Properties

Label 10.28.a.c.1.1
Level $10$
Weight $28$
Character 10.1
Self dual yes
Analytic conductor $46.186$
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] = [10,28,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, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 28 \)
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: \(46.1855574838\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{711649}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 177912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(422.296\) 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)
 
\(f(q)\) \(=\) \(q+8192.00 q^{2} -5.48964e6 q^{3} +6.71089e7 q^{4} -1.22070e9 q^{5} -4.49712e10 q^{6} -8.15752e10 q^{7} +5.49756e11 q^{8} +2.25106e13 q^{9} -1.00000e13 q^{10} +1.33950e13 q^{11} -3.68404e14 q^{12} +5.64686e14 q^{13} -6.68264e14 q^{14} +6.70123e15 q^{15} +4.50360e15 q^{16} -1.28911e16 q^{17} +1.84407e17 q^{18} +3.07151e17 q^{19} -8.19200e16 q^{20} +4.47819e17 q^{21} +1.09732e17 q^{22} -1.99871e18 q^{23} -3.01796e18 q^{24} +1.49012e18 q^{25} +4.62590e18 q^{26} -8.17133e19 q^{27} -5.47442e18 q^{28} -8.80094e19 q^{29} +5.48964e19 q^{30} +2.22760e20 q^{31} +3.68935e19 q^{32} -7.35337e19 q^{33} -1.05604e20 q^{34} +9.95791e19 q^{35} +1.51066e21 q^{36} -1.61882e21 q^{37} +2.51618e21 q^{38} -3.09992e21 q^{39} -6.71089e20 q^{40} -9.68899e21 q^{41} +3.66853e21 q^{42} +9.33483e21 q^{43} +8.98922e20 q^{44} -2.74787e22 q^{45} -1.63734e22 q^{46} +1.20564e22 q^{47} -2.47232e22 q^{48} -5.90579e22 q^{49} +1.22070e22 q^{50} +7.07676e22 q^{51} +3.78954e22 q^{52} -8.75697e21 q^{53} -6.69395e23 q^{54} -1.63513e22 q^{55} -4.48464e22 q^{56} -1.68615e24 q^{57} -7.20973e23 q^{58} +3.03895e23 q^{59} +4.49712e23 q^{60} +1.15214e24 q^{61} +1.82485e24 q^{62} -1.83631e24 q^{63} +3.02231e23 q^{64} -6.89313e23 q^{65} -6.02388e23 q^{66} -1.86933e24 q^{67} -8.65108e23 q^{68} +1.09722e25 q^{69} +8.15752e23 q^{70} -8.93880e24 q^{71} +1.23753e25 q^{72} -2.18673e25 q^{73} -1.32614e25 q^{74} -8.18021e24 q^{75} +2.06126e25 q^{76} -1.09270e24 q^{77} -2.53946e25 q^{78} +1.05795e25 q^{79} -5.49756e24 q^{80} +2.76920e26 q^{81} -7.93722e25 q^{82} +2.25868e25 q^{83} +3.00526e25 q^{84} +1.57362e25 q^{85} +7.64710e25 q^{86} +4.83140e26 q^{87} +7.36397e24 q^{88} -1.78891e26 q^{89} -2.25106e26 q^{90} -4.60643e25 q^{91} -1.34131e26 q^{92} -1.22287e27 q^{93} +9.87663e25 q^{94} -3.74941e26 q^{95} -2.02532e26 q^{96} +8.87997e26 q^{97} -4.83802e26 q^{98} +3.01529e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16384 q^{2} - 4702956 q^{3} + 134217728 q^{4} - 2441406250 q^{5} - 38526615552 q^{6} - 57185041508 q^{7} + 1099511627776 q^{8} + 15503869642194 q^{9} - 20000000000000 q^{10} + 169851430699104 q^{11}+ \cdots - 79\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8192.00 0.707107
\(3\) −5.48964e6 −1.98796 −0.993979 0.109571i \(-0.965052\pi\)
−0.993979 + 0.109571i \(0.965052\pi\)
\(4\) 6.71089e7 0.500000
\(5\) −1.22070e9 −0.447214
\(6\) −4.49712e10 −1.40570
\(7\) −8.15752e10 −0.318225 −0.159113 0.987260i \(-0.550863\pi\)
−0.159113 + 0.987260i \(0.550863\pi\)
\(8\) 5.49756e11 0.353553
\(9\) 2.25106e13 2.95198
\(10\) −1.00000e13 −0.316228
\(11\) 1.33950e13 0.116988 0.0584939 0.998288i \(-0.481370\pi\)
0.0584939 + 0.998288i \(0.481370\pi\)
\(12\) −3.68404e14 −0.993979
\(13\) 5.64686e14 0.517096 0.258548 0.965998i \(-0.416756\pi\)
0.258548 + 0.965998i \(0.416756\pi\)
\(14\) −6.68264e14 −0.225019
\(15\) 6.70123e15 0.889042
\(16\) 4.50360e15 0.250000
\(17\) −1.28911e16 −0.315668 −0.157834 0.987466i \(-0.550451\pi\)
−0.157834 + 0.987466i \(0.550451\pi\)
\(18\) 1.84407e17 2.08736
\(19\) 3.07151e17 1.67563 0.837816 0.545952i \(-0.183832\pi\)
0.837816 + 0.545952i \(0.183832\pi\)
\(20\) −8.19200e16 −0.223607
\(21\) 4.47819e17 0.632618
\(22\) 1.09732e17 0.0827229
\(23\) −1.99871e18 −0.826843 −0.413422 0.910540i \(-0.635666\pi\)
−0.413422 + 0.910540i \(0.635666\pi\)
\(24\) −3.01796e18 −0.702849
\(25\) 1.49012e18 0.200000
\(26\) 4.62590e18 0.365642
\(27\) −8.17133e19 −3.88045
\(28\) −5.47442e18 −0.159113
\(29\) −8.80094e19 −1.59278 −0.796390 0.604783i \(-0.793260\pi\)
−0.796390 + 0.604783i \(0.793260\pi\)
\(30\) 5.48964e19 0.628648
\(31\) 2.22760e20 1.63853 0.819265 0.573415i \(-0.194382\pi\)
0.819265 + 0.573415i \(0.194382\pi\)
\(32\) 3.68935e19 0.176777
\(33\) −7.35337e19 −0.232567
\(34\) −1.05604e20 −0.223211
\(35\) 9.95791e19 0.142315
\(36\) 1.51066e21 1.47599
\(37\) −1.61882e21 −1.09264 −0.546319 0.837577i \(-0.683971\pi\)
−0.546319 + 0.837577i \(0.683971\pi\)
\(38\) 2.51618e21 1.18485
\(39\) −3.09992e21 −1.02797
\(40\) −6.71089e20 −0.158114
\(41\) −9.68899e21 −1.63567 −0.817837 0.575450i \(-0.804827\pi\)
−0.817837 + 0.575450i \(0.804827\pi\)
\(42\) 3.66853e21 0.447329
\(43\) 9.33483e21 0.828481 0.414240 0.910167i \(-0.364047\pi\)
0.414240 + 0.910167i \(0.364047\pi\)
\(44\) 8.98922e20 0.0584939
\(45\) −2.74787e22 −1.32016
\(46\) −1.63734e22 −0.584667
\(47\) 1.20564e22 0.322031 0.161016 0.986952i \(-0.448523\pi\)
0.161016 + 0.986952i \(0.448523\pi\)
\(48\) −2.47232e22 −0.496990
\(49\) −5.90579e22 −0.898733
\(50\) 1.22070e22 0.141421
\(51\) 7.07676e22 0.627534
\(52\) 3.78954e22 0.258548
\(53\) −8.75697e21 −0.0461987 −0.0230993 0.999733i \(-0.507353\pi\)
−0.0230993 + 0.999733i \(0.507353\pi\)
\(54\) −6.69395e23 −2.74389
\(55\) −1.63513e22 −0.0523186
\(56\) −4.48464e22 −0.112510
\(57\) −1.68615e24 −3.33109
\(58\) −7.20973e23 −1.12627
\(59\) 3.03895e23 0.376896 0.188448 0.982083i \(-0.439654\pi\)
0.188448 + 0.982083i \(0.439654\pi\)
\(60\) 4.49712e23 0.444521
\(61\) 1.15214e24 0.911070 0.455535 0.890218i \(-0.349448\pi\)
0.455535 + 0.890218i \(0.349448\pi\)
\(62\) 1.82485e24 1.15862
\(63\) −1.83631e24 −0.939393
\(64\) 3.02231e23 0.125000
\(65\) −6.89313e23 −0.231253
\(66\) −6.02388e23 −0.164450
\(67\) −1.86933e24 −0.416559 −0.208280 0.978069i \(-0.566786\pi\)
−0.208280 + 0.978069i \(0.566786\pi\)
\(68\) −8.65108e23 −0.157834
\(69\) 1.09722e25 1.64373
\(70\) 8.15752e23 0.100632
\(71\) −8.93880e24 −0.910524 −0.455262 0.890357i \(-0.650454\pi\)
−0.455262 + 0.890357i \(0.650454\pi\)
\(72\) 1.23753e25 1.04368
\(73\) −2.18673e25 −1.53086 −0.765432 0.643517i \(-0.777475\pi\)
−0.765432 + 0.643517i \(0.777475\pi\)
\(74\) −1.32614e25 −0.772612
\(75\) −8.18021e24 −0.397592
\(76\) 2.06126e25 0.837816
\(77\) −1.09270e24 −0.0372285
\(78\) −2.53946e25 −0.726882
\(79\) 1.05795e25 0.254975 0.127488 0.991840i \(-0.459309\pi\)
0.127488 + 0.991840i \(0.459309\pi\)
\(80\) −5.49756e24 −0.111803
\(81\) 2.76920e26 4.76219
\(82\) −7.93722e25 −1.15660
\(83\) 2.25868e25 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(84\) 3.00526e25 0.316309
\(85\) 1.57362e25 0.141171
\(86\) 7.64710e25 0.585824
\(87\) 4.83140e26 3.16638
\(88\) 7.36397e24 0.0413615
\(89\) −1.78891e26 −0.862626 −0.431313 0.902202i \(-0.641950\pi\)
−0.431313 + 0.902202i \(0.641950\pi\)
\(90\) −2.25106e26 −0.933497
\(91\) −4.60643e25 −0.164553
\(92\) −1.34131e26 −0.413422
\(93\) −1.22287e27 −3.25733
\(94\) 9.87663e25 0.227710
\(95\) −3.74941e26 −0.749366
\(96\) −2.02532e26 −0.351425
\(97\) 8.87997e26 1.33966 0.669828 0.742517i \(-0.266368\pi\)
0.669828 + 0.742517i \(0.266368\pi\)
\(98\) −4.83802e26 −0.635500
\(99\) 3.01529e26 0.345346
\(100\) 1.00000e26 0.100000
\(101\) 5.69733e26 0.498119 0.249059 0.968488i \(-0.419879\pi\)
0.249059 + 0.968488i \(0.419879\pi\)
\(102\) 5.79728e26 0.443733
\(103\) −1.77162e27 −1.18869 −0.594343 0.804212i \(-0.702588\pi\)
−0.594343 + 0.804212i \(0.702588\pi\)
\(104\) 3.10439e26 0.182821
\(105\) −5.46654e26 −0.282915
\(106\) −7.17371e25 −0.0326674
\(107\) −2.46061e27 −0.987102 −0.493551 0.869717i \(-0.664301\pi\)
−0.493551 + 0.869717i \(0.664301\pi\)
\(108\) −5.48369e27 −1.94022
\(109\) 3.19335e27 0.997674 0.498837 0.866696i \(-0.333761\pi\)
0.498837 + 0.866696i \(0.333761\pi\)
\(110\) −1.33950e26 −0.0369948
\(111\) 8.88677e27 2.17212
\(112\) −3.67382e26 −0.0795563
\(113\) −7.75073e26 −0.148862 −0.0744310 0.997226i \(-0.523714\pi\)
−0.0744310 + 0.997226i \(0.523714\pi\)
\(114\) −1.38130e28 −2.35543
\(115\) 2.43983e27 0.369776
\(116\) −5.90621e27 −0.796390
\(117\) 1.27114e28 1.52646
\(118\) 2.48951e27 0.266506
\(119\) 1.05159e27 0.100453
\(120\) 3.68404e27 0.314324
\(121\) −1.29306e28 −0.986314
\(122\) 9.43831e27 0.644224
\(123\) 5.31891e28 3.25165
\(124\) 1.49492e28 0.819265
\(125\) −1.81899e27 −0.0894427
\(126\) −1.50430e28 −0.664251
\(127\) 1.11798e28 0.443695 0.221848 0.975081i \(-0.428791\pi\)
0.221848 + 0.975081i \(0.428791\pi\)
\(128\) 2.47588e27 0.0883883
\(129\) −5.12449e28 −1.64699
\(130\) −5.64686e27 −0.163520
\(131\) 8.48344e27 0.221518 0.110759 0.993847i \(-0.464672\pi\)
0.110759 + 0.993847i \(0.464672\pi\)
\(132\) −4.93476e27 −0.116283
\(133\) −2.50559e28 −0.533228
\(134\) −1.53136e28 −0.294552
\(135\) 9.97477e28 1.73539
\(136\) −7.08696e27 −0.111605
\(137\) −8.14420e28 −1.16177 −0.580886 0.813985i \(-0.697294\pi\)
−0.580886 + 0.813985i \(0.697294\pi\)
\(138\) 8.98842e28 1.16229
\(139\) −4.14318e28 −0.485997 −0.242999 0.970027i \(-0.578131\pi\)
−0.242999 + 0.970027i \(0.578131\pi\)
\(140\) 6.68264e27 0.0711573
\(141\) −6.61855e28 −0.640184
\(142\) −7.32267e28 −0.643838
\(143\) 7.56395e27 0.0604940
\(144\) 1.01379e29 0.737994
\(145\) 1.07433e29 0.712313
\(146\) −1.79137e29 −1.08248
\(147\) 3.24207e29 1.78664
\(148\) −1.08637e29 −0.546319
\(149\) −2.60623e29 −1.19674 −0.598368 0.801221i \(-0.704184\pi\)
−0.598368 + 0.801221i \(0.704184\pi\)
\(150\) −6.70123e28 −0.281140
\(151\) −2.46658e29 −0.946034 −0.473017 0.881053i \(-0.656835\pi\)
−0.473017 + 0.881053i \(0.656835\pi\)
\(152\) 1.68858e29 0.592426
\(153\) −2.90187e29 −0.931843
\(154\) −8.95138e27 −0.0263245
\(155\) −2.71924e29 −0.732773
\(156\) −2.08032e29 −0.513983
\(157\) −6.53188e29 −1.48045 −0.740223 0.672361i \(-0.765280\pi\)
−0.740223 + 0.672361i \(0.765280\pi\)
\(158\) 8.66671e28 0.180295
\(159\) 4.80726e28 0.0918410
\(160\) −4.50360e28 −0.0790569
\(161\) 1.63045e29 0.263122
\(162\) 2.26853e30 3.36738
\(163\) −4.95480e29 −0.676851 −0.338426 0.940993i \(-0.609894\pi\)
−0.338426 + 0.940993i \(0.609894\pi\)
\(164\) −6.50217e29 −0.817837
\(165\) 8.97628e28 0.104007
\(166\) 1.85031e29 0.197600
\(167\) −3.99121e29 −0.393036 −0.196518 0.980500i \(-0.562963\pi\)
−0.196518 + 0.980500i \(0.562963\pi\)
\(168\) 2.46191e29 0.223664
\(169\) −8.73663e29 −0.732611
\(170\) 1.28911e29 0.0998228
\(171\) 6.91416e30 4.94643
\(172\) 6.26450e29 0.414240
\(173\) −1.45903e30 −0.892160 −0.446080 0.894993i \(-0.647180\pi\)
−0.446080 + 0.894993i \(0.647180\pi\)
\(174\) 3.95788e30 2.23897
\(175\) −1.21556e29 −0.0636450
\(176\) 6.03256e28 0.0292470
\(177\) −1.66828e30 −0.749254
\(178\) −1.46547e30 −0.609969
\(179\) 3.95662e29 0.152689 0.0763446 0.997081i \(-0.475675\pi\)
0.0763446 + 0.997081i \(0.475675\pi\)
\(180\) −1.84407e30 −0.660082
\(181\) −1.52183e30 −0.505481 −0.252740 0.967534i \(-0.581332\pi\)
−0.252740 + 0.967534i \(0.581332\pi\)
\(182\) −3.77359e29 −0.116357
\(183\) −6.32483e30 −1.81117
\(184\) −1.09880e30 −0.292333
\(185\) 1.97610e30 0.488643
\(186\) −1.00178e31 −2.30328
\(187\) −1.72676e29 −0.0369293
\(188\) 8.09094e29 0.161016
\(189\) 6.66578e30 1.23486
\(190\) −3.07151e30 −0.529882
\(191\) 2.06556e29 0.0331961 0.0165980 0.999862i \(-0.494716\pi\)
0.0165980 + 0.999862i \(0.494716\pi\)
\(192\) −1.65914e30 −0.248495
\(193\) 8.12733e30 1.13481 0.567407 0.823438i \(-0.307947\pi\)
0.567407 + 0.823438i \(0.307947\pi\)
\(194\) 7.27447e30 0.947279
\(195\) 3.78409e30 0.459720
\(196\) −3.96331e30 −0.449366
\(197\) −1.15879e31 −1.22662 −0.613311 0.789841i \(-0.710163\pi\)
−0.613311 + 0.789841i \(0.710163\pi\)
\(198\) 2.47012e30 0.244196
\(199\) −1.61914e31 −1.49543 −0.747717 0.664018i \(-0.768850\pi\)
−0.747717 + 0.664018i \(0.768850\pi\)
\(200\) 8.19200e29 0.0707107
\(201\) 1.02620e31 0.828103
\(202\) 4.66725e30 0.352223
\(203\) 7.17938e30 0.506863
\(204\) 4.74913e30 0.313767
\(205\) 1.18274e31 0.731495
\(206\) −1.45131e31 −0.840528
\(207\) −4.49921e31 −2.44082
\(208\) 2.54312e30 0.129274
\(209\) 4.11429e30 0.196029
\(210\) −4.47819e30 −0.200051
\(211\) −8.13808e30 −0.340964 −0.170482 0.985361i \(-0.554532\pi\)
−0.170482 + 0.985361i \(0.554532\pi\)
\(212\) −5.87670e29 −0.0230993
\(213\) 4.90708e31 1.81008
\(214\) −2.01573e31 −0.697986
\(215\) −1.13951e31 −0.370508
\(216\) −4.49224e31 −1.37195
\(217\) −1.81717e31 −0.521422
\(218\) 2.61599e31 0.705462
\(219\) 1.20044e32 3.04329
\(220\) −1.09732e30 −0.0261593
\(221\) −7.27943e30 −0.163231
\(222\) 7.28004e31 1.53592
\(223\) −4.12421e31 −0.818890 −0.409445 0.912335i \(-0.634278\pi\)
−0.409445 + 0.912335i \(0.634278\pi\)
\(224\) −3.00959e30 −0.0562548
\(225\) 3.35434e31 0.590395
\(226\) −6.34940e30 −0.105261
\(227\) 8.27912e31 1.29311 0.646555 0.762868i \(-0.276209\pi\)
0.646555 + 0.762868i \(0.276209\pi\)
\(228\) −1.13156e32 −1.66554
\(229\) 9.00491e31 1.24940 0.624698 0.780866i \(-0.285222\pi\)
0.624698 + 0.780866i \(0.285222\pi\)
\(230\) 1.99871e31 0.261471
\(231\) 5.99852e30 0.0740087
\(232\) −4.83837e31 −0.563133
\(233\) 3.39840e31 0.373224 0.186612 0.982434i \(-0.440249\pi\)
0.186612 + 0.982434i \(0.440249\pi\)
\(234\) 1.04132e32 1.07937
\(235\) −1.47173e31 −0.144017
\(236\) 2.03941e31 0.188448
\(237\) −5.80775e31 −0.506880
\(238\) 8.61467e30 0.0710313
\(239\) −1.37775e32 −1.07349 −0.536747 0.843743i \(-0.680347\pi\)
−0.536747 + 0.843743i \(0.680347\pi\)
\(240\) 3.01796e31 0.222260
\(241\) −8.67658e31 −0.604114 −0.302057 0.953290i \(-0.597673\pi\)
−0.302057 + 0.953290i \(0.597673\pi\)
\(242\) −1.05927e32 −0.697429
\(243\) −8.97081e32 −5.58659
\(244\) 7.73187e31 0.455535
\(245\) 7.20921e31 0.401926
\(246\) 4.35725e32 2.29926
\(247\) 1.73444e32 0.866464
\(248\) 1.22464e32 0.579308
\(249\) −1.23994e32 −0.555531
\(250\) −1.49012e31 −0.0632456
\(251\) 4.76708e32 1.91716 0.958578 0.284829i \(-0.0919367\pi\)
0.958578 + 0.284829i \(0.0919367\pi\)
\(252\) −1.23232e32 −0.469697
\(253\) −2.67726e31 −0.0967307
\(254\) 9.15852e31 0.313740
\(255\) −8.63862e31 −0.280642
\(256\) 2.02824e31 0.0625000
\(257\) 3.36363e32 0.983357 0.491678 0.870777i \(-0.336384\pi\)
0.491678 + 0.870777i \(0.336384\pi\)
\(258\) −4.19798e32 −1.16459
\(259\) 1.32056e32 0.347705
\(260\) −4.62590e31 −0.115626
\(261\) −1.98114e33 −4.70185
\(262\) 6.94964e31 0.156637
\(263\) −1.96857e32 −0.421453 −0.210726 0.977545i \(-0.567583\pi\)
−0.210726 + 0.977545i \(0.567583\pi\)
\(264\) −4.04256e31 −0.0822248
\(265\) 1.06897e31 0.0206607
\(266\) −2.05258e32 −0.377049
\(267\) 9.82046e32 1.71486
\(268\) −1.25449e32 −0.208280
\(269\) 9.12489e32 1.44069 0.720347 0.693613i \(-0.243982\pi\)
0.720347 + 0.693613i \(0.243982\pi\)
\(270\) 8.17133e32 1.22711
\(271\) 3.57579e32 0.510841 0.255421 0.966830i \(-0.417786\pi\)
0.255421 + 0.966830i \(0.417786\pi\)
\(272\) −5.80564e31 −0.0789169
\(273\) 2.52877e32 0.327125
\(274\) −6.67173e32 −0.821496
\(275\) 1.99601e31 0.0233976
\(276\) 7.36331e32 0.821865
\(277\) −1.49389e33 −1.58797 −0.793986 0.607936i \(-0.791998\pi\)
−0.793986 + 0.607936i \(0.791998\pi\)
\(278\) −3.39409e32 −0.343652
\(279\) 5.01446e33 4.83690
\(280\) 5.47442e31 0.0503158
\(281\) −6.43291e32 −0.563472 −0.281736 0.959492i \(-0.590910\pi\)
−0.281736 + 0.959492i \(0.590910\pi\)
\(282\) −5.42192e32 −0.452679
\(283\) 4.79228e32 0.381439 0.190720 0.981645i \(-0.438918\pi\)
0.190720 + 0.981645i \(0.438918\pi\)
\(284\) −5.99873e32 −0.455262
\(285\) 2.05829e33 1.48971
\(286\) 6.19639e31 0.0427757
\(287\) 7.90381e32 0.520512
\(288\) 8.30494e32 0.521841
\(289\) −1.50153e33 −0.900354
\(290\) 8.80094e32 0.503682
\(291\) −4.87479e33 −2.66318
\(292\) −1.46749e33 −0.765432
\(293\) −3.27376e32 −0.163055 −0.0815275 0.996671i \(-0.525980\pi\)
−0.0815275 + 0.996671i \(0.525980\pi\)
\(294\) 2.65590e33 1.26335
\(295\) −3.70966e32 −0.168553
\(296\) −8.89958e32 −0.386306
\(297\) −1.09455e33 −0.453965
\(298\) −2.13503e33 −0.846221
\(299\) −1.12864e33 −0.427558
\(300\) −5.48964e32 −0.198796
\(301\) −7.61491e32 −0.263643
\(302\) −2.02062e33 −0.668947
\(303\) −3.12763e33 −0.990239
\(304\) 1.38329e33 0.418908
\(305\) −1.40642e33 −0.407443
\(306\) −2.37721e33 −0.658913
\(307\) 8.73502e32 0.231684 0.115842 0.993268i \(-0.463043\pi\)
0.115842 + 0.993268i \(0.463043\pi\)
\(308\) −7.33297e31 −0.0186142
\(309\) 9.72554e33 2.36306
\(310\) −2.22760e33 −0.518149
\(311\) 3.78077e33 0.842005 0.421002 0.907059i \(-0.361678\pi\)
0.421002 + 0.907059i \(0.361678\pi\)
\(312\) −1.70420e33 −0.363441
\(313\) −1.46656e33 −0.299537 −0.149769 0.988721i \(-0.547853\pi\)
−0.149769 + 0.988721i \(0.547853\pi\)
\(314\) −5.35091e33 −1.04683
\(315\) 2.24158e33 0.420110
\(316\) 7.09976e32 0.127488
\(317\) −7.26388e31 −0.0124988 −0.00624940 0.999980i \(-0.501989\pi\)
−0.00624940 + 0.999980i \(0.501989\pi\)
\(318\) 3.93811e32 0.0649414
\(319\) −1.17888e33 −0.186336
\(320\) −3.68935e32 −0.0559017
\(321\) 1.35079e34 1.96232
\(322\) 1.33566e33 0.186056
\(323\) −3.95952e33 −0.528943
\(324\) 1.85838e34 2.38110
\(325\) 8.41447e32 0.103419
\(326\) −4.05897e33 −0.478606
\(327\) −1.75303e34 −1.98333
\(328\) −5.32658e33 −0.578298
\(329\) −9.83506e32 −0.102478
\(330\) 7.35337e32 0.0735441
\(331\) 1.22753e34 1.17857 0.589284 0.807926i \(-0.299410\pi\)
0.589284 + 0.807926i \(0.299410\pi\)
\(332\) 1.51578e33 0.139724
\(333\) −3.64407e34 −3.22544
\(334\) −3.26960e33 −0.277918
\(335\) 2.28190e33 0.186291
\(336\) 2.01680e33 0.158155
\(337\) 3.74086e33 0.281817 0.140909 0.990023i \(-0.454998\pi\)
0.140909 + 0.990023i \(0.454998\pi\)
\(338\) −7.15705e33 −0.518034
\(339\) 4.25487e33 0.295931
\(340\) 1.05604e33 0.0705854
\(341\) 2.98387e33 0.191688
\(342\) 5.66408e34 3.49765
\(343\) 1.01782e34 0.604225
\(344\) 5.13188e33 0.292912
\(345\) −1.33938e34 −0.735098
\(346\) −1.19524e34 −0.630852
\(347\) −5.08016e33 −0.257887 −0.128943 0.991652i \(-0.541159\pi\)
−0.128943 + 0.991652i \(0.541159\pi\)
\(348\) 3.24230e34 1.58319
\(349\) 3.72143e34 1.74810 0.874050 0.485835i \(-0.161485\pi\)
0.874050 + 0.485835i \(0.161485\pi\)
\(350\) −9.95791e32 −0.0450038
\(351\) −4.61423e34 −2.00657
\(352\) 4.94188e32 0.0206807
\(353\) 1.32758e34 0.534690 0.267345 0.963601i \(-0.413854\pi\)
0.267345 + 0.963601i \(0.413854\pi\)
\(354\) −1.36665e34 −0.529803
\(355\) 1.09116e34 0.407199
\(356\) −1.20052e34 −0.431313
\(357\) −5.77288e33 −0.199697
\(358\) 3.24126e33 0.107968
\(359\) −4.06074e34 −1.30266 −0.651329 0.758795i \(-0.725788\pi\)
−0.651329 + 0.758795i \(0.725788\pi\)
\(360\) −1.51066e34 −0.466749
\(361\) 6.07413e34 1.80774
\(362\) −1.24668e34 −0.357429
\(363\) 7.09842e34 1.96075
\(364\) −3.09132e33 −0.0822765
\(365\) 2.66935e34 0.684623
\(366\) −5.18130e34 −1.28069
\(367\) 4.81487e33 0.114708 0.0573539 0.998354i \(-0.481734\pi\)
0.0573539 + 0.998354i \(0.481734\pi\)
\(368\) −9.00137e33 −0.206711
\(369\) −2.18105e35 −4.82847
\(370\) 1.61882e34 0.345523
\(371\) 7.14351e32 0.0147016
\(372\) −8.20657e34 −1.62866
\(373\) −2.48819e34 −0.476228 −0.238114 0.971237i \(-0.576529\pi\)
−0.238114 + 0.971237i \(0.576529\pi\)
\(374\) −1.41456e33 −0.0261129
\(375\) 9.98560e33 0.177808
\(376\) 6.62810e33 0.113855
\(377\) −4.96976e34 −0.823621
\(378\) 5.46061e34 0.873175
\(379\) 2.13394e34 0.329270 0.164635 0.986355i \(-0.447355\pi\)
0.164635 + 0.986355i \(0.447355\pi\)
\(380\) −2.51618e34 −0.374683
\(381\) −6.13733e34 −0.882048
\(382\) 1.69210e33 0.0234732
\(383\) 9.26670e34 1.24091 0.620457 0.784241i \(-0.286947\pi\)
0.620457 + 0.784241i \(0.286947\pi\)
\(384\) −1.35917e34 −0.175712
\(385\) 1.33386e33 0.0166491
\(386\) 6.65791e34 0.802434
\(387\) 2.10133e35 2.44566
\(388\) 5.95925e34 0.669828
\(389\) 2.66544e34 0.289368 0.144684 0.989478i \(-0.453784\pi\)
0.144684 + 0.989478i \(0.453784\pi\)
\(390\) 3.09992e34 0.325071
\(391\) 2.57656e34 0.261008
\(392\) −3.24674e34 −0.317750
\(393\) −4.65711e34 −0.440369
\(394\) −9.49281e34 −0.867353
\(395\) −1.29144e34 −0.114028
\(396\) 2.02353e34 0.172673
\(397\) 6.20560e34 0.511814 0.255907 0.966701i \(-0.417626\pi\)
0.255907 + 0.966701i \(0.417626\pi\)
\(398\) −1.32640e35 −1.05743
\(399\) 1.37548e35 1.06004
\(400\) 6.71089e33 0.0500000
\(401\) −3.83625e34 −0.276349 −0.138174 0.990408i \(-0.544123\pi\)
−0.138174 + 0.990408i \(0.544123\pi\)
\(402\) 8.40661e34 0.585557
\(403\) 1.25789e35 0.847278
\(404\) 3.82341e34 0.249059
\(405\) −3.38037e35 −2.12972
\(406\) 5.88135e34 0.358406
\(407\) −2.16841e34 −0.127825
\(408\) 3.89049e34 0.221867
\(409\) 6.77841e33 0.0373993 0.0186996 0.999825i \(-0.494047\pi\)
0.0186996 + 0.999825i \(0.494047\pi\)
\(410\) 9.68899e34 0.517245
\(411\) 4.47087e35 2.30955
\(412\) −1.18891e35 −0.594343
\(413\) −2.47903e34 −0.119938
\(414\) −3.68575e35 −1.72592
\(415\) −2.75718e34 −0.124973
\(416\) 2.08332e34 0.0914106
\(417\) 2.27446e35 0.966142
\(418\) 3.37042e34 0.138613
\(419\) −1.94310e35 −0.773760 −0.386880 0.922130i \(-0.626447\pi\)
−0.386880 + 0.922130i \(0.626447\pi\)
\(420\) −3.66853e34 −0.141458
\(421\) 1.96151e35 0.732457 0.366229 0.930525i \(-0.380649\pi\)
0.366229 + 0.930525i \(0.380649\pi\)
\(422\) −6.66671e34 −0.241098
\(423\) 2.71397e35 0.950629
\(424\) −4.81419e33 −0.0163337
\(425\) −1.92093e34 −0.0631335
\(426\) 4.01988e35 1.27992
\(427\) −9.39859e34 −0.289925
\(428\) −1.65129e35 −0.493551
\(429\) −4.15234e34 −0.120260
\(430\) −9.33483e34 −0.261989
\(431\) −4.47018e33 −0.0121585 −0.00607927 0.999982i \(-0.501935\pi\)
−0.00607927 + 0.999982i \(0.501935\pi\)
\(432\) −3.68004e35 −0.970112
\(433\) −3.75519e35 −0.959501 −0.479751 0.877405i \(-0.659273\pi\)
−0.479751 + 0.877405i \(0.659273\pi\)
\(434\) −1.48863e35 −0.368701
\(435\) −5.89771e35 −1.41605
\(436\) 2.14302e35 0.498837
\(437\) −6.13905e35 −1.38549
\(438\) 9.83398e35 2.15193
\(439\) −2.32360e35 −0.493050 −0.246525 0.969136i \(-0.579289\pi\)
−0.246525 + 0.969136i \(0.579289\pi\)
\(440\) −8.98922e33 −0.0184974
\(441\) −1.32943e36 −2.65304
\(442\) −5.96331e34 −0.115421
\(443\) −9.07441e35 −1.70360 −0.851801 0.523865i \(-0.824490\pi\)
−0.851801 + 0.523865i \(0.824490\pi\)
\(444\) 5.96381e35 1.08606
\(445\) 2.18372e35 0.385778
\(446\) −3.37855e35 −0.579042
\(447\) 1.43073e36 2.37906
\(448\) −2.46546e34 −0.0397781
\(449\) −7.86327e35 −1.23106 −0.615528 0.788115i \(-0.711057\pi\)
−0.615528 + 0.788115i \(0.711057\pi\)
\(450\) 2.74787e35 0.417473
\(451\) −1.29784e35 −0.191354
\(452\) −5.20143e34 −0.0744310
\(453\) 1.35407e36 1.88068
\(454\) 6.78226e35 0.914366
\(455\) 5.62309e34 0.0735904
\(456\) −9.26972e35 −1.17772
\(457\) −7.49955e35 −0.925053 −0.462526 0.886606i \(-0.653057\pi\)
−0.462526 + 0.886606i \(0.653057\pi\)
\(458\) 7.37682e35 0.883457
\(459\) 1.05338e36 1.22493
\(460\) 1.63734e35 0.184888
\(461\) −1.56743e35 −0.171880 −0.0859399 0.996300i \(-0.527389\pi\)
−0.0859399 + 0.996300i \(0.527389\pi\)
\(462\) 4.91399e34 0.0523320
\(463\) −5.52219e35 −0.571173 −0.285587 0.958353i \(-0.592188\pi\)
−0.285587 + 0.958353i \(0.592188\pi\)
\(464\) −3.96359e35 −0.398195
\(465\) 1.49277e36 1.45672
\(466\) 2.78397e35 0.263909
\(467\) −1.27404e36 −1.17329 −0.586643 0.809845i \(-0.699551\pi\)
−0.586643 + 0.809845i \(0.699551\pi\)
\(468\) 8.53048e35 0.763228
\(469\) 1.52491e35 0.132560
\(470\) −1.20564e35 −0.101835
\(471\) 3.58577e36 2.94306
\(472\) 1.67068e35 0.133253
\(473\) 1.25040e35 0.0969222
\(474\) −4.75771e35 −0.358419
\(475\) 4.57691e35 0.335127
\(476\) 7.05713e34 0.0502267
\(477\) −1.97125e35 −0.136377
\(478\) −1.12866e36 −0.759076
\(479\) −9.37975e35 −0.613285 −0.306642 0.951825i \(-0.599206\pi\)
−0.306642 + 0.951825i \(0.599206\pi\)
\(480\) 2.47232e35 0.157162
\(481\) −9.14127e35 −0.564999
\(482\) −7.10786e35 −0.427173
\(483\) −8.95058e35 −0.523076
\(484\) −8.67756e35 −0.493157
\(485\) −1.08398e36 −0.599112
\(486\) −7.34888e36 −3.95032
\(487\) 1.24632e35 0.0651610 0.0325805 0.999469i \(-0.489627\pi\)
0.0325805 + 0.999469i \(0.489627\pi\)
\(488\) 6.33395e35 0.322112
\(489\) 2.72001e36 1.34555
\(490\) 5.90579e35 0.284204
\(491\) 2.13166e36 0.997972 0.498986 0.866610i \(-0.333706\pi\)
0.498986 + 0.866610i \(0.333706\pi\)
\(492\) 3.56946e36 1.62583
\(493\) 1.13454e36 0.502789
\(494\) 1.42085e36 0.612682
\(495\) −3.68077e35 −0.154443
\(496\) 1.00322e36 0.409633
\(497\) 7.29184e35 0.289752
\(498\) −1.01576e36 −0.392820
\(499\) 1.17967e36 0.444020 0.222010 0.975044i \(-0.428738\pi\)
0.222010 + 0.975044i \(0.428738\pi\)
\(500\) −1.22070e35 −0.0447214
\(501\) 2.19103e36 0.781338
\(502\) 3.90520e36 1.35563
\(503\) −4.40912e36 −1.48999 −0.744996 0.667069i \(-0.767548\pi\)
−0.744996 + 0.667069i \(0.767548\pi\)
\(504\) −1.00952e36 −0.332126
\(505\) −6.95475e35 −0.222765
\(506\) −2.19321e35 −0.0683989
\(507\) 4.79610e36 1.45640
\(508\) 7.50266e35 0.221848
\(509\) −1.66203e36 −0.478574 −0.239287 0.970949i \(-0.576914\pi\)
−0.239287 + 0.970949i \(0.576914\pi\)
\(510\) −7.07676e35 −0.198444
\(511\) 1.78383e36 0.487160
\(512\) 1.66153e35 0.0441942
\(513\) −2.50984e37 −6.50221
\(514\) 2.75549e36 0.695338
\(515\) 2.16262e36 0.531596
\(516\) −3.43899e36 −0.823493
\(517\) 1.61496e35 0.0376737
\(518\) 1.08180e36 0.245865
\(519\) 8.00957e36 1.77358
\(520\) −3.78954e35 −0.0817601
\(521\) −1.69142e36 −0.355583 −0.177792 0.984068i \(-0.556895\pi\)
−0.177792 + 0.984068i \(0.556895\pi\)
\(522\) −1.62295e37 −3.32471
\(523\) 4.63272e36 0.924832 0.462416 0.886663i \(-0.346983\pi\)
0.462416 + 0.886663i \(0.346983\pi\)
\(524\) 5.69314e35 0.110759
\(525\) 6.67302e35 0.126524
\(526\) −1.61265e36 −0.298012
\(527\) −2.87163e36 −0.517231
\(528\) −3.31166e35 −0.0581417
\(529\) −1.84838e36 −0.316330
\(530\) 8.75697e34 0.0146093
\(531\) 6.84086e36 1.11259
\(532\) −1.68147e36 −0.266614
\(533\) −5.47124e36 −0.845801
\(534\) 8.04492e36 1.21259
\(535\) 3.00368e36 0.441445
\(536\) −1.02768e36 −0.147276
\(537\) −2.17204e36 −0.303540
\(538\) 7.47511e36 1.01873
\(539\) −7.91079e35 −0.105141
\(540\) 6.69395e36 0.867695
\(541\) 1.22619e37 1.55022 0.775112 0.631823i \(-0.217693\pi\)
0.775112 + 0.631823i \(0.217693\pi\)
\(542\) 2.92928e36 0.361219
\(543\) 8.35429e36 1.00487
\(544\) −4.75598e35 −0.0558027
\(545\) −3.89813e36 −0.446173
\(546\) 2.07157e36 0.231312
\(547\) 1.26741e37 1.38066 0.690330 0.723494i \(-0.257465\pi\)
0.690330 + 0.723494i \(0.257465\pi\)
\(548\) −5.46548e36 −0.580886
\(549\) 2.59353e37 2.68946
\(550\) 1.63513e35 0.0165446
\(551\) −2.70322e37 −2.66892
\(552\) 6.03202e36 0.581146
\(553\) −8.63023e35 −0.0811396
\(554\) −1.22380e37 −1.12287
\(555\) −1.08481e37 −0.971401
\(556\) −2.78044e36 −0.242999
\(557\) −5.14205e36 −0.438623 −0.219312 0.975655i \(-0.570381\pi\)
−0.219312 + 0.975655i \(0.570381\pi\)
\(558\) 4.10785e37 3.42021
\(559\) 5.27125e36 0.428404
\(560\) 4.48464e35 0.0355787
\(561\) 9.47931e35 0.0734139
\(562\) −5.26984e36 −0.398435
\(563\) 1.08130e37 0.798147 0.399074 0.916919i \(-0.369332\pi\)
0.399074 + 0.916919i \(0.369332\pi\)
\(564\) −4.44164e36 −0.320092
\(565\) 9.46134e35 0.0665731
\(566\) 3.92583e36 0.269718
\(567\) −2.25898e37 −1.51545
\(568\) −4.91416e36 −0.321919
\(569\) −1.71706e37 −1.09843 −0.549213 0.835682i \(-0.685073\pi\)
−0.549213 + 0.835682i \(0.685073\pi\)
\(570\) 1.68615e37 1.05338
\(571\) −2.73444e37 −1.66833 −0.834163 0.551518i \(-0.814049\pi\)
−0.834163 + 0.551518i \(0.814049\pi\)
\(572\) 5.07608e35 0.0302470
\(573\) −1.13392e36 −0.0659924
\(574\) 6.47480e36 0.368058
\(575\) −2.97831e36 −0.165369
\(576\) 6.80341e36 0.368997
\(577\) −3.20240e36 −0.169669 −0.0848345 0.996395i \(-0.527036\pi\)
−0.0848345 + 0.996395i \(0.527036\pi\)
\(578\) −1.23005e37 −0.636646
\(579\) −4.46162e37 −2.25596
\(580\) 7.20973e36 0.356157
\(581\) −1.84253e36 −0.0889275
\(582\) −3.99343e37 −1.88315
\(583\) −1.17299e35 −0.00540469
\(584\) −1.20217e37 −0.541242
\(585\) −1.55169e37 −0.682652
\(586\) −2.68186e36 −0.115297
\(587\) −5.30209e36 −0.222758 −0.111379 0.993778i \(-0.535527\pi\)
−0.111379 + 0.993778i \(0.535527\pi\)
\(588\) 2.17571e37 0.893321
\(589\) 6.84211e37 2.74558
\(590\) −3.03895e36 −0.119185
\(591\) 6.36135e37 2.43847
\(592\) −7.29054e36 −0.273160
\(593\) −1.26381e37 −0.462855 −0.231428 0.972852i \(-0.574340\pi\)
−0.231428 + 0.972852i \(0.574340\pi\)
\(594\) −8.96654e36 −0.321002
\(595\) −1.28369e36 −0.0449241
\(596\) −1.74901e37 −0.598368
\(597\) 8.88849e37 2.97286
\(598\) −9.24583e36 −0.302329
\(599\) −2.17161e37 −0.694255 −0.347128 0.937818i \(-0.612843\pi\)
−0.347128 + 0.937818i \(0.612843\pi\)
\(600\) −4.49712e36 −0.140570
\(601\) −2.82821e37 −0.864382 −0.432191 0.901782i \(-0.642259\pi\)
−0.432191 + 0.901782i \(0.642259\pi\)
\(602\) −6.23813e36 −0.186424
\(603\) −4.20798e37 −1.22967
\(604\) −1.65529e37 −0.473017
\(605\) 1.57844e37 0.441093
\(606\) −2.56215e37 −0.700204
\(607\) 4.82470e37 1.28951 0.644753 0.764391i \(-0.276960\pi\)
0.644753 + 0.764391i \(0.276960\pi\)
\(608\) 1.13319e37 0.296213
\(609\) −3.94122e37 −1.00762
\(610\) −1.15214e37 −0.288106
\(611\) 6.80810e36 0.166521
\(612\) −1.94741e37 −0.465922
\(613\) −4.26300e37 −0.997698 −0.498849 0.866689i \(-0.666244\pi\)
−0.498849 + 0.866689i \(0.666244\pi\)
\(614\) 7.15573e36 0.163825
\(615\) −6.49281e37 −1.45418
\(616\) −6.00717e35 −0.0131623
\(617\) 7.03658e37 1.50838 0.754192 0.656655i \(-0.228029\pi\)
0.754192 + 0.656655i \(0.228029\pi\)
\(618\) 7.96716e37 1.67093
\(619\) −6.05562e37 −1.24261 −0.621305 0.783569i \(-0.713397\pi\)
−0.621305 + 0.783569i \(0.713397\pi\)
\(620\) −1.82485e37 −0.366387
\(621\) 1.63321e38 3.20852
\(622\) 3.09720e37 0.595387
\(623\) 1.45930e37 0.274509
\(624\) −1.39608e37 −0.256991
\(625\) 2.22045e36 0.0400000
\(626\) −1.20140e37 −0.211805
\(627\) −2.25860e37 −0.389697
\(628\) −4.38347e37 −0.740223
\(629\) 2.08684e37 0.344910
\(630\) 1.83631e37 0.297062
\(631\) 9.89033e37 1.56608 0.783041 0.621970i \(-0.213668\pi\)
0.783041 + 0.621970i \(0.213668\pi\)
\(632\) 5.81613e36 0.0901474
\(633\) 4.46751e37 0.677822
\(634\) −5.95057e35 −0.00883799
\(635\) −1.36473e37 −0.198427
\(636\) 3.22610e36 0.0459205
\(637\) −3.33491e37 −0.464731
\(638\) −9.65742e36 −0.131759
\(639\) −2.01218e38 −2.68785
\(640\) −3.02231e36 −0.0395285
\(641\) 5.95420e37 0.762500 0.381250 0.924472i \(-0.375494\pi\)
0.381250 + 0.924472i \(0.375494\pi\)
\(642\) 1.10657e38 1.38757
\(643\) 2.01207e36 0.0247056 0.0123528 0.999924i \(-0.496068\pi\)
0.0123528 + 0.999924i \(0.496068\pi\)
\(644\) 1.09418e37 0.131561
\(645\) 6.25548e37 0.736554
\(646\) −3.24364e37 −0.374019
\(647\) 3.99414e37 0.451040 0.225520 0.974239i \(-0.427592\pi\)
0.225520 + 0.974239i \(0.427592\pi\)
\(648\) 1.52238e38 1.68369
\(649\) 4.07067e36 0.0440923
\(650\) 6.89313e36 0.0731285
\(651\) 9.97561e37 1.03656
\(652\) −3.32511e37 −0.338426
\(653\) 1.78987e37 0.178440 0.0892201 0.996012i \(-0.471563\pi\)
0.0892201 + 0.996012i \(0.471563\pi\)
\(654\) −1.43609e38 −1.40243
\(655\) −1.03558e37 −0.0990660
\(656\) −4.36353e37 −0.408918
\(657\) −4.92246e38 −4.51908
\(658\) −8.05688e36 −0.0724632
\(659\) −5.06340e36 −0.0446159 −0.0223079 0.999751i \(-0.507101\pi\)
−0.0223079 + 0.999751i \(0.507101\pi\)
\(660\) 6.02388e36 0.0520036
\(661\) −2.02410e38 −1.71203 −0.856016 0.516949i \(-0.827068\pi\)
−0.856016 + 0.516949i \(0.827068\pi\)
\(662\) 1.00559e38 0.833373
\(663\) 3.99615e37 0.324495
\(664\) 1.24173e37 0.0987999
\(665\) 3.05858e37 0.238467
\(666\) −2.98522e38 −2.28073
\(667\) 1.75905e38 1.31698
\(668\) −2.67845e37 −0.196518
\(669\) 2.26405e38 1.62792
\(670\) 1.86933e37 0.131728
\(671\) 1.54329e37 0.106584
\(672\) 1.65216e37 0.111832
\(673\) 7.53924e36 0.0500178 0.0250089 0.999687i \(-0.492039\pi\)
0.0250089 + 0.999687i \(0.492039\pi\)
\(674\) 3.06451e37 0.199275
\(675\) −1.21762e38 −0.776090
\(676\) −5.86306e37 −0.366306
\(677\) 1.83489e38 1.12373 0.561866 0.827228i \(-0.310084\pi\)
0.561866 + 0.827228i \(0.310084\pi\)
\(678\) 3.48559e37 0.209255
\(679\) −7.24385e37 −0.426312
\(680\) 8.65108e36 0.0499114
\(681\) −4.54494e38 −2.57065
\(682\) 2.44438e37 0.135544
\(683\) 2.34491e38 1.27481 0.637407 0.770527i \(-0.280007\pi\)
0.637407 + 0.770527i \(0.280007\pi\)
\(684\) 4.64001e38 2.47321
\(685\) 9.94165e37 0.519560
\(686\) 8.33794e37 0.427251
\(687\) −4.94337e38 −2.48375
\(688\) 4.20404e37 0.207120
\(689\) −4.94493e36 −0.0238892
\(690\) −1.09722e38 −0.519793
\(691\) 6.34902e37 0.294953 0.147477 0.989066i \(-0.452885\pi\)
0.147477 + 0.989066i \(0.452885\pi\)
\(692\) −9.79141e37 −0.446080
\(693\) −2.45973e37 −0.109898
\(694\) −4.16166e37 −0.182354
\(695\) 5.05759e37 0.217345
\(696\) 2.65609e38 1.11948
\(697\) 1.24902e38 0.516329
\(698\) 3.04859e38 1.23609
\(699\) −1.86560e38 −0.741953
\(700\) −8.15752e36 −0.0318225
\(701\) 2.26958e38 0.868465 0.434233 0.900801i \(-0.357020\pi\)
0.434233 + 0.900801i \(0.357020\pi\)
\(702\) −3.77998e38 −1.41886
\(703\) −4.97224e38 −1.83086
\(704\) 4.04838e36 0.0146235
\(705\) 8.07929e37 0.286299
\(706\) 1.08755e38 0.378083
\(707\) −4.64761e37 −0.158514
\(708\) −1.11956e38 −0.374627
\(709\) −4.92739e37 −0.161768 −0.0808840 0.996724i \(-0.525774\pi\)
−0.0808840 + 0.996724i \(0.525774\pi\)
\(710\) 8.93880e37 0.287933
\(711\) 2.38150e38 0.752682
\(712\) −9.83462e37 −0.304984
\(713\) −4.45232e38 −1.35481
\(714\) −4.72914e37 −0.141207
\(715\) −9.23334e36 −0.0270537
\(716\) 2.65524e37 0.0763446
\(717\) 7.56338e38 2.13406
\(718\) −3.32656e38 −0.921118
\(719\) −1.73047e38 −0.470246 −0.235123 0.971966i \(-0.575549\pi\)
−0.235123 + 0.971966i \(0.575549\pi\)
\(720\) −1.23753e38 −0.330041
\(721\) 1.44520e38 0.378270
\(722\) 4.97593e38 1.27827
\(723\) 4.76314e38 1.20095
\(724\) −1.02128e38 −0.252740
\(725\) −1.31144e38 −0.318556
\(726\) 5.81503e38 1.38646
\(727\) 7.52465e37 0.176105 0.0880525 0.996116i \(-0.471936\pi\)
0.0880525 + 0.996116i \(0.471936\pi\)
\(728\) −2.53241e37 −0.0581783
\(729\) 2.81297e39 6.34371
\(730\) 2.18673e38 0.484102
\(731\) −1.20336e38 −0.261525
\(732\) −4.24452e38 −0.905584
\(733\) −5.84393e37 −0.122406 −0.0612028 0.998125i \(-0.519494\pi\)
−0.0612028 + 0.998125i \(0.519494\pi\)
\(734\) 3.94434e37 0.0811106
\(735\) −3.95760e38 −0.799011
\(736\) −7.37393e37 −0.146167
\(737\) −2.50397e37 −0.0487324
\(738\) −1.78672e39 −3.41424
\(739\) −2.75465e38 −0.516851 −0.258426 0.966031i \(-0.583204\pi\)
−0.258426 + 0.966031i \(0.583204\pi\)
\(740\) 1.32614e38 0.244321
\(741\) −9.52145e38 −1.72249
\(742\) 5.85197e36 0.0103956
\(743\) 6.91625e38 1.20648 0.603242 0.797558i \(-0.293875\pi\)
0.603242 + 0.797558i \(0.293875\pi\)
\(744\) −6.72282e38 −1.15164
\(745\) 3.18144e38 0.535197
\(746\) −2.03833e38 −0.336744
\(747\) 5.08443e38 0.824925
\(748\) −1.15881e37 −0.0184646
\(749\) 2.00725e38 0.314121
\(750\) 8.18021e37 0.125730
\(751\) −9.44712e38 −1.42613 −0.713067 0.701096i \(-0.752694\pi\)
−0.713067 + 0.701096i \(0.752694\pi\)
\(752\) 5.42974e37 0.0805078
\(753\) −2.61696e39 −3.81123
\(754\) −4.07123e38 −0.582388
\(755\) 3.01096e38 0.423079
\(756\) 4.47333e38 0.617428
\(757\) −1.05837e39 −1.43497 −0.717484 0.696575i \(-0.754706\pi\)
−0.717484 + 0.696575i \(0.754706\pi\)
\(758\) 1.74812e38 0.232829
\(759\) 1.46972e38 0.192296
\(760\) −2.06126e38 −0.264941
\(761\) 6.48621e38 0.819026 0.409513 0.912304i \(-0.365699\pi\)
0.409513 + 0.912304i \(0.365699\pi\)
\(762\) −5.02770e38 −0.623702
\(763\) −2.60498e38 −0.317485
\(764\) 1.38617e37 0.0165980
\(765\) 3.54232e38 0.416733
\(766\) 7.59128e38 0.877458
\(767\) 1.71605e38 0.194892
\(768\) −1.11343e38 −0.124247
\(769\) −5.73159e38 −0.628449 −0.314224 0.949349i \(-0.601744\pi\)
−0.314224 + 0.949349i \(0.601744\pi\)
\(770\) 1.09270e37 0.0117727
\(771\) −1.84651e39 −1.95487
\(772\) 5.45416e38 0.567407
\(773\) −7.89580e38 −0.807185 −0.403592 0.914939i \(-0.632239\pi\)
−0.403592 + 0.914939i \(0.632239\pi\)
\(774\) 1.72141e39 1.72934
\(775\) 3.31938e38 0.327706
\(776\) 4.88181e38 0.473640
\(777\) −7.24940e38 −0.691223
\(778\) 2.18353e38 0.204614
\(779\) −2.97599e39 −2.74079
\(780\) 2.53946e38 0.229860
\(781\) −1.19735e38 −0.106520
\(782\) 2.11071e38 0.184560
\(783\) 7.19154e39 6.18070
\(784\) −2.65973e38 −0.224683
\(785\) 7.97348e38 0.662076
\(786\) −3.81510e38 −0.311388
\(787\) 1.73852e39 1.39483 0.697414 0.716668i \(-0.254334\pi\)
0.697414 + 0.716668i \(0.254334\pi\)
\(788\) −7.77651e38 −0.613311
\(789\) 1.08068e39 0.837830
\(790\) −1.05795e38 −0.0806303
\(791\) 6.32267e37 0.0473716
\(792\) 1.65767e38 0.122098
\(793\) 6.50596e38 0.471111
\(794\) 5.08363e38 0.361907
\(795\) −5.86824e37 −0.0410726
\(796\) −1.08658e39 −0.747717
\(797\) 1.88301e39 1.27399 0.636994 0.770869i \(-0.280178\pi\)
0.636994 + 0.770869i \(0.280178\pi\)
\(798\) 1.12679e39 0.749559
\(799\) −1.55421e38 −0.101655
\(800\) 5.49756e37 0.0353553
\(801\) −4.02693e39 −2.54645
\(802\) −3.14265e38 −0.195408
\(803\) −2.92912e38 −0.179093
\(804\) 6.88669e38 0.414051
\(805\) −1.99029e38 −0.117672
\(806\) 1.03047e39 0.599116
\(807\) −5.00924e39 −2.86404
\(808\) 3.13214e38 0.176111
\(809\) −1.60495e38 −0.0887474 −0.0443737 0.999015i \(-0.514129\pi\)
−0.0443737 + 0.999015i \(0.514129\pi\)
\(810\) −2.76920e39 −1.50594
\(811\) 2.12672e39 1.13744 0.568721 0.822531i \(-0.307439\pi\)
0.568721 + 0.822531i \(0.307439\pi\)
\(812\) 4.81800e38 0.253431
\(813\) −1.96298e39 −1.01553
\(814\) −1.77636e38 −0.0903862
\(815\) 6.04834e38 0.302697
\(816\) 3.18709e38 0.156883
\(817\) 2.86721e39 1.38823
\(818\) 5.55287e37 0.0264453
\(819\) −1.03694e39 −0.485757
\(820\) 7.93722e38 0.365748
\(821\) −2.13403e39 −0.967317 −0.483658 0.875257i \(-0.660692\pi\)
−0.483658 + 0.875257i \(0.660692\pi\)
\(822\) 3.66254e39 1.63310
\(823\) 1.46819e39 0.643997 0.321998 0.946740i \(-0.395645\pi\)
0.321998 + 0.946740i \(0.395645\pi\)
\(824\) −9.73956e38 −0.420264
\(825\) −1.09574e38 −0.0465134
\(826\) −2.03082e38 −0.0848089
\(827\) 1.74523e39 0.717016 0.358508 0.933527i \(-0.383286\pi\)
0.358508 + 0.933527i \(0.383286\pi\)
\(828\) −3.01937e39 −1.22041
\(829\) −1.03362e39 −0.411031 −0.205515 0.978654i \(-0.565887\pi\)
−0.205515 + 0.978654i \(0.565887\pi\)
\(830\) −2.25868e38 −0.0883693
\(831\) 8.20094e39 3.15682
\(832\) 1.70666e38 0.0646370
\(833\) 7.61321e38 0.283701
\(834\) 1.86323e39 0.683165
\(835\) 4.87208e38 0.175771
\(836\) 2.76105e38 0.0980144
\(837\) −1.82025e40 −6.35823
\(838\) −1.59179e39 −0.547131
\(839\) 1.58461e39 0.535966 0.267983 0.963424i \(-0.413643\pi\)
0.267983 + 0.963424i \(0.413643\pi\)
\(840\) −3.00526e38 −0.100026
\(841\) 4.69252e39 1.53695
\(842\) 1.60687e39 0.517926
\(843\) 3.53144e39 1.12016
\(844\) −5.46137e38 −0.170482
\(845\) 1.06648e39 0.327634
\(846\) 2.22329e39 0.672196
\(847\) 1.05481e39 0.313870
\(848\) −3.94379e37 −0.0115497
\(849\) −2.63079e39 −0.758285
\(850\) −1.57362e38 −0.0446421
\(851\) 3.23556e39 0.903441
\(852\) 3.29309e39 0.905042
\(853\) −4.84028e39 −1.30936 −0.654679 0.755907i \(-0.727196\pi\)
−0.654679 + 0.755907i \(0.727196\pi\)
\(854\) −7.69932e38 −0.205008
\(855\) −8.44013e39 −2.21211
\(856\) −1.35274e39 −0.348993
\(857\) 1.36798e38 0.0347406 0.0173703 0.999849i \(-0.494471\pi\)
0.0173703 + 0.999849i \(0.494471\pi\)
\(858\) −3.40160e38 −0.0850363
\(859\) −4.42921e39 −1.08998 −0.544990 0.838442i \(-0.683467\pi\)
−0.544990 + 0.838442i \(0.683467\pi\)
\(860\) −7.64710e38 −0.185254
\(861\) −4.33891e39 −1.03476
\(862\) −3.66197e37 −0.00859739
\(863\) −7.25613e39 −1.67710 −0.838550 0.544824i \(-0.816596\pi\)
−0.838550 + 0.544824i \(0.816596\pi\)
\(864\) −3.01469e39 −0.685973
\(865\) 1.78105e39 0.398986
\(866\) −3.07625e39 −0.678470
\(867\) 8.24287e39 1.78987
\(868\) −1.21948e39 −0.260711
\(869\) 1.41712e38 0.0298290
\(870\) −4.83140e39 −1.00130
\(871\) −1.05559e39 −0.215401
\(872\) 1.75556e39 0.352731
\(873\) 1.99893e40 3.95463
\(874\) −5.02911e39 −0.979686
\(875\) 1.48384e38 0.0284629
\(876\) 8.05600e39 1.52165
\(877\) 7.76731e39 1.44470 0.722348 0.691530i \(-0.243063\pi\)
0.722348 + 0.691530i \(0.243063\pi\)
\(878\) −1.90350e39 −0.348639
\(879\) 1.79718e39 0.324146
\(880\) −7.36397e37 −0.0130796
\(881\) −3.34645e39 −0.585341 −0.292671 0.956213i \(-0.594544\pi\)
−0.292671 + 0.956213i \(0.594544\pi\)
\(882\) −1.08907e40 −1.87598
\(883\) 5.35163e39 0.907856 0.453928 0.891038i \(-0.350022\pi\)
0.453928 + 0.891038i \(0.350022\pi\)
\(884\) −4.88514e38 −0.0816153
\(885\) 2.03647e39 0.335077
\(886\) −7.43376e39 −1.20463
\(887\) 9.27354e38 0.148005 0.0740026 0.997258i \(-0.476423\pi\)
0.0740026 + 0.997258i \(0.476423\pi\)
\(888\) 4.88555e39 0.767960
\(889\) −9.11997e38 −0.141195
\(890\) 1.78891e39 0.272786
\(891\) 3.70934e39 0.557119
\(892\) −2.76771e39 −0.409445
\(893\) 3.70315e39 0.539606
\(894\) 1.17205e40 1.68225
\(895\) −4.82986e38 −0.0682847
\(896\) −2.01970e38 −0.0281274
\(897\) 6.19584e39 0.849967
\(898\) −6.44159e39 −0.870488
\(899\) −1.96050e40 −2.60982
\(900\) 2.25106e39 0.295198
\(901\) 1.12887e38 0.0145834
\(902\) −1.06319e39 −0.135308
\(903\) 4.18031e39 0.524112
\(904\) −4.26101e38 −0.0526306
\(905\) 1.85770e39 0.226058
\(906\) 1.10925e40 1.32984
\(907\) 1.06013e40 1.25217 0.626083 0.779756i \(-0.284657\pi\)
0.626083 + 0.779756i \(0.284657\pi\)
\(908\) 5.55602e39 0.646555
\(909\) 1.28250e40 1.47043
\(910\) 4.60643e38 0.0520362
\(911\) −1.34212e40 −1.49381 −0.746903 0.664933i \(-0.768460\pi\)
−0.746903 + 0.664933i \(0.768460\pi\)
\(912\) −7.59375e39 −0.832772
\(913\) 3.02550e38 0.0326921
\(914\) −6.14363e39 −0.654111
\(915\) 7.72074e39 0.809979
\(916\) 6.04309e39 0.624698
\(917\) −6.92038e38 −0.0704927
\(918\) 8.62925e39 0.866158
\(919\) 6.48496e39 0.641428 0.320714 0.947176i \(-0.396077\pi\)
0.320714 + 0.947176i \(0.396077\pi\)
\(920\) 1.34131e39 0.130735
\(921\) −4.79521e39 −0.460578
\(922\) −1.28404e39 −0.121537
\(923\) −5.04761e39 −0.470829
\(924\) 4.02554e38 0.0370043
\(925\) −2.41224e39 −0.218528
\(926\) −4.52378e39 −0.403881
\(927\) −3.98801e40 −3.50897
\(928\) −3.24697e39 −0.281567
\(929\) 4.31118e39 0.368454 0.184227 0.982884i \(-0.441022\pi\)
0.184227 + 0.982884i \(0.441022\pi\)
\(930\) 1.22287e40 1.03006
\(931\) −1.81397e40 −1.50595
\(932\) 2.28063e39 0.186612
\(933\) −2.07551e40 −1.67387
\(934\) −1.04369e40 −0.829639
\(935\) 2.10786e38 0.0165153
\(936\) 6.98817e39 0.539684
\(937\) 1.66692e40 1.26891 0.634453 0.772962i \(-0.281226\pi\)
0.634453 + 0.772962i \(0.281226\pi\)
\(938\) 1.24921e39 0.0937339
\(939\) 8.05088e39 0.595467
\(940\) −9.87663e38 −0.0720084
\(941\) −2.51285e40 −1.80596 −0.902978 0.429687i \(-0.858624\pi\)
−0.902978 + 0.429687i \(0.858624\pi\)
\(942\) 2.93746e40 2.08106
\(943\) 1.93655e40 1.35245
\(944\) 1.36862e39 0.0942241
\(945\) −8.13694e39 −0.552245
\(946\) 1.02433e39 0.0685344
\(947\) 1.74731e40 1.15251 0.576257 0.817268i \(-0.304513\pi\)
0.576257 + 0.817268i \(0.304513\pi\)
\(948\) −3.89752e39 −0.253440
\(949\) −1.23482e40 −0.791604
\(950\) 3.74941e39 0.236970
\(951\) 3.98761e38 0.0248471
\(952\) 5.78120e38 0.0355156
\(953\) −1.34477e40 −0.814504 −0.407252 0.913316i \(-0.633513\pi\)
−0.407252 + 0.913316i \(0.633513\pi\)
\(954\) −1.61484e39 −0.0964334
\(955\) −2.52143e38 −0.0148457
\(956\) −9.24595e39 −0.536747
\(957\) 6.47165e39 0.370428
\(958\) −7.68389e39 −0.433658
\(959\) 6.64365e39 0.369705
\(960\) 2.02532e39 0.111130
\(961\) 3.11393e40 1.68478
\(962\) −7.48853e39 −0.399515
\(963\) −5.53898e40 −2.91390
\(964\) −5.82276e39 −0.302057
\(965\) −9.92106e39 −0.507504
\(966\) −7.33232e39 −0.369871
\(967\) −6.46684e38 −0.0321688 −0.0160844 0.999871i \(-0.505120\pi\)
−0.0160844 + 0.999871i \(0.505120\pi\)
\(968\) −7.10866e39 −0.348715
\(969\) 2.17364e40 1.05152
\(970\) −8.87997e39 −0.423636
\(971\) 9.04161e38 0.0425389 0.0212694 0.999774i \(-0.493229\pi\)
0.0212694 + 0.999774i \(0.493229\pi\)
\(972\) −6.02021e40 −2.79329
\(973\) 3.37980e39 0.154657
\(974\) 1.02098e39 0.0460758
\(975\) −4.61924e39 −0.205593
\(976\) 5.18877e39 0.227767
\(977\) −2.22532e40 −0.963420 −0.481710 0.876331i \(-0.659984\pi\)
−0.481710 + 0.876331i \(0.659984\pi\)
\(978\) 2.22823e40 0.951449
\(979\) −2.39624e39 −0.100917
\(980\) 4.83802e39 0.200963
\(981\) 7.18842e40 2.94511
\(982\) 1.74626e40 0.705673
\(983\) −2.74028e40 −1.09225 −0.546126 0.837703i \(-0.683898\pi\)
−0.546126 + 0.837703i \(0.683898\pi\)
\(984\) 2.92410e40 1.14963
\(985\) 1.41454e40 0.548562
\(986\) 9.29414e39 0.355526
\(987\) 5.39910e39 0.203723
\(988\) 1.16396e40 0.433232
\(989\) −1.86576e40 −0.685024
\(990\) −3.01529e39 −0.109208
\(991\) 1.63958e40 0.585786 0.292893 0.956145i \(-0.405382\pi\)
0.292893 + 0.956145i \(0.405382\pi\)
\(992\) 8.21840e39 0.289654
\(993\) −6.73871e40 −2.34294
\(994\) 5.97348e39 0.204885
\(995\) 1.97649e40 0.668778
\(996\) −8.32108e39 −0.277766
\(997\) 3.56689e40 1.17464 0.587320 0.809355i \(-0.300183\pi\)
0.587320 + 0.809355i \(0.300183\pi\)
\(998\) 9.66385e39 0.313970
\(999\) 1.32279e41 4.23993
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 10.28.a.c.1.1 2
5.2 odd 4 50.28.b.c.49.4 4
5.3 odd 4 50.28.b.c.49.1 4
5.4 even 2 50.28.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.28.a.c.1.1 2 1.1 even 1 trivial
50.28.a.c.1.2 2 5.4 even 2
50.28.b.c.49.1 4 5.3 odd 4
50.28.b.c.49.4 4 5.2 odd 4