Properties

Label 960.4.a.w.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, 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("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} +90.0000 q^{13} -15.0000 q^{15} -70.0000 q^{17} -40.0000 q^{19} -12.0000 q^{21} +108.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -166.000 q^{29} -40.0000 q^{31} -120.000 q^{33} +20.0000 q^{35} +130.000 q^{37} +270.000 q^{39} -310.000 q^{41} +268.000 q^{43} -45.0000 q^{45} -556.000 q^{47} -327.000 q^{49} -210.000 q^{51} +370.000 q^{53} +200.000 q^{55} -120.000 q^{57} -240.000 q^{59} +130.000 q^{61} -36.0000 q^{63} -450.000 q^{65} -876.000 q^{67} +324.000 q^{69} -840.000 q^{71} +250.000 q^{73} +75.0000 q^{75} +160.000 q^{77} -880.000 q^{79} +81.0000 q^{81} +188.000 q^{83} +350.000 q^{85} -498.000 q^{87} -726.000 q^{89} -360.000 q^{91} -120.000 q^{93} +200.000 q^{95} -1550.00 q^{97} -360.000 q^{99} +O(q^{100})\)

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
\(5\) −5.00000 −0.447214
\(6\) 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
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −70.0000 −0.998676 −0.499338 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 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
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) −40.0000 −0.231749 −0.115874 0.993264i \(-0.536967\pi\)
−0.115874 + 0.993264i \(0.536967\pi\)
\(32\) 0 0
\(33\) −120.000 −0.633010
\(34\) 0 0
\(35\) 20.0000 0.0965891
\(36\) 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 \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 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
\(55\) 200.000 0.490327
\(56\) 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 \(-0.456436\pi\)
0.136433 + 0.990649i \(0.456436\pi\)
\(62\) 0 0
\(63\) −36.0000 −0.0719932
\(64\) 0 0
\(65\) −450.000 −0.858702
\(66\) 0 0
\(67\) −876.000 −1.59732 −0.798660 0.601783i \(-0.794457\pi\)
−0.798660 + 0.601783i \(0.794457\pi\)
\(68\) 0 0
\(69\) 324.000 0.565290
\(70\) 0 0
\(71\) −840.000 −1.40408 −0.702040 0.712138i \(-0.747727\pi\)
−0.702040 + 0.712138i \(0.747727\pi\)
\(72\) 0 0
\(73\) 250.000 0.400826 0.200413 0.979712i \(-0.435772\pi\)
0.200413 + 0.979712i \(0.435772\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) 160.000 0.236801
\(78\) 0 0
\(79\) −880.000 −1.25326 −0.626631 0.779316i \(-0.715567\pi\)
−0.626631 + 0.779316i \(0.715567\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 188.000 0.248623 0.124311 0.992243i \(-0.460328\pi\)
0.124311 + 0.992243i \(0.460328\pi\)
\(84\) 0 0
\(85\) 350.000 0.446622
\(86\) 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
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −1550.00 −1.62246 −0.811230 0.584727i \(-0.801202\pi\)
−0.811230 + 0.584727i \(0.801202\pi\)
\(98\) 0 0
\(99\) −360.000 −0.365468
\(100\) 0 0
\(101\) 898.000 0.884696 0.442348 0.896843i \(-0.354146\pi\)
0.442348 + 0.896843i \(0.354146\pi\)
\(102\) 0 0
\(103\) −1148.00 −1.09821 −0.549106 0.835753i \(-0.685032\pi\)
−0.549106 + 0.835753i \(0.685032\pi\)
\(104\) 0 0
\(105\) 60.0000 0.0557657
\(106\) 0 0
\(107\) 276.000 0.249364 0.124682 0.992197i \(-0.460209\pi\)
0.124682 + 0.992197i \(0.460209\pi\)
\(108\) 0 0
\(109\) 530.000 0.465732 0.232866 0.972509i \(-0.425190\pi\)
0.232866 + 0.972509i \(0.425190\pi\)
\(110\) 0 0
\(111\) 390.000 0.333488
\(112\) 0 0
\(113\) 1050.00 0.874121 0.437061 0.899432i \(-0.356020\pi\)
0.437061 + 0.899432i \(0.356020\pi\)
\(114\) 0 0
\(115\) −540.000 −0.437872
\(116\) 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
\(125\) −125.000 −0.0894427
\(126\) 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
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) −190.000 −0.118488 −0.0592438 0.998244i \(-0.518869\pi\)
−0.0592438 + 0.998244i \(0.518869\pi\)
\(138\) 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
\(145\) 830.000 0.475364
\(146\) 0 0
\(147\) −981.000 −0.550418
\(148\) 0 0
\(149\) −3310.00 −1.81990 −0.909952 0.414713i \(-0.863882\pi\)
−0.909952 + 0.414713i \(0.863882\pi\)
\(150\) 0 0
\(151\) −1160.00 −0.625162 −0.312581 0.949891i \(-0.601194\pi\)
−0.312581 + 0.949891i \(0.601194\pi\)
\(152\) 0 0
\(153\) −630.000 −0.332892
\(154\) 0 0
\(155\) 200.000 0.103641
\(156\) 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 \(-0.854017\pi\)
−0.896665 + 0.442710i \(0.854017\pi\)
\(164\) 0 0
\(165\) 600.000 0.283091
\(166\) 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
\(175\) −100.000 −0.0431959
\(176\) 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 \(-0.497255\pi\)
0.00862385 + 0.999963i \(0.497255\pi\)
\(182\) 0 0
\(183\) 390.000 0.157539
\(184\) 0 0
\(185\) −650.000 −0.258319
\(186\) 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 \(-0.541114\pi\)
−0.128804 + 0.991670i \(0.541114\pi\)
\(192\) 0 0
\(193\) 2210.00 0.824245 0.412122 0.911128i \(-0.364788\pi\)
0.412122 + 0.911128i \(0.364788\pi\)
\(194\) 0 0
\(195\) −1350.00 −0.495772
\(196\) 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 \(-0.317873\pi\)
0.541457 + 0.840728i \(0.317873\pi\)
\(200\) 0 0
\(201\) −2628.00 −0.922213
\(202\) 0 0
\(203\) 664.000 0.229575
\(204\) 0 0
\(205\) 1550.00 0.528081
\(206\) 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
\(215\) −1340.00 −0.425057
\(216\) 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
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −2364.00 −0.691208 −0.345604 0.938380i \(-0.612326\pi\)
−0.345604 + 0.938380i \(0.612326\pi\)
\(228\) 0 0
\(229\) −1334.00 −0.384948 −0.192474 0.981302i \(-0.561651\pi\)
−0.192474 + 0.981302i \(0.561651\pi\)
\(230\) 0 0
\(231\) 480.000 0.136717
\(232\) 0 0
\(233\) 5570.00 1.56611 0.783053 0.621955i \(-0.213661\pi\)
0.783053 + 0.621955i \(0.213661\pi\)
\(234\) 0 0
\(235\) 2780.00 0.771690
\(236\) 0 0
\(237\) −2640.00 −0.723571
\(238\) 0 0
\(239\) −3520.00 −0.952677 −0.476339 0.879262i \(-0.658036\pi\)
−0.476339 + 0.879262i \(0.658036\pi\)
\(240\) 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
\(245\) 1635.00 0.426352
\(246\) 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
\(255\) 1050.00 0.257857
\(256\) 0 0
\(257\) −7510.00 −1.82281 −0.911403 0.411516i \(-0.864999\pi\)
−0.911403 + 0.411516i \(0.864999\pi\)
\(258\) 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
\(265\) −1850.00 −0.428848
\(266\) 0 0
\(267\) −2178.00 −0.499219
\(268\) 0 0
\(269\) −3750.00 −0.849969 −0.424984 0.905201i \(-0.639720\pi\)
−0.424984 + 0.905201i \(0.639720\pi\)
\(270\) 0 0
\(271\) −1000.00 −0.224154 −0.112077 0.993700i \(-0.535750\pi\)
−0.112077 + 0.993700i \(0.535750\pi\)
\(272\) 0 0
\(273\) −1080.00 −0.239431
\(274\) 0 0
\(275\) −1000.00 −0.219281
\(276\) 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 \(-0.583459\pi\)
−0.259200 + 0.965824i \(0.583459\pi\)
\(284\) 0 0
\(285\) 600.000 0.124705
\(286\) 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
\(295\) 1200.00 0.236836
\(296\) 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
\(305\) −650.000 −0.122029
\(306\) 0 0
\(307\) −5116.00 −0.951093 −0.475546 0.879691i \(-0.657750\pi\)
−0.475546 + 0.879691i \(0.657750\pi\)
\(308\) 0 0
\(309\) −3444.00 −0.634053
\(310\) 0 0
\(311\) −3640.00 −0.663683 −0.331842 0.943335i \(-0.607670\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(312\) 0 0
\(313\) 3930.00 0.709702 0.354851 0.934923i \(-0.384532\pi\)
0.354851 + 0.934923i \(0.384532\pi\)
\(314\) 0 0
\(315\) 180.000 0.0321964
\(316\) 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
\(325\) 2250.00 0.384023
\(326\) 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
\(335\) 4380.00 0.714343
\(336\) 0 0
\(337\) 610.000 0.0986018 0.0493009 0.998784i \(-0.484301\pi\)
0.0493009 + 0.998784i \(0.484301\pi\)
\(338\) 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
\(345\) −1620.00 −0.252805
\(346\) 0 0
\(347\) 8004.00 1.23826 0.619131 0.785287i \(-0.287485\pi\)
0.619131 + 0.785287i \(0.287485\pi\)
\(348\) 0 0
\(349\) −5614.00 −0.861062 −0.430531 0.902576i \(-0.641674\pi\)
−0.430531 + 0.902576i \(0.641674\pi\)
\(350\) 0 0
\(351\) 2430.00 0.369527
\(352\) 0 0
\(353\) −3270.00 −0.493044 −0.246522 0.969137i \(-0.579288\pi\)
−0.246522 + 0.969137i \(0.579288\pi\)
\(354\) 0 0
\(355\) 4200.00 0.627924
\(356\) 0 0
\(357\) 840.000 0.124531
\(358\) 0 0
\(359\) 2960.00 0.435161 0.217581 0.976042i \(-0.430184\pi\)
0.217581 + 0.976042i \(0.430184\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) 807.000 0.116685
\(364\) 0 0
\(365\) −1250.00 −0.179255
\(366\) 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
\(375\) −375.000 −0.0516398
\(376\) 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
\(385\) −800.000 −0.105901
\(386\) 0 0
\(387\) 2412.00 0.316819
\(388\) 0 0
\(389\) −5630.00 −0.733811 −0.366905 0.930258i \(-0.619583\pi\)
−0.366905 + 0.930258i \(0.619583\pi\)
\(390\) 0 0
\(391\) −7560.00 −0.977815
\(392\) 0 0
\(393\) 1560.00 0.200233
\(394\) 0 0
\(395\) 4400.00 0.560476
\(396\) 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
\(405\) −405.000 −0.0496904
\(406\) 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
\(415\) −940.000 −0.111187
\(416\) 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 \(-0.354806\pi\)
0.440485 + 0.897760i \(0.354806\pi\)
\(422\) 0 0
\(423\) −5004.00 −0.575184
\(424\) 0 0
\(425\) −1750.00 −0.199735
\(426\) 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 \(-0.196273\pi\)
0.815844 + 0.578273i \(0.196273\pi\)
\(432\) 0 0
\(433\) 13970.0 1.55047 0.775237 0.631670i \(-0.217630\pi\)
0.775237 + 0.631670i \(0.217630\pi\)
\(434\) 0 0
\(435\) 2490.00 0.274451
\(436\) 0 0
\(437\) −4320.00 −0.472892
\(438\) 0 0
\(439\) −800.000 −0.0869748 −0.0434874 0.999054i \(-0.513847\pi\)
−0.0434874 + 0.999054i \(0.513847\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) −13572.0 −1.45559 −0.727794 0.685796i \(-0.759454\pi\)
−0.727794 + 0.685796i \(0.759454\pi\)
\(444\) 0 0
\(445\) 3630.00 0.386693
\(446\) 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
\(455\) 1800.00 0.185462
\(456\) 0 0
\(457\) −7110.00 −0.727772 −0.363886 0.931444i \(-0.618550\pi\)
−0.363886 + 0.931444i \(0.618550\pi\)
\(458\) 0 0
\(459\) −1890.00 −0.192195
\(460\) 0 0
\(461\) 282.000 0.0284903 0.0142452 0.999899i \(-0.495465\pi\)
0.0142452 + 0.999899i \(0.495465\pi\)
\(462\) 0 0
\(463\) −3868.00 −0.388253 −0.194127 0.980976i \(-0.562187\pi\)
−0.194127 + 0.980976i \(0.562187\pi\)
\(464\) 0 0
\(465\) 600.000 0.0598373
\(466\) 0 0
\(467\) 4164.00 0.412606 0.206303 0.978488i \(-0.433857\pi\)
0.206303 + 0.978488i \(0.433857\pi\)
\(468\) 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
\(475\) −1000.00 −0.0965961
\(476\) 0 0
\(477\) 3330.00 0.319644
\(478\) 0 0
\(479\) 16800.0 1.60253 0.801265 0.598310i \(-0.204161\pi\)
0.801265 + 0.598310i \(0.204161\pi\)
\(480\) 0 0
\(481\) 11700.0 1.10909
\(482\) 0 0
\(483\) −1296.00 −0.122091
\(484\) 0 0
\(485\) 7750.00 0.725586
\(486\) 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
\(495\) 1800.00 0.163442
\(496\) 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
\(505\) −4490.00 −0.395648
\(506\) 0 0
\(507\) 17709.0 1.55125
\(508\) 0 0
\(509\) 17754.0 1.54604 0.773018 0.634384i \(-0.218746\pi\)
0.773018 + 0.634384i \(0.218746\pi\)
\(510\) 0 0
\(511\) −1000.00 −0.0865702
\(512\) 0 0
\(513\) −1080.00 −0.0929496
\(514\) 0 0
\(515\) 5740.00 0.491135
\(516\) 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 \(-0.362542\pi\)
0.418541 + 0.908198i \(0.362542\pi\)
\(524\) 0 0
\(525\) −300.000 −0.0249392
\(526\) 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
\(535\) −1380.00 −0.111519
\(536\) 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 \(-0.541579\pi\)
−0.130252 + 0.991481i \(0.541579\pi\)
\(542\) 0 0
\(543\) 126.000 0.00995797
\(544\) 0 0
\(545\) −2650.00 −0.208282
\(546\) 0 0
\(547\) −21404.0 −1.67307 −0.836535 0.547914i \(-0.815422\pi\)
−0.836535 + 0.547914i \(0.815422\pi\)
\(548\) 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
\(555\) −1950.00 −0.149140
\(556\) 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 \(-0.949972\pi\)
−0.987675 + 0.156521i \(0.949972\pi\)
\(564\) 0 0
\(565\) −5250.00 −0.390919
\(566\) 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
\(575\) 2700.00 0.195822
\(576\) 0 0
\(577\) 6370.00 0.459595 0.229798 0.973238i \(-0.426194\pi\)
0.229798 + 0.973238i \(0.426194\pi\)
\(578\) 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
\(585\) −4050.00 −0.286234
\(586\) 0 0
\(587\) −5084.00 −0.357477 −0.178739 0.983897i \(-0.557202\pi\)
−0.178739 + 0.983897i \(0.557202\pi\)
\(588\) 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 \(-0.483129\pi\)
0.0529760 + 0.998596i \(0.483129\pi\)
\(594\) 0 0
\(595\) −1400.00 −0.0964612
\(596\) 0 0
\(597\) 9120.00 0.625221
\(598\) 0 0
\(599\) 11040.0 0.753059 0.376529 0.926405i \(-0.377117\pi\)
0.376529 + 0.926405i \(0.377117\pi\)
\(600\) 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
\(605\) −1345.00 −0.0903835
\(606\) 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
\(615\) 4650.00 0.304888
\(616\) 0 0
\(617\) −24030.0 −1.56793 −0.783964 0.620806i \(-0.786805\pi\)
−0.783964 + 0.620806i \(0.786805\pi\)
\(618\) 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
\(625\) 625.000 0.0400000
\(626\) 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 \(-0.824432\pi\)
−0.851706 + 0.524020i \(0.824432\pi\)
\(632\) 0 0
\(633\) −1680.00 −0.105488
\(634\) 0 0
\(635\) −11580.0 −0.723682
\(636\) 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 \(-0.688805\pi\)
−0.558975 + 0.829184i \(0.688805\pi\)
\(644\) 0 0
\(645\) −4020.00 −0.245407
\(646\) 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
\(655\) −2600.00 −0.155100
\(656\) 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 \(-0.473654\pi\)
0.0826750 + 0.996577i \(0.473654\pi\)
\(662\) 0 0
\(663\) −18900.0 −1.10711
\(664\) 0 0
\(665\) −800.000 −0.0466506
\(666\) 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 \(-0.822526\pi\)
−0.848553 + 0.529110i \(0.822526\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 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 \(-0.205597\pi\)
0.798557 + 0.601920i \(0.205597\pi\)
\(684\) 0 0
\(685\) 950.000 0.0529892
\(686\) 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
\(695\) 13400.0 0.731354
\(696\) 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 \(-0.563447\pi\)
−0.198007 + 0.980201i \(0.563447\pi\)
\(702\) 0 0
\(703\) −5200.00 −0.278978
\(704\) 0 0
\(705\) 8340.00 0.445536
\(706\) 0 0
\(707\) −3592.00 −0.191076
\(708\) 0 0
\(709\) −25046.0 −1.32669 −0.663344 0.748314i \(-0.730864\pi\)
−0.663344 + 0.748314i \(0.730864\pi\)
\(710\) 0 0
\(711\) −7920.00 −0.417754
\(712\) 0 0
\(713\) −4320.00 −0.226908
\(714\) 0 0
\(715\) 18000.0 0.941485
\(716\) 0 0
\(717\) −10560.0 −0.550028
\(718\) 0 0
\(719\) 22960.0 1.19091 0.595454 0.803389i \(-0.296972\pi\)
0.595454 + 0.803389i \(0.296972\pi\)
\(720\) 0 0
\(721\) 4592.00 0.237191
\(722\) 0 0
\(723\) 6390.00 0.328695
\(724\) 0 0
\(725\) −4150.00 −0.212589
\(726\) 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
\(735\) 4905.00 0.246155
\(736\) 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
\(745\) 16550.0 0.813886
\(746\) 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 \(-0.680592\pi\)
−0.537397 + 0.843329i \(0.680592\pi\)
\(752\) 0 0
\(753\) 120.000 0.00580749
\(754\) 0 0
\(755\) 5800.00 0.279581
\(756\) 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
\(765\) 3150.00 0.148874
\(766\) 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
\(775\) −1000.00 −0.0463498
\(776\) 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
\(785\) −5650.00 −0.256888
\(786\) 0 0
\(787\) 19396.0 0.878517 0.439258 0.898361i \(-0.355241\pi\)
0.439258 + 0.898361i \(0.355241\pi\)
\(788\) 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
\(795\) −5550.00 −0.247595
\(796\) 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
\(805\) 2160.00 0.0945714
\(806\) 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
\(815\) 18660.0 0.802002
\(816\) 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 \(-0.462227\pi\)
0.118389 + 0.992967i \(0.462227\pi\)
\(822\) 0 0
\(823\) −11772.0 −0.498598 −0.249299 0.968427i \(-0.580200\pi\)
−0.249299 + 0.968427i \(0.580200\pi\)
\(824\) 0 0
\(825\) −3000.00 −0.126602
\(826\) 0 0
\(827\) −3196.00 −0.134384 −0.0671921 0.997740i \(-0.521404\pi\)
−0.0671921 + 0.997740i \(0.521404\pi\)
\(828\) 0 0
\(829\) 33730.0 1.41314 0.706569 0.707644i \(-0.250242\pi\)
0.706569 + 0.707644i \(0.250242\pi\)
\(830\) 0 0
\(831\) 16950.0 0.707568
\(832\) 0 0
\(833\) 22890.0 0.952091
\(834\) 0 0
\(835\) 18220.0 0.755125
\(836\) 0 0
\(837\) −1080.00 −0.0446001
\(838\) 0 0
\(839\) −24960.0 −1.02707 −0.513537 0.858068i \(-0.671665\pi\)
−0.513537 + 0.858068i \(0.671665\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 0 0
\(843\) 11310.0 0.462084
\(844\) 0 0
\(845\) −29515.0 −1.20159
\(846\) 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
\(855\) 1800.00 0.0719985
\(856\) 0 0
\(857\) −21630.0 −0.862155 −0.431077 0.902315i \(-0.641866\pi\)
−0.431077 + 0.902315i \(0.641866\pi\)
\(858\) 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
\(865\) −6450.00 −0.253534
\(866\) 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
\(875\) 500.000 0.0193178
\(876\) 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 \(-0.861506\pi\)
−0.906831 + 0.421494i \(0.861506\pi\)
\(884\) 0 0
\(885\) 3600.00 0.136737
\(886\) 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
\(895\) −9600.00 −0.358539
\(896\) 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
\(905\) −210.000 −0.00771341
\(906\) 0 0
\(907\) −21196.0 −0.775967 −0.387983 0.921666i \(-0.626828\pi\)
−0.387983 + 0.921666i \(0.626828\pi\)
\(908\) 0 0
\(909\) 8082.00 0.294899
\(910\) 0 0
\(911\) −2120.00 −0.0771007 −0.0385503 0.999257i \(-0.512274\pi\)
−0.0385503 + 0.999257i \(0.512274\pi\)
\(912\) 0 0
\(913\) −7520.00 −0.272591
\(914\) 0 0
\(915\) −1950.00 −0.0704536
\(916\) 0 0
\(917\) −2080.00 −0.0749047
\(918\) 0 0
\(919\) 33760.0 1.21180 0.605898 0.795543i \(-0.292814\pi\)
0.605898 + 0.795543i \(0.292814\pi\)
\(920\) 0 0
\(921\) −15348.0 −0.549114
\(922\) 0 0
\(923\) −75600.0 −2.69600
\(924\) 0 0
\(925\) 3250.00 0.115524
\(926\) 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
\(935\) −14000.0 −0.489678
\(936\) 0 0
\(937\) 43210.0 1.50652 0.753260 0.657723i \(-0.228480\pi\)
0.753260 + 0.657723i \(0.228480\pi\)
\(938\) 0 0
\(939\) 11790.0 0.409747
\(940\) 0 0
\(941\) 20122.0 0.697087 0.348543 0.937293i \(-0.386676\pi\)
0.348543 + 0.937293i \(0.386676\pi\)
\(942\) 0 0
\(943\) −33480.0 −1.15616
\(944\) 0 0
\(945\) 540.000 0.0185886
\(946\) 0 0
\(947\) −6716.00 −0.230455 −0.115227 0.993339i \(-0.536760\pi\)
−0.115227 + 0.993339i \(0.536760\pi\)
\(948\) 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 \(-0.479808\pi\)
0.0633927 + 0.997989i \(0.479808\pi\)
\(954\) 0 0
\(955\) 3400.00 0.115206
\(956\) 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
\(965\) −11050.0 −0.368614
\(966\) 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
\(975\) 6750.00 0.221716
\(976\) 0 0
\(977\) 30330.0 0.993186 0.496593 0.867984i \(-0.334584\pi\)
0.496593 + 0.867984i \(0.334584\pi\)
\(978\) 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
\(985\) −650.000 −0.0210261
\(986\) 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 \(-0.320677\pi\)
0.534029 + 0.845466i \(0.320677\pi\)
\(992\) 0 0
\(993\) −13200.0 −0.421842
\(994\) 0 0
\(995\) −15200.0 −0.484294
\(996\) 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 960.4.a.w.1.1 1
4.3 odd 2 960.4.a.f.1.1 1
8.3 odd 2 480.4.a.k.1.1 yes 1
8.5 even 2 480.4.a.d.1.1 1
24.5 odd 2 1440.4.a.e.1.1 1
24.11 even 2 1440.4.a.f.1.1 1
40.19 odd 2 2400.4.a.d.1.1 1
40.29 even 2 2400.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.d.1.1 1 8.5 even 2
480.4.a.k.1.1 yes 1 8.3 odd 2
960.4.a.f.1.1 1 4.3 odd 2
960.4.a.w.1.1 1 1.1 even 1 trivial
1440.4.a.e.1.1 1 24.5 odd 2
1440.4.a.f.1.1 1 24.11 even 2
2400.4.a.d.1.1 1 40.19 odd 2
2400.4.a.s.1.1 1 40.29 even 2