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Properties

Label 8664.2.a.h.1.1
Level $8664$
Weight $2$
Character 8664.1
Self dual yes
Analytic conductor $69.182$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8664,2,Mod(1,8664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,-2,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.1823883112\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8664.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -5.00000 q^{13} -2.00000 q^{15} -5.00000 q^{21} -6.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{31} -4.00000 q^{33} +10.0000 q^{35} -7.00000 q^{37} -5.00000 q^{39} -11.0000 q^{43} -2.00000 q^{45} +10.0000 q^{47} +18.0000 q^{49} -6.00000 q^{53} +8.00000 q^{55} -8.00000 q^{59} -1.00000 q^{61} -5.00000 q^{63} +10.0000 q^{65} -5.00000 q^{67} -6.00000 q^{69} -6.00000 q^{71} +1.00000 q^{73} -1.00000 q^{75} +20.0000 q^{77} +13.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -8.00000 q^{87} +12.0000 q^{89} +25.0000 q^{91} +1.00000 q^{93} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 − 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 10.0000 1.69031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −5.00000 −0.629941
\(64\) 0 0
\(65\) 10.0000 1.24035
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 20.0000 2.27921
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 25.0000 2.62071
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 − 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) 10.0000 0.975900
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 − 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) −5.00000 −0.462250
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 − 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 − 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 30.0000 2.36433
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 − 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 14.0000 1.02930
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −26.0000 −1.88129 −0.940647 − 0.339387i \(-0.889781\pi\)
−0.940647 + 0.339387i \(0.889781\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 10.0000 0.716115
\(196\) 0 0
\(197\) −20.0000 −1.42494 −0.712470 − 0.701702i \(-0.752424\pi\)
−0.712470 + 0.701702i \(0.752424\pi\)
\(198\) 0 0
\(199\) 3.00000 0.212664 0.106332 − 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) 40.0000 2.80745
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 21.0000 1.44570 0.722850 − 0.691005i \(-0.242832\pi\)
0.722850 + 0.691005i \(0.242832\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 22.0000 1.50039
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 0 0
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −27.0000 −1.78421 −0.892105 − 0.451828i \(-0.850772\pi\)
−0.892105 + 0.451828i \(0.850772\pi\)
\(230\) 0 0
\(231\) 20.0000 1.31590
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −20.0000 −1.30466
\(236\) 0 0
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 − 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 11.0000 0.708572 0.354286 − 0.935137i \(-0.384724\pi\)
0.354286 + 0.935137i \(0.384724\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −36.0000 −2.29996
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 26.0000 1.64111 0.820553 − 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 − 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 35.0000 2.17479
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 − 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 25.0000 1.51307
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 30.0000 1.73494
\(300\) 0 0
\(301\) 55.0000 3.17015
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 10.0000 0.563436
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 32.0000 1.79166
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) −50.0000 −2.75659
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 − 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) 7.00000 0.381314 0.190657 − 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −30.0000 −1.61048 −0.805242 − 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 − 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 − 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −15.0000 −0.782994 −0.391497 − 0.920179i \(-0.628043\pi\)
−0.391497 + 0.920179i \(0.628043\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 30.0000 1.55752
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 40.0000 2.06010
\(378\) 0 0
\(379\) −13.0000 −0.667765 −0.333883 − 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) −40.0000 −2.03859
\(386\) 0 0
\(387\) −11.0000 −0.559161
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 10.0000 0.504433
\(394\) 0 0
\(395\) −26.0000 −1.30820
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 − 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −5.00000 −0.249068
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 28.0000 1.38791
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 10.0000 0.486217
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) 0 0
\(429\) 20.0000 0.965609
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 − 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 − 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) 16.0000 0.767141
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −13.0000 −0.620456 −0.310228 − 0.950662i \(-0.600405\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 0 0
\(447\) 8.00000 0.378387
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −50.0000 −2.34404
\(456\) 0 0
\(457\) −15.0000 −0.701670 −0.350835 − 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.0000 −0.745194 −0.372597 − 0.927993i \(-0.621533\pi\)
−0.372597 + 0.927993i \(0.621533\pi\)
\(462\) 0 0
\(463\) 25.0000 1.16185 0.580924 − 0.813958i \(-0.302691\pi\)
0.580924 + 0.813958i \(0.302691\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 25.0000 1.15439
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 44.0000 2.02312
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 35.0000 1.59586
\(482\) 0 0
\(483\) 30.0000 1.36505
\(484\) 0 0
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 − 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 30.0000 1.34568
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 − 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 0 0
\(523\) 9.00000 0.393543 0.196771 − 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) −72.0000 −3.10126
\(540\) 0 0
\(541\) 1.00000 0.0429934 0.0214967 − 0.999769i \(-0.493157\pi\)
0.0214967 + 0.999769i \(0.493157\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −17.0000 −0.726868 −0.363434 − 0.931620i \(-0.618396\pi\)
−0.363434 + 0.931620i \(0.618396\pi\)
\(548\) 0 0
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −65.0000 −2.76408
\(554\) 0 0
\(555\) 14.0000 0.594267
\(556\) 0 0
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) 55.0000 2.32625
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.0000 −0.590030 −0.295015 − 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 0 0
\(567\) −5.00000 −0.209980
\(568\) 0 0
\(569\) −12.0000 −0.503066 −0.251533 − 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) 11.0000 0.460336 0.230168 − 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) 0 0
\(573\) −26.0000 −1.08617
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 5.00000 0.207793
\(580\) 0 0
\(581\) 20.0000 0.829740
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 10.0000 0.413449
\(586\) 0 0
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.00000 0.122782
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 − 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 − 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 5.00000 0.202944 0.101472 − 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) 0 0
\(609\) 40.0000 1.62088
\(610\) 0 0
\(611\) −50.0000 −2.02278
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 − 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 − 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) −45.0000 −1.80870 −0.904351 − 0.426789i \(-0.859645\pi\)
−0.904351 + 0.426789i \(0.859645\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −60.0000 −2.40385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −29.0000 −1.15447 −0.577236 − 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 0 0
\(633\) 21.0000 0.834675
\(634\) 0 0
\(635\) 40.0000 1.58735
\(636\) 0 0
\(637\) −90.0000 −3.56593
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −46.0000 −1.81689 −0.908445 − 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 − 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) 22.0000 0.866249
\(646\) 0 0
\(647\) 40.0000 1.57256 0.786281 − 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) −5.00000 −0.195965
\(652\) 0 0
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) 0 0
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 1.00000 0.0386622
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 − 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 − 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 0 0
\(683\) −10.0000 −0.382639 −0.191320 − 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) −27.0000 −1.03011
\(688\) 0 0
\(689\) 30.0000 1.14291
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) 20.0000 0.759737
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −20.0000 −0.753244
\(706\) 0 0
\(707\) 30.0000 1.12827
\(708\) 0 0
\(709\) −27.0000 −1.01401 −0.507003 − 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) 0 0
\(711\) 13.0000 0.487538
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −40.0000 −1.49592
\(716\) 0 0
\(717\) 2.00000 0.0746914
\(718\) 0 0
\(719\) −42.0000 −1.56634 −0.783168 − 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) −25.0000 −0.931049
\(722\) 0 0
\(723\) 11.0000 0.409094
\(724\) 0 0
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 − 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 − 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) −36.0000 −1.32788
\(736\) 0 0
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −3.00000 −0.110357 −0.0551784 − 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 90.0000 3.28853
\(750\) 0 0
\(751\) 29.0000 1.05823 0.529113 − 0.848552i \(-0.322525\pi\)
0.529113 + 0.848552i \(0.322525\pi\)
\(752\) 0 0
\(753\) 26.0000 0.947493
\(754\) 0 0
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −15.0000 −0.545184 −0.272592 − 0.962130i \(-0.587881\pi\)
−0.272592 + 0.962130i \(0.587881\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 43.0000 1.55062 0.775310 − 0.631581i \(-0.217594\pi\)
0.775310 + 0.631581i \(0.217594\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) 0 0
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) 35.0000 1.25562
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 26.0000 0.927980
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 − 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 0 0
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) −20.0000 −0.711118
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) −60.0000 −2.11472
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 25.0000 0.873571
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 − 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −45.0000 −1.56291 −0.781457 − 0.623959i \(-0.785523\pi\)
−0.781457 + 0.623959i \(0.785523\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) −24.0000 −0.825625
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) −1.00000 −0.0342393 −0.0171197 − 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 − 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34.0000 −1.15737 −0.578687 − 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −52.0000 −1.76398
\(870\) 0 0
\(871\) 25.0000 0.847093
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −60.0000 −2.02837
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 − 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −35.0000 −1.17784 −0.588922 − 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 0 0
\(885\) 16.0000 0.537834
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 − 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) 100.000 3.35389
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 30.0000 1.00167
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 55.0000 1.83029
\(904\) 0 0
\(905\) −36.0000 −1.19668
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −50.0000 −1.65115
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 − 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) 7.00000 0.230159
\(926\) 0 0
\(927\) 5.00000 0.164222
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0000 0.751377 0.375689 − 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 56.0000 1.82555 0.912774 − 0.408465i \(-0.133936\pi\)
0.912774 + 0.408465i \(0.133936\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 10.0000 0.325300
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) −5.00000 −0.162307
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −60.0000 −1.94359 −0.971795 − 0.235826i \(-0.924220\pi\)
−0.971795 + 0.235826i \(0.924220\pi\)
\(954\) 0 0
\(955\) 52.0000 1.68268
\(956\) 0 0
\(957\) 32.0000 1.03441
\(958\) 0 0
\(959\) −50.0000 −1.61458
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −23.0000 −0.739630 −0.369815 − 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 − 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) 25.0000 0.801463
\(974\) 0 0
\(975\) 5.00000 0.160128
\(976\) 0 0
\(977\) −50.0000 −1.59964 −0.799821 − 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −40.0000 −1.27580 −0.637901 − 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 0 0
\(985\) 40.0000 1.27451
\(986\) 0 0
\(987\) −50.0000 −1.59152
\(988\) 0 0
\(989\) 66.0000 2.09868
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 − 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 0 0
\(993\) −5.00000 −0.158670
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) −25.0000 −0.791758 −0.395879 − 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) 0 0
\(999\) −7.00000 −0.221470
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 8664.2.a.h.1.1 1
19.8 odd 6 456.2.q.c.121.1 yes 2
19.12 odd 6 456.2.q.c.49.1 ✓ 2
19.18 odd 2 8664.2.a.b.1.1 1
57.8 even 6 1368.2.s.b.577.1 2
57.50 even 6 1368.2.s.b.505.1 2
76.27 even 6 912.2.q.c.577.1 2
76.31 even 6 912.2.q.c.49.1 2
228.107 odd 6 2736.2.s.f.1873.1 2
228.179 odd 6 2736.2.s.f.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.q.c.49.1 ✓ 2 19.12 odd 6
456.2.q.c.121.1 yes 2 19.8 odd 6
912.2.q.c.49.1 2 76.31 even 6
912.2.q.c.577.1 2 76.27 even 6
1368.2.s.b.505.1 2 57.50 even 6
1368.2.s.b.577.1 2 57.8 even 6
2736.2.s.f.577.1 2 228.179 odd 6
2736.2.s.f.1873.1 2 228.107 odd 6
8664.2.a.b.1.1 1 19.18 odd 2
8664.2.a.h.1.1 1 1.1 even 1 trivial