Properties

Label 98.13.b.c.97.5
Level $98$
Weight $13$
Character 98.97
Analytic conductor $89.571$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,13,Mod(97,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.97");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 98.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(89.5713940931\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 6613776 x^{14} + 17532494948988 x^{12} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{8}\cdot 7^{24} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.5
Root \(39.0012i\) of defining polynomial
Character \(\chi\) \(=\) 98.97
Dual form 98.13.b.c.97.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-45.2548 q^{2} +39.0012i q^{3} +2048.00 q^{4} +2865.82i q^{5} -1764.99i q^{6} -92681.9 q^{8} +529920. q^{9} +O(q^{10})\) \(q-45.2548 q^{2} +39.0012i q^{3} +2048.00 q^{4} +2865.82i q^{5} -1764.99i q^{6} -92681.9 q^{8} +529920. q^{9} -129692. i q^{10} +3.41289e6 q^{11} +79874.6i q^{12} -6.97895e6i q^{13} -111771. q^{15} +4.19430e6 q^{16} +2.85574e7i q^{17} -2.39814e7 q^{18} -5.25546e7i q^{19} +5.86920e6i q^{20} -1.54450e8 q^{22} +3.11954e7 q^{23} -3.61471e6i q^{24} +2.35928e8 q^{25} +3.15831e8i q^{26} +4.13944e7i q^{27} +4.56209e8 q^{29} +5.05816e6 q^{30} -1.75305e9i q^{31} -1.89813e8 q^{32} +1.33107e8i q^{33} -1.29236e9i q^{34} +1.08528e9 q^{36} -1.81676e9 q^{37} +2.37835e9i q^{38} +2.72188e8 q^{39} -2.65610e8i q^{40} +5.14132e9i q^{41} -7.27073e9 q^{43} +6.98959e9 q^{44} +1.51866e9i q^{45} -1.41174e9 q^{46} +4.28207e9i q^{47} +1.63583e8i q^{48} -1.06769e10 q^{50} -1.11377e9 q^{51} -1.42929e10i q^{52} -7.22753e9 q^{53} -1.87330e9i q^{54} +9.78073e9i q^{55} +2.04970e9 q^{57} -2.06457e10 q^{58} +5.15674e9i q^{59} -2.28906e8 q^{60} +5.11141e9i q^{61} +7.93338e10i q^{62} +8.58993e9 q^{64} +2.00004e10 q^{65} -6.02373e9i q^{66} -7.59045e10 q^{67} +5.84855e10i q^{68} +1.21666e9i q^{69} -1.13092e11 q^{71} -4.91140e10 q^{72} -1.74577e11i q^{73} +8.22171e10 q^{74} +9.20147e9i q^{75} -1.07632e11i q^{76} -1.23178e10 q^{78} +2.26089e11 q^{79} +1.20201e10i q^{80} +2.80007e11 q^{81} -2.32670e11i q^{82} +4.13229e11i q^{83} -8.18404e10 q^{85} +3.29036e11 q^{86} +1.77927e10i q^{87} -3.16313e11 q^{88} -2.60295e11i q^{89} -6.87265e10i q^{90} +6.38882e10 q^{92} +6.83710e10 q^{93} -1.93784e11i q^{94} +1.50612e11 q^{95} -7.40293e9i q^{96} -4.10959e11i q^{97} +1.80856e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32768 q^{4} - 4724496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32768 q^{4} - 4724496 q^{9} - 4144176 q^{11} - 27163728 q^{15} + 67108864 q^{16} + 9185280 q^{18} + 62776320 q^{22} - 51261120 q^{23} - 1042411616 q^{25} + 532360944 q^{29} + 4303835136 q^{30} - 9675767808 q^{36} - 11529048080 q^{37} - 21052166544 q^{39} - 66929432000 q^{43} - 8487272448 q^{44} - 14406426624 q^{46} - 99248080896 q^{50} + 46598512752 q^{51} - 78268323600 q^{53} - 328243960080 q^{57} - 59274792960 q^{58} - 55631314944 q^{60} + 137438953472 q^{64} - 316645407792 q^{65} - 215534239840 q^{67} - 1150259029344 q^{71} + 18811453440 q^{72} - 9805556736 q^{74} - 786088888320 q^{78} - 455264129536 q^{79} + 783968356800 q^{81} + 710209696080 q^{85} + 76210384896 q^{86} + 128565903360 q^{88} - 104982773760 q^{92} + 917337537360 q^{93} + 373007725920 q^{95} + 3921180354576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −45.2548 −0.707107
\(3\) 39.0012i 0.0534997i 0.999642 + 0.0267498i \(0.00851575\pi\)
−0.999642 + 0.0267498i \(0.991484\pi\)
\(4\) 2048.00 0.500000
\(5\) 2865.82i 0.183413i 0.995786 + 0.0917063i \(0.0292321\pi\)
−0.995786 + 0.0917063i \(0.970768\pi\)
\(6\) − 1764.99i − 0.0378300i
\(7\) 0 0
\(8\) −92681.9 −0.353553
\(9\) 529920. 0.997138
\(10\) − 129692.i − 0.129692i
\(11\) 3.41289e6 1.92649 0.963243 0.268631i \(-0.0865712\pi\)
0.963243 + 0.268631i \(0.0865712\pi\)
\(12\) 79874.6i 0.0267498i
\(13\) − 6.97895e6i − 1.44587i −0.690915 0.722936i \(-0.742792\pi\)
0.690915 0.722936i \(-0.257208\pi\)
\(14\) 0 0
\(15\) −111771. −0.00981251
\(16\) 4.19430e6 0.250000
\(17\) 2.85574e7i 1.18311i 0.806265 + 0.591555i \(0.201486\pi\)
−0.806265 + 0.591555i \(0.798514\pi\)
\(18\) −2.39814e7 −0.705083
\(19\) − 5.25546e7i − 1.11709i −0.829473 0.558546i \(-0.811359\pi\)
0.829473 0.558546i \(-0.188641\pi\)
\(20\) 5.86920e6i 0.0917063i
\(21\) 0 0
\(22\) −1.54450e8 −1.36223
\(23\) 3.11954e7 0.210729 0.105364 0.994434i \(-0.466399\pi\)
0.105364 + 0.994434i \(0.466399\pi\)
\(24\) − 3.61471e6i − 0.0189150i
\(25\) 2.35928e8 0.966360
\(26\) 3.15831e8i 1.02239i
\(27\) 4.13944e7i 0.106846i
\(28\) 0 0
\(29\) 4.56209e8 0.766966 0.383483 0.923548i \(-0.374724\pi\)
0.383483 + 0.923548i \(0.374724\pi\)
\(30\) 5.05816e6 0.00693849
\(31\) − 1.75305e9i − 1.97526i −0.156817 0.987628i \(-0.550123\pi\)
0.156817 0.987628i \(-0.449877\pi\)
\(32\) −1.89813e8 −0.176777
\(33\) 1.33107e8i 0.103066i
\(34\) − 1.29236e9i − 0.836585i
\(35\) 0 0
\(36\) 1.08528e9 0.498569
\(37\) −1.81676e9 −0.708087 −0.354043 0.935229i \(-0.615194\pi\)
−0.354043 + 0.935229i \(0.615194\pi\)
\(38\) 2.37835e9i 0.789904i
\(39\) 2.72188e8 0.0773536
\(40\) − 2.65610e8i − 0.0648462i
\(41\) 5.14132e9i 1.08236i 0.840907 + 0.541180i \(0.182022\pi\)
−0.840907 + 0.541180i \(0.817978\pi\)
\(42\) 0 0
\(43\) −7.27073e9 −1.15018 −0.575092 0.818089i \(-0.695034\pi\)
−0.575092 + 0.818089i \(0.695034\pi\)
\(44\) 6.98959e9 0.963243
\(45\) 1.51866e9i 0.182888i
\(46\) −1.41174e9 −0.149008
\(47\) 4.28207e9i 0.397252i 0.980075 + 0.198626i \(0.0636479\pi\)
−0.980075 + 0.198626i \(0.936352\pi\)
\(48\) 1.63583e8i 0.0133749i
\(49\) 0 0
\(50\) −1.06769e10 −0.683320
\(51\) −1.11377e9 −0.0632959
\(52\) − 1.42929e10i − 0.722936i
\(53\) −7.22753e9 −0.326088 −0.163044 0.986619i \(-0.552131\pi\)
−0.163044 + 0.986619i \(0.552131\pi\)
\(54\) − 1.87330e9i − 0.0755517i
\(55\) 9.78073e9i 0.353342i
\(56\) 0 0
\(57\) 2.04970e9 0.0597641
\(58\) −2.06457e10 −0.542327
\(59\) 5.15674e9i 0.122254i 0.998130 + 0.0611270i \(0.0194695\pi\)
−0.998130 + 0.0611270i \(0.980531\pi\)
\(60\) −2.28906e8 −0.00490626
\(61\) 5.11141e9i 0.0992114i 0.998769 + 0.0496057i \(0.0157965\pi\)
−0.998769 + 0.0496057i \(0.984204\pi\)
\(62\) 7.93338e10i 1.39672i
\(63\) 0 0
\(64\) 8.58993e9 0.125000
\(65\) 2.00004e10 0.265191
\(66\) − 6.02373e9i − 0.0728789i
\(67\) −7.59045e10 −0.839109 −0.419555 0.907730i \(-0.637814\pi\)
−0.419555 + 0.907730i \(0.637814\pi\)
\(68\) 5.84855e10i 0.591555i
\(69\) 1.21666e9i 0.0112739i
\(70\) 0 0
\(71\) −1.13092e11 −0.882838 −0.441419 0.897301i \(-0.645525\pi\)
−0.441419 + 0.897301i \(0.645525\pi\)
\(72\) −4.91140e10 −0.352541
\(73\) − 1.74577e11i − 1.15359i −0.816890 0.576794i \(-0.804304\pi\)
0.816890 0.576794i \(-0.195696\pi\)
\(74\) 8.22171e10 0.500693
\(75\) 9.20147e9i 0.0516999i
\(76\) − 1.07632e11i − 0.558546i
\(77\) 0 0
\(78\) −1.23178e10 −0.0546973
\(79\) 2.26089e11 0.930071 0.465036 0.885292i \(-0.346042\pi\)
0.465036 + 0.885292i \(0.346042\pi\)
\(80\) 1.20201e10i 0.0458532i
\(81\) 2.80007e11 0.991422
\(82\) − 2.32670e11i − 0.765344i
\(83\) 4.13229e11i 1.26393i 0.774998 + 0.631964i \(0.217751\pi\)
−0.774998 + 0.631964i \(0.782249\pi\)
\(84\) 0 0
\(85\) −8.18404e10 −0.216997
\(86\) 3.29036e11 0.813303
\(87\) 1.77927e10i 0.0410324i
\(88\) −3.16313e11 −0.681116
\(89\) − 2.60295e11i − 0.523753i −0.965101 0.261876i \(-0.915659\pi\)
0.965101 0.261876i \(-0.0843413\pi\)
\(90\) − 6.87265e10i − 0.129321i
\(91\) 0 0
\(92\) 6.38882e10 0.105364
\(93\) 6.83710e10 0.105675
\(94\) − 1.93784e11i − 0.280900i
\(95\) 1.50612e11 0.204889
\(96\) − 7.40293e9i − 0.00945749i
\(97\) − 4.10959e11i − 0.493364i −0.969096 0.246682i \(-0.920660\pi\)
0.969096 0.246682i \(-0.0793404\pi\)
\(98\) 0 0
\(99\) 1.80856e12 1.92097
\(100\) 4.83180e11 0.483180
\(101\) − 1.39405e12i − 1.31326i −0.754214 0.656628i \(-0.771982\pi\)
0.754214 0.656628i \(-0.228018\pi\)
\(102\) 5.04036e10 0.0447570
\(103\) − 1.44553e12i − 1.21061i −0.795993 0.605306i \(-0.793051\pi\)
0.795993 0.605306i \(-0.206949\pi\)
\(104\) 6.46822e11i 0.511193i
\(105\) 0 0
\(106\) 3.27081e11 0.230579
\(107\) 5.70567e11 0.380193 0.190097 0.981765i \(-0.439120\pi\)
0.190097 + 0.981765i \(0.439120\pi\)
\(108\) 8.47757e10i 0.0534231i
\(109\) 2.82818e11 0.168635 0.0843176 0.996439i \(-0.473129\pi\)
0.0843176 + 0.996439i \(0.473129\pi\)
\(110\) − 4.42625e11i − 0.249850i
\(111\) − 7.08558e10i − 0.0378824i
\(112\) 0 0
\(113\) −1.58024e12 −0.759017 −0.379509 0.925188i \(-0.623907\pi\)
−0.379509 + 0.925188i \(0.623907\pi\)
\(114\) −9.27586e10 −0.0422596
\(115\) 8.94005e10i 0.0386503i
\(116\) 9.34317e11 0.383483
\(117\) − 3.69828e12i − 1.44173i
\(118\) − 2.33367e11i − 0.0864466i
\(119\) 0 0
\(120\) 1.03591e10 0.00346925
\(121\) 8.50938e12 2.71135
\(122\) − 2.31316e11i − 0.0701530i
\(123\) −2.00518e11 −0.0579059
\(124\) − 3.59024e12i − 0.987628i
\(125\) 1.37579e12i 0.360655i
\(126\) 0 0
\(127\) 5.21870e12 1.24377 0.621885 0.783109i \(-0.286367\pi\)
0.621885 + 0.783109i \(0.286367\pi\)
\(128\) −3.88736e11 −0.0883883
\(129\) − 2.83568e11i − 0.0615345i
\(130\) −9.05116e11 −0.187518
\(131\) − 4.57350e12i − 0.904942i −0.891779 0.452471i \(-0.850543\pi\)
0.891779 0.452471i \(-0.149457\pi\)
\(132\) 2.72603e11i 0.0515332i
\(133\) 0 0
\(134\) 3.43504e12 0.593340
\(135\) −1.18629e11 −0.0195969
\(136\) − 2.64675e12i − 0.418292i
\(137\) 5.69134e12 0.860778 0.430389 0.902644i \(-0.358376\pi\)
0.430389 + 0.902644i \(0.358376\pi\)
\(138\) − 5.50597e10i − 0.00797186i
\(139\) − 3.77435e12i − 0.523304i −0.965162 0.261652i \(-0.915733\pi\)
0.965162 0.261652i \(-0.0842672\pi\)
\(140\) 0 0
\(141\) −1.67006e11 −0.0212529
\(142\) 5.11795e12 0.624261
\(143\) − 2.38184e13i − 2.78545i
\(144\) 2.22265e12 0.249284
\(145\) 1.30741e12i 0.140671i
\(146\) 7.90047e12i 0.815710i
\(147\) 0 0
\(148\) −3.72072e12 −0.354043
\(149\) 6.68941e12 0.611323 0.305661 0.952140i \(-0.401122\pi\)
0.305661 + 0.952140i \(0.401122\pi\)
\(150\) − 4.16411e11i − 0.0365574i
\(151\) 5.23680e11 0.0441778 0.0220889 0.999756i \(-0.492968\pi\)
0.0220889 + 0.999756i \(0.492968\pi\)
\(152\) 4.87086e12i 0.394952i
\(153\) 1.51331e13i 1.17972i
\(154\) 0 0
\(155\) 5.02392e12 0.362287
\(156\) 5.57440e11 0.0386768
\(157\) 1.56303e13i 1.04369i 0.853041 + 0.521844i \(0.174756\pi\)
−0.853041 + 0.521844i \(0.825244\pi\)
\(158\) −1.02316e13 −0.657660
\(159\) − 2.81883e11i − 0.0174456i
\(160\) − 5.43969e11i − 0.0324231i
\(161\) 0 0
\(162\) −1.26717e13 −0.701041
\(163\) 2.43293e12 0.129719 0.0648597 0.997894i \(-0.479340\pi\)
0.0648597 + 0.997894i \(0.479340\pi\)
\(164\) 1.05294e13i 0.541180i
\(165\) −3.81461e11 −0.0189037
\(166\) − 1.87006e13i − 0.893732i
\(167\) 5.39380e12i 0.248654i 0.992241 + 0.124327i \(0.0396772\pi\)
−0.992241 + 0.124327i \(0.960323\pi\)
\(168\) 0 0
\(169\) −2.54076e13 −1.09055
\(170\) 3.70367e12 0.153440
\(171\) − 2.78497e13i − 1.11390i
\(172\) −1.48905e13 −0.575092
\(173\) − 2.54933e13i − 0.950934i −0.879734 0.475467i \(-0.842279\pi\)
0.879734 0.475467i \(-0.157721\pi\)
\(174\) − 8.05207e11i − 0.0290143i
\(175\) 0 0
\(176\) 1.43147e13 0.481622
\(177\) −2.01119e11 −0.00654055
\(178\) 1.17796e13i 0.370349i
\(179\) 3.59883e13 1.09407 0.547033 0.837111i \(-0.315757\pi\)
0.547033 + 0.837111i \(0.315757\pi\)
\(180\) 3.11021e12i 0.0914438i
\(181\) − 1.12708e13i − 0.320542i −0.987073 0.160271i \(-0.948763\pi\)
0.987073 0.160271i \(-0.0512368\pi\)
\(182\) 0 0
\(183\) −1.99351e11 −0.00530777
\(184\) −2.89125e12 −0.0745038
\(185\) − 5.20650e12i − 0.129872i
\(186\) −3.09412e12 −0.0747238
\(187\) 9.74632e13i 2.27924i
\(188\) 8.76967e12i 0.198626i
\(189\) 0 0
\(190\) −6.81593e12 −0.144878
\(191\) −9.89329e12 −0.203770 −0.101885 0.994796i \(-0.532487\pi\)
−0.101885 + 0.994796i \(0.532487\pi\)
\(192\) 3.35018e11i 0.00668746i
\(193\) −4.39071e13 −0.849555 −0.424777 0.905298i \(-0.639648\pi\)
−0.424777 + 0.905298i \(0.639648\pi\)
\(194\) 1.85979e13i 0.348861i
\(195\) 7.80041e11i 0.0141876i
\(196\) 0 0
\(197\) 3.88271e13 0.664259 0.332129 0.943234i \(-0.392233\pi\)
0.332129 + 0.943234i \(0.392233\pi\)
\(198\) −8.18460e13 −1.35833
\(199\) 1.89326e13i 0.304854i 0.988315 + 0.152427i \(0.0487089\pi\)
−0.988315 + 0.152427i \(0.951291\pi\)
\(200\) −2.18662e13 −0.341660
\(201\) − 2.96037e12i − 0.0448921i
\(202\) 6.30874e13i 0.928613i
\(203\) 0 0
\(204\) −2.28101e12 −0.0316480
\(205\) −1.47341e13 −0.198518
\(206\) 6.54174e13i 0.856031i
\(207\) 1.65311e13 0.210125
\(208\) − 2.92718e13i − 0.361468i
\(209\) − 1.79363e14i − 2.15206i
\(210\) 0 0
\(211\) −3.19733e13 −0.362320 −0.181160 0.983454i \(-0.557985\pi\)
−0.181160 + 0.983454i \(0.557985\pi\)
\(212\) −1.48020e13 −0.163044
\(213\) − 4.41072e12i − 0.0472315i
\(214\) −2.58209e13 −0.268837
\(215\) − 2.08366e13i − 0.210958i
\(216\) − 3.83651e12i − 0.0377758i
\(217\) 0 0
\(218\) −1.27989e13 −0.119243
\(219\) 6.80873e12 0.0617166
\(220\) 2.00309e13i 0.176671i
\(221\) 1.99301e14 1.71063
\(222\) 3.20657e12i 0.0267869i
\(223\) − 1.78662e14i − 1.45279i −0.687279 0.726394i \(-0.741195\pi\)
0.687279 0.726394i \(-0.258805\pi\)
\(224\) 0 0
\(225\) 1.25023e14 0.963594
\(226\) 7.15134e13 0.536706
\(227\) 1.63219e14i 1.19293i 0.802639 + 0.596465i \(0.203429\pi\)
−0.802639 + 0.596465i \(0.796571\pi\)
\(228\) 4.19778e12 0.0298820
\(229\) 1.97880e14i 1.37211i 0.727549 + 0.686056i \(0.240659\pi\)
−0.727549 + 0.686056i \(0.759341\pi\)
\(230\) − 4.04580e12i − 0.0273299i
\(231\) 0 0
\(232\) −4.22823e13 −0.271163
\(233\) −1.90911e14 −1.19315 −0.596576 0.802557i \(-0.703473\pi\)
−0.596576 + 0.802557i \(0.703473\pi\)
\(234\) 1.67365e14i 1.01946i
\(235\) −1.22716e13 −0.0728611
\(236\) 1.05610e13i 0.0611270i
\(237\) 8.81774e12i 0.0497585i
\(238\) 0 0
\(239\) −1.79102e14 −0.960979 −0.480490 0.877000i \(-0.659541\pi\)
−0.480490 + 0.877000i \(0.659541\pi\)
\(240\) −4.68800e11 −0.00245313
\(241\) − 1.98275e14i − 1.01197i −0.862543 0.505984i \(-0.831130\pi\)
0.862543 0.505984i \(-0.168870\pi\)
\(242\) −3.85090e14 −1.91721
\(243\) 3.29193e13i 0.159887i
\(244\) 1.04682e13i 0.0496057i
\(245\) 0 0
\(246\) 9.07440e12 0.0409456
\(247\) −3.66776e14 −1.61517
\(248\) 1.62476e14i 0.698358i
\(249\) −1.61165e13 −0.0676197
\(250\) − 6.22612e13i − 0.255022i
\(251\) 2.18276e14i 0.872897i 0.899729 + 0.436448i \(0.143764\pi\)
−0.899729 + 0.436448i \(0.856236\pi\)
\(252\) 0 0
\(253\) 1.06466e14 0.405966
\(254\) −2.36171e14 −0.879478
\(255\) − 3.19188e12i − 0.0116093i
\(256\) 1.75922e13 0.0625000
\(257\) 4.14883e14i 1.43988i 0.694036 + 0.719941i \(0.255831\pi\)
−0.694036 + 0.719941i \(0.744169\pi\)
\(258\) 1.28328e13i 0.0435114i
\(259\) 0 0
\(260\) 4.09609e13 0.132596
\(261\) 2.41754e14 0.764771
\(262\) 2.06973e14i 0.639891i
\(263\) 1.17695e14 0.355651 0.177825 0.984062i \(-0.443094\pi\)
0.177825 + 0.984062i \(0.443094\pi\)
\(264\) − 1.23366e13i − 0.0364395i
\(265\) − 2.07128e13i − 0.0598087i
\(266\) 0 0
\(267\) 1.01518e13 0.0280206
\(268\) −1.55452e14 −0.419555
\(269\) − 3.96686e14i − 1.04697i −0.852036 0.523483i \(-0.824632\pi\)
0.852036 0.523483i \(-0.175368\pi\)
\(270\) 5.36853e12 0.0138571
\(271\) − 2.07647e14i − 0.524215i −0.965039 0.262107i \(-0.915583\pi\)
0.965039 0.262107i \(-0.0844174\pi\)
\(272\) 1.19778e14i 0.295777i
\(273\) 0 0
\(274\) −2.57561e14 −0.608662
\(275\) 8.05195e14 1.86168
\(276\) 2.49172e12i 0.00563695i
\(277\) 3.15993e14 0.699518 0.349759 0.936840i \(-0.386263\pi\)
0.349759 + 0.936840i \(0.386263\pi\)
\(278\) 1.70808e14i 0.370032i
\(279\) − 9.28974e14i − 1.96960i
\(280\) 0 0
\(281\) −3.27034e14 −0.664286 −0.332143 0.943229i \(-0.607772\pi\)
−0.332143 + 0.943229i \(0.607772\pi\)
\(282\) 7.55783e12 0.0150280
\(283\) − 8.02664e13i − 0.156248i −0.996944 0.0781242i \(-0.975107\pi\)
0.996944 0.0781242i \(-0.0248931\pi\)
\(284\) −2.31612e14 −0.441419
\(285\) 5.87406e12i 0.0109615i
\(286\) 1.07790e15i 1.96961i
\(287\) 0 0
\(288\) −1.00585e14 −0.176271
\(289\) −2.32902e14 −0.399748
\(290\) − 5.91668e13i − 0.0994696i
\(291\) 1.60279e13 0.0263948
\(292\) − 3.57534e14i − 0.576794i
\(293\) 3.68606e14i 0.582581i 0.956635 + 0.291290i \(0.0940846\pi\)
−0.956635 + 0.291290i \(0.905915\pi\)
\(294\) 0 0
\(295\) −1.47783e13 −0.0224229
\(296\) 1.68381e14 0.250347
\(297\) 1.41274e14i 0.205838i
\(298\) −3.02728e14 −0.432270
\(299\) − 2.17711e14i − 0.304687i
\(300\) 1.88446e13i 0.0258500i
\(301\) 0 0
\(302\) −2.36990e13 −0.0312384
\(303\) 5.43696e13 0.0702588
\(304\) − 2.20430e14i − 0.279273i
\(305\) −1.46484e13 −0.0181966
\(306\) − 6.84847e14i − 0.834190i
\(307\) 6.36629e14i 0.760425i 0.924899 + 0.380212i \(0.124149\pi\)
−0.924899 + 0.380212i \(0.875851\pi\)
\(308\) 0 0
\(309\) 5.63776e13 0.0647673
\(310\) −2.27357e14 −0.256175
\(311\) − 1.31752e15i − 1.45612i −0.685515 0.728059i \(-0.740423\pi\)
0.685515 0.728059i \(-0.259577\pi\)
\(312\) −2.52269e13 −0.0273486
\(313\) 7.06353e14i 0.751200i 0.926782 + 0.375600i \(0.122563\pi\)
−0.926782 + 0.375600i \(0.877437\pi\)
\(314\) − 7.07348e14i − 0.737999i
\(315\) 0 0
\(316\) 4.63030e14 0.465036
\(317\) 7.46432e14 0.735588 0.367794 0.929907i \(-0.380113\pi\)
0.367794 + 0.929907i \(0.380113\pi\)
\(318\) 1.27566e13i 0.0123359i
\(319\) 1.55699e15 1.47755
\(320\) 2.46172e13i 0.0229266i
\(321\) 2.22528e13i 0.0203402i
\(322\) 0 0
\(323\) 1.50082e15 1.32164
\(324\) 5.73454e14 0.495711
\(325\) − 1.64653e15i − 1.39723i
\(326\) −1.10102e14 −0.0917255
\(327\) 1.10303e13i 0.00902193i
\(328\) − 4.76507e14i − 0.382672i
\(329\) 0 0
\(330\) 1.72629e13 0.0133669
\(331\) 1.13325e15 0.861701 0.430850 0.902423i \(-0.358214\pi\)
0.430850 + 0.902423i \(0.358214\pi\)
\(332\) 8.46293e14i 0.631964i
\(333\) −9.62736e14 −0.706060
\(334\) − 2.44095e14i − 0.175825i
\(335\) − 2.17529e14i − 0.153903i
\(336\) 0 0
\(337\) −5.02783e13 −0.0343242 −0.0171621 0.999853i \(-0.505463\pi\)
−0.0171621 + 0.999853i \(0.505463\pi\)
\(338\) 1.14982e15 0.771132
\(339\) − 6.16312e13i − 0.0406072i
\(340\) −1.67609e14 −0.108499
\(341\) − 5.98295e15i − 3.80530i
\(342\) 1.26034e15i 0.787643i
\(343\) 0 0
\(344\) 6.73865e14 0.406652
\(345\) −3.48673e12 −0.00206778
\(346\) 1.15370e15i 0.672412i
\(347\) 8.40065e14 0.481212 0.240606 0.970623i \(-0.422654\pi\)
0.240606 + 0.970623i \(0.422654\pi\)
\(348\) 3.64395e13i 0.0205162i
\(349\) 4.32312e14i 0.239246i 0.992819 + 0.119623i \(0.0381686\pi\)
−0.992819 + 0.119623i \(0.961831\pi\)
\(350\) 0 0
\(351\) 2.88889e14 0.154486
\(352\) −6.47809e14 −0.340558
\(353\) 2.41123e15i 1.24621i 0.782140 + 0.623103i \(0.214128\pi\)
−0.782140 + 0.623103i \(0.785872\pi\)
\(354\) 9.10162e12 0.00462487
\(355\) − 3.24101e14i − 0.161924i
\(356\) − 5.33085e14i − 0.261876i
\(357\) 0 0
\(358\) −1.62865e15 −0.773622
\(359\) 6.44763e14 0.301185 0.150593 0.988596i \(-0.451882\pi\)
0.150593 + 0.988596i \(0.451882\pi\)
\(360\) − 1.40752e14i − 0.0646605i
\(361\) −5.48673e14 −0.247896
\(362\) 5.10060e14i 0.226658i
\(363\) 3.31876e14i 0.145056i
\(364\) 0 0
\(365\) 5.00308e14 0.211583
\(366\) 9.02161e12 0.00375316
\(367\) − 1.41994e15i − 0.581130i −0.956855 0.290565i \(-0.906157\pi\)
0.956855 0.290565i \(-0.0938432\pi\)
\(368\) 1.30843e14 0.0526822
\(369\) 2.72449e15i 1.07926i
\(370\) 2.35619e14i 0.0918334i
\(371\) 0 0
\(372\) 1.40024e14 0.0528377
\(373\) 2.91410e15 1.08206 0.541030 0.841003i \(-0.318034\pi\)
0.541030 + 0.841003i \(0.318034\pi\)
\(374\) − 4.41068e15i − 1.61167i
\(375\) −5.36575e13 −0.0192949
\(376\) − 3.96870e14i − 0.140450i
\(377\) − 3.18386e15i − 1.10893i
\(378\) 0 0
\(379\) 1.27572e15 0.430446 0.215223 0.976565i \(-0.430952\pi\)
0.215223 + 0.976565i \(0.430952\pi\)
\(380\) 3.08454e14 0.102444
\(381\) 2.03536e14i 0.0665413i
\(382\) 4.47719e14 0.144087
\(383\) 2.58299e15i 0.818332i 0.912460 + 0.409166i \(0.134180\pi\)
−0.912460 + 0.409166i \(0.865820\pi\)
\(384\) − 1.51612e13i − 0.00472875i
\(385\) 0 0
\(386\) 1.98701e15 0.600726
\(387\) −3.85291e15 −1.14689
\(388\) − 8.41643e14i − 0.246682i
\(389\) 3.87640e15 1.11875 0.559373 0.828916i \(-0.311042\pi\)
0.559373 + 0.828916i \(0.311042\pi\)
\(390\) − 3.53006e13i − 0.0100322i
\(391\) 8.90859e14i 0.249315i
\(392\) 0 0
\(393\) 1.78372e14 0.0484141
\(394\) −1.75711e15 −0.469702
\(395\) 6.47930e14i 0.170587i
\(396\) 3.70393e15 0.960486
\(397\) 7.38279e13i 0.0188572i 0.999956 + 0.00942861i \(0.00300126\pi\)
−0.999956 + 0.00942861i \(0.996999\pi\)
\(398\) − 8.56791e14i − 0.215564i
\(399\) 0 0
\(400\) 9.89552e14 0.241590
\(401\) −4.02524e15 −0.968111 −0.484055 0.875037i \(-0.660837\pi\)
−0.484055 + 0.875037i \(0.660837\pi\)
\(402\) 1.33971e14i 0.0317435i
\(403\) −1.22344e16 −2.85597
\(404\) − 2.85501e15i − 0.656628i
\(405\) 8.02449e14i 0.181839i
\(406\) 0 0
\(407\) −6.20039e15 −1.36412
\(408\) 1.03227e14 0.0223785
\(409\) 5.75990e15i 1.23048i 0.788340 + 0.615240i \(0.210941\pi\)
−0.788340 + 0.615240i \(0.789059\pi\)
\(410\) 6.66790e14 0.140374
\(411\) 2.21969e14i 0.0460513i
\(412\) − 2.96045e15i − 0.605306i
\(413\) 0 0
\(414\) −7.48110e14 −0.148581
\(415\) −1.18424e15 −0.231820
\(416\) 1.32469e15i 0.255596i
\(417\) 1.47205e14 0.0279966
\(418\) 8.11704e15i 1.52174i
\(419\) 9.49384e15i 1.75452i 0.480018 + 0.877259i \(0.340630\pi\)
−0.480018 + 0.877259i \(0.659370\pi\)
\(420\) 0 0
\(421\) −8.68328e15 −1.55952 −0.779761 0.626078i \(-0.784659\pi\)
−0.779761 + 0.626078i \(0.784659\pi\)
\(422\) 1.44695e15 0.256199
\(423\) 2.26915e15i 0.396115i
\(424\) 6.69861e14 0.115290
\(425\) 6.73748e15i 1.14331i
\(426\) 1.99606e14i 0.0333977i
\(427\) 0 0
\(428\) 1.16852e15 0.190097
\(429\) 9.28946e14 0.149021
\(430\) 9.42958e14i 0.149170i
\(431\) 2.69513e14 0.0420451 0.0210226 0.999779i \(-0.493308\pi\)
0.0210226 + 0.999779i \(0.493308\pi\)
\(432\) 1.73621e14i 0.0267115i
\(433\) − 5.06968e15i − 0.769224i −0.923078 0.384612i \(-0.874335\pi\)
0.923078 0.384612i \(-0.125665\pi\)
\(434\) 0 0
\(435\) −5.09908e13 −0.00752586
\(436\) 5.79212e14 0.0843176
\(437\) − 1.63946e15i − 0.235403i
\(438\) −3.08128e14 −0.0436402
\(439\) 1.63813e14i 0.0228855i 0.999935 + 0.0114427i \(0.00364242\pi\)
−0.999935 + 0.0114427i \(0.996358\pi\)
\(440\) − 9.06497e14i − 0.124925i
\(441\) 0 0
\(442\) −9.01931e15 −1.20959
\(443\) −8.71487e15 −1.15302 −0.576512 0.817089i \(-0.695587\pi\)
−0.576512 + 0.817089i \(0.695587\pi\)
\(444\) − 1.45113e14i − 0.0189412i
\(445\) 7.45960e14 0.0960629
\(446\) 8.08530e15i 1.02728i
\(447\) 2.60895e14i 0.0327055i
\(448\) 0 0
\(449\) 5.38504e15 0.657220 0.328610 0.944466i \(-0.393420\pi\)
0.328610 + 0.944466i \(0.393420\pi\)
\(450\) −5.65789e15 −0.681364
\(451\) 1.75468e16i 2.08515i
\(452\) −3.23633e15 −0.379509
\(453\) 2.04242e13i 0.00236350i
\(454\) − 7.38644e15i − 0.843529i
\(455\) 0 0
\(456\) −1.89970e14 −0.0211298
\(457\) −1.32562e16 −1.45519 −0.727596 0.686005i \(-0.759363\pi\)
−0.727596 + 0.686005i \(0.759363\pi\)
\(458\) − 8.95504e15i − 0.970230i
\(459\) −1.18212e15 −0.126411
\(460\) 1.83092e14i 0.0193251i
\(461\) 1.36310e16i 1.42011i 0.704144 + 0.710057i \(0.251331\pi\)
−0.704144 + 0.710057i \(0.748669\pi\)
\(462\) 0 0
\(463\) 1.32894e16 1.34902 0.674511 0.738265i \(-0.264354\pi\)
0.674511 + 0.738265i \(0.264354\pi\)
\(464\) 1.91348e15 0.191742
\(465\) 1.95939e14i 0.0193822i
\(466\) 8.63965e15 0.843686
\(467\) − 6.87464e15i − 0.662748i −0.943499 0.331374i \(-0.892488\pi\)
0.943499 0.331374i \(-0.107512\pi\)
\(468\) − 7.57408e15i − 0.720867i
\(469\) 0 0
\(470\) 5.55351e14 0.0515206
\(471\) −6.09603e14 −0.0558369
\(472\) − 4.77936e14i − 0.0432233i
\(473\) −2.48142e16 −2.21581
\(474\) − 3.99045e14i − 0.0351846i
\(475\) − 1.23991e16i − 1.07951i
\(476\) 0 0
\(477\) −3.83001e15 −0.325155
\(478\) 8.10525e15 0.679515
\(479\) − 3.42398e15i − 0.283478i −0.989904 0.141739i \(-0.954731\pi\)
0.989904 0.141739i \(-0.0452693\pi\)
\(480\) 2.12155e13 0.00173462
\(481\) 1.26791e16i 1.02380i
\(482\) 8.97292e15i 0.715569i
\(483\) 0 0
\(484\) 1.74272e16 1.35567
\(485\) 1.17773e15 0.0904892
\(486\) − 1.48976e15i − 0.113057i
\(487\) 1.58272e16 1.18640 0.593200 0.805055i \(-0.297864\pi\)
0.593200 + 0.805055i \(0.297864\pi\)
\(488\) − 4.73735e14i − 0.0350765i
\(489\) 9.48875e13i 0.00693994i
\(490\) 0 0
\(491\) −2.26555e16 −1.61690 −0.808452 0.588563i \(-0.799694\pi\)
−0.808452 + 0.588563i \(0.799694\pi\)
\(492\) −4.10661e14 −0.0289529
\(493\) 1.30281e16i 0.907405i
\(494\) 1.65984e16 1.14210
\(495\) 5.18300e15i 0.352331i
\(496\) − 7.35281e15i − 0.493814i
\(497\) 0 0
\(498\) 7.29347e14 0.0478144
\(499\) 9.07153e15 0.587594 0.293797 0.955868i \(-0.405081\pi\)
0.293797 + 0.955868i \(0.405081\pi\)
\(500\) 2.81762e15i 0.180328i
\(501\) −2.10365e14 −0.0133029
\(502\) − 9.87803e15i − 0.617231i
\(503\) − 2.00202e16i − 1.23612i −0.786131 0.618060i \(-0.787919\pi\)
0.786131 0.618060i \(-0.212081\pi\)
\(504\) 0 0
\(505\) 3.99509e15 0.240868
\(506\) −4.81812e15 −0.287061
\(507\) − 9.90929e14i − 0.0583438i
\(508\) 1.06879e16 0.621885
\(509\) − 2.26064e16i − 1.29995i −0.759957 0.649973i \(-0.774780\pi\)
0.759957 0.649973i \(-0.225220\pi\)
\(510\) 1.44448e14i 0.00820900i
\(511\) 0 0
\(512\) −7.96131e14 −0.0441942
\(513\) 2.17547e15 0.119357
\(514\) − 1.87755e16i − 1.01815i
\(515\) 4.14264e15 0.222041
\(516\) − 5.80747e14i − 0.0307672i
\(517\) 1.46142e16i 0.765301i
\(518\) 0 0
\(519\) 9.94272e14 0.0508746
\(520\) −1.85368e15 −0.0937592
\(521\) − 4.83379e15i − 0.241692i −0.992671 0.120846i \(-0.961439\pi\)
0.992671 0.120846i \(-0.0385606\pi\)
\(522\) −1.09406e16 −0.540775
\(523\) − 2.06681e16i − 1.00993i −0.863141 0.504964i \(-0.831506\pi\)
0.863141 0.504964i \(-0.168494\pi\)
\(524\) − 9.36653e15i − 0.452471i
\(525\) 0 0
\(526\) −5.32627e15 −0.251483
\(527\) 5.00624e16 2.33694
\(528\) 5.58291e14i 0.0257666i
\(529\) −2.09415e16 −0.955593
\(530\) 9.37355e14i 0.0422911i
\(531\) 2.73266e15i 0.121904i
\(532\) 0 0
\(533\) 3.58810e16 1.56495
\(534\) −4.59420e14 −0.0198135
\(535\) 1.63514e15i 0.0697322i
\(536\) 7.03497e15 0.296670
\(537\) 1.40359e15i 0.0585322i
\(538\) 1.79519e16i 0.740317i
\(539\) 0 0
\(540\) −2.42952e14 −0.00979847
\(541\) −4.12580e16 −1.64560 −0.822801 0.568329i \(-0.807590\pi\)
−0.822801 + 0.568329i \(0.807590\pi\)
\(542\) 9.39701e15i 0.370676i
\(543\) 4.39577e14 0.0171489
\(544\) − 5.42055e15i − 0.209146i
\(545\) 8.10507e14i 0.0309298i
\(546\) 0 0
\(547\) 1.32093e16 0.493124 0.246562 0.969127i \(-0.420699\pi\)
0.246562 + 0.969127i \(0.420699\pi\)
\(548\) 1.16559e16 0.430389
\(549\) 2.70864e15i 0.0989274i
\(550\) −3.64390e16 −1.31641
\(551\) − 2.39759e16i − 0.856772i
\(552\) − 1.12762e14i − 0.00398593i
\(553\) 0 0
\(554\) −1.43002e16 −0.494634
\(555\) 2.03060e14 0.00694811
\(556\) − 7.72988e15i − 0.261652i
\(557\) 3.67951e16 1.23214 0.616069 0.787692i \(-0.288724\pi\)
0.616069 + 0.787692i \(0.288724\pi\)
\(558\) 4.20406e16i 1.39272i
\(559\) 5.07421e16i 1.66302i
\(560\) 0 0
\(561\) −3.80118e15 −0.121939
\(562\) 1.47999e16 0.469721
\(563\) − 5.22593e16i − 1.64102i −0.571635 0.820508i \(-0.693690\pi\)
0.571635 0.820508i \(-0.306310\pi\)
\(564\) −3.42028e14 −0.0106264
\(565\) − 4.52868e15i − 0.139213i
\(566\) 3.63244e15i 0.110484i
\(567\) 0 0
\(568\) 1.04816e16 0.312130
\(569\) 6.20823e16 1.82934 0.914669 0.404204i \(-0.132451\pi\)
0.914669 + 0.404204i \(0.132451\pi\)
\(570\) − 2.65830e14i − 0.00775094i
\(571\) 9.20520e15 0.265593 0.132796 0.991143i \(-0.457604\pi\)
0.132796 + 0.991143i \(0.457604\pi\)
\(572\) − 4.87800e16i − 1.39273i
\(573\) − 3.85851e14i − 0.0109016i
\(574\) 0 0
\(575\) 7.35986e15 0.203640
\(576\) 4.55198e15 0.124642
\(577\) − 5.92325e16i − 1.60511i −0.596577 0.802556i \(-0.703473\pi\)
0.596577 0.802556i \(-0.296527\pi\)
\(578\) 1.05400e16 0.282665
\(579\) − 1.71243e15i − 0.0454509i
\(580\) 2.67759e15i 0.0703356i
\(581\) 0 0
\(582\) −7.25340e14 −0.0186640
\(583\) −2.46668e16 −0.628204
\(584\) 1.61802e16i 0.407855i
\(585\) 1.05986e16 0.264432
\(586\) − 1.66812e16i − 0.411947i
\(587\) 1.82380e16i 0.445808i 0.974840 + 0.222904i \(0.0715536\pi\)
−0.974840 + 0.222904i \(0.928446\pi\)
\(588\) 0 0
\(589\) −9.21307e16 −2.20654
\(590\) 6.68789e14 0.0158554
\(591\) 1.51430e15i 0.0355376i
\(592\) −7.62003e15 −0.177022
\(593\) 2.59009e16i 0.595644i 0.954621 + 0.297822i \(0.0962601\pi\)
−0.954621 + 0.297822i \(0.903740\pi\)
\(594\) − 6.39335e15i − 0.145549i
\(595\) 0 0
\(596\) 1.36999e16 0.305661
\(597\) −7.38394e14 −0.0163096
\(598\) 9.85248e15i 0.215446i
\(599\) −4.89381e16 −1.05946 −0.529732 0.848165i \(-0.677707\pi\)
−0.529732 + 0.848165i \(0.677707\pi\)
\(600\) − 8.52810e14i − 0.0182787i
\(601\) − 2.83221e16i − 0.601005i −0.953781 0.300503i \(-0.902846\pi\)
0.953781 0.300503i \(-0.0971544\pi\)
\(602\) 0 0
\(603\) −4.02233e16 −0.836708
\(604\) 1.07250e15 0.0220889
\(605\) 2.43864e16i 0.497296i
\(606\) −2.46049e15 −0.0496804
\(607\) 6.25489e15i 0.125051i 0.998043 + 0.0625256i \(0.0199155\pi\)
−0.998043 + 0.0625256i \(0.980084\pi\)
\(608\) 9.97552e15i 0.197476i
\(609\) 0 0
\(610\) 6.62910e14 0.0128670
\(611\) 2.98843e16 0.574376
\(612\) 3.09926e16i 0.589862i
\(613\) 5.55586e16 1.04710 0.523551 0.851995i \(-0.324607\pi\)
0.523551 + 0.851995i \(0.324607\pi\)
\(614\) − 2.88105e16i − 0.537701i
\(615\) − 5.74649e14i − 0.0106207i
\(616\) 0 0
\(617\) 8.06869e16 1.46249 0.731243 0.682117i \(-0.238940\pi\)
0.731243 + 0.682117i \(0.238940\pi\)
\(618\) −2.55136e15 −0.0457974
\(619\) 3.93731e15i 0.0699931i 0.999387 + 0.0349966i \(0.0111420\pi\)
−0.999387 + 0.0349966i \(0.988858\pi\)
\(620\) 1.02890e16 0.181143
\(621\) 1.29131e15i 0.0225155i
\(622\) 5.96244e16i 1.02963i
\(623\) 0 0
\(624\) 1.14164e15 0.0193384
\(625\) 5.36568e16 0.900211
\(626\) − 3.19659e16i − 0.531179i
\(627\) 6.99538e15 0.115135
\(628\) 3.20109e16i 0.521844i
\(629\) − 5.18818e16i − 0.837744i
\(630\) 0 0
\(631\) 6.92232e16 1.09667 0.548334 0.836259i \(-0.315262\pi\)
0.548334 + 0.836259i \(0.315262\pi\)
\(632\) −2.09543e16 −0.328830
\(633\) − 1.24700e15i − 0.0193840i
\(634\) −3.37797e16 −0.520139
\(635\) 1.49559e16i 0.228123i
\(636\) − 5.77296e14i − 0.00872280i
\(637\) 0 0
\(638\) −7.04614e16 −1.04479
\(639\) −5.99296e16 −0.880311
\(640\) − 1.11405e15i − 0.0162115i
\(641\) 3.16865e16 0.456800 0.228400 0.973567i \(-0.426651\pi\)
0.228400 + 0.973567i \(0.426651\pi\)
\(642\) − 1.00705e15i − 0.0143827i
\(643\) − 5.19233e16i − 0.734678i −0.930087 0.367339i \(-0.880269\pi\)
0.930087 0.367339i \(-0.119731\pi\)
\(644\) 0 0
\(645\) 8.12655e14 0.0112862
\(646\) −6.79195e16 −0.934543
\(647\) − 1.20952e17i − 1.64888i −0.565951 0.824439i \(-0.691491\pi\)
0.565951 0.824439i \(-0.308509\pi\)
\(648\) −2.59516e16 −0.350520
\(649\) 1.75994e16i 0.235521i
\(650\) 7.45133e16i 0.987993i
\(651\) 0 0
\(652\) 4.98265e15 0.0648597
\(653\) −4.65159e16 −0.599960 −0.299980 0.953945i \(-0.596980\pi\)
−0.299980 + 0.953945i \(0.596980\pi\)
\(654\) − 4.99173e14i − 0.00637947i
\(655\) 1.31068e16 0.165978
\(656\) 2.15643e16i 0.270590i
\(657\) − 9.25120e16i − 1.15029i
\(658\) 0 0
\(659\) 3.88240e16 0.474011 0.237005 0.971508i \(-0.423834\pi\)
0.237005 + 0.971508i \(0.423834\pi\)
\(660\) −7.81231e14 −0.00945183
\(661\) 1.08911e17i 1.30576i 0.757463 + 0.652878i \(0.226439\pi\)
−0.757463 + 0.652878i \(0.773561\pi\)
\(662\) −5.12849e16 −0.609314
\(663\) 7.77297e15i 0.0915178i
\(664\) − 3.82989e16i − 0.446866i
\(665\) 0 0
\(666\) 4.35685e16 0.499260
\(667\) 1.42316e16 0.161622
\(668\) 1.10465e16i 0.124327i
\(669\) 6.96803e15 0.0777236
\(670\) 9.84423e15i 0.108826i
\(671\) 1.74447e16i 0.191129i
\(672\) 0 0
\(673\) −4.67477e16 −0.503118 −0.251559 0.967842i \(-0.580943\pi\)
−0.251559 + 0.967842i \(0.580943\pi\)
\(674\) 2.27534e15 0.0242709
\(675\) 9.76608e15i 0.103252i
\(676\) −5.20348e16 −0.545273
\(677\) 6.23836e16i 0.647945i 0.946066 + 0.323973i \(0.105019\pi\)
−0.946066 + 0.323973i \(0.894981\pi\)
\(678\) 2.78911e15i 0.0287136i
\(679\) 0 0
\(680\) 7.58512e15 0.0767201
\(681\) −6.36574e15 −0.0638214
\(682\) 2.70757e17i 2.69075i
\(683\) 6.73352e16 0.663312 0.331656 0.943400i \(-0.392393\pi\)
0.331656 + 0.943400i \(0.392393\pi\)
\(684\) − 5.70363e16i − 0.556948i
\(685\) 1.63104e16i 0.157877i
\(686\) 0 0
\(687\) −7.71758e15 −0.0734075
\(688\) −3.04957e16 −0.287546
\(689\) 5.04406e16i 0.471482i
\(690\) 1.57791e14 0.00146214
\(691\) 2.39079e16i 0.219621i 0.993953 + 0.109810i \(0.0350244\pi\)
−0.993953 + 0.109810i \(0.964976\pi\)
\(692\) − 5.22104e16i − 0.475467i
\(693\) 0 0
\(694\) −3.80170e16 −0.340268
\(695\) 1.08166e16 0.0959805
\(696\) − 1.64906e15i − 0.0145072i
\(697\) −1.46823e17 −1.28055
\(698\) − 1.95642e16i − 0.169172i
\(699\) − 7.44577e15i − 0.0638332i
\(700\) 0 0
\(701\) −1.42561e17 −1.20141 −0.600707 0.799469i \(-0.705114\pi\)
−0.600707 + 0.799469i \(0.705114\pi\)
\(702\) −1.30736e16 −0.109238
\(703\) 9.54790e16i 0.790999i
\(704\) 2.93165e16 0.240811
\(705\) − 4.78609e14i − 0.00389804i
\(706\) − 1.09120e17i − 0.881201i
\(707\) 0 0
\(708\) −4.11892e14 −0.00327027
\(709\) −2.16853e17 −1.70721 −0.853607 0.520917i \(-0.825590\pi\)
−0.853607 + 0.520917i \(0.825590\pi\)
\(710\) 1.46671e16i 0.114497i
\(711\) 1.19809e17 0.927409
\(712\) 2.41247e16i 0.185175i
\(713\) − 5.46870e16i − 0.416243i
\(714\) 0 0
\(715\) 6.82592e16 0.510887
\(716\) 7.37041e16 0.547033
\(717\) − 6.98522e15i − 0.0514121i
\(718\) −2.91786e16 −0.212970
\(719\) 1.72963e16i 0.125193i 0.998039 + 0.0625966i \(0.0199381\pi\)
−0.998039 + 0.0625966i \(0.980062\pi\)
\(720\) 6.36971e15i 0.0457219i
\(721\) 0 0
\(722\) 2.48301e16 0.175289
\(723\) 7.73299e15 0.0541399
\(724\) − 2.30827e16i − 0.160271i
\(725\) 1.07632e17 0.741165
\(726\) − 1.50190e16i − 0.102570i
\(727\) − 1.17362e17i − 0.794915i −0.917621 0.397458i \(-0.869893\pi\)
0.917621 0.397458i \(-0.130107\pi\)
\(728\) 0 0
\(729\) 1.47523e17 0.982868
\(730\) −2.26413e16 −0.149611
\(731\) − 2.07633e17i − 1.36079i
\(732\) −4.08271e14 −0.00265389
\(733\) 1.72659e17i 1.11318i 0.830787 + 0.556591i \(0.187891\pi\)
−0.830787 + 0.556591i \(0.812109\pi\)
\(734\) 6.42591e16i 0.410921i
\(735\) 0 0
\(736\) −5.92128e15 −0.0372519
\(737\) −2.59053e17 −1.61653
\(738\) − 1.23296e17i − 0.763153i
\(739\) −1.90886e16 −0.117195 −0.0585974 0.998282i \(-0.518663\pi\)
−0.0585974 + 0.998282i \(0.518663\pi\)
\(740\) − 1.06629e16i − 0.0649360i
\(741\) − 1.43047e16i − 0.0864112i
\(742\) 0 0
\(743\) −5.56299e16 −0.330655 −0.165328 0.986239i \(-0.552868\pi\)
−0.165328 + 0.986239i \(0.552868\pi\)
\(744\) −6.33675e15 −0.0373619
\(745\) 1.91707e16i 0.112124i
\(746\) −1.31877e17 −0.765132
\(747\) 2.18978e17i 1.26031i
\(748\) 1.99605e17i 1.13962i
\(749\) 0 0
\(750\) 2.42826e15 0.0136436
\(751\) 2.73102e17 1.52225 0.761125 0.648605i \(-0.224647\pi\)
0.761125 + 0.648605i \(0.224647\pi\)
\(752\) 1.79603e16i 0.0993131i
\(753\) −8.51302e15 −0.0466997
\(754\) 1.44085e17i 0.784135i
\(755\) 1.50077e15i 0.00810276i
\(756\) 0 0
\(757\) −9.53508e16 −0.506698 −0.253349 0.967375i \(-0.581532\pi\)
−0.253349 + 0.967375i \(0.581532\pi\)
\(758\) −5.77323e16 −0.304371
\(759\) 4.15232e15i 0.0217190i
\(760\) −1.39590e16 −0.0724392
\(761\) 8.72482e16i 0.449209i 0.974450 + 0.224605i \(0.0721090\pi\)
−0.974450 + 0.224605i \(0.927891\pi\)
\(762\) − 9.21098e15i − 0.0470518i
\(763\) 0 0
\(764\) −2.02615e16 −0.101885
\(765\) −4.33689e16 −0.216376
\(766\) − 1.16893e17i − 0.578648i
\(767\) 3.59886e16 0.176764
\(768\) 6.86117e14i 0.00334373i
\(769\) 2.35317e17i 1.13787i 0.822381 + 0.568937i \(0.192645\pi\)
−0.822381 + 0.568937i \(0.807355\pi\)
\(770\) 0 0
\(771\) −1.61809e16 −0.0770331
\(772\) −8.99218e16 −0.424777
\(773\) 2.20992e17i 1.03586i 0.855423 + 0.517929i \(0.173297\pi\)
−0.855423 + 0.517929i \(0.826703\pi\)
\(774\) 1.74363e17 0.810975
\(775\) − 4.13592e17i − 1.90881i
\(776\) 3.80884e16i 0.174431i
\(777\) 0 0
\(778\) −1.75426e17 −0.791072
\(779\) 2.70200e17 1.20910
\(780\) 1.59752e15i 0.00709382i
\(781\) −3.85970e17 −1.70077
\(782\) − 4.03157e16i − 0.176292i
\(783\) 1.88845e16i 0.0819474i
\(784\) 0 0
\(785\) −4.47938e16 −0.191425
\(786\) −8.07220e15 −0.0342339
\(787\) 1.83402e16i 0.0771892i 0.999255 + 0.0385946i \(0.0122881\pi\)
−0.999255 + 0.0385946i \(0.987712\pi\)
\(788\) 7.95179e16 0.332129
\(789\) 4.59025e15i 0.0190272i
\(790\) − 2.93220e16i − 0.120623i
\(791\) 0 0
\(792\) −1.67621e17 −0.679166
\(793\) 3.56722e16 0.143447
\(794\) − 3.34107e15i − 0.0133341i
\(795\) 8.07826e14 0.00319974
\(796\) 3.87739e16i 0.152427i
\(797\) − 9.39084e16i − 0.366399i −0.983076 0.183200i \(-0.941355\pi\)
0.983076 0.183200i \(-0.0586454\pi\)
\(798\) 0 0
\(799\) −1.22285e17 −0.469993
\(800\) −4.47820e16 −0.170830
\(801\) − 1.37936e17i − 0.522254i
\(802\) 1.82161e17 0.684558
\(803\) − 5.95813e17i − 2.22237i
\(804\) − 6.06284e15i − 0.0224460i
\(805\) 0 0
\(806\) 5.53667e17 2.01947
\(807\) 1.54712e16 0.0560123
\(808\) 1.29203e17i 0.464306i
\(809\) 1.00865e17 0.359789 0.179895 0.983686i \(-0.442424\pi\)
0.179895 + 0.983686i \(0.442424\pi\)
\(810\) − 3.63147e16i − 0.128580i
\(811\) 2.30062e17i 0.808572i 0.914633 + 0.404286i \(0.132480\pi\)
−0.914633 + 0.404286i \(0.867520\pi\)
\(812\) 0 0
\(813\) 8.09848e15 0.0280453
\(814\) 2.80598e17 0.964578
\(815\) 6.97236e15i 0.0237922i
\(816\) −4.67151e15 −0.0158240
\(817\) 3.82111e17i 1.28486i
\(818\) − 2.60663e17i − 0.870081i
\(819\) 0 0
\(820\) −3.01755e16 −0.0992592
\(821\) 1.51990e17 0.496315 0.248157 0.968720i \(-0.420175\pi\)
0.248157 + 0.968720i \(0.420175\pi\)
\(822\) − 1.00452e16i − 0.0325632i
\(823\) −1.64859e17 −0.530535 −0.265268 0.964175i \(-0.585460\pi\)
−0.265268 + 0.964175i \(0.585460\pi\)
\(824\) 1.33975e17i 0.428016i
\(825\) 3.14036e16i 0.0995992i
\(826\) 0 0
\(827\) 6.48794e16 0.202803 0.101401 0.994846i \(-0.467667\pi\)
0.101401 + 0.994846i \(0.467667\pi\)
\(828\) 3.38556e16 0.105063
\(829\) − 4.54965e17i − 1.40169i −0.713315 0.700844i \(-0.752807\pi\)
0.713315 0.700844i \(-0.247193\pi\)
\(830\) 5.35927e16 0.163922
\(831\) 1.23241e16i 0.0374240i
\(832\) − 5.99487e16i − 0.180734i
\(833\) 0 0
\(834\) −6.66172e15 −0.0197966
\(835\) −1.54577e16 −0.0456063
\(836\) − 3.67335e17i − 1.07603i
\(837\) 7.25663e16 0.211048
\(838\) − 4.29642e17i − 1.24063i
\(839\) 4.60417e17i 1.32002i 0.751259 + 0.660008i \(0.229447\pi\)
−0.751259 + 0.660008i \(0.770553\pi\)
\(840\) 0 0
\(841\) −1.45688e17 −0.411763
\(842\) 3.92961e17 1.10275
\(843\) − 1.27547e16i − 0.0355391i
\(844\) −6.54813e16 −0.181160
\(845\) − 7.28138e16i − 0.200020i
\(846\) − 1.02690e17i − 0.280096i
\(847\) 0 0
\(848\) −3.03145e16 −0.0815220
\(849\) 3.13049e15 0.00835923
\(850\) − 3.04903e17i − 0.808442i
\(851\) −5.66745e16 −0.149214
\(852\) − 9.03315e15i − 0.0236158i
\(853\) 4.56236e17i 1.18439i 0.805794 + 0.592196i \(0.201739\pi\)
−0.805794 + 0.592196i \(0.798261\pi\)
\(854\) 0 0
\(855\) 7.98124e16 0.204302
\(856\) −5.28813e16 −0.134419
\(857\) 1.79216e17i 0.452369i 0.974084 + 0.226184i \(0.0726252\pi\)
−0.974084 + 0.226184i \(0.927375\pi\)
\(858\) −4.20393e16 −0.105374
\(859\) − 6.75475e17i − 1.68132i −0.541563 0.840660i \(-0.682167\pi\)
0.541563 0.840660i \(-0.317833\pi\)
\(860\) − 4.26734e16i − 0.105479i
\(861\) 0 0
\(862\) −1.21968e16 −0.0297304
\(863\) −2.08990e17 −0.505896 −0.252948 0.967480i \(-0.581400\pi\)
−0.252948 + 0.967480i \(0.581400\pi\)
\(864\) − 7.85718e15i − 0.0188879i
\(865\) 7.30594e16 0.174413
\(866\) 2.29427e17i 0.543923i
\(867\) − 9.08348e15i − 0.0213864i
\(868\) 0 0
\(869\) 7.71615e17 1.79177
\(870\) 2.30758e15 0.00532159
\(871\) 5.29733e17i 1.21324i
\(872\) −2.62121e16 −0.0596216
\(873\) − 2.17775e17i − 0.491952i
\(874\) 7.41936e16i 0.166455i
\(875\) 0 0
\(876\) 1.39443e16 0.0308583
\(877\) −2.23561e17 −0.491358 −0.245679 0.969351i \(-0.579011\pi\)
−0.245679 + 0.969351i \(0.579011\pi\)
\(878\) − 7.41331e15i − 0.0161825i
\(879\) −1.43761e16 −0.0311679
\(880\) 4.10234e16i 0.0883355i
\(881\) − 9.10277e17i − 1.94678i −0.229144 0.973392i \(-0.573593\pi\)
0.229144 0.973392i \(-0.426407\pi\)
\(882\) 0 0
\(883\) 4.97000e17 1.04856 0.524279 0.851547i \(-0.324335\pi\)
0.524279 + 0.851547i \(0.324335\pi\)
\(884\) 4.08167e17 0.855313
\(885\) − 5.76372e14i − 0.00119962i
\(886\) 3.94390e17 0.815311
\(887\) − 2.86381e17i − 0.588033i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949921\pi\)
\(888\) 6.56705e15i 0.0133935i
\(889\) 0 0
\(890\) −3.37583e16 −0.0679267
\(891\) 9.55632e17 1.90996
\(892\) − 3.65899e17i − 0.726394i
\(893\) 2.25042e17 0.443768
\(894\) − 1.18068e16i − 0.0231263i
\(895\) 1.03136e17i 0.200666i
\(896\) 0 0
\(897\) 8.49100e15 0.0163006
\(898\) −2.43699e17 −0.464725
\(899\) − 7.99756e17i − 1.51495i
\(900\) 2.56047e17 0.481797
\(901\) − 2.06399e17i − 0.385798i
\(902\) − 7.94075e17i − 1.47442i
\(903\) 0 0
\(904\) 1.46459e17 0.268353
\(905\) 3.23002e16 0.0587915
\(906\) − 9.24292e14i − 0.00167124i
\(907\) −3.30810e17 −0.594203 −0.297102 0.954846i \(-0.596020\pi\)
−0.297102 + 0.954846i \(0.596020\pi\)
\(908\) 3.34272e17i 0.596465i
\(909\) − 7.38734e17i − 1.30950i
\(910\) 0 0
\(911\) −7.13090e17 −1.24748 −0.623741 0.781631i \(-0.714388\pi\)
−0.623741 + 0.781631i \(0.714388\pi\)
\(912\) 8.59705e15 0.0149410
\(913\) 1.41031e18i 2.43494i
\(914\) 5.99905e17 1.02898
\(915\) − 5.71305e14i 0 0.000973513i
\(916\) 4.05259e17i 0.686056i
\(917\) 0 0
\(918\) 5.34965e16 0.0893859
\(919\) 3.82752e17 0.635367 0.317684 0.948197i \(-0.397095\pi\)
0.317684 + 0.948197i \(0.397095\pi\)
\(920\) − 8.28580e15i − 0.0136649i
\(921\) −2.48293e16 −0.0406825
\(922\) − 6.16870e17i − 1.00417i
\(923\) 7.89262e17i 1.27647i
\(924\) 0 0
\(925\) −4.28623e17 −0.684267
\(926\) −6.01409e17 −0.953903
\(927\) − 7.66017e17i − 1.20715i
\(928\) −8.65942e16 −0.135582
\(929\) 5.12377e17i 0.797068i 0.917154 + 0.398534i \(0.130481\pi\)
−0.917154 + 0.398534i \(0.869519\pi\)
\(930\) − 8.86719e15i − 0.0137053i
\(931\) 0 0
\(932\) −3.90986e17 −0.596576
\(933\) 5.13851e16 0.0779018
\(934\) 3.11111e17i 0.468634i
\(935\) −2.79312e17 −0.418042
\(936\) 3.42764e17i 0.509730i
\(937\) 1.03036e18i 1.52248i 0.648471 + 0.761239i \(0.275409\pi\)
−0.648471 + 0.761239i \(0.724591\pi\)
\(938\) 0 0
\(939\) −2.75486e16 −0.0401889
\(940\) −2.51323e16 −0.0364305
\(941\) 9.64166e17i 1.38872i 0.719629 + 0.694359i \(0.244312\pi\)
−0.719629 + 0.694359i \(0.755688\pi\)
\(942\) 2.75875e16 0.0394827
\(943\) 1.60386e17i 0.228084i
\(944\) 2.16289e16i 0.0305635i
\(945\) 0 0
\(946\) 1.12296e18 1.56682
\(947\) −2.47776e17 −0.343526 −0.171763 0.985138i \(-0.554946\pi\)
−0.171763 + 0.985138i \(0.554946\pi\)
\(948\) 1.80587e16i 0.0248792i
\(949\) −1.21837e18 −1.66794
\(950\) 5.61119e17i 0.763331i
\(951\) 2.91118e16i 0.0393537i
\(952\) 0 0
\(953\) −1.21407e18 −1.62064 −0.810320 0.585988i \(-0.800707\pi\)
−0.810320 + 0.585988i \(0.800707\pi\)
\(954\) 1.73327e17 0.229919
\(955\) − 2.83524e16i − 0.0373740i
\(956\) −3.66802e17 −0.480490
\(957\) 6.07246e16i 0.0790484i
\(958\) 1.54952e17i 0.200449i
\(959\) 0 0
\(960\) −9.60102e14 −0.00122656
\(961\) −2.28551e18 −2.90163
\(962\) − 5.73789e17i − 0.723938i
\(963\) 3.02355e17 0.379105
\(964\) − 4.06068e17i − 0.505984i
\(965\) − 1.25830e17i − 0.155819i
\(966\) 0 0
\(967\) 1.21549e18 1.48660 0.743298 0.668960i \(-0.233260\pi\)
0.743298 + 0.668960i \(0.233260\pi\)
\(968\) −7.88665e17 −0.958607
\(969\) 5.85339e16i 0.0707074i
\(970\) −5.32982e16 −0.0639856
\(971\) 1.32675e18i 1.58297i 0.611187 + 0.791486i \(0.290692\pi\)
−0.611187 + 0.791486i \(0.709308\pi\)
\(972\) 6.74187e16i 0.0799434i
\(973\) 0 0
\(974\) −7.16259e17 −0.838911
\(975\) 6.42166e16 0.0747515
\(976\) 2.14388e16i 0.0248028i
\(977\) −4.26902e17 −0.490863 −0.245432 0.969414i \(-0.578930\pi\)
−0.245432 + 0.969414i \(0.578930\pi\)
\(978\) − 4.29412e15i − 0.00490728i
\(979\) − 8.88359e17i − 1.00900i
\(980\) 0 0
\(981\) 1.49871e17 0.168153
\(982\) 1.02527e18 1.14332
\(983\) − 6.02837e17i − 0.668157i −0.942545 0.334079i \(-0.891575\pi\)
0.942545 0.334079i \(-0.108425\pi\)
\(984\) 1.85844e16 0.0204728
\(985\) 1.11272e17i 0.121833i
\(986\) − 5.89587e17i − 0.641632i
\(987\) 0 0
\(988\) −7.51157e17 −0.807587
\(989\) −2.26813e17 −0.242377
\(990\) − 2.34556e17i − 0.249135i
\(991\) 6.49021e17 0.685200 0.342600 0.939481i \(-0.388693\pi\)
0.342600 + 0.939481i \(0.388693\pi\)
\(992\) 3.32750e17i 0.349179i
\(993\) 4.41980e16i 0.0461007i
\(994\) 0 0
\(995\) −5.42574e16 −0.0559140
\(996\) −3.30065e16 −0.0338099
\(997\) 1.12875e18i 1.14928i 0.818407 + 0.574639i \(0.194858\pi\)
−0.818407 + 0.574639i \(0.805142\pi\)
\(998\) −4.10530e17 −0.415491
\(999\) − 7.52036e16i − 0.0756564i
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 98.13.b.c.97.5 16
7.2 even 3 14.13.d.a.3.7 16
7.3 odd 6 14.13.d.a.5.7 yes 16
7.4 even 3 98.13.d.a.19.6 16
7.5 odd 6 98.13.d.a.31.6 16
7.6 odd 2 inner 98.13.b.c.97.4 16
21.2 odd 6 126.13.n.a.73.2 16
21.17 even 6 126.13.n.a.19.2 16
28.3 even 6 112.13.s.c.33.4 16
28.23 odd 6 112.13.s.c.17.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.13.d.a.3.7 16 7.2 even 3
14.13.d.a.5.7 yes 16 7.3 odd 6
98.13.b.c.97.4 16 7.6 odd 2 inner
98.13.b.c.97.5 16 1.1 even 1 trivial
98.13.d.a.19.6 16 7.4 even 3
98.13.d.a.31.6 16 7.5 odd 6
112.13.s.c.17.4 16 28.23 odd 6
112.13.s.c.33.4 16 28.3 even 6
126.13.n.a.19.2 16 21.17 even 6
126.13.n.a.73.2 16 21.2 odd 6