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Properties

Label 720.4.a.t.1.1
Level $720$
Weight $4$
Character 720.1
Self dual yes
Analytic conductor $42.481$
Analytic rank $0$
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] = [720,4,Mod(1,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, 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("720.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -14.0000 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} -14.0000 q^{7} +6.00000 q^{11} +68.0000 q^{13} -78.0000 q^{17} -44.0000 q^{19} +120.000 q^{23} +25.0000 q^{25} -126.000 q^{29} +244.000 q^{31} -70.0000 q^{35} -304.000 q^{37} +480.000 q^{41} -104.000 q^{43} +600.000 q^{47} -147.000 q^{49} +258.000 q^{53} +30.0000 q^{55} +534.000 q^{59} +362.000 q^{61} +340.000 q^{65} +268.000 q^{67} -972.000 q^{71} +470.000 q^{73} -84.0000 q^{77} -1244.00 q^{79} +396.000 q^{83} -390.000 q^{85} +972.000 q^{89} -952.000 q^{91} -220.000 q^{95} -46.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −14.0000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 0.164461 0.0822304 − 0.996613i \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(12\) 0 0
\(13\) 68.0000 1.45075 0.725377 − 0.688352i \(-0.241665\pi\)
0.725377 + 0.688352i \(0.241665\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −78.0000 −1.11281 −0.556405 − 0.830911i \(-0.687820\pi\)
−0.556405 + 0.830911i \(0.687820\pi\)
\(18\) 0 0
\(19\) −44.0000 −0.531279 −0.265639 − 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 120.000 1.08790 0.543951 − 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −126.000 −0.806814 −0.403407 − 0.915021i \(-0.632174\pi\)
−0.403407 + 0.915021i \(0.632174\pi\)
\(30\) 0 0
\(31\) 244.000 1.41367 0.706834 − 0.707380i \(-0.250123\pi\)
0.706834 + 0.707380i \(0.250123\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −70.0000 −0.338062
\(36\) 0 0
\(37\) −304.000 −1.35074 −0.675369 − 0.737480i \(-0.736016\pi\)
−0.675369 + 0.737480i \(0.736016\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 480.000 1.82838 0.914188 − 0.405291i \(-0.132830\pi\)
0.914188 + 0.405291i \(0.132830\pi\)
\(42\) 0 0
\(43\) −104.000 −0.368834 −0.184417 − 0.982848i \(-0.559040\pi\)
−0.184417 + 0.982848i \(0.559040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 600.000 1.86211 0.931053 − 0.364884i \(-0.118891\pi\)
0.931053 + 0.364884i \(0.118891\pi\)
\(48\) 0 0
\(49\) −147.000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 258.000 0.668661 0.334330 − 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) 0 0
\(55\) 30.0000 0.0735491
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 534.000 1.17832 0.589160 − 0.808016i \(-0.299459\pi\)
0.589160 + 0.808016i \(0.299459\pi\)
\(60\) 0 0
\(61\) 362.000 0.759825 0.379913 − 0.925022i \(-0.375954\pi\)
0.379913 + 0.925022i \(0.375954\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 340.000 0.648797
\(66\) 0 0
\(67\) 268.000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −972.000 −1.62472 −0.812360 − 0.583156i \(-0.801818\pi\)
−0.812360 + 0.583156i \(0.801818\pi\)
\(72\) 0 0
\(73\) 470.000 0.753553 0.376776 − 0.926304i \(-0.377033\pi\)
0.376776 + 0.926304i \(0.377033\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −84.0000 −0.124321
\(78\) 0 0
\(79\) −1244.00 −1.77166 −0.885829 − 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 396.000 0.523695 0.261847 − 0.965109i \(-0.415668\pi\)
0.261847 + 0.965109i \(0.415668\pi\)
\(84\) 0 0
\(85\) −390.000 −0.497664
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 972.000 1.15766 0.578830 − 0.815448i \(-0.303509\pi\)
0.578830 + 0.815448i \(0.303509\pi\)
\(90\) 0 0
\(91\) −952.000 −1.09667
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −220.000 −0.237595
\(96\) 0 0
\(97\) −46.0000 −0.0481504 −0.0240752 − 0.999710i \(-0.507664\pi\)
−0.0240752 + 0.999710i \(0.507664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1506.00 1.48369 0.741845 − 0.670572i \(-0.233951\pi\)
0.741845 + 0.670572i \(0.233951\pi\)
\(102\) 0 0
\(103\) 1474.00 1.41007 0.705037 − 0.709171i \(-0.250931\pi\)
0.705037 + 0.709171i \(0.250931\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 924.000 0.834827 0.417413 − 0.908717i \(-0.362937\pi\)
0.417413 + 0.908717i \(0.362937\pi\)
\(108\) 0 0
\(109\) 698.000 0.613360 0.306680 − 0.951813i \(-0.400782\pi\)
0.306680 + 0.951813i \(0.400782\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 222.000 0.184814 0.0924071 − 0.995721i \(-0.470544\pi\)
0.0924071 + 0.995721i \(0.470544\pi\)
\(114\) 0 0
\(115\) 600.000 0.486524
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1092.00 0.841206
\(120\) 0 0
\(121\) −1295.00 −0.972953
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1906.00 1.33173 0.665867 − 0.746071i \(-0.268062\pi\)
0.665867 + 0.746071i \(0.268062\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2874.00 1.91681 0.958407 − 0.285406i \(-0.0921285\pi\)
0.958407 + 0.285406i \(0.0921285\pi\)
\(132\) 0 0
\(133\) 616.000 0.401609
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 798.000 0.497648 0.248824 − 0.968549i \(-0.419956\pi\)
0.248824 + 0.968549i \(0.419956\pi\)
\(138\) 0 0
\(139\) 700.000 0.427146 0.213573 − 0.976927i \(-0.431490\pi\)
0.213573 + 0.976927i \(0.431490\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 408.000 0.238592
\(144\) 0 0
\(145\) −630.000 −0.360818
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −114.000 −0.0626795 −0.0313397 − 0.999509i \(-0.509977\pi\)
−0.0313397 + 0.999509i \(0.509977\pi\)
\(150\) 0 0
\(151\) −1064.00 −0.573424 −0.286712 − 0.958017i \(-0.592562\pi\)
−0.286712 + 0.958017i \(0.592562\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1220.00 0.632211
\(156\) 0 0
\(157\) −1948.00 −0.990238 −0.495119 − 0.868825i \(-0.664875\pi\)
−0.495119 + 0.868825i \(0.664875\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1680.00 −0.822376
\(162\) 0 0
\(163\) −2060.00 −0.989887 −0.494944 − 0.868925i \(-0.664811\pi\)
−0.494944 + 0.868925i \(0.664811\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1248.00 −0.578282 −0.289141 − 0.957286i \(-0.593370\pi\)
−0.289141 + 0.957286i \(0.593370\pi\)
\(168\) 0 0
\(169\) 2427.00 1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1146.00 0.503634 0.251817 − 0.967775i \(-0.418972\pi\)
0.251817 + 0.967775i \(0.418972\pi\)
\(174\) 0 0
\(175\) −350.000 −0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1146.00 −0.478525 −0.239263 − 0.970955i \(-0.576906\pi\)
−0.239263 + 0.970955i \(0.576906\pi\)
\(180\) 0 0
\(181\) −118.000 −0.0484579 −0.0242289 − 0.999706i \(-0.507713\pi\)
−0.0242289 + 0.999706i \(0.507713\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1520.00 −0.604068
\(186\) 0 0
\(187\) −468.000 −0.183014
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1692.00 0.640989 0.320494 − 0.947250i \(-0.396151\pi\)
0.320494 + 0.947250i \(0.396151\pi\)
\(192\) 0 0
\(193\) 3350.00 1.24942 0.624711 − 0.780856i \(-0.285217\pi\)
0.624711 + 0.780856i \(0.285217\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3606.00 1.30415 0.652073 − 0.758156i \(-0.273899\pi\)
0.652073 + 0.758156i \(0.273899\pi\)
\(198\) 0 0
\(199\) −2696.00 −0.960374 −0.480187 − 0.877166i \(-0.659431\pi\)
−0.480187 + 0.877166i \(0.659431\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1764.00 0.609894
\(204\) 0 0
\(205\) 2400.00 0.817674
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −264.000 −0.0873745
\(210\) 0 0
\(211\) 4.00000 0.00130508 0.000652539 − 1.00000i \(-0.499792\pi\)
0.000652539 1.00000i \(0.499792\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −520.000 −0.164947
\(216\) 0 0
\(217\) −3416.00 −1.06863
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5304.00 −1.61441
\(222\) 0 0
\(223\) 1162.00 0.348938 0.174469 − 0.984663i \(-0.444179\pi\)
0.174469 + 0.984663i \(0.444179\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2400.00 −0.701734 −0.350867 − 0.936425i \(-0.614113\pi\)
−0.350867 + 0.936425i \(0.614113\pi\)
\(228\) 0 0
\(229\) −2314.00 −0.667744 −0.333872 − 0.942618i \(-0.608355\pi\)
−0.333872 + 0.942618i \(0.608355\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 0.00506103 0.00253051 − 0.999997i \(-0.499195\pi\)
0.00253051 + 0.999997i \(0.499195\pi\)
\(234\) 0 0
\(235\) 3000.00 0.832759
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5868.00 −1.58816 −0.794078 − 0.607816i \(-0.792046\pi\)
−0.794078 + 0.607816i \(0.792046\pi\)
\(240\) 0 0
\(241\) −4330.00 −1.15734 −0.578672 − 0.815560i \(-0.696429\pi\)
−0.578672 + 0.815560i \(0.696429\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −735.000 −0.191663
\(246\) 0 0
\(247\) −2992.00 −0.770755
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 498.000 0.125233 0.0626165 − 0.998038i \(-0.480056\pi\)
0.0626165 + 0.998038i \(0.480056\pi\)
\(252\) 0 0
\(253\) 720.000 0.178917
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −642.000 −0.155824 −0.0779122 − 0.996960i \(-0.524825\pi\)
−0.0779122 + 0.996960i \(0.524825\pi\)
\(258\) 0 0
\(259\) 4256.00 1.02106
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7968.00 −1.86817 −0.934084 − 0.357055i \(-0.883781\pi\)
−0.934084 + 0.357055i \(0.883781\pi\)
\(264\) 0 0
\(265\) 1290.00 0.299034
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4218.00 0.956045 0.478022 − 0.878348i \(-0.341354\pi\)
0.478022 + 0.878348i \(0.341354\pi\)
\(270\) 0 0
\(271\) −848.000 −0.190082 −0.0950412 − 0.995473i \(-0.530298\pi\)
−0.0950412 + 0.995473i \(0.530298\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 150.000 0.0328921
\(276\) 0 0
\(277\) −1504.00 −0.326233 −0.163117 − 0.986607i \(-0.552155\pi\)
−0.163117 + 0.986607i \(0.552155\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1308.00 0.277682 0.138841 − 0.990315i \(-0.455662\pi\)
0.138841 + 0.990315i \(0.455662\pi\)
\(282\) 0 0
\(283\) 5932.00 1.24601 0.623005 − 0.782218i \(-0.285912\pi\)
0.623005 + 0.782218i \(0.285912\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6720.00 −1.38212
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5226.00 −1.04200 −0.521000 − 0.853556i \(-0.674441\pi\)
−0.521000 + 0.853556i \(0.674441\pi\)
\(294\) 0 0
\(295\) 2670.00 0.526961
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8160.00 1.57828
\(300\) 0 0
\(301\) 1456.00 0.278812
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1810.00 0.339804
\(306\) 0 0
\(307\) −4448.00 −0.826908 −0.413454 − 0.910525i \(-0.635678\pi\)
−0.413454 + 0.910525i \(0.635678\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9132.00 −1.66504 −0.832521 − 0.553993i \(-0.813103\pi\)
−0.832521 + 0.553993i \(0.813103\pi\)
\(312\) 0 0
\(313\) −2170.00 −0.391871 −0.195936 − 0.980617i \(-0.562774\pi\)
−0.195936 + 0.980617i \(0.562774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7674.00 −1.35967 −0.679834 − 0.733366i \(-0.737948\pi\)
−0.679834 + 0.733366i \(0.737948\pi\)
\(318\) 0 0
\(319\) −756.000 −0.132689
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3432.00 0.591212
\(324\) 0 0
\(325\) 1700.00 0.290151
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8400.00 −1.40762
\(330\) 0 0
\(331\) −9596.00 −1.59349 −0.796743 − 0.604318i \(-0.793446\pi\)
−0.796743 + 0.604318i \(0.793446\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1340.00 0.218543
\(336\) 0 0
\(337\) 12158.0 1.96525 0.982624 − 0.185608i \(-0.0594255\pi\)
0.982624 + 0.185608i \(0.0594255\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1464.00 0.232493
\(342\) 0 0
\(343\) 6860.00 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10320.0 1.59656 0.798280 − 0.602286i \(-0.205743\pi\)
0.798280 + 0.602286i \(0.205743\pi\)
\(348\) 0 0
\(349\) −2158.00 −0.330989 −0.165494 − 0.986211i \(-0.552922\pi\)
−0.165494 + 0.986211i \(0.552922\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 330.000 0.0497567 0.0248784 − 0.999690i \(-0.492080\pi\)
0.0248784 + 0.999690i \(0.492080\pi\)
\(354\) 0 0
\(355\) −4860.00 −0.726597
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8664.00 −1.27373 −0.636864 − 0.770976i \(-0.719769\pi\)
−0.636864 + 0.770976i \(0.719769\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2350.00 0.336999
\(366\) 0 0
\(367\) −3782.00 −0.537926 −0.268963 − 0.963151i \(-0.586681\pi\)
−0.268963 + 0.963151i \(0.586681\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3612.00 −0.505460
\(372\) 0 0
\(373\) 11276.0 1.56528 0.782640 − 0.622475i \(-0.213873\pi\)
0.782640 + 0.622475i \(0.213873\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8568.00 −1.17049
\(378\) 0 0
\(379\) −980.000 −0.132821 −0.0664106 − 0.997792i \(-0.521155\pi\)
−0.0664106 + 0.997792i \(0.521155\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4200.00 −0.560339 −0.280170 − 0.959950i \(-0.590391\pi\)
−0.280170 + 0.959950i \(0.590391\pi\)
\(384\) 0 0
\(385\) −420.000 −0.0555979
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13338.0 −1.73847 −0.869233 − 0.494402i \(-0.835387\pi\)
−0.869233 + 0.494402i \(0.835387\pi\)
\(390\) 0 0
\(391\) −9360.00 −1.21063
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6220.00 −0.792309
\(396\) 0 0
\(397\) −7192.00 −0.909209 −0.454605 − 0.890693i \(-0.650219\pi\)
−0.454605 + 0.890693i \(0.650219\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2316.00 0.288418 0.144209 − 0.989547i \(-0.453936\pi\)
0.144209 + 0.989547i \(0.453936\pi\)
\(402\) 0 0
\(403\) 16592.0 2.05088
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1824.00 −0.222143
\(408\) 0 0
\(409\) −12358.0 −1.49404 −0.747022 − 0.664800i \(-0.768517\pi\)
−0.747022 + 0.664800i \(0.768517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7476.00 −0.890726
\(414\) 0 0
\(415\) 1980.00 0.234203
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3306.00 0.385462 0.192731 − 0.981252i \(-0.438265\pi\)
0.192731 + 0.981252i \(0.438265\pi\)
\(420\) 0 0
\(421\) −14506.0 −1.67929 −0.839643 − 0.543139i \(-0.817236\pi\)
−0.839643 + 0.543139i \(0.817236\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1950.00 −0.222562
\(426\) 0 0
\(427\) −5068.00 −0.574374
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6480.00 −0.724201 −0.362100 − 0.932139i \(-0.617940\pi\)
−0.362100 + 0.932139i \(0.617940\pi\)
\(432\) 0 0
\(433\) 11894.0 1.32007 0.660034 − 0.751236i \(-0.270542\pi\)
0.660034 + 0.751236i \(0.270542\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5280.00 −0.577979
\(438\) 0 0
\(439\) 12688.0 1.37942 0.689710 − 0.724086i \(-0.257738\pi\)
0.689710 + 0.724086i \(0.257738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4968.00 0.532814 0.266407 − 0.963861i \(-0.414163\pi\)
0.266407 + 0.963861i \(0.414163\pi\)
\(444\) 0 0
\(445\) 4860.00 0.517722
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11508.0 1.20957 0.604784 − 0.796389i \(-0.293259\pi\)
0.604784 + 0.796389i \(0.293259\pi\)
\(450\) 0 0
\(451\) 2880.00 0.300696
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4760.00 −0.490444
\(456\) 0 0
\(457\) 1082.00 0.110752 0.0553762 − 0.998466i \(-0.482364\pi\)
0.0553762 + 0.998466i \(0.482364\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11238.0 1.13537 0.567685 − 0.823246i \(-0.307839\pi\)
0.567685 + 0.823246i \(0.307839\pi\)
\(462\) 0 0
\(463\) 2302.00 0.231065 0.115532 − 0.993304i \(-0.463143\pi\)
0.115532 + 0.993304i \(0.463143\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15876.0 1.57313 0.786567 − 0.617505i \(-0.211856\pi\)
0.786567 + 0.617505i \(0.211856\pi\)
\(468\) 0 0
\(469\) −3752.00 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −624.000 −0.0606587
\(474\) 0 0
\(475\) −1100.00 −0.106256
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4644.00 0.442985 0.221492 − 0.975162i \(-0.428907\pi\)
0.221492 + 0.975162i \(0.428907\pi\)
\(480\) 0 0
\(481\) −20672.0 −1.95959
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −230.000 −0.0215335
\(486\) 0 0
\(487\) −2426.00 −0.225734 −0.112867 − 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 234.000 0.0215077 0.0107538 − 0.999942i \(-0.496577\pi\)
0.0107538 + 0.999942i \(0.496577\pi\)
\(492\) 0 0
\(493\) 9828.00 0.897831
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13608.0 1.22817
\(498\) 0 0
\(499\) −14204.0 −1.27427 −0.637133 − 0.770754i \(-0.719880\pi\)
−0.637133 + 0.770754i \(0.719880\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4920.00 0.436127 0.218064 − 0.975935i \(-0.430026\pi\)
0.218064 + 0.975935i \(0.430026\pi\)
\(504\) 0 0
\(505\) 7530.00 0.663526
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4458.00 0.388207 0.194104 − 0.980981i \(-0.437820\pi\)
0.194104 + 0.980981i \(0.437820\pi\)
\(510\) 0 0
\(511\) −6580.00 −0.569632
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7370.00 0.630604
\(516\) 0 0
\(517\) 3600.00 0.306243
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4212.00 −0.354186 −0.177093 − 0.984194i \(-0.556669\pi\)
−0.177093 + 0.984194i \(0.556669\pi\)
\(522\) 0 0
\(523\) 11212.0 0.937412 0.468706 − 0.883354i \(-0.344720\pi\)
0.468706 + 0.883354i \(0.344720\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19032.0 −1.57314
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32640.0 2.65252
\(534\) 0 0
\(535\) 4620.00 0.373346
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −882.000 −0.0704832
\(540\) 0 0
\(541\) 14018.0 1.11401 0.557006 − 0.830508i \(-0.311950\pi\)
0.557006 + 0.830508i \(0.311950\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3490.00 0.274303
\(546\) 0 0
\(547\) −18200.0 −1.42262 −0.711312 − 0.702876i \(-0.751899\pi\)
−0.711312 + 0.702876i \(0.751899\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5544.00 0.428643
\(552\) 0 0
\(553\) 17416.0 1.33925
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11826.0 0.899612 0.449806 − 0.893126i \(-0.351493\pi\)
0.449806 + 0.893126i \(0.351493\pi\)
\(558\) 0 0
\(559\) −7072.00 −0.535087
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2952.00 −0.220980 −0.110490 − 0.993877i \(-0.535242\pi\)
−0.110490 + 0.993877i \(0.535242\pi\)
\(564\) 0 0
\(565\) 1110.00 0.0826514
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3084.00 −0.227220 −0.113610 − 0.993525i \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(570\) 0 0
\(571\) 4756.00 0.348568 0.174284 − 0.984695i \(-0.444239\pi\)
0.174284 + 0.984695i \(0.444239\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3000.00 0.217580
\(576\) 0 0
\(577\) −11014.0 −0.794660 −0.397330 − 0.917676i \(-0.630063\pi\)
−0.397330 + 0.917676i \(0.630063\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5544.00 −0.395876
\(582\) 0 0
\(583\) 1548.00 0.109968
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −852.000 −0.0599077 −0.0299538 − 0.999551i \(-0.509536\pi\)
−0.0299538 + 0.999551i \(0.509536\pi\)
\(588\) 0 0
\(589\) −10736.0 −0.751051
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15546.0 −1.07656 −0.538278 − 0.842767i \(-0.680925\pi\)
−0.538278 + 0.842767i \(0.680925\pi\)
\(594\) 0 0
\(595\) 5460.00 0.376199
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8616.00 −0.587713 −0.293857 − 0.955850i \(-0.594939\pi\)
−0.293857 + 0.955850i \(0.594939\pi\)
\(600\) 0 0
\(601\) 17510.0 1.18843 0.594216 − 0.804305i \(-0.297462\pi\)
0.594216 + 0.804305i \(0.297462\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6475.00 −0.435118
\(606\) 0 0
\(607\) 13894.0 0.929061 0.464531 − 0.885557i \(-0.346223\pi\)
0.464531 + 0.885557i \(0.346223\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40800.0 2.70146
\(612\) 0 0
\(613\) −6496.00 −0.428011 −0.214006 − 0.976832i \(-0.568651\pi\)
−0.214006 + 0.976832i \(0.568651\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 570.000 0.0371918 0.0185959 − 0.999827i \(-0.494080\pi\)
0.0185959 + 0.999827i \(0.494080\pi\)
\(618\) 0 0
\(619\) 2140.00 0.138956 0.0694781 − 0.997583i \(-0.477867\pi\)
0.0694781 + 0.997583i \(0.477867\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13608.0 −0.875109
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23712.0 1.50312
\(630\) 0 0
\(631\) −14660.0 −0.924890 −0.462445 − 0.886648i \(-0.653028\pi\)
−0.462445 + 0.886648i \(0.653028\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9530.00 0.595569
\(636\) 0 0
\(637\) −9996.00 −0.621752
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 456.000 0.0280982 0.0140491 − 0.999901i \(-0.495528\pi\)
0.0140491 + 0.999901i \(0.495528\pi\)
\(642\) 0 0
\(643\) 23452.0 1.43835 0.719173 − 0.694831i \(-0.244521\pi\)
0.719173 + 0.694831i \(0.244521\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7224.00 0.438956 0.219478 − 0.975617i \(-0.429565\pi\)
0.219478 + 0.975617i \(0.429565\pi\)
\(648\) 0 0
\(649\) 3204.00 0.193787
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19146.0 −1.14738 −0.573691 − 0.819072i \(-0.694489\pi\)
−0.573691 + 0.819072i \(0.694489\pi\)
\(654\) 0 0
\(655\) 14370.0 0.857225
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27810.0 1.64389 0.821945 − 0.569567i \(-0.192889\pi\)
0.821945 + 0.569567i \(0.192889\pi\)
\(660\) 0 0
\(661\) −30598.0 −1.80049 −0.900245 − 0.435383i \(-0.856613\pi\)
−0.900245 + 0.435383i \(0.856613\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3080.00 0.179605
\(666\) 0 0
\(667\) −15120.0 −0.877734
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2172.00 0.124961
\(672\) 0 0
\(673\) −3778.00 −0.216391 −0.108196 − 0.994130i \(-0.534507\pi\)
−0.108196 + 0.994130i \(0.534507\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27198.0 1.54402 0.772012 − 0.635608i \(-0.219251\pi\)
0.772012 + 0.635608i \(0.219251\pi\)
\(678\) 0 0
\(679\) 644.000 0.0363983
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32316.0 1.81045 0.905225 − 0.424933i \(-0.139702\pi\)
0.905225 + 0.424933i \(0.139702\pi\)
\(684\) 0 0
\(685\) 3990.00 0.222555
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 17544.0 0.970063
\(690\) 0 0
\(691\) −29324.0 −1.61438 −0.807191 − 0.590291i \(-0.799013\pi\)
−0.807191 + 0.590291i \(0.799013\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3500.00 0.191025
\(696\) 0 0
\(697\) −37440.0 −2.03464
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22782.0 −1.22748 −0.613741 − 0.789508i \(-0.710336\pi\)
−0.613741 + 0.789508i \(0.710336\pi\)
\(702\) 0 0
\(703\) 13376.0 0.717618
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −21084.0 −1.12156
\(708\) 0 0
\(709\) 26054.0 1.38008 0.690041 − 0.723770i \(-0.257592\pi\)
0.690041 + 0.723770i \(0.257592\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29280.0 1.53793
\(714\) 0 0
\(715\) 2040.00 0.106702
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5976.00 −0.309968 −0.154984 − 0.987917i \(-0.549533\pi\)
−0.154984 + 0.987917i \(0.549533\pi\)
\(720\) 0 0
\(721\) −20636.0 −1.06592
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3150.00 −0.161363
\(726\) 0 0
\(727\) 5110.00 0.260687 0.130343 − 0.991469i \(-0.458392\pi\)
0.130343 + 0.991469i \(0.458392\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8112.00 0.410442
\(732\) 0 0
\(733\) 17336.0 0.873560 0.436780 − 0.899568i \(-0.356119\pi\)
0.436780 + 0.899568i \(0.356119\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1608.00 0.0803683
\(738\) 0 0
\(739\) 13660.0 0.679961 0.339981 − 0.940432i \(-0.389580\pi\)
0.339981 + 0.940432i \(0.389580\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1320.00 0.0651765 0.0325882 − 0.999469i \(-0.489625\pi\)
0.0325882 + 0.999469i \(0.489625\pi\)
\(744\) 0 0
\(745\) −570.000 −0.0280311
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12936.0 −0.631070
\(750\) 0 0
\(751\) −15860.0 −0.770625 −0.385313 − 0.922786i \(-0.625906\pi\)
−0.385313 + 0.922786i \(0.625906\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5320.00 −0.256443
\(756\) 0 0
\(757\) 22160.0 1.06396 0.531981 − 0.846756i \(-0.321448\pi\)
0.531981 + 0.846756i \(0.321448\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13116.0 −0.624776 −0.312388 − 0.949955i \(-0.601129\pi\)
−0.312388 + 0.949955i \(0.601129\pi\)
\(762\) 0 0
\(763\) −9772.00 −0.463657
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36312.0 1.70945
\(768\) 0 0
\(769\) 32846.0 1.54026 0.770128 − 0.637889i \(-0.220192\pi\)
0.770128 + 0.637889i \(0.220192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11982.0 0.557520 0.278760 − 0.960361i \(-0.410077\pi\)
0.278760 + 0.960361i \(0.410077\pi\)
\(774\) 0 0
\(775\) 6100.00 0.282734
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −21120.0 −0.971377
\(780\) 0 0
\(781\) −5832.00 −0.267203
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9740.00 −0.442848
\(786\) 0 0
\(787\) 21076.0 0.954610 0.477305 − 0.878738i \(-0.341614\pi\)
0.477305 + 0.878738i \(0.341614\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3108.00 −0.139706
\(792\) 0 0
\(793\) 24616.0 1.10232
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22086.0 −0.981589 −0.490794 − 0.871275i \(-0.663293\pi\)
−0.490794 + 0.871275i \(0.663293\pi\)
\(798\) 0 0
\(799\) −46800.0 −2.07217
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2820.00 0.123930
\(804\) 0 0
\(805\) −8400.00 −0.367778
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21384.0 −0.929322 −0.464661 − 0.885489i \(-0.653824\pi\)
−0.464661 + 0.885489i \(0.653824\pi\)
\(810\) 0 0
\(811\) −5228.00 −0.226362 −0.113181 − 0.993574i \(-0.536104\pi\)
−0.113181 + 0.993574i \(0.536104\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10300.0 −0.442691
\(816\) 0 0
\(817\) 4576.00 0.195953
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38010.0 1.61578 0.807892 − 0.589331i \(-0.200609\pi\)
0.807892 + 0.589331i \(0.200609\pi\)
\(822\) 0 0
\(823\) −38642.0 −1.63667 −0.818333 − 0.574745i \(-0.805101\pi\)
−0.818333 + 0.574745i \(0.805101\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15432.0 −0.648879 −0.324440 − 0.945906i \(-0.605176\pi\)
−0.324440 + 0.945906i \(0.605176\pi\)
\(828\) 0 0
\(829\) −3886.00 −0.162806 −0.0814031 − 0.996681i \(-0.525940\pi\)
−0.0814031 + 0.996681i \(0.525940\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 11466.0 0.476919
\(834\) 0 0
\(835\) −6240.00 −0.258616
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27552.0 1.13373 0.566866 − 0.823810i \(-0.308156\pi\)
0.566866 + 0.823810i \(0.308156\pi\)
\(840\) 0 0
\(841\) −8513.00 −0.349051
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12135.0 0.494032
\(846\) 0 0
\(847\) 18130.0 0.735483
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −36480.0 −1.46947
\(852\) 0 0
\(853\) 15104.0 0.606273 0.303137 − 0.952947i \(-0.401966\pi\)
0.303137 + 0.952947i \(0.401966\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12306.0 0.490508 0.245254 − 0.969459i \(-0.421129\pi\)
0.245254 + 0.969459i \(0.421129\pi\)
\(858\) 0 0
\(859\) 47500.0 1.88670 0.943352 − 0.331793i \(-0.107654\pi\)
0.943352 + 0.331793i \(0.107654\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4272.00 0.168506 0.0842529 − 0.996444i \(-0.473150\pi\)
0.0842529 + 0.996444i \(0.473150\pi\)
\(864\) 0 0
\(865\) 5730.00 0.225232
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7464.00 −0.291368
\(870\) 0 0
\(871\) 18224.0 0.708951
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1750.00 −0.0676123
\(876\) 0 0
\(877\) −27796.0 −1.07024 −0.535122 − 0.844775i \(-0.679734\pi\)
−0.535122 + 0.844775i \(0.679734\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39996.0 1.52951 0.764756 − 0.644320i \(-0.222860\pi\)
0.764756 + 0.644320i \(0.222860\pi\)
\(882\) 0 0
\(883\) 3772.00 0.143758 0.0718788 − 0.997413i \(-0.477101\pi\)
0.0718788 + 0.997413i \(0.477101\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5784.00 −0.218949 −0.109474 − 0.993990i \(-0.534917\pi\)
−0.109474 + 0.993990i \(0.534917\pi\)
\(888\) 0 0
\(889\) −26684.0 −1.00670
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −26400.0 −0.989297
\(894\) 0 0
\(895\) −5730.00 −0.214003
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −30744.0 −1.14057
\(900\) 0 0
\(901\) −20124.0 −0.744093
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −590.000 −0.0216710
\(906\) 0 0
\(907\) 8440.00 0.308981 0.154490 − 0.987994i \(-0.450626\pi\)
0.154490 + 0.987994i \(0.450626\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31920.0 −1.16087 −0.580437 − 0.814305i \(-0.697118\pi\)
−0.580437 + 0.814305i \(0.697118\pi\)
\(912\) 0 0
\(913\) 2376.00 0.0861272
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −40236.0 −1.44897
\(918\) 0 0
\(919\) −34652.0 −1.24381 −0.621906 − 0.783092i \(-0.713642\pi\)
−0.621906 + 0.783092i \(0.713642\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −66096.0 −2.35707
\(924\) 0 0
\(925\) −7600.00 −0.270148
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1404.00 −0.0495842 −0.0247921 − 0.999693i \(-0.507892\pi\)
−0.0247921 + 0.999693i \(0.507892\pi\)
\(930\) 0 0
\(931\) 6468.00 0.227691
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2340.00 −0.0818462
\(936\) 0 0
\(937\) −7654.00 −0.266857 −0.133429 − 0.991058i \(-0.542599\pi\)
−0.133429 + 0.991058i \(0.542599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11298.0 −0.391397 −0.195698 − 0.980664i \(-0.562697\pi\)
−0.195698 + 0.980664i \(0.562697\pi\)
\(942\) 0 0
\(943\) 57600.0 1.98909
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28968.0 0.994016 0.497008 − 0.867746i \(-0.334432\pi\)
0.497008 + 0.867746i \(0.334432\pi\)
\(948\) 0 0
\(949\) 31960.0 1.09322
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −46410.0 −1.57751 −0.788755 − 0.614707i \(-0.789274\pi\)
−0.788755 + 0.614707i \(0.789274\pi\)
\(954\) 0 0
\(955\) 8460.00 0.286659
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11172.0 −0.376186
\(960\) 0 0
\(961\) 29745.0 0.998456
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16750.0 0.558758
\(966\) 0 0
\(967\) 41506.0 1.38029 0.690146 − 0.723670i \(-0.257546\pi\)
0.690146 + 0.723670i \(0.257546\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18246.0 −0.603030 −0.301515 − 0.953461i \(-0.597492\pi\)
−0.301515 + 0.953461i \(0.597492\pi\)
\(972\) 0 0
\(973\) −9800.00 −0.322892
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25998.0 −0.851330 −0.425665 − 0.904881i \(-0.639960\pi\)
−0.425665 + 0.904881i \(0.639960\pi\)
\(978\) 0 0
\(979\) 5832.00 0.190390
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14616.0 0.474240 0.237120 − 0.971480i \(-0.423797\pi\)
0.237120 + 0.971480i \(0.423797\pi\)
\(984\) 0 0
\(985\) 18030.0 0.583232
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12480.0 −0.401255
\(990\) 0 0
\(991\) 2968.00 0.0951379 0.0475689 − 0.998868i \(-0.484853\pi\)
0.0475689 + 0.998868i \(0.484853\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13480.0 −0.429492
\(996\) 0 0
\(997\) −9052.00 −0.287542 −0.143771 − 0.989611i \(-0.545923\pi\)
−0.143771 + 0.989611i \(0.545923\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 720.4.a.t.1.1 1
3.2 odd 2 720.4.a.e.1.1 1
4.3 odd 2 90.4.a.e.1.1 yes 1
12.11 even 2 90.4.a.b.1.1 ✓ 1
20.3 even 4 450.4.c.f.199.1 2
20.7 even 4 450.4.c.f.199.2 2
20.19 odd 2 450.4.a.c.1.1 1
36.7 odd 6 810.4.e.a.271.1 2
36.11 even 6 810.4.e.u.271.1 2
36.23 even 6 810.4.e.u.541.1 2
36.31 odd 6 810.4.e.a.541.1 2
60.23 odd 4 450.4.c.g.199.2 2
60.47 odd 4 450.4.c.g.199.1 2
60.59 even 2 450.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 ✓ 1 12.11 even 2
90.4.a.e.1.1 yes 1 4.3 odd 2
450.4.a.c.1.1 1 20.19 odd 2
450.4.a.m.1.1 1 60.59 even 2
450.4.c.f.199.1 2 20.3 even 4
450.4.c.f.199.2 2 20.7 even 4
450.4.c.g.199.1 2 60.47 odd 4
450.4.c.g.199.2 2 60.23 odd 4
720.4.a.e.1.1 1 3.2 odd 2
720.4.a.t.1.1 1 1.1 even 1 trivial
810.4.e.a.271.1 2 36.7 odd 6
810.4.e.a.541.1 2 36.31 odd 6
810.4.e.u.271.1 2 36.11 even 6
810.4.e.u.541.1 2 36.23 even 6