Properties

Label 1911.4.a.f.1.1
Level $1911$
Weight $4$
Character 1911.1
Self dual yes
Analytic conductor $112.753$
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] = [1911,4,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1911.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(112.752650021\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -8.00000 q^{4} +12.0000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -8.00000 q^{4} +12.0000 q^{5} +9.00000 q^{9} -36.0000 q^{11} -24.0000 q^{12} -13.0000 q^{13} +36.0000 q^{15} +64.0000 q^{16} +78.0000 q^{17} -74.0000 q^{19} -96.0000 q^{20} -96.0000 q^{23} +19.0000 q^{25} +27.0000 q^{27} +18.0000 q^{29} +214.000 q^{31} -108.000 q^{33} -72.0000 q^{36} -286.000 q^{37} -39.0000 q^{39} +384.000 q^{41} +524.000 q^{43} +288.000 q^{44} +108.000 q^{45} -300.000 q^{47} +192.000 q^{48} +234.000 q^{51} +104.000 q^{52} +558.000 q^{53} -432.000 q^{55} -222.000 q^{57} -576.000 q^{59} -288.000 q^{60} -74.0000 q^{61} -512.000 q^{64} -156.000 q^{65} +38.0000 q^{67} -624.000 q^{68} -288.000 q^{69} -456.000 q^{71} +682.000 q^{73} +57.0000 q^{75} +592.000 q^{76} +704.000 q^{79} +768.000 q^{80} +81.0000 q^{81} +888.000 q^{83} +936.000 q^{85} +54.0000 q^{87} +1020.00 q^{89} +768.000 q^{92} +642.000 q^{93} -888.000 q^{95} -110.000 q^{97} -324.000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 3.00000 0.577350
\(4\) −8.00000 −1.00000
\(5\) 12.0000 1.07331 0.536656 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) −24.0000 −0.577350
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 36.0000 0.619677
\(16\) 64.0000 1.00000
\(17\) 78.0000 1.11281 0.556405 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) −74.0000 −0.893514 −0.446757 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) −96.0000 −1.07331
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 −0.870321 −0.435161 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 18.0000 0.115259 0.0576296 0.998338i \(-0.481646\pi\)
0.0576296 + 0.998338i \(0.481646\pi\)
\(30\) 0 0
\(31\) 214.000 1.23986 0.619928 0.784659i \(-0.287162\pi\)
0.619928 + 0.784659i \(0.287162\pi\)
\(32\) 0 0
\(33\) −108.000 −0.569709
\(34\) 0 0
\(35\) 0 0
\(36\) −72.0000 −0.333333
\(37\) −286.000 −1.27076 −0.635380 0.772200i \(-0.719156\pi\)
−0.635380 + 0.772200i \(0.719156\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 384.000 1.46270 0.731350 0.682002i \(-0.238890\pi\)
0.731350 + 0.682002i \(0.238890\pi\)
\(42\) 0 0
\(43\) 524.000 1.85835 0.929177 0.369634i \(-0.120517\pi\)
0.929177 + 0.369634i \(0.120517\pi\)
\(44\) 288.000 0.986764
\(45\) 108.000 0.357771
\(46\) 0 0
\(47\) −300.000 −0.931053 −0.465527 0.885034i \(-0.654135\pi\)
−0.465527 + 0.885034i \(0.654135\pi\)
\(48\) 192.000 0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) 234.000 0.642481
\(52\) 104.000 0.277350
\(53\) 558.000 1.44617 0.723087 0.690757i \(-0.242723\pi\)
0.723087 + 0.690757i \(0.242723\pi\)
\(54\) 0 0
\(55\) −432.000 −1.05911
\(56\) 0 0
\(57\) −222.000 −0.515870
\(58\) 0 0
\(59\) −576.000 −1.27100 −0.635498 0.772102i \(-0.719205\pi\)
−0.635498 + 0.772102i \(0.719205\pi\)
\(60\) −288.000 −0.619677
\(61\) −74.0000 −0.155323 −0.0776617 0.996980i \(-0.524745\pi\)
−0.0776617 + 0.996980i \(0.524745\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) −156.000 −0.297683
\(66\) 0 0
\(67\) 38.0000 0.0692901 0.0346451 0.999400i \(-0.488970\pi\)
0.0346451 + 0.999400i \(0.488970\pi\)
\(68\) −624.000 −1.11281
\(69\) −288.000 −0.502480
\(70\) 0 0
\(71\) −456.000 −0.762215 −0.381107 0.924531i \(-0.624457\pi\)
−0.381107 + 0.924531i \(0.624457\pi\)
\(72\) 0 0
\(73\) 682.000 1.09345 0.546726 0.837311i \(-0.315874\pi\)
0.546726 + 0.837311i \(0.315874\pi\)
\(74\) 0 0
\(75\) 57.0000 0.0877572
\(76\) 592.000 0.893514
\(77\) 0 0
\(78\) 0 0
\(79\) 704.000 1.00261 0.501305 0.865271i \(-0.332853\pi\)
0.501305 + 0.865271i \(0.332853\pi\)
\(80\) 768.000 1.07331
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 888.000 1.17435 0.587173 0.809462i \(-0.300241\pi\)
0.587173 + 0.809462i \(0.300241\pi\)
\(84\) 0 0
\(85\) 936.000 1.19439
\(86\) 0 0
\(87\) 54.0000 0.0665449
\(88\) 0 0
\(89\) 1020.00 1.21483 0.607415 0.794385i \(-0.292207\pi\)
0.607415 + 0.794385i \(0.292207\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 768.000 0.870321
\(93\) 642.000 0.715831
\(94\) 0 0
\(95\) −888.000 −0.959020
\(96\) 0 0
\(97\) −110.000 −0.115142 −0.0575712 0.998341i \(-0.518336\pi\)
−0.0575712 + 0.998341i \(0.518336\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) −152.000 −0.152000
\(101\) 990.000 0.975333 0.487667 0.873030i \(-0.337848\pi\)
0.487667 + 0.873030i \(0.337848\pi\)
\(102\) 0 0
\(103\) −1208.00 −1.15561 −0.577805 0.816175i \(-0.696090\pi\)
−0.577805 + 0.816175i \(0.696090\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 996.000 0.899878 0.449939 0.893059i \(-0.351446\pi\)
0.449939 + 0.893059i \(0.351446\pi\)
\(108\) −216.000 −0.192450
\(109\) −1402.00 −1.23199 −0.615997 0.787749i \(-0.711246\pi\)
−0.615997 + 0.787749i \(0.711246\pi\)
\(110\) 0 0
\(111\) −858.000 −0.733673
\(112\) 0 0
\(113\) 1926.00 1.60339 0.801694 0.597735i \(-0.203932\pi\)
0.801694 + 0.597735i \(0.203932\pi\)
\(114\) 0 0
\(115\) −1152.00 −0.934127
\(116\) −144.000 −0.115259
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 1152.00 0.844491
\(124\) −1712.00 −1.23986
\(125\) −1272.00 −0.910169
\(126\) 0 0
\(127\) −988.000 −0.690321 −0.345161 0.938544i \(-0.612176\pi\)
−0.345161 + 0.938544i \(0.612176\pi\)
\(128\) 0 0
\(129\) 1572.00 1.07292
\(130\) 0 0
\(131\) 2100.00 1.40059 0.700297 0.713851i \(-0.253051\pi\)
0.700297 + 0.713851i \(0.253051\pi\)
\(132\) 864.000 0.569709
\(133\) 0 0
\(134\) 0 0
\(135\) 324.000 0.206559
\(136\) 0 0
\(137\) −2496.00 −1.55655 −0.778276 0.627922i \(-0.783906\pi\)
−0.778276 + 0.627922i \(0.783906\pi\)
\(138\) 0 0
\(139\) 2464.00 1.50355 0.751776 0.659418i \(-0.229197\pi\)
0.751776 + 0.659418i \(0.229197\pi\)
\(140\) 0 0
\(141\) −900.000 −0.537544
\(142\) 0 0
\(143\) 468.000 0.273679
\(144\) 576.000 0.333333
\(145\) 216.000 0.123709
\(146\) 0 0
\(147\) 0 0
\(148\) 2288.00 1.27076
\(149\) 216.000 0.118761 0.0593806 0.998235i \(-0.481087\pi\)
0.0593806 + 0.998235i \(0.481087\pi\)
\(150\) 0 0
\(151\) −898.000 −0.483962 −0.241981 0.970281i \(-0.577797\pi\)
−0.241981 + 0.970281i \(0.577797\pi\)
\(152\) 0 0
\(153\) 702.000 0.370937
\(154\) 0 0
\(155\) 2568.00 1.33075
\(156\) 312.000 0.160128
\(157\) 1510.00 0.767587 0.383793 0.923419i \(-0.374617\pi\)
0.383793 + 0.923419i \(0.374617\pi\)
\(158\) 0 0
\(159\) 1674.00 0.834949
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −394.000 −0.189328 −0.0946640 0.995509i \(-0.530178\pi\)
−0.0946640 + 0.995509i \(0.530178\pi\)
\(164\) −3072.00 −1.46270
\(165\) −1296.00 −0.611476
\(166\) 0 0
\(167\) −84.0000 −0.0389228 −0.0194614 0.999811i \(-0.506195\pi\)
−0.0194614 + 0.999811i \(0.506195\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −666.000 −0.297838
\(172\) −4192.00 −1.85835
\(173\) −1194.00 −0.524729 −0.262365 0.964969i \(-0.584502\pi\)
−0.262365 + 0.964969i \(0.584502\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2304.00 −0.986764
\(177\) −1728.00 −0.733810
\(178\) 0 0
\(179\) 3156.00 1.31782 0.658912 0.752220i \(-0.271017\pi\)
0.658912 + 0.752220i \(0.271017\pi\)
\(180\) −864.000 −0.357771
\(181\) 1078.00 0.442691 0.221346 0.975195i \(-0.428955\pi\)
0.221346 + 0.975195i \(0.428955\pi\)
\(182\) 0 0
\(183\) −222.000 −0.0896760
\(184\) 0 0
\(185\) −3432.00 −1.36392
\(186\) 0 0
\(187\) −2808.00 −1.09808
\(188\) 2400.00 0.931053
\(189\) 0 0
\(190\) 0 0
\(191\) 3192.00 1.20924 0.604620 0.796514i \(-0.293325\pi\)
0.604620 + 0.796514i \(0.293325\pi\)
\(192\) −1536.00 −0.577350
\(193\) 722.000 0.269278 0.134639 0.990895i \(-0.457012\pi\)
0.134639 + 0.990895i \(0.457012\pi\)
\(194\) 0 0
\(195\) −468.000 −0.171868
\(196\) 0 0
\(197\) 2796.00 1.01120 0.505601 0.862767i \(-0.331271\pi\)
0.505601 + 0.862767i \(0.331271\pi\)
\(198\) 0 0
\(199\) 340.000 0.121115 0.0605577 0.998165i \(-0.480712\pi\)
0.0605577 + 0.998165i \(0.480712\pi\)
\(200\) 0 0
\(201\) 114.000 0.0400047
\(202\) 0 0
\(203\) 0 0
\(204\) −1872.00 −0.642481
\(205\) 4608.00 1.56994
\(206\) 0 0
\(207\) −864.000 −0.290107
\(208\) −832.000 −0.277350
\(209\) 2664.00 0.881688
\(210\) 0 0
\(211\) −1924.00 −0.627742 −0.313871 0.949466i \(-0.601626\pi\)
−0.313871 + 0.949466i \(0.601626\pi\)
\(212\) −4464.00 −1.44617
\(213\) −1368.00 −0.440065
\(214\) 0 0
\(215\) 6288.00 1.99460
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2046.00 0.631305
\(220\) 3456.00 1.05911
\(221\) −1014.00 −0.308638
\(222\) 0 0
\(223\) −5042.00 −1.51407 −0.757034 0.653375i \(-0.773352\pi\)
−0.757034 + 0.653375i \(0.773352\pi\)
\(224\) 0 0
\(225\) 171.000 0.0506667
\(226\) 0 0
\(227\) 2676.00 0.782433 0.391217 0.920299i \(-0.372054\pi\)
0.391217 + 0.920299i \(0.372054\pi\)
\(228\) 1776.00 0.515870
\(229\) 2410.00 0.695447 0.347723 0.937597i \(-0.386955\pi\)
0.347723 + 0.937597i \(0.386955\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3726.00 1.04763 0.523816 0.851831i \(-0.324508\pi\)
0.523816 + 0.851831i \(0.324508\pi\)
\(234\) 0 0
\(235\) −3600.00 −0.999311
\(236\) 4608.00 1.27100
\(237\) 2112.00 0.578857
\(238\) 0 0
\(239\) 1248.00 0.337767 0.168884 0.985636i \(-0.445984\pi\)
0.168884 + 0.985636i \(0.445984\pi\)
\(240\) 2304.00 0.619677
\(241\) 4210.00 1.12527 0.562635 0.826706i \(-0.309788\pi\)
0.562635 + 0.826706i \(0.309788\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 592.000 0.155323
\(245\) 0 0
\(246\) 0 0
\(247\) 962.000 0.247816
\(248\) 0 0
\(249\) 2664.00 0.678009
\(250\) 0 0
\(251\) 7692.00 1.93432 0.967161 0.254165i \(-0.0818007\pi\)
0.967161 + 0.254165i \(0.0818007\pi\)
\(252\) 0 0
\(253\) 3456.00 0.858802
\(254\) 0 0
\(255\) 2808.00 0.689583
\(256\) 4096.00 1.00000
\(257\) −1326.00 −0.321843 −0.160921 0.986967i \(-0.551447\pi\)
−0.160921 + 0.986967i \(0.551447\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1248.00 0.297683
\(261\) 162.000 0.0384197
\(262\) 0 0
\(263\) −6048.00 −1.41801 −0.709003 0.705205i \(-0.750855\pi\)
−0.709003 + 0.705205i \(0.750855\pi\)
\(264\) 0 0
\(265\) 6696.00 1.55220
\(266\) 0 0
\(267\) 3060.00 0.701382
\(268\) −304.000 −0.0692901
\(269\) −6474.00 −1.46739 −0.733693 0.679481i \(-0.762205\pi\)
−0.733693 + 0.679481i \(0.762205\pi\)
\(270\) 0 0
\(271\) −5978.00 −1.33999 −0.669996 0.742365i \(-0.733704\pi\)
−0.669996 + 0.742365i \(0.733704\pi\)
\(272\) 4992.00 1.11281
\(273\) 0 0
\(274\) 0 0
\(275\) −684.000 −0.149988
\(276\) 2304.00 0.502480
\(277\) 8750.00 1.89797 0.948983 0.315327i \(-0.102114\pi\)
0.948983 + 0.315327i \(0.102114\pi\)
\(278\) 0 0
\(279\) 1926.00 0.413285
\(280\) 0 0
\(281\) 8976.00 1.90556 0.952782 0.303656i \(-0.0982075\pi\)
0.952782 + 0.303656i \(0.0982075\pi\)
\(282\) 0 0
\(283\) 592.000 0.124349 0.0621745 0.998065i \(-0.480196\pi\)
0.0621745 + 0.998065i \(0.480196\pi\)
\(284\) 3648.00 0.762215
\(285\) −2664.00 −0.553690
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) −330.000 −0.0664775
\(292\) −5456.00 −1.09345
\(293\) 4608.00 0.918779 0.459389 0.888235i \(-0.348068\pi\)
0.459389 + 0.888235i \(0.348068\pi\)
\(294\) 0 0
\(295\) −6912.00 −1.36418
\(296\) 0 0
\(297\) −972.000 −0.189903
\(298\) 0 0
\(299\) 1248.00 0.241384
\(300\) −456.000 −0.0877572
\(301\) 0 0
\(302\) 0 0
\(303\) 2970.00 0.563109
\(304\) −4736.00 −0.893514
\(305\) −888.000 −0.166711
\(306\) 0 0
\(307\) 3166.00 0.588577 0.294289 0.955717i \(-0.404917\pi\)
0.294289 + 0.955717i \(0.404917\pi\)
\(308\) 0 0
\(309\) −3624.00 −0.667191
\(310\) 0 0
\(311\) −2472.00 −0.450721 −0.225361 0.974275i \(-0.572356\pi\)
−0.225361 + 0.974275i \(0.572356\pi\)
\(312\) 0 0
\(313\) 3094.00 0.558732 0.279366 0.960185i \(-0.409876\pi\)
0.279366 + 0.960185i \(0.409876\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −5632.00 −1.00261
\(317\) 2316.00 0.410345 0.205173 0.978726i \(-0.434224\pi\)
0.205173 + 0.978726i \(0.434224\pi\)
\(318\) 0 0
\(319\) −648.000 −0.113734
\(320\) −6144.00 −1.07331
\(321\) 2988.00 0.519545
\(322\) 0 0
\(323\) −5772.00 −0.994312
\(324\) −648.000 −0.111111
\(325\) −247.000 −0.0421572
\(326\) 0 0
\(327\) −4206.00 −0.711292
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4426.00 −0.734970 −0.367485 0.930030i \(-0.619781\pi\)
−0.367485 + 0.930030i \(0.619781\pi\)
\(332\) −7104.00 −1.17435
\(333\) −2574.00 −0.423587
\(334\) 0 0
\(335\) 456.000 0.0743700
\(336\) 0 0
\(337\) 866.000 0.139982 0.0699911 0.997548i \(-0.477703\pi\)
0.0699911 + 0.997548i \(0.477703\pi\)
\(338\) 0 0
\(339\) 5778.00 0.925716
\(340\) −7488.00 −1.19439
\(341\) −7704.00 −1.22345
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3456.00 −0.539318
\(346\) 0 0
\(347\) −2556.00 −0.395427 −0.197714 0.980260i \(-0.563352\pi\)
−0.197714 + 0.980260i \(0.563352\pi\)
\(348\) −432.000 −0.0665449
\(349\) 11014.0 1.68930 0.844650 0.535318i \(-0.179808\pi\)
0.844650 + 0.535318i \(0.179808\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 9720.00 1.46556 0.732781 0.680465i \(-0.238222\pi\)
0.732781 + 0.680465i \(0.238222\pi\)
\(354\) 0 0
\(355\) −5472.00 −0.818095
\(356\) −8160.00 −1.21483
\(357\) 0 0
\(358\) 0 0
\(359\) −2988.00 −0.439277 −0.219639 0.975581i \(-0.570488\pi\)
−0.219639 + 0.975581i \(0.570488\pi\)
\(360\) 0 0
\(361\) −1383.00 −0.201633
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 8184.00 1.17362
\(366\) 0 0
\(367\) 2068.00 0.294138 0.147069 0.989126i \(-0.453016\pi\)
0.147069 + 0.989126i \(0.453016\pi\)
\(368\) −6144.00 −0.870321
\(369\) 3456.00 0.487567
\(370\) 0 0
\(371\) 0 0
\(372\) −5136.00 −0.715831
\(373\) 902.000 0.125211 0.0626056 0.998038i \(-0.480059\pi\)
0.0626056 + 0.998038i \(0.480059\pi\)
\(374\) 0 0
\(375\) −3816.00 −0.525486
\(376\) 0 0
\(377\) −234.000 −0.0319671
\(378\) 0 0
\(379\) 12818.0 1.73725 0.868623 0.495473i \(-0.165005\pi\)
0.868623 + 0.495473i \(0.165005\pi\)
\(380\) 7104.00 0.959020
\(381\) −2964.00 −0.398557
\(382\) 0 0
\(383\) 1332.00 0.177708 0.0888538 0.996045i \(-0.471680\pi\)
0.0888538 + 0.996045i \(0.471680\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4716.00 0.619452
\(388\) 880.000 0.115142
\(389\) 3054.00 0.398056 0.199028 0.979994i \(-0.436221\pi\)
0.199028 + 0.979994i \(0.436221\pi\)
\(390\) 0 0
\(391\) −7488.00 −0.968502
\(392\) 0 0
\(393\) 6300.00 0.808633
\(394\) 0 0
\(395\) 8448.00 1.07611
\(396\) 2592.00 0.328921
\(397\) −11162.0 −1.41110 −0.705548 0.708663i \(-0.749299\pi\)
−0.705548 + 0.708663i \(0.749299\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1216.00 0.152000
\(401\) −14820.0 −1.84557 −0.922787 0.385310i \(-0.874095\pi\)
−0.922787 + 0.385310i \(0.874095\pi\)
\(402\) 0 0
\(403\) −2782.00 −0.343874
\(404\) −7920.00 −0.975333
\(405\) 972.000 0.119257
\(406\) 0 0
\(407\) 10296.0 1.25394
\(408\) 0 0
\(409\) 9682.00 1.17052 0.585262 0.810844i \(-0.300992\pi\)
0.585262 + 0.810844i \(0.300992\pi\)
\(410\) 0 0
\(411\) −7488.00 −0.898676
\(412\) 9664.00 1.15561
\(413\) 0 0
\(414\) 0 0
\(415\) 10656.0 1.26044
\(416\) 0 0
\(417\) 7392.00 0.868076
\(418\) 0 0
\(419\) 348.000 0.0405750 0.0202875 0.999794i \(-0.493542\pi\)
0.0202875 + 0.999794i \(0.493542\pi\)
\(420\) 0 0
\(421\) 2486.00 0.287792 0.143896 0.989593i \(-0.454037\pi\)
0.143896 + 0.989593i \(0.454037\pi\)
\(422\) 0 0
\(423\) −2700.00 −0.310351
\(424\) 0 0
\(425\) 1482.00 0.169147
\(426\) 0 0
\(427\) 0 0
\(428\) −7968.00 −0.899878
\(429\) 1404.00 0.158009
\(430\) 0 0
\(431\) −1812.00 −0.202508 −0.101254 0.994861i \(-0.532285\pi\)
−0.101254 + 0.994861i \(0.532285\pi\)
\(432\) 1728.00 0.192450
\(433\) 6226.00 0.690999 0.345499 0.938419i \(-0.387710\pi\)
0.345499 + 0.938419i \(0.387710\pi\)
\(434\) 0 0
\(435\) 648.000 0.0714235
\(436\) 11216.0 1.23199
\(437\) 7104.00 0.777644
\(438\) 0 0
\(439\) 12544.0 1.36376 0.681882 0.731462i \(-0.261162\pi\)
0.681882 + 0.731462i \(0.261162\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8556.00 −0.917625 −0.458812 0.888533i \(-0.651725\pi\)
−0.458812 + 0.888533i \(0.651725\pi\)
\(444\) 6864.00 0.733673
\(445\) 12240.0 1.30389
\(446\) 0 0
\(447\) 648.000 0.0685668
\(448\) 0 0
\(449\) 4116.00 0.432619 0.216310 0.976325i \(-0.430598\pi\)
0.216310 + 0.976325i \(0.430598\pi\)
\(450\) 0 0
\(451\) −13824.0 −1.44334
\(452\) −15408.0 −1.60339
\(453\) −2694.00 −0.279415
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6514.00 −0.666766 −0.333383 0.942791i \(-0.608190\pi\)
−0.333383 + 0.942791i \(0.608190\pi\)
\(458\) 0 0
\(459\) 2106.00 0.214160
\(460\) 9216.00 0.934127
\(461\) −10500.0 −1.06081 −0.530405 0.847744i \(-0.677960\pi\)
−0.530405 + 0.847744i \(0.677960\pi\)
\(462\) 0 0
\(463\) −5542.00 −0.556282 −0.278141 0.960540i \(-0.589718\pi\)
−0.278141 + 0.960540i \(0.589718\pi\)
\(464\) 1152.00 0.115259
\(465\) 7704.00 0.768311
\(466\) 0 0
\(467\) 5220.00 0.517244 0.258622 0.965979i \(-0.416732\pi\)
0.258622 + 0.965979i \(0.416732\pi\)
\(468\) 936.000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 4530.00 0.443166
\(472\) 0 0
\(473\) −18864.0 −1.83376
\(474\) 0 0
\(475\) −1406.00 −0.135814
\(476\) 0 0
\(477\) 5022.00 0.482058
\(478\) 0 0
\(479\) −11592.0 −1.10575 −0.552873 0.833266i \(-0.686468\pi\)
−0.552873 + 0.833266i \(0.686468\pi\)
\(480\) 0 0
\(481\) 3718.00 0.352445
\(482\) 0 0
\(483\) 0 0
\(484\) 280.000 0.0262960
\(485\) −1320.00 −0.123584
\(486\) 0 0
\(487\) 12170.0 1.13239 0.566196 0.824270i \(-0.308414\pi\)
0.566196 + 0.824270i \(0.308414\pi\)
\(488\) 0 0
\(489\) −1182.00 −0.109309
\(490\) 0 0
\(491\) 1812.00 0.166547 0.0832733 0.996527i \(-0.473463\pi\)
0.0832733 + 0.996527i \(0.473463\pi\)
\(492\) −9216.00 −0.844491
\(493\) 1404.00 0.128262
\(494\) 0 0
\(495\) −3888.00 −0.353036
\(496\) 13696.0 1.23986
\(497\) 0 0
\(498\) 0 0
\(499\) −1330.00 −0.119317 −0.0596583 0.998219i \(-0.519001\pi\)
−0.0596583 + 0.998219i \(0.519001\pi\)
\(500\) 10176.0 0.910169
\(501\) −252.000 −0.0224721
\(502\) 0 0
\(503\) 2688.00 0.238274 0.119137 0.992878i \(-0.461987\pi\)
0.119137 + 0.992878i \(0.461987\pi\)
\(504\) 0 0
\(505\) 11880.0 1.04684
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 7904.00 0.690321
\(509\) −5124.00 −0.446203 −0.223101 0.974795i \(-0.571618\pi\)
−0.223101 + 0.974795i \(0.571618\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1998.00 −0.171957
\(514\) 0 0
\(515\) −14496.0 −1.24033
\(516\) −12576.0 −1.07292
\(517\) 10800.0 0.918730
\(518\) 0 0
\(519\) −3582.00 −0.302953
\(520\) 0 0
\(521\) 882.000 0.0741672 0.0370836 0.999312i \(-0.488193\pi\)
0.0370836 + 0.999312i \(0.488193\pi\)
\(522\) 0 0
\(523\) 2320.00 0.193970 0.0969852 0.995286i \(-0.469080\pi\)
0.0969852 + 0.995286i \(0.469080\pi\)
\(524\) −16800.0 −1.40059
\(525\) 0 0
\(526\) 0 0
\(527\) 16692.0 1.37972
\(528\) −6912.00 −0.569709
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) −5184.00 −0.423666
\(532\) 0 0
\(533\) −4992.00 −0.405680
\(534\) 0 0
\(535\) 11952.0 0.965851
\(536\) 0 0
\(537\) 9468.00 0.760846
\(538\) 0 0
\(539\) 0 0
\(540\) −2592.00 −0.206559
\(541\) 21422.0 1.70241 0.851205 0.524833i \(-0.175872\pi\)
0.851205 + 0.524833i \(0.175872\pi\)
\(542\) 0 0
\(543\) 3234.00 0.255588
\(544\) 0 0
\(545\) −16824.0 −1.32231
\(546\) 0 0
\(547\) 7040.00 0.550290 0.275145 0.961403i \(-0.411274\pi\)
0.275145 + 0.961403i \(0.411274\pi\)
\(548\) 19968.0 1.55655
\(549\) −666.000 −0.0517745
\(550\) 0 0
\(551\) −1332.00 −0.102986
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10296.0 −0.787461
\(556\) −19712.0 −1.50355
\(557\) −8400.00 −0.638994 −0.319497 0.947587i \(-0.603514\pi\)
−0.319497 + 0.947587i \(0.603514\pi\)
\(558\) 0 0
\(559\) −6812.00 −0.515415
\(560\) 0 0
\(561\) −8424.00 −0.633978
\(562\) 0 0
\(563\) −19044.0 −1.42559 −0.712797 0.701371i \(-0.752572\pi\)
−0.712797 + 0.701371i \(0.752572\pi\)
\(564\) 7200.00 0.537544
\(565\) 23112.0 1.72094
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4698.00 −0.346134 −0.173067 0.984910i \(-0.555368\pi\)
−0.173067 + 0.984910i \(0.555368\pi\)
\(570\) 0 0
\(571\) −8728.00 −0.639677 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(572\) −3744.00 −0.273679
\(573\) 9576.00 0.698156
\(574\) 0 0
\(575\) −1824.00 −0.132289
\(576\) −4608.00 −0.333333
\(577\) −2018.00 −0.145599 −0.0727993 0.997347i \(-0.523193\pi\)
−0.0727993 + 0.997347i \(0.523193\pi\)
\(578\) 0 0
\(579\) 2166.00 0.155468
\(580\) −1728.00 −0.123709
\(581\) 0 0
\(582\) 0 0
\(583\) −20088.0 −1.42703
\(584\) 0 0
\(585\) −1404.00 −0.0992278
\(586\) 0 0
\(587\) −11376.0 −0.799894 −0.399947 0.916538i \(-0.630971\pi\)
−0.399947 + 0.916538i \(0.630971\pi\)
\(588\) 0 0
\(589\) −15836.0 −1.10783
\(590\) 0 0
\(591\) 8388.00 0.583818
\(592\) −18304.0 −1.27076
\(593\) 25596.0 1.77252 0.886258 0.463192i \(-0.153296\pi\)
0.886258 + 0.463192i \(0.153296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1728.00 −0.118761
\(597\) 1020.00 0.0699260
\(598\) 0 0
\(599\) 3480.00 0.237377 0.118689 0.992932i \(-0.462131\pi\)
0.118689 + 0.992932i \(0.462131\pi\)
\(600\) 0 0
\(601\) −10010.0 −0.679395 −0.339698 0.940535i \(-0.610325\pi\)
−0.339698 + 0.940535i \(0.610325\pi\)
\(602\) 0 0
\(603\) 342.000 0.0230967
\(604\) 7184.00 0.483962
\(605\) −420.000 −0.0282238
\(606\) 0 0
\(607\) −3764.00 −0.251690 −0.125845 0.992050i \(-0.540164\pi\)
−0.125845 + 0.992050i \(0.540164\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3900.00 0.258228
\(612\) −5616.00 −0.370937
\(613\) 13610.0 0.896742 0.448371 0.893848i \(-0.352004\pi\)
0.448371 + 0.893848i \(0.352004\pi\)
\(614\) 0 0
\(615\) 13824.0 0.906402
\(616\) 0 0
\(617\) 6408.00 0.418114 0.209057 0.977903i \(-0.432961\pi\)
0.209057 + 0.977903i \(0.432961\pi\)
\(618\) 0 0
\(619\) 6694.00 0.434660 0.217330 0.976098i \(-0.430265\pi\)
0.217330 + 0.976098i \(0.430265\pi\)
\(620\) −20544.0 −1.33075
\(621\) −2592.00 −0.167493
\(622\) 0 0
\(623\) 0 0
\(624\) −2496.00 −0.160128
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 7992.00 0.509043
\(628\) −12080.0 −0.767587
\(629\) −22308.0 −1.41411
\(630\) 0 0
\(631\) −27250.0 −1.71918 −0.859592 0.510981i \(-0.829282\pi\)
−0.859592 + 0.510981i \(0.829282\pi\)
\(632\) 0 0
\(633\) −5772.00 −0.362427
\(634\) 0 0
\(635\) −11856.0 −0.740931
\(636\) −13392.0 −0.834949
\(637\) 0 0
\(638\) 0 0
\(639\) −4104.00 −0.254072
\(640\) 0 0
\(641\) −12630.0 −0.778245 −0.389122 0.921186i \(-0.627222\pi\)
−0.389122 + 0.921186i \(0.627222\pi\)
\(642\) 0 0
\(643\) −14798.0 −0.907583 −0.453792 0.891108i \(-0.649929\pi\)
−0.453792 + 0.891108i \(0.649929\pi\)
\(644\) 0 0
\(645\) 18864.0 1.15158
\(646\) 0 0
\(647\) −26232.0 −1.59395 −0.796976 0.604012i \(-0.793568\pi\)
−0.796976 + 0.604012i \(0.793568\pi\)
\(648\) 0 0
\(649\) 20736.0 1.25417
\(650\) 0 0
\(651\) 0 0
\(652\) 3152.00 0.189328
\(653\) −30390.0 −1.82121 −0.910607 0.413274i \(-0.864385\pi\)
−0.910607 + 0.413274i \(0.864385\pi\)
\(654\) 0 0
\(655\) 25200.0 1.50328
\(656\) 24576.0 1.46270
\(657\) 6138.00 0.364484
\(658\) 0 0
\(659\) −28740.0 −1.69886 −0.849432 0.527698i \(-0.823055\pi\)
−0.849432 + 0.527698i \(0.823055\pi\)
\(660\) 10368.0 0.611476
\(661\) 9214.00 0.542183 0.271092 0.962554i \(-0.412615\pi\)
0.271092 + 0.962554i \(0.412615\pi\)
\(662\) 0 0
\(663\) −3042.00 −0.178192
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1728.00 −0.100312
\(668\) 672.000 0.0389228
\(669\) −15126.0 −0.874148
\(670\) 0 0
\(671\) 2664.00 0.153268
\(672\) 0 0
\(673\) 16598.0 0.950677 0.475339 0.879803i \(-0.342326\pi\)
0.475339 + 0.879803i \(0.342326\pi\)
\(674\) 0 0
\(675\) 513.000 0.0292524
\(676\) −1352.00 −0.0769231
\(677\) 8610.00 0.488788 0.244394 0.969676i \(-0.421411\pi\)
0.244394 + 0.969676i \(0.421411\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8028.00 0.451738
\(682\) 0 0
\(683\) 804.000 0.0450428 0.0225214 0.999746i \(-0.492831\pi\)
0.0225214 + 0.999746i \(0.492831\pi\)
\(684\) 5328.00 0.297838
\(685\) −29952.0 −1.67067
\(686\) 0 0
\(687\) 7230.00 0.401516
\(688\) 33536.0 1.85835
\(689\) −7254.00 −0.401096
\(690\) 0 0
\(691\) −2270.00 −0.124971 −0.0624854 0.998046i \(-0.519903\pi\)
−0.0624854 + 0.998046i \(0.519903\pi\)
\(692\) 9552.00 0.524729
\(693\) 0 0
\(694\) 0 0
\(695\) 29568.0 1.61378
\(696\) 0 0
\(697\) 29952.0 1.62771
\(698\) 0 0
\(699\) 11178.0 0.604851
\(700\) 0 0
\(701\) 1782.00 0.0960131 0.0480066 0.998847i \(-0.484713\pi\)
0.0480066 + 0.998847i \(0.484713\pi\)
\(702\) 0 0
\(703\) 21164.0 1.13544
\(704\) 18432.0 0.986764
\(705\) −10800.0 −0.576953
\(706\) 0 0
\(707\) 0 0
\(708\) 13824.0 0.733810
\(709\) −10690.0 −0.566250 −0.283125 0.959083i \(-0.591371\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(710\) 0 0
\(711\) 6336.00 0.334203
\(712\) 0 0
\(713\) −20544.0 −1.07907
\(714\) 0 0
\(715\) 5616.00 0.293743
\(716\) −25248.0 −1.31782
\(717\) 3744.00 0.195010
\(718\) 0 0
\(719\) −11568.0 −0.600019 −0.300009 0.953936i \(-0.596990\pi\)
−0.300009 + 0.953936i \(0.596990\pi\)
\(720\) 6912.00 0.357771
\(721\) 0 0
\(722\) 0 0
\(723\) 12630.0 0.649675
\(724\) −8624.00 −0.442691
\(725\) 342.000 0.0175194
\(726\) 0 0
\(727\) 11644.0 0.594019 0.297010 0.954874i \(-0.404011\pi\)
0.297010 + 0.954874i \(0.404011\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 40872.0 2.06800
\(732\) 1776.00 0.0896760
\(733\) 15010.0 0.756353 0.378177 0.925733i \(-0.376551\pi\)
0.378177 + 0.925733i \(0.376551\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1368.00 −0.0683730
\(738\) 0 0
\(739\) 33410.0 1.66307 0.831534 0.555474i \(-0.187463\pi\)
0.831534 + 0.555474i \(0.187463\pi\)
\(740\) 27456.0 1.36392
\(741\) 2886.00 0.143077
\(742\) 0 0
\(743\) −6504.00 −0.321142 −0.160571 0.987024i \(-0.551334\pi\)
−0.160571 + 0.987024i \(0.551334\pi\)
\(744\) 0 0
\(745\) 2592.00 0.127468
\(746\) 0 0
\(747\) 7992.00 0.391448
\(748\) 22464.0 1.09808
\(749\) 0 0
\(750\) 0 0
\(751\) −13912.0 −0.675973 −0.337987 0.941151i \(-0.609746\pi\)
−0.337987 + 0.941151i \(0.609746\pi\)
\(752\) −19200.0 −0.931053
\(753\) 23076.0 1.11678
\(754\) 0 0
\(755\) −10776.0 −0.519442
\(756\) 0 0
\(757\) −23974.0 −1.15106 −0.575528 0.817782i \(-0.695204\pi\)
−0.575528 + 0.817782i \(0.695204\pi\)
\(758\) 0 0
\(759\) 10368.0 0.495829
\(760\) 0 0
\(761\) 288.000 0.0137188 0.00685939 0.999976i \(-0.497817\pi\)
0.00685939 + 0.999976i \(0.497817\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −25536.0 −1.20924
\(765\) 8424.00 0.398131
\(766\) 0 0
\(767\) 7488.00 0.352511
\(768\) 12288.0 0.577350
\(769\) −1514.00 −0.0709964 −0.0354982 0.999370i \(-0.511302\pi\)
−0.0354982 + 0.999370i \(0.511302\pi\)
\(770\) 0 0
\(771\) −3978.00 −0.185816
\(772\) −5776.00 −0.269278
\(773\) −15816.0 −0.735915 −0.367957 0.929843i \(-0.619943\pi\)
−0.367957 + 0.929843i \(0.619943\pi\)
\(774\) 0 0
\(775\) 4066.00 0.188458
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28416.0 −1.30694
\(780\) 3744.00 0.171868
\(781\) 16416.0 0.752126
\(782\) 0 0
\(783\) 486.000 0.0221816
\(784\) 0 0
\(785\) 18120.0 0.823861
\(786\) 0 0
\(787\) −10154.0 −0.459912 −0.229956 0.973201i \(-0.573858\pi\)
−0.229956 + 0.973201i \(0.573858\pi\)
\(788\) −22368.0 −1.01120
\(789\) −18144.0 −0.818686
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 962.000 0.0430790
\(794\) 0 0
\(795\) 20088.0 0.896161
\(796\) −2720.00 −0.121115
\(797\) 17442.0 0.775191 0.387596 0.921830i \(-0.373306\pi\)
0.387596 + 0.921830i \(0.373306\pi\)
\(798\) 0 0
\(799\) −23400.0 −1.03609
\(800\) 0 0
\(801\) 9180.00 0.404943
\(802\) 0 0
\(803\) −24552.0 −1.07898
\(804\) −912.000 −0.0400047
\(805\) 0 0
\(806\) 0 0
\(807\) −19422.0 −0.847196
\(808\) 0 0
\(809\) −8778.00 −0.381481 −0.190740 0.981641i \(-0.561089\pi\)
−0.190740 + 0.981641i \(0.561089\pi\)
\(810\) 0 0
\(811\) 430.000 0.0186182 0.00930909 0.999957i \(-0.497037\pi\)
0.00930909 + 0.999957i \(0.497037\pi\)
\(812\) 0 0
\(813\) −17934.0 −0.773644
\(814\) 0 0
\(815\) −4728.00 −0.203208
\(816\) 14976.0 0.642481
\(817\) −38776.0 −1.66047
\(818\) 0 0
\(819\) 0 0
\(820\) −36864.0 −1.56994
\(821\) −32976.0 −1.40179 −0.700895 0.713264i \(-0.747216\pi\)
−0.700895 + 0.713264i \(0.747216\pi\)
\(822\) 0 0
\(823\) −1168.00 −0.0494701 −0.0247351 0.999694i \(-0.507874\pi\)
−0.0247351 + 0.999694i \(0.507874\pi\)
\(824\) 0 0
\(825\) −2052.00 −0.0865957
\(826\) 0 0
\(827\) −17172.0 −0.722042 −0.361021 0.932558i \(-0.617572\pi\)
−0.361021 + 0.932558i \(0.617572\pi\)
\(828\) 6912.00 0.290107
\(829\) −27146.0 −1.13730 −0.568649 0.822580i \(-0.692534\pi\)
−0.568649 + 0.822580i \(0.692534\pi\)
\(830\) 0 0
\(831\) 26250.0 1.09579
\(832\) 6656.00 0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) −1008.00 −0.0417764
\(836\) −21312.0 −0.881688
\(837\) 5778.00 0.238610
\(838\) 0 0
\(839\) −30696.0 −1.26310 −0.631552 0.775334i \(-0.717582\pi\)
−0.631552 + 0.775334i \(0.717582\pi\)
\(840\) 0 0
\(841\) −24065.0 −0.986715
\(842\) 0 0
\(843\) 26928.0 1.10018
\(844\) 15392.0 0.627742
\(845\) 2028.00 0.0825625
\(846\) 0 0
\(847\) 0 0
\(848\) 35712.0 1.44617
\(849\) 1776.00 0.0717929
\(850\) 0 0
\(851\) 27456.0 1.10597
\(852\) 10944.0 0.440065
\(853\) −24842.0 −0.997156 −0.498578 0.866845i \(-0.666144\pi\)
−0.498578 + 0.866845i \(0.666144\pi\)
\(854\) 0 0
\(855\) −7992.00 −0.319673
\(856\) 0 0
\(857\) −11406.0 −0.454634 −0.227317 0.973821i \(-0.572995\pi\)
−0.227317 + 0.973821i \(0.572995\pi\)
\(858\) 0 0
\(859\) −20540.0 −0.815851 −0.407925 0.913015i \(-0.633748\pi\)
−0.407925 + 0.913015i \(0.633748\pi\)
\(860\) −50304.0 −1.99460
\(861\) 0 0
\(862\) 0 0
\(863\) 9108.00 0.359258 0.179629 0.983734i \(-0.442510\pi\)
0.179629 + 0.983734i \(0.442510\pi\)
\(864\) 0 0
\(865\) −14328.0 −0.563198
\(866\) 0 0
\(867\) 3513.00 0.137610
\(868\) 0 0
\(869\) −25344.0 −0.989340
\(870\) 0 0
\(871\) −494.000 −0.0192176
\(872\) 0 0
\(873\) −990.000 −0.0383808
\(874\) 0 0
\(875\) 0 0
\(876\) −16368.0 −0.631305
\(877\) −24046.0 −0.925856 −0.462928 0.886396i \(-0.653201\pi\)
−0.462928 + 0.886396i \(0.653201\pi\)
\(878\) 0 0
\(879\) 13824.0 0.530457
\(880\) −27648.0 −1.05911
\(881\) −7998.00 −0.305856 −0.152928 0.988237i \(-0.548870\pi\)
−0.152928 + 0.988237i \(0.548870\pi\)
\(882\) 0 0
\(883\) 24032.0 0.915902 0.457951 0.888978i \(-0.348584\pi\)
0.457951 + 0.888978i \(0.348584\pi\)
\(884\) 8112.00 0.308638
\(885\) −20736.0 −0.787608
\(886\) 0 0
\(887\) 15648.0 0.592343 0.296172 0.955135i \(-0.404290\pi\)
0.296172 + 0.955135i \(0.404290\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 40336.0 1.51407
\(893\) 22200.0 0.831909
\(894\) 0 0
\(895\) 37872.0 1.41444
\(896\) 0 0
\(897\) 3744.00 0.139363
\(898\) 0 0
\(899\) 3852.00 0.142905
\(900\) −1368.00 −0.0506667
\(901\) 43524.0 1.60932
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12936.0 0.475146
\(906\) 0 0
\(907\) −808.000 −0.0295802 −0.0147901 0.999891i \(-0.504708\pi\)
−0.0147901 + 0.999891i \(0.504708\pi\)
\(908\) −21408.0 −0.782433
\(909\) 8910.00 0.325111
\(910\) 0 0
\(911\) 39144.0 1.42360 0.711799 0.702383i \(-0.247880\pi\)
0.711799 + 0.702383i \(0.247880\pi\)
\(912\) −14208.0 −0.515870
\(913\) −31968.0 −1.15880
\(914\) 0 0
\(915\) −2664.00 −0.0962504
\(916\) −19280.0 −0.695447
\(917\) 0 0
\(918\) 0 0
\(919\) −38248.0 −1.37289 −0.686445 0.727182i \(-0.740830\pi\)
−0.686445 + 0.727182i \(0.740830\pi\)
\(920\) 0 0
\(921\) 9498.00 0.339815
\(922\) 0 0
\(923\) 5928.00 0.211400
\(924\) 0 0
\(925\) −5434.00 −0.193155
\(926\) 0 0
\(927\) −10872.0 −0.385203
\(928\) 0 0
\(929\) 54264.0 1.91641 0.958205 0.286084i \(-0.0923536\pi\)
0.958205 + 0.286084i \(0.0923536\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −29808.0 −1.04763
\(933\) −7416.00 −0.260224
\(934\) 0 0
\(935\) −33696.0 −1.17859
\(936\) 0 0
\(937\) −12206.0 −0.425563 −0.212782 0.977100i \(-0.568252\pi\)
−0.212782 + 0.977100i \(0.568252\pi\)
\(938\) 0 0
\(939\) 9282.00 0.322584
\(940\) 28800.0 0.999311
\(941\) −17664.0 −0.611934 −0.305967 0.952042i \(-0.598980\pi\)
−0.305967 + 0.952042i \(0.598980\pi\)
\(942\) 0 0
\(943\) −36864.0 −1.27302
\(944\) −36864.0 −1.27100
\(945\) 0 0
\(946\) 0 0
\(947\) −51984.0 −1.78379 −0.891897 0.452238i \(-0.850626\pi\)
−0.891897 + 0.452238i \(0.850626\pi\)
\(948\) −16896.0 −0.578857
\(949\) −8866.00 −0.303269
\(950\) 0 0
\(951\) 6948.00 0.236913
\(952\) 0 0
\(953\) 13782.0 0.468460 0.234230 0.972181i \(-0.424743\pi\)
0.234230 + 0.972181i \(0.424743\pi\)
\(954\) 0 0
\(955\) 38304.0 1.29789
\(956\) −9984.00 −0.337767
\(957\) −1944.00 −0.0656642
\(958\) 0 0
\(959\) 0 0
\(960\) −18432.0 −0.619677
\(961\) 16005.0 0.537243
\(962\) 0 0
\(963\) 8964.00 0.299959
\(964\) −33680.0 −1.12527
\(965\) 8664.00 0.289020
\(966\) 0 0
\(967\) 14618.0 0.486125 0.243063 0.970011i \(-0.421848\pi\)
0.243063 + 0.970011i \(0.421848\pi\)
\(968\) 0 0
\(969\) −17316.0 −0.574066
\(970\) 0 0
\(971\) 18708.0 0.618299 0.309149 0.951013i \(-0.399956\pi\)
0.309149 + 0.951013i \(0.399956\pi\)
\(972\) −1944.00 −0.0641500
\(973\) 0 0
\(974\) 0 0
\(975\) −741.000 −0.0243395
\(976\) −4736.00 −0.155323
\(977\) −48804.0 −1.59814 −0.799068 0.601241i \(-0.794673\pi\)
−0.799068 + 0.601241i \(0.794673\pi\)
\(978\) 0 0
\(979\) −36720.0 −1.19875
\(980\) 0 0
\(981\) −12618.0 −0.410664
\(982\) 0 0
\(983\) 44736.0 1.45153 0.725766 0.687941i \(-0.241485\pi\)
0.725766 + 0.687941i \(0.241485\pi\)
\(984\) 0 0
\(985\) 33552.0 1.08534
\(986\) 0 0
\(987\) 0 0
\(988\) −7696.00 −0.247816
\(989\) −50304.0 −1.61737
\(990\) 0 0
\(991\) −21004.0 −0.673274 −0.336637 0.941635i \(-0.609289\pi\)
−0.336637 + 0.941635i \(0.609289\pi\)
\(992\) 0 0
\(993\) −13278.0 −0.424335
\(994\) 0 0
\(995\) 4080.00 0.129995
\(996\) −21312.0 −0.678009
\(997\) −9038.00 −0.287098 −0.143549 0.989643i \(-0.545851\pi\)
−0.143549 + 0.989643i \(0.545851\pi\)
\(998\) 0 0
\(999\) −7722.00 −0.244558
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 1911.4.a.f.1.1 1
7.6 odd 2 39.4.a.a.1.1 1
21.20 even 2 117.4.a.a.1.1 1
28.27 even 2 624.4.a.g.1.1 1
35.34 odd 2 975.4.a.e.1.1 1
56.13 odd 2 2496.4.a.o.1.1 1
56.27 even 2 2496.4.a.f.1.1 1
84.83 odd 2 1872.4.a.m.1.1 1
91.34 even 4 507.4.b.b.337.1 2
91.83 even 4 507.4.b.b.337.2 2
91.90 odd 2 507.4.a.c.1.1 1
273.272 even 2 1521.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.a.1.1 1 7.6 odd 2
117.4.a.a.1.1 1 21.20 even 2
507.4.a.c.1.1 1 91.90 odd 2
507.4.b.b.337.1 2 91.34 even 4
507.4.b.b.337.2 2 91.83 even 4
624.4.a.g.1.1 1 28.27 even 2
975.4.a.e.1.1 1 35.34 odd 2
1521.4.a.f.1.1 1 273.272 even 2
1872.4.a.m.1.1 1 84.83 odd 2
1911.4.a.f.1.1 1 1.1 even 1 trivial
2496.4.a.f.1.1 1 56.27 even 2
2496.4.a.o.1.1 1 56.13 odd 2