Properties

Label 117.4.a.a.1.1
Level $117$
Weight $4$
Character 117.1
Self dual yes
Analytic conductor $6.903$
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] = [117,4,Mod(1,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.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: \(6.90322347067\)
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 \(\chi\) \(=\) 117.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{4} +12.0000 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-8.00000 q^{4} +12.0000 q^{5} +2.00000 q^{7} +36.0000 q^{11} +13.0000 q^{13} +64.0000 q^{16} +78.0000 q^{17} +74.0000 q^{19} -96.0000 q^{20} +96.0000 q^{23} +19.0000 q^{25} -16.0000 q^{28} -18.0000 q^{29} -214.000 q^{31} +24.0000 q^{35} -286.000 q^{37} +384.000 q^{41} +524.000 q^{43} -288.000 q^{44} -300.000 q^{47} -339.000 q^{49} -104.000 q^{52} -558.000 q^{53} +432.000 q^{55} -576.000 q^{59} +74.0000 q^{61} -512.000 q^{64} +156.000 q^{65} +38.0000 q^{67} -624.000 q^{68} +456.000 q^{71} -682.000 q^{73} -592.000 q^{76} +72.0000 q^{77} +704.000 q^{79} +768.000 q^{80} +888.000 q^{83} +936.000 q^{85} +1020.00 q^{89} +26.0000 q^{91} -768.000 q^{92} +888.000 q^{95} +110.000 q^{97} +O(q^{100})\)

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 \(0.5\pi\)
\(3\) 0 0
\(4\) −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
\(7\) 2.00000 0.107990 0.0539949 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 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 \(-0.352579\pi\)
0.446757 + 0.894655i \(0.352579\pi\)
\(20\) −96.0000 −1.07331
\(21\) 0 0
\(22\) 0 0
\(23\) 96.0000 0.870321 0.435161 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 0 0
\(28\) −16.0000 −0.107990
\(29\) −18.0000 −0.115259 −0.0576296 0.998338i \(-0.518354\pi\)
−0.0576296 + 0.998338i \(0.518354\pi\)
\(30\) 0 0
\(31\) −214.000 −1.23986 −0.619928 0.784659i \(-0.712838\pi\)
−0.619928 + 0.784659i \(0.712838\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 24.0000 0.115907
\(36\) 0 0
\(37\) −286.000 −1.27076 −0.635380 0.772200i \(-0.719156\pi\)
−0.635380 + 0.772200i \(0.719156\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 0 0
\(47\) −300.000 −0.931053 −0.465527 0.885034i \(-0.654135\pi\)
−0.465527 + 0.885034i \(0.654135\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) −104.000 −0.277350
\(53\) −558.000 −1.44617 −0.723087 0.690757i \(-0.757277\pi\)
−0.723087 + 0.690757i \(0.757277\pi\)
\(54\) 0 0
\(55\) 432.000 1.05911
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −576.000 −1.27100 −0.635498 0.772102i \(-0.719205\pi\)
−0.635498 + 0.772102i \(0.719205\pi\)
\(60\) 0 0
\(61\) 74.0000 0.155323 0.0776617 0.996980i \(-0.475255\pi\)
0.0776617 + 0.996980i \(0.475255\pi\)
\(62\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 456.000 0.762215 0.381107 0.924531i \(-0.375543\pi\)
0.381107 + 0.924531i \(0.375543\pi\)
\(72\) 0 0
\(73\) −682.000 −1.09345 −0.546726 0.837311i \(-0.684126\pi\)
−0.546726 + 0.837311i \(0.684126\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −592.000 −0.893514
\(77\) 72.0000 0.106561
\(78\) 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
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) 1020.00 1.21483 0.607415 0.794385i \(-0.292207\pi\)
0.607415 + 0.794385i \(0.292207\pi\)
\(90\) 0 0
\(91\) 26.0000 0.0299510
\(92\) −768.000 −0.870321
\(93\) 0 0
\(94\) 0 0
\(95\) 888.000 0.959020
\(96\) 0 0
\(97\) 110.000 0.115142 0.0575712 0.998341i \(-0.481664\pi\)
0.0575712 + 0.998341i \(0.481664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −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 \(-0.303910\pi\)
0.577805 + 0.816175i \(0.303910\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −996.000 −0.899878 −0.449939 0.893059i \(-0.648554\pi\)
−0.449939 + 0.893059i \(0.648554\pi\)
\(108\) 0 0
\(109\) −1402.00 −1.23199 −0.615997 0.787749i \(-0.711246\pi\)
−0.615997 + 0.787749i \(0.711246\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 128.000 0.107990
\(113\) −1926.00 −1.60339 −0.801694 0.597735i \(-0.796068\pi\)
−0.801694 + 0.597735i \(0.796068\pi\)
\(114\) 0 0
\(115\) 1152.00 0.934127
\(116\) 144.000 0.115259
\(117\) 0 0
\(118\) 0 0
\(119\) 156.000 0.120172
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 0 0
\(131\) 2100.00 1.40059 0.700297 0.713851i \(-0.253051\pi\)
0.700297 + 0.713851i \(0.253051\pi\)
\(132\) 0 0
\(133\) 148.000 0.0964904
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2496.00 1.55655 0.778276 0.627922i \(-0.216094\pi\)
0.778276 + 0.627922i \(0.216094\pi\)
\(138\) 0 0
\(139\) −2464.00 −1.50355 −0.751776 0.659418i \(-0.770803\pi\)
−0.751776 + 0.659418i \(0.770803\pi\)
\(140\) −192.000 −0.115907
\(141\) 0 0
\(142\) 0 0
\(143\) 468.000 0.273679
\(144\) 0 0
\(145\) −216.000 −0.123709
\(146\) 0 0
\(147\) 0 0
\(148\) 2288.00 1.27076
\(149\) −216.000 −0.118761 −0.0593806 0.998235i \(-0.518913\pi\)
−0.0593806 + 0.998235i \(0.518913\pi\)
\(150\) 0 0
\(151\) −898.000 −0.483962 −0.241981 0.970281i \(-0.577797\pi\)
−0.241981 + 0.970281i \(0.577797\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2568.00 −1.33075
\(156\) 0 0
\(157\) −1510.00 −0.767587 −0.383793 0.923419i \(-0.625383\pi\)
−0.383793 + 0.923419i \(0.625383\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 192.000 0.0939858
\(162\) 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
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) −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
\(175\) 38.0000 0.0164145
\(176\) 2304.00 0.986764
\(177\) 0 0
\(178\) 0 0
\(179\) −3156.00 −1.31782 −0.658912 0.752220i \(-0.728983\pi\)
−0.658912 + 0.752220i \(0.728983\pi\)
\(180\) 0 0
\(181\) −1078.00 −0.442691 −0.221346 0.975195i \(-0.571045\pi\)
−0.221346 + 0.975195i \(0.571045\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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 \(-0.706675\pi\)
−0.604620 + 0.796514i \(0.706675\pi\)
\(192\) 0 0
\(193\) 722.000 0.269278 0.134639 0.990895i \(-0.457012\pi\)
0.134639 + 0.990895i \(0.457012\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2712.00 0.988338
\(197\) −2796.00 −1.01120 −0.505601 0.862767i \(-0.668729\pi\)
−0.505601 + 0.862767i \(0.668729\pi\)
\(198\) 0 0
\(199\) −340.000 −0.121115 −0.0605577 0.998165i \(-0.519288\pi\)
−0.0605577 + 0.998165i \(0.519288\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −36.0000 −0.0124468
\(204\) 0 0
\(205\) 4608.00 1.56994
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 6288.00 1.99460
\(216\) 0 0
\(217\) −428.000 −0.133892
\(218\) 0 0
\(219\) 0 0
\(220\) −3456.00 −1.05911
\(221\) 1014.00 0.308638
\(222\) 0 0
\(223\) 5042.00 1.51407 0.757034 0.653375i \(-0.226648\pi\)
0.757034 + 0.653375i \(0.226648\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2676.00 0.782433 0.391217 0.920299i \(-0.372054\pi\)
0.391217 + 0.920299i \(0.372054\pi\)
\(228\) 0 0
\(229\) −2410.00 −0.695447 −0.347723 0.937597i \(-0.613045\pi\)
−0.347723 + 0.937597i \(0.613045\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3726.00 −1.04763 −0.523816 0.851831i \(-0.675492\pi\)
−0.523816 + 0.851831i \(0.675492\pi\)
\(234\) 0 0
\(235\) −3600.00 −0.999311
\(236\) 4608.00 1.27100
\(237\) 0 0
\(238\) 0 0
\(239\) −1248.00 −0.337767 −0.168884 0.985636i \(-0.554016\pi\)
−0.168884 + 0.985636i \(0.554016\pi\)
\(240\) 0 0
\(241\) −4210.00 −1.12527 −0.562635 0.826706i \(-0.690212\pi\)
−0.562635 + 0.826706i \(0.690212\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −592.000 −0.155323
\(245\) −4068.00 −1.06080
\(246\) 0 0
\(247\) 962.000 0.247816
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) 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
\(259\) −572.000 −0.137229
\(260\) −1248.00 −0.297683
\(261\) 0 0
\(262\) 0 0
\(263\) 6048.00 1.41801 0.709003 0.705205i \(-0.249145\pi\)
0.709003 + 0.705205i \(0.249145\pi\)
\(264\) 0 0
\(265\) −6696.00 −1.55220
\(266\) 0 0
\(267\) 0 0
\(268\) −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 \(-0.266296\pi\)
0.669996 + 0.742365i \(0.266296\pi\)
\(272\) 4992.00 1.11281
\(273\) 0 0
\(274\) 0 0
\(275\) 684.000 0.149988
\(276\) 0 0
\(277\) 8750.00 1.89797 0.948983 0.315327i \(-0.102114\pi\)
0.948983 + 0.315327i \(0.102114\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8976.00 −1.90556 −0.952782 0.303656i \(-0.901793\pi\)
−0.952782 + 0.303656i \(0.901793\pi\)
\(282\) 0 0
\(283\) −592.000 −0.124349 −0.0621745 0.998065i \(-0.519804\pi\)
−0.0621745 + 0.998065i \(0.519804\pi\)
\(284\) −3648.00 −0.762215
\(285\) 0 0
\(286\) 0 0
\(287\) 768.000 0.157957
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) 1248.00 0.241384
\(300\) 0 0
\(301\) 1048.00 0.200683
\(302\) 0 0
\(303\) 0 0
\(304\) 4736.00 0.893514
\(305\) 888.000 0.166711
\(306\) 0 0
\(307\) −3166.00 −0.588577 −0.294289 0.955717i \(-0.595083\pi\)
−0.294289 + 0.955717i \(0.595083\pi\)
\(308\) −576.000 −0.106561
\(309\) 0 0
\(310\) 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 \(-0.590124\pi\)
−0.279366 + 0.960185i \(0.590124\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −5632.00 −1.00261
\(317\) −2316.00 −0.410345 −0.205173 0.978726i \(-0.565776\pi\)
−0.205173 + 0.978726i \(0.565776\pi\)
\(318\) 0 0
\(319\) −648.000 −0.113734
\(320\) −6144.00 −1.07331
\(321\) 0 0
\(322\) 0 0
\(323\) 5772.00 0.994312
\(324\) 0 0
\(325\) 247.000 0.0421572
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −600.000 −0.100544
\(330\) 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
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) −7488.00 −1.19439
\(341\) −7704.00 −1.22345
\(342\) 0 0
\(343\) −1364.00 −0.214720
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2556.00 0.395427 0.197714 0.980260i \(-0.436648\pi\)
0.197714 + 0.980260i \(0.436648\pi\)
\(348\) 0 0
\(349\) −11014.0 −1.68930 −0.844650 0.535318i \(-0.820192\pi\)
−0.844650 + 0.535318i \(0.820192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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 \(-0.429512\pi\)
0.219639 + 0.975581i \(0.429512\pi\)
\(360\) 0 0
\(361\) −1383.00 −0.201633
\(362\) 0 0
\(363\) 0 0
\(364\) −208.000 −0.0299510
\(365\) −8184.00 −1.17362
\(366\) 0 0
\(367\) −2068.00 −0.294138 −0.147069 0.989126i \(-0.546984\pi\)
−0.147069 + 0.989126i \(0.546984\pi\)
\(368\) 6144.00 0.870321
\(369\) 0 0
\(370\) 0 0
\(371\) −1116.00 −0.156172
\(372\) 0 0
\(373\) 902.000 0.125211 0.0626056 0.998038i \(-0.480059\pi\)
0.0626056 + 0.998038i \(0.480059\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 0 0
\(383\) 1332.00 0.177708 0.0888538 0.996045i \(-0.471680\pi\)
0.0888538 + 0.996045i \(0.471680\pi\)
\(384\) 0 0
\(385\) 864.000 0.114373
\(386\) 0 0
\(387\) 0 0
\(388\) −880.000 −0.115142
\(389\) −3054.00 −0.398056 −0.199028 0.979994i \(-0.563779\pi\)
−0.199028 + 0.979994i \(0.563779\pi\)
\(390\) 0 0
\(391\) 7488.00 0.968502
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8448.00 1.07611
\(396\) 0 0
\(397\) 11162.0 1.41110 0.705548 0.708663i \(-0.250701\pi\)
0.705548 + 0.708663i \(0.250701\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1216.00 0.152000
\(401\) 14820.0 1.84557 0.922787 0.385310i \(-0.125905\pi\)
0.922787 + 0.385310i \(0.125905\pi\)
\(402\) 0 0
\(403\) −2782.00 −0.343874
\(404\) −7920.00 −0.975333
\(405\) 0 0
\(406\) 0 0
\(407\) −10296.0 −1.25394
\(408\) 0 0
\(409\) −9682.00 −1.17052 −0.585262 0.810844i \(-0.699008\pi\)
−0.585262 + 0.810844i \(0.699008\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9664.00 −1.15561
\(413\) −1152.00 −0.137255
\(414\) 0 0
\(415\) 10656.0 1.26044
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 1482.00 0.169147
\(426\) 0 0
\(427\) 148.000 0.0167734
\(428\) 7968.00 0.899878
\(429\) 0 0
\(430\) 0 0
\(431\) 1812.00 0.202508 0.101254 0.994861i \(-0.467715\pi\)
0.101254 + 0.994861i \(0.467715\pi\)
\(432\) 0 0
\(433\) −6226.00 −0.690999 −0.345499 0.938419i \(-0.612290\pi\)
−0.345499 + 0.938419i \(0.612290\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 11216.0 1.23199
\(437\) 7104.00 0.777644
\(438\) 0 0
\(439\) −12544.0 −1.36376 −0.681882 0.731462i \(-0.738838\pi\)
−0.681882 + 0.731462i \(0.738838\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8556.00 0.917625 0.458812 0.888533i \(-0.348275\pi\)
0.458812 + 0.888533i \(0.348275\pi\)
\(444\) 0 0
\(445\) 12240.0 1.30389
\(446\) 0 0
\(447\) 0 0
\(448\) −1024.00 −0.107990
\(449\) −4116.00 −0.432619 −0.216310 0.976325i \(-0.569402\pi\)
−0.216310 + 0.976325i \(0.569402\pi\)
\(450\) 0 0
\(451\) 13824.0 1.44334
\(452\) 15408.0 1.60339
\(453\) 0 0
\(454\) 0 0
\(455\) 312.000 0.0321468
\(456\) 0 0
\(457\) −6514.00 −0.666766 −0.333383 0.942791i \(-0.608190\pi\)
−0.333383 + 0.942791i \(0.608190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) 0 0
\(467\) 5220.00 0.517244 0.258622 0.965979i \(-0.416732\pi\)
0.258622 + 0.965979i \(0.416732\pi\)
\(468\) 0 0
\(469\) 76.0000 0.00748263
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18864.0 1.83376
\(474\) 0 0
\(475\) 1406.00 0.135814
\(476\) −1248.00 −0.120172
\(477\) 0 0
\(478\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) −1812.00 −0.166547 −0.0832733 0.996527i \(-0.526537\pi\)
−0.0832733 + 0.996527i \(0.526537\pi\)
\(492\) 0 0
\(493\) −1404.00 −0.128262
\(494\) 0 0
\(495\) 0 0
\(496\) −13696.0 −1.23986
\(497\) 912.000 0.0823115
\(498\) 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
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 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
\(511\) −1364.00 −0.118082
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14496.0 1.24033
\(516\) 0 0
\(517\) −10800.0 −0.918730
\(518\) 0 0
\(519\) 0 0
\(520\) 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 \(-0.530920\pi\)
−0.0969852 + 0.995286i \(0.530920\pi\)
\(524\) −16800.0 −1.40059
\(525\) 0 0
\(526\) 0 0
\(527\) −16692.0 −1.37972
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) 0 0
\(532\) −1184.00 −0.0964904
\(533\) 4992.00 0.405680
\(534\) 0 0
\(535\) −11952.0 −0.965851
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12204.0 −0.975257
\(540\) 0 0
\(541\) 21422.0 1.70241 0.851205 0.524833i \(-0.175872\pi\)
0.851205 + 0.524833i \(0.175872\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) −1332.00 −0.102986
\(552\) 0 0
\(553\) 1408.00 0.108272
\(554\) 0 0
\(555\) 0 0
\(556\) 19712.0 1.50355
\(557\) 8400.00 0.638994 0.319497 0.947587i \(-0.396486\pi\)
0.319497 + 0.947587i \(0.396486\pi\)
\(558\) 0 0
\(559\) 6812.00 0.515415
\(560\) 1536.00 0.115907
\(561\) 0 0
\(562\) 0 0
\(563\) −19044.0 −1.42559 −0.712797 0.701371i \(-0.752572\pi\)
−0.712797 + 0.701371i \(0.752572\pi\)
\(564\) 0 0
\(565\) −23112.0 −1.72094
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4698.00 0.346134 0.173067 0.984910i \(-0.444632\pi\)
0.173067 + 0.984910i \(0.444632\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 0 0
\(575\) 1824.00 0.132289
\(576\) 0 0
\(577\) 2018.00 0.145599 0.0727993 0.997347i \(-0.476807\pi\)
0.0727993 + 0.997347i \(0.476807\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 1728.00 0.123709
\(581\) 1776.00 0.126817
\(582\) 0 0
\(583\) −20088.0 −1.42703
\(584\) 0 0
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) −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
\(595\) 1872.00 0.128982
\(596\) 1728.00 0.118761
\(597\) 0 0
\(598\) 0 0
\(599\) −3480.00 −0.237377 −0.118689 0.992932i \(-0.537869\pi\)
−0.118689 + 0.992932i \(0.537869\pi\)
\(600\) 0 0
\(601\) 10010.0 0.679395 0.339698 0.940535i \(-0.389675\pi\)
0.339698 + 0.940535i \(0.389675\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 7184.00 0.483962
\(605\) −420.000 −0.0282238
\(606\) 0 0
\(607\) 3764.00 0.251690 0.125845 0.992050i \(-0.459836\pi\)
0.125845 + 0.992050i \(0.459836\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3900.00 −0.258228
\(612\) 0 0
\(613\) 13610.0 0.896742 0.448371 0.893848i \(-0.352004\pi\)
0.448371 + 0.893848i \(0.352004\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6408.00 −0.418114 −0.209057 0.977903i \(-0.567039\pi\)
−0.209057 + 0.977903i \(0.567039\pi\)
\(618\) 0 0
\(619\) −6694.00 −0.434660 −0.217330 0.976098i \(-0.569735\pi\)
−0.217330 + 0.976098i \(0.569735\pi\)
\(620\) 20544.0 1.33075
\(621\) 0 0
\(622\) 0 0
\(623\) 2040.00 0.131189
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) −11856.0 −0.740931
\(636\) 0 0
\(637\) −4407.00 −0.274116
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12630.0 0.778245 0.389122 0.921186i \(-0.372778\pi\)
0.389122 + 0.921186i \(0.372778\pi\)
\(642\) 0 0
\(643\) 14798.0 0.907583 0.453792 0.891108i \(-0.350071\pi\)
0.453792 + 0.891108i \(0.350071\pi\)
\(644\) −1536.00 −0.0939858
\(645\) 0 0
\(646\) 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 \(-0.135615\pi\)
0.910607 + 0.413274i \(0.135615\pi\)
\(654\) 0 0
\(655\) 25200.0 1.50328
\(656\) 24576.0 1.46270
\(657\) 0 0
\(658\) 0 0
\(659\) 28740.0 1.69886 0.849432 0.527698i \(-0.176945\pi\)
0.849432 + 0.527698i \(0.176945\pi\)
\(660\) 0 0
\(661\) −9214.00 −0.542183 −0.271092 0.962554i \(-0.587385\pi\)
−0.271092 + 0.962554i \(0.587385\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1776.00 0.103564
\(666\) 0 0
\(667\) −1728.00 −0.100312
\(668\) 672.000 0.0389228
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) −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
\(679\) 220.000 0.0124342
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −804.000 −0.0450428 −0.0225214 0.999746i \(-0.507169\pi\)
−0.0225214 + 0.999746i \(0.507169\pi\)
\(684\) 0 0
\(685\) 29952.0 1.67067
\(686\) 0 0
\(687\) 0 0
\(688\) 33536.0 1.85835
\(689\) −7254.00 −0.401096
\(690\) 0 0
\(691\) 2270.00 0.124971 0.0624854 0.998046i \(-0.480097\pi\)
0.0624854 + 0.998046i \(0.480097\pi\)
\(692\) 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
\(699\) 0 0
\(700\) −304.000 −0.0164145
\(701\) −1782.00 −0.0960131 −0.0480066 0.998847i \(-0.515287\pi\)
−0.0480066 + 0.998847i \(0.515287\pi\)
\(702\) 0 0
\(703\) −21164.0 −1.13544
\(704\) −18432.0 −0.986764
\(705\) 0 0
\(706\) 0 0
\(707\) 1980.00 0.105326
\(708\) 0 0
\(709\) −10690.0 −0.566250 −0.283125 0.959083i \(-0.591371\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20544.0 −1.07907
\(714\) 0 0
\(715\) 5616.00 0.293743
\(716\) 25248.0 1.31782
\(717\) 0 0
\(718\) 0 0
\(719\) −11568.0 −0.600019 −0.300009 0.953936i \(-0.596990\pi\)
−0.300009 + 0.953936i \(0.596990\pi\)
\(720\) 0 0
\(721\) 2416.00 0.124794
\(722\) 0 0
\(723\) 0 0
\(724\) 8624.00 0.442691
\(725\) −342.000 −0.0175194
\(726\) 0 0
\(727\) −11644.0 −0.594019 −0.297010 0.954874i \(-0.595989\pi\)
−0.297010 + 0.954874i \(0.595989\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 40872.0 2.06800
\(732\) 0 0
\(733\) −15010.0 −0.756353 −0.378177 0.925733i \(-0.623449\pi\)
−0.378177 + 0.925733i \(0.623449\pi\)
\(734\) 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
\(741\) 0 0
\(742\) 0 0
\(743\) 6504.00 0.321142 0.160571 0.987024i \(-0.448666\pi\)
0.160571 + 0.987024i \(0.448666\pi\)
\(744\) 0 0
\(745\) −2592.00 −0.127468
\(746\) 0 0
\(747\) 0 0
\(748\) −22464.0 −1.09808
\(749\) −1992.00 −0.0971777
\(750\) 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
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 0 0
\(761\) 288.000 0.0137188 0.00685939 0.999976i \(-0.497817\pi\)
0.00685939 + 0.999976i \(0.497817\pi\)
\(762\) 0 0
\(763\) −2804.00 −0.133043
\(764\) 25536.0 1.20924
\(765\) 0 0
\(766\) 0 0
\(767\) −7488.00 −0.352511
\(768\) 0 0
\(769\) 1514.00 0.0709964 0.0354982 0.999370i \(-0.488698\pi\)
0.0354982 + 0.999370i \(0.488698\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −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
\(780\) 0 0
\(781\) 16416.0 0.752126
\(782\) 0 0
\(783\) 0 0
\(784\) −21696.0 −0.988338
\(785\) −18120.0 −0.823861
\(786\) 0 0
\(787\) 10154.0 0.459912 0.229956 0.973201i \(-0.426142\pi\)
0.229956 + 0.973201i \(0.426142\pi\)
\(788\) 22368.0 1.01120
\(789\) 0 0
\(790\) 0 0
\(791\) −3852.00 −0.173150
\(792\) 0 0
\(793\) 962.000 0.0430790
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) −24552.0 −1.07898
\(804\) 0 0
\(805\) 2304.00 0.100876
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8778.00 0.381481 0.190740 0.981641i \(-0.438911\pi\)
0.190740 + 0.981641i \(0.438911\pi\)
\(810\) 0 0
\(811\) −430.000 −0.0186182 −0.00930909 0.999957i \(-0.502963\pi\)
−0.00930909 + 0.999957i \(0.502963\pi\)
\(812\) 288.000 0.0124468
\(813\) 0 0
\(814\) 0 0
\(815\) −4728.00 −0.203208
\(816\) 0 0
\(817\) 38776.0 1.66047
\(818\) 0 0
\(819\) 0 0
\(820\) −36864.0 −1.56994
\(821\) 32976.0 1.40179 0.700895 0.713264i \(-0.252784\pi\)
0.700895 + 0.713264i \(0.252784\pi\)
\(822\) 0 0
\(823\) −1168.00 −0.0494701 −0.0247351 0.999694i \(-0.507874\pi\)
−0.0247351 + 0.999694i \(0.507874\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17172.0 0.722042 0.361021 0.932558i \(-0.382428\pi\)
0.361021 + 0.932558i \(0.382428\pi\)
\(828\) 0 0
\(829\) 27146.0 1.13730 0.568649 0.822580i \(-0.307466\pi\)
0.568649 + 0.822580i \(0.307466\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6656.00 −0.277350
\(833\) −26442.0 −1.09983
\(834\) 0 0
\(835\) −1008.00 −0.0417764
\(836\) −21312.0 −0.881688
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 15392.0 0.627742
\(845\) 2028.00 0.0825625
\(846\) 0 0
\(847\) −70.0000 −0.00283970
\(848\) −35712.0 −1.44617
\(849\) 0 0
\(850\) 0 0
\(851\) −27456.0 −1.10597
\(852\) 0 0
\(853\) 24842.0 0.997156 0.498578 0.866845i \(-0.333856\pi\)
0.498578 + 0.866845i \(0.333856\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 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 \(-0.366252\pi\)
0.407925 + 0.913015i \(0.366252\pi\)
\(860\) −50304.0 −1.99460
\(861\) 0 0
\(862\) 0 0
\(863\) −9108.00 −0.359258 −0.179629 0.983734i \(-0.557490\pi\)
−0.179629 + 0.983734i \(0.557490\pi\)
\(864\) 0 0
\(865\) −14328.0 −0.563198
\(866\) 0 0
\(867\) 0 0
\(868\) 3424.00 0.133892
\(869\) 25344.0 0.989340
\(870\) 0 0
\(871\) 494.000 0.0192176
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2544.00 −0.0982890
\(876\) 0 0
\(877\) −24046.0 −0.925856 −0.462928 0.886396i \(-0.653201\pi\)
−0.462928 + 0.886396i \(0.653201\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) 0 0
\(887\) 15648.0 0.592343 0.296172 0.955135i \(-0.404290\pi\)
0.296172 + 0.955135i \(0.404290\pi\)
\(888\) 0 0
\(889\) −1976.00 −0.0745477
\(890\) 0 0
\(891\) 0 0
\(892\) −40336.0 −1.51407
\(893\) −22200.0 −0.831909
\(894\) 0 0
\(895\) −37872.0 −1.41444
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3852.00 0.142905
\(900\) 0 0
\(901\) −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
\(909\) 0 0
\(910\) 0 0
\(911\) −39144.0 −1.42360 −0.711799 0.702383i \(-0.752120\pi\)
−0.711799 + 0.702383i \(0.752120\pi\)
\(912\) 0 0
\(913\) 31968.0 1.15880
\(914\) 0 0
\(915\) 0 0
\(916\) 19280.0 0.695447
\(917\) 4200.00 0.151250
\(918\) 0 0
\(919\) −38248.0 −1.37289 −0.686445 0.727182i \(-0.740830\pi\)
−0.686445 + 0.727182i \(0.740830\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5928.00 0.211400
\(924\) 0 0
\(925\) −5434.00 −0.193155
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 54264.0 1.91641 0.958205 0.286084i \(-0.0923536\pi\)
0.958205 + 0.286084i \(0.0923536\pi\)
\(930\) 0 0
\(931\) −25086.0 −0.883094
\(932\) 29808.0 1.04763
\(933\) 0 0
\(934\) 0 0
\(935\) 33696.0 1.17859
\(936\) 0 0
\(937\) 12206.0 0.425563 0.212782 0.977100i \(-0.431748\pi\)
0.212782 + 0.977100i \(0.431748\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 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 \(-0.149374\pi\)
0.891897 + 0.452238i \(0.149374\pi\)
\(948\) 0 0
\(949\) −8866.00 −0.303269
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −13782.0 −0.468460 −0.234230 0.972181i \(-0.575257\pi\)
−0.234230 + 0.972181i \(0.575257\pi\)
\(954\) 0 0
\(955\) −38304.0 −1.29789
\(956\) 9984.00 0.337767
\(957\) 0 0
\(958\) 0 0
\(959\) 4992.00 0.168092
\(960\) 0 0
\(961\) 16005.0 0.537243
\(962\) 0 0
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 0 0
\(971\) 18708.0 0.618299 0.309149 0.951013i \(-0.399956\pi\)
0.309149 + 0.951013i \(0.399956\pi\)
\(972\) 0 0
\(973\) −4928.00 −0.162368
\(974\) 0 0
\(975\) 0 0
\(976\) 4736.00 0.155323
\(977\) 48804.0 1.59814 0.799068 0.601241i \(-0.205327\pi\)
0.799068 + 0.601241i \(0.205327\pi\)
\(978\) 0 0
\(979\) 36720.0 1.19875
\(980\) 32544.0 1.06080
\(981\) 0 0
\(982\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) −4080.00 −0.129995
\(996\) 0 0
\(997\) 9038.00 0.287098 0.143549 0.989643i \(-0.454149\pi\)
0.143549 + 0.989643i \(0.454149\pi\)
\(998\) 0 0
\(999\) 0 0
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 117.4.a.a.1.1 1
3.2 odd 2 39.4.a.a.1.1 1
4.3 odd 2 1872.4.a.m.1.1 1
12.11 even 2 624.4.a.g.1.1 1
13.12 even 2 1521.4.a.f.1.1 1
15.14 odd 2 975.4.a.e.1.1 1
21.20 even 2 1911.4.a.f.1.1 1
24.5 odd 2 2496.4.a.o.1.1 1
24.11 even 2 2496.4.a.f.1.1 1
39.5 even 4 507.4.b.b.337.2 2
39.8 even 4 507.4.b.b.337.1 2
39.38 odd 2 507.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.a.1.1 1 3.2 odd 2
117.4.a.a.1.1 1 1.1 even 1 trivial
507.4.a.c.1.1 1 39.38 odd 2
507.4.b.b.337.1 2 39.8 even 4
507.4.b.b.337.2 2 39.5 even 4
624.4.a.g.1.1 1 12.11 even 2
975.4.a.e.1.1 1 15.14 odd 2
1521.4.a.f.1.1 1 13.12 even 2
1872.4.a.m.1.1 1 4.3 odd 2
1911.4.a.f.1.1 1 21.20 even 2
2496.4.a.f.1.1 1 24.11 even 2
2496.4.a.o.1.1 1 24.5 odd 2