Properties

Label 7400.2.a.j.1.1
Level $7400$
Weight $2$
Character 7400.1
Self dual yes
Analytic conductor $59.089$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7400,2,Mod(1,7400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7400.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.0892974957\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{7} +6.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} +4.00000 q^{17} -4.00000 q^{19} +15.0000 q^{21} +4.00000 q^{23} +9.00000 q^{27} -8.00000 q^{29} -2.00000 q^{31} -9.00000 q^{33} -1.00000 q^{37} +12.0000 q^{39} -5.00000 q^{41} +10.0000 q^{43} -7.00000 q^{47} +18.0000 q^{49} +12.0000 q^{51} -3.00000 q^{53} -12.0000 q^{57} +4.00000 q^{59} +2.00000 q^{61} +30.0000 q^{63} +12.0000 q^{67} +12.0000 q^{69} -1.00000 q^{71} +11.0000 q^{73} -15.0000 q^{77} +14.0000 q^{79} +9.00000 q^{81} -9.00000 q^{83} -24.0000 q^{87} -14.0000 q^{89} +20.0000 q^{91} -6.00000 q^{93} +2.00000 q^{97} -18.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 15.0000 3.27327
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −9.00000 −1.56670
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 30.0000 3.77964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −24.0000 −2.57307
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 20.0000 2.09657
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) −18.0000 −1.77359 −0.886796 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 24.0000 2.21880
\(118\) 0 0
\(119\) 20.0000 1.83340
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −15.0000 −1.35250
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) 30.0000 2.64135
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −20.0000 −1.73422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −21.0000 −1.76852
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.0000 4.45384
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 24.0000 1.94029
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.0000 1.51637 0.758183 0.652042i \(-0.226088\pi\)
0.758183 + 0.652042i \(0.226088\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −24.0000 −1.83533
\(172\) 0 0
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 45.0000 3.27327
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.0000 −0.926212 −0.463106 0.886303i \(-0.653265\pi\)
−0.463106 + 0.886303i \(0.653265\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 36.0000 2.53924
\(202\) 0 0
\(203\) −40.0000 −2.80745
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 24.0000 1.66812
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 0 0
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 33.0000 2.22993
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) −45.0000 −2.96078
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 42.0000 2.72819
\(238\) 0 0
\(239\) 14.0000 0.905585 0.452792 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) −27.0000 −1.71106
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) −48.0000 −2.97113
\(262\) 0 0
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −42.0000 −2.57036
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 0 0
\(273\) 60.0000 3.63137
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −25.0000 −1.47570
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −27.0000 −1.56670
\(298\) 0 0
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 50.0000 2.88195
\(302\) 0 0
\(303\) −15.0000 −0.861727
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −15.0000 −0.856095 −0.428048 0.903756i \(-0.640798\pi\)
−0.428048 + 0.903756i \(0.640798\pi\)
\(308\) 0 0
\(309\) −54.0000 −3.07195
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 30.0000 1.65900
\(328\) 0 0
\(329\) −35.0000 −1.92961
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 55.0000 2.96972
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 36.0000 1.92154
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 60.0000 3.17554
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 5.00000 0.258890 0.129445 0.991587i \(-0.458680\pi\)
0.129445 + 0.991587i \(0.458680\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −32.0000 −1.64808
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −39.0000 −1.99803
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 60.0000 3.04997
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −54.0000 −2.72394
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −37.0000 −1.85698 −0.928488 0.371361i \(-0.878891\pi\)
−0.928488 + 0.371361i \(0.878891\pi\)
\(398\) 0 0
\(399\) −60.0000 −3.00376
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −60.0000 −2.93821
\(418\) 0 0
\(419\) 37.0000 1.80757 0.903784 0.427989i \(-0.140778\pi\)
0.903784 + 0.427989i \(0.140778\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 0 0
\(423\) −42.0000 −2.04211
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) −36.0000 −1.73810
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 108.000 5.14286
\(442\) 0 0
\(443\) −13.0000 −0.617649 −0.308824 0.951119i \(-0.599936\pi\)
−0.308824 + 0.951119i \(0.599936\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 45.0000 2.12843
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) 0 0
\(453\) −24.0000 −1.12762
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 36.0000 1.68034
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 60.0000 2.77054
\(470\) 0 0
\(471\) 57.0000 2.62642
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 60.0000 2.73009
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.00000 −0.224281
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 66.0000 2.94866
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) 55.0000 2.43306
\(512\) 0 0
\(513\) −36.0000 −1.58944
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 21.0000 0.923579
\(518\) 0 0
\(519\) 39.0000 1.71191
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) −2.00000 −0.0874539 −0.0437269 0.999044i \(-0.513923\pi\)
−0.0437269 + 0.999044i \(0.513923\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 36.0000 1.55351
\(538\) 0 0
\(539\) −54.0000 −2.32594
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) −57.0000 −2.44610
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 70.0000 2.97670
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 0 0
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 45.0000 1.88982
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 0 0
\(573\) −36.0000 −1.50392
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −45.0000 −1.86691
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −39.0000 −1.60425
\(592\) 0 0
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 30.0000 1.22782
\(598\) 0 0
\(599\) 39.0000 1.59350 0.796748 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 72.0000 2.93207
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 0 0
\(609\) −120.000 −4.86265
\(610\) 0 0
\(611\) −28.0000 −1.13276
\(612\) 0 0
\(613\) −33.0000 −1.33286 −0.666429 0.745569i \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 0 0
\(619\) −27.0000 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(620\) 0 0
\(621\) 36.0000 1.44463
\(622\) 0 0
\(623\) −70.0000 −2.80449
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 36.0000 1.43770
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) −81.0000 −3.21946
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 72.0000 2.85274
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −25.0000 −0.987441 −0.493720 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 66.0000 2.57491
\(658\) 0 0
\(659\) −45.0000 −1.75295 −0.876476 0.481446i \(-0.840112\pi\)
−0.876476 + 0.481446i \(0.840112\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) 0 0
\(663\) 48.0000 1.86417
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 27.0000 1.04388
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.0000 −0.576497 −0.288248 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) −32.0000 −1.22445 −0.612223 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −51.0000 −1.94577
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) −90.0000 −3.41882
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 0 0
\(699\) −30.0000 −1.13470
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.0000 −0.940222
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 84.0000 3.15025
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 42.0000 1.56852
\(718\) 0 0
\(719\) −11.0000 −0.410231 −0.205115 0.978738i \(-0.565757\pi\)
−0.205115 + 0.978738i \(0.565757\pi\)
\(720\) 0 0
\(721\) −90.0000 −3.35178
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) −39.0000 −1.43464 −0.717319 0.696745i \(-0.754631\pi\)
−0.717319 + 0.696745i \(0.754631\pi\)
\(740\) 0 0
\(741\) −48.0000 −1.76332
\(742\) 0 0
\(743\) −13.0000 −0.476924 −0.238462 0.971152i \(-0.576643\pi\)
−0.238462 + 0.971152i \(0.576643\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −54.0000 −1.97576
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) −36.0000 −1.30672
\(760\) 0 0
\(761\) −7.00000 −0.253750 −0.126875 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(762\) 0 0
\(763\) 50.0000 1.81012
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 66.0000 2.37693
\(772\) 0 0
\(773\) −21.0000 −0.755318 −0.377659 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −15.0000 −0.538122
\(778\) 0 0
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 3.00000 0.107348
\(782\) 0 0
\(783\) −72.0000 −2.57307
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −11.0000 −0.392108 −0.196054 0.980593i \(-0.562813\pi\)
−0.196054 + 0.980593i \(0.562813\pi\)
\(788\) 0 0
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.0000 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) −84.0000 −2.96799
\(802\) 0 0
\(803\) −33.0000 −1.16454
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) 49.0000 1.72062 0.860311 0.509769i \(-0.170269\pi\)
0.860311 + 0.509769i \(0.170269\pi\)
\(812\) 0 0
\(813\) −39.0000 −1.36779
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) 0 0
\(819\) 120.000 4.19314
\(820\) 0 0
\(821\) 37.0000 1.29131 0.645654 0.763630i \(-0.276585\pi\)
0.645654 + 0.763630i \(0.276585\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) 0 0
\(833\) 72.0000 2.49465
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −18.0000 −0.622171
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 96.0000 3.29471
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) −75.0000 −2.55599
\(862\) 0 0
\(863\) 56.0000 1.90626 0.953131 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.0000 0.906571 0.453286 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) 0 0
\(889\) −65.0000 −2.18003
\(890\) 0 0
\(891\) −27.0000 −0.904534
\(892\) 0 0
\(893\) 28.0000 0.936984
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 0 0
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 150.000 4.99169
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 0 0
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 27.0000 0.893570
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −90.0000 −2.97206
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −45.0000 −1.48280
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −108.000 −3.54719
\(928\) 0 0
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) −72.0000 −2.35970
\(932\) 0 0
\(933\) −36.0000 −1.17859
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −15.0000 −0.490029 −0.245014 0.969519i \(-0.578793\pi\)
−0.245014 + 0.969519i \(0.578793\pi\)
\(938\) 0 0
\(939\) 90.0000 2.93704
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) −20.0000 −0.651290
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) 44.0000 1.42830
\(950\) 0 0
\(951\) 54.0000 1.75107
\(952\) 0 0
\(953\) 33.0000 1.06897 0.534487 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 72.0000 2.32743
\(958\) 0 0
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 48.0000 1.54678
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) −100.000 −3.20585
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) 42.0000 1.34233
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −105.000 −3.34219
\(988\) 0 0
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 0 0
\(993\) −96.0000 −3.04647
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) 0 0
\(999\) −9.00000 −0.284747
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 7400.2.a.j.1.1 1
5.4 even 2 1480.2.a.a.1.1 1
20.19 odd 2 2960.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.a.1.1 1 5.4 even 2
2960.2.a.n.1.1 1 20.19 odd 2
7400.2.a.j.1.1 1 1.1 even 1 trivial