Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00i q^{2} -715531. i q^{3} -1.67772e7 q^{4} -2.93082e9 q^{6} +6.40133e10i q^{7} +6.87195e10i q^{8} +3.35303e11 q^{9} +1.40584e13 q^{11} +1.20046e13i q^{12} +1.57637e13i q^{13} +2.62199e14 q^{14} +2.81475e14 q^{16} +1.00698e15i q^{17} -1.37340e15i q^{18} +8.66632e15 q^{19} +4.58035e16 q^{21} -5.75832e16i q^{22} +1.43338e17i q^{23} +4.91709e16 q^{24} +6.45682e16 q^{26} -8.46182e17i q^{27} -1.07397e18i q^{28} +3.60779e17 q^{29} +8.72952e17 q^{31} -1.15292e18i q^{32} -1.00592e19i q^{33} +4.12457e18 q^{34} -5.62546e18 q^{36} -7.37033e19i q^{37} -3.54972e19i q^{38} +1.12794e19 q^{39} -6.57824e19 q^{41} -1.87611e20i q^{42} +4.50969e20i q^{43} -2.35861e20 q^{44} +5.87113e20 q^{46} +5.55700e20i q^{47} -2.01404e20i q^{48} -2.75664e21 q^{49} +7.20523e20 q^{51} -2.64471e20i q^{52} +4.75642e21i q^{53} -3.46596e21 q^{54} -4.39896e21 q^{56} -6.20102e21i q^{57} -1.47775e21i q^{58} +1.95774e22 q^{59} -2.95892e21 q^{61} -3.57561e21i q^{62} +2.14639e22i q^{63} -4.72237e21 q^{64} -4.12026e22 q^{66} -4.26809e22i q^{67} -1.68942e22i q^{68} +1.02563e23 q^{69} -1.93990e23 q^{71} +2.30419e22i q^{72} -2.11325e22i q^{73} -3.01889e23 q^{74} -1.45397e23 q^{76} +8.99925e23i q^{77} -4.62006e22i q^{78} +4.99401e23 q^{79} -3.21371e23 q^{81} +2.69445e23i q^{82} +1.79998e24i q^{83} -7.68456e23 q^{84} +1.84717e24 q^{86} -2.58149e23i q^{87} +9.66086e23i q^{88} -2.07122e24 q^{89} -1.00909e24 q^{91} -2.40482e24i q^{92} -6.24624e23i q^{93} +2.27615e24 q^{94} -8.24952e23 q^{96} +3.44576e24i q^{97} +1.12912e25i q^{98} +4.71383e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 67108864 q^{4} - 795377664 q^{6} + 1600249078908 q^{9} + 28833201602688 q^{11} + 132195615735808 q^{14} + 11\!\cdots\!24 q^{16} + 76\!\cdots\!40 q^{19} + 15\!\cdots\!08 q^{21} + 13\!\cdots\!24 q^{24}+ \cdots + 97\!\cdots\!76 q^{99}+O(q^{100}) \) 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 \(-0.872934\pi\)
0.921376 0.388672i \(-0.127066\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) −2.93082e9 −0.549665
\(7\) 6.40133e10i 1.74801i 0.485914 + 0.874007i \(0.338487\pi\)
−0.485914 + 0.874007i \(0.661513\pi\)
\(8\) 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 \(0.0299107\pi\)
−0.995588 + 0.0938289i \(0.970089\pi\)
\(14\) 2.62199e14 1.23603
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 1.00698e15i 0.419187i 0.977789 + 0.209593i \(0.0672140\pi\)
−0.977789 + 0.209593i \(0.932786\pi\)
\(18\) − 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 \(0.238855\pi\)
−0.731426 + 0.681921i \(0.761145\pi\)
\(24\) 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 \(-0.627934\pi\)
0.391183 0.920313i \(-0.372066\pi\)
\(38\) − 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 \(0.329861\pi\)
−0.509418 + 0.860519i \(0.670139\pi\)
\(44\) −2.35861e20 −0.675301
\(45\) 0 0
\(46\) 5.87113e20 0.964382
\(47\) 5.55700e20i 0.697617i 0.937194 + 0.348809i \(0.113414\pi\)
−0.937194 + 0.348809i \(0.886586\pi\)
\(48\) − 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 \(0.231555\pi\)
−0.746871 + 0.664969i \(0.768445\pi\)
\(54\) −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 \(-0.896782\pi\)
0.947884 0.318617i \(-0.103218\pi\)
\(68\) − 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 \(-0.982803\pi\)
0.998541 0.0539990i \(-0.0171968\pi\)
\(74\) −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 \(0.375257\pi\)
−0.381937 + 0.924189i \(0.624743\pi\)
\(84\) −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 \(0.0811281\pi\)
−0.967696 + 0.252121i \(0.918872\pi\)
\(98\) 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 \(-0.784430\pi\)
0.779310 0.626639i \(-0.215570\pi\)
\(104\) −1.08327e24 −0.0663471
\(105\) 0 0
\(106\) 1.94823e25 0.940409
\(107\) 4.09213e25i 1.75652i 0.478186 + 0.878259i \(0.341295\pi\)
−0.478186 + 0.878259i \(0.658705\pi\)
\(108\) 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 \(-0.908561\pi\)
0.959023 0.283329i \(-0.0914390\pi\)
\(114\) −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 \(-0.975323\pi\)
0.996996 0.0774477i \(-0.0246771\pi\)
\(128\) 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 \(-0.857082\pi\)
0.900886 0.434056i \(-0.142918\pi\)
\(138\) − 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 \(-0.845431\pi\)
0.884398 0.466733i \(-0.154569\pi\)
\(158\) − 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 \(0.0421133\pi\)
−0.991261 + 0.131917i \(0.957887\pi\)
\(164\) 1.10365e27 0.227657
\(165\) 0 0
\(166\) 7.37270e27 1.30700
\(167\) − 7.47611e27i − 1.22948i −0.788732 0.614738i \(-0.789262\pi\)
0.788732 0.614738i \(-0.210738\pi\)
\(168\) 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 \(-0.943163\pi\)
0.984101 0.177611i \(-0.0568370\pi\)
\(174\) −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 \(0.0127844\pi\)
−0.999194 + 0.0401527i \(0.987216\pi\)
\(194\) 1.41138e28 0.356553
\(195\) 0 0
\(196\) 4.62487e28 1.02778
\(197\) − 4.10890e28i − 0.856836i −0.903581 0.428418i \(-0.859071\pi\)
0.903581 0.428418i \(-0.140929\pi\)
\(198\) − 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 \(0.445968\pi\)
−0.168934 + 0.985627i \(0.554032\pi\)
\(224\) 7.38023e28 0.309008
\(225\) 0 0
\(226\) −1.06945e29 −0.400688
\(227\) 2.11324e29i 0.749248i 0.927177 + 0.374624i \(0.122228\pi\)
−0.927177 + 0.374624i \(0.877772\pi\)
\(228\) 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 \(0.232393\pi\)
−0.745118 + 0.666932i \(0.767607\pi\)
\(234\) 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 \(0.125671\pi\)
−0.923070 + 0.384632i \(0.874329\pi\)
\(258\) − 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 \(-0.909688\pi\)
0.960020 0.279933i \(-0.0903121\pi\)
\(264\) 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 \(-0.679398\pi\)
0.534230 0.845339i \(-0.320602\pi\)
\(278\) 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 \(0.189564\pi\)
−0.827850 + 0.560950i \(0.810436\pi\)
\(284\) 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 \(-0.883674\pi\)
0.933964 0.357368i \(-0.116326\pi\)
\(294\) 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 \(0.402825\pi\)
−0.300563 + 0.953762i \(0.597175\pi\)
\(308\) − 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 \(0.256528\pi\)
−0.692458 + 0.721458i \(0.743472\pi\)
\(314\) −1.07449e31 −0.660060
\(315\) 0 0
\(316\) −8.37856e30 −0.475424
\(317\) − 4.83401e29i − 0.0263674i −0.999913 0.0131837i \(-0.995803\pi\)
0.999913 0.0131837i \(-0.00419662\pi\)
\(318\) − 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 \(-0.842318\pi\)
0.879792 0.475359i \(-0.157682\pi\)
\(338\) − 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 \(-0.964481\pi\)
0.993781 0.111356i \(-0.0355193\pi\)
\(348\) 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 \(-0.962232\pi\)
0.992969 0.118374i \(-0.0377683\pi\)
\(354\) −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 \(0.251590\pi\)
−0.703567 + 0.710629i \(0.748410\pi\)
\(368\) 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 \(0.0999562\pi\)
−0.951099 + 0.308886i \(0.900044\pi\)
\(374\) 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 \(-0.973930\pi\)
0.996648 0.0818087i \(-0.0260696\pi\)
\(384\) 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 \(-0.923759\pi\)
0.971452 0.237236i \(-0.0762414\pi\)
\(398\) − 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 \(0.117133\pi\)
−0.933054 + 0.359736i \(0.882867\pi\)
\(434\) 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 \(0.0475515\pi\)
−0.988862 + 0.148832i \(0.952449\pi\)
\(444\) 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 \(-0.743487\pi\)
0.692491 0.721427i \(-0.256513\pi\)
\(458\) 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 \(0.0252510\pi\)
−0.996855 + 0.0792452i \(0.974749\pi\)
\(464\) 1.01550e32 0.0473376
\(465\) 0 0
\(466\) 2.13511e33 0.943185
\(467\) 2.96201e33i 1.27387i 0.770917 + 0.636936i \(0.219798\pi\)
−0.770917 + 0.636936i \(0.780202\pi\)
\(468\) − 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 \(0.278707\pi\)
−0.640549 + 0.767917i \(0.721293\pi\)
\(488\) − 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 \(0.251357\pi\)
−0.704086 + 0.710115i \(0.748643\pi\)
\(504\) −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 \(-0.757604\pi\)
0.723796 0.690014i \(-0.242396\pi\)
\(524\) −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 \(0.0905127\pi\)
−0.959843 + 0.280538i \(0.909487\pi\)
\(548\) 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 \(0.152371\pi\)
−0.887601 + 0.460614i \(0.847629\pi\)
\(558\) − 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 \(-0.852756\pi\)
0.894903 0.446260i \(-0.147244\pi\)
\(564\) −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 \(0.349544\pi\)
−0.455266 + 0.890356i \(0.650456\pi\)
\(578\) − 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 \(-0.980566\pi\)
0.998137 0.0610151i \(-0.0194338\pi\)
\(588\) − 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 \(-0.786980\pi\)
0.784304 0.620376i \(-0.213020\pi\)
\(594\) −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 \(0.0656062\pi\)
−0.978835 + 0.204652i \(0.934394\pi\)
\(608\) − 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 \(0.219745\pi\)
−0.771023 + 0.636808i \(0.780255\pi\)
\(614\) 9.59532e34 1.34882
\(615\) 0 0
\(616\) −6.18424e34 −0.834694
\(617\) − 3.04657e34i − 0.402946i −0.979494 0.201473i \(-0.935427\pi\)
0.979494 0.201473i \(-0.0645728\pi\)
\(618\) 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 \(0.0328577\pi\)
−0.994677 + 0.103042i \(0.967142\pi\)
\(644\) 1.53940e35 1.19201
\(645\) 0 0
\(646\) 3.57449e34 0.266261
\(647\) − 1.90009e35i − 1.38826i −0.719848 0.694131i \(-0.755789\pi\)
0.719848 0.694131i \(-0.244211\pi\)
\(648\) − 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 \(0.185448\pi\)
−0.835034 + 0.550198i \(0.814552\pi\)
\(654\) 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 \(-0.895377\pi\)
0.946468 0.322796i \(-0.104623\pi\)
\(674\) −1.53386e35 −0.672259
\(675\) 0 0
\(676\) −1.14218e35 −0.482392
\(677\) 2.51992e35i 1.04479i 0.852704 + 0.522394i \(0.174961\pi\)
−0.852704 + 0.522394i \(0.825039\pi\)
\(678\) 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 \(0.0848806\pi\)
−0.964656 + 0.263511i \(0.915119\pi\)
\(684\) −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 \(-0.997644\pi\)
0.999973 0.00740221i \(-0.00235622\pi\)
\(728\) − 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 \(-0.952870\pi\)
0.989059 0.147523i \(-0.0471301\pi\)
\(734\) 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 \(-0.798261\pi\)
0.805794 0.592197i \(-0.201739\pi\)
\(744\) 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 \(0.193693\pi\)
−0.820504 + 0.571640i \(0.806307\pi\)
\(758\) − 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 \(0.207864\pi\)
−0.794251 + 0.607590i \(0.792136\pi\)
\(774\) 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 \(-0.809663\pi\)
0.826484 0.562960i \(-0.190337\pi\)
\(788\) 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 \(-0.964690\pi\)
0.993854 0.110703i \(-0.0353103\pi\)
\(798\) − 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 \(0.225535\pi\)
−0.759312 + 0.650726i \(0.774465\pi\)
\(824\) 1.24621e36 0.443101
\(825\) 0 0
\(826\) 5.13315e36 1.77065
\(827\) 1.03992e36i 0.353331i 0.984271 + 0.176665i \(0.0565311\pi\)
−0.984271 + 0.176665i \(0.943469\pi\)
\(828\) − 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 \(-0.801600\pi\)
0.811962 0.583710i \(-0.198400\pi\)
\(854\) −7.75823e35 −0.176417
\(855\) 0 0
\(856\) −2.81209e36 −0.621023
\(857\) 2.18945e36i 0.476513i 0.971202 + 0.238257i \(0.0765760\pi\)
−0.971202 + 0.238257i \(0.923424\pi\)
\(858\) − 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 \(0.125618\pi\)
−0.923135 + 0.384476i \(0.874382\pi\)
\(864\) −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 \(0.135080\pi\)
−0.911299 + 0.411744i \(0.864920\pi\)
\(878\) − 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 \(0.334841\pi\)
−0.495893 + 0.868384i \(0.665159\pi\)
\(884\) 2.66316e35 0.0393318
\(885\) 0 0
\(886\) 1.46600e36 0.210481
\(887\) − 1.11924e37i − 1.58445i −0.610228 0.792226i \(-0.708922\pi\)
0.610228 0.792226i \(-0.291078\pi\)
\(888\) − 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 \(-0.970950\pi\)
0.995838 0.0911381i \(-0.0290505\pi\)
\(908\) − 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 \(0.0873440\pi\)
−0.962588 + 0.270969i \(0.912656\pi\)
\(938\) − 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 \(-0.796896\pi\)
0.803247 0.595645i \(-0.203104\pi\)
\(948\) 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 \(-0.984543\pi\)
0.998821 0.0485396i \(-0.0154567\pi\)
\(954\) 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 \(0.0317970\pi\)
−0.995015 + 0.0997271i \(0.968203\pi\)
\(968\) 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 \(0.0785681\pi\)
−0.969692 + 0.244330i \(0.921432\pi\)
\(978\) − 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 \(-0.966345\pi\)
0.994416 0.105533i \(-0.0336547\pi\)
\(984\) −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 \(-0.740833\pi\)
0.686453 0.727174i \(-0.259167\pi\)
\(998\) − 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.49.1 4
5.2 odd 4 10.26.a.d.1.1 2
5.3 odd 4 50.26.a.d.1.2 2
5.4 even 2 inner 50.26.b.d.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.a.d.1.1 2 5.2 odd 4
50.26.a.d.1.2 2 5.3 odd 4
50.26.b.d.49.1 4 1.1 even 1 trivial
50.26.b.d.49.4 4 5.4 even 2 inner