Properties

Label 624.4.a.g.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -12.0000 q^{5} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -12.0000 q^{5} -2.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} +13.0000 q^{13} -36.0000 q^{15} -78.0000 q^{17} -74.0000 q^{19} -6.00000 q^{21} +96.0000 q^{23} +19.0000 q^{25} +27.0000 q^{27} +18.0000 q^{29} +214.000 q^{31} +108.000 q^{33} +24.0000 q^{35} -286.000 q^{37} +39.0000 q^{39} -384.000 q^{41} -524.000 q^{43} -108.000 q^{45} -300.000 q^{47} -339.000 q^{49} -234.000 q^{51} +558.000 q^{53} -432.000 q^{55} -222.000 q^{57} -576.000 q^{59} +74.0000 q^{61} -18.0000 q^{63} -156.000 q^{65} -38.0000 q^{67} +288.000 q^{69} +456.000 q^{71} -682.000 q^{73} +57.0000 q^{75} -72.0000 q^{77} -704.000 q^{79} +81.0000 q^{81} +888.000 q^{83} +936.000 q^{85} +54.0000 q^{87} -1020.00 q^{89} -26.0000 q^{91} +642.000 q^{93} +888.000 q^{95} +110.000 q^{97} +324.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −12.0000 −1.07331 −0.536656 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.107990 −0.0539949 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 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
\(15\) −36.0000 −0.619677
\(16\) 0 0
\(17\) −78.0000 −1.11281 −0.556405 0.830911i \(-0.687820\pi\)
−0.556405 + 0.830911i \(0.687820\pi\)
\(18\) 0 0
\(19\) −74.0000 −0.893514 −0.446757 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.0623480
\(22\) 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
\(27\) 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
\(35\) 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
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −384.000 −1.46270 −0.731350 0.682002i \(-0.761110\pi\)
−0.731350 + 0.682002i \(0.761110\pi\)
\(42\) 0 0
\(43\) −524.000 −1.85835 −0.929177 0.369634i \(-0.879483\pi\)
−0.929177 + 0.369634i \(0.879483\pi\)
\(44\) 0 0
\(45\) −108.000 −0.357771
\(46\) 0 0
\(47\) −300.000 −0.931053 −0.465527 0.885034i \(-0.654135\pi\)
−0.465527 + 0.885034i \(0.654135\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) 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 \(0.719205\pi\)
\(60\) 0 0
\(61\) 74.0000 0.155323 0.0776617 0.996980i \(-0.475255\pi\)
0.0776617 + 0.996980i \(0.475255\pi\)
\(62\) 0 0
\(63\) −18.0000 −0.0359966
\(64\) 0 0
\(65\) −156.000 −0.297683
\(66\) 0 0
\(67\) −38.0000 −0.0692901 −0.0346451 0.999400i \(-0.511030\pi\)
−0.0346451 + 0.999400i \(0.511030\pi\)
\(68\) 0 0
\(69\) 288.000 0.502480
\(70\) 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
\(75\) 57.0000 0.0877572
\(76\) 0 0
\(77\) −72.0000 −0.106561
\(78\) 0 0
\(79\) −704.000 −1.00261 −0.501305 0.865271i \(-0.667147\pi\)
−0.501305 + 0.865271i \(0.667147\pi\)
\(80\) 0 0
\(81\) 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 \(-0.707793\pi\)
−0.607415 + 0.794385i \(0.707793\pi\)
\(90\) 0 0
\(91\) −26.0000 −0.0299510
\(92\) 0 0
\(93\) 642.000 0.715831
\(94\) 0 0
\(95\) 888.000 0.959020
\(96\) 0 0
\(97\) 110.000 0.115142 0.0575712 0.998341i \(-0.481664\pi\)
0.0575712 + 0.998341i \(0.481664\pi\)
\(98\) 0 0
\(99\) 324.000 0.328921
\(100\) 0 0
\(101\) −990.000 −0.975333 −0.487667 0.873030i \(-0.662152\pi\)
−0.487667 + 0.873030i \(0.662152\pi\)
\(102\) 0 0
\(103\) −1208.00 −1.15561 −0.577805 0.816175i \(-0.696090\pi\)
−0.577805 + 0.816175i \(0.696090\pi\)
\(104\) 0 0
\(105\) 72.0000 0.0669189
\(106\) 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
\(111\) −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
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 156.000 0.120172
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) −1152.00 −0.844491
\(124\) 0 0
\(125\) 1272.00 0.910169
\(126\) 0 0
\(127\) 988.000 0.690321 0.345161 0.938544i \(-0.387824\pi\)
0.345161 + 0.938544i \(0.387824\pi\)
\(128\) 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 \(0.253051\pi\)
\(132\) 0 0
\(133\) 148.000 0.0964904
\(134\) 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
\(144\) 0 0
\(145\) −216.000 −0.123709
\(146\) 0 0
\(147\) −1017.00 −0.570617
\(148\) 0 0
\(149\) 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 \(-0.422203\pi\)
0.241981 + 0.970281i \(0.422203\pi\)
\(152\) 0 0
\(153\) −702.000 −0.370937
\(154\) 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
\(159\) 1674.00 0.834949
\(160\) 0 0
\(161\) −192.000 −0.0939858
\(162\) 0 0
\(163\) 394.000 0.189328 0.0946640 0.995509i \(-0.469822\pi\)
0.0946640 + 0.995509i \(0.469822\pi\)
\(164\) 0 0
\(165\) −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
\(172\) 0 0
\(173\) 1194.00 0.524729 0.262365 0.964969i \(-0.415498\pi\)
0.262365 + 0.964969i \(0.415498\pi\)
\(174\) 0 0
\(175\) −38.0000 −0.0164145
\(176\) 0 0
\(177\) −1728.00 −0.733810
\(178\) 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
\(183\) 222.000 0.0896760
\(184\) 0 0
\(185\) 3432.00 1.36392
\(186\) 0 0
\(187\) −2808.00 −1.09808
\(188\) 0 0
\(189\) −54.0000 −0.0207827
\(190\) 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
\(195\) −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
\(203\) −36.0000 −0.0124468
\(204\) 0 0
\(205\) 4608.00 1.56994
\(206\) 0 0
\(207\) 864.000 0.290107
\(208\) 0 0
\(209\) −2664.00 −0.881688
\(210\) 0 0
\(211\) 1924.00 0.627742 0.313871 0.949466i \(-0.398374\pi\)
0.313871 + 0.949466i \(0.398374\pi\)
\(212\) 0 0
\(213\) 1368.00 0.440065
\(214\) 0 0
\(215\) 6288.00 1.99460
\(216\) 0 0
\(217\) −428.000 −0.133892
\(218\) 0 0
\(219\) −2046.00 −0.631305
\(220\) 0 0
\(221\) −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 \(0.372054\pi\)
\(228\) 0 0
\(229\) −2410.00 −0.695447 −0.347723 0.937597i \(-0.613045\pi\)
−0.347723 + 0.937597i \(0.613045\pi\)
\(230\) 0 0
\(231\) −216.000 −0.0615228
\(232\) 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
\(236\) 0 0
\(237\) −2112.00 −0.578857
\(238\) 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
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 4068.00 1.06080
\(246\) 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
\(256\) 0 0
\(257\) 1326.00 0.321843 0.160921 0.986967i \(-0.448553\pi\)
0.160921 + 0.986967i \(0.448553\pi\)
\(258\) 0 0
\(259\) 572.000 0.137229
\(260\) 0 0
\(261\) 162.000 0.0384197
\(262\) 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
\(267\) −3060.00 −0.701382
\(268\) 0 0
\(269\) 6474.00 1.46739 0.733693 0.679481i \(-0.237795\pi\)
0.733693 + 0.679481i \(0.237795\pi\)
\(270\) 0 0
\(271\) −5978.00 −1.33999 −0.669996 0.742365i \(-0.733704\pi\)
−0.669996 + 0.742365i \(0.733704\pi\)
\(272\) 0 0
\(273\) −78.0000 −0.0172922
\(274\) 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
\(279\) 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 \(0.480196\pi\)
\(284\) 0 0
\(285\) 2664.00 0.553690
\(286\) 0 0
\(287\) 768.000 0.157957
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 330.000 0.0664775
\(292\) 0 0
\(293\) −4608.00 −0.918779 −0.459389 0.888235i \(-0.651932\pi\)
−0.459389 + 0.888235i \(0.651932\pi\)
\(294\) 0 0
\(295\) 6912.00 1.36418
\(296\) 0 0
\(297\) 972.000 0.189903
\(298\) 0 0
\(299\) 1248.00 0.241384
\(300\) 0 0
\(301\) 1048.00 0.200683
\(302\) 0 0
\(303\) −2970.00 −0.563109
\(304\) 0 0
\(305\) −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 \(-0.590124\pi\)
−0.279366 + 0.960185i \(0.590124\pi\)
\(314\) 0 0
\(315\) 216.000 0.0386356
\(316\) 0 0
\(317\) 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
\(320\) 0 0
\(321\) −2988.00 −0.519545
\(322\) 0 0
\(323\) 5772.00 0.994312
\(324\) 0 0
\(325\) 247.000 0.0421572
\(326\) 0 0
\(327\) −4206.00 −0.711292
\(328\) 0 0
\(329\) 600.000 0.100544
\(330\) 0 0
\(331\) 4426.00 0.734970 0.367485 0.930030i \(-0.380219\pi\)
0.367485 + 0.930030i \(0.380219\pi\)
\(332\) 0 0
\(333\) −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
\(340\) 0 0
\(341\) 7704.00 1.22345
\(342\) 0 0
\(343\) 1364.00 0.214720
\(344\) 0 0
\(345\) −3456.00 −0.539318
\(346\) 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
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −9720.00 −1.46556 −0.732781 0.680465i \(-0.761778\pi\)
−0.732781 + 0.680465i \(0.761778\pi\)
\(354\) 0 0
\(355\) −5472.00 −0.818095
\(356\) 0 0
\(357\) 468.000 0.0693815
\(358\) 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
\(363\) −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 \(0.453016\pi\)
\(368\) 0 0
\(369\) −3456.00 −0.487567
\(370\) 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
\(375\) 3816.00 0.525486
\(376\) 0 0
\(377\) 234.000 0.0319671
\(378\) 0 0
\(379\) −12818.0 −1.73725 −0.868623 0.495473i \(-0.834995\pi\)
−0.868623 + 0.495473i \(0.834995\pi\)
\(380\) 0 0
\(381\) 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
\(385\) 864.000 0.114373
\(386\) 0 0
\(387\) −4716.00 −0.619452
\(388\) 0 0
\(389\) 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
\(396\) 0 0
\(397\) 11162.0 1.41110 0.705548 0.708663i \(-0.250701\pi\)
0.705548 + 0.708663i \(0.250701\pi\)
\(398\) 0 0
\(399\) 444.000 0.0557088
\(400\) 0 0
\(401\) −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
\(404\) 0 0
\(405\) −972.000 −0.119257
\(406\) 0 0
\(407\) −10296.0 −1.25394
\(408\) 0 0
\(409\) −9682.00 −1.17052 −0.585262 0.810844i \(-0.699008\pi\)
−0.585262 + 0.810844i \(0.699008\pi\)
\(410\) 0 0
\(411\) −7488.00 −0.898676
\(412\) 0 0
\(413\) 1152.00 0.137255
\(414\) 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
\(427\) −148.000 −0.0167734
\(428\) 0 0
\(429\) 1404.00 0.158009
\(430\) 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
\(435\) −648.000 −0.0714235
\(436\) 0 0
\(437\) −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
\(441\) −3051.00 −0.329446
\(442\) 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
\(447\) 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
\(452\) 0 0
\(453\) 2694.00 0.279415
\(454\) 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
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) 10500.0 1.06081 0.530405 0.847744i \(-0.322040\pi\)
0.530405 + 0.847744i \(0.322040\pi\)
\(462\) 0 0
\(463\) 5542.00 0.556282 0.278141 0.960540i \(-0.410282\pi\)
0.278141 + 0.960540i \(0.410282\pi\)
\(464\) 0 0
\(465\) −7704.00 −0.768311
\(466\) 0 0
\(467\) 5220.00 0.517244 0.258622 0.965979i \(-0.416732\pi\)
0.258622 + 0.965979i \(0.416732\pi\)
\(468\) 0 0
\(469\) 76.0000 0.00748263
\(470\) 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
\(483\) −576.000 −0.0542627
\(484\) 0 0
\(485\) −1320.00 −0.123584
\(486\) 0 0
\(487\) −12170.0 −1.13239 −0.566196 0.824270i \(-0.691586\pi\)
−0.566196 + 0.824270i \(0.691586\pi\)
\(488\) 0 0
\(489\) 1182.00 0.109309
\(490\) 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
\(495\) −3888.00 −0.353036
\(496\) 0 0
\(497\) −912.000 −0.0823115
\(498\) 0 0
\(499\) 1330.00 0.119317 0.0596583 0.998219i \(-0.480999\pi\)
0.0596583 + 0.998219i \(0.480999\pi\)
\(500\) 0 0
\(501\) −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
\(508\) 0 0
\(509\) 5124.00 0.446203 0.223101 0.974795i \(-0.428382\pi\)
0.223101 + 0.974795i \(0.428382\pi\)
\(510\) 0 0
\(511\) 1364.00 0.118082
\(512\) 0 0
\(513\) −1998.00 −0.171957
\(514\) 0 0
\(515\) 14496.0 1.24033
\(516\) 0 0
\(517\) −10800.0 −0.918730
\(518\) 0 0
\(519\) 3582.00 0.302953
\(520\) 0 0
\(521\) −882.000 −0.0741672 −0.0370836 0.999312i \(-0.511807\pi\)
−0.0370836 + 0.999312i \(0.511807\pi\)
\(522\) 0 0
\(523\) 2320.00 0.193970 0.0969852 0.995286i \(-0.469080\pi\)
0.0969852 + 0.995286i \(0.469080\pi\)
\(524\) 0 0
\(525\) −114.000 −0.00947689
\(526\) 0 0
\(527\) −16692.0 −1.37972
\(528\) 0 0
\(529\) −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
\(539\) −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
\(543\) −3234.00 −0.255588
\(544\) 0 0
\(545\) 16824.0 1.32231
\(546\) 0 0
\(547\) −7040.00 −0.550290 −0.275145 0.961403i \(-0.588726\pi\)
−0.275145 + 0.961403i \(0.588726\pi\)
\(548\) 0 0
\(549\) 666.000 0.0517745
\(550\) 0 0
\(551\) −1332.00 −0.102986
\(552\) 0 0
\(553\) 1408.00 0.108272
\(554\) 0 0
\(555\) 10296.0 0.787461
\(556\) 0 0
\(557\) −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 \(0.752572\pi\)
\(564\) 0 0
\(565\) −23112.0 −1.72094
\(566\) 0 0
\(567\) −162.000 −0.0119989
\(568\) 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 \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(572\) 0 0
\(573\) −9576.00 −0.698156
\(574\) 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
\(579\) 2166.00 0.155468
\(580\) 0 0
\(581\) −1776.00 −0.126817
\(582\) 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
\(592\) 0 0
\(593\) −25596.0 −1.77252 −0.886258 0.463192i \(-0.846704\pi\)
−0.886258 + 0.463192i \(0.846704\pi\)
\(594\) 0 0
\(595\) −1872.00 −0.128982
\(596\) 0 0
\(597\) 1020.00 0.0699260
\(598\) 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
\(603\) −342.000 −0.0230967
\(604\) 0 0
\(605\) 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
\(609\) −108.000 −0.00718618
\(610\) 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
\(615\) 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 \(0.430265\pi\)
\(620\) 0 0
\(621\) 2592.00 0.167493
\(622\) 0 0
\(623\) 2040.00 0.131189
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) −7992.00 −0.509043
\(628\) 0 0
\(629\) 22308.0 1.41411
\(630\) 0 0
\(631\) 27250.0 1.71918 0.859592 0.510981i \(-0.170718\pi\)
0.859592 + 0.510981i \(0.170718\pi\)
\(632\) 0 0
\(633\) 5772.00 0.362427
\(634\) 0 0
\(635\) −11856.0 −0.740931
\(636\) 0 0
\(637\) −4407.00 −0.274116
\(638\) 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
\(651\) −1284.00 −0.0773025
\(652\) 0 0
\(653\) −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
\(656\) 0 0
\(657\) −6138.00 −0.364484
\(658\) 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
\(663\) −3042.00 −0.178192
\(664\) 0 0
\(665\) −1776.00 −0.103564
\(666\) 0 0
\(667\) 1728.00 0.100312
\(668\) 0 0
\(669\) −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
\(676\) 0 0
\(677\) −8610.00 −0.488788 −0.244394 0.969676i \(-0.578589\pi\)
−0.244394 + 0.969676i \(0.578589\pi\)
\(678\) 0 0
\(679\) −220.000 −0.0124342
\(680\) 0 0
\(681\) 8028.00 0.451738
\(682\) 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
\(687\) −7230.00 −0.401516
\(688\) 0 0
\(689\) 7254.00 0.401096
\(690\) 0 0
\(691\) −2270.00 −0.124971 −0.0624854 0.998046i \(-0.519903\pi\)
−0.0624854 + 0.998046i \(0.519903\pi\)
\(692\) 0 0
\(693\) −648.000 −0.0355202
\(694\) 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
\(704\) 0 0
\(705\) 10800.0 0.576953
\(706\) 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
\(711\) −6336.00 −0.334203
\(712\) 0 0
\(713\) 20544.0 1.07907
\(714\) 0 0
\(715\) −5616.00 −0.293743
\(716\) 0 0
\(717\) −3744.00 −0.195010
\(718\) 0 0
\(719\) −11568.0 −0.600019 −0.300009 0.953936i \(-0.596990\pi\)
−0.300009 + 0.953936i \(0.596990\pi\)
\(720\) 0 0
\(721\) 2416.00 0.124794
\(722\) 0 0
\(723\) −12630.0 −0.649675
\(724\) 0 0
\(725\) 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
\(732\) 0 0
\(733\) −15010.0 −0.756353 −0.378177 0.925733i \(-0.623449\pi\)
−0.378177 + 0.925733i \(0.623449\pi\)
\(734\) 0 0
\(735\) 12204.0 0.612451
\(736\) 0 0
\(737\) −1368.00 −0.0683730
\(738\) 0 0
\(739\) −33410.0 −1.66307 −0.831534 0.555474i \(-0.812537\pi\)
−0.831534 + 0.555474i \(0.812537\pi\)
\(740\) 0 0
\(741\) −2886.00 −0.143077
\(742\) 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
\(747\) 7992.00 0.391448
\(748\) 0 0
\(749\) 1992.00 0.0971777
\(750\) 0 0
\(751\) 13912.0 0.675973 0.337987 0.941151i \(-0.390254\pi\)
0.337987 + 0.941151i \(0.390254\pi\)
\(752\) 0 0
\(753\) 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 \(-0.502183\pi\)
−0.00685939 + 0.999976i \(0.502183\pi\)
\(762\) 0 0
\(763\) 2804.00 0.133043
\(764\) 0 0
\(765\) 8424.00 0.398131
\(766\) 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
\(771\) 3978.00 0.185816
\(772\) 0 0
\(773\) 15816.0 0.735915 0.367957 0.929843i \(-0.380057\pi\)
0.367957 + 0.929843i \(0.380057\pi\)
\(774\) 0 0
\(775\) 4066.00 0.188458
\(776\) 0 0
\(777\) 1716.00 0.0792293
\(778\) 0 0
\(779\) 28416.0 1.30694
\(780\) 0 0
\(781\) 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 \(0.573858\pi\)
\(788\) 0 0
\(789\) 18144.0 0.818686
\(790\) 0 0
\(791\) −3852.00 −0.173150
\(792\) 0 0
\(793\) 962.000 0.0430790
\(794\) 0 0
\(795\) −20088.0 −0.896161
\(796\) 0 0
\(797\) −17442.0 −0.775191 −0.387596 0.921830i \(-0.626694\pi\)
−0.387596 + 0.921830i \(0.626694\pi\)
\(798\) 0 0
\(799\) 23400.0 1.03609
\(800\) 0 0
\(801\) −9180.00 −0.404943
\(802\) 0 0
\(803\) −24552.0 −1.07898
\(804\) 0 0
\(805\) 2304.00 0.100876
\(806\) 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
\(816\) 0 0
\(817\) 38776.0 1.66047
\(818\) 0 0
\(819\) −234.000 −0.00998367
\(820\) 0 0
\(821\) −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 \(-0.492126\pi\)
0.0247351 + 0.999694i \(0.492126\pi\)
\(824\) 0 0
\(825\) 2052.00 0.0865957
\(826\) 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
\(831\) 26250.0 1.09579
\(832\) 0 0
\(833\) 26442.0 1.09983
\(834\) 0 0
\(835\) 1008.00 0.0417764
\(836\) 0 0
\(837\) 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
\(844\) 0 0
\(845\) −2028.00 −0.0825625
\(846\) 0 0
\(847\) 70.0000 0.00283970
\(848\) 0 0
\(849\) 1776.00 0.0717929
\(850\) 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
\(855\) 7992.00 0.319673
\(856\) 0 0
\(857\) 11406.0 0.454634 0.227317 0.973821i \(-0.427005\pi\)
0.227317 + 0.973821i \(0.427005\pi\)
\(858\) 0 0
\(859\) −20540.0 −0.815851 −0.407925 0.913015i \(-0.633748\pi\)
−0.407925 + 0.913015i \(0.633748\pi\)
\(860\) 0 0
\(861\) 2304.00 0.0911964
\(862\) 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
\(867\) 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
\(875\) −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
\(879\) −13824.0 −0.530457
\(880\) 0 0
\(881\) 7998.00 0.305856 0.152928 0.988237i \(-0.451130\pi\)
0.152928 + 0.988237i \(0.451130\pi\)
\(882\) 0 0
\(883\) −24032.0 −0.915902 −0.457951 0.888978i \(-0.651416\pi\)
−0.457951 + 0.888978i \(0.651416\pi\)
\(884\) 0 0
\(885\) 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
\(889\) −1976.00 −0.0745477
\(890\) 0 0
\(891\) 2916.00 0.109640
\(892\) 0 0
\(893\) 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
\(900\) 0 0
\(901\) −43524.0 −1.60932
\(902\) 0 0
\(903\) 3144.00 0.115865
\(904\) 0 0
\(905\) 12936.0 0.475146
\(906\) 0 0
\(907\) 808.000 0.0295802 0.0147901 0.999891i \(-0.495292\pi\)
0.0147901 + 0.999891i \(0.495292\pi\)
\(908\) 0 0
\(909\) −8910.00 −0.325111
\(910\) 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
\(915\) −2664.00 −0.0962504
\(916\) 0 0
\(917\) −4200.00 −0.151250
\(918\) 0 0
\(919\) 38248.0 1.37289 0.686445 0.727182i \(-0.259170\pi\)
0.686445 + 0.727182i \(0.259170\pi\)
\(920\) 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 \(-0.907646\pi\)
−0.958205 + 0.286084i \(0.907646\pi\)
\(930\) 0 0
\(931\) 25086.0 0.883094
\(932\) 0 0
\(933\) −7416.00 −0.260224
\(934\) 0 0
\(935\) 33696.0 1.17859
\(936\) 0 0
\(937\) 12206.0 0.425563 0.212782 0.977100i \(-0.431748\pi\)
0.212782 + 0.977100i \(0.431748\pi\)
\(938\) 0 0
\(939\) −9282.00 −0.322584
\(940\) 0 0
\(941\) 17664.0 0.611934 0.305967 0.952042i \(-0.401020\pi\)
0.305967 + 0.952042i \(0.401020\pi\)
\(942\) 0 0
\(943\) −36864.0 −1.27302
\(944\) 0 0
\(945\) 648.000 0.0223063
\(946\) 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
\(951\) 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
\(956\) 0 0
\(957\) 1944.00 0.0656642
\(958\) 0 0
\(959\) 4992.00 0.168092
\(960\) 0 0
\(961\) 16005.0 0.537243
\(962\) 0 0
\(963\) −8964.00 −0.299959
\(964\) 0 0
\(965\) −8664.00 −0.289020
\(966\) 0 0
\(967\) −14618.0 −0.486125 −0.243063 0.970011i \(-0.578152\pi\)
−0.243063 + 0.970011i \(0.578152\pi\)
\(968\) 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 \(0.399956\pi\)
\(972\) 0 0
\(973\) −4928.00 −0.162368
\(974\) 0 0
\(975\) 741.000 0.0243395
\(976\) 0 0
\(977\) −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
\(987\) 1800.00 0.0580493
\(988\) 0 0
\(989\) −50304.0 −1.61737
\(990\) 0 0
\(991\) 21004.0 0.673274 0.336637 0.941635i \(-0.390711\pi\)
0.336637 + 0.941635i \(0.390711\pi\)
\(992\) 0 0
\(993\) 13278.0 0.424335
\(994\) 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
\(999\) −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 624.4.a.g.1.1 1
3.2 odd 2 1872.4.a.m.1.1 1
4.3 odd 2 39.4.a.a.1.1 1
8.3 odd 2 2496.4.a.o.1.1 1
8.5 even 2 2496.4.a.f.1.1 1
12.11 even 2 117.4.a.a.1.1 1
20.19 odd 2 975.4.a.e.1.1 1
28.27 even 2 1911.4.a.f.1.1 1
52.31 even 4 507.4.b.b.337.2 2
52.47 even 4 507.4.b.b.337.1 2
52.51 odd 2 507.4.a.c.1.1 1
156.155 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 4.3 odd 2
117.4.a.a.1.1 1 12.11 even 2
507.4.a.c.1.1 1 52.51 odd 2
507.4.b.b.337.1 2 52.47 even 4
507.4.b.b.337.2 2 52.31 even 4
624.4.a.g.1.1 1 1.1 even 1 trivial
975.4.a.e.1.1 1 20.19 odd 2
1521.4.a.f.1.1 1 156.155 even 2
1872.4.a.m.1.1 1 3.2 odd 2
1911.4.a.f.1.1 1 28.27 even 2
2496.4.a.f.1.1 1 8.5 even 2
2496.4.a.o.1.1 1 8.3 odd 2