Properties

Label 50.26.b.d.49.4
Level $50$
Weight $26$
Character 50.49
Analytic conductor $197.998$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,26,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(197.998389976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 47675x^{2} + 568250244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(154.395 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.26.b.d.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4096.00i q^{2} +715531. i q^{3} -1.67772e7 q^{4} -2.93082e9 q^{6} -6.40133e10i q^{7} -6.87195e10i q^{8} +3.35303e11 q^{9} +1.40584e13 q^{11} -1.20046e13i q^{12} -1.57637e13i q^{13} +2.62199e14 q^{14} +2.81475e14 q^{16} -1.00698e15i q^{17} +1.37340e15i q^{18} +8.66632e15 q^{19} +4.58035e16 q^{21} +5.75832e16i q^{22} -1.43338e17i q^{23} +4.91709e16 q^{24} +6.45682e16 q^{26} +8.46182e17i q^{27} +1.07397e18i q^{28} +3.60779e17 q^{29} +8.72952e17 q^{31} +1.15292e18i q^{32} +1.00592e19i q^{33} +4.12457e18 q^{34} -5.62546e18 q^{36} +7.37033e19i q^{37} +3.54972e19i q^{38} +1.12794e19 q^{39} -6.57824e19 q^{41} +1.87611e20i q^{42} -4.50969e20i q^{43} -2.35861e20 q^{44} +5.87113e20 q^{46} -5.55700e20i q^{47} +2.01404e20i q^{48} -2.75664e21 q^{49} +7.20523e20 q^{51} +2.64471e20i q^{52} -4.75642e21i q^{53} -3.46596e21 q^{54} -4.39896e21 q^{56} +6.20102e21i q^{57} +1.47775e21i q^{58} +1.95774e22 q^{59} -2.95892e21 q^{61} +3.57561e21i q^{62} -2.14639e22i q^{63} -4.72237e21 q^{64} -4.12026e22 q^{66} +4.26809e22i q^{67} +1.68942e22i q^{68} +1.02563e23 q^{69} -1.93990e23 q^{71} -2.30419e22i q^{72} +2.11325e22i q^{73} -3.01889e23 q^{74} -1.45397e23 q^{76} -8.99925e23i q^{77} +4.62006e22i q^{78} +4.99401e23 q^{79} -3.21371e23 q^{81} -2.69445e23i q^{82} -1.79998e24i q^{83} -7.68456e23 q^{84} +1.84717e24 q^{86} +2.58149e23i q^{87} -9.66086e23i q^{88} -2.07122e24 q^{89} -1.00909e24 q^{91} +2.40482e24i q^{92} +6.24624e23i q^{93} +2.27615e24 q^{94} -8.24952e23 q^{96} -3.44576e24i q^{97} -1.12912e25i q^{98} +4.71383e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 67108864 q^{4} - 795377664 q^{6} + 1600249078908 q^{9} + 28833201602688 q^{11} + 132195615735808 q^{14} + 11\!\cdots\!24 q^{16} + 76\!\cdots\!40 q^{19} + 15\!\cdots\!08 q^{21} + 13\!\cdots\!24 q^{24}+ \cdots + 97\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00i 0.707107i
\(3\) 715531.i 0.777344i 0.921376 + 0.388672i \(0.127066\pi\)
−0.921376 + 0.388672i \(0.872934\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) −2.93082e9 −0.549665
\(7\) − 6.40133e10i − 1.74801i −0.485914 0.874007i \(-0.661513\pi\)
0.485914 0.874007i \(-0.338487\pi\)
\(8\) − 6.87195e10i − 0.353553i
\(9\) 3.35303e11 0.395737
\(10\) 0 0
\(11\) 1.40584e13 1.35060 0.675301 0.737542i \(-0.264014\pi\)
0.675301 + 0.737542i \(0.264014\pi\)
\(12\) − 1.20046e13i − 0.388672i
\(13\) − 1.57637e13i − 0.187658i −0.995588 0.0938289i \(-0.970089\pi\)
0.995588 0.0938289i \(-0.0299107\pi\)
\(14\) 2.62199e14 1.23603
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) − 1.00698e15i − 0.419187i −0.977789 0.209593i \(-0.932786\pi\)
0.977789 0.209593i \(-0.0672140\pi\)
\(18\) 1.37340e15i 0.279828i
\(19\) 8.66632e15 0.898286 0.449143 0.893460i \(-0.351729\pi\)
0.449143 + 0.893460i \(0.351729\pi\)
\(20\) 0 0
\(21\) 4.58035e16 1.35881
\(22\) 5.75832e16i 0.955020i
\(23\) − 1.43338e17i − 1.36384i −0.731426 0.681921i \(-0.761145\pi\)
0.731426 0.681921i \(-0.238855\pi\)
\(24\) 4.91709e16 0.274832
\(25\) 0 0
\(26\) 6.45682e16 0.132694
\(27\) 8.46182e17i 1.08497i
\(28\) 1.07397e18i 0.874007i
\(29\) 3.60779e17 0.189351 0.0946753 0.995508i \(-0.469819\pi\)
0.0946753 + 0.995508i \(0.469819\pi\)
\(30\) 0 0
\(31\) 8.72952e17 0.199053 0.0995266 0.995035i \(-0.468267\pi\)
0.0995266 + 0.995035i \(0.468267\pi\)
\(32\) 1.15292e18i 0.176777i
\(33\) 1.00592e19i 1.04988i
\(34\) 4.12457e18 0.296410
\(35\) 0 0
\(36\) −5.62546e18 −0.197868
\(37\) 7.37033e19i 1.84063i 0.391183 + 0.920313i \(0.372066\pi\)
−0.391183 + 0.920313i \(0.627934\pi\)
\(38\) 3.54972e19i 0.635184i
\(39\) 1.12794e19 0.145875
\(40\) 0 0
\(41\) −6.57824e19 −0.455315 −0.227657 0.973741i \(-0.573107\pi\)
−0.227657 + 0.973741i \(0.573107\pi\)
\(42\) 1.87611e20i 0.960822i
\(43\) − 4.50969e20i − 1.72104i −0.509418 0.860519i \(-0.670139\pi\)
0.509418 0.860519i \(-0.329861\pi\)
\(44\) −2.35861e20 −0.675301
\(45\) 0 0
\(46\) 5.87113e20 0.964382
\(47\) − 5.55700e20i − 0.697617i −0.937194 0.348809i \(-0.886586\pi\)
0.937194 0.348809i \(-0.113414\pi\)
\(48\) 2.01404e20i 0.194336i
\(49\) −2.75664e21 −2.05555
\(50\) 0 0
\(51\) 7.20523e20 0.325852
\(52\) 2.64471e20i 0.0938289i
\(53\) − 4.75642e21i − 1.32994i −0.746871 0.664969i \(-0.768445\pi\)
0.746871 0.664969i \(-0.231555\pi\)
\(54\) −3.46596e21 −0.767188
\(55\) 0 0
\(56\) −4.39896e21 −0.618016
\(57\) 6.20102e21i 0.698277i
\(58\) 1.47775e21i 0.133891i
\(59\) 1.95774e22 1.43253 0.716265 0.697828i \(-0.245850\pi\)
0.716265 + 0.697828i \(0.245850\pi\)
\(60\) 0 0
\(61\) −2.95892e21 −0.142728 −0.0713641 0.997450i \(-0.522735\pi\)
−0.0713641 + 0.997450i \(0.522735\pi\)
\(62\) 3.57561e21i 0.140752i
\(63\) − 2.14639e22i − 0.691753i
\(64\) −4.72237e21 −0.125000
\(65\) 0 0
\(66\) −4.12026e22 −0.742379
\(67\) 4.26809e22i 0.637234i 0.947884 + 0.318617i \(0.103218\pi\)
−0.947884 + 0.318617i \(0.896782\pi\)
\(68\) 1.68942e22i 0.209593i
\(69\) 1.02563e23 1.06017
\(70\) 0 0
\(71\) −1.93990e23 −1.40297 −0.701486 0.712683i \(-0.747480\pi\)
−0.701486 + 0.712683i \(0.747480\pi\)
\(72\) − 2.30419e22i − 0.139914i
\(73\) 2.11325e22i 0.107998i 0.998541 + 0.0539990i \(0.0171968\pi\)
−0.998541 + 0.0539990i \(0.982803\pi\)
\(74\) −3.01889e23 −1.30152
\(75\) 0 0
\(76\) −1.45397e23 −0.449143
\(77\) − 8.99925e23i − 2.36087i
\(78\) 4.62006e22i 0.103149i
\(79\) 4.99401e23 0.950848 0.475424 0.879757i \(-0.342295\pi\)
0.475424 + 0.879757i \(0.342295\pi\)
\(80\) 0 0
\(81\) −3.21371e23 −0.447656
\(82\) − 2.69445e23i − 0.321956i
\(83\) − 1.79998e24i − 1.84838i −0.381937 0.924189i \(-0.624743\pi\)
0.381937 0.924189i \(-0.375257\pi\)
\(84\) −7.68456e23 −0.679404
\(85\) 0 0
\(86\) 1.84717e24 1.21696
\(87\) 2.58149e23i 0.147190i
\(88\) − 9.66086e23i − 0.477510i
\(89\) −2.07122e24 −0.888898 −0.444449 0.895804i \(-0.646600\pi\)
−0.444449 + 0.895804i \(0.646600\pi\)
\(90\) 0 0
\(91\) −1.00909e24 −0.328029
\(92\) 2.40482e24i 0.681921i
\(93\) 6.24624e23i 0.154733i
\(94\) 2.27615e24 0.493290
\(95\) 0 0
\(96\) −8.24952e23 −0.137416
\(97\) − 3.44576e24i − 0.504242i −0.967696 0.252121i \(-0.918872\pi\)
0.967696 0.252121i \(-0.0811281\pi\)
\(98\) − 1.12912e25i − 1.45349i
\(99\) 4.71383e24 0.534483
\(100\) 0 0
\(101\) 1.35804e24 0.119921 0.0599607 0.998201i \(-0.480902\pi\)
0.0599607 + 0.998201i \(0.480902\pi\)
\(102\) 2.95126e24i 0.230412i
\(103\) 1.81348e25i 1.25328i 0.779310 + 0.626639i \(0.215570\pi\)
−0.779310 + 0.626639i \(0.784430\pi\)
\(104\) −1.08327e24 −0.0663471
\(105\) 0 0
\(106\) 1.94823e25 0.940409
\(107\) − 4.09213e25i − 1.75652i −0.478186 0.878259i \(-0.658705\pi\)
0.478186 0.878259i \(-0.341295\pi\)
\(108\) − 1.41966e25i − 0.542484i
\(109\) −4.31829e25 −1.47055 −0.735277 0.677767i \(-0.762948\pi\)
−0.735277 + 0.677767i \(0.762948\pi\)
\(110\) 0 0
\(111\) −5.27370e25 −1.43080
\(112\) − 1.80181e25i − 0.437003i
\(113\) 2.61097e25i 0.566659i 0.959023 + 0.283329i \(0.0914390\pi\)
−0.959023 + 0.283329i \(0.908561\pi\)
\(114\) −2.53994e25 −0.493756
\(115\) 0 0
\(116\) −6.05287e24 −0.0946753
\(117\) − 5.28563e24i − 0.0742631i
\(118\) 8.01888e25i 1.01295i
\(119\) −6.44599e25 −0.732744
\(120\) 0 0
\(121\) 8.92916e25 0.824126
\(122\) − 1.21197e25i − 0.100924i
\(123\) − 4.70694e25i − 0.353936i
\(124\) −1.46457e25 −0.0995266
\(125\) 0 0
\(126\) 8.79160e25 0.489144
\(127\) 3.07316e25i 0.154895i 0.996996 + 0.0774477i \(0.0246771\pi\)
−0.996996 + 0.0774477i \(0.975323\pi\)
\(128\) − 1.93428e25i − 0.0883883i
\(129\) 3.22682e26 1.33784
\(130\) 0 0
\(131\) 1.55205e26 0.530902 0.265451 0.964124i \(-0.414479\pi\)
0.265451 + 0.964124i \(0.414479\pi\)
\(132\) − 1.68766e26i − 0.524941i
\(133\) − 5.54760e26i − 1.57022i
\(134\) −1.74821e26 −0.450593
\(135\) 0 0
\(136\) −6.91988e25 −0.148205
\(137\) 4.44205e26i 0.868112i 0.900886 + 0.434056i \(0.142918\pi\)
−0.900886 + 0.434056i \(0.857082\pi\)
\(138\) 4.20098e26i 0.749656i
\(139\) −9.86493e26 −1.60845 −0.804227 0.594322i \(-0.797421\pi\)
−0.804227 + 0.594322i \(0.797421\pi\)
\(140\) 0 0
\(141\) 3.97621e26 0.542288
\(142\) − 7.94582e26i − 0.992051i
\(143\) − 2.21613e26i − 0.253451i
\(144\) 9.43795e25 0.0989342
\(145\) 0 0
\(146\) −8.65589e25 −0.0763661
\(147\) − 1.97246e27i − 1.59787i
\(148\) − 1.23654e27i − 0.920313i
\(149\) −1.38635e27 −0.948518 −0.474259 0.880385i \(-0.657284\pi\)
−0.474259 + 0.880385i \(0.657284\pi\)
\(150\) 0 0
\(151\) −9.02182e26 −0.522495 −0.261248 0.965272i \(-0.584134\pi\)
−0.261248 + 0.965272i \(0.584134\pi\)
\(152\) − 5.95545e26i − 0.317592i
\(153\) − 3.37642e26i − 0.165888i
\(154\) 3.68609e27 1.66939
\(155\) 0 0
\(156\) −1.89238e26 −0.0729373
\(157\) 2.62328e27i 0.933466i 0.884398 + 0.466733i \(0.154569\pi\)
−0.884398 + 0.466733i \(0.845431\pi\)
\(158\) 2.04555e27i 0.672351i
\(159\) 3.40337e27 1.03382
\(160\) 0 0
\(161\) −9.17555e27 −2.38401
\(162\) − 1.31634e27i − 0.316540i
\(163\) − 1.18488e27i − 0.263834i −0.991261 0.131917i \(-0.957887\pi\)
0.991261 0.131917i \(-0.0421133\pi\)
\(164\) 1.10365e27 0.227657
\(165\) 0 0
\(166\) 7.37270e27 1.30700
\(167\) 7.47611e27i 1.22948i 0.788732 + 0.614738i \(0.210738\pi\)
−0.788732 + 0.614738i \(0.789262\pi\)
\(168\) − 3.14760e27i − 0.480411i
\(169\) 6.80792e27 0.964785
\(170\) 0 0
\(171\) 2.90585e27 0.355485
\(172\) 7.56600e27i 0.860519i
\(173\) 3.35797e27i 0.355223i 0.984101 + 0.177611i \(0.0568370\pi\)
−0.984101 + 0.177611i \(0.943163\pi\)
\(174\) −1.05738e27 −0.104079
\(175\) 0 0
\(176\) 3.95709e27 0.337651
\(177\) 1.40082e28i 1.11357i
\(178\) − 8.48373e27i − 0.628546i
\(179\) −6.30645e27 −0.435634 −0.217817 0.975990i \(-0.569894\pi\)
−0.217817 + 0.975990i \(0.569894\pi\)
\(180\) 0 0
\(181\) 1.20387e28 0.723764 0.361882 0.932224i \(-0.382134\pi\)
0.361882 + 0.932224i \(0.382134\pi\)
\(182\) − 4.13322e27i − 0.231951i
\(183\) − 2.11720e27i − 0.110949i
\(184\) −9.85012e27 −0.482191
\(185\) 0 0
\(186\) −2.55846e27 −0.109413
\(187\) − 1.41565e28i − 0.566154i
\(188\) 9.32310e27i 0.348809i
\(189\) 5.41669e28 1.89654
\(190\) 0 0
\(191\) 2.13809e28 0.656309 0.328155 0.944624i \(-0.393573\pi\)
0.328155 + 0.944624i \(0.393573\pi\)
\(192\) − 3.37900e27i − 0.0971680i
\(193\) − 2.97996e27i − 0.0803054i −0.999194 0.0401527i \(-0.987216\pi\)
0.999194 0.0401527i \(-0.0127844\pi\)
\(194\) 1.41138e28 0.356553
\(195\) 0 0
\(196\) 4.62487e28 1.02778
\(197\) 4.10890e28i 0.856836i 0.903581 + 0.428418i \(0.140929\pi\)
−0.903581 + 0.428418i \(0.859071\pi\)
\(198\) 1.93078e28i 0.377937i
\(199\) 5.34406e28 0.982218 0.491109 0.871098i \(-0.336592\pi\)
0.491109 + 0.871098i \(0.336592\pi\)
\(200\) 0 0
\(201\) −3.05396e28 −0.495350
\(202\) 5.56255e27i 0.0847973i
\(203\) − 2.30947e28i − 0.330987i
\(204\) −1.20884e28 −0.162926
\(205\) 0 0
\(206\) −7.42801e28 −0.886202
\(207\) − 4.80618e28i − 0.539722i
\(208\) − 4.43709e27i − 0.0469145i
\(209\) 1.21835e29 1.21323
\(210\) 0 0
\(211\) 4.77946e28 0.422521 0.211260 0.977430i \(-0.432243\pi\)
0.211260 + 0.977430i \(0.432243\pi\)
\(212\) 7.97994e28i 0.664969i
\(213\) − 1.38806e29i − 1.09059i
\(214\) 1.67614e29 1.24205
\(215\) 0 0
\(216\) 5.81492e28 0.383594
\(217\) − 5.58805e28i − 0.347948i
\(218\) − 1.76877e29i − 1.03984i
\(219\) −1.51210e28 −0.0839516
\(220\) 0 0
\(221\) −1.58737e28 −0.0786637
\(222\) − 2.16011e29i − 1.01173i
\(223\) − 4.45198e29i − 1.97125i −0.168934 0.985627i \(-0.554032\pi\)
0.168934 0.985627i \(-0.445968\pi\)
\(224\) 7.38023e28 0.309008
\(225\) 0 0
\(226\) −1.06945e29 −0.400688
\(227\) − 2.11324e29i − 0.749248i −0.927177 0.374624i \(-0.877772\pi\)
0.927177 0.374624i \(-0.122228\pi\)
\(228\) − 1.04036e29i − 0.349138i
\(229\) −4.64740e29 −1.47661 −0.738307 0.674465i \(-0.764374\pi\)
−0.738307 + 0.674465i \(0.764374\pi\)
\(230\) 0 0
\(231\) 6.43925e29 1.83521
\(232\) − 2.47926e28i − 0.0669455i
\(233\) − 5.21268e29i − 1.33386i −0.745118 0.666932i \(-0.767607\pi\)
0.745118 0.666932i \(-0.232393\pi\)
\(234\) 2.16499e28 0.0525120
\(235\) 0 0
\(236\) −3.28453e29 −0.716265
\(237\) 3.57337e29i 0.739135i
\(238\) − 2.64028e29i − 0.518128i
\(239\) 9.68016e29 1.80264 0.901319 0.433155i \(-0.142600\pi\)
0.901319 + 0.433155i \(0.142600\pi\)
\(240\) 0 0
\(241\) −1.30231e29 −0.218525 −0.109263 0.994013i \(-0.534849\pi\)
−0.109263 + 0.994013i \(0.534849\pi\)
\(242\) 3.65738e29i 0.582745i
\(243\) 4.87009e29i 0.736985i
\(244\) 4.96424e28 0.0713641
\(245\) 0 0
\(246\) 1.92796e29 0.250271
\(247\) − 1.36613e29i − 0.168570i
\(248\) − 5.99888e28i − 0.0703759i
\(249\) 1.28794e30 1.43682
\(250\) 0 0
\(251\) 1.40543e30 1.41869 0.709345 0.704862i \(-0.248991\pi\)
0.709345 + 0.704862i \(0.248991\pi\)
\(252\) 3.60104e29i 0.345877i
\(253\) − 2.01511e30i − 1.84201i
\(254\) −1.25877e29 −0.109528
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) − 1.02386e30i − 0.769263i −0.923070 0.384632i \(-0.874329\pi\)
0.923070 0.384632i \(-0.125671\pi\)
\(258\) 1.32171e30i 0.945995i
\(259\) 4.71799e30 3.21744
\(260\) 0 0
\(261\) 1.20970e29 0.0749330
\(262\) 6.35719e29i 0.375404i
\(263\) 9.94329e29i 0.559865i 0.960020 + 0.279933i \(0.0903121\pi\)
−0.960020 + 0.279933i \(0.909688\pi\)
\(264\) 6.91265e29 0.371189
\(265\) 0 0
\(266\) 2.27230e30 1.11031
\(267\) − 1.48203e30i − 0.690979i
\(268\) − 7.16067e29i − 0.318617i
\(269\) −9.41553e29 −0.399891 −0.199945 0.979807i \(-0.564076\pi\)
−0.199945 + 0.979807i \(0.564076\pi\)
\(270\) 0 0
\(271\) −2.84694e29 −0.110220 −0.0551101 0.998480i \(-0.517551\pi\)
−0.0551101 + 0.998480i \(0.517551\pi\)
\(272\) − 2.83438e29i − 0.104797i
\(273\) − 7.22034e29i − 0.254991i
\(274\) −1.81946e30 −0.613848
\(275\) 0 0
\(276\) −1.72072e30 −0.530087
\(277\) 5.74193e30i 1.69068i 0.534230 + 0.845339i \(0.320602\pi\)
−0.534230 + 0.845339i \(0.679398\pi\)
\(278\) − 4.04067e30i − 1.13735i
\(279\) 2.92704e29 0.0787726
\(280\) 0 0
\(281\) 7.10013e29 0.174758 0.0873790 0.996175i \(-0.472151\pi\)
0.0873790 + 0.996175i \(0.472151\pi\)
\(282\) 1.62865e30i 0.383456i
\(283\) − 4.98063e30i − 1.12190i −0.827850 0.560950i \(-0.810436\pi\)
0.827850 0.560950i \(-0.189564\pi\)
\(284\) 3.25461e30 0.701486
\(285\) 0 0
\(286\) 9.07726e29 0.179217
\(287\) 4.21095e30i 0.795896i
\(288\) 3.86578e29i 0.0699571i
\(289\) 4.75663e30 0.824283
\(290\) 0 0
\(291\) 2.46555e30 0.391969
\(292\) − 3.54545e29i − 0.0539990i
\(293\) 4.89769e30i 0.714736i 0.933964 + 0.357368i \(0.116326\pi\)
−0.933964 + 0.357368i \(0.883674\pi\)
\(294\) 8.07920e30 1.12987
\(295\) 0 0
\(296\) 5.06485e30 0.650759
\(297\) 1.18960e31i 1.46536i
\(298\) − 5.67850e30i − 0.670704i
\(299\) −2.25954e30 −0.255936
\(300\) 0 0
\(301\) −2.88680e31 −3.00840
\(302\) − 3.69534e30i − 0.369460i
\(303\) 9.71724e29i 0.0932202i
\(304\) 2.43935e30 0.224572
\(305\) 0 0
\(306\) 1.38298e30 0.117300
\(307\) − 2.34261e31i − 1.90752i −0.300563 0.953762i \(-0.597175\pi\)
0.300563 0.953762i \(-0.402825\pi\)
\(308\) 1.50982e31i 1.18044i
\(309\) −1.29760e31 −0.974228
\(310\) 0 0
\(311\) 1.34425e31 0.931054 0.465527 0.885034i \(-0.345865\pi\)
0.465527 + 0.885034i \(0.345865\pi\)
\(312\) − 7.75117e29i − 0.0515745i
\(313\) − 2.25707e31i − 1.44292i −0.692458 0.721458i \(-0.743472\pi\)
0.692458 0.721458i \(-0.256528\pi\)
\(314\) −1.07449e31 −0.660060
\(315\) 0 0
\(316\) −8.37856e30 −0.475424
\(317\) 4.83401e29i 0.0263674i 0.999913 + 0.0131837i \(0.00419662\pi\)
−0.999913 + 0.0131837i \(0.995803\pi\)
\(318\) 1.39402e31i 0.731021i
\(319\) 5.07198e30 0.255737
\(320\) 0 0
\(321\) 2.92805e31 1.36542
\(322\) − 3.75831e31i − 1.68575i
\(323\) − 8.72677e30i − 0.376550i
\(324\) 5.39171e30 0.223828
\(325\) 0 0
\(326\) 4.85329e30 0.186559
\(327\) − 3.08987e31i − 1.14313i
\(328\) 4.52053e30i 0.160978i
\(329\) −3.55722e31 −1.21944
\(330\) 0 0
\(331\) −4.89856e31 −1.55675 −0.778375 0.627800i \(-0.783956\pi\)
−0.778375 + 0.627800i \(0.783956\pi\)
\(332\) 3.01986e31i 0.924189i
\(333\) 2.47130e31i 0.728403i
\(334\) −3.06221e31 −0.869370
\(335\) 0 0
\(336\) 1.28926e31 0.339702
\(337\) 3.74477e31i 0.950718i 0.879792 + 0.475359i \(0.157682\pi\)
−0.879792 + 0.475359i \(0.842318\pi\)
\(338\) 2.78852e31i 0.682206i
\(339\) −1.86823e31 −0.440489
\(340\) 0 0
\(341\) 1.22723e31 0.268842
\(342\) 1.19023e31i 0.251366i
\(343\) 9.06152e31i 1.84512i
\(344\) −3.09903e31 −0.608479
\(345\) 0 0
\(346\) −1.37543e31 −0.251181
\(347\) 1.26433e31i 0.222711i 0.993781 + 0.111356i \(0.0355193\pi\)
−0.993781 + 0.111356i \(0.964481\pi\)
\(348\) − 4.33102e30i − 0.0735952i
\(349\) −5.64423e31 −0.925308 −0.462654 0.886539i \(-0.653103\pi\)
−0.462654 + 0.886539i \(0.653103\pi\)
\(350\) 0 0
\(351\) 1.33390e31 0.203603
\(352\) 1.62082e31i 0.238755i
\(353\) 1.66522e31i 0.236749i 0.992969 + 0.118374i \(0.0377683\pi\)
−0.992969 + 0.118374i \(0.962232\pi\)
\(354\) −5.73776e31 −0.787412
\(355\) 0 0
\(356\) 3.47494e31 0.444449
\(357\) − 4.61231e31i − 0.569594i
\(358\) − 2.58312e31i − 0.308040i
\(359\) −1.12363e32 −1.29403 −0.647014 0.762478i \(-0.723982\pi\)
−0.647014 + 0.762478i \(0.723982\pi\)
\(360\) 0 0
\(361\) −1.79714e31 −0.193082
\(362\) 4.93104e31i 0.511778i
\(363\) 6.38910e31i 0.640629i
\(364\) 1.69297e31 0.164014
\(365\) 0 0
\(366\) 8.67204e30 0.0784527
\(367\) − 1.62554e32i − 1.42126i −0.703567 0.710629i \(-0.748410\pi\)
0.703567 0.710629i \(-0.251590\pi\)
\(368\) − 4.03461e31i − 0.340960i
\(369\) −2.20571e31 −0.180185
\(370\) 0 0
\(371\) −3.04474e32 −2.32475
\(372\) − 1.04795e31i − 0.0773663i
\(373\) − 8.65344e31i − 0.617772i −0.951099 0.308886i \(-0.900044\pi\)
0.951099 0.308886i \(-0.0999562\pi\)
\(374\) 5.79849e31 0.400332
\(375\) 0 0
\(376\) −3.81874e31 −0.246645
\(377\) − 5.68722e30i − 0.0355331i
\(378\) 2.21868e32i 1.34105i
\(379\) 1.30491e32 0.763117 0.381558 0.924345i \(-0.375388\pi\)
0.381558 + 0.924345i \(0.375388\pi\)
\(380\) 0 0
\(381\) −2.19894e31 −0.120407
\(382\) 8.75761e31i 0.464081i
\(383\) 3.19017e31i 0.163617i 0.996648 + 0.0818087i \(0.0260696\pi\)
−0.996648 + 0.0818087i \(0.973930\pi\)
\(384\) 1.38404e31 0.0687081
\(385\) 0 0
\(386\) 1.22059e31 0.0567845
\(387\) − 1.51211e32i − 0.681078i
\(388\) 5.78103e31i 0.252121i
\(389\) −1.71823e32 −0.725622 −0.362811 0.931863i \(-0.618183\pi\)
−0.362811 + 0.931863i \(0.618183\pi\)
\(390\) 0 0
\(391\) −1.44338e32 −0.571704
\(392\) 1.89435e32i 0.726747i
\(393\) 1.11054e32i 0.412693i
\(394\) −1.68301e32 −0.605874
\(395\) 0 0
\(396\) −7.90849e31 −0.267241
\(397\) 1.44908e32i 0.474471i 0.971452 + 0.237236i \(0.0762414\pi\)
−0.971452 + 0.237236i \(0.923759\pi\)
\(398\) 2.18893e32i 0.694533i
\(399\) 3.96948e32 1.22060
\(400\) 0 0
\(401\) 6.47108e32 1.86927 0.934635 0.355609i \(-0.115726\pi\)
0.934635 + 0.355609i \(0.115726\pi\)
\(402\) − 1.25090e32i − 0.350265i
\(403\) − 1.37610e31i − 0.0373539i
\(404\) −2.27842e31 −0.0599607
\(405\) 0 0
\(406\) 9.45958e31 0.234043
\(407\) 1.03615e33i 2.48595i
\(408\) − 4.95140e31i − 0.115206i
\(409\) 1.49705e32 0.337828 0.168914 0.985631i \(-0.445974\pi\)
0.168914 + 0.985631i \(0.445974\pi\)
\(410\) 0 0
\(411\) −3.17842e32 −0.674822
\(412\) − 3.04251e32i − 0.626639i
\(413\) − 1.25321e33i − 2.50408i
\(414\) 1.96861e32 0.381641
\(415\) 0 0
\(416\) 1.81743e31 0.0331735
\(417\) − 7.05867e32i − 1.25032i
\(418\) 4.99035e32i 0.857881i
\(419\) −6.23538e32 −1.04037 −0.520185 0.854054i \(-0.674137\pi\)
−0.520185 + 0.854054i \(0.674137\pi\)
\(420\) 0 0
\(421\) 4.35871e32 0.685223 0.342611 0.939477i \(-0.388689\pi\)
0.342611 + 0.939477i \(0.388689\pi\)
\(422\) 1.95767e32i 0.298767i
\(423\) − 1.86328e32i − 0.276073i
\(424\) −3.26859e32 −0.470204
\(425\) 0 0
\(426\) 5.68548e32 0.771165
\(427\) 1.89410e32i 0.249491i
\(428\) 6.86546e32i 0.878259i
\(429\) 1.58571e32 0.197019
\(430\) 0 0
\(431\) −9.60606e32 −1.12611 −0.563053 0.826420i \(-0.690373\pi\)
−0.563053 + 0.826420i \(0.690373\pi\)
\(432\) 2.38179e32i 0.271242i
\(433\) − 6.50298e32i − 0.719472i −0.933054 0.359736i \(-0.882867\pi\)
0.933054 0.359736i \(-0.117133\pi\)
\(434\) 2.28887e32 0.246036
\(435\) 0 0
\(436\) 7.24489e32 0.735277
\(437\) − 1.24221e33i − 1.22512i
\(438\) − 6.19356e31i − 0.0593627i
\(439\) 1.26999e33 1.18303 0.591515 0.806294i \(-0.298530\pi\)
0.591515 + 0.806294i \(0.298530\pi\)
\(440\) 0 0
\(441\) −9.24309e32 −0.813458
\(442\) − 6.50186e31i − 0.0556236i
\(443\) − 3.57910e32i − 0.297665i −0.988862 0.148832i \(-0.952449\pi\)
0.988862 0.148832i \(-0.0475515\pi\)
\(444\) 8.84781e32 0.715399
\(445\) 0 0
\(446\) 1.82353e33 1.39389
\(447\) − 9.91980e32i − 0.737325i
\(448\) 3.02294e32i 0.218502i
\(449\) 2.89265e32 0.203337 0.101669 0.994818i \(-0.467582\pi\)
0.101669 + 0.994818i \(0.467582\pi\)
\(450\) 0 0
\(451\) −9.24796e32 −0.614949
\(452\) − 4.38048e32i − 0.283329i
\(453\) − 6.45540e32i − 0.406158i
\(454\) 8.65583e32 0.529798
\(455\) 0 0
\(456\) 4.26131e32 0.246878
\(457\) 2.55962e33i 1.44285i 0.692491 + 0.721427i \(0.256513\pi\)
−0.692491 + 0.721427i \(0.743487\pi\)
\(458\) − 1.90358e33i − 1.04412i
\(459\) 8.52085e32 0.454804
\(460\) 0 0
\(461\) −7.93005e31 −0.0400880 −0.0200440 0.999799i \(-0.506381\pi\)
−0.0200440 + 0.999799i \(0.506381\pi\)
\(462\) 2.63752e33i 1.29769i
\(463\) − 3.30952e32i − 0.158490i −0.996855 0.0792452i \(-0.974749\pi\)
0.996855 0.0792452i \(-0.0252510\pi\)
\(464\) 1.01550e32 0.0473376
\(465\) 0 0
\(466\) 2.13511e33 0.943185
\(467\) − 2.96201e33i − 1.27387i −0.770917 0.636936i \(-0.780202\pi\)
0.770917 0.636936i \(-0.219798\pi\)
\(468\) 8.86781e31i 0.0371316i
\(469\) 2.73215e33 1.11389
\(470\) 0 0
\(471\) −1.87704e33 −0.725624
\(472\) − 1.34535e33i − 0.506476i
\(473\) − 6.33990e33i − 2.32444i
\(474\) −1.46365e33 −0.522648
\(475\) 0 0
\(476\) 1.08146e33 0.366372
\(477\) − 1.59484e33i − 0.526306i
\(478\) 3.96499e33i 1.27466i
\(479\) −1.66525e33 −0.521537 −0.260769 0.965401i \(-0.583976\pi\)
−0.260769 + 0.965401i \(0.583976\pi\)
\(480\) 0 0
\(481\) 1.16184e33 0.345408
\(482\) − 5.33427e32i − 0.154521i
\(483\) − 6.56540e33i − 1.85320i
\(484\) −1.49806e33 −0.412063
\(485\) 0 0
\(486\) −1.99479e33 −0.521127
\(487\) − 6.03193e33i − 1.53583i −0.640549 0.767917i \(-0.721293\pi\)
0.640549 0.767917i \(-0.278707\pi\)
\(488\) 2.03335e32i 0.0504620i
\(489\) 8.47822e32 0.205090
\(490\) 0 0
\(491\) 2.62144e33 0.602590 0.301295 0.953531i \(-0.402581\pi\)
0.301295 + 0.953531i \(0.402581\pi\)
\(492\) 7.89693e32i 0.176968i
\(493\) − 3.63296e32i − 0.0793732i
\(494\) 5.59568e32 0.119197
\(495\) 0 0
\(496\) 2.45714e32 0.0497633
\(497\) 1.24179e34i 2.45242i
\(498\) 5.27540e33i 1.01599i
\(499\) 8.04276e33 1.51060 0.755298 0.655382i \(-0.227492\pi\)
0.755298 + 0.655382i \(0.227492\pi\)
\(500\) 0 0
\(501\) −5.34939e33 −0.955725
\(502\) 5.75664e33i 1.00316i
\(503\) − 8.35523e33i − 1.42023i −0.704086 0.710115i \(-0.748643\pi\)
0.704086 0.710115i \(-0.251357\pi\)
\(504\) −1.47499e33 −0.244572
\(505\) 0 0
\(506\) 8.25387e33 1.30250
\(507\) 4.87128e33i 0.749969i
\(508\) − 5.15590e32i − 0.0774477i
\(509\) 7.38135e33 1.08184 0.540920 0.841074i \(-0.318076\pi\)
0.540920 + 0.841074i \(0.318076\pi\)
\(510\) 0 0
\(511\) 1.35276e33 0.188782
\(512\) 3.24519e32i 0.0441942i
\(513\) 7.33328e33i 0.974611i
\(514\) 4.19371e33 0.543951
\(515\) 0 0
\(516\) −5.41371e33 −0.668919
\(517\) − 7.81225e33i − 0.942203i
\(518\) 1.93249e34i 2.27507i
\(519\) −2.40274e33 −0.276130
\(520\) 0 0
\(521\) −1.16894e34 −1.28032 −0.640161 0.768241i \(-0.721133\pi\)
−0.640161 + 0.768241i \(0.721133\pi\)
\(522\) 4.95495e32i 0.0529856i
\(523\) 1.32178e34i 1.38003i 0.723796 + 0.690014i \(0.242396\pi\)
−0.723796 + 0.690014i \(0.757604\pi\)
\(524\) −2.60391e33 −0.265451
\(525\) 0 0
\(526\) −4.07277e33 −0.395884
\(527\) − 8.79041e32i − 0.0834404i
\(528\) 2.83142e33i 0.262470i
\(529\) −9.50007e33 −0.860064
\(530\) 0 0
\(531\) 6.56435e33 0.566905
\(532\) 9.30733e33i 0.785108i
\(533\) 1.03698e33i 0.0854434i
\(534\) 6.07037e33 0.488596
\(535\) 0 0
\(536\) 2.93301e33 0.225296
\(537\) − 4.51247e33i − 0.338638i
\(538\) − 3.85660e33i − 0.282765i
\(539\) −3.87539e34 −2.77623
\(540\) 0 0
\(541\) −1.02274e34 −0.699522 −0.349761 0.936839i \(-0.613737\pi\)
−0.349761 + 0.936839i \(0.613737\pi\)
\(542\) − 1.16610e33i − 0.0779375i
\(543\) 8.61406e33i 0.562613i
\(544\) 1.16096e33 0.0741024
\(545\) 0 0
\(546\) 2.95745e33 0.180306
\(547\) − 9.41592e33i − 0.561075i −0.959843 0.280538i \(-0.909487\pi\)
0.959843 0.280538i \(-0.0905127\pi\)
\(548\) − 7.45252e33i − 0.434056i
\(549\) −9.92134e32 −0.0564828
\(550\) 0 0
\(551\) 3.12663e33 0.170091
\(552\) − 7.04807e33i − 0.374828i
\(553\) − 3.19683e34i − 1.66209i
\(554\) −2.35190e34 −1.19549
\(555\) 0 0
\(556\) 1.65506e34 0.804227
\(557\) − 1.93891e34i − 0.921227i −0.887601 0.460614i \(-0.847629\pi\)
0.887601 0.460614i \(-0.152371\pi\)
\(558\) 1.19891e33i 0.0557007i
\(559\) −7.10894e33 −0.322966
\(560\) 0 0
\(561\) 1.01294e34 0.440097
\(562\) 2.90821e33i 0.123573i
\(563\) 2.14770e34i 0.892520i 0.894903 + 0.446260i \(0.147244\pi\)
−0.894903 + 0.446260i \(0.852756\pi\)
\(564\) −6.67097e33 −0.271144
\(565\) 0 0
\(566\) 2.04007e34 0.793303
\(567\) 2.05720e34i 0.782508i
\(568\) 1.33309e34i 0.496026i
\(569\) 1.35765e33 0.0494180 0.0247090 0.999695i \(-0.492134\pi\)
0.0247090 + 0.999695i \(0.492134\pi\)
\(570\) 0 0
\(571\) −1.57397e34 −0.548334 −0.274167 0.961682i \(-0.588402\pi\)
−0.274167 + 0.961682i \(0.588402\pi\)
\(572\) 3.71804e33i 0.126726i
\(573\) 1.52987e34i 0.510178i
\(574\) −1.72481e34 −0.562784
\(575\) 0 0
\(576\) −1.58343e33 −0.0494671
\(577\) − 5.82494e34i − 1.78071i −0.455266 0.890356i \(-0.650456\pi\)
0.455266 0.890356i \(-0.349544\pi\)
\(578\) 1.94831e34i 0.582856i
\(579\) 2.13226e33 0.0624249
\(580\) 0 0
\(581\) −1.15222e35 −3.23099
\(582\) 1.00989e34i 0.277164i
\(583\) − 6.68676e34i − 1.79622i
\(584\) 1.45222e33 0.0381831
\(585\) 0 0
\(586\) −2.00609e34 −0.505395
\(587\) 4.94816e33i 0.122030i 0.998137 + 0.0610151i \(0.0194338\pi\)
−0.998137 + 0.0610151i \(0.980566\pi\)
\(588\) 3.30924e34i 0.798935i
\(589\) 7.56528e33 0.178807
\(590\) 0 0
\(591\) −2.94005e34 −0.666056
\(592\) 2.07456e34i 0.460156i
\(593\) 5.71306e34i 1.24075i 0.784304 + 0.620376i \(0.213020\pi\)
−0.784304 + 0.620376i \(0.786980\pi\)
\(594\) −4.87259e34 −1.03617
\(595\) 0 0
\(596\) 2.32592e34 0.474259
\(597\) 3.82384e34i 0.763521i
\(598\) − 9.25509e33i − 0.180974i
\(599\) −4.92727e34 −0.943564 −0.471782 0.881715i \(-0.656389\pi\)
−0.471782 + 0.881715i \(0.656389\pi\)
\(600\) 0 0
\(601\) 7.24167e34 1.33017 0.665086 0.746767i \(-0.268395\pi\)
0.665086 + 0.746767i \(0.268395\pi\)
\(602\) − 1.18243e35i − 2.12726i
\(603\) 1.43111e34i 0.252177i
\(604\) 1.51361e34 0.261248
\(605\) 0 0
\(606\) −3.98018e33 −0.0659166
\(607\) − 2.52292e34i − 0.409304i −0.978835 0.204652i \(-0.934394\pi\)
0.978835 0.204652i \(-0.0656062\pi\)
\(608\) 9.99158e33i 0.158796i
\(609\) 1.65250e34 0.257291
\(610\) 0 0
\(611\) −8.75990e33 −0.130913
\(612\) 5.66470e33i 0.0829438i
\(613\) − 8.87757e34i − 1.27362i −0.771023 0.636808i \(-0.780255\pi\)
0.771023 0.636808i \(-0.219745\pi\)
\(614\) 9.59532e34 1.34882
\(615\) 0 0
\(616\) −6.18424e34 −0.834694
\(617\) 3.04657e34i 0.402946i 0.979494 + 0.201473i \(0.0645728\pi\)
−0.979494 + 0.201473i \(0.935427\pi\)
\(618\) − 5.31498e34i − 0.688883i
\(619\) −8.61565e34 −1.09435 −0.547173 0.837019i \(-0.684296\pi\)
−0.547173 + 0.837019i \(0.684296\pi\)
\(620\) 0 0
\(621\) 1.21290e35 1.47972
\(622\) 5.50605e34i 0.658355i
\(623\) 1.32586e35i 1.55381i
\(624\) 3.17488e33 0.0364687
\(625\) 0 0
\(626\) 9.24496e34 1.02030
\(627\) 8.71765e34i 0.943094i
\(628\) − 4.40113e34i − 0.466733i
\(629\) 7.42174e34 0.771566
\(630\) 0 0
\(631\) −3.11716e34 −0.311452 −0.155726 0.987800i \(-0.549772\pi\)
−0.155726 + 0.987800i \(0.549772\pi\)
\(632\) − 3.43186e34i − 0.336175i
\(633\) 3.41985e34i 0.328444i
\(634\) −1.98001e33 −0.0186446
\(635\) 0 0
\(636\) −5.70990e34 −0.516910
\(637\) 4.34548e34i 0.385741i
\(638\) 2.07748e34i 0.180834i
\(639\) −6.50454e34 −0.555208
\(640\) 0 0
\(641\) 1.40132e35 1.15030 0.575150 0.818048i \(-0.304944\pi\)
0.575150 + 0.818048i \(0.304944\pi\)
\(642\) 1.19933e35i 0.965496i
\(643\) − 2.61025e34i − 0.206085i −0.994677 0.103042i \(-0.967142\pi\)
0.994677 0.103042i \(-0.0328577\pi\)
\(644\) 1.53940e35 1.19201
\(645\) 0 0
\(646\) 3.57449e34 0.266261
\(647\) 1.90009e35i 1.38826i 0.719848 + 0.694131i \(0.244211\pi\)
−0.719848 + 0.694131i \(0.755789\pi\)
\(648\) 2.20844e34i 0.158270i
\(649\) 2.75226e35 1.93478
\(650\) 0 0
\(651\) 3.99843e34 0.270475
\(652\) 1.98791e34i 0.131917i
\(653\) − 1.69030e35i − 1.10040i −0.835034 0.550198i \(-0.814552\pi\)
0.835034 0.550198i \(-0.185448\pi\)
\(654\) 1.26561e35 0.808312
\(655\) 0 0
\(656\) −1.85161e34 −0.113829
\(657\) 7.08581e33i 0.0427388i
\(658\) − 1.45704e35i − 0.862277i
\(659\) 2.20649e35 1.28125 0.640627 0.767853i \(-0.278675\pi\)
0.640627 + 0.767853i \(0.278675\pi\)
\(660\) 0 0
\(661\) 9.29705e34 0.519789 0.259894 0.965637i \(-0.416312\pi\)
0.259894 + 0.965637i \(0.416312\pi\)
\(662\) − 2.00645e35i − 1.10079i
\(663\) − 1.13581e34i − 0.0611487i
\(664\) −1.23693e35 −0.653500
\(665\) 0 0
\(666\) −1.01224e35 −0.515059
\(667\) − 5.17134e34i − 0.258244i
\(668\) − 1.25428e35i − 0.614738i
\(669\) 3.18553e35 1.53234
\(670\) 0 0
\(671\) −4.15976e34 −0.192769
\(672\) 5.28079e34i 0.240205i
\(673\) 1.44593e35i 0.645593i 0.946468 + 0.322796i \(0.104623\pi\)
−0.946468 + 0.322796i \(0.895377\pi\)
\(674\) −1.53386e35 −0.672259
\(675\) 0 0
\(676\) −1.14218e35 −0.482392
\(677\) − 2.51992e35i − 1.04479i −0.852704 0.522394i \(-0.825039\pi\)
0.852704 0.522394i \(-0.174961\pi\)
\(678\) − 7.65228e34i − 0.311472i
\(679\) −2.20575e35 −0.881422
\(680\) 0 0
\(681\) 1.51209e35 0.582423
\(682\) 5.02674e34i 0.190100i
\(683\) − 1.41934e35i − 0.527022i −0.964656 0.263511i \(-0.915119\pi\)
0.964656 0.263511i \(-0.0848806\pi\)
\(684\) −4.87520e34 −0.177742
\(685\) 0 0
\(686\) −3.71160e35 −1.30470
\(687\) − 3.32536e35i − 1.14784i
\(688\) − 1.26936e35i − 0.430260i
\(689\) −7.49788e34 −0.249573
\(690\) 0 0
\(691\) 4.26858e34 0.137028 0.0685138 0.997650i \(-0.478174\pi\)
0.0685138 + 0.997650i \(0.478174\pi\)
\(692\) − 5.63374e34i − 0.177611i
\(693\) − 3.01748e35i − 0.934284i
\(694\) −5.17870e34 −0.157481
\(695\) 0 0
\(696\) 1.77399e34 0.0520397
\(697\) 6.62413e34i 0.190862i
\(698\) − 2.31187e35i − 0.654292i
\(699\) 3.72984e35 1.03687
\(700\) 0 0
\(701\) −7.85276e34 −0.210643 −0.105322 0.994438i \(-0.533587\pi\)
−0.105322 + 0.994438i \(0.533587\pi\)
\(702\) 5.46364e34i 0.143969i
\(703\) 6.38736e35i 1.65341i
\(704\) −6.63889e34 −0.168825
\(705\) 0 0
\(706\) −6.82074e34 −0.167407
\(707\) − 8.69330e34i − 0.209624i
\(708\) − 2.35019e35i − 0.556784i
\(709\) 3.06083e35 0.712461 0.356231 0.934398i \(-0.384062\pi\)
0.356231 + 0.934398i \(0.384062\pi\)
\(710\) 0 0
\(711\) 1.67451e35 0.376285
\(712\) 1.42333e35i 0.314273i
\(713\) − 1.25127e35i − 0.271477i
\(714\) 1.88920e35 0.402764
\(715\) 0 0
\(716\) 1.05805e35 0.217817
\(717\) 6.92646e35i 1.40127i
\(718\) − 4.60240e35i − 0.915016i
\(719\) −2.20247e35 −0.430327 −0.215163 0.976578i \(-0.569028\pi\)
−0.215163 + 0.976578i \(0.569028\pi\)
\(720\) 0 0
\(721\) 1.16087e36 2.19075
\(722\) − 7.36109e34i − 0.136530i
\(723\) − 9.31844e34i − 0.169869i
\(724\) −2.01976e35 −0.361882
\(725\) 0 0
\(726\) −2.61697e35 −0.452993
\(727\) 8.70105e33i 0.0148044i 0.999973 + 0.00740221i \(0.00235622\pi\)
−0.999973 + 0.00740221i \(0.997644\pi\)
\(728\) 6.93440e34i 0.115976i
\(729\) −6.20764e35 −1.02055
\(730\) 0 0
\(731\) −4.54115e35 −0.721437
\(732\) 3.55207e34i 0.0554744i
\(733\) 1.92172e35i 0.295046i 0.989059 + 0.147523i \(0.0471301\pi\)
−0.989059 + 0.147523i \(0.952870\pi\)
\(734\) 6.65823e35 1.00498
\(735\) 0 0
\(736\) 1.65258e35 0.241095
\(737\) 6.00026e35i 0.860650i
\(738\) − 9.03457e34i − 0.127410i
\(739\) −9.08434e35 −1.25961 −0.629807 0.776751i \(-0.716866\pi\)
−0.629807 + 0.776751i \(0.716866\pi\)
\(740\) 0 0
\(741\) 9.77512e34 0.131037
\(742\) − 1.24713e36i − 1.64385i
\(743\) 9.13810e35i 1.18439i 0.805794 + 0.592197i \(0.201739\pi\)
−0.805794 + 0.592197i \(0.798261\pi\)
\(744\) 4.29239e34 0.0547063
\(745\) 0 0
\(746\) 3.54445e35 0.436831
\(747\) − 6.03538e35i − 0.731471i
\(748\) 2.37506e35i 0.283077i
\(749\) −2.61951e36 −3.07042
\(750\) 0 0
\(751\) 3.01650e35 0.341982 0.170991 0.985273i \(-0.445303\pi\)
0.170991 + 0.985273i \(0.445303\pi\)
\(752\) − 1.56416e35i − 0.174404i
\(753\) 1.00563e36i 1.10281i
\(754\) 2.32949e34 0.0251257
\(755\) 0 0
\(756\) −9.08770e35 −0.948269
\(757\) − 1.11391e36i − 1.14328i −0.820504 0.571640i \(-0.806307\pi\)
0.820504 0.571640i \(-0.193693\pi\)
\(758\) 5.34492e35i 0.539605i
\(759\) 1.44187e36 1.43187
\(760\) 0 0
\(761\) −8.13224e35 −0.781451 −0.390726 0.920507i \(-0.627776\pi\)
−0.390726 + 0.920507i \(0.627776\pi\)
\(762\) − 9.00686e34i − 0.0851405i
\(763\) 2.76428e36i 2.57055i
\(764\) −3.58712e35 −0.328155
\(765\) 0 0
\(766\) −1.30669e35 −0.115695
\(767\) − 3.08612e35i − 0.268826i
\(768\) 5.66902e34i 0.0485840i
\(769\) 9.19378e35 0.775202 0.387601 0.921827i \(-0.373304\pi\)
0.387601 + 0.921827i \(0.373304\pi\)
\(770\) 0 0
\(771\) 7.32601e35 0.597982
\(772\) 4.99955e34i 0.0401527i
\(773\) − 1.53774e36i − 1.21518i −0.794251 0.607590i \(-0.792136\pi\)
0.794251 0.607590i \(-0.207864\pi\)
\(774\) 6.19361e35 0.481595
\(775\) 0 0
\(776\) −2.36791e35 −0.178276
\(777\) 3.37587e36i 2.50106i
\(778\) − 7.03786e35i − 0.513093i
\(779\) −5.70091e35 −0.409003
\(780\) 0 0
\(781\) −2.72718e36 −1.89486
\(782\) − 5.91209e35i − 0.404256i
\(783\) 3.05285e35i 0.205439i
\(784\) −7.75924e35 −0.513888
\(785\) 0 0
\(786\) −4.54877e35 −0.291818
\(787\) 1.78317e36i 1.12592i 0.826484 + 0.562960i \(0.190337\pi\)
−0.826484 + 0.562960i \(0.809663\pi\)
\(788\) − 6.89359e35i − 0.428418i
\(789\) −7.11474e35 −0.435208
\(790\) 0 0
\(791\) 1.67137e36 0.990527
\(792\) − 3.23932e35i − 0.188968i
\(793\) 4.66435e34i 0.0267841i
\(794\) −5.93542e35 −0.335502
\(795\) 0 0
\(796\) −8.96584e35 −0.491109
\(797\) 4.10600e35i 0.221406i 0.993854 + 0.110703i \(0.0353103\pi\)
−0.993854 + 0.110703i \(0.964690\pi\)
\(798\) 1.62590e36i 0.863093i
\(799\) −5.59576e35 −0.292432
\(800\) 0 0
\(801\) −6.94488e35 −0.351770
\(802\) 2.65055e36i 1.32177i
\(803\) 2.97090e35i 0.145862i
\(804\) 5.12369e35 0.247675
\(805\) 0 0
\(806\) 5.63649e34 0.0264132
\(807\) − 6.73711e35i − 0.310853i
\(808\) − 9.33241e34i − 0.0423986i
\(809\) 2.45266e36 1.09719 0.548593 0.836090i \(-0.315164\pi\)
0.548593 + 0.836090i \(0.315164\pi\)
\(810\) 0 0
\(811\) −3.62915e36 −1.57414 −0.787072 0.616861i \(-0.788404\pi\)
−0.787072 + 0.616861i \(0.788404\pi\)
\(812\) 3.87464e35i 0.165494i
\(813\) − 2.03707e35i − 0.0856790i
\(814\) −4.24407e36 −1.75783
\(815\) 0 0
\(816\) 2.02809e35 0.0814630
\(817\) − 3.90824e36i − 1.54599i
\(818\) 6.13193e35i 0.238880i
\(819\) −3.38351e35 −0.129813
\(820\) 0 0
\(821\) 4.84702e36 1.80379 0.901894 0.431957i \(-0.142177\pi\)
0.901894 + 0.431957i \(0.142177\pi\)
\(822\) − 1.30188e36i − 0.477171i
\(823\) − 3.60517e36i − 1.30145i −0.759312 0.650726i \(-0.774465\pi\)
0.759312 0.650726i \(-0.225535\pi\)
\(824\) 1.24621e36 0.443101
\(825\) 0 0
\(826\) 5.13315e36 1.77065
\(827\) − 1.03992e36i − 0.353331i −0.984271 0.176665i \(-0.943469\pi\)
0.984271 0.176665i \(-0.0565311\pi\)
\(828\) 8.06343e35i 0.269861i
\(829\) −3.59238e36 −1.18427 −0.592134 0.805839i \(-0.701714\pi\)
−0.592134 + 0.805839i \(0.701714\pi\)
\(830\) 0 0
\(831\) −4.10853e36 −1.31424
\(832\) 7.44421e34i 0.0234572i
\(833\) 2.77587e36i 0.861660i
\(834\) 2.89123e36 0.884111
\(835\) 0 0
\(836\) −2.04405e36 −0.606613
\(837\) 7.38676e35i 0.215966i
\(838\) − 2.55401e36i − 0.735652i
\(839\) −4.65321e36 −1.32047 −0.660234 0.751060i \(-0.729543\pi\)
−0.660234 + 0.751060i \(0.729543\pi\)
\(840\) 0 0
\(841\) −3.50020e36 −0.964146
\(842\) 1.78533e36i 0.484526i
\(843\) 5.08037e35i 0.135847i
\(844\) −8.01860e35 −0.211260
\(845\) 0 0
\(846\) 7.63200e35 0.195213
\(847\) − 5.71585e36i − 1.44058i
\(848\) − 1.33881e36i − 0.332485i
\(849\) 3.56380e36 0.872102
\(850\) 0 0
\(851\) 1.05645e37 2.51032
\(852\) 2.32877e36i 0.545296i
\(853\) 5.05930e36i 1.16742i 0.811962 + 0.583710i \(0.198400\pi\)
−0.811962 + 0.583710i \(0.801600\pi\)
\(854\) −7.75823e35 −0.176417
\(855\) 0 0
\(856\) −2.81209e36 −0.621023
\(857\) − 2.18945e36i − 0.476513i −0.971202 0.238257i \(-0.923424\pi\)
0.971202 0.238257i \(-0.0765760\pi\)
\(858\) 6.49506e35i 0.139313i
\(859\) 5.91491e36 1.25035 0.625177 0.780483i \(-0.285027\pi\)
0.625177 + 0.780483i \(0.285027\pi\)
\(860\) 0 0
\(861\) −3.01307e36 −0.618685
\(862\) − 3.93464e36i − 0.796278i
\(863\) − 3.85509e36i − 0.768952i −0.923135 0.384476i \(-0.874382\pi\)
0.923135 0.384476i \(-0.125618\pi\)
\(864\) −9.75581e35 −0.191797
\(865\) 0 0
\(866\) 2.66362e36 0.508744
\(867\) 3.40352e36i 0.640751i
\(868\) 9.37520e35i 0.173974i
\(869\) 7.02078e36 1.28422
\(870\) 0 0
\(871\) 6.72810e35 0.119582
\(872\) 2.96751e36i 0.519919i
\(873\) − 1.15538e36i − 0.199547i
\(874\) 5.08811e36 0.866291
\(875\) 0 0
\(876\) 2.53688e35 0.0419758
\(877\) − 5.04839e36i − 0.823489i −0.911299 0.411744i \(-0.864920\pi\)
0.911299 0.411744i \(-0.135080\pi\)
\(878\) 5.20190e36i 0.836528i
\(879\) −3.50445e36 −0.555596
\(880\) 0 0
\(881\) 7.50642e36 1.15673 0.578367 0.815776i \(-0.303690\pi\)
0.578367 + 0.815776i \(0.303690\pi\)
\(882\) − 3.78597e36i − 0.575201i
\(883\) − 1.15945e37i − 1.73677i −0.495893 0.868384i \(-0.665159\pi\)
0.495893 0.868384i \(-0.334841\pi\)
\(884\) 2.66316e35 0.0393318
\(885\) 0 0
\(886\) 1.46600e36 0.210481
\(887\) 1.11924e37i 1.58445i 0.610228 + 0.792226i \(0.291078\pi\)
−0.610228 + 0.792226i \(0.708922\pi\)
\(888\) 3.62406e36i 0.505864i
\(889\) 1.96723e36 0.270759
\(890\) 0 0
\(891\) −4.51796e36 −0.604604
\(892\) 7.46919e36i 0.985627i
\(893\) − 4.81587e36i − 0.626660i
\(894\) 4.06315e36 0.521367
\(895\) 0 0
\(896\) −1.23820e36 −0.154504
\(897\) − 1.61677e36i − 0.198950i
\(898\) 1.18483e36i 0.143781i
\(899\) 3.14943e35 0.0376908
\(900\) 0 0
\(901\) −4.78960e36 −0.557493
\(902\) − 3.78796e36i − 0.434835i
\(903\) − 2.06560e37i − 2.33856i
\(904\) 1.79425e36 0.200344
\(905\) 0 0
\(906\) 2.64413e36 0.287197
\(907\) 1.70146e36i 0.182276i 0.995838 + 0.0911381i \(0.0290505\pi\)
−0.995838 + 0.0911381i \(0.970950\pi\)
\(908\) 3.54543e36i 0.374624i
\(909\) 4.55357e35 0.0474573
\(910\) 0 0
\(911\) −1.90683e36 −0.193344 −0.0966721 0.995316i \(-0.530820\pi\)
−0.0966721 + 0.995316i \(0.530820\pi\)
\(912\) 1.74543e36i 0.174569i
\(913\) − 2.53048e37i − 2.49642i
\(914\) −1.04842e37 −1.02025
\(915\) 0 0
\(916\) 7.79705e36 0.738307
\(917\) − 9.93518e36i − 0.928024i
\(918\) 3.49014e36i 0.321595i
\(919\) 1.39528e37 1.26829 0.634145 0.773214i \(-0.281352\pi\)
0.634145 + 0.773214i \(0.281352\pi\)
\(920\) 0 0
\(921\) 1.67621e37 1.48280
\(922\) − 3.24815e35i − 0.0283465i
\(923\) 3.05800e36i 0.263279i
\(924\) −1.08033e37 −0.917604
\(925\) 0 0
\(926\) 1.35558e36 0.112070
\(927\) 6.08066e36i 0.495968i
\(928\) 4.15950e35i 0.0334728i
\(929\) −3.87552e36 −0.307704 −0.153852 0.988094i \(-0.549168\pi\)
−0.153852 + 0.988094i \(0.549168\pi\)
\(930\) 0 0
\(931\) −2.38899e37 −1.84647
\(932\) 8.74542e36i 0.666932i
\(933\) 9.61853e36i 0.723749i
\(934\) 1.21324e37 0.900763
\(935\) 0 0
\(936\) −3.63226e35 −0.0262560
\(937\) − 7.59791e36i − 0.541938i −0.962588 0.270969i \(-0.912656\pi\)
0.962588 0.270969i \(-0.0873440\pi\)
\(938\) 1.11909e37i 0.787642i
\(939\) 1.61500e37 1.12164
\(940\) 0 0
\(941\) −2.56881e37 −1.73725 −0.868624 0.495472i \(-0.834995\pi\)
−0.868624 + 0.495472i \(0.834995\pi\)
\(942\) − 7.68835e36i − 0.513094i
\(943\) 9.42913e36i 0.620977i
\(944\) 5.51053e36 0.358133
\(945\) 0 0
\(946\) 2.59682e37 1.64363
\(947\) 1.90718e37i 1.19129i 0.803247 + 0.595645i \(0.203104\pi\)
−0.803247 + 0.595645i \(0.796896\pi\)
\(948\) − 5.99512e36i − 0.369568i
\(949\) 3.33127e35 0.0202667
\(950\) 0 0
\(951\) −3.45889e35 −0.0204965
\(952\) 4.42965e36i 0.259064i
\(953\) 1.68185e36i 0.0970791i 0.998821 + 0.0485396i \(0.0154567\pi\)
−0.998821 + 0.0485396i \(0.984543\pi\)
\(954\) 6.53247e36 0.372154
\(955\) 0 0
\(956\) −1.62406e37 −0.901319
\(957\) 3.62916e36i 0.198796i
\(958\) − 6.82086e36i − 0.368783i
\(959\) 2.84350e37 1.51747
\(960\) 0 0
\(961\) −1.84707e37 −0.960378
\(962\) 4.75889e36i 0.244240i
\(963\) − 1.37211e37i − 0.695119i
\(964\) 2.18492e36 0.109263
\(965\) 0 0
\(966\) 2.68919e37 1.31041
\(967\) − 4.14643e36i − 0.199454i −0.995015 0.0997271i \(-0.968203\pi\)
0.995015 0.0997271i \(-0.0317970\pi\)
\(968\) − 6.13607e36i − 0.291373i
\(969\) 6.24428e36 0.292708
\(970\) 0 0
\(971\) 2.14239e37 0.978720 0.489360 0.872082i \(-0.337230\pi\)
0.489360 + 0.872082i \(0.337230\pi\)
\(972\) − 8.17066e36i − 0.368493i
\(973\) 6.31487e37i 2.81160i
\(974\) 2.47068e37 1.08600
\(975\) 0 0
\(976\) −8.32861e35 −0.0356820
\(977\) − 1.15529e37i − 0.488660i −0.969692 0.244330i \(-0.921432\pi\)
0.969692 0.244330i \(-0.0785681\pi\)
\(978\) 3.47268e36i 0.145020i
\(979\) −2.91181e37 −1.20055
\(980\) 0 0
\(981\) −1.44794e37 −0.581952
\(982\) 1.07374e37i 0.426096i
\(983\) 5.38686e36i 0.211065i 0.994416 + 0.105533i \(0.0336547\pi\)
−0.994416 + 0.105533i \(0.966345\pi\)
\(984\) −3.23458e36 −0.125135
\(985\) 0 0
\(986\) 1.48806e36 0.0561254
\(987\) − 2.54530e37i − 0.947927i
\(988\) 2.29199e36i 0.0842852i
\(989\) −6.46410e37 −2.34722
\(990\) 0 0
\(991\) 3.50366e37 1.24051 0.620256 0.784399i \(-0.287029\pi\)
0.620256 + 0.784399i \(0.287029\pi\)
\(992\) 1.00644e36i 0.0351880i
\(993\) − 3.50508e37i − 1.21013i
\(994\) −5.08638e37 −1.73412
\(995\) 0 0
\(996\) −2.16080e37 −0.718412
\(997\) 4.42953e37i 1.45435i 0.686453 + 0.727174i \(0.259167\pi\)
−0.686453 + 0.727174i \(0.740833\pi\)
\(998\) 3.29431e37i 1.06815i
\(999\) −6.23664e37 −1.99702
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 50.26.b.d.49.4 4
5.2 odd 4 50.26.a.d.1.2 2
5.3 odd 4 10.26.a.d.1.1 2
5.4 even 2 inner 50.26.b.d.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.a.d.1.1 2 5.3 odd 4
50.26.a.d.1.2 2 5.2 odd 4
50.26.b.d.49.1 4 5.4 even 2 inner
50.26.b.d.49.4 4 1.1 even 1 trivial