Properties

Label 234.4.a.h.1.1
Level $234$
Weight $4$
Character 234.1
Self dual yes
Analytic conductor $13.806$
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] = [234,4,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, 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("234.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(13.8064469413\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -10.0000 q^{5} -8.00000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -10.0000 q^{5} -8.00000 q^{7} +8.00000 q^{8} -20.0000 q^{10} -40.0000 q^{11} +13.0000 q^{13} -16.0000 q^{14} +16.0000 q^{16} -130.000 q^{17} -20.0000 q^{19} -40.0000 q^{20} -80.0000 q^{22} -25.0000 q^{25} +26.0000 q^{26} -32.0000 q^{28} +18.0000 q^{29} -184.000 q^{31} +32.0000 q^{32} -260.000 q^{34} +80.0000 q^{35} -74.0000 q^{37} -40.0000 q^{38} -80.0000 q^{40} +362.000 q^{41} +76.0000 q^{43} -160.000 q^{44} +452.000 q^{47} -279.000 q^{49} -50.0000 q^{50} +52.0000 q^{52} -382.000 q^{53} +400.000 q^{55} -64.0000 q^{56} +36.0000 q^{58} -464.000 q^{59} +358.000 q^{61} -368.000 q^{62} +64.0000 q^{64} -130.000 q^{65} -700.000 q^{67} -520.000 q^{68} +160.000 q^{70} +748.000 q^{71} +1058.00 q^{73} -148.000 q^{74} -80.0000 q^{76} +320.000 q^{77} -976.000 q^{79} -160.000 q^{80} +724.000 q^{82} +1008.00 q^{83} +1300.00 q^{85} +152.000 q^{86} -320.000 q^{88} +386.000 q^{89} -104.000 q^{91} +904.000 q^{94} +200.000 q^{95} -614.000 q^{97} -558.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −10.0000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −20.0000 −0.632456
\(11\) −40.0000 −1.09640 −0.548202 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) −16.0000 −0.305441
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −130.000 −1.85468 −0.927342 0.374215i \(-0.877912\pi\)
−0.927342 + 0.374215i \(0.877912\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) −40.0000 −0.447214
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 26.0000 0.196116
\(27\) 0 0
\(28\) −32.0000 −0.215980
\(29\) 18.0000 0.115259 0.0576296 0.998338i \(-0.481646\pi\)
0.0576296 + 0.998338i \(0.481646\pi\)
\(30\) 0 0
\(31\) −184.000 −1.06604 −0.533022 0.846101i \(-0.678944\pi\)
−0.533022 + 0.846101i \(0.678944\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −260.000 −1.31146
\(35\) 80.0000 0.386356
\(36\) 0 0
\(37\) −74.0000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −40.0000 −0.170759
\(39\) 0 0
\(40\) −80.0000 −0.316228
\(41\) 362.000 1.37890 0.689450 0.724333i \(-0.257852\pi\)
0.689450 + 0.724333i \(0.257852\pi\)
\(42\) 0 0
\(43\) 76.0000 0.269532 0.134766 0.990877i \(-0.456972\pi\)
0.134766 + 0.990877i \(0.456972\pi\)
\(44\) −160.000 −0.548202
\(45\) 0 0
\(46\) 0 0
\(47\) 452.000 1.40279 0.701393 0.712774i \(-0.252562\pi\)
0.701393 + 0.712774i \(0.252562\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) −50.0000 −0.141421
\(51\) 0 0
\(52\) 52.0000 0.138675
\(53\) −382.000 −0.990033 −0.495016 0.868884i \(-0.664838\pi\)
−0.495016 + 0.868884i \(0.664838\pi\)
\(54\) 0 0
\(55\) 400.000 0.980654
\(56\) −64.0000 −0.152721
\(57\) 0 0
\(58\) 36.0000 0.0815005
\(59\) −464.000 −1.02386 −0.511929 0.859028i \(-0.671069\pi\)
−0.511929 + 0.859028i \(0.671069\pi\)
\(60\) 0 0
\(61\) 358.000 0.751430 0.375715 0.926735i \(-0.377397\pi\)
0.375715 + 0.926735i \(0.377397\pi\)
\(62\) −368.000 −0.753807
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −130.000 −0.248069
\(66\) 0 0
\(67\) −700.000 −1.27640 −0.638199 0.769872i \(-0.720320\pi\)
−0.638199 + 0.769872i \(0.720320\pi\)
\(68\) −520.000 −0.927342
\(69\) 0 0
\(70\) 160.000 0.273195
\(71\) 748.000 1.25030 0.625150 0.780505i \(-0.285038\pi\)
0.625150 + 0.780505i \(0.285038\pi\)
\(72\) 0 0
\(73\) 1058.00 1.69629 0.848147 0.529760i \(-0.177718\pi\)
0.848147 + 0.529760i \(0.177718\pi\)
\(74\) −148.000 −0.232495
\(75\) 0 0
\(76\) −80.0000 −0.120745
\(77\) 320.000 0.473602
\(78\) 0 0
\(79\) −976.000 −1.38998 −0.694991 0.719018i \(-0.744592\pi\)
−0.694991 + 0.719018i \(0.744592\pi\)
\(80\) −160.000 −0.223607
\(81\) 0 0
\(82\) 724.000 0.975030
\(83\) 1008.00 1.33304 0.666520 0.745487i \(-0.267783\pi\)
0.666520 + 0.745487i \(0.267783\pi\)
\(84\) 0 0
\(85\) 1300.00 1.65888
\(86\) 152.000 0.190588
\(87\) 0 0
\(88\) −320.000 −0.387638
\(89\) 386.000 0.459729 0.229865 0.973223i \(-0.426172\pi\)
0.229865 + 0.973223i \(0.426172\pi\)
\(90\) 0 0
\(91\) −104.000 −0.119804
\(92\) 0 0
\(93\) 0 0
\(94\) 904.000 0.991920
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −614.000 −0.642704 −0.321352 0.946960i \(-0.604137\pi\)
−0.321352 + 0.946960i \(0.604137\pi\)
\(98\) −558.000 −0.575168
\(99\) 0 0
\(100\) −100.000 −0.100000
\(101\) −518.000 −0.510326 −0.255163 0.966898i \(-0.582129\pi\)
−0.255163 + 0.966898i \(0.582129\pi\)
\(102\) 0 0
\(103\) 112.000 0.107143 0.0535713 0.998564i \(-0.482940\pi\)
0.0535713 + 0.998564i \(0.482940\pi\)
\(104\) 104.000 0.0980581
\(105\) 0 0
\(106\) −764.000 −0.700059
\(107\) 372.000 0.336099 0.168050 0.985779i \(-0.446253\pi\)
0.168050 + 0.985779i \(0.446253\pi\)
\(108\) 0 0
\(109\) 934.000 0.820743 0.410371 0.911918i \(-0.365399\pi\)
0.410371 + 0.911918i \(0.365399\pi\)
\(110\) 800.000 0.693427
\(111\) 0 0
\(112\) −128.000 −0.107990
\(113\) −1914.00 −1.59340 −0.796699 0.604376i \(-0.793422\pi\)
−0.796699 + 0.604376i \(0.793422\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 72.0000 0.0576296
\(117\) 0 0
\(118\) −928.000 −0.723977
\(119\) 1040.00 0.801148
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 716.000 0.531341
\(123\) 0 0
\(124\) −736.000 −0.533022
\(125\) 1500.00 1.07331
\(126\) 0 0
\(127\) 1296.00 0.905523 0.452761 0.891632i \(-0.350439\pi\)
0.452761 + 0.891632i \(0.350439\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −260.000 −0.175412
\(131\) 892.000 0.594919 0.297460 0.954734i \(-0.403861\pi\)
0.297460 + 0.954734i \(0.403861\pi\)
\(132\) 0 0
\(133\) 160.000 0.104314
\(134\) −1400.00 −0.902549
\(135\) 0 0
\(136\) −1040.00 −0.655730
\(137\) −2326.00 −1.45054 −0.725269 0.688466i \(-0.758284\pi\)
−0.725269 + 0.688466i \(0.758284\pi\)
\(138\) 0 0
\(139\) 1932.00 1.17892 0.589461 0.807797i \(-0.299340\pi\)
0.589461 + 0.807797i \(0.299340\pi\)
\(140\) 320.000 0.193178
\(141\) 0 0
\(142\) 1496.00 0.884095
\(143\) −520.000 −0.304088
\(144\) 0 0
\(145\) −180.000 −0.103091
\(146\) 2116.00 1.19946
\(147\) 0 0
\(148\) −296.000 −0.164399
\(149\) −882.000 −0.484941 −0.242471 0.970159i \(-0.577958\pi\)
−0.242471 + 0.970159i \(0.577958\pi\)
\(150\) 0 0
\(151\) −1776.00 −0.957145 −0.478572 0.878048i \(-0.658846\pi\)
−0.478572 + 0.878048i \(0.658846\pi\)
\(152\) −160.000 −0.0853797
\(153\) 0 0
\(154\) 640.000 0.334887
\(155\) 1840.00 0.953499
\(156\) 0 0
\(157\) −2410.00 −1.22509 −0.612544 0.790436i \(-0.709854\pi\)
−0.612544 + 0.790436i \(0.709854\pi\)
\(158\) −1952.00 −0.982866
\(159\) 0 0
\(160\) −320.000 −0.158114
\(161\) 0 0
\(162\) 0 0
\(163\) 3212.00 1.54346 0.771728 0.635953i \(-0.219393\pi\)
0.771728 + 0.635953i \(0.219393\pi\)
\(164\) 1448.00 0.689450
\(165\) 0 0
\(166\) 2016.00 0.942602
\(167\) −1668.00 −0.772896 −0.386448 0.922311i \(-0.626298\pi\)
−0.386448 + 0.922311i \(0.626298\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 2600.00 1.17301
\(171\) 0 0
\(172\) 304.000 0.134766
\(173\) −3598.00 −1.58122 −0.790609 0.612321i \(-0.790236\pi\)
−0.790609 + 0.612321i \(0.790236\pi\)
\(174\) 0 0
\(175\) 200.000 0.0863919
\(176\) −640.000 −0.274101
\(177\) 0 0
\(178\) 772.000 0.325078
\(179\) −1068.00 −0.445956 −0.222978 0.974824i \(-0.571578\pi\)
−0.222978 + 0.974824i \(0.571578\pi\)
\(180\) 0 0
\(181\) −4786.00 −1.96542 −0.982709 0.185158i \(-0.940720\pi\)
−0.982709 + 0.185158i \(0.940720\pi\)
\(182\) −208.000 −0.0847142
\(183\) 0 0
\(184\) 0 0
\(185\) 740.000 0.294086
\(186\) 0 0
\(187\) 5200.00 2.03348
\(188\) 1808.00 0.701393
\(189\) 0 0
\(190\) 400.000 0.152732
\(191\) 1312.00 0.497031 0.248516 0.968628i \(-0.420057\pi\)
0.248516 + 0.968628i \(0.420057\pi\)
\(192\) 0 0
\(193\) −350.000 −0.130537 −0.0652683 0.997868i \(-0.520790\pi\)
−0.0652683 + 0.997868i \(0.520790\pi\)
\(194\) −1228.00 −0.454460
\(195\) 0 0
\(196\) −1116.00 −0.406706
\(197\) 342.000 0.123688 0.0618439 0.998086i \(-0.480302\pi\)
0.0618439 + 0.998086i \(0.480302\pi\)
\(198\) 0 0
\(199\) −3368.00 −1.19975 −0.599877 0.800092i \(-0.704784\pi\)
−0.599877 + 0.800092i \(0.704784\pi\)
\(200\) −200.000 −0.0707107
\(201\) 0 0
\(202\) −1036.00 −0.360855
\(203\) −144.000 −0.0497873
\(204\) 0 0
\(205\) −3620.00 −1.23333
\(206\) 224.000 0.0757613
\(207\) 0 0
\(208\) 208.000 0.0693375
\(209\) 800.000 0.264771
\(210\) 0 0
\(211\) −2004.00 −0.653844 −0.326922 0.945051i \(-0.606011\pi\)
−0.326922 + 0.945051i \(0.606011\pi\)
\(212\) −1528.00 −0.495016
\(213\) 0 0
\(214\) 744.000 0.237658
\(215\) −760.000 −0.241077
\(216\) 0 0
\(217\) 1472.00 0.460488
\(218\) 1868.00 0.580353
\(219\) 0 0
\(220\) 1600.00 0.490327
\(221\) −1690.00 −0.514397
\(222\) 0 0
\(223\) −5608.00 −1.68403 −0.842017 0.539451i \(-0.818632\pi\)
−0.842017 + 0.539451i \(0.818632\pi\)
\(224\) −256.000 −0.0763604
\(225\) 0 0
\(226\) −3828.00 −1.12670
\(227\) 1928.00 0.563726 0.281863 0.959455i \(-0.409048\pi\)
0.281863 + 0.959455i \(0.409048\pi\)
\(228\) 0 0
\(229\) −3938.00 −1.13638 −0.568189 0.822898i \(-0.692356\pi\)
−0.568189 + 0.822898i \(0.692356\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 144.000 0.0407503
\(233\) −2562.00 −0.720353 −0.360176 0.932884i \(-0.617283\pi\)
−0.360176 + 0.932884i \(0.617283\pi\)
\(234\) 0 0
\(235\) −4520.00 −1.25469
\(236\) −1856.00 −0.511929
\(237\) 0 0
\(238\) 2080.00 0.566497
\(239\) −7164.00 −1.93891 −0.969457 0.245260i \(-0.921127\pi\)
−0.969457 + 0.245260i \(0.921127\pi\)
\(240\) 0 0
\(241\) −6182.00 −1.65236 −0.826178 0.563410i \(-0.809489\pi\)
−0.826178 + 0.563410i \(0.809489\pi\)
\(242\) 538.000 0.142909
\(243\) 0 0
\(244\) 1432.00 0.375715
\(245\) 2790.00 0.727537
\(246\) 0 0
\(247\) −260.000 −0.0669773
\(248\) −1472.00 −0.376904
\(249\) 0 0
\(250\) 3000.00 0.758947
\(251\) 1396.00 0.351055 0.175527 0.984475i \(-0.443837\pi\)
0.175527 + 0.984475i \(0.443837\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2592.00 0.640301
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −6906.00 −1.67620 −0.838102 0.545514i \(-0.816335\pi\)
−0.838102 + 0.545514i \(0.816335\pi\)
\(258\) 0 0
\(259\) 592.000 0.142027
\(260\) −520.000 −0.124035
\(261\) 0 0
\(262\) 1784.00 0.420671
\(263\) 6848.00 1.60557 0.802787 0.596266i \(-0.203350\pi\)
0.802787 + 0.596266i \(0.203350\pi\)
\(264\) 0 0
\(265\) 3820.00 0.885512
\(266\) 320.000 0.0737611
\(267\) 0 0
\(268\) −2800.00 −0.638199
\(269\) 6034.00 1.36766 0.683828 0.729643i \(-0.260314\pi\)
0.683828 + 0.729643i \(0.260314\pi\)
\(270\) 0 0
\(271\) 4832.00 1.08311 0.541556 0.840665i \(-0.317836\pi\)
0.541556 + 0.840665i \(0.317836\pi\)
\(272\) −2080.00 −0.463671
\(273\) 0 0
\(274\) −4652.00 −1.02568
\(275\) 1000.00 0.219281
\(276\) 0 0
\(277\) −4082.00 −0.885428 −0.442714 0.896663i \(-0.645984\pi\)
−0.442714 + 0.896663i \(0.645984\pi\)
\(278\) 3864.00 0.833623
\(279\) 0 0
\(280\) 640.000 0.136598
\(281\) −3350.00 −0.711189 −0.355595 0.934640i \(-0.615722\pi\)
−0.355595 + 0.934640i \(0.615722\pi\)
\(282\) 0 0
\(283\) 7796.00 1.63754 0.818770 0.574121i \(-0.194656\pi\)
0.818770 + 0.574121i \(0.194656\pi\)
\(284\) 2992.00 0.625150
\(285\) 0 0
\(286\) −1040.00 −0.215023
\(287\) −2896.00 −0.595629
\(288\) 0 0
\(289\) 11987.0 2.43985
\(290\) −360.000 −0.0728963
\(291\) 0 0
\(292\) 4232.00 0.848147
\(293\) −3922.00 −0.781999 −0.390999 0.920391i \(-0.627871\pi\)
−0.390999 + 0.920391i \(0.627871\pi\)
\(294\) 0 0
\(295\) 4640.00 0.915767
\(296\) −592.000 −0.116248
\(297\) 0 0
\(298\) −1764.00 −0.342905
\(299\) 0 0
\(300\) 0 0
\(301\) −608.000 −0.116427
\(302\) −3552.00 −0.676803
\(303\) 0 0
\(304\) −320.000 −0.0603726
\(305\) −3580.00 −0.672099
\(306\) 0 0
\(307\) 5956.00 1.10725 0.553627 0.832765i \(-0.313243\pi\)
0.553627 + 0.832765i \(0.313243\pi\)
\(308\) 1280.00 0.236801
\(309\) 0 0
\(310\) 3680.00 0.674226
\(311\) −2352.00 −0.428841 −0.214421 0.976741i \(-0.568786\pi\)
−0.214421 + 0.976741i \(0.568786\pi\)
\(312\) 0 0
\(313\) 8442.00 1.52450 0.762252 0.647280i \(-0.224093\pi\)
0.762252 + 0.647280i \(0.224093\pi\)
\(314\) −4820.00 −0.866269
\(315\) 0 0
\(316\) −3904.00 −0.694991
\(317\) 5550.00 0.983341 0.491670 0.870781i \(-0.336386\pi\)
0.491670 + 0.870781i \(0.336386\pi\)
\(318\) 0 0
\(319\) −720.000 −0.126371
\(320\) −640.000 −0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 2600.00 0.447888
\(324\) 0 0
\(325\) −325.000 −0.0554700
\(326\) 6424.00 1.09139
\(327\) 0 0
\(328\) 2896.00 0.487515
\(329\) −3616.00 −0.605947
\(330\) 0 0
\(331\) 140.000 0.0232480 0.0116240 0.999932i \(-0.496300\pi\)
0.0116240 + 0.999932i \(0.496300\pi\)
\(332\) 4032.00 0.666520
\(333\) 0 0
\(334\) −3336.00 −0.546520
\(335\) 7000.00 1.14164
\(336\) 0 0
\(337\) −6174.00 −0.997980 −0.498990 0.866608i \(-0.666296\pi\)
−0.498990 + 0.866608i \(0.666296\pi\)
\(338\) 338.000 0.0543928
\(339\) 0 0
\(340\) 5200.00 0.829440
\(341\) 7360.00 1.16882
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 608.000 0.0952941
\(345\) 0 0
\(346\) −7196.00 −1.11809
\(347\) 2988.00 0.462260 0.231130 0.972923i \(-0.425758\pi\)
0.231130 + 0.972923i \(0.425758\pi\)
\(348\) 0 0
\(349\) −162.000 −0.0248472 −0.0124236 0.999923i \(-0.503955\pi\)
−0.0124236 + 0.999923i \(0.503955\pi\)
\(350\) 400.000 0.0610883
\(351\) 0 0
\(352\) −1280.00 −0.193819
\(353\) 10754.0 1.62147 0.810733 0.585416i \(-0.199069\pi\)
0.810733 + 0.585416i \(0.199069\pi\)
\(354\) 0 0
\(355\) −7480.00 −1.11830
\(356\) 1544.00 0.229865
\(357\) 0 0
\(358\) −2136.00 −0.315338
\(359\) −3588.00 −0.527486 −0.263743 0.964593i \(-0.584957\pi\)
−0.263743 + 0.964593i \(0.584957\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) −9572.00 −1.38976
\(363\) 0 0
\(364\) −416.000 −0.0599020
\(365\) −10580.0 −1.51721
\(366\) 0 0
\(367\) 11272.0 1.60325 0.801626 0.597826i \(-0.203968\pi\)
0.801626 + 0.597826i \(0.203968\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 1480.00 0.207950
\(371\) 3056.00 0.427654
\(372\) 0 0
\(373\) −10914.0 −1.51503 −0.757514 0.652819i \(-0.773586\pi\)
−0.757514 + 0.652819i \(0.773586\pi\)
\(374\) 10400.0 1.43789
\(375\) 0 0
\(376\) 3616.00 0.495960
\(377\) 234.000 0.0319671
\(378\) 0 0
\(379\) 8100.00 1.09781 0.548904 0.835886i \(-0.315045\pi\)
0.548904 + 0.835886i \(0.315045\pi\)
\(380\) 800.000 0.107998
\(381\) 0 0
\(382\) 2624.00 0.351454
\(383\) −6180.00 −0.824499 −0.412250 0.911071i \(-0.635257\pi\)
−0.412250 + 0.911071i \(0.635257\pi\)
\(384\) 0 0
\(385\) −3200.00 −0.423603
\(386\) −700.000 −0.0923033
\(387\) 0 0
\(388\) −2456.00 −0.321352
\(389\) 7522.00 0.980413 0.490206 0.871606i \(-0.336921\pi\)
0.490206 + 0.871606i \(0.336921\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2232.00 −0.287584
\(393\) 0 0
\(394\) 684.000 0.0874605
\(395\) 9760.00 1.24324
\(396\) 0 0
\(397\) 6078.00 0.768378 0.384189 0.923254i \(-0.374481\pi\)
0.384189 + 0.923254i \(0.374481\pi\)
\(398\) −6736.00 −0.848355
\(399\) 0 0
\(400\) −400.000 −0.0500000
\(401\) −1830.00 −0.227895 −0.113947 0.993487i \(-0.536350\pi\)
−0.113947 + 0.993487i \(0.536350\pi\)
\(402\) 0 0
\(403\) −2392.00 −0.295668
\(404\) −2072.00 −0.255163
\(405\) 0 0
\(406\) −288.000 −0.0352049
\(407\) 2960.00 0.360496
\(408\) 0 0
\(409\) 12434.0 1.50323 0.751616 0.659601i \(-0.229275\pi\)
0.751616 + 0.659601i \(0.229275\pi\)
\(410\) −7240.00 −0.872093
\(411\) 0 0
\(412\) 448.000 0.0535713
\(413\) 3712.00 0.442265
\(414\) 0 0
\(415\) −10080.0 −1.19231
\(416\) 416.000 0.0490290
\(417\) 0 0
\(418\) 1600.00 0.187221
\(419\) 14188.0 1.65425 0.827123 0.562021i \(-0.189976\pi\)
0.827123 + 0.562021i \(0.189976\pi\)
\(420\) 0 0
\(421\) 8638.00 0.999977 0.499989 0.866032i \(-0.333338\pi\)
0.499989 + 0.866032i \(0.333338\pi\)
\(422\) −4008.00 −0.462337
\(423\) 0 0
\(424\) −3056.00 −0.350029
\(425\) 3250.00 0.370937
\(426\) 0 0
\(427\) −2864.00 −0.324587
\(428\) 1488.00 0.168050
\(429\) 0 0
\(430\) −1520.00 −0.170467
\(431\) −4292.00 −0.479671 −0.239836 0.970813i \(-0.577094\pi\)
−0.239836 + 0.970813i \(0.577094\pi\)
\(432\) 0 0
\(433\) −5982.00 −0.663918 −0.331959 0.943294i \(-0.607710\pi\)
−0.331959 + 0.943294i \(0.607710\pi\)
\(434\) 2944.00 0.325614
\(435\) 0 0
\(436\) 3736.00 0.410371
\(437\) 0 0
\(438\) 0 0
\(439\) 256.000 0.0278319 0.0139160 0.999903i \(-0.495570\pi\)
0.0139160 + 0.999903i \(0.495570\pi\)
\(440\) 3200.00 0.346714
\(441\) 0 0
\(442\) −3380.00 −0.363733
\(443\) −12556.0 −1.34662 −0.673311 0.739359i \(-0.735128\pi\)
−0.673311 + 0.739359i \(0.735128\pi\)
\(444\) 0 0
\(445\) −3860.00 −0.411194
\(446\) −11216.0 −1.19079
\(447\) 0 0
\(448\) −512.000 −0.0539949
\(449\) −5574.00 −0.585865 −0.292932 0.956133i \(-0.594631\pi\)
−0.292932 + 0.956133i \(0.594631\pi\)
\(450\) 0 0
\(451\) −14480.0 −1.51183
\(452\) −7656.00 −0.796699
\(453\) 0 0
\(454\) 3856.00 0.398615
\(455\) 1040.00 0.107156
\(456\) 0 0
\(457\) 1266.00 0.129586 0.0647932 0.997899i \(-0.479361\pi\)
0.0647932 + 0.997899i \(0.479361\pi\)
\(458\) −7876.00 −0.803540
\(459\) 0 0
\(460\) 0 0
\(461\) −7554.00 −0.763178 −0.381589 0.924332i \(-0.624623\pi\)
−0.381589 + 0.924332i \(0.624623\pi\)
\(462\) 0 0
\(463\) −6752.00 −0.677737 −0.338868 0.940834i \(-0.610044\pi\)
−0.338868 + 0.940834i \(0.610044\pi\)
\(464\) 288.000 0.0288148
\(465\) 0 0
\(466\) −5124.00 −0.509366
\(467\) −7924.00 −0.785180 −0.392590 0.919714i \(-0.628421\pi\)
−0.392590 + 0.919714i \(0.628421\pi\)
\(468\) 0 0
\(469\) 5600.00 0.551352
\(470\) −9040.00 −0.887200
\(471\) 0 0
\(472\) −3712.00 −0.361989
\(473\) −3040.00 −0.295517
\(474\) 0 0
\(475\) 500.000 0.0482980
\(476\) 4160.00 0.400574
\(477\) 0 0
\(478\) −14328.0 −1.37102
\(479\) 11084.0 1.05729 0.528644 0.848844i \(-0.322701\pi\)
0.528644 + 0.848844i \(0.322701\pi\)
\(480\) 0 0
\(481\) −962.000 −0.0911922
\(482\) −12364.0 −1.16839
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) 6140.00 0.574852
\(486\) 0 0
\(487\) 4432.00 0.412388 0.206194 0.978511i \(-0.433892\pi\)
0.206194 + 0.978511i \(0.433892\pi\)
\(488\) 2864.00 0.265670
\(489\) 0 0
\(490\) 5580.00 0.514446
\(491\) 1140.00 0.104781 0.0523905 0.998627i \(-0.483316\pi\)
0.0523905 + 0.998627i \(0.483316\pi\)
\(492\) 0 0
\(493\) −2340.00 −0.213769
\(494\) −520.000 −0.0473601
\(495\) 0 0
\(496\) −2944.00 −0.266511
\(497\) −5984.00 −0.540079
\(498\) 0 0
\(499\) 1764.00 0.158251 0.0791257 0.996865i \(-0.474787\pi\)
0.0791257 + 0.996865i \(0.474787\pi\)
\(500\) 6000.00 0.536656
\(501\) 0 0
\(502\) 2792.00 0.248233
\(503\) −16976.0 −1.50482 −0.752408 0.658697i \(-0.771108\pi\)
−0.752408 + 0.658697i \(0.771108\pi\)
\(504\) 0 0
\(505\) 5180.00 0.456449
\(506\) 0 0
\(507\) 0 0
\(508\) 5184.00 0.452761
\(509\) −9474.00 −0.825005 −0.412503 0.910956i \(-0.635345\pi\)
−0.412503 + 0.910956i \(0.635345\pi\)
\(510\) 0 0
\(511\) −8464.00 −0.732731
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −13812.0 −1.18526
\(515\) −1120.00 −0.0958313
\(516\) 0 0
\(517\) −18080.0 −1.53802
\(518\) 1184.00 0.100429
\(519\) 0 0
\(520\) −1040.00 −0.0877058
\(521\) −14114.0 −1.18684 −0.593422 0.804892i \(-0.702223\pi\)
−0.593422 + 0.804892i \(0.702223\pi\)
\(522\) 0 0
\(523\) 20284.0 1.69590 0.847952 0.530074i \(-0.177836\pi\)
0.847952 + 0.530074i \(0.177836\pi\)
\(524\) 3568.00 0.297460
\(525\) 0 0
\(526\) 13696.0 1.13531
\(527\) 23920.0 1.97718
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 7640.00 0.626152
\(531\) 0 0
\(532\) 640.000 0.0521570
\(533\) 4706.00 0.382438
\(534\) 0 0
\(535\) −3720.00 −0.300616
\(536\) −5600.00 −0.451275
\(537\) 0 0
\(538\) 12068.0 0.967079
\(539\) 11160.0 0.891828
\(540\) 0 0
\(541\) −14362.0 −1.14135 −0.570675 0.821176i \(-0.693318\pi\)
−0.570675 + 0.821176i \(0.693318\pi\)
\(542\) 9664.00 0.765875
\(543\) 0 0
\(544\) −4160.00 −0.327865
\(545\) −9340.00 −0.734095
\(546\) 0 0
\(547\) −20956.0 −1.63805 −0.819025 0.573757i \(-0.805485\pi\)
−0.819025 + 0.573757i \(0.805485\pi\)
\(548\) −9304.00 −0.725269
\(549\) 0 0
\(550\) 2000.00 0.155055
\(551\) −360.000 −0.0278340
\(552\) 0 0
\(553\) 7808.00 0.600416
\(554\) −8164.00 −0.626092
\(555\) 0 0
\(556\) 7728.00 0.589461
\(557\) 4134.00 0.314476 0.157238 0.987561i \(-0.449741\pi\)
0.157238 + 0.987561i \(0.449741\pi\)
\(558\) 0 0
\(559\) 988.000 0.0747548
\(560\) 1280.00 0.0965891
\(561\) 0 0
\(562\) −6700.00 −0.502887
\(563\) 16228.0 1.21479 0.607397 0.794399i \(-0.292214\pi\)
0.607397 + 0.794399i \(0.292214\pi\)
\(564\) 0 0
\(565\) 19140.0 1.42518
\(566\) 15592.0 1.15792
\(567\) 0 0
\(568\) 5984.00 0.442048
\(569\) −2514.00 −0.185224 −0.0926119 0.995702i \(-0.529522\pi\)
−0.0926119 + 0.995702i \(0.529522\pi\)
\(570\) 0 0
\(571\) −11612.0 −0.851046 −0.425523 0.904948i \(-0.639910\pi\)
−0.425523 + 0.904948i \(0.639910\pi\)
\(572\) −2080.00 −0.152044
\(573\) 0 0
\(574\) −5792.00 −0.421173
\(575\) 0 0
\(576\) 0 0
\(577\) 6354.00 0.458441 0.229221 0.973375i \(-0.426382\pi\)
0.229221 + 0.973375i \(0.426382\pi\)
\(578\) 23974.0 1.72524
\(579\) 0 0
\(580\) −720.000 −0.0515455
\(581\) −8064.00 −0.575819
\(582\) 0 0
\(583\) 15280.0 1.08548
\(584\) 8464.00 0.599731
\(585\) 0 0
\(586\) −7844.00 −0.552957
\(587\) 13240.0 0.930960 0.465480 0.885059i \(-0.345882\pi\)
0.465480 + 0.885059i \(0.345882\pi\)
\(588\) 0 0
\(589\) 3680.00 0.257439
\(590\) 9280.00 0.647545
\(591\) 0 0
\(592\) −1184.00 −0.0821995
\(593\) 1146.00 0.0793602 0.0396801 0.999212i \(-0.487366\pi\)
0.0396801 + 0.999212i \(0.487366\pi\)
\(594\) 0 0
\(595\) −10400.0 −0.716569
\(596\) −3528.00 −0.242471
\(597\) 0 0
\(598\) 0 0
\(599\) −10464.0 −0.713769 −0.356884 0.934149i \(-0.616161\pi\)
−0.356884 + 0.934149i \(0.616161\pi\)
\(600\) 0 0
\(601\) 6650.00 0.451346 0.225673 0.974203i \(-0.427542\pi\)
0.225673 + 0.974203i \(0.427542\pi\)
\(602\) −1216.00 −0.0823263
\(603\) 0 0
\(604\) −7104.00 −0.478572
\(605\) −2690.00 −0.180767
\(606\) 0 0
\(607\) −6664.00 −0.445607 −0.222803 0.974863i \(-0.571521\pi\)
−0.222803 + 0.974863i \(0.571521\pi\)
\(608\) −640.000 −0.0426898
\(609\) 0 0
\(610\) −7160.00 −0.475246
\(611\) 5876.00 0.389063
\(612\) 0 0
\(613\) 2134.00 0.140606 0.0703030 0.997526i \(-0.477603\pi\)
0.0703030 + 0.997526i \(0.477603\pi\)
\(614\) 11912.0 0.782947
\(615\) 0 0
\(616\) 2560.00 0.167444
\(617\) 714.000 0.0465876 0.0232938 0.999729i \(-0.492585\pi\)
0.0232938 + 0.999729i \(0.492585\pi\)
\(618\) 0 0
\(619\) 29228.0 1.89786 0.948928 0.315494i \(-0.102170\pi\)
0.948928 + 0.315494i \(0.102170\pi\)
\(620\) 7360.00 0.476750
\(621\) 0 0
\(622\) −4704.00 −0.303237
\(623\) −3088.00 −0.198584
\(624\) 0 0
\(625\) −11875.0 −0.760000
\(626\) 16884.0 1.07799
\(627\) 0 0
\(628\) −9640.00 −0.612544
\(629\) 9620.00 0.609816
\(630\) 0 0
\(631\) −13536.0 −0.853977 −0.426989 0.904257i \(-0.640426\pi\)
−0.426989 + 0.904257i \(0.640426\pi\)
\(632\) −7808.00 −0.491433
\(633\) 0 0
\(634\) 11100.0 0.695327
\(635\) −12960.0 −0.809924
\(636\) 0 0
\(637\) −3627.00 −0.225600
\(638\) −1440.00 −0.0893576
\(639\) 0 0
\(640\) −1280.00 −0.0790569
\(641\) −17218.0 −1.06095 −0.530476 0.847700i \(-0.677987\pi\)
−0.530476 + 0.847700i \(0.677987\pi\)
\(642\) 0 0
\(643\) 15044.0 0.922671 0.461335 0.887226i \(-0.347370\pi\)
0.461335 + 0.887226i \(0.347370\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5200.00 0.316705
\(647\) −25176.0 −1.52978 −0.764892 0.644158i \(-0.777208\pi\)
−0.764892 + 0.644158i \(0.777208\pi\)
\(648\) 0 0
\(649\) 18560.0 1.12256
\(650\) −650.000 −0.0392232
\(651\) 0 0
\(652\) 12848.0 0.771728
\(653\) 16034.0 0.960887 0.480443 0.877026i \(-0.340476\pi\)
0.480443 + 0.877026i \(0.340476\pi\)
\(654\) 0 0
\(655\) −8920.00 −0.532112
\(656\) 5792.00 0.344725
\(657\) 0 0
\(658\) −7232.00 −0.428469
\(659\) −25356.0 −1.49883 −0.749415 0.662100i \(-0.769665\pi\)
−0.749415 + 0.662100i \(0.769665\pi\)
\(660\) 0 0
\(661\) 18310.0 1.07742 0.538711 0.842490i \(-0.318911\pi\)
0.538711 + 0.842490i \(0.318911\pi\)
\(662\) 280.000 0.0164388
\(663\) 0 0
\(664\) 8064.00 0.471301
\(665\) −1600.00 −0.0933013
\(666\) 0 0
\(667\) 0 0
\(668\) −6672.00 −0.386448
\(669\) 0 0
\(670\) 14000.0 0.807264
\(671\) −14320.0 −0.823871
\(672\) 0 0
\(673\) 24802.0 1.42057 0.710287 0.703912i \(-0.248565\pi\)
0.710287 + 0.703912i \(0.248565\pi\)
\(674\) −12348.0 −0.705678
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) 22706.0 1.28901 0.644507 0.764598i \(-0.277063\pi\)
0.644507 + 0.764598i \(0.277063\pi\)
\(678\) 0 0
\(679\) 4912.00 0.277622
\(680\) 10400.0 0.586503
\(681\) 0 0
\(682\) 14720.0 0.826478
\(683\) 14792.0 0.828697 0.414349 0.910118i \(-0.364009\pi\)
0.414349 + 0.910118i \(0.364009\pi\)
\(684\) 0 0
\(685\) 23260.0 1.29740
\(686\) 9952.00 0.553891
\(687\) 0 0
\(688\) 1216.00 0.0673831
\(689\) −4966.00 −0.274586
\(690\) 0 0
\(691\) −1148.00 −0.0632011 −0.0316006 0.999501i \(-0.510060\pi\)
−0.0316006 + 0.999501i \(0.510060\pi\)
\(692\) −14392.0 −0.790609
\(693\) 0 0
\(694\) 5976.00 0.326867
\(695\) −19320.0 −1.05446
\(696\) 0 0
\(697\) −47060.0 −2.55742
\(698\) −324.000 −0.0175696
\(699\) 0 0
\(700\) 800.000 0.0431959
\(701\) −14870.0 −0.801187 −0.400594 0.916256i \(-0.631196\pi\)
−0.400594 + 0.916256i \(0.631196\pi\)
\(702\) 0 0
\(703\) 1480.00 0.0794015
\(704\) −2560.00 −0.137051
\(705\) 0 0
\(706\) 21508.0 1.14655
\(707\) 4144.00 0.220440
\(708\) 0 0
\(709\) −6354.00 −0.336572 −0.168286 0.985738i \(-0.553823\pi\)
−0.168286 + 0.985738i \(0.553823\pi\)
\(710\) −14960.0 −0.790759
\(711\) 0 0
\(712\) 3088.00 0.162539
\(713\) 0 0
\(714\) 0 0
\(715\) 5200.00 0.271985
\(716\) −4272.00 −0.222978
\(717\) 0 0
\(718\) −7176.00 −0.372989
\(719\) −9288.00 −0.481758 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(720\) 0 0
\(721\) −896.000 −0.0462813
\(722\) −12918.0 −0.665870
\(723\) 0 0
\(724\) −19144.0 −0.982709
\(725\) −450.000 −0.0230518
\(726\) 0 0
\(727\) −21544.0 −1.09907 −0.549534 0.835471i \(-0.685195\pi\)
−0.549534 + 0.835471i \(0.685195\pi\)
\(728\) −832.000 −0.0423571
\(729\) 0 0
\(730\) −21160.0 −1.07283
\(731\) −9880.00 −0.499897
\(732\) 0 0
\(733\) 19990.0 1.00730 0.503648 0.863909i \(-0.331991\pi\)
0.503648 + 0.863909i \(0.331991\pi\)
\(734\) 22544.0 1.13367
\(735\) 0 0
\(736\) 0 0
\(737\) 28000.0 1.39945
\(738\) 0 0
\(739\) 532.000 0.0264816 0.0132408 0.999912i \(-0.495785\pi\)
0.0132408 + 0.999912i \(0.495785\pi\)
\(740\) 2960.00 0.147043
\(741\) 0 0
\(742\) 6112.00 0.302397
\(743\) 25452.0 1.25672 0.628360 0.777922i \(-0.283726\pi\)
0.628360 + 0.777922i \(0.283726\pi\)
\(744\) 0 0
\(745\) 8820.00 0.433745
\(746\) −21828.0 −1.07129
\(747\) 0 0
\(748\) 20800.0 1.01674
\(749\) −2976.00 −0.145181
\(750\) 0 0
\(751\) 6440.00 0.312915 0.156457 0.987685i \(-0.449993\pi\)
0.156457 + 0.987685i \(0.449993\pi\)
\(752\) 7232.00 0.350697
\(753\) 0 0
\(754\) 468.000 0.0226042
\(755\) 17760.0 0.856096
\(756\) 0 0
\(757\) −786.000 −0.0377380 −0.0188690 0.999822i \(-0.506007\pi\)
−0.0188690 + 0.999822i \(0.506007\pi\)
\(758\) 16200.0 0.776267
\(759\) 0 0
\(760\) 1600.00 0.0763659
\(761\) 1498.00 0.0713567 0.0356784 0.999363i \(-0.488641\pi\)
0.0356784 + 0.999363i \(0.488641\pi\)
\(762\) 0 0
\(763\) −7472.00 −0.354528
\(764\) 5248.00 0.248516
\(765\) 0 0
\(766\) −12360.0 −0.583009
\(767\) −6032.00 −0.283967
\(768\) 0 0
\(769\) 14738.0 0.691113 0.345556 0.938398i \(-0.387690\pi\)
0.345556 + 0.938398i \(0.387690\pi\)
\(770\) −6400.00 −0.299532
\(771\) 0 0
\(772\) −1400.00 −0.0652683
\(773\) 3822.00 0.177837 0.0889184 0.996039i \(-0.471659\pi\)
0.0889184 + 0.996039i \(0.471659\pi\)
\(774\) 0 0
\(775\) 4600.00 0.213209
\(776\) −4912.00 −0.227230
\(777\) 0 0
\(778\) 15044.0 0.693256
\(779\) −7240.00 −0.332991
\(780\) 0 0
\(781\) −29920.0 −1.37083
\(782\) 0 0
\(783\) 0 0
\(784\) −4464.00 −0.203353
\(785\) 24100.0 1.09575
\(786\) 0 0
\(787\) −11900.0 −0.538995 −0.269498 0.963001i \(-0.586858\pi\)
−0.269498 + 0.963001i \(0.586858\pi\)
\(788\) 1368.00 0.0618439
\(789\) 0 0
\(790\) 19520.0 0.879102
\(791\) 15312.0 0.688283
\(792\) 0 0
\(793\) 4654.00 0.208409
\(794\) 12156.0 0.543325
\(795\) 0 0
\(796\) −13472.0 −0.599877
\(797\) 21274.0 0.945500 0.472750 0.881197i \(-0.343261\pi\)
0.472750 + 0.881197i \(0.343261\pi\)
\(798\) 0 0
\(799\) −58760.0 −2.60173
\(800\) −800.000 −0.0353553
\(801\) 0 0
\(802\) −3660.00 −0.161146
\(803\) −42320.0 −1.85983
\(804\) 0 0
\(805\) 0 0
\(806\) −4784.00 −0.209069
\(807\) 0 0
\(808\) −4144.00 −0.180427
\(809\) 27566.0 1.19798 0.598992 0.800755i \(-0.295568\pi\)
0.598992 + 0.800755i \(0.295568\pi\)
\(810\) 0 0
\(811\) −11244.0 −0.486844 −0.243422 0.969921i \(-0.578270\pi\)
−0.243422 + 0.969921i \(0.578270\pi\)
\(812\) −576.000 −0.0248936
\(813\) 0 0
\(814\) 5920.00 0.254909
\(815\) −32120.0 −1.38051
\(816\) 0 0
\(817\) −1520.00 −0.0650894
\(818\) 24868.0 1.06295
\(819\) 0 0
\(820\) −14480.0 −0.616663
\(821\) −13554.0 −0.576173 −0.288086 0.957604i \(-0.593019\pi\)
−0.288086 + 0.957604i \(0.593019\pi\)
\(822\) 0 0
\(823\) 14384.0 0.609228 0.304614 0.952476i \(-0.401473\pi\)
0.304614 + 0.952476i \(0.401473\pi\)
\(824\) 896.000 0.0378806
\(825\) 0 0
\(826\) 7424.00 0.312729
\(827\) 2488.00 0.104615 0.0523073 0.998631i \(-0.483342\pi\)
0.0523073 + 0.998631i \(0.483342\pi\)
\(828\) 0 0
\(829\) −20858.0 −0.873858 −0.436929 0.899496i \(-0.643934\pi\)
−0.436929 + 0.899496i \(0.643934\pi\)
\(830\) −20160.0 −0.843089
\(831\) 0 0
\(832\) 832.000 0.0346688
\(833\) 36270.0 1.50862
\(834\) 0 0
\(835\) 16680.0 0.691300
\(836\) 3200.00 0.132386
\(837\) 0 0
\(838\) 28376.0 1.16973
\(839\) −23116.0 −0.951195 −0.475598 0.879663i \(-0.657768\pi\)
−0.475598 + 0.879663i \(0.657768\pi\)
\(840\) 0 0
\(841\) −24065.0 −0.986715
\(842\) 17276.0 0.707091
\(843\) 0 0
\(844\) −8016.00 −0.326922
\(845\) −1690.00 −0.0688021
\(846\) 0 0
\(847\) −2152.00 −0.0873006
\(848\) −6112.00 −0.247508
\(849\) 0 0
\(850\) 6500.00 0.262292
\(851\) 0 0
\(852\) 0 0
\(853\) 934.000 0.0374907 0.0187453 0.999824i \(-0.494033\pi\)
0.0187453 + 0.999824i \(0.494033\pi\)
\(854\) −5728.00 −0.229518
\(855\) 0 0
\(856\) 2976.00 0.118829
\(857\) −12642.0 −0.503900 −0.251950 0.967740i \(-0.581072\pi\)
−0.251950 + 0.967740i \(0.581072\pi\)
\(858\) 0 0
\(859\) −22796.0 −0.905459 −0.452730 0.891648i \(-0.649550\pi\)
−0.452730 + 0.891648i \(0.649550\pi\)
\(860\) −3040.00 −0.120539
\(861\) 0 0
\(862\) −8584.00 −0.339179
\(863\) 76.0000 0.00299776 0.00149888 0.999999i \(-0.499523\pi\)
0.00149888 + 0.999999i \(0.499523\pi\)
\(864\) 0 0
\(865\) 35980.0 1.41429
\(866\) −11964.0 −0.469461
\(867\) 0 0
\(868\) 5888.00 0.230244
\(869\) 39040.0 1.52398
\(870\) 0 0
\(871\) −9100.00 −0.354009
\(872\) 7472.00 0.290176
\(873\) 0 0
\(874\) 0 0
\(875\) −12000.0 −0.463627
\(876\) 0 0
\(877\) −46130.0 −1.77617 −0.888084 0.459681i \(-0.847964\pi\)
−0.888084 + 0.459681i \(0.847964\pi\)
\(878\) 512.000 0.0196801
\(879\) 0 0
\(880\) 6400.00 0.245164
\(881\) −6682.00 −0.255530 −0.127765 0.991804i \(-0.540780\pi\)
−0.127765 + 0.991804i \(0.540780\pi\)
\(882\) 0 0
\(883\) 47404.0 1.80665 0.903325 0.428957i \(-0.141119\pi\)
0.903325 + 0.428957i \(0.141119\pi\)
\(884\) −6760.00 −0.257198
\(885\) 0 0
\(886\) −25112.0 −0.952206
\(887\) −33672.0 −1.27463 −0.637314 0.770604i \(-0.719955\pi\)
−0.637314 + 0.770604i \(0.719955\pi\)
\(888\) 0 0
\(889\) −10368.0 −0.391149
\(890\) −7720.00 −0.290758
\(891\) 0 0
\(892\) −22432.0 −0.842017
\(893\) −9040.00 −0.338759
\(894\) 0 0
\(895\) 10680.0 0.398875
\(896\) −1024.00 −0.0381802
\(897\) 0 0
\(898\) −11148.0 −0.414269
\(899\) −3312.00 −0.122871
\(900\) 0 0
\(901\) 49660.0 1.83620
\(902\) −28960.0 −1.06903
\(903\) 0 0
\(904\) −15312.0 −0.563351
\(905\) 47860.0 1.75792
\(906\) 0 0
\(907\) −14540.0 −0.532296 −0.266148 0.963932i \(-0.585751\pi\)
−0.266148 + 0.963932i \(0.585751\pi\)
\(908\) 7712.00 0.281863
\(909\) 0 0
\(910\) 2080.00 0.0757707
\(911\) 7840.00 0.285127 0.142564 0.989786i \(-0.454465\pi\)
0.142564 + 0.989786i \(0.454465\pi\)
\(912\) 0 0
\(913\) −40320.0 −1.46155
\(914\) 2532.00 0.0916314
\(915\) 0 0
\(916\) −15752.0 −0.568189
\(917\) −7136.00 −0.256981
\(918\) 0 0
\(919\) 47720.0 1.71288 0.856440 0.516246i \(-0.172671\pi\)
0.856440 + 0.516246i \(0.172671\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −15108.0 −0.539648
\(923\) 9724.00 0.346771
\(924\) 0 0
\(925\) 1850.00 0.0657596
\(926\) −13504.0 −0.479232
\(927\) 0 0
\(928\) 576.000 0.0203751
\(929\) −7502.00 −0.264944 −0.132472 0.991187i \(-0.542291\pi\)
−0.132472 + 0.991187i \(0.542291\pi\)
\(930\) 0 0
\(931\) 5580.00 0.196431
\(932\) −10248.0 −0.360176
\(933\) 0 0
\(934\) −15848.0 −0.555206
\(935\) −52000.0 −1.81880
\(936\) 0 0
\(937\) 22058.0 0.769054 0.384527 0.923114i \(-0.374365\pi\)
0.384527 + 0.923114i \(0.374365\pi\)
\(938\) 11200.0 0.389865
\(939\) 0 0
\(940\) −18080.0 −0.627345
\(941\) −23338.0 −0.808498 −0.404249 0.914649i \(-0.632467\pi\)
−0.404249 + 0.914649i \(0.632467\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −7424.00 −0.255965
\(945\) 0 0
\(946\) −6080.00 −0.208962
\(947\) 30488.0 1.04617 0.523087 0.852279i \(-0.324780\pi\)
0.523087 + 0.852279i \(0.324780\pi\)
\(948\) 0 0
\(949\) 13754.0 0.470468
\(950\) 1000.00 0.0341519
\(951\) 0 0
\(952\) 8320.00 0.283249
\(953\) −9522.00 −0.323660 −0.161830 0.986819i \(-0.551740\pi\)
−0.161830 + 0.986819i \(0.551740\pi\)
\(954\) 0 0
\(955\) −13120.0 −0.444558
\(956\) −28656.0 −0.969457
\(957\) 0 0
\(958\) 22168.0 0.747615
\(959\) 18608.0 0.626573
\(960\) 0 0
\(961\) 4065.00 0.136451
\(962\) −1924.00 −0.0644826
\(963\) 0 0
\(964\) −24728.0 −0.826178
\(965\) 3500.00 0.116755
\(966\) 0 0
\(967\) −7616.00 −0.253272 −0.126636 0.991949i \(-0.540418\pi\)
−0.126636 + 0.991949i \(0.540418\pi\)
\(968\) 2152.00 0.0714544
\(969\) 0 0
\(970\) 12280.0 0.406481
\(971\) −51316.0 −1.69599 −0.847996 0.530002i \(-0.822191\pi\)
−0.847996 + 0.530002i \(0.822191\pi\)
\(972\) 0 0
\(973\) −15456.0 −0.509246
\(974\) 8864.00 0.291603
\(975\) 0 0
\(976\) 5728.00 0.187857
\(977\) 48666.0 1.59362 0.796808 0.604232i \(-0.206520\pi\)
0.796808 + 0.604232i \(0.206520\pi\)
\(978\) 0 0
\(979\) −15440.0 −0.504050
\(980\) 11160.0 0.363768
\(981\) 0 0
\(982\) 2280.00 0.0740914
\(983\) −17388.0 −0.564182 −0.282091 0.959388i \(-0.591028\pi\)
−0.282091 + 0.959388i \(0.591028\pi\)
\(984\) 0 0
\(985\) −3420.00 −0.110630
\(986\) −4680.00 −0.151158
\(987\) 0 0
\(988\) −1040.00 −0.0334887
\(989\) 0 0
\(990\) 0 0
\(991\) 11496.0 0.368499 0.184249 0.982880i \(-0.441015\pi\)
0.184249 + 0.982880i \(0.441015\pi\)
\(992\) −5888.00 −0.188452
\(993\) 0 0
\(994\) −11968.0 −0.381893
\(995\) 33680.0 1.07309
\(996\) 0 0
\(997\) 48862.0 1.55213 0.776066 0.630652i \(-0.217212\pi\)
0.776066 + 0.630652i \(0.217212\pi\)
\(998\) 3528.00 0.111901
\(999\) 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 234.4.a.h.1.1 1
3.2 odd 2 78.4.a.c.1.1 1
4.3 odd 2 1872.4.a.d.1.1 1
12.11 even 2 624.4.a.d.1.1 1
15.14 odd 2 1950.4.a.l.1.1 1
24.5 odd 2 2496.4.a.a.1.1 1
24.11 even 2 2496.4.a.j.1.1 1
39.5 even 4 1014.4.b.h.337.2 2
39.8 even 4 1014.4.b.h.337.1 2
39.38 odd 2 1014.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 3.2 odd 2
234.4.a.h.1.1 1 1.1 even 1 trivial
624.4.a.d.1.1 1 12.11 even 2
1014.4.a.j.1.1 1 39.38 odd 2
1014.4.b.h.337.1 2 39.8 even 4
1014.4.b.h.337.2 2 39.5 even 4
1872.4.a.d.1.1 1 4.3 odd 2
1950.4.a.l.1.1 1 15.14 odd 2
2496.4.a.a.1.1 1 24.5 odd 2
2496.4.a.j.1.1 1 24.11 even 2