
\(824\) 3.32308e44 0.211188
\(825\) −2.73410e44 −0.170521
\(826\) −1.19852e45 −0.733590
\(827\) 4.60750e44 0.276777 0.138389 0.990378i \(-0.455808\pi\)
0.138389 + 0.990378i \(0.455808\pi\)
\(828\) 4.78921e44 0.282354
\(829\) −3.33913e45 −1.93214 −0.966069 0.258284i \(-0.916843\pi\)
−0.966069 + 0.258284i \(0.916843\pi\)
\(830\) 1.06637e44 0.0605617
\(831\) 1.84589e45 1.02894
\(832\) 1.38773e44 0.0759263
\(833\) −2.65151e45 −1.42396
\(834\) −3.72434e44 −0.196325
\(835\) −4.94362e44 −0.255803
\(836\) 1.76943e45 0.898742
\(837\) −2.37527e45 −1.18432
\(838\) 3.24203e44 0.158684
\(839\) −3.12484e45 −1.50147 −0.750737 0.660601i \(-0.770302\pi\)
−0.750737 + 0.660601i \(0.770302\pi\)
\(840\) −3.88129e44 −0.183083
\(841\) 1.68626e45 0.780886
\(842\) 1.31319e45 0.597021
\(843\) −5.36003e43 −0.0239243
\(844\) 3.40812e44 0.149351
\(845\) −6.55933e44 −0.282215
\(846\) −9.77030e44 −0.412731
\(847\) 8.96929e43 0.0372019
\(848\) 2.42913e43 0.00989270
\(849\) −2.42916e45 −0.971375
\(850\) 5.73506e44 0.225188
\(851\) −3.78962e45 −1.46113
\(852\) −1.51980e45 −0.575402
\(853\) −1.87985e44 −0.0698896 −0.0349448 0.999389i \(-0.511126\pi\)
−0.0349448 + 0.999389i \(0.511126\pi\)
\(854\) 1.44401e45 0.527198
\(855\) 6.46455e44 0.231773
\(856\) −1.19687e45 −0.421408
\(857\) 1.66635e45 0.576186 0.288093 0.957602i \(-0.406979\pi\)
0.288093 + 0.957602i \(0.406979\pi\)
\(858\) 1.07837e45 0.366197
\(859\) 1.58190e45 0.527576 0.263788 0.964581i \(-0.415028\pi\)
0.263788 + 0.964581i \(0.415028\pi\)
\(860\) 4.04086e43 0.0132357
\(861\) −2.40223e45 −0.772796
\(862\) 4.33589e45 1.36998
\(863\) 4.50883e45 1.39925 0.699627 0.714508i \(-0.253350\pi\)
0.699627 + 0.714508i \(0.253350\pi\)
\(864\) 6.30482e44 0.192181
\(865\) 2.54265e45 0.761268
\(866\) 2.52281e45 0.741922
\(867\) 4.47192e45 1.29181
\(868\) 2.64198e45 0.749678
\(869\) 2.99576e45 0.835026
\(870\) −1.29665e45 −0.355038
\(871\) 2.89422e45 0.778486
\(872\) −3.96278e44 −0.104712
\(873\) −5.39904e44 −0.140151
\(874\) −9.50462e45 −2.42386
\(875\) 4.91349e44 0.123102
\(876\) −1.32128e45 −0.325222
\(877\) −3.33310e45 −0.806036 −0.403018 0.915192i \(-0.632039\pi\)
−0.403018 + 0.915192i \(0.632039\pi\)
\(878\) 2.08719e45 0.495903
\(879\) −2.40316e45 −0.560990
\(880\) −4.94001e44 −0.113304
\(881\) 8.52925e45 1.92214 0.961068 0.276311i \(-0.0891121\pi\)
0.961068 + 0.276311i \(0.0891121\pi\)
\(882\) 8.34430e44 0.184768
\(883\) −3.84364e45 −0.836280 −0.418140 0.908383i \(-0.637318\pi\)
−0.418140 + 0.908383i \(0.637318\pi\)
\(884\) −2.26200e45 −0.483595
\(885\) 1.35002e45 0.283609
\(886\) −2.08784e44 −0.0430997
\(887\) 2.95824e45 0.600092 0.300046 0.953925i \(-0.402998\pi\)
0.300046 + 0.953925i \(0.402998\pi\)
\(888\) −1.12811e45 −0.224880
\(889\) −1.34854e46 −2.64172
\(890\) 5.07280e44 0.0976569
\(891\) 3.33413e45 0.630781
\(892\) 1.27087e45 0.236291
\(893\) 1.93900e46 3.54308
\(894\) 2.48894e45 0.446975
\(895\) −3.66104e45 −0.646171
\(896\) −7.01277e44 −0.121651
\(897\) −5.79256e45 −0.987614
\(898\) 4.68132e45 0.784485
\(899\) 8.82627e45 1.45379
\(900\) −1.80482e44 −0.0292196
\(901\) −3.95948e44 −0.0630092
\(902\) −3.05750e45 −0.478261
\(903\) 4.45759e44 0.0685393
\(904\) 6.02428e44 0.0910531
\(905\) −2.81586e45 −0.418367
\(906\) −4.14940e45 −0.606036
\(907\) −8.06932e44 −0.115858 −0.0579288 0.998321i \(-0.518450\pi\)
−0.0579288 + 0.998321i \(0.518450\pi\)
\(908\) 1.85768e43 0.00262204
\(909\) −1.00372e45 −0.139275
\(910\) −1.93796e45 −0.264364
\(911\) −5.29138e45 −0.709634 −0.354817 0.934936i \(-0.615457\pi\)
−0.354817 + 0.934936i \(0.615457\pi\)
\(912\) −2.82937e45 −0.373053
\(913\) 1.49722e45 0.194084
\(914\) 1.74558e45 0.222471
\(915\) −1.62655e45 −0.203817
\(916\) −1.68721e45 −0.207868
\(917\) 1.24229e46 1.50487
\(918\) −1.02769e46 −1.22405
\(919\) −1.26770e46 −1.48465 −0.742326 0.670039i \(-0.766278\pi\)
−0.742326 + 0.670039i \(0.766278\pi\)
\(920\) 2.65357e45 0.305576
\(921\) 2.13793e45 0.242086
\(922\) 5.10089e45 0.567958
\(923\) −7.58846e45 −0.830859
\(924\) −5.44947e45 −0.586730
\(925\) 1.42812e45 0.151206
\(926\) 9.51332e45 0.990517
\(927\) 1.70461e45 0.174538
\(928\) −2.34281e45 −0.235908
\(929\) −1.42045e46 −1.40664 −0.703320 0.710873i \(-0.748300\pi\)
−0.703320 + 0.710873i \(0.748300\pi\)
\(930\) −2.97595e45 −0.289828
\(931\) −1.65600e46 −1.58614
\(932\) −6.84839e45 −0.645122
\(933\) 1.14836e46 1.06393
\(934\) 4.89824e45 0.446337
\(935\) 8.05223e45 0.721666
\(936\) 7.11849e44 0.0627498
\(937\) −5.11233e45 −0.443256 −0.221628 0.975131i \(-0.571137\pi\)
−0.221628 + 0.975131i \(0.571137\pi\)
\(938\) −1.46257e46 −1.24731
\(939\) 6.65242e45 0.558037
\(940\) −5.41345e45 −0.446676
\(941\) −3.09574e45 −0.251262 −0.125631 0.992077i \(-0.540095\pi\)
−0.125631 + 0.992077i \(0.540095\pi\)
\(942\) −1.51731e45 −0.121139
\(943\) 1.64236e46 1.28984
\(944\) 2.43924e45 0.188447
\(945\) −8.80468e45 −0.669145
\(946\) 5.67351e44 0.0424169
\(947\) 9.70650e45 0.713901 0.356951 0.934123i \(-0.383816\pi\)
0.356951 + 0.934123i \(0.383816\pi\)
\(948\) −4.79032e45 −0.346605
\(949\) −6.59723e45 −0.469608
\(950\) 3.58183e45 0.250835
\(951\) 1.01348e46 0.698261
\(952\) 1.14308e46 0.774827
\(953\) −1.66762e46 −1.11213 −0.556067 0.831137i \(-0.687690\pi\)
−0.556067 + 0.831137i \(0.687690\pi\)
\(954\) 1.24605e44 0.00817588
\(955\) 2.81024e45 0.181422
\(956\) 3.27191e45 0.207828
\(957\) −1.82054e46 −1.13780
\(958\) −2.95184e45 −0.181521
\(959\) 2.55392e46 1.54532
\(960\) 7.89924e44 0.0470307
\(961\) 3.18808e45 0.186774
\(962\) −5.63274e45 −0.324718
\(963\) −6.13945e45 −0.348275
\(964\) 1.30244e45 0.0727047
\(965\) −1.24251e46 −0.682536
\(966\) 2.92722e46 1.58238
\(967\) −1.87173e46 −0.995712 −0.497856 0.867260i \(-0.665879\pi\)
−0.497856 + 0.867260i \(0.665879\pi\)
\(968\) −1.82544e44 −0.00955652
\(969\) 4.61188e46 2.37607
\(970\) −2.99146e45 −0.151677
\(971\) −7.35947e45 −0.367239 −0.183619 0.982997i \(-0.558781\pi\)
−0.183619 + 0.982997i \(0.558781\pi\)
\(972\) 5.73694e45 0.281743
\(973\) 9.39732e45 0.454208
\(974\) −6.84454e45 −0.325597
\(975\) 2.18293e45 0.102204
\(976\) −2.93887e45 −0.135428
\(977\) 2.82554e46 1.28155 0.640775 0.767728i \(-0.278613\pi\)
0.640775 + 0.767728i \(0.278613\pi\)
\(978\) 8.63044e45 0.385284
\(979\) 7.12240e45 0.312964
\(980\) 4.62334e45 0.199964
\(981\) −2.03275e45 −0.0865395
\(982\) −1.70001e46 −0.712399
\(983\) −7.33102e45 −0.302402 −0.151201 0.988503i \(-0.548314\pi\)
−0.151201 + 0.988503i \(0.548314\pi\)
\(984\) 4.88904e45 0.198518
\(985\) 6.02833e45 0.240955
\(986\) 3.81878e46 1.50256
\(987\) −5.97173e46 −2.31305
\(988\) −1.41273e46 −0.538674
\(989\) −3.04757e45 −0.114396
\(990\) −2.53403e45 −0.0936411
\(991\) −4.55867e46 −1.65843 −0.829213 0.558933i \(-0.811211\pi\)
−0.829213 + 0.558933i \(0.811211\pi\)
\(992\) −5.37700e45 −0.192579
\(993\) 2.45935e46 0.867174
\(994\) 3.83477e46 1.33122
\(995\) 1.36192e46 0.465472
\(996\) −2.39411e45 −0.0805609
\(997\) 9.50456e45 0.314889 0.157444 0.987528i \(-0.449674\pi\)
0.157444 + 0.987528i \(0.449674\pi\)
\(998\) 1.04168e46 0.339789
\(999\) −2.55911e46 −0.821909
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 10.32.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.32.a.a.1.2 2 1.1 even 1 trivial