Properties

Label 68.4.a.a.1.1
Level $68$
Weight $4$
Character 68.1
Self dual yes
Analytic conductor $4.012$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,4,Mod(1,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 68.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: \(4.01212988039\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 68.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} -23.0000 q^{9} -10.0000 q^{11} -38.0000 q^{13} +16.0000 q^{15} -17.0000 q^{17} +4.00000 q^{19} +24.0000 q^{21} +120.000 q^{23} -61.0000 q^{25} +100.000 q^{27} +56.0000 q^{29} +164.000 q^{31} +20.0000 q^{33} +96.0000 q^{35} -236.000 q^{37} +76.0000 q^{39} +70.0000 q^{41} -144.000 q^{43} +184.000 q^{45} +48.0000 q^{47} -199.000 q^{49} +34.0000 q^{51} -366.000 q^{53} +80.0000 q^{55} -8.00000 q^{57} -504.000 q^{59} -460.000 q^{61} +276.000 q^{63} +304.000 q^{65} -768.000 q^{67} -240.000 q^{69} +72.0000 q^{71} -734.000 q^{73} +122.000 q^{75} +120.000 q^{77} +736.000 q^{79} +421.000 q^{81} +856.000 q^{83} +136.000 q^{85} -112.000 q^{87} +906.000 q^{89} +456.000 q^{91} -328.000 q^{93} -32.0000 q^{95} +46.0000 q^{97} +230.000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) −8.00000 −0.715542 −0.357771 0.933809i \(-0.616463\pi\)
−0.357771 + 0.933809i \(0.616463\pi\)
\(6\) 0 0
\(7\) −12.0000 −0.647939 −0.323970 0.946068i \(-0.605018\pi\)
−0.323970 + 0.946068i \(0.605018\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −10.0000 −0.274101 −0.137051 0.990564i \(-0.543762\pi\)
−0.137051 + 0.990564i \(0.543762\pi\)
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) 16.0000 0.275412
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 4.00000 0.0482980 0.0241490 0.999708i \(-0.492312\pi\)
0.0241490 + 0.999708i \(0.492312\pi\)
\(20\) 0 0
\(21\) 24.0000 0.249392
\(22\) 0 0
\(23\) 120.000 1.08790 0.543951 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) −61.0000 −0.488000
\(26\) 0 0
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) 56.0000 0.358584 0.179292 0.983796i \(-0.442619\pi\)
0.179292 + 0.983796i \(0.442619\pi\)
\(30\) 0 0
\(31\) 164.000 0.950170 0.475085 0.879940i \(-0.342417\pi\)
0.475085 + 0.879940i \(0.342417\pi\)
\(32\) 0 0
\(33\) 20.0000 0.105502
\(34\) 0 0
\(35\) 96.0000 0.463627
\(36\) 0 0
\(37\) −236.000 −1.04860 −0.524299 0.851534i \(-0.675673\pi\)
−0.524299 + 0.851534i \(0.675673\pi\)
\(38\) 0 0
\(39\) 76.0000 0.312045
\(40\) 0 0
\(41\) 70.0000 0.266638 0.133319 0.991073i \(-0.457436\pi\)
0.133319 + 0.991073i \(0.457436\pi\)
\(42\) 0 0
\(43\) −144.000 −0.510693 −0.255346 0.966850i \(-0.582190\pi\)
−0.255346 + 0.966850i \(0.582190\pi\)
\(44\) 0 0
\(45\) 184.000 0.609536
\(46\) 0 0
\(47\) 48.0000 0.148969 0.0744843 0.997222i \(-0.476269\pi\)
0.0744843 + 0.997222i \(0.476269\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) 34.0000 0.0933520
\(52\) 0 0
\(53\) −366.000 −0.948565 −0.474283 0.880373i \(-0.657293\pi\)
−0.474283 + 0.880373i \(0.657293\pi\)
\(54\) 0 0
\(55\) 80.0000 0.196131
\(56\) 0 0
\(57\) −8.00000 −0.0185899
\(58\) 0 0
\(59\) −504.000 −1.11212 −0.556061 0.831141i \(-0.687688\pi\)
−0.556061 + 0.831141i \(0.687688\pi\)
\(60\) 0 0
\(61\) −460.000 −0.965524 −0.482762 0.875752i \(-0.660366\pi\)
−0.482762 + 0.875752i \(0.660366\pi\)
\(62\) 0 0
\(63\) 276.000 0.551948
\(64\) 0 0
\(65\) 304.000 0.580101
\(66\) 0 0
\(67\) −768.000 −1.40039 −0.700195 0.713952i \(-0.746904\pi\)
−0.700195 + 0.713952i \(0.746904\pi\)
\(68\) 0 0
\(69\) −240.000 −0.418733
\(70\) 0 0
\(71\) 72.0000 0.120350 0.0601748 0.998188i \(-0.480834\pi\)
0.0601748 + 0.998188i \(0.480834\pi\)
\(72\) 0 0
\(73\) −734.000 −1.17682 −0.588412 0.808561i \(-0.700247\pi\)
−0.588412 + 0.808561i \(0.700247\pi\)
\(74\) 0 0
\(75\) 122.000 0.187831
\(76\) 0 0
\(77\) 120.000 0.177601
\(78\) 0 0
\(79\) 736.000 1.04818 0.524092 0.851662i \(-0.324405\pi\)
0.524092 + 0.851662i \(0.324405\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 856.000 1.13203 0.566013 0.824396i \(-0.308485\pi\)
0.566013 + 0.824396i \(0.308485\pi\)
\(84\) 0 0
\(85\) 136.000 0.173544
\(86\) 0 0
\(87\) −112.000 −0.138019
\(88\) 0 0
\(89\) 906.000 1.07905 0.539527 0.841968i \(-0.318603\pi\)
0.539527 + 0.841968i \(0.318603\pi\)
\(90\) 0 0
\(91\) 456.000 0.525294
\(92\) 0 0
\(93\) −328.000 −0.365721
\(94\) 0 0
\(95\) −32.0000 −0.0345593
\(96\) 0 0
\(97\) 46.0000 0.0481504 0.0240752 0.999710i \(-0.492336\pi\)
0.0240752 + 0.999710i \(0.492336\pi\)
\(98\) 0 0
\(99\) 230.000 0.233494
\(100\) 0 0
\(101\) 1178.00 1.16055 0.580274 0.814421i \(-0.302945\pi\)
0.580274 + 0.814421i \(0.302945\pi\)
\(102\) 0 0
\(103\) −608.000 −0.581631 −0.290816 0.956779i \(-0.593927\pi\)
−0.290816 + 0.956779i \(0.593927\pi\)
\(104\) 0 0
\(105\) −192.000 −0.178450
\(106\) 0 0
\(107\) −1798.00 −1.62448 −0.812239 0.583324i \(-0.801752\pi\)
−0.812239 + 0.583324i \(0.801752\pi\)
\(108\) 0 0
\(109\) 200.000 0.175748 0.0878740 0.996132i \(-0.471993\pi\)
0.0878740 + 0.996132i \(0.471993\pi\)
\(110\) 0 0
\(111\) 472.000 0.403606
\(112\) 0 0
\(113\) 2330.00 1.93972 0.969858 0.243670i \(-0.0783513\pi\)
0.969858 + 0.243670i \(0.0783513\pi\)
\(114\) 0 0
\(115\) −960.000 −0.778439
\(116\) 0 0
\(117\) 874.000 0.690610
\(118\) 0 0
\(119\) 204.000 0.157148
\(120\) 0 0
\(121\) −1231.00 −0.924869
\(122\) 0 0
\(123\) −140.000 −0.102629
\(124\) 0 0
\(125\) 1488.00 1.06473
\(126\) 0 0
\(127\) −2016.00 −1.40859 −0.704296 0.709907i \(-0.748737\pi\)
−0.704296 + 0.709907i \(0.748737\pi\)
\(128\) 0 0
\(129\) 288.000 0.196566
\(130\) 0 0
\(131\) −102.000 −0.0680289 −0.0340144 0.999421i \(-0.510829\pi\)
−0.0340144 + 0.999421i \(0.510829\pi\)
\(132\) 0 0
\(133\) −48.0000 −0.0312942
\(134\) 0 0
\(135\) −800.000 −0.510022
\(136\) 0 0
\(137\) −2186.00 −1.36323 −0.681615 0.731711i \(-0.738722\pi\)
−0.681615 + 0.731711i \(0.738722\pi\)
\(138\) 0 0
\(139\) −2590.00 −1.58044 −0.790219 0.612824i \(-0.790033\pi\)
−0.790219 + 0.612824i \(0.790033\pi\)
\(140\) 0 0
\(141\) −96.0000 −0.0573380
\(142\) 0 0
\(143\) 380.000 0.222218
\(144\) 0 0
\(145\) −448.000 −0.256582
\(146\) 0 0
\(147\) 398.000 0.223309
\(148\) 0 0
\(149\) −1282.00 −0.704869 −0.352435 0.935836i \(-0.614646\pi\)
−0.352435 + 0.935836i \(0.614646\pi\)
\(150\) 0 0
\(151\) 3312.00 1.78495 0.892473 0.451102i \(-0.148969\pi\)
0.892473 + 0.451102i \(0.148969\pi\)
\(152\) 0 0
\(153\) 391.000 0.206604
\(154\) 0 0
\(155\) −1312.00 −0.679886
\(156\) 0 0
\(157\) 310.000 0.157584 0.0787920 0.996891i \(-0.474894\pi\)
0.0787920 + 0.996891i \(0.474894\pi\)
\(158\) 0 0
\(159\) 732.000 0.365103
\(160\) 0 0
\(161\) −1440.00 −0.704894
\(162\) 0 0
\(163\) −1838.00 −0.883210 −0.441605 0.897210i \(-0.645591\pi\)
−0.441605 + 0.897210i \(0.645591\pi\)
\(164\) 0 0
\(165\) −160.000 −0.0754908
\(166\) 0 0
\(167\) 1684.00 0.780310 0.390155 0.920749i \(-0.372421\pi\)
0.390155 + 0.920749i \(0.372421\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) −92.0000 −0.0411428
\(172\) 0 0
\(173\) −1104.00 −0.485177 −0.242588 0.970129i \(-0.577996\pi\)
−0.242588 + 0.970129i \(0.577996\pi\)
\(174\) 0 0
\(175\) 732.000 0.316194
\(176\) 0 0
\(177\) 1008.00 0.428056
\(178\) 0 0
\(179\) −3660.00 −1.52828 −0.764138 0.645053i \(-0.776835\pi\)
−0.764138 + 0.645053i \(0.776835\pi\)
\(180\) 0 0
\(181\) 4724.00 1.93996 0.969978 0.243191i \(-0.0781943\pi\)
0.969978 + 0.243191i \(0.0781943\pi\)
\(182\) 0 0
\(183\) 920.000 0.371630
\(184\) 0 0
\(185\) 1888.00 0.750316
\(186\) 0 0
\(187\) 170.000 0.0664793
\(188\) 0 0
\(189\) −1200.00 −0.461837
\(190\) 0 0
\(191\) 4280.00 1.62141 0.810707 0.585453i \(-0.199083\pi\)
0.810707 + 0.585453i \(0.199083\pi\)
\(192\) 0 0
\(193\) 630.000 0.234966 0.117483 0.993075i \(-0.462517\pi\)
0.117483 + 0.993075i \(0.462517\pi\)
\(194\) 0 0
\(195\) −608.000 −0.223281
\(196\) 0 0
\(197\) −1204.00 −0.435439 −0.217719 0.976011i \(-0.569862\pi\)
−0.217719 + 0.976011i \(0.569862\pi\)
\(198\) 0 0
\(199\) 4632.00 1.65002 0.825009 0.565119i \(-0.191170\pi\)
0.825009 + 0.565119i \(0.191170\pi\)
\(200\) 0 0
\(201\) 1536.00 0.539010
\(202\) 0 0
\(203\) −672.000 −0.232341
\(204\) 0 0
\(205\) −560.000 −0.190791
\(206\) 0 0
\(207\) −2760.00 −0.926731
\(208\) 0 0
\(209\) −40.0000 −0.0132386
\(210\) 0 0
\(211\) 3590.00 1.17131 0.585654 0.810561i \(-0.300838\pi\)
0.585654 + 0.810561i \(0.300838\pi\)
\(212\) 0 0
\(213\) −144.000 −0.0463226
\(214\) 0 0
\(215\) 1152.00 0.365422
\(216\) 0 0
\(217\) −1968.00 −0.615652
\(218\) 0 0
\(219\) 1468.00 0.452960
\(220\) 0 0
\(221\) 646.000 0.196627
\(222\) 0 0
\(223\) 2344.00 0.703883 0.351941 0.936022i \(-0.385522\pi\)
0.351941 + 0.936022i \(0.385522\pi\)
\(224\) 0 0
\(225\) 1403.00 0.415704
\(226\) 0 0
\(227\) 1258.00 0.367826 0.183913 0.982943i \(-0.441124\pi\)
0.183913 + 0.982943i \(0.441124\pi\)
\(228\) 0 0
\(229\) −5258.00 −1.51729 −0.758643 0.651507i \(-0.774137\pi\)
−0.758643 + 0.651507i \(0.774137\pi\)
\(230\) 0 0
\(231\) −240.000 −0.0683586
\(232\) 0 0
\(233\) −3694.00 −1.03864 −0.519318 0.854581i \(-0.673814\pi\)
−0.519318 + 0.854581i \(0.673814\pi\)
\(234\) 0 0
\(235\) −384.000 −0.106593
\(236\) 0 0
\(237\) −1472.00 −0.403446
\(238\) 0 0
\(239\) −1944.00 −0.526138 −0.263069 0.964777i \(-0.584735\pi\)
−0.263069 + 0.964777i \(0.584735\pi\)
\(240\) 0 0
\(241\) −1162.00 −0.310585 −0.155293 0.987869i \(-0.549632\pi\)
−0.155293 + 0.987869i \(0.549632\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) 0 0
\(245\) 1592.00 0.415139
\(246\) 0 0
\(247\) −152.000 −0.0391560
\(248\) 0 0
\(249\) −1712.00 −0.435717
\(250\) 0 0
\(251\) −176.000 −0.0442590 −0.0221295 0.999755i \(-0.507045\pi\)
−0.0221295 + 0.999755i \(0.507045\pi\)
\(252\) 0 0
\(253\) −1200.00 −0.298195
\(254\) 0 0
\(255\) −272.000 −0.0667973
\(256\) 0 0
\(257\) −4474.00 −1.08592 −0.542958 0.839760i \(-0.682696\pi\)
−0.542958 + 0.839760i \(0.682696\pi\)
\(258\) 0 0
\(259\) 2832.00 0.679428
\(260\) 0 0
\(261\) −1288.00 −0.305461
\(262\) 0 0
\(263\) −2696.00 −0.632101 −0.316050 0.948742i \(-0.602357\pi\)
−0.316050 + 0.948742i \(0.602357\pi\)
\(264\) 0 0
\(265\) 2928.00 0.678738
\(266\) 0 0
\(267\) −1812.00 −0.415328
\(268\) 0 0
\(269\) 2304.00 0.522221 0.261110 0.965309i \(-0.415911\pi\)
0.261110 + 0.965309i \(0.415911\pi\)
\(270\) 0 0
\(271\) 1720.00 0.385544 0.192772 0.981244i \(-0.438252\pi\)
0.192772 + 0.981244i \(0.438252\pi\)
\(272\) 0 0
\(273\) −912.000 −0.202186
\(274\) 0 0
\(275\) 610.000 0.133761
\(276\) 0 0
\(277\) −4224.00 −0.916229 −0.458115 0.888893i \(-0.651475\pi\)
−0.458115 + 0.888893i \(0.651475\pi\)
\(278\) 0 0
\(279\) −3772.00 −0.809404
\(280\) 0 0
\(281\) 3126.00 0.663635 0.331818 0.943344i \(-0.392338\pi\)
0.331818 + 0.943344i \(0.392338\pi\)
\(282\) 0 0
\(283\) 5118.00 1.07503 0.537515 0.843254i \(-0.319363\pi\)
0.537515 + 0.843254i \(0.319363\pi\)
\(284\) 0 0
\(285\) 64.0000 0.0133019
\(286\) 0 0
\(287\) −840.000 −0.172765
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) −92.0000 −0.0185331
\(292\) 0 0
\(293\) −4418.00 −0.880895 −0.440448 0.897778i \(-0.645180\pi\)
−0.440448 + 0.897778i \(0.645180\pi\)
\(294\) 0 0
\(295\) 4032.00 0.795770
\(296\) 0 0
\(297\) −1000.00 −0.195373
\(298\) 0 0
\(299\) −4560.00 −0.881979
\(300\) 0 0
\(301\) 1728.00 0.330898
\(302\) 0 0
\(303\) −2356.00 −0.446695
\(304\) 0 0
\(305\) 3680.00 0.690873
\(306\) 0 0
\(307\) 4476.00 0.832113 0.416057 0.909339i \(-0.363412\pi\)
0.416057 + 0.909339i \(0.363412\pi\)
\(308\) 0 0
\(309\) 1216.00 0.223870
\(310\) 0 0
\(311\) −300.000 −0.0546992 −0.0273496 0.999626i \(-0.508707\pi\)
−0.0273496 + 0.999626i \(0.508707\pi\)
\(312\) 0 0
\(313\) −1154.00 −0.208396 −0.104198 0.994557i \(-0.533228\pi\)
−0.104198 + 0.994557i \(0.533228\pi\)
\(314\) 0 0
\(315\) −2208.00 −0.394942
\(316\) 0 0
\(317\) 684.000 0.121190 0.0605951 0.998162i \(-0.480700\pi\)
0.0605951 + 0.998162i \(0.480700\pi\)
\(318\) 0 0
\(319\) −560.000 −0.0982883
\(320\) 0 0
\(321\) 3596.00 0.625262
\(322\) 0 0
\(323\) −68.0000 −0.0117140
\(324\) 0 0
\(325\) 2318.00 0.395629
\(326\) 0 0
\(327\) −400.000 −0.0676454
\(328\) 0 0
\(329\) −576.000 −0.0965225
\(330\) 0 0
\(331\) 9400.00 1.56094 0.780469 0.625194i \(-0.214980\pi\)
0.780469 + 0.625194i \(0.214980\pi\)
\(332\) 0 0
\(333\) 5428.00 0.893251
\(334\) 0 0
\(335\) 6144.00 1.00204
\(336\) 0 0
\(337\) −470.000 −0.0759719 −0.0379860 0.999278i \(-0.512094\pi\)
−0.0379860 + 0.999278i \(0.512094\pi\)
\(338\) 0 0
\(339\) −4660.00 −0.746597
\(340\) 0 0
\(341\) −1640.00 −0.260443
\(342\) 0 0
\(343\) 6504.00 1.02386
\(344\) 0 0
\(345\) 1920.00 0.299621
\(346\) 0 0
\(347\) −6302.00 −0.974954 −0.487477 0.873136i \(-0.662083\pi\)
−0.487477 + 0.873136i \(0.662083\pi\)
\(348\) 0 0
\(349\) 10946.0 1.67887 0.839435 0.543459i \(-0.182886\pi\)
0.839435 + 0.543459i \(0.182886\pi\)
\(350\) 0 0
\(351\) −3800.00 −0.577860
\(352\) 0 0
\(353\) 7782.00 1.17335 0.586677 0.809821i \(-0.300436\pi\)
0.586677 + 0.809821i \(0.300436\pi\)
\(354\) 0 0
\(355\) −576.000 −0.0861152
\(356\) 0 0
\(357\) −408.000 −0.0604864
\(358\) 0 0
\(359\) −4920.00 −0.723308 −0.361654 0.932312i \(-0.617788\pi\)
−0.361654 + 0.932312i \(0.617788\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 2462.00 0.355982
\(364\) 0 0
\(365\) 5872.00 0.842067
\(366\) 0 0
\(367\) −5976.00 −0.849985 −0.424993 0.905197i \(-0.639723\pi\)
−0.424993 + 0.905197i \(0.639723\pi\)
\(368\) 0 0
\(369\) −1610.00 −0.227136
\(370\) 0 0
\(371\) 4392.00 0.614613
\(372\) 0 0
\(373\) −3150.00 −0.437268 −0.218634 0.975807i \(-0.570160\pi\)
−0.218634 + 0.975807i \(0.570160\pi\)
\(374\) 0 0
\(375\) −2976.00 −0.409813
\(376\) 0 0
\(377\) −2128.00 −0.290710
\(378\) 0 0
\(379\) 2414.00 0.327174 0.163587 0.986529i \(-0.447694\pi\)
0.163587 + 0.986529i \(0.447694\pi\)
\(380\) 0 0
\(381\) 4032.00 0.542167
\(382\) 0 0
\(383\) −10120.0 −1.35015 −0.675076 0.737749i \(-0.735889\pi\)
−0.675076 + 0.737749i \(0.735889\pi\)
\(384\) 0 0
\(385\) −960.000 −0.127081
\(386\) 0 0
\(387\) 3312.00 0.435035
\(388\) 0 0
\(389\) −11450.0 −1.49239 −0.746193 0.665730i \(-0.768120\pi\)
−0.746193 + 0.665730i \(0.768120\pi\)
\(390\) 0 0
\(391\) −2040.00 −0.263855
\(392\) 0 0
\(393\) 204.000 0.0261843
\(394\) 0 0
\(395\) −5888.00 −0.750019
\(396\) 0 0
\(397\) −6164.00 −0.779250 −0.389625 0.920974i \(-0.627395\pi\)
−0.389625 + 0.920974i \(0.627395\pi\)
\(398\) 0 0
\(399\) 96.0000 0.0120451
\(400\) 0 0
\(401\) 5758.00 0.717059 0.358530 0.933518i \(-0.383278\pi\)
0.358530 + 0.933518i \(0.383278\pi\)
\(402\) 0 0
\(403\) −6232.00 −0.770318
\(404\) 0 0
\(405\) −3368.00 −0.413228
\(406\) 0 0
\(407\) 2360.00 0.287422
\(408\) 0 0
\(409\) 3302.00 0.399201 0.199601 0.979877i \(-0.436035\pi\)
0.199601 + 0.979877i \(0.436035\pi\)
\(410\) 0 0
\(411\) 4372.00 0.524708
\(412\) 0 0
\(413\) 6048.00 0.720587
\(414\) 0 0
\(415\) −6848.00 −0.810012
\(416\) 0 0
\(417\) 5180.00 0.608311
\(418\) 0 0
\(419\) −9998.00 −1.16571 −0.582857 0.812575i \(-0.698065\pi\)
−0.582857 + 0.812575i \(0.698065\pi\)
\(420\) 0 0
\(421\) −5846.00 −0.676762 −0.338381 0.941009i \(-0.609879\pi\)
−0.338381 + 0.941009i \(0.609879\pi\)
\(422\) 0 0
\(423\) −1104.00 −0.126899
\(424\) 0 0
\(425\) 1037.00 0.118357
\(426\) 0 0
\(427\) 5520.00 0.625601
\(428\) 0 0
\(429\) −760.000 −0.0855318
\(430\) 0 0
\(431\) −13712.0 −1.53245 −0.766223 0.642575i \(-0.777866\pi\)
−0.766223 + 0.642575i \(0.777866\pi\)
\(432\) 0 0
\(433\) −62.0000 −0.00688113 −0.00344057 0.999994i \(-0.501095\pi\)
−0.00344057 + 0.999994i \(0.501095\pi\)
\(434\) 0 0
\(435\) 896.000 0.0987584
\(436\) 0 0
\(437\) 480.000 0.0525435
\(438\) 0 0
\(439\) −9392.00 −1.02108 −0.510542 0.859853i \(-0.670555\pi\)
−0.510542 + 0.859853i \(0.670555\pi\)
\(440\) 0 0
\(441\) 4577.00 0.494223
\(442\) 0 0
\(443\) 8528.00 0.914622 0.457311 0.889307i \(-0.348813\pi\)
0.457311 + 0.889307i \(0.348813\pi\)
\(444\) 0 0
\(445\) −7248.00 −0.772108
\(446\) 0 0
\(447\) 2564.00 0.271304
\(448\) 0 0
\(449\) 4238.00 0.445442 0.222721 0.974882i \(-0.428506\pi\)
0.222721 + 0.974882i \(0.428506\pi\)
\(450\) 0 0
\(451\) −700.000 −0.0730858
\(452\) 0 0
\(453\) −6624.00 −0.687026
\(454\) 0 0
\(455\) −3648.00 −0.375870
\(456\) 0 0
\(457\) −19382.0 −1.98392 −0.991960 0.126549i \(-0.959610\pi\)
−0.991960 + 0.126549i \(0.959610\pi\)
\(458\) 0 0
\(459\) −1700.00 −0.172874
\(460\) 0 0
\(461\) 18882.0 1.90764 0.953820 0.300377i \(-0.0971127\pi\)
0.953820 + 0.300377i \(0.0971127\pi\)
\(462\) 0 0
\(463\) 6224.00 0.624738 0.312369 0.949961i \(-0.398877\pi\)
0.312369 + 0.949961i \(0.398877\pi\)
\(464\) 0 0
\(465\) 2624.00 0.261688
\(466\) 0 0
\(467\) 10068.0 0.997626 0.498813 0.866710i \(-0.333769\pi\)
0.498813 + 0.866710i \(0.333769\pi\)
\(468\) 0 0
\(469\) 9216.00 0.907367
\(470\) 0 0
\(471\) −620.000 −0.0606541
\(472\) 0 0
\(473\) 1440.00 0.139982
\(474\) 0 0
\(475\) −244.000 −0.0235694
\(476\) 0 0
\(477\) 8418.00 0.808037
\(478\) 0 0
\(479\) 20064.0 1.91388 0.956939 0.290289i \(-0.0937515\pi\)
0.956939 + 0.290289i \(0.0937515\pi\)
\(480\) 0 0
\(481\) 8968.00 0.850116
\(482\) 0 0
\(483\) 2880.00 0.271314
\(484\) 0 0
\(485\) −368.000 −0.0344536
\(486\) 0 0
\(487\) −15616.0 −1.45304 −0.726518 0.687147i \(-0.758863\pi\)
−0.726518 + 0.687147i \(0.758863\pi\)
\(488\) 0 0
\(489\) 3676.00 0.339948
\(490\) 0 0
\(491\) 5868.00 0.539347 0.269673 0.962952i \(-0.413084\pi\)
0.269673 + 0.962952i \(0.413084\pi\)
\(492\) 0 0
\(493\) −952.000 −0.0869694
\(494\) 0 0
\(495\) −1840.00 −0.167074
\(496\) 0 0
\(497\) −864.000 −0.0779793
\(498\) 0 0
\(499\) −962.000 −0.0863027 −0.0431513 0.999069i \(-0.513740\pi\)
−0.0431513 + 0.999069i \(0.513740\pi\)
\(500\) 0 0
\(501\) −3368.00 −0.300342
\(502\) 0 0
\(503\) −4128.00 −0.365921 −0.182961 0.983120i \(-0.558568\pi\)
−0.182961 + 0.983120i \(0.558568\pi\)
\(504\) 0 0
\(505\) −9424.00 −0.830421
\(506\) 0 0
\(507\) 1506.00 0.131921
\(508\) 0 0
\(509\) 12006.0 1.04549 0.522747 0.852488i \(-0.324907\pi\)
0.522747 + 0.852488i \(0.324907\pi\)
\(510\) 0 0
\(511\) 8808.00 0.762511
\(512\) 0 0
\(513\) 400.000 0.0344258
\(514\) 0 0
\(515\) 4864.00 0.416181
\(516\) 0 0
\(517\) −480.000 −0.0408324
\(518\) 0 0
\(519\) 2208.00 0.186745
\(520\) 0 0
\(521\) −20146.0 −1.69407 −0.847037 0.531534i \(-0.821616\pi\)
−0.847037 + 0.531534i \(0.821616\pi\)
\(522\) 0 0
\(523\) 1496.00 0.125077 0.0625387 0.998043i \(-0.480080\pi\)
0.0625387 + 0.998043i \(0.480080\pi\)
\(524\) 0 0
\(525\) −1464.00 −0.121703
\(526\) 0 0
\(527\) −2788.00 −0.230450
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) 11592.0 0.947363
\(532\) 0 0
\(533\) −2660.00 −0.216168
\(534\) 0 0
\(535\) 14384.0 1.16238
\(536\) 0 0
\(537\) 7320.00 0.588233
\(538\) 0 0
\(539\) 1990.00 0.159027
\(540\) 0 0
\(541\) −16756.0 −1.33160 −0.665801 0.746129i \(-0.731910\pi\)
−0.665801 + 0.746129i \(0.731910\pi\)
\(542\) 0 0
\(543\) −9448.00 −0.746690
\(544\) 0 0
\(545\) −1600.00 −0.125755
\(546\) 0 0
\(547\) 11010.0 0.860610 0.430305 0.902684i \(-0.358406\pi\)
0.430305 + 0.902684i \(0.358406\pi\)
\(548\) 0 0
\(549\) 10580.0 0.822483
\(550\) 0 0
\(551\) 224.000 0.0173189
\(552\) 0 0
\(553\) −8832.00 −0.679159
\(554\) 0 0
\(555\) −3776.00 −0.288797
\(556\) 0 0
\(557\) −1606.00 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(558\) 0 0
\(559\) 5472.00 0.414027
\(560\) 0 0
\(561\) −340.000 −0.0255879
\(562\) 0 0
\(563\) 24480.0 1.83252 0.916260 0.400584i \(-0.131193\pi\)
0.916260 + 0.400584i \(0.131193\pi\)
\(564\) 0 0
\(565\) −18640.0 −1.38795
\(566\) 0 0
\(567\) −5052.00 −0.374187
\(568\) 0 0
\(569\) 7026.00 0.517654 0.258827 0.965924i \(-0.416664\pi\)
0.258827 + 0.965924i \(0.416664\pi\)
\(570\) 0 0
\(571\) −15542.0 −1.13908 −0.569538 0.821965i \(-0.692878\pi\)
−0.569538 + 0.821965i \(0.692878\pi\)
\(572\) 0 0
\(573\) −8560.00 −0.624082
\(574\) 0 0
\(575\) −7320.00 −0.530896
\(576\) 0 0
\(577\) 7886.00 0.568975 0.284487 0.958680i \(-0.408177\pi\)
0.284487 + 0.958680i \(0.408177\pi\)
\(578\) 0 0
\(579\) −1260.00 −0.0904384
\(580\) 0 0
\(581\) −10272.0 −0.733484
\(582\) 0 0
\(583\) 3660.00 0.260003
\(584\) 0 0
\(585\) −6992.00 −0.494160
\(586\) 0 0
\(587\) 1732.00 0.121784 0.0608921 0.998144i \(-0.480605\pi\)
0.0608921 + 0.998144i \(0.480605\pi\)
\(588\) 0 0
\(589\) 656.000 0.0458914
\(590\) 0 0
\(591\) 2408.00 0.167600
\(592\) 0 0
\(593\) −21182.0 −1.46685 −0.733424 0.679772i \(-0.762079\pi\)
−0.733424 + 0.679772i \(0.762079\pi\)
\(594\) 0 0
\(595\) −1632.00 −0.112446
\(596\) 0 0
\(597\) −9264.00 −0.635093
\(598\) 0 0
\(599\) −22280.0 −1.51976 −0.759880 0.650063i \(-0.774742\pi\)
−0.759880 + 0.650063i \(0.774742\pi\)
\(600\) 0 0
\(601\) −26190.0 −1.77756 −0.888779 0.458336i \(-0.848446\pi\)
−0.888779 + 0.458336i \(0.848446\pi\)
\(602\) 0 0
\(603\) 17664.0 1.19292
\(604\) 0 0
\(605\) 9848.00 0.661782
\(606\) 0 0
\(607\) 12444.0 0.832103 0.416051 0.909341i \(-0.363414\pi\)
0.416051 + 0.909341i \(0.363414\pi\)
\(608\) 0 0
\(609\) 1344.00 0.0894280
\(610\) 0 0
\(611\) −1824.00 −0.120771
\(612\) 0 0
\(613\) −3418.00 −0.225207 −0.112603 0.993640i \(-0.535919\pi\)
−0.112603 + 0.993640i \(0.535919\pi\)
\(614\) 0 0
\(615\) 1120.00 0.0734354
\(616\) 0 0
\(617\) 5346.00 0.348820 0.174410 0.984673i \(-0.444198\pi\)
0.174410 + 0.984673i \(0.444198\pi\)
\(618\) 0 0
\(619\) −17846.0 −1.15879 −0.579395 0.815047i \(-0.696711\pi\)
−0.579395 + 0.815047i \(0.696711\pi\)
\(620\) 0 0
\(621\) 12000.0 0.775432
\(622\) 0 0
\(623\) −10872.0 −0.699161
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) 0 0
\(627\) 80.0000 0.00509552
\(628\) 0 0
\(629\) 4012.00 0.254323
\(630\) 0 0
\(631\) −13416.0 −0.846407 −0.423203 0.906035i \(-0.639094\pi\)
−0.423203 + 0.906035i \(0.639094\pi\)
\(632\) 0 0
\(633\) −7180.00 −0.450836
\(634\) 0 0
\(635\) 16128.0 1.00791
\(636\) 0 0
\(637\) 7562.00 0.470357
\(638\) 0 0
\(639\) −1656.00 −0.102520
\(640\) 0 0
\(641\) −7762.00 −0.478285 −0.239142 0.970985i \(-0.576866\pi\)
−0.239142 + 0.970985i \(0.576866\pi\)
\(642\) 0 0
\(643\) −9106.00 −0.558485 −0.279242 0.960221i \(-0.590083\pi\)
−0.279242 + 0.960221i \(0.590083\pi\)
\(644\) 0 0
\(645\) −2304.00 −0.140651
\(646\) 0 0
\(647\) −5488.00 −0.333471 −0.166735 0.986002i \(-0.553323\pi\)
−0.166735 + 0.986002i \(0.553323\pi\)
\(648\) 0 0
\(649\) 5040.00 0.304834
\(650\) 0 0
\(651\) 3936.00 0.236965
\(652\) 0 0
\(653\) 2640.00 0.158210 0.0791050 0.996866i \(-0.474794\pi\)
0.0791050 + 0.996866i \(0.474794\pi\)
\(654\) 0 0
\(655\) 816.000 0.0486775
\(656\) 0 0
\(657\) 16882.0 1.00248
\(658\) 0 0
\(659\) 7548.00 0.446173 0.223087 0.974799i \(-0.428387\pi\)
0.223087 + 0.974799i \(0.428387\pi\)
\(660\) 0 0
\(661\) −5718.00 −0.336467 −0.168233 0.985747i \(-0.553806\pi\)
−0.168233 + 0.985747i \(0.553806\pi\)
\(662\) 0 0
\(663\) −1292.00 −0.0756819
\(664\) 0 0
\(665\) 384.000 0.0223923
\(666\) 0 0
\(667\) 6720.00 0.390104
\(668\) 0 0
\(669\) −4688.00 −0.270925
\(670\) 0 0
\(671\) 4600.00 0.264651
\(672\) 0 0
\(673\) 26150.0 1.49778 0.748892 0.662692i \(-0.230586\pi\)
0.748892 + 0.662692i \(0.230586\pi\)
\(674\) 0 0
\(675\) −6100.00 −0.347836
\(676\) 0 0
\(677\) 19276.0 1.09429 0.547147 0.837037i \(-0.315714\pi\)
0.547147 + 0.837037i \(0.315714\pi\)
\(678\) 0 0
\(679\) −552.000 −0.0311986
\(680\) 0 0
\(681\) −2516.00 −0.141576
\(682\) 0 0
\(683\) 17758.0 0.994862 0.497431 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) 17488.0 0.975448
\(686\) 0 0
\(687\) 10516.0 0.584004
\(688\) 0 0
\(689\) 13908.0 0.769017
\(690\) 0 0
\(691\) −3994.00 −0.219883 −0.109941 0.993938i \(-0.535066\pi\)
−0.109941 + 0.993938i \(0.535066\pi\)
\(692\) 0 0
\(693\) −2760.00 −0.151290
\(694\) 0 0
\(695\) 20720.0 1.13087
\(696\) 0 0
\(697\) −1190.00 −0.0646692
\(698\) 0 0
\(699\) 7388.00 0.399771
\(700\) 0 0
\(701\) −14486.0 −0.780497 −0.390249 0.920709i \(-0.627611\pi\)
−0.390249 + 0.920709i \(0.627611\pi\)
\(702\) 0 0
\(703\) −944.000 −0.0506453
\(704\) 0 0
\(705\) 768.000 0.0410277
\(706\) 0 0
\(707\) −14136.0 −0.751965
\(708\) 0 0
\(709\) −7256.00 −0.384351 −0.192175 0.981361i \(-0.561554\pi\)
−0.192175 + 0.981361i \(0.561554\pi\)
\(710\) 0 0
\(711\) −16928.0 −0.892897
\(712\) 0 0
\(713\) 19680.0 1.03369
\(714\) 0 0
\(715\) −3040.00 −0.159006
\(716\) 0 0
\(717\) 3888.00 0.202510
\(718\) 0 0
\(719\) −37448.0 −1.94238 −0.971192 0.238297i \(-0.923411\pi\)
−0.971192 + 0.238297i \(0.923411\pi\)
\(720\) 0 0
\(721\) 7296.00 0.376862
\(722\) 0 0
\(723\) 2324.00 0.119544
\(724\) 0 0
\(725\) −3416.00 −0.174989
\(726\) 0 0
\(727\) 30856.0 1.57412 0.787060 0.616876i \(-0.211602\pi\)
0.787060 + 0.616876i \(0.211602\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) 2448.00 0.123861
\(732\) 0 0
\(733\) −5294.00 −0.266764 −0.133382 0.991065i \(-0.542584\pi\)
−0.133382 + 0.991065i \(0.542584\pi\)
\(734\) 0 0
\(735\) −3184.00 −0.159787
\(736\) 0 0
\(737\) 7680.00 0.383849
\(738\) 0 0
\(739\) −836.000 −0.0416140 −0.0208070 0.999784i \(-0.506624\pi\)
−0.0208070 + 0.999784i \(0.506624\pi\)
\(740\) 0 0
\(741\) 304.000 0.0150711
\(742\) 0 0
\(743\) 24452.0 1.20734 0.603672 0.797233i \(-0.293704\pi\)
0.603672 + 0.797233i \(0.293704\pi\)
\(744\) 0 0
\(745\) 10256.0 0.504363
\(746\) 0 0
\(747\) −19688.0 −0.964319
\(748\) 0 0
\(749\) 21576.0 1.05256
\(750\) 0 0
\(751\) −5680.00 −0.275987 −0.137993 0.990433i \(-0.544065\pi\)
−0.137993 + 0.990433i \(0.544065\pi\)
\(752\) 0 0
\(753\) 352.000 0.0170353
\(754\) 0 0
\(755\) −26496.0 −1.27720
\(756\) 0 0
\(757\) 10622.0 0.509991 0.254995 0.966942i \(-0.417926\pi\)
0.254995 + 0.966942i \(0.417926\pi\)
\(758\) 0 0
\(759\) 2400.00 0.114775
\(760\) 0 0
\(761\) −28842.0 −1.37388 −0.686939 0.726715i \(-0.741046\pi\)
−0.686939 + 0.726715i \(0.741046\pi\)
\(762\) 0 0
\(763\) −2400.00 −0.113874
\(764\) 0 0
\(765\) −3128.00 −0.147834
\(766\) 0 0
\(767\) 19152.0 0.901615
\(768\) 0 0
\(769\) 35598.0 1.66931 0.834653 0.550776i \(-0.185668\pi\)
0.834653 + 0.550776i \(0.185668\pi\)
\(770\) 0 0
\(771\) 8948.00 0.417969
\(772\) 0 0
\(773\) −32962.0 −1.53371 −0.766857 0.641818i \(-0.778180\pi\)
−0.766857 + 0.641818i \(0.778180\pi\)
\(774\) 0 0
\(775\) −10004.0 −0.463683
\(776\) 0 0
\(777\) −5664.00 −0.261512
\(778\) 0 0
\(779\) 280.000 0.0128781
\(780\) 0 0
\(781\) −720.000 −0.0329880
\(782\) 0 0
\(783\) 5600.00 0.255591
\(784\) 0 0
\(785\) −2480.00 −0.112758
\(786\) 0 0
\(787\) −26318.0 −1.19204 −0.596020 0.802970i \(-0.703252\pi\)
−0.596020 + 0.802970i \(0.703252\pi\)
\(788\) 0 0
\(789\) 5392.00 0.243296
\(790\) 0 0
\(791\) −27960.0 −1.25682
\(792\) 0 0
\(793\) 17480.0 0.782765
\(794\) 0 0
\(795\) −5856.00 −0.261246
\(796\) 0 0
\(797\) −17134.0 −0.761502 −0.380751 0.924678i \(-0.624335\pi\)
−0.380751 + 0.924678i \(0.624335\pi\)
\(798\) 0 0
\(799\) −816.000 −0.0361302
\(800\) 0 0
\(801\) −20838.0 −0.919194
\(802\) 0 0
\(803\) 7340.00 0.322569
\(804\) 0 0
\(805\) 11520.0 0.504381
\(806\) 0 0
\(807\) −4608.00 −0.201003
\(808\) 0 0
\(809\) −21826.0 −0.948531 −0.474265 0.880382i \(-0.657286\pi\)
−0.474265 + 0.880382i \(0.657286\pi\)
\(810\) 0 0
\(811\) −3386.00 −0.146607 −0.0733037 0.997310i \(-0.523354\pi\)
−0.0733037 + 0.997310i \(0.523354\pi\)
\(812\) 0 0
\(813\) −3440.00 −0.148396
\(814\) 0 0
\(815\) 14704.0 0.631974
\(816\) 0 0
\(817\) −576.000 −0.0246655
\(818\) 0 0
\(819\) −10488.0 −0.447473
\(820\) 0 0
\(821\) −17284.0 −0.734733 −0.367366 0.930076i \(-0.619741\pi\)
−0.367366 + 0.930076i \(0.619741\pi\)
\(822\) 0 0
\(823\) −5656.00 −0.239557 −0.119779 0.992801i \(-0.538219\pi\)
−0.119779 + 0.992801i \(0.538219\pi\)
\(824\) 0 0
\(825\) −1220.00 −0.0514848
\(826\) 0 0
\(827\) −8714.00 −0.366403 −0.183202 0.983075i \(-0.558646\pi\)
−0.183202 + 0.983075i \(0.558646\pi\)
\(828\) 0 0
\(829\) 11198.0 0.469147 0.234573 0.972098i \(-0.424631\pi\)
0.234573 + 0.972098i \(0.424631\pi\)
\(830\) 0 0
\(831\) 8448.00 0.352657
\(832\) 0 0
\(833\) 3383.00 0.140713
\(834\) 0 0
\(835\) −13472.0 −0.558345
\(836\) 0 0
\(837\) 16400.0 0.677260
\(838\) 0 0
\(839\) −6636.00 −0.273063 −0.136532 0.990636i \(-0.543596\pi\)
−0.136532 + 0.990636i \(0.543596\pi\)
\(840\) 0 0
\(841\) −21253.0 −0.871417
\(842\) 0 0
\(843\) −6252.00 −0.255433
\(844\) 0 0
\(845\) 6024.00 0.245245
\(846\) 0 0
\(847\) 14772.0 0.599258
\(848\) 0 0
\(849\) −10236.0 −0.413779
\(850\) 0 0
\(851\) −28320.0 −1.14077
\(852\) 0 0
\(853\) −16112.0 −0.646734 −0.323367 0.946274i \(-0.604815\pi\)
−0.323367 + 0.946274i \(0.604815\pi\)
\(854\) 0 0
\(855\) 736.000 0.0294394
\(856\) 0 0
\(857\) 41742.0 1.66380 0.831902 0.554923i \(-0.187252\pi\)
0.831902 + 0.554923i \(0.187252\pi\)
\(858\) 0 0
\(859\) −22248.0 −0.883693 −0.441846 0.897091i \(-0.645676\pi\)
−0.441846 + 0.897091i \(0.645676\pi\)
\(860\) 0 0
\(861\) 1680.00 0.0664974
\(862\) 0 0
\(863\) −11888.0 −0.468913 −0.234457 0.972127i \(-0.575331\pi\)
−0.234457 + 0.972127i \(0.575331\pi\)
\(864\) 0 0
\(865\) 8832.00 0.347164
\(866\) 0 0
\(867\) −578.000 −0.0226412
\(868\) 0 0
\(869\) −7360.00 −0.287308
\(870\) 0 0
\(871\) 29184.0 1.13532
\(872\) 0 0
\(873\) −1058.00 −0.0410170
\(874\) 0 0
\(875\) −17856.0 −0.689878
\(876\) 0 0
\(877\) 40024.0 1.54107 0.770533 0.637400i \(-0.219990\pi\)
0.770533 + 0.637400i \(0.219990\pi\)
\(878\) 0 0
\(879\) 8836.00 0.339057
\(880\) 0 0
\(881\) 36818.0 1.40798 0.703990 0.710210i \(-0.251400\pi\)
0.703990 + 0.710210i \(0.251400\pi\)
\(882\) 0 0
\(883\) −38424.0 −1.46441 −0.732203 0.681086i \(-0.761508\pi\)
−0.732203 + 0.681086i \(0.761508\pi\)
\(884\) 0 0
\(885\) −8064.00 −0.306292
\(886\) 0 0
\(887\) −13248.0 −0.501493 −0.250747 0.968053i \(-0.580676\pi\)
−0.250747 + 0.968053i \(0.580676\pi\)
\(888\) 0 0
\(889\) 24192.0 0.912681
\(890\) 0 0
\(891\) −4210.00 −0.158294
\(892\) 0 0
\(893\) 192.000 0.00719489
\(894\) 0 0
\(895\) 29280.0 1.09354
\(896\) 0 0
\(897\) 9120.00 0.339474
\(898\) 0 0
\(899\) 9184.00 0.340716
\(900\) 0 0
\(901\) 6222.00 0.230061
\(902\) 0 0
\(903\) −3456.00 −0.127363
\(904\) 0 0
\(905\) −37792.0 −1.38812
\(906\) 0 0
\(907\) 29014.0 1.06218 0.531088 0.847317i \(-0.321783\pi\)
0.531088 + 0.847317i \(0.321783\pi\)
\(908\) 0 0
\(909\) −27094.0 −0.988615
\(910\) 0 0
\(911\) −16156.0 −0.587565 −0.293783 0.955872i \(-0.594914\pi\)
−0.293783 + 0.955872i \(0.594914\pi\)
\(912\) 0 0
\(913\) −8560.00 −0.310290
\(914\) 0 0
\(915\) −7360.00 −0.265917
\(916\) 0 0
\(917\) 1224.00 0.0440786
\(918\) 0 0
\(919\) 46584.0 1.67210 0.836052 0.548650i \(-0.184858\pi\)
0.836052 + 0.548650i \(0.184858\pi\)
\(920\) 0 0
\(921\) −8952.00 −0.320281
\(922\) 0 0
\(923\) −2736.00 −0.0975694
\(924\) 0 0
\(925\) 14396.0 0.511716
\(926\) 0 0
\(927\) 13984.0 0.495464
\(928\) 0 0
\(929\) 34766.0 1.22781 0.613905 0.789380i \(-0.289598\pi\)
0.613905 + 0.789380i \(0.289598\pi\)
\(930\) 0 0
\(931\) −796.000 −0.0280213
\(932\) 0 0
\(933\) 600.000 0.0210537
\(934\) 0 0
\(935\) −1360.00 −0.0475687
\(936\) 0 0
\(937\) −19690.0 −0.686493 −0.343247 0.939245i \(-0.611527\pi\)
−0.343247 + 0.939245i \(0.611527\pi\)
\(938\) 0 0
\(939\) 2308.00 0.0802116
\(940\) 0 0
\(941\) 8564.00 0.296683 0.148341 0.988936i \(-0.452607\pi\)
0.148341 + 0.988936i \(0.452607\pi\)
\(942\) 0 0
\(943\) 8400.00 0.290076
\(944\) 0 0
\(945\) 9600.00 0.330464
\(946\) 0 0
\(947\) 42578.0 1.46103 0.730517 0.682895i \(-0.239279\pi\)
0.730517 + 0.682895i \(0.239279\pi\)
\(948\) 0 0
\(949\) 27892.0 0.954070
\(950\) 0 0
\(951\) −1368.00 −0.0466461
\(952\) 0 0
\(953\) −47406.0 −1.61137 −0.805683 0.592348i \(-0.798201\pi\)
−0.805683 + 0.592348i \(0.798201\pi\)
\(954\) 0 0
\(955\) −34240.0 −1.16019
\(956\) 0 0
\(957\) 1120.00 0.0378312
\(958\) 0 0
\(959\) 26232.0 0.883290
\(960\) 0 0
\(961\) −2895.00 −0.0971770
\(962\) 0 0
\(963\) 41354.0 1.38382
\(964\) 0 0
\(965\) −5040.00 −0.168128
\(966\) 0 0
\(967\) 51400.0 1.70932 0.854660 0.519188i \(-0.173766\pi\)
0.854660 + 0.519188i \(0.173766\pi\)
\(968\) 0 0
\(969\) 136.000 0.00450872
\(970\) 0 0
\(971\) 5840.00 0.193012 0.0965059 0.995332i \(-0.469233\pi\)
0.0965059 + 0.995332i \(0.469233\pi\)
\(972\) 0 0
\(973\) 31080.0 1.02403
\(974\) 0 0
\(975\) −4636.00 −0.152278
\(976\) 0 0
\(977\) −18358.0 −0.601151 −0.300575 0.953758i \(-0.597179\pi\)
−0.300575 + 0.953758i \(0.597179\pi\)
\(978\) 0 0
\(979\) −9060.00 −0.295770
\(980\) 0 0
\(981\) −4600.00 −0.149711
\(982\) 0 0
\(983\) 54888.0 1.78093 0.890466 0.455051i \(-0.150379\pi\)
0.890466 + 0.455051i \(0.150379\pi\)
\(984\) 0 0
\(985\) 9632.00 0.311575
\(986\) 0 0
\(987\) 1152.00 0.0371515
\(988\) 0 0
\(989\) −17280.0 −0.555583
\(990\) 0 0
\(991\) −6472.00 −0.207457 −0.103728 0.994606i \(-0.533077\pi\)
−0.103728 + 0.994606i \(0.533077\pi\)
\(992\) 0 0
\(993\) −18800.0 −0.600806
\(994\) 0 0
\(995\) −37056.0 −1.18066
\(996\) 0 0
\(997\) 16444.0 0.522354 0.261177 0.965291i \(-0.415889\pi\)
0.261177 + 0.965291i \(0.415889\pi\)
\(998\) 0 0
\(999\) −23600.0 −0.747418
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 68.4.a.a.1.1 1
3.2 odd 2 612.4.a.c.1.1 1
4.3 odd 2 272.4.a.b.1.1 1
5.2 odd 4 1700.4.e.c.749.2 2
5.3 odd 4 1700.4.e.c.749.1 2
5.4 even 2 1700.4.a.b.1.1 1
8.3 odd 2 1088.4.a.e.1.1 1
8.5 even 2 1088.4.a.h.1.1 1
12.11 even 2 2448.4.a.l.1.1 1
17.4 even 4 1156.4.b.b.577.2 2
17.13 even 4 1156.4.b.b.577.1 2
17.16 even 2 1156.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.4.a.a.1.1 1 1.1 even 1 trivial
272.4.a.b.1.1 1 4.3 odd 2
612.4.a.c.1.1 1 3.2 odd 2
1088.4.a.e.1.1 1 8.3 odd 2
1088.4.a.h.1.1 1 8.5 even 2
1156.4.a.a.1.1 1 17.16 even 2
1156.4.b.b.577.1 2 17.13 even 4
1156.4.b.b.577.2 2 17.4 even 4
1700.4.a.b.1.1 1 5.4 even 2
1700.4.e.c.749.1 2 5.3 odd 4
1700.4.e.c.749.2 2 5.2 odd 4
2448.4.a.l.1.1 1 12.11 even 2