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Properties

Label 2400.4.a.d.1.1
Level $2400$
Weight $4$
Character 2400.1
Self dual yes
Analytic conductor $141.605$
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] = [2400,4,Mod(1,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2400.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: \(141.604584014\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} +90.0000 q^{13} +70.0000 q^{17} -40.0000 q^{19} +12.0000 q^{21} +108.000 q^{23} -27.0000 q^{27} +166.000 q^{29} +40.0000 q^{31} +120.000 q^{33} +130.000 q^{37} -270.000 q^{39} -310.000 q^{41} -268.000 q^{43} -556.000 q^{47} -327.000 q^{49} -210.000 q^{51} +370.000 q^{53} +120.000 q^{57} -240.000 q^{59} -130.000 q^{61} -36.0000 q^{63} +876.000 q^{67} -324.000 q^{69} +840.000 q^{71} -250.000 q^{73} +160.000 q^{77} +880.000 q^{79} +81.0000 q^{81} -188.000 q^{83} -498.000 q^{87} -726.000 q^{89} -360.000 q^{91} -120.000 q^{93} +1550.00 q^{97} -360.000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −0.215980 −0.107990 − 0.994152i \(-0.534441\pi\)
−0.107990 + 0.994152i \(0.534441\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −40.0000 −1.09640 −0.548202 − 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) 90.0000 1.92012 0.960058 − 0.279801i \(-0.0902685\pi\)
0.960058 + 0.279801i \(0.0902685\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 70.0000 0.998676 0.499338 − 0.866407i \(-0.333577\pi\)
0.499338 + 0.866407i \(0.333577\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) 12.0000 0.124696
\(22\) 0 0
\(23\) 108.000 0.979111 0.489556 − 0.871972i \(-0.337159\pi\)
0.489556 + 0.871972i \(0.337159\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 166.000 1.06295 0.531473 − 0.847075i \(-0.321639\pi\)
0.531473 + 0.847075i \(0.321639\pi\)
\(30\) 0 0
\(31\) 40.0000 0.231749 0.115874 − 0.993264i \(-0.463033\pi\)
0.115874 + 0.993264i \(0.463033\pi\)
\(32\) 0 0
\(33\) 120.000 0.633010
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 130.000 0.577618 0.288809 − 0.957387i \(-0.406741\pi\)
0.288809 + 0.957387i \(0.406741\pi\)
\(38\) 0 0
\(39\) −270.000 −1.10858
\(40\) 0 0
\(41\) −310.000 −1.18083 −0.590413 − 0.807101i \(-0.701035\pi\)
−0.590413 + 0.807101i \(0.701035\pi\)
\(42\) 0 0
\(43\) −268.000 −0.950456 −0.475228 − 0.879863i \(-0.657634\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −556.000 −1.72555 −0.862776 − 0.505587i \(-0.831276\pi\)
−0.862776 + 0.505587i \(0.831276\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) −210.000 −0.576586
\(52\) 0 0
\(53\) 370.000 0.958932 0.479466 − 0.877560i \(-0.340830\pi\)
0.479466 + 0.877560i \(0.340830\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 120.000 0.278849
\(58\) 0 0
\(59\) −240.000 −0.529582 −0.264791 − 0.964306i \(-0.585303\pi\)
−0.264791 + 0.964306i \(0.585303\pi\)
\(60\) 0 0
\(61\) −130.000 −0.272865 −0.136433 − 0.990649i \(-0.543564\pi\)
−0.136433 + 0.990649i \(0.543564\pi\)
\(62\) 0 0
\(63\) −36.0000 −0.0719932
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 876.000 1.59732 0.798660 − 0.601783i \(-0.205543\pi\)
0.798660 + 0.601783i \(0.205543\pi\)
\(68\) 0 0
\(69\) −324.000 −0.565290
\(70\) 0 0
\(71\) 840.000 1.40408 0.702040 − 0.712138i \(-0.252273\pi\)
0.702040 + 0.712138i \(0.252273\pi\)
\(72\) 0 0
\(73\) −250.000 −0.400826 −0.200413 − 0.979712i \(-0.564228\pi\)
−0.200413 + 0.979712i \(0.564228\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 160.000 0.236801
\(78\) 0 0
\(79\) 880.000 1.25326 0.626631 − 0.779316i \(-0.284433\pi\)
0.626631 + 0.779316i \(0.284433\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −188.000 −0.248623 −0.124311 − 0.992243i \(-0.539672\pi\)
−0.124311 + 0.992243i \(0.539672\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −498.000 −0.613692
\(88\) 0 0
\(89\) −726.000 −0.864672 −0.432336 − 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) 0 0
\(91\) −360.000 −0.414706
\(92\) 0 0
\(93\) −120.000 −0.133800
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1550.00 1.62246 0.811230 − 0.584727i \(-0.198798\pi\)
0.811230 + 0.584727i \(0.198798\pi\)
\(98\) 0 0
\(99\) −360.000 −0.365468
\(100\) 0 0
\(101\) −898.000 −0.884696 −0.442348 − 0.896843i \(-0.645854\pi\)
−0.442348 + 0.896843i \(0.645854\pi\)
\(102\) 0 0
\(103\) −1148.00 −1.09821 −0.549106 − 0.835753i \(-0.685032\pi\)
−0.549106 + 0.835753i \(0.685032\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −276.000 −0.249364 −0.124682 − 0.992197i \(-0.539791\pi\)
−0.124682 + 0.992197i \(0.539791\pi\)
\(108\) 0 0
\(109\) −530.000 −0.465732 −0.232866 − 0.972509i \(-0.574810\pi\)
−0.232866 + 0.972509i \(0.574810\pi\)
\(110\) 0 0
\(111\) −390.000 −0.333488
\(112\) 0 0
\(113\) −1050.00 −0.874121 −0.437061 − 0.899432i \(-0.643980\pi\)
−0.437061 + 0.899432i \(0.643980\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 810.000 0.640039
\(118\) 0 0
\(119\) −280.000 −0.215694
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) 930.000 0.681750
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2316.00 1.61820 0.809101 − 0.587669i \(-0.199954\pi\)
0.809101 + 0.587669i \(0.199954\pi\)
\(128\) 0 0
\(129\) 804.000 0.548746
\(130\) 0 0
\(131\) 520.000 0.346814 0.173407 − 0.984850i \(-0.444522\pi\)
0.173407 + 0.984850i \(0.444522\pi\)
\(132\) 0 0
\(133\) 160.000 0.104314
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 190.000 0.118488 0.0592438 − 0.998244i \(-0.481131\pi\)
0.0592438 + 0.998244i \(0.481131\pi\)
\(138\) 0 0
\(139\) −2680.00 −1.63536 −0.817679 − 0.575675i \(-0.804739\pi\)
−0.817679 + 0.575675i \(0.804739\pi\)
\(140\) 0 0
\(141\) 1668.00 0.996248
\(142\) 0 0
\(143\) −3600.00 −2.10522
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 981.000 0.550418
\(148\) 0 0
\(149\) 3310.00 1.81990 0.909952 − 0.414713i \(-0.136118\pi\)
0.909952 + 0.414713i \(0.136118\pi\)
\(150\) 0 0
\(151\) 1160.00 0.625162 0.312581 − 0.949891i \(-0.398806\pi\)
0.312581 + 0.949891i \(0.398806\pi\)
\(152\) 0 0
\(153\) 630.000 0.332892
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1130.00 0.574419 0.287210 − 0.957868i \(-0.407272\pi\)
0.287210 + 0.957868i \(0.407272\pi\)
\(158\) 0 0
\(159\) −1110.00 −0.553640
\(160\) 0 0
\(161\) −432.000 −0.211468
\(162\) 0 0
\(163\) 3732.00 1.79333 0.896665 − 0.442710i \(-0.145983\pi\)
0.896665 + 0.442710i \(0.145983\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3644.00 −1.68851 −0.844255 − 0.535942i \(-0.819957\pi\)
−0.844255 + 0.535942i \(0.819957\pi\)
\(168\) 0 0
\(169\) 5903.00 2.68685
\(170\) 0 0
\(171\) −360.000 −0.160993
\(172\) 0 0
\(173\) 1290.00 0.566918 0.283459 − 0.958984i \(-0.408518\pi\)
0.283459 + 0.958984i \(0.408518\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 720.000 0.305754
\(178\) 0 0
\(179\) 1920.00 0.801718 0.400859 − 0.916140i \(-0.368712\pi\)
0.400859 + 0.916140i \(0.368712\pi\)
\(180\) 0 0
\(181\) −42.0000 −0.0172477 −0.00862385 − 0.999963i \(-0.502745\pi\)
−0.00862385 + 0.999963i \(0.502745\pi\)
\(182\) 0 0
\(183\) 390.000 0.157539
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2800.00 −1.09495
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) 680.000 0.257608 0.128804 − 0.991670i \(-0.458886\pi\)
0.128804 + 0.991670i \(0.458886\pi\)
\(192\) 0 0
\(193\) −2210.00 −0.824245 −0.412122 − 0.911128i \(-0.635212\pi\)
−0.412122 + 0.911128i \(0.635212\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 130.000 0.0470158 0.0235079 − 0.999724i \(-0.492517\pi\)
0.0235079 + 0.999724i \(0.492517\pi\)
\(198\) 0 0
\(199\) −3040.00 −1.08291 −0.541457 − 0.840728i \(-0.682127\pi\)
−0.541457 + 0.840728i \(0.682127\pi\)
\(200\) 0 0
\(201\) −2628.00 −0.922213
\(202\) 0 0
\(203\) −664.000 −0.229575
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 972.000 0.326370
\(208\) 0 0
\(209\) 1600.00 0.529542
\(210\) 0 0
\(211\) −560.000 −0.182711 −0.0913554 − 0.995818i \(-0.529120\pi\)
−0.0913554 + 0.995818i \(0.529120\pi\)
\(212\) 0 0
\(213\) −2520.00 −0.810646
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −160.000 −0.0500530
\(218\) 0 0
\(219\) 750.000 0.231417
\(220\) 0 0
\(221\) 6300.00 1.91757
\(222\) 0 0
\(223\) −332.000 −0.0996967 −0.0498484 − 0.998757i \(-0.515874\pi\)
−0.0498484 + 0.998757i \(0.515874\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2364.00 0.691208 0.345604 − 0.938380i \(-0.387674\pi\)
0.345604 + 0.938380i \(0.387674\pi\)
\(228\) 0 0
\(229\) 1334.00 0.384948 0.192474 − 0.981302i \(-0.438349\pi\)
0.192474 + 0.981302i \(0.438349\pi\)
\(230\) 0 0
\(231\) −480.000 −0.136717
\(232\) 0 0
\(233\) −5570.00 −1.56611 −0.783053 − 0.621955i \(-0.786339\pi\)
−0.783053 + 0.621955i \(0.786339\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2640.00 −0.723571
\(238\) 0 0
\(239\) 3520.00 0.952677 0.476339 − 0.879262i \(-0.341964\pi\)
0.476339 + 0.879262i \(0.341964\pi\)
\(240\) 0 0
\(241\) 2130.00 0.569317 0.284658 − 0.958629i \(-0.408120\pi\)
0.284658 + 0.958629i \(0.408120\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3600.00 −0.927379
\(248\) 0 0
\(249\) 564.000 0.143542
\(250\) 0 0
\(251\) 40.0000 0.0100589 0.00502944 − 0.999987i \(-0.498399\pi\)
0.00502944 + 0.999987i \(0.498399\pi\)
\(252\) 0 0
\(253\) −4320.00 −1.07350
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7510.00 1.82281 0.911403 − 0.411516i \(-0.135001\pi\)
0.911403 + 0.411516i \(0.135001\pi\)
\(258\) 0 0
\(259\) −520.000 −0.124754
\(260\) 0 0
\(261\) 1494.00 0.354315
\(262\) 0 0
\(263\) −2228.00 −0.522374 −0.261187 − 0.965288i \(-0.584114\pi\)
−0.261187 + 0.965288i \(0.584114\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2178.00 0.499219
\(268\) 0 0
\(269\) 3750.00 0.849969 0.424984 − 0.905201i \(-0.360280\pi\)
0.424984 + 0.905201i \(0.360280\pi\)
\(270\) 0 0
\(271\) 1000.00 0.224154 0.112077 − 0.993700i \(-0.464250\pi\)
0.112077 + 0.993700i \(0.464250\pi\)
\(272\) 0 0
\(273\) 1080.00 0.239431
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5650.00 1.22554 0.612772 − 0.790260i \(-0.290054\pi\)
0.612772 + 0.790260i \(0.290054\pi\)
\(278\) 0 0
\(279\) 360.000 0.0772496
\(280\) 0 0
\(281\) 3770.00 0.800354 0.400177 − 0.916438i \(-0.368949\pi\)
0.400177 + 0.916438i \(0.368949\pi\)
\(282\) 0 0
\(283\) 2468.00 0.518401 0.259200 − 0.965824i \(-0.416541\pi\)
0.259200 + 0.965824i \(0.416541\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1240.00 0.255034
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) −4650.00 −0.936728
\(292\) 0 0
\(293\) −2910.00 −0.580218 −0.290109 − 0.956994i \(-0.593692\pi\)
−0.290109 + 0.956994i \(0.593692\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1080.00 0.211003
\(298\) 0 0
\(299\) 9720.00 1.88001
\(300\) 0 0
\(301\) 1072.00 0.205279
\(302\) 0 0
\(303\) 2694.00 0.510780
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5116.00 0.951093 0.475546 − 0.879691i \(-0.342250\pi\)
0.475546 + 0.879691i \(0.342250\pi\)
\(308\) 0 0
\(309\) 3444.00 0.634053
\(310\) 0 0
\(311\) 3640.00 0.663683 0.331842 − 0.943335i \(-0.392330\pi\)
0.331842 + 0.943335i \(0.392330\pi\)
\(312\) 0 0
\(313\) −3930.00 −0.709702 −0.354851 − 0.934923i \(-0.615468\pi\)
−0.354851 + 0.934923i \(0.615468\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8890.00 1.57512 0.787559 − 0.616240i \(-0.211345\pi\)
0.787559 + 0.616240i \(0.211345\pi\)
\(318\) 0 0
\(319\) −6640.00 −1.16542
\(320\) 0 0
\(321\) 828.000 0.143970
\(322\) 0 0
\(323\) −2800.00 −0.482341
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1590.00 0.268891
\(328\) 0 0
\(329\) 2224.00 0.372684
\(330\) 0 0
\(331\) −4400.00 −0.730652 −0.365326 − 0.930880i \(-0.619043\pi\)
−0.365326 + 0.930880i \(0.619043\pi\)
\(332\) 0 0
\(333\) 1170.00 0.192539
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −610.000 −0.0986018 −0.0493009 − 0.998784i \(-0.515699\pi\)
−0.0493009 + 0.998784i \(0.515699\pi\)
\(338\) 0 0
\(339\) 3150.00 0.504674
\(340\) 0 0
\(341\) −1600.00 −0.254090
\(342\) 0 0
\(343\) 2680.00 0.421885
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8004.00 −1.23826 −0.619131 − 0.785287i \(-0.712515\pi\)
−0.619131 + 0.785287i \(0.712515\pi\)
\(348\) 0 0
\(349\) 5614.00 0.861062 0.430531 − 0.902576i \(-0.358326\pi\)
0.430531 + 0.902576i \(0.358326\pi\)
\(350\) 0 0
\(351\) −2430.00 −0.369527
\(352\) 0 0
\(353\) 3270.00 0.493044 0.246522 − 0.969137i \(-0.420712\pi\)
0.246522 + 0.969137i \(0.420712\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 840.000 0.124531
\(358\) 0 0
\(359\) −2960.00 −0.435161 −0.217581 − 0.976042i \(-0.569816\pi\)
−0.217581 + 0.976042i \(0.569816\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) −807.000 −0.116685
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11964.0 1.70168 0.850839 − 0.525427i \(-0.176094\pi\)
0.850839 + 0.525427i \(0.176094\pi\)
\(368\) 0 0
\(369\) −2790.00 −0.393609
\(370\) 0 0
\(371\) −1480.00 −0.207110
\(372\) 0 0
\(373\) 12770.0 1.77267 0.886334 − 0.463046i \(-0.153243\pi\)
0.886334 + 0.463046i \(0.153243\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14940.0 2.04098
\(378\) 0 0
\(379\) 12600.0 1.70770 0.853850 − 0.520519i \(-0.174261\pi\)
0.853850 + 0.520519i \(0.174261\pi\)
\(380\) 0 0
\(381\) −6948.00 −0.934270
\(382\) 0 0
\(383\) 8828.00 1.17778 0.588890 − 0.808213i \(-0.299565\pi\)
0.588890 + 0.808213i \(0.299565\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2412.00 −0.316819
\(388\) 0 0
\(389\) 5630.00 0.733811 0.366905 − 0.930258i \(-0.380417\pi\)
0.366905 + 0.930258i \(0.380417\pi\)
\(390\) 0 0
\(391\) 7560.00 0.977815
\(392\) 0 0
\(393\) −1560.00 −0.200233
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14410.0 1.82171 0.910853 − 0.412731i \(-0.135425\pi\)
0.910853 + 0.412731i \(0.135425\pi\)
\(398\) 0 0
\(399\) −480.000 −0.0602257
\(400\) 0 0
\(401\) −9102.00 −1.13350 −0.566748 − 0.823891i \(-0.691799\pi\)
−0.566748 + 0.823891i \(0.691799\pi\)
\(402\) 0 0
\(403\) 3600.00 0.444985
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5200.00 −0.633303
\(408\) 0 0
\(409\) 10010.0 1.21018 0.605089 − 0.796158i \(-0.293138\pi\)
0.605089 + 0.796158i \(0.293138\pi\)
\(410\) 0 0
\(411\) −570.000 −0.0684088
\(412\) 0 0
\(413\) 960.000 0.114379
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8040.00 0.944174
\(418\) 0 0
\(419\) 720.000 0.0839482 0.0419741 − 0.999119i \(-0.486635\pi\)
0.0419741 + 0.999119i \(0.486635\pi\)
\(420\) 0 0
\(421\) −7610.00 −0.880971 −0.440485 − 0.897760i \(-0.645194\pi\)
−0.440485 + 0.897760i \(0.645194\pi\)
\(422\) 0 0
\(423\) −5004.00 −0.575184
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 520.000 0.0589334
\(428\) 0 0
\(429\) 10800.0 1.21545
\(430\) 0 0
\(431\) −14600.0 −1.63169 −0.815844 − 0.578273i \(-0.803727\pi\)
−0.815844 + 0.578273i \(0.803727\pi\)
\(432\) 0 0
\(433\) −13970.0 −1.55047 −0.775237 − 0.631670i \(-0.782370\pi\)
−0.775237 + 0.631670i \(0.782370\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4320.00 −0.472892
\(438\) 0 0
\(439\) 800.000 0.0869748 0.0434874 − 0.999054i \(-0.486153\pi\)
0.0434874 + 0.999054i \(0.486153\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) 13572.0 1.45559 0.727794 − 0.685796i \(-0.240546\pi\)
0.727794 + 0.685796i \(0.240546\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −9930.00 −1.05072
\(448\) 0 0
\(449\) 5650.00 0.593853 0.296926 − 0.954900i \(-0.404038\pi\)
0.296926 + 0.954900i \(0.404038\pi\)
\(450\) 0 0
\(451\) 12400.0 1.29466
\(452\) 0 0
\(453\) −3480.00 −0.360937
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7110.00 0.727772 0.363886 − 0.931444i \(-0.381450\pi\)
0.363886 + 0.931444i \(0.381450\pi\)
\(458\) 0 0
\(459\) −1890.00 −0.192195
\(460\) 0 0
\(461\) −282.000 −0.0284903 −0.0142452 − 0.999899i \(-0.504535\pi\)
−0.0142452 + 0.999899i \(0.504535\pi\)
\(462\) 0 0
\(463\) −3868.00 −0.388253 −0.194127 − 0.980976i \(-0.562187\pi\)
−0.194127 + 0.980976i \(0.562187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4164.00 −0.412606 −0.206303 − 0.978488i \(-0.566143\pi\)
−0.206303 + 0.978488i \(0.566143\pi\)
\(468\) 0 0
\(469\) −3504.00 −0.344989
\(470\) 0 0
\(471\) −3390.00 −0.331641
\(472\) 0 0
\(473\) 10720.0 1.04208
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3330.00 0.319644
\(478\) 0 0
\(479\) −16800.0 −1.60253 −0.801265 − 0.598310i \(-0.795839\pi\)
−0.801265 + 0.598310i \(0.795839\pi\)
\(480\) 0 0
\(481\) 11700.0 1.10909
\(482\) 0 0
\(483\) 1296.00 0.122091
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10204.0 0.949461 0.474730 − 0.880131i \(-0.342546\pi\)
0.474730 + 0.880131i \(0.342546\pi\)
\(488\) 0 0
\(489\) −11196.0 −1.03538
\(490\) 0 0
\(491\) 7720.00 0.709570 0.354785 − 0.934948i \(-0.384554\pi\)
0.354785 + 0.934948i \(0.384554\pi\)
\(492\) 0 0
\(493\) 11620.0 1.06154
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3360.00 −0.303253
\(498\) 0 0
\(499\) 7160.00 0.642336 0.321168 − 0.947022i \(-0.395925\pi\)
0.321168 + 0.947022i \(0.395925\pi\)
\(500\) 0 0
\(501\) 10932.0 0.974862
\(502\) 0 0
\(503\) −21268.0 −1.88527 −0.942637 − 0.333818i \(-0.891663\pi\)
−0.942637 + 0.333818i \(0.891663\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −17709.0 −1.55125
\(508\) 0 0
\(509\) −17754.0 −1.54604 −0.773018 − 0.634384i \(-0.781254\pi\)
−0.773018 + 0.634384i \(0.781254\pi\)
\(510\) 0 0
\(511\) 1000.00 0.0865702
\(512\) 0 0
\(513\) 1080.00 0.0929496
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22240.0 1.89190
\(518\) 0 0
\(519\) −3870.00 −0.327310
\(520\) 0 0
\(521\) 1962.00 0.164984 0.0824921 − 0.996592i \(-0.473712\pi\)
0.0824921 + 0.996592i \(0.473712\pi\)
\(522\) 0 0
\(523\) −10012.0 −0.837083 −0.418541 − 0.908198i \(-0.637458\pi\)
−0.418541 + 0.908198i \(0.637458\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2800.00 0.231442
\(528\) 0 0
\(529\) −503.000 −0.0413413
\(530\) 0 0
\(531\) −2160.00 −0.176527
\(532\) 0 0
\(533\) −27900.0 −2.26732
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5760.00 −0.462872
\(538\) 0 0
\(539\) 13080.0 1.04526
\(540\) 0 0
\(541\) 3278.00 0.260503 0.130252 − 0.991481i \(-0.458421\pi\)
0.130252 + 0.991481i \(0.458421\pi\)
\(542\) 0 0
\(543\) 126.000 0.00995797
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21404.0 1.67307 0.836535 − 0.547914i \(-0.184578\pi\)
0.836535 + 0.547914i \(0.184578\pi\)
\(548\) 0 0
\(549\) −1170.00 −0.0909552
\(550\) 0 0
\(551\) −6640.00 −0.513382
\(552\) 0 0
\(553\) −3520.00 −0.270679
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5270.00 −0.400892 −0.200446 − 0.979705i \(-0.564239\pi\)
−0.200446 + 0.979705i \(0.564239\pi\)
\(558\) 0 0
\(559\) −24120.0 −1.82499
\(560\) 0 0
\(561\) 8400.00 0.632172
\(562\) 0 0
\(563\) 26388.0 1.97535 0.987675 − 0.156521i \(-0.0500279\pi\)
0.987675 + 0.156521i \(0.0500279\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −324.000 −0.0239977
\(568\) 0 0
\(569\) 13770.0 1.01453 0.507266 − 0.861790i \(-0.330656\pi\)
0.507266 + 0.861790i \(0.330656\pi\)
\(570\) 0 0
\(571\) 23440.0 1.71792 0.858961 − 0.512041i \(-0.171110\pi\)
0.858961 + 0.512041i \(0.171110\pi\)
\(572\) 0 0
\(573\) −2040.00 −0.148730
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −6370.00 −0.459595 −0.229798 − 0.973238i \(-0.573806\pi\)
−0.229798 + 0.973238i \(0.573806\pi\)
\(578\) 0 0
\(579\) 6630.00 0.475878
\(580\) 0 0
\(581\) 752.000 0.0536974
\(582\) 0 0
\(583\) −14800.0 −1.05138
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5084.00 0.357477 0.178739 − 0.983897i \(-0.442798\pi\)
0.178739 + 0.983897i \(0.442798\pi\)
\(588\) 0 0
\(589\) −1600.00 −0.111930
\(590\) 0 0
\(591\) −390.000 −0.0271446
\(592\) 0 0
\(593\) −1530.00 −0.105952 −0.0529760 − 0.998596i \(-0.516871\pi\)
−0.0529760 + 0.998596i \(0.516871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9120.00 0.625221
\(598\) 0 0
\(599\) −11040.0 −0.753059 −0.376529 − 0.926405i \(-0.622883\pi\)
−0.376529 + 0.926405i \(0.622883\pi\)
\(600\) 0 0
\(601\) 16810.0 1.14092 0.570461 − 0.821325i \(-0.306765\pi\)
0.570461 + 0.821325i \(0.306765\pi\)
\(602\) 0 0
\(603\) 7884.00 0.532440
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −10756.0 −0.719230 −0.359615 − 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(608\) 0 0
\(609\) 1992.00 0.132545
\(610\) 0 0
\(611\) −50040.0 −3.31326
\(612\) 0 0
\(613\) −16190.0 −1.06673 −0.533367 − 0.845884i \(-0.679074\pi\)
−0.533367 + 0.845884i \(0.679074\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24030.0 1.56793 0.783964 − 0.620806i \(-0.213195\pi\)
0.783964 + 0.620806i \(0.213195\pi\)
\(618\) 0 0
\(619\) 24920.0 1.61812 0.809062 − 0.587723i \(-0.199975\pi\)
0.809062 + 0.587723i \(0.199975\pi\)
\(620\) 0 0
\(621\) −2916.00 −0.188430
\(622\) 0 0
\(623\) 2904.00 0.186752
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4800.00 −0.305731
\(628\) 0 0
\(629\) 9100.00 0.576853
\(630\) 0 0
\(631\) 27000.0 1.70341 0.851706 − 0.524020i \(-0.175568\pi\)
0.851706 + 0.524020i \(0.175568\pi\)
\(632\) 0 0
\(633\) 1680.00 0.105488
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −29430.0 −1.83055
\(638\) 0 0
\(639\) 7560.00 0.468027
\(640\) 0 0
\(641\) −20190.0 −1.24408 −0.622041 − 0.782984i \(-0.713696\pi\)
−0.622041 + 0.782984i \(0.713696\pi\)
\(642\) 0 0
\(643\) 18228.0 1.11795 0.558975 − 0.829184i \(-0.311195\pi\)
0.558975 + 0.829184i \(0.311195\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14964.0 0.909267 0.454633 − 0.890679i \(-0.349770\pi\)
0.454633 + 0.890679i \(0.349770\pi\)
\(648\) 0 0
\(649\) 9600.00 0.580636
\(650\) 0 0
\(651\) 480.000 0.0288981
\(652\) 0 0
\(653\) −2070.00 −0.124051 −0.0620255 − 0.998075i \(-0.519756\pi\)
−0.0620255 + 0.998075i \(0.519756\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2250.00 −0.133609
\(658\) 0 0
\(659\) −12880.0 −0.761356 −0.380678 − 0.924708i \(-0.624309\pi\)
−0.380678 + 0.924708i \(0.624309\pi\)
\(660\) 0 0
\(661\) −2810.00 −0.165350 −0.0826750 − 0.996577i \(-0.526346\pi\)
−0.0826750 + 0.996577i \(0.526346\pi\)
\(662\) 0 0
\(663\) −18900.0 −1.10711
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17928.0 1.04074
\(668\) 0 0
\(669\) 996.000 0.0575599
\(670\) 0 0
\(671\) 5200.00 0.299171
\(672\) 0 0
\(673\) 29630.0 1.69711 0.848553 − 0.529110i \(-0.177474\pi\)
0.848553 + 0.529110i \(0.177474\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18110.0 −1.02810 −0.514050 − 0.857760i \(-0.671855\pi\)
−0.514050 + 0.857760i \(0.671855\pi\)
\(678\) 0 0
\(679\) −6200.00 −0.350419
\(680\) 0 0
\(681\) −7092.00 −0.399069
\(682\) 0 0
\(683\) −28508.0 −1.59711 −0.798557 − 0.601920i \(-0.794403\pi\)
−0.798557 + 0.601920i \(0.794403\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4002.00 −0.222250
\(688\) 0 0
\(689\) 33300.0 1.84126
\(690\) 0 0
\(691\) 18000.0 0.990958 0.495479 − 0.868620i \(-0.334992\pi\)
0.495479 + 0.868620i \(0.334992\pi\)
\(692\) 0 0
\(693\) 1440.00 0.0789337
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −21700.0 −1.17926
\(698\) 0 0
\(699\) 16710.0 0.904192
\(700\) 0 0
\(701\) 7350.00 0.396014 0.198007 − 0.980201i \(-0.436553\pi\)
0.198007 + 0.980201i \(0.436553\pi\)
\(702\) 0 0
\(703\) −5200.00 −0.278978
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3592.00 0.191076
\(708\) 0 0
\(709\) 25046.0 1.32669 0.663344 − 0.748314i \(-0.269136\pi\)
0.663344 + 0.748314i \(0.269136\pi\)
\(710\) 0 0
\(711\) 7920.00 0.417754
\(712\) 0 0
\(713\) 4320.00 0.226908
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10560.0 −0.550028
\(718\) 0 0
\(719\) −22960.0 −1.19091 −0.595454 − 0.803389i \(-0.703028\pi\)
−0.595454 + 0.803389i \(0.703028\pi\)
\(720\) 0 0
\(721\) 4592.00 0.237191
\(722\) 0 0
\(723\) −6390.00 −0.328695
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10324.0 −0.526679 −0.263340 − 0.964703i \(-0.584824\pi\)
−0.263340 + 0.964703i \(0.584824\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −18760.0 −0.949198
\(732\) 0 0
\(733\) −25830.0 −1.30157 −0.650786 − 0.759261i \(-0.725561\pi\)
−0.650786 + 0.759261i \(0.725561\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −35040.0 −1.75131
\(738\) 0 0
\(739\) −28280.0 −1.40771 −0.703854 − 0.710344i \(-0.748539\pi\)
−0.703854 + 0.710344i \(0.748539\pi\)
\(740\) 0 0
\(741\) 10800.0 0.535422
\(742\) 0 0
\(743\) −4692.00 −0.231673 −0.115836 − 0.993268i \(-0.536955\pi\)
−0.115836 + 0.993268i \(0.536955\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1692.00 −0.0828742
\(748\) 0 0
\(749\) 1104.00 0.0538575
\(750\) 0 0
\(751\) 22120.0 1.07479 0.537397 − 0.843329i \(-0.319408\pi\)
0.537397 + 0.843329i \(0.319408\pi\)
\(752\) 0 0
\(753\) −120.000 −0.00580749
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23570.0 1.13166 0.565830 − 0.824522i \(-0.308556\pi\)
0.565830 + 0.824522i \(0.308556\pi\)
\(758\) 0 0
\(759\) 12960.0 0.619787
\(760\) 0 0
\(761\) 30682.0 1.46153 0.730763 − 0.682631i \(-0.239164\pi\)
0.730763 + 0.682631i \(0.239164\pi\)
\(762\) 0 0
\(763\) 2120.00 0.100589
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −21600.0 −1.01686
\(768\) 0 0
\(769\) −21294.0 −0.998545 −0.499273 − 0.866445i \(-0.666399\pi\)
−0.499273 + 0.866445i \(0.666399\pi\)
\(770\) 0 0
\(771\) −22530.0 −1.05240
\(772\) 0 0
\(773\) 23810.0 1.10787 0.553937 − 0.832559i \(-0.313125\pi\)
0.553937 + 0.832559i \(0.313125\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1560.00 0.0720266
\(778\) 0 0
\(779\) 12400.0 0.570316
\(780\) 0 0
\(781\) −33600.0 −1.53944
\(782\) 0 0
\(783\) −4482.00 −0.204564
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −19396.0 −0.878517 −0.439258 − 0.898361i \(-0.644759\pi\)
−0.439258 + 0.898361i \(0.644759\pi\)
\(788\) 0 0
\(789\) 6684.00 0.301593
\(790\) 0 0
\(791\) 4200.00 0.188792
\(792\) 0 0
\(793\) −11700.0 −0.523933
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8070.00 −0.358663 −0.179331 − 0.983789i \(-0.557393\pi\)
−0.179331 + 0.983789i \(0.557393\pi\)
\(798\) 0 0
\(799\) −38920.0 −1.72327
\(800\) 0 0
\(801\) −6534.00 −0.288224
\(802\) 0 0
\(803\) 10000.0 0.439467
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11250.0 −0.490730
\(808\) 0 0
\(809\) −34854.0 −1.51471 −0.757356 − 0.653003i \(-0.773509\pi\)
−0.757356 + 0.653003i \(0.773509\pi\)
\(810\) 0 0
\(811\) −36080.0 −1.56220 −0.781098 − 0.624409i \(-0.785340\pi\)
−0.781098 + 0.624409i \(0.785340\pi\)
\(812\) 0 0
\(813\) −3000.00 −0.129415
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 10720.0 0.459052
\(818\) 0 0
\(819\) −3240.00 −0.138235
\(820\) 0 0
\(821\) −5570.00 −0.236778 −0.118389 − 0.992967i \(-0.537773\pi\)
−0.118389 + 0.992967i \(0.537773\pi\)
\(822\) 0 0
\(823\) −11772.0 −0.498598 −0.249299 − 0.968427i \(-0.580200\pi\)
−0.249299 + 0.968427i \(0.580200\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3196.00 0.134384 0.0671921 − 0.997740i \(-0.478596\pi\)
0.0671921 + 0.997740i \(0.478596\pi\)
\(828\) 0 0
\(829\) −33730.0 −1.41314 −0.706569 − 0.707644i \(-0.749758\pi\)
−0.706569 + 0.707644i \(0.749758\pi\)
\(830\) 0 0
\(831\) −16950.0 −0.707568
\(832\) 0 0
\(833\) −22890.0 −0.952091
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1080.00 −0.0446001
\(838\) 0 0
\(839\) 24960.0 1.02707 0.513537 − 0.858068i \(-0.328335\pi\)
0.513537 + 0.858068i \(0.328335\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 0 0
\(843\) −11310.0 −0.462084
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1076.00 −0.0436503
\(848\) 0 0
\(849\) −7404.00 −0.299299
\(850\) 0 0
\(851\) 14040.0 0.565552
\(852\) 0 0
\(853\) 5330.00 0.213946 0.106973 − 0.994262i \(-0.465884\pi\)
0.106973 + 0.994262i \(0.465884\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21630.0 0.862155 0.431077 − 0.902315i \(-0.358134\pi\)
0.431077 + 0.902315i \(0.358134\pi\)
\(858\) 0 0
\(859\) −18040.0 −0.716550 −0.358275 − 0.933616i \(-0.616635\pi\)
−0.358275 + 0.933616i \(0.616635\pi\)
\(860\) 0 0
\(861\) −3720.00 −0.147244
\(862\) 0 0
\(863\) −17732.0 −0.699426 −0.349713 − 0.936857i \(-0.613721\pi\)
−0.349713 + 0.936857i \(0.613721\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 39.0000 0.00152769
\(868\) 0 0
\(869\) −35200.0 −1.37408
\(870\) 0 0
\(871\) 78840.0 3.06704
\(872\) 0 0
\(873\) 13950.0 0.540820
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −31750.0 −1.22249 −0.611244 − 0.791442i \(-0.709330\pi\)
−0.611244 + 0.791442i \(0.709330\pi\)
\(878\) 0 0
\(879\) 8730.00 0.334989
\(880\) 0 0
\(881\) 11570.0 0.442455 0.221228 − 0.975222i \(-0.428994\pi\)
0.221228 + 0.975222i \(0.428994\pi\)
\(882\) 0 0
\(883\) 47588.0 1.81366 0.906831 − 0.421494i \(-0.138494\pi\)
0.906831 + 0.421494i \(0.138494\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10924.0 −0.413520 −0.206760 − 0.978392i \(-0.566292\pi\)
−0.206760 + 0.978392i \(0.566292\pi\)
\(888\) 0 0
\(889\) −9264.00 −0.349499
\(890\) 0 0
\(891\) −3240.00 −0.121823
\(892\) 0 0
\(893\) 22240.0 0.833408
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −29160.0 −1.08542
\(898\) 0 0
\(899\) 6640.00 0.246336
\(900\) 0 0
\(901\) 25900.0 0.957663
\(902\) 0 0
\(903\) −3216.00 −0.118518
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 21196.0 0.775967 0.387983 − 0.921666i \(-0.373172\pi\)
0.387983 + 0.921666i \(0.373172\pi\)
\(908\) 0 0
\(909\) −8082.00 −0.294899
\(910\) 0 0
\(911\) 2120.00 0.0771007 0.0385503 − 0.999257i \(-0.487726\pi\)
0.0385503 + 0.999257i \(0.487726\pi\)
\(912\) 0 0
\(913\) 7520.00 0.272591
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2080.00 −0.0749047
\(918\) 0 0
\(919\) −33760.0 −1.21180 −0.605898 − 0.795543i \(-0.707186\pi\)
−0.605898 + 0.795543i \(0.707186\pi\)
\(920\) 0 0
\(921\) −15348.0 −0.549114
\(922\) 0 0
\(923\) 75600.0 2.69600
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10332.0 −0.366071
\(928\) 0 0
\(929\) −54990.0 −1.94205 −0.971024 − 0.238980i \(-0.923187\pi\)
−0.971024 + 0.238980i \(0.923187\pi\)
\(930\) 0 0
\(931\) 13080.0 0.460451
\(932\) 0 0
\(933\) −10920.0 −0.383178
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −43210.0 −1.50652 −0.753260 − 0.657723i \(-0.771520\pi\)
−0.753260 + 0.657723i \(0.771520\pi\)
\(938\) 0 0
\(939\) 11790.0 0.409747
\(940\) 0 0
\(941\) −20122.0 −0.697087 −0.348543 − 0.937293i \(-0.613324\pi\)
−0.348543 + 0.937293i \(0.613324\pi\)
\(942\) 0 0
\(943\) −33480.0 −1.15616
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6716.00 0.230455 0.115227 − 0.993339i \(-0.463240\pi\)
0.115227 + 0.993339i \(0.463240\pi\)
\(948\) 0 0
\(949\) −22500.0 −0.769632
\(950\) 0 0
\(951\) −26670.0 −0.909394
\(952\) 0 0
\(953\) −3730.00 −0.126785 −0.0633927 − 0.997989i \(-0.520192\pi\)
−0.0633927 + 0.997989i \(0.520192\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 19920.0 0.672855
\(958\) 0 0
\(959\) −760.000 −0.0255909
\(960\) 0 0
\(961\) −28191.0 −0.946293
\(962\) 0 0
\(963\) −2484.00 −0.0831213
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37244.0 1.23856 0.619279 − 0.785171i \(-0.287425\pi\)
0.619279 + 0.785171i \(0.287425\pi\)
\(968\) 0 0
\(969\) 8400.00 0.278480
\(970\) 0 0
\(971\) −56520.0 −1.86798 −0.933992 − 0.357293i \(-0.883700\pi\)
−0.933992 + 0.357293i \(0.883700\pi\)
\(972\) 0 0
\(973\) 10720.0 0.353204
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30330.0 −0.993186 −0.496593 − 0.867984i \(-0.665416\pi\)
−0.496593 + 0.867984i \(0.665416\pi\)
\(978\) 0 0
\(979\) 29040.0 0.948031
\(980\) 0 0
\(981\) −4770.00 −0.155244
\(982\) 0 0
\(983\) −68.0000 −0.00220637 −0.00110319 − 0.999999i \(-0.500351\pi\)
−0.00110319 + 0.999999i \(0.500351\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6672.00 −0.215169
\(988\) 0 0
\(989\) −28944.0 −0.930602
\(990\) 0 0
\(991\) −33320.0 −1.06806 −0.534029 − 0.845466i \(-0.679323\pi\)
−0.534029 + 0.845466i \(0.679323\pi\)
\(992\) 0 0
\(993\) 13200.0 0.421842
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −750.000 −0.0238242 −0.0119121 − 0.999929i \(-0.503792\pi\)
−0.0119121 + 0.999929i \(0.503792\pi\)
\(998\) 0 0
\(999\) −3510.00 −0.111163
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 2400.4.a.d.1.1 1
4.3 odd 2 2400.4.a.s.1.1 1
5.4 even 2 480.4.a.k.1.1 yes 1
15.14 odd 2 1440.4.a.f.1.1 1
20.19 odd 2 480.4.a.d.1.1 ✓ 1
40.19 odd 2 960.4.a.w.1.1 1
40.29 even 2 960.4.a.f.1.1 1
60.59 even 2 1440.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.d.1.1 ✓ 1 20.19 odd 2
480.4.a.k.1.1 yes 1 5.4 even 2
960.4.a.f.1.1 1 40.29 even 2
960.4.a.w.1.1 1 40.19 odd 2
1440.4.a.e.1.1 1 60.59 even 2
1440.4.a.f.1.1 1 15.14 odd 2
2400.4.a.d.1.1 1 1.1 even 1 trivial
2400.4.a.s.1.1 1 4.3 odd 2