Properties

Label 490.4.a.o.1.1
Level $490$
Weight $4$
Character 490.1
Self dual yes
Analytic conductor $28.911$
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] = [490,4,Mod(1,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, 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("490.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.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: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 490.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +16.0000 q^{6} +8.00000 q^{8} +37.0000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +16.0000 q^{6} +8.00000 q^{8} +37.0000 q^{9} -10.0000 q^{10} +12.0000 q^{11} +32.0000 q^{12} +58.0000 q^{13} -40.0000 q^{15} +16.0000 q^{16} -66.0000 q^{17} +74.0000 q^{18} +100.000 q^{19} -20.0000 q^{20} +24.0000 q^{22} +132.000 q^{23} +64.0000 q^{24} +25.0000 q^{25} +116.000 q^{26} +80.0000 q^{27} -90.0000 q^{29} -80.0000 q^{30} -152.000 q^{31} +32.0000 q^{32} +96.0000 q^{33} -132.000 q^{34} +148.000 q^{36} -34.0000 q^{37} +200.000 q^{38} +464.000 q^{39} -40.0000 q^{40} +438.000 q^{41} +32.0000 q^{43} +48.0000 q^{44} -185.000 q^{45} +264.000 q^{46} +204.000 q^{47} +128.000 q^{48} +50.0000 q^{50} -528.000 q^{51} +232.000 q^{52} +222.000 q^{53} +160.000 q^{54} -60.0000 q^{55} +800.000 q^{57} -180.000 q^{58} -420.000 q^{59} -160.000 q^{60} -902.000 q^{61} -304.000 q^{62} +64.0000 q^{64} -290.000 q^{65} +192.000 q^{66} -1024.00 q^{67} -264.000 q^{68} +1056.00 q^{69} +432.000 q^{71} +296.000 q^{72} -362.000 q^{73} -68.0000 q^{74} +200.000 q^{75} +400.000 q^{76} +928.000 q^{78} -160.000 q^{79} -80.0000 q^{80} -359.000 q^{81} +876.000 q^{82} -72.0000 q^{83} +330.000 q^{85} +64.0000 q^{86} -720.000 q^{87} +96.0000 q^{88} -810.000 q^{89} -370.000 q^{90} +528.000 q^{92} -1216.00 q^{93} +408.000 q^{94} -500.000 q^{95} +256.000 q^{96} -1106.00 q^{97} +444.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 8.00000 1.53960 0.769800 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 16.0000 1.08866
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 37.0000 1.37037
\(10\) −10.0000 −0.316228
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 32.0000 0.769800
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) −40.0000 −0.688530
\(16\) 16.0000 0.250000
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 74.0000 0.968998
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) 24.0000 0.232583
\(23\) 132.000 1.19669 0.598346 0.801238i \(-0.295825\pi\)
0.598346 + 0.801238i \(0.295825\pi\)
\(24\) 64.0000 0.544331
\(25\) 25.0000 0.200000
\(26\) 116.000 0.874980
\(27\) 80.0000 0.570222
\(28\) 0 0
\(29\) −90.0000 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) −80.0000 −0.486864
\(31\) −152.000 −0.880645 −0.440323 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) 32.0000 0.176777
\(33\) 96.0000 0.506408
\(34\) −132.000 −0.665818
\(35\) 0 0
\(36\) 148.000 0.685185
\(37\) −34.0000 −0.151069 −0.0755347 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 200.000 0.853797
\(39\) 464.000 1.90511
\(40\) −40.0000 −0.158114
\(41\) 438.000 1.66839 0.834196 0.551467i \(-0.185932\pi\)
0.834196 + 0.551467i \(0.185932\pi\)
\(42\) 0 0
\(43\) 32.0000 0.113487 0.0567437 0.998389i \(-0.481928\pi\)
0.0567437 + 0.998389i \(0.481928\pi\)
\(44\) 48.0000 0.164461
\(45\) −185.000 −0.612848
\(46\) 264.000 0.846189
\(47\) 204.000 0.633116 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(48\) 128.000 0.384900
\(49\) 0 0
\(50\) 50.0000 0.141421
\(51\) −528.000 −1.44970
\(52\) 232.000 0.618704
\(53\) 222.000 0.575359 0.287680 0.957727i \(-0.407116\pi\)
0.287680 + 0.957727i \(0.407116\pi\)
\(54\) 160.000 0.403208
\(55\) −60.0000 −0.147098
\(56\) 0 0
\(57\) 800.000 1.85899
\(58\) −180.000 −0.407503
\(59\) −420.000 −0.926769 −0.463384 0.886157i \(-0.653365\pi\)
−0.463384 + 0.886157i \(0.653365\pi\)
\(60\) −160.000 −0.344265
\(61\) −902.000 −1.89327 −0.946633 0.322312i \(-0.895540\pi\)
−0.946633 + 0.322312i \(0.895540\pi\)
\(62\) −304.000 −0.622710
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −290.000 −0.553386
\(66\) 192.000 0.358084
\(67\) −1024.00 −1.86719 −0.933593 0.358334i \(-0.883345\pi\)
−0.933593 + 0.358334i \(0.883345\pi\)
\(68\) −264.000 −0.470804
\(69\) 1056.00 1.84243
\(70\) 0 0
\(71\) 432.000 0.722098 0.361049 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) 296.000 0.484499
\(73\) −362.000 −0.580396 −0.290198 0.956967i \(-0.593721\pi\)
−0.290198 + 0.956967i \(0.593721\pi\)
\(74\) −68.0000 −0.106822
\(75\) 200.000 0.307920
\(76\) 400.000 0.603726
\(77\) 0 0
\(78\) 928.000 1.34712
\(79\) −160.000 −0.227866 −0.113933 0.993488i \(-0.536345\pi\)
−0.113933 + 0.993488i \(0.536345\pi\)
\(80\) −80.0000 −0.111803
\(81\) −359.000 −0.492455
\(82\) 876.000 1.17973
\(83\) −72.0000 −0.0952172 −0.0476086 0.998866i \(-0.515160\pi\)
−0.0476086 + 0.998866i \(0.515160\pi\)
\(84\) 0 0
\(85\) 330.000 0.421100
\(86\) 64.0000 0.0802476
\(87\) −720.000 −0.887266
\(88\) 96.0000 0.116291
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) −370.000 −0.433349
\(91\) 0 0
\(92\) 528.000 0.598346
\(93\) −1216.00 −1.35584
\(94\) 408.000 0.447681
\(95\) −500.000 −0.539989
\(96\) 256.000 0.272166
\(97\) −1106.00 −1.15770 −0.578852 0.815433i \(-0.696499\pi\)
−0.578852 + 0.815433i \(0.696499\pi\)
\(98\) 0 0
\(99\) 444.000 0.450744
\(100\) 100.000 0.100000
\(101\) 258.000 0.254178 0.127089 0.991891i \(-0.459437\pi\)
0.127089 + 0.991891i \(0.459437\pi\)
\(102\) −1056.00 −1.02509
\(103\) 988.000 0.945151 0.472575 0.881290i \(-0.343324\pi\)
0.472575 + 0.881290i \(0.343324\pi\)
\(104\) 464.000 0.437490
\(105\) 0 0
\(106\) 444.000 0.406840
\(107\) −24.0000 −0.0216838 −0.0108419 0.999941i \(-0.503451\pi\)
−0.0108419 + 0.999941i \(0.503451\pi\)
\(108\) 320.000 0.285111
\(109\) 950.000 0.834803 0.417401 0.908722i \(-0.362941\pi\)
0.417401 + 0.908722i \(0.362941\pi\)
\(110\) −120.000 −0.104014
\(111\) −272.000 −0.232586
\(112\) 0 0
\(113\) −1038.00 −0.864131 −0.432066 0.901842i \(-0.642215\pi\)
−0.432066 + 0.901842i \(0.642215\pi\)
\(114\) 1600.00 1.31451
\(115\) −660.000 −0.535177
\(116\) −360.000 −0.288148
\(117\) 2146.00 1.69571
\(118\) −840.000 −0.655324
\(119\) 0 0
\(120\) −320.000 −0.243432
\(121\) −1187.00 −0.891811
\(122\) −1804.00 −1.33874
\(123\) 3504.00 2.56866
\(124\) −608.000 −0.440323
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −124.000 −0.0866395 −0.0433198 0.999061i \(-0.513793\pi\)
−0.0433198 + 0.999061i \(0.513793\pi\)
\(128\) 128.000 0.0883883
\(129\) 256.000 0.174725
\(130\) −580.000 −0.391303
\(131\) −132.000 −0.0880374 −0.0440187 0.999031i \(-0.514016\pi\)
−0.0440187 + 0.999031i \(0.514016\pi\)
\(132\) 384.000 0.253204
\(133\) 0 0
\(134\) −2048.00 −1.32030
\(135\) −400.000 −0.255011
\(136\) −528.000 −0.332909
\(137\) −1254.00 −0.782018 −0.391009 0.920387i \(-0.627874\pi\)
−0.391009 + 0.920387i \(0.627874\pi\)
\(138\) 2112.00 1.30279
\(139\) 2860.00 1.74519 0.872597 0.488440i \(-0.162434\pi\)
0.872597 + 0.488440i \(0.162434\pi\)
\(140\) 0 0
\(141\) 1632.00 0.974746
\(142\) 864.000 0.510600
\(143\) 696.000 0.407010
\(144\) 592.000 0.342593
\(145\) 450.000 0.257727
\(146\) −724.000 −0.410402
\(147\) 0 0
\(148\) −136.000 −0.0755347
\(149\) 750.000 0.412365 0.206183 0.978514i \(-0.433896\pi\)
0.206183 + 0.978514i \(0.433896\pi\)
\(150\) 400.000 0.217732
\(151\) −448.000 −0.241442 −0.120721 0.992686i \(-0.538521\pi\)
−0.120721 + 0.992686i \(0.538521\pi\)
\(152\) 800.000 0.426898
\(153\) −2442.00 −1.29035
\(154\) 0 0
\(155\) 760.000 0.393837
\(156\) 1856.00 0.952557
\(157\) −2246.00 −1.14172 −0.570861 0.821047i \(-0.693390\pi\)
−0.570861 + 0.821047i \(0.693390\pi\)
\(158\) −320.000 −0.161126
\(159\) 1776.00 0.885824
\(160\) −160.000 −0.0790569
\(161\) 0 0
\(162\) −718.000 −0.348219
\(163\) −568.000 −0.272940 −0.136470 0.990644i \(-0.543576\pi\)
−0.136470 + 0.990644i \(0.543576\pi\)
\(164\) 1752.00 0.834196
\(165\) −480.000 −0.226472
\(166\) −144.000 −0.0673287
\(167\) 1524.00 0.706172 0.353086 0.935591i \(-0.385132\pi\)
0.353086 + 0.935591i \(0.385132\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 660.000 0.297763
\(171\) 3700.00 1.65466
\(172\) 128.000 0.0567437
\(173\) −3702.00 −1.62692 −0.813462 0.581618i \(-0.802420\pi\)
−0.813462 + 0.581618i \(0.802420\pi\)
\(174\) −1440.00 −0.627391
\(175\) 0 0
\(176\) 192.000 0.0822304
\(177\) −3360.00 −1.42685
\(178\) −1620.00 −0.682158
\(179\) 3180.00 1.32785 0.663923 0.747801i \(-0.268890\pi\)
0.663923 + 0.747801i \(0.268890\pi\)
\(180\) −740.000 −0.306424
\(181\) 2098.00 0.861564 0.430782 0.902456i \(-0.358238\pi\)
0.430782 + 0.902456i \(0.358238\pi\)
\(182\) 0 0
\(183\) −7216.00 −2.91487
\(184\) 1056.00 0.423094
\(185\) 170.000 0.0675603
\(186\) −2432.00 −0.958725
\(187\) −792.000 −0.309715
\(188\) 816.000 0.316558
\(189\) 0 0
\(190\) −1000.00 −0.381830
\(191\) 4392.00 1.66384 0.831921 0.554894i \(-0.187241\pi\)
0.831921 + 0.554894i \(0.187241\pi\)
\(192\) 512.000 0.192450
\(193\) −2158.00 −0.804851 −0.402425 0.915453i \(-0.631833\pi\)
−0.402425 + 0.915453i \(0.631833\pi\)
\(194\) −2212.00 −0.818620
\(195\) −2320.00 −0.851993
\(196\) 0 0
\(197\) −1074.00 −0.388423 −0.194212 0.980960i \(-0.562215\pi\)
−0.194212 + 0.980960i \(0.562215\pi\)
\(198\) 888.000 0.318724
\(199\) −2840.00 −1.01167 −0.505835 0.862630i \(-0.668815\pi\)
−0.505835 + 0.862630i \(0.668815\pi\)
\(200\) 200.000 0.0707107
\(201\) −8192.00 −2.87472
\(202\) 516.000 0.179731
\(203\) 0 0
\(204\) −2112.00 −0.724851
\(205\) −2190.00 −0.746128
\(206\) 1976.00 0.668323
\(207\) 4884.00 1.63991
\(208\) 928.000 0.309352
\(209\) 1200.00 0.397157
\(210\) 0 0
\(211\) −2668.00 −0.870487 −0.435243 0.900313i \(-0.643338\pi\)
−0.435243 + 0.900313i \(0.643338\pi\)
\(212\) 888.000 0.287680
\(213\) 3456.00 1.11174
\(214\) −48.0000 −0.0153328
\(215\) −160.000 −0.0507531
\(216\) 640.000 0.201604
\(217\) 0 0
\(218\) 1900.00 0.590295
\(219\) −2896.00 −0.893578
\(220\) −240.000 −0.0735491
\(221\) −3828.00 −1.16515
\(222\) −544.000 −0.164463
\(223\) −1772.00 −0.532116 −0.266058 0.963957i \(-0.585721\pi\)
−0.266058 + 0.963957i \(0.585721\pi\)
\(224\) 0 0
\(225\) 925.000 0.274074
\(226\) −2076.00 −0.611033
\(227\) 2784.00 0.814011 0.407006 0.913426i \(-0.366573\pi\)
0.407006 + 0.913426i \(0.366573\pi\)
\(228\) 3200.00 0.929496
\(229\) −350.000 −0.100998 −0.0504992 0.998724i \(-0.516081\pi\)
−0.0504992 + 0.998724i \(0.516081\pi\)
\(230\) −1320.00 −0.378427
\(231\) 0 0
\(232\) −720.000 −0.203751
\(233\) 1962.00 0.551652 0.275826 0.961208i \(-0.411049\pi\)
0.275826 + 0.961208i \(0.411049\pi\)
\(234\) 4292.00 1.19905
\(235\) −1020.00 −0.283138
\(236\) −1680.00 −0.463384
\(237\) −1280.00 −0.350823
\(238\) 0 0
\(239\) −4320.00 −1.16919 −0.584597 0.811324i \(-0.698748\pi\)
−0.584597 + 0.811324i \(0.698748\pi\)
\(240\) −640.000 −0.172133
\(241\) 478.000 0.127762 0.0638811 0.997958i \(-0.479652\pi\)
0.0638811 + 0.997958i \(0.479652\pi\)
\(242\) −2374.00 −0.630605
\(243\) −5032.00 −1.32841
\(244\) −3608.00 −0.946633
\(245\) 0 0
\(246\) 7008.00 1.81632
\(247\) 5800.00 1.49411
\(248\) −1216.00 −0.311355
\(249\) −576.000 −0.146596
\(250\) −250.000 −0.0632456
\(251\) −2652.00 −0.666903 −0.333452 0.942767i \(-0.608213\pi\)
−0.333452 + 0.942767i \(0.608213\pi\)
\(252\) 0 0
\(253\) 1584.00 0.393617
\(254\) −248.000 −0.0612634
\(255\) 2640.00 0.648326
\(256\) 256.000 0.0625000
\(257\) 2334.00 0.566502 0.283251 0.959046i \(-0.408587\pi\)
0.283251 + 0.959046i \(0.408587\pi\)
\(258\) 512.000 0.123549
\(259\) 0 0
\(260\) −1160.00 −0.276693
\(261\) −3330.00 −0.789739
\(262\) −264.000 −0.0622518
\(263\) −3948.00 −0.925643 −0.462822 0.886451i \(-0.653163\pi\)
−0.462822 + 0.886451i \(0.653163\pi\)
\(264\) 768.000 0.179042
\(265\) −1110.00 −0.257309
\(266\) 0 0
\(267\) −6480.00 −1.48528
\(268\) −4096.00 −0.933593
\(269\) −1590.00 −0.360387 −0.180193 0.983631i \(-0.557672\pi\)
−0.180193 + 0.983631i \(0.557672\pi\)
\(270\) −800.000 −0.180320
\(271\) −4952.00 −1.11001 −0.555005 0.831847i \(-0.687284\pi\)
−0.555005 + 0.831847i \(0.687284\pi\)
\(272\) −1056.00 −0.235402
\(273\) 0 0
\(274\) −2508.00 −0.552970
\(275\) 300.000 0.0657843
\(276\) 4224.00 0.921213
\(277\) 1646.00 0.357034 0.178517 0.983937i \(-0.442870\pi\)
0.178517 + 0.983937i \(0.442870\pi\)
\(278\) 5720.00 1.23404
\(279\) −5624.00 −1.20681
\(280\) 0 0
\(281\) −1158.00 −0.245838 −0.122919 0.992417i \(-0.539226\pi\)
−0.122919 + 0.992417i \(0.539226\pi\)
\(282\) 3264.00 0.689250
\(283\) −6992.00 −1.46866 −0.734331 0.678792i \(-0.762504\pi\)
−0.734331 + 0.678792i \(0.762504\pi\)
\(284\) 1728.00 0.361049
\(285\) −4000.00 −0.831367
\(286\) 1392.00 0.287800
\(287\) 0 0
\(288\) 1184.00 0.242250
\(289\) −557.000 −0.113373
\(290\) 900.000 0.182241
\(291\) −8848.00 −1.78240
\(292\) −1448.00 −0.290198
\(293\) 258.000 0.0514421 0.0257210 0.999669i \(-0.491812\pi\)
0.0257210 + 0.999669i \(0.491812\pi\)
\(294\) 0 0
\(295\) 2100.00 0.414463
\(296\) −272.000 −0.0534111
\(297\) 960.000 0.187558
\(298\) 1500.00 0.291586
\(299\) 7656.00 1.48080
\(300\) 800.000 0.153960
\(301\) 0 0
\(302\) −896.000 −0.170725
\(303\) 2064.00 0.391332
\(304\) 1600.00 0.301863
\(305\) 4510.00 0.846695
\(306\) −4884.00 −0.912417
\(307\) 8944.00 1.66274 0.831370 0.555720i \(-0.187557\pi\)
0.831370 + 0.555720i \(0.187557\pi\)
\(308\) 0 0
\(309\) 7904.00 1.45515
\(310\) 1520.00 0.278485
\(311\) −1392.00 −0.253804 −0.126902 0.991915i \(-0.540503\pi\)
−0.126902 + 0.991915i \(0.540503\pi\)
\(312\) 3712.00 0.673560
\(313\) 5878.00 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(314\) −4492.00 −0.807319
\(315\) 0 0
\(316\) −640.000 −0.113933
\(317\) 10326.0 1.82955 0.914773 0.403969i \(-0.132370\pi\)
0.914773 + 0.403969i \(0.132370\pi\)
\(318\) 3552.00 0.626372
\(319\) −1080.00 −0.189556
\(320\) −320.000 −0.0559017
\(321\) −192.000 −0.0333844
\(322\) 0 0
\(323\) −6600.00 −1.13695
\(324\) −1436.00 −0.246228
\(325\) 1450.00 0.247482
\(326\) −1136.00 −0.192998
\(327\) 7600.00 1.28526
\(328\) 3504.00 0.589866
\(329\) 0 0
\(330\) −960.000 −0.160140
\(331\) −4228.00 −0.702090 −0.351045 0.936359i \(-0.614174\pi\)
−0.351045 + 0.936359i \(0.614174\pi\)
\(332\) −288.000 −0.0476086
\(333\) −1258.00 −0.207021
\(334\) 3048.00 0.499339
\(335\) 5120.00 0.835031
\(336\) 0 0
\(337\) 1106.00 0.178776 0.0893882 0.995997i \(-0.471509\pi\)
0.0893882 + 0.995997i \(0.471509\pi\)
\(338\) 2334.00 0.375600
\(339\) −8304.00 −1.33042
\(340\) 1320.00 0.210550
\(341\) −1824.00 −0.289663
\(342\) 7400.00 1.17002
\(343\) 0 0
\(344\) 256.000 0.0401238
\(345\) −5280.00 −0.823958
\(346\) −7404.00 −1.15041
\(347\) 9336.00 1.44433 0.722165 0.691720i \(-0.243147\pi\)
0.722165 + 0.691720i \(0.243147\pi\)
\(348\) −2880.00 −0.443633
\(349\) 11770.0 1.80525 0.902627 0.430424i \(-0.141636\pi\)
0.902627 + 0.430424i \(0.141636\pi\)
\(350\) 0 0
\(351\) 4640.00 0.705598
\(352\) 384.000 0.0581456
\(353\) −8322.00 −1.25477 −0.627387 0.778707i \(-0.715876\pi\)
−0.627387 + 0.778707i \(0.715876\pi\)
\(354\) −6720.00 −1.00894
\(355\) −2160.00 −0.322932
\(356\) −3240.00 −0.482359
\(357\) 0 0
\(358\) 6360.00 0.938929
\(359\) 10680.0 1.57011 0.785054 0.619427i \(-0.212635\pi\)
0.785054 + 0.619427i \(0.212635\pi\)
\(360\) −1480.00 −0.216675
\(361\) 3141.00 0.457938
\(362\) 4196.00 0.609218
\(363\) −9496.00 −1.37303
\(364\) 0 0
\(365\) 1810.00 0.259561
\(366\) −14432.0 −2.06113
\(367\) 5884.00 0.836900 0.418450 0.908240i \(-0.362574\pi\)
0.418450 + 0.908240i \(0.362574\pi\)
\(368\) 2112.00 0.299173
\(369\) 16206.0 2.28632
\(370\) 340.000 0.0477723
\(371\) 0 0
\(372\) −4864.00 −0.677921
\(373\) −2098.00 −0.291234 −0.145617 0.989341i \(-0.546517\pi\)
−0.145617 + 0.989341i \(0.546517\pi\)
\(374\) −1584.00 −0.219002
\(375\) −1000.00 −0.137706
\(376\) 1632.00 0.223840
\(377\) −5220.00 −0.713113
\(378\) 0 0
\(379\) 3860.00 0.523153 0.261576 0.965183i \(-0.415758\pi\)
0.261576 + 0.965183i \(0.415758\pi\)
\(380\) −2000.00 −0.269994
\(381\) −992.000 −0.133390
\(382\) 8784.00 1.17651
\(383\) 9588.00 1.27917 0.639587 0.768718i \(-0.279105\pi\)
0.639587 + 0.768718i \(0.279105\pi\)
\(384\) 1024.00 0.136083
\(385\) 0 0
\(386\) −4316.00 −0.569116
\(387\) 1184.00 0.155520
\(388\) −4424.00 −0.578852
\(389\) −13410.0 −1.74785 −0.873925 0.486060i \(-0.838434\pi\)
−0.873925 + 0.486060i \(0.838434\pi\)
\(390\) −4640.00 −0.602450
\(391\) −8712.00 −1.12682
\(392\) 0 0
\(393\) −1056.00 −0.135542
\(394\) −2148.00 −0.274657
\(395\) 800.000 0.101905
\(396\) 1776.00 0.225372
\(397\) 13114.0 1.65787 0.828933 0.559348i \(-0.188948\pi\)
0.828933 + 0.559348i \(0.188948\pi\)
\(398\) −5680.00 −0.715358
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) −5838.00 −0.727022 −0.363511 0.931590i \(-0.618422\pi\)
−0.363511 + 0.931590i \(0.618422\pi\)
\(402\) −16384.0 −2.03274
\(403\) −8816.00 −1.08972
\(404\) 1032.00 0.127089
\(405\) 1795.00 0.220233
\(406\) 0 0
\(407\) −408.000 −0.0496899
\(408\) −4224.00 −0.512547
\(409\) −9530.00 −1.15215 −0.576074 0.817398i \(-0.695416\pi\)
−0.576074 + 0.817398i \(0.695416\pi\)
\(410\) −4380.00 −0.527592
\(411\) −10032.0 −1.20400
\(412\) 3952.00 0.472575
\(413\) 0 0
\(414\) 9768.00 1.15959
\(415\) 360.000 0.0425824
\(416\) 1856.00 0.218745
\(417\) 22880.0 2.68690
\(418\) 2400.00 0.280832
\(419\) −7260.00 −0.846478 −0.423239 0.906018i \(-0.639107\pi\)
−0.423239 + 0.906018i \(0.639107\pi\)
\(420\) 0 0
\(421\) 12062.0 1.39636 0.698178 0.715924i \(-0.253994\pi\)
0.698178 + 0.715924i \(0.253994\pi\)
\(422\) −5336.00 −0.615527
\(423\) 7548.00 0.867604
\(424\) 1776.00 0.203420
\(425\) −1650.00 −0.188322
\(426\) 6912.00 0.786121
\(427\) 0 0
\(428\) −96.0000 −0.0108419
\(429\) 5568.00 0.626633
\(430\) −320.000 −0.0358878
\(431\) −13608.0 −1.52082 −0.760411 0.649442i \(-0.775002\pi\)
−0.760411 + 0.649442i \(0.775002\pi\)
\(432\) 1280.00 0.142556
\(433\) 3838.00 0.425964 0.212982 0.977056i \(-0.431682\pi\)
0.212982 + 0.977056i \(0.431682\pi\)
\(434\) 0 0
\(435\) 3600.00 0.396797
\(436\) 3800.00 0.417401
\(437\) 13200.0 1.44495
\(438\) −5792.00 −0.631855
\(439\) −7400.00 −0.804516 −0.402258 0.915526i \(-0.631775\pi\)
−0.402258 + 0.915526i \(0.631775\pi\)
\(440\) −480.000 −0.0520071
\(441\) 0 0
\(442\) −7656.00 −0.823889
\(443\) 8352.00 0.895746 0.447873 0.894097i \(-0.352182\pi\)
0.447873 + 0.894097i \(0.352182\pi\)
\(444\) −1088.00 −0.116293
\(445\) 4050.00 0.431435
\(446\) −3544.00 −0.376263
\(447\) 6000.00 0.634878
\(448\) 0 0
\(449\) 10770.0 1.13200 0.566000 0.824405i \(-0.308490\pi\)
0.566000 + 0.824405i \(0.308490\pi\)
\(450\) 1850.00 0.193800
\(451\) 5256.00 0.548770
\(452\) −4152.00 −0.432066
\(453\) −3584.00 −0.371724
\(454\) 5568.00 0.575593
\(455\) 0 0
\(456\) 6400.00 0.657253
\(457\) −6694.00 −0.685191 −0.342595 0.939483i \(-0.611306\pi\)
−0.342595 + 0.939483i \(0.611306\pi\)
\(458\) −700.000 −0.0714167
\(459\) −5280.00 −0.536927
\(460\) −2640.00 −0.267588
\(461\) 3018.00 0.304907 0.152454 0.988311i \(-0.451283\pi\)
0.152454 + 0.988311i \(0.451283\pi\)
\(462\) 0 0
\(463\) 14492.0 1.45464 0.727322 0.686296i \(-0.240765\pi\)
0.727322 + 0.686296i \(0.240765\pi\)
\(464\) −1440.00 −0.144074
\(465\) 6080.00 0.606351
\(466\) 3924.00 0.390077
\(467\) −7776.00 −0.770515 −0.385257 0.922809i \(-0.625887\pi\)
−0.385257 + 0.922809i \(0.625887\pi\)
\(468\) 8584.00 0.847854
\(469\) 0 0
\(470\) −2040.00 −0.200209
\(471\) −17968.0 −1.75780
\(472\) −3360.00 −0.327662
\(473\) 384.000 0.0373284
\(474\) −2560.00 −0.248069
\(475\) 2500.00 0.241490
\(476\) 0 0
\(477\) 8214.00 0.788455
\(478\) −8640.00 −0.826746
\(479\) 13680.0 1.30492 0.652458 0.757825i \(-0.273738\pi\)
0.652458 + 0.757825i \(0.273738\pi\)
\(480\) −1280.00 −0.121716
\(481\) −1972.00 −0.186934
\(482\) 956.000 0.0903415
\(483\) 0 0
\(484\) −4748.00 −0.445905
\(485\) 5530.00 0.517741
\(486\) −10064.0 −0.939326
\(487\) 7916.00 0.736567 0.368284 0.929714i \(-0.379946\pi\)
0.368284 + 0.929714i \(0.379946\pi\)
\(488\) −7216.00 −0.669371
\(489\) −4544.00 −0.420218
\(490\) 0 0
\(491\) 13932.0 1.28053 0.640267 0.768152i \(-0.278824\pi\)
0.640267 + 0.768152i \(0.278824\pi\)
\(492\) 14016.0 1.28433
\(493\) 5940.00 0.542645
\(494\) 11600.0 1.05650
\(495\) −2220.00 −0.201579
\(496\) −2432.00 −0.220161
\(497\) 0 0
\(498\) −1152.00 −0.103659
\(499\) −8260.00 −0.741019 −0.370509 0.928829i \(-0.620817\pi\)
−0.370509 + 0.928829i \(0.620817\pi\)
\(500\) −500.000 −0.0447214
\(501\) 12192.0 1.08722
\(502\) −5304.00 −0.471572
\(503\) 11148.0 0.988200 0.494100 0.869405i \(-0.335498\pi\)
0.494100 + 0.869405i \(0.335498\pi\)
\(504\) 0 0
\(505\) −1290.00 −0.113672
\(506\) 3168.00 0.278330
\(507\) 9336.00 0.817803
\(508\) −496.000 −0.0433198
\(509\) 9690.00 0.843815 0.421907 0.906639i \(-0.361361\pi\)
0.421907 + 0.906639i \(0.361361\pi\)
\(510\) 5280.00 0.458436
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 8000.00 0.688516
\(514\) 4668.00 0.400577
\(515\) −4940.00 −0.422684
\(516\) 1024.00 0.0873626
\(517\) 2448.00 0.208245
\(518\) 0 0
\(519\) −29616.0 −2.50481
\(520\) −2320.00 −0.195651
\(521\) 16038.0 1.34863 0.674316 0.738443i \(-0.264438\pi\)
0.674316 + 0.738443i \(0.264438\pi\)
\(522\) −6660.00 −0.558430
\(523\) −992.000 −0.0829391 −0.0414695 0.999140i \(-0.513204\pi\)
−0.0414695 + 0.999140i \(0.513204\pi\)
\(524\) −528.000 −0.0440187
\(525\) 0 0
\(526\) −7896.00 −0.654528
\(527\) 10032.0 0.829223
\(528\) 1536.00 0.126602
\(529\) 5257.00 0.432070
\(530\) −2220.00 −0.181945
\(531\) −15540.0 −1.27002
\(532\) 0 0
\(533\) 25404.0 2.06448
\(534\) −12960.0 −1.05025
\(535\) 120.000 0.00969729
\(536\) −8192.00 −0.660150
\(537\) 25440.0 2.04435
\(538\) −3180.00 −0.254832
\(539\) 0 0
\(540\) −1600.00 −0.127506
\(541\) 7142.00 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(542\) −9904.00 −0.784895
\(543\) 16784.0 1.32646
\(544\) −2112.00 −0.166455
\(545\) −4750.00 −0.373335
\(546\) 0 0
\(547\) 7616.00 0.595314 0.297657 0.954673i \(-0.403795\pi\)
0.297657 + 0.954673i \(0.403795\pi\)
\(548\) −5016.00 −0.391009
\(549\) −33374.0 −2.59448
\(550\) 600.000 0.0465165
\(551\) −9000.00 −0.695849
\(552\) 8448.00 0.651396
\(553\) 0 0
\(554\) 3292.00 0.252462
\(555\) 1360.00 0.104016
\(556\) 11440.0 0.872597
\(557\) −10314.0 −0.784593 −0.392296 0.919839i \(-0.628319\pi\)
−0.392296 + 0.919839i \(0.628319\pi\)
\(558\) −11248.0 −0.853344
\(559\) 1856.00 0.140430
\(560\) 0 0
\(561\) −6336.00 −0.476838
\(562\) −2316.00 −0.173834
\(563\) 7128.00 0.533587 0.266793 0.963754i \(-0.414036\pi\)
0.266793 + 0.963754i \(0.414036\pi\)
\(564\) 6528.00 0.487373
\(565\) 5190.00 0.386451
\(566\) −13984.0 −1.03850
\(567\) 0 0
\(568\) 3456.00 0.255300
\(569\) 2010.00 0.148091 0.0740453 0.997255i \(-0.476409\pi\)
0.0740453 + 0.997255i \(0.476409\pi\)
\(570\) −8000.00 −0.587865
\(571\) −23188.0 −1.69945 −0.849726 0.527224i \(-0.823233\pi\)
−0.849726 + 0.527224i \(0.823233\pi\)
\(572\) 2784.00 0.203505
\(573\) 35136.0 2.56165
\(574\) 0 0
\(575\) 3300.00 0.239338
\(576\) 2368.00 0.171296
\(577\) −22466.0 −1.62092 −0.810461 0.585793i \(-0.800783\pi\)
−0.810461 + 0.585793i \(0.800783\pi\)
\(578\) −1114.00 −0.0801666
\(579\) −17264.0 −1.23915
\(580\) 1800.00 0.128864
\(581\) 0 0
\(582\) −17696.0 −1.26035
\(583\) 2664.00 0.189248
\(584\) −2896.00 −0.205201
\(585\) −10730.0 −0.758343
\(586\) 516.000 0.0363750
\(587\) −22776.0 −1.60148 −0.800738 0.599015i \(-0.795559\pi\)
−0.800738 + 0.599015i \(0.795559\pi\)
\(588\) 0 0
\(589\) −15200.0 −1.06334
\(590\) 4200.00 0.293070
\(591\) −8592.00 −0.598016
\(592\) −544.000 −0.0377673
\(593\) 21198.0 1.46796 0.733978 0.679174i \(-0.237662\pi\)
0.733978 + 0.679174i \(0.237662\pi\)
\(594\) 1920.00 0.132624
\(595\) 0 0
\(596\) 3000.00 0.206183
\(597\) −22720.0 −1.55757
\(598\) 15312.0 1.04708
\(599\) 15960.0 1.08866 0.544330 0.838871i \(-0.316784\pi\)
0.544330 + 0.838871i \(0.316784\pi\)
\(600\) 1600.00 0.108866
\(601\) −5882.00 −0.399221 −0.199610 0.979875i \(-0.563968\pi\)
−0.199610 + 0.979875i \(0.563968\pi\)
\(602\) 0 0
\(603\) −37888.0 −2.55874
\(604\) −1792.00 −0.120721
\(605\) 5935.00 0.398830
\(606\) 4128.00 0.276714
\(607\) −8516.00 −0.569446 −0.284723 0.958610i \(-0.591902\pi\)
−0.284723 + 0.958610i \(0.591902\pi\)
\(608\) 3200.00 0.213449
\(609\) 0 0
\(610\) 9020.00 0.598703
\(611\) 11832.0 0.783423
\(612\) −9768.00 −0.645176
\(613\) 8462.00 0.557548 0.278774 0.960357i \(-0.410072\pi\)
0.278774 + 0.960357i \(0.410072\pi\)
\(614\) 17888.0 1.17573
\(615\) −17520.0 −1.14874
\(616\) 0 0
\(617\) −11094.0 −0.723870 −0.361935 0.932203i \(-0.617884\pi\)
−0.361935 + 0.932203i \(0.617884\pi\)
\(618\) 15808.0 1.02895
\(619\) −2180.00 −0.141553 −0.0707767 0.997492i \(-0.522548\pi\)
−0.0707767 + 0.997492i \(0.522548\pi\)
\(620\) 3040.00 0.196918
\(621\) 10560.0 0.682380
\(622\) −2784.00 −0.179467
\(623\) 0 0
\(624\) 7424.00 0.476279
\(625\) 625.000 0.0400000
\(626\) 11756.0 0.750582
\(627\) 9600.00 0.611463
\(628\) −8984.00 −0.570861
\(629\) 2244.00 0.142248
\(630\) 0 0
\(631\) −26848.0 −1.69382 −0.846911 0.531734i \(-0.821541\pi\)
−0.846911 + 0.531734i \(0.821541\pi\)
\(632\) −1280.00 −0.0805628
\(633\) −21344.0 −1.34020
\(634\) 20652.0 1.29368
\(635\) 620.000 0.0387464
\(636\) 7104.00 0.442912
\(637\) 0 0
\(638\) −2160.00 −0.134036
\(639\) 15984.0 0.989542
\(640\) −640.000 −0.0395285
\(641\) 26322.0 1.62193 0.810965 0.585095i \(-0.198943\pi\)
0.810965 + 0.585095i \(0.198943\pi\)
\(642\) −384.000 −0.0236063
\(643\) 10168.0 0.623619 0.311809 0.950145i \(-0.399065\pi\)
0.311809 + 0.950145i \(0.399065\pi\)
\(644\) 0 0
\(645\) −1280.00 −0.0781395
\(646\) −13200.0 −0.803943
\(647\) 23604.0 1.43426 0.717132 0.696937i \(-0.245454\pi\)
0.717132 + 0.696937i \(0.245454\pi\)
\(648\) −2872.00 −0.174109
\(649\) −5040.00 −0.304834
\(650\) 2900.00 0.174996
\(651\) 0 0
\(652\) −2272.00 −0.136470
\(653\) 16422.0 0.984139 0.492069 0.870556i \(-0.336241\pi\)
0.492069 + 0.870556i \(0.336241\pi\)
\(654\) 15200.0 0.908818
\(655\) 660.000 0.0393715
\(656\) 7008.00 0.417098
\(657\) −13394.0 −0.795357
\(658\) 0 0
\(659\) −26100.0 −1.54281 −0.771405 0.636345i \(-0.780446\pi\)
−0.771405 + 0.636345i \(0.780446\pi\)
\(660\) −1920.00 −0.113236
\(661\) 3058.00 0.179943 0.0899716 0.995944i \(-0.471322\pi\)
0.0899716 + 0.995944i \(0.471322\pi\)
\(662\) −8456.00 −0.496453
\(663\) −30624.0 −1.79387
\(664\) −576.000 −0.0336644
\(665\) 0 0
\(666\) −2516.00 −0.146386
\(667\) −11880.0 −0.689648
\(668\) 6096.00 0.353086
\(669\) −14176.0 −0.819246
\(670\) 10240.0 0.590456
\(671\) −10824.0 −0.622736
\(672\) 0 0
\(673\) 10802.0 0.618702 0.309351 0.950948i \(-0.399888\pi\)
0.309351 + 0.950948i \(0.399888\pi\)
\(674\) 2212.00 0.126414
\(675\) 2000.00 0.114044
\(676\) 4668.00 0.265589
\(677\) 10674.0 0.605960 0.302980 0.952997i \(-0.402018\pi\)
0.302980 + 0.952997i \(0.402018\pi\)
\(678\) −16608.0 −0.940747
\(679\) 0 0
\(680\) 2640.00 0.148881
\(681\) 22272.0 1.25325
\(682\) −3648.00 −0.204823
\(683\) −28608.0 −1.60272 −0.801358 0.598185i \(-0.795889\pi\)
−0.801358 + 0.598185i \(0.795889\pi\)
\(684\) 14800.0 0.827328
\(685\) 6270.00 0.349729
\(686\) 0 0
\(687\) −2800.00 −0.155497
\(688\) 512.000 0.0283718
\(689\) 12876.0 0.711954
\(690\) −10560.0 −0.582627
\(691\) 2428.00 0.133669 0.0668346 0.997764i \(-0.478710\pi\)
0.0668346 + 0.997764i \(0.478710\pi\)
\(692\) −14808.0 −0.813462
\(693\) 0 0
\(694\) 18672.0 1.02130
\(695\) −14300.0 −0.780475
\(696\) −5760.00 −0.313696
\(697\) −28908.0 −1.57097
\(698\) 23540.0 1.27651
\(699\) 15696.0 0.849324
\(700\) 0 0
\(701\) −6618.00 −0.356574 −0.178287 0.983979i \(-0.557056\pi\)
−0.178287 + 0.983979i \(0.557056\pi\)
\(702\) 9280.00 0.498933
\(703\) −3400.00 −0.182409
\(704\) 768.000 0.0411152
\(705\) −8160.00 −0.435920
\(706\) −16644.0 −0.887259
\(707\) 0 0
\(708\) −13440.0 −0.713427
\(709\) 20510.0 1.08642 0.543208 0.839598i \(-0.317209\pi\)
0.543208 + 0.839598i \(0.317209\pi\)
\(710\) −4320.00 −0.228347
\(711\) −5920.00 −0.312261
\(712\) −6480.00 −0.341079
\(713\) −20064.0 −1.05386
\(714\) 0 0
\(715\) −3480.00 −0.182020
\(716\) 12720.0 0.663923
\(717\) −34560.0 −1.80009
\(718\) 21360.0 1.11023
\(719\) −31680.0 −1.64321 −0.821603 0.570061i \(-0.806920\pi\)
−0.821603 + 0.570061i \(0.806920\pi\)
\(720\) −2960.00 −0.153212
\(721\) 0 0
\(722\) 6282.00 0.323811
\(723\) 3824.00 0.196703
\(724\) 8392.00 0.430782
\(725\) −2250.00 −0.115259
\(726\) −18992.0 −0.970880
\(727\) −13196.0 −0.673195 −0.336597 0.941649i \(-0.609276\pi\)
−0.336597 + 0.941649i \(0.609276\pi\)
\(728\) 0 0
\(729\) −30563.0 −1.55276
\(730\) 3620.00 0.183537
\(731\) −2112.00 −0.106861
\(732\) −28864.0 −1.45744
\(733\) −8102.00 −0.408259 −0.204130 0.978944i \(-0.565436\pi\)
−0.204130 + 0.978944i \(0.565436\pi\)
\(734\) 11768.0 0.591778
\(735\) 0 0
\(736\) 4224.00 0.211547
\(737\) −12288.0 −0.614158
\(738\) 32412.0 1.61667
\(739\) −12580.0 −0.626201 −0.313101 0.949720i \(-0.601368\pi\)
−0.313101 + 0.949720i \(0.601368\pi\)
\(740\) 680.000 0.0337801
\(741\) 46400.0 2.30033
\(742\) 0 0
\(743\) 29892.0 1.47595 0.737975 0.674828i \(-0.235782\pi\)
0.737975 + 0.674828i \(0.235782\pi\)
\(744\) −9728.00 −0.479363
\(745\) −3750.00 −0.184415
\(746\) −4196.00 −0.205934
\(747\) −2664.00 −0.130483
\(748\) −3168.00 −0.154858
\(749\) 0 0
\(750\) −2000.00 −0.0973729
\(751\) −40408.0 −1.96339 −0.981697 0.190450i \(-0.939005\pi\)
−0.981697 + 0.190450i \(0.939005\pi\)
\(752\) 3264.00 0.158279
\(753\) −21216.0 −1.02676
\(754\) −10440.0 −0.504247
\(755\) 2240.00 0.107976
\(756\) 0 0
\(757\) 32366.0 1.55398 0.776990 0.629513i \(-0.216746\pi\)
0.776990 + 0.629513i \(0.216746\pi\)
\(758\) 7720.00 0.369925
\(759\) 12672.0 0.606014
\(760\) −4000.00 −0.190915
\(761\) 17238.0 0.821126 0.410563 0.911832i \(-0.365332\pi\)
0.410563 + 0.911832i \(0.365332\pi\)
\(762\) −1984.00 −0.0943212
\(763\) 0 0
\(764\) 17568.0 0.831921
\(765\) 12210.0 0.577063
\(766\) 19176.0 0.904513
\(767\) −24360.0 −1.14679
\(768\) 2048.00 0.0962250
\(769\) −10850.0 −0.508792 −0.254396 0.967100i \(-0.581877\pi\)
−0.254396 + 0.967100i \(0.581877\pi\)
\(770\) 0 0
\(771\) 18672.0 0.872186
\(772\) −8632.00 −0.402425
\(773\) −9102.00 −0.423514 −0.211757 0.977322i \(-0.567919\pi\)
−0.211757 + 0.977322i \(0.567919\pi\)
\(774\) 2368.00 0.109969
\(775\) −3800.00 −0.176129
\(776\) −8848.00 −0.409310
\(777\) 0 0
\(778\) −26820.0 −1.23592
\(779\) 43800.0 2.01450
\(780\) −9280.00 −0.425997
\(781\) 5184.00 0.237514
\(782\) −17424.0 −0.796779
\(783\) −7200.00 −0.328617
\(784\) 0 0
\(785\) 11230.0 0.510593
\(786\) −2112.00 −0.0958429
\(787\) 25504.0 1.15517 0.577585 0.816330i \(-0.303995\pi\)
0.577585 + 0.816330i \(0.303995\pi\)
\(788\) −4296.00 −0.194212
\(789\) −31584.0 −1.42512
\(790\) 1600.00 0.0720575
\(791\) 0 0
\(792\) 3552.00 0.159362
\(793\) −52316.0 −2.34274
\(794\) 26228.0 1.17229
\(795\) −8880.00 −0.396152
\(796\) −11360.0 −0.505835
\(797\) −14166.0 −0.629593 −0.314796 0.949159i \(-0.601936\pi\)
−0.314796 + 0.949159i \(0.601936\pi\)
\(798\) 0 0
\(799\) −13464.0 −0.596148
\(800\) 800.000 0.0353553
\(801\) −29970.0 −1.32202
\(802\) −11676.0 −0.514082
\(803\) −4344.00 −0.190905
\(804\) −32768.0 −1.43736
\(805\) 0 0
\(806\) −17632.0 −0.770547
\(807\) −12720.0 −0.554852
\(808\) 2064.00 0.0898654
\(809\) 33210.0 1.44327 0.721633 0.692276i \(-0.243392\pi\)
0.721633 + 0.692276i \(0.243392\pi\)
\(810\) 3590.00 0.155728
\(811\) −39212.0 −1.69780 −0.848902 0.528550i \(-0.822736\pi\)
−0.848902 + 0.528550i \(0.822736\pi\)
\(812\) 0 0
\(813\) −39616.0 −1.70897
\(814\) −816.000 −0.0351361
\(815\) 2840.00 0.122062
\(816\) −8448.00 −0.362425
\(817\) 3200.00 0.137030
\(818\) −19060.0 −0.814691
\(819\) 0 0
\(820\) −8760.00 −0.373064
\(821\) 6222.00 0.264494 0.132247 0.991217i \(-0.457781\pi\)
0.132247 + 0.991217i \(0.457781\pi\)
\(822\) −20064.0 −0.851353
\(823\) 31172.0 1.32028 0.660138 0.751144i \(-0.270498\pi\)
0.660138 + 0.751144i \(0.270498\pi\)
\(824\) 7904.00 0.334161
\(825\) 2400.00 0.101282
\(826\) 0 0
\(827\) −264.000 −0.0111006 −0.00555029 0.999985i \(-0.501767\pi\)
−0.00555029 + 0.999985i \(0.501767\pi\)
\(828\) 19536.0 0.819955
\(829\) 29050.0 1.21707 0.608533 0.793528i \(-0.291758\pi\)
0.608533 + 0.793528i \(0.291758\pi\)
\(830\) 720.000 0.0301103
\(831\) 13168.0 0.549691
\(832\) 3712.00 0.154676
\(833\) 0 0
\(834\) 45760.0 1.89993
\(835\) −7620.00 −0.315810
\(836\) 4800.00 0.198578
\(837\) −12160.0 −0.502164
\(838\) −14520.0 −0.598550
\(839\) 21720.0 0.893752 0.446876 0.894596i \(-0.352537\pi\)
0.446876 + 0.894596i \(0.352537\pi\)
\(840\) 0 0
\(841\) −16289.0 −0.667883
\(842\) 24124.0 0.987373
\(843\) −9264.00 −0.378492
\(844\) −10672.0 −0.435243
\(845\) −5835.00 −0.237550
\(846\) 15096.0 0.613488
\(847\) 0 0
\(848\) 3552.00 0.143840
\(849\) −55936.0 −2.26115
\(850\) −3300.00 −0.133164
\(851\) −4488.00 −0.180783
\(852\) 13824.0 0.555871
\(853\) 6658.00 0.267252 0.133626 0.991032i \(-0.457338\pi\)
0.133626 + 0.991032i \(0.457338\pi\)
\(854\) 0 0
\(855\) −18500.0 −0.739984
\(856\) −192.000 −0.00766638
\(857\) 13974.0 0.556993 0.278496 0.960437i \(-0.410164\pi\)
0.278496 + 0.960437i \(0.410164\pi\)
\(858\) 11136.0 0.443096
\(859\) −23780.0 −0.944544 −0.472272 0.881453i \(-0.656566\pi\)
−0.472272 + 0.881453i \(0.656566\pi\)
\(860\) −640.000 −0.0253765
\(861\) 0 0
\(862\) −27216.0 −1.07538
\(863\) −12228.0 −0.482324 −0.241162 0.970485i \(-0.577529\pi\)
−0.241162 + 0.970485i \(0.577529\pi\)
\(864\) 2560.00 0.100802
\(865\) 18510.0 0.727583
\(866\) 7676.00 0.301202
\(867\) −4456.00 −0.174549
\(868\) 0 0
\(869\) −1920.00 −0.0749500
\(870\) 7200.00 0.280578
\(871\) −59392.0 −2.31047
\(872\) 7600.00 0.295147
\(873\) −40922.0 −1.58648
\(874\) 26400.0 1.02173
\(875\) 0 0
\(876\) −11584.0 −0.446789
\(877\) 11606.0 0.446872 0.223436 0.974719i \(-0.428273\pi\)
0.223436 + 0.974719i \(0.428273\pi\)
\(878\) −14800.0 −0.568879
\(879\) 2064.00 0.0792002
\(880\) −960.000 −0.0367745
\(881\) 32958.0 1.26037 0.630183 0.776446i \(-0.282980\pi\)
0.630183 + 0.776446i \(0.282980\pi\)
\(882\) 0 0
\(883\) 8072.00 0.307638 0.153819 0.988099i \(-0.450843\pi\)
0.153819 + 0.988099i \(0.450843\pi\)
\(884\) −15312.0 −0.582577
\(885\) 16800.0 0.638108
\(886\) 16704.0 0.633388
\(887\) −15756.0 −0.596431 −0.298216 0.954498i \(-0.596391\pi\)
−0.298216 + 0.954498i \(0.596391\pi\)
\(888\) −2176.00 −0.0822317
\(889\) 0 0
\(890\) 8100.00 0.305070
\(891\) −4308.00 −0.161979
\(892\) −7088.00 −0.266058
\(893\) 20400.0 0.764457
\(894\) 12000.0 0.448926
\(895\) −15900.0 −0.593831
\(896\) 0 0
\(897\) 61248.0 2.27983
\(898\) 21540.0 0.800444
\(899\) 13680.0 0.507512
\(900\) 3700.00 0.137037
\(901\) −14652.0 −0.541763
\(902\) 10512.0 0.388039
\(903\) 0 0
\(904\) −8304.00 −0.305517
\(905\) −10490.0 −0.385303
\(906\) −7168.00 −0.262849
\(907\) 18776.0 0.687372 0.343686 0.939085i \(-0.388324\pi\)
0.343686 + 0.939085i \(0.388324\pi\)
\(908\) 11136.0 0.407006
\(909\) 9546.00 0.348318
\(910\) 0 0
\(911\) −20568.0 −0.748022 −0.374011 0.927424i \(-0.622018\pi\)
−0.374011 + 0.927424i \(0.622018\pi\)
\(912\) 12800.0 0.464748
\(913\) −864.000 −0.0313190
\(914\) −13388.0 −0.484503
\(915\) 36080.0 1.30357
\(916\) −1400.00 −0.0504992
\(917\) 0 0
\(918\) −10560.0 −0.379664
\(919\) −6280.00 −0.225417 −0.112708 0.993628i \(-0.535953\pi\)
−0.112708 + 0.993628i \(0.535953\pi\)
\(920\) −5280.00 −0.189214
\(921\) 71552.0 2.55996
\(922\) 6036.00 0.215602
\(923\) 25056.0 0.893530
\(924\) 0 0
\(925\) −850.000 −0.0302139
\(926\) 28984.0 1.02859
\(927\) 36556.0 1.29521
\(928\) −2880.00 −0.101876
\(929\) 20430.0 0.721514 0.360757 0.932660i \(-0.382518\pi\)
0.360757 + 0.932660i \(0.382518\pi\)
\(930\) 12160.0 0.428755
\(931\) 0 0
\(932\) 7848.00 0.275826
\(933\) −11136.0 −0.390757
\(934\) −15552.0 −0.544836
\(935\) 3960.00 0.138509
\(936\) 17168.0 0.599523
\(937\) −8906.00 −0.310508 −0.155254 0.987875i \(-0.549620\pi\)
−0.155254 + 0.987875i \(0.549620\pi\)
\(938\) 0 0
\(939\) 47024.0 1.63426
\(940\) −4080.00 −0.141569
\(941\) 17418.0 0.603412 0.301706 0.953401i \(-0.402444\pi\)
0.301706 + 0.953401i \(0.402444\pi\)
\(942\) −35936.0 −1.24295
\(943\) 57816.0 1.99655
\(944\) −6720.00 −0.231692
\(945\) 0 0
\(946\) 768.000 0.0263952
\(947\) −2544.00 −0.0872956 −0.0436478 0.999047i \(-0.513898\pi\)
−0.0436478 + 0.999047i \(0.513898\pi\)
\(948\) −5120.00 −0.175411
\(949\) −20996.0 −0.718187
\(950\) 5000.00 0.170759
\(951\) 82608.0 2.81677
\(952\) 0 0
\(953\) 15402.0 0.523525 0.261763 0.965132i \(-0.415696\pi\)
0.261763 + 0.965132i \(0.415696\pi\)
\(954\) 16428.0 0.557522
\(955\) −21960.0 −0.744093
\(956\) −17280.0 −0.584597
\(957\) −8640.00 −0.291841
\(958\) 27360.0 0.922716
\(959\) 0 0
\(960\) −2560.00 −0.0860663
\(961\) −6687.00 −0.224464
\(962\) −3944.00 −0.132183
\(963\) −888.000 −0.0297148
\(964\) 1912.00 0.0638811
\(965\) 10790.0 0.359940
\(966\) 0 0
\(967\) −49444.0 −1.64427 −0.822136 0.569291i \(-0.807218\pi\)
−0.822136 + 0.569291i \(0.807218\pi\)
\(968\) −9496.00 −0.315303
\(969\) −52800.0 −1.75044
\(970\) 11060.0 0.366098
\(971\) 25188.0 0.832463 0.416231 0.909259i \(-0.363351\pi\)
0.416231 + 0.909259i \(0.363351\pi\)
\(972\) −20128.0 −0.664204
\(973\) 0 0
\(974\) 15832.0 0.520832
\(975\) 11600.0 0.381023
\(976\) −14432.0 −0.473317
\(977\) 2946.00 0.0964697 0.0482348 0.998836i \(-0.484640\pi\)
0.0482348 + 0.998836i \(0.484640\pi\)
\(978\) −9088.00 −0.297139
\(979\) −9720.00 −0.317316
\(980\) 0 0
\(981\) 35150.0 1.14399
\(982\) 27864.0 0.905475
\(983\) −15012.0 −0.487089 −0.243544 0.969890i \(-0.578310\pi\)
−0.243544 + 0.969890i \(0.578310\pi\)
\(984\) 28032.0 0.908158
\(985\) 5370.00 0.173708
\(986\) 11880.0 0.383708
\(987\) 0 0
\(988\) 23200.0 0.747055
\(989\) 4224.00 0.135809
\(990\) −4440.00 −0.142538
\(991\) −5128.00 −0.164376 −0.0821878 0.996617i \(-0.526191\pi\)
−0.0821878 + 0.996617i \(0.526191\pi\)
\(992\) −4864.00 −0.155678
\(993\) −33824.0 −1.08094
\(994\) 0 0
\(995\) 14200.0 0.452432
\(996\) −2304.00 −0.0732982
\(997\) 49714.0 1.57920 0.789598 0.613625i \(-0.210289\pi\)
0.789598 + 0.613625i \(0.210289\pi\)
\(998\) −16520.0 −0.523979
\(999\) −2720.00 −0.0861431
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 490.4.a.o.1.1 1
5.4 even 2 2450.4.a.b.1.1 1
7.2 even 3 490.4.e.a.361.1 2
7.3 odd 6 490.4.e.i.471.1 2
7.4 even 3 490.4.e.a.471.1 2
7.5 odd 6 490.4.e.i.361.1 2
7.6 odd 2 10.4.a.a.1.1 1
21.20 even 2 90.4.a.a.1.1 1
28.27 even 2 80.4.a.f.1.1 1
35.13 even 4 50.4.b.a.49.1 2
35.27 even 4 50.4.b.a.49.2 2
35.34 odd 2 50.4.a.c.1.1 1
56.13 odd 2 320.4.a.m.1.1 1
56.27 even 2 320.4.a.b.1.1 1
63.13 odd 6 810.4.e.c.541.1 2
63.20 even 6 810.4.e.w.271.1 2
63.34 odd 6 810.4.e.c.271.1 2
63.41 even 6 810.4.e.w.541.1 2
77.76 even 2 1210.4.a.b.1.1 1
84.83 odd 2 720.4.a.j.1.1 1
91.90 odd 2 1690.4.a.a.1.1 1
105.62 odd 4 450.4.c.d.199.1 2
105.83 odd 4 450.4.c.d.199.2 2
105.104 even 2 450.4.a.q.1.1 1
112.13 odd 4 1280.4.d.j.641.1 2
112.27 even 4 1280.4.d.g.641.1 2
112.69 odd 4 1280.4.d.j.641.2 2
112.83 even 4 1280.4.d.g.641.2 2
140.27 odd 4 400.4.c.c.49.1 2
140.83 odd 4 400.4.c.c.49.2 2
140.139 even 2 400.4.a.b.1.1 1
280.69 odd 2 1600.4.a.d.1.1 1
280.139 even 2 1600.4.a.bx.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 7.6 odd 2
50.4.a.c.1.1 1 35.34 odd 2
50.4.b.a.49.1 2 35.13 even 4
50.4.b.a.49.2 2 35.27 even 4
80.4.a.f.1.1 1 28.27 even 2
90.4.a.a.1.1 1 21.20 even 2
320.4.a.b.1.1 1 56.27 even 2
320.4.a.m.1.1 1 56.13 odd 2
400.4.a.b.1.1 1 140.139 even 2
400.4.c.c.49.1 2 140.27 odd 4
400.4.c.c.49.2 2 140.83 odd 4
450.4.a.q.1.1 1 105.104 even 2
450.4.c.d.199.1 2 105.62 odd 4
450.4.c.d.199.2 2 105.83 odd 4
490.4.a.o.1.1 1 1.1 even 1 trivial
490.4.e.a.361.1 2 7.2 even 3
490.4.e.a.471.1 2 7.4 even 3
490.4.e.i.361.1 2 7.5 odd 6
490.4.e.i.471.1 2 7.3 odd 6
720.4.a.j.1.1 1 84.83 odd 2
810.4.e.c.271.1 2 63.34 odd 6
810.4.e.c.541.1 2 63.13 odd 6
810.4.e.w.271.1 2 63.20 even 6
810.4.e.w.541.1 2 63.41 even 6
1210.4.a.b.1.1 1 77.76 even 2
1280.4.d.g.641.1 2 112.27 even 4
1280.4.d.g.641.2 2 112.83 even 4
1280.4.d.j.641.1 2 112.13 odd 4
1280.4.d.j.641.2 2 112.69 odd 4
1600.4.a.d.1.1 1 280.69 odd 2
1600.4.a.bx.1.1 1 280.139 even 2
1690.4.a.a.1.1 1 91.90 odd 2
2450.4.a.b.1.1 1 5.4 even 2