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Properties

Label 432.4.a.j.1.1
Level $432$
Weight $4$
Character 432.1
Self dual yes
Analytic conductor $25.489$
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] = [432,4,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("432.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(25.4888251225\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 432.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{5} -29.0000 q^{7} +O(q^{10})\) \(q+3.00000 q^{5} -29.0000 q^{7} +57.0000 q^{11} +20.0000 q^{13} -72.0000 q^{17} +106.000 q^{19} -174.000 q^{23} -116.000 q^{25} -210.000 q^{29} -47.0000 q^{31} -87.0000 q^{35} +2.00000 q^{37} -6.00000 q^{41} -218.000 q^{43} -474.000 q^{47} +498.000 q^{49} +81.0000 q^{53} +171.000 q^{55} -84.0000 q^{59} +56.0000 q^{61} +60.0000 q^{65} +142.000 q^{67} -360.000 q^{71} -1159.00 q^{73} -1653.00 q^{77} +160.000 q^{79} -735.000 q^{83} -216.000 q^{85} -954.000 q^{89} -580.000 q^{91} +318.000 q^{95} +191.000 q^{97} +O(q^{100})\)

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
\(5\) 3.00000 0.268328 0.134164 − 0.990959i \(-0.457165\pi\)
0.134164 + 0.990959i \(0.457165\pi\)
\(6\) 0 0
\(7\) −29.0000 −1.56585 −0.782926 − 0.622114i \(-0.786274\pi\)
−0.782926 + 0.622114i \(0.786274\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 57.0000 1.56238 0.781188 − 0.624295i \(-0.214614\pi\)
0.781188 + 0.624295i \(0.214614\pi\)
\(12\) 0 0
\(13\) 20.0000 0.426692 0.213346 − 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −72.0000 −1.02721 −0.513605 − 0.858027i \(-0.671690\pi\)
−0.513605 + 0.858027i \(0.671690\pi\)
\(18\) 0 0
\(19\) 106.000 1.27990 0.639949 − 0.768417i \(-0.278955\pi\)
0.639949 + 0.768417i \(0.278955\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −174.000 −1.57746 −0.788728 − 0.614742i \(-0.789260\pi\)
−0.788728 + 0.614742i \(0.789260\pi\)
\(24\) 0 0
\(25\) −116.000 −0.928000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −210.000 −1.34469 −0.672345 − 0.740238i \(-0.734713\pi\)
−0.672345 + 0.740238i \(0.734713\pi\)
\(30\) 0 0
\(31\) −47.0000 −0.272305 −0.136152 − 0.990688i \(-0.543474\pi\)
−0.136152 + 0.990688i \(0.543474\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −87.0000 −0.420162
\(36\) 0 0
\(37\) 2.00000 0.00888643 0.00444322 − 0.999990i \(-0.498586\pi\)
0.00444322 + 0.999990i \(0.498586\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) 0 0
\(43\) −218.000 −0.773132 −0.386566 − 0.922262i \(-0.626339\pi\)
−0.386566 + 0.922262i \(0.626339\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −474.000 −1.47106 −0.735532 − 0.677490i \(-0.763068\pi\)
−0.735532 + 0.677490i \(0.763068\pi\)
\(48\) 0 0
\(49\) 498.000 1.45190
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 81.0000 0.209928 0.104964 − 0.994476i \(-0.466527\pi\)
0.104964 + 0.994476i \(0.466527\pi\)
\(54\) 0 0
\(55\) 171.000 0.419230
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −84.0000 −0.185354 −0.0926769 − 0.995696i \(-0.529542\pi\)
−0.0926769 + 0.995696i \(0.529542\pi\)
\(60\) 0 0
\(61\) 56.0000 0.117542 0.0587710 − 0.998271i \(-0.481282\pi\)
0.0587710 + 0.998271i \(0.481282\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 60.0000 0.114494
\(66\) 0 0
\(67\) 142.000 0.258926 0.129463 − 0.991584i \(-0.458675\pi\)
0.129463 + 0.991584i \(0.458675\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −360.000 −0.601748 −0.300874 − 0.953664i \(-0.597278\pi\)
−0.300874 + 0.953664i \(0.597278\pi\)
\(72\) 0 0
\(73\) −1159.00 −1.85823 −0.929114 − 0.369793i \(-0.879429\pi\)
−0.929114 + 0.369793i \(0.879429\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1653.00 −2.44645
\(78\) 0 0
\(79\) 160.000 0.227866 0.113933 − 0.993488i \(-0.463655\pi\)
0.113933 + 0.993488i \(0.463655\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −735.000 −0.972009 −0.486004 − 0.873956i \(-0.661546\pi\)
−0.486004 + 0.873956i \(0.661546\pi\)
\(84\) 0 0
\(85\) −216.000 −0.275629
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −954.000 −1.13622 −0.568111 − 0.822952i \(-0.692326\pi\)
−0.568111 + 0.822952i \(0.692326\pi\)
\(90\) 0 0
\(91\) −580.000 −0.668138
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 318.000 0.343433
\(96\) 0 0
\(97\) 191.000 0.199929 0.0999645 − 0.994991i \(-0.468127\pi\)
0.0999645 + 0.994991i \(0.468127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −363.000 −0.357622 −0.178811 − 0.983883i \(-0.557225\pi\)
−0.178811 + 0.983883i \(0.557225\pi\)
\(102\) 0 0
\(103\) 628.000 0.600764 0.300382 − 0.953819i \(-0.402886\pi\)
0.300382 + 0.953819i \(0.402886\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −675.000 −0.609857 −0.304929 − 0.952375i \(-0.598633\pi\)
−0.304929 + 0.952375i \(0.598633\pi\)
\(108\) 0 0
\(109\) 1730.00 1.52022 0.760110 − 0.649795i \(-0.225145\pi\)
0.760110 + 0.649795i \(0.225145\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1866.00 1.55344 0.776719 − 0.629847i \(-0.216882\pi\)
0.776719 + 0.629847i \(0.216882\pi\)
\(114\) 0 0
\(115\) −522.000 −0.423276
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2088.00 1.60846
\(120\) 0 0
\(121\) 1918.00 1.44102
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −723.000 −0.517337
\(126\) 0 0
\(127\) −1379.00 −0.963515 −0.481758 − 0.876304i \(-0.660001\pi\)
−0.481758 + 0.876304i \(0.660001\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −579.000 −0.386164 −0.193082 − 0.981183i \(-0.561848\pi\)
−0.193082 + 0.981183i \(0.561848\pi\)
\(132\) 0 0
\(133\) −3074.00 −2.00413
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 654.000 0.407847 0.203923 − 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(138\) 0 0
\(139\) 3004.00 1.83306 0.916532 − 0.399961i \(-0.130976\pi\)
0.916532 + 0.399961i \(0.130976\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1140.00 0.666654
\(144\) 0 0
\(145\) −630.000 −0.360818
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1803.00 0.991326 0.495663 − 0.868515i \(-0.334925\pi\)
0.495663 + 0.868515i \(0.334925\pi\)
\(150\) 0 0
\(151\) −2459.00 −1.32524 −0.662618 − 0.748958i \(-0.730555\pi\)
−0.662618 + 0.748958i \(0.730555\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −141.000 −0.0730670
\(156\) 0 0
\(157\) −196.000 −0.0996338 −0.0498169 − 0.998758i \(-0.515864\pi\)
−0.0498169 + 0.998758i \(0.515864\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5046.00 2.47007
\(162\) 0 0
\(163\) 1564.00 0.751546 0.375773 − 0.926712i \(-0.377377\pi\)
0.375773 + 0.926712i \(0.377377\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1974.00 −0.914687 −0.457343 − 0.889290i \(-0.651199\pi\)
−0.457343 + 0.889290i \(0.651199\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2217.00 −0.974309 −0.487154 − 0.873316i \(-0.661965\pi\)
−0.487154 + 0.873316i \(0.661965\pi\)
\(174\) 0 0
\(175\) 3364.00 1.45311
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2475.00 1.03346 0.516732 − 0.856147i \(-0.327148\pi\)
0.516732 + 0.856147i \(0.327148\pi\)
\(180\) 0 0
\(181\) 1568.00 0.643914 0.321957 − 0.946754i \(-0.395659\pi\)
0.321957 + 0.946754i \(0.395659\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.00238448
\(186\) 0 0
\(187\) −4104.00 −1.60489
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1140.00 −0.431872 −0.215936 − 0.976408i \(-0.569280\pi\)
−0.215936 + 0.976408i \(0.569280\pi\)
\(192\) 0 0
\(193\) 2045.00 0.762706 0.381353 − 0.924429i \(-0.375458\pi\)
0.381353 + 0.924429i \(0.375458\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3735.00 −1.35080 −0.675400 − 0.737451i \(-0.736029\pi\)
−0.675400 + 0.737451i \(0.736029\pi\)
\(198\) 0 0
\(199\) −1163.00 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6090.00 2.10559
\(204\) 0 0
\(205\) −18.0000 −0.00613256
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6042.00 1.99968
\(210\) 0 0
\(211\) −2126.00 −0.693649 −0.346824 − 0.937930i \(-0.612740\pi\)
−0.346824 + 0.937930i \(0.612740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −654.000 −0.207453
\(216\) 0 0
\(217\) 1363.00 0.426389
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1440.00 −0.438303
\(222\) 0 0
\(223\) 2752.00 0.826402 0.413201 − 0.910640i \(-0.364411\pi\)
0.413201 + 0.910640i \(0.364411\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3972.00 1.16137 0.580685 − 0.814128i \(-0.302785\pi\)
0.580685 + 0.814128i \(0.302785\pi\)
\(228\) 0 0
\(229\) 4502.00 1.29913 0.649564 − 0.760307i \(-0.274951\pi\)
0.649564 + 0.760307i \(0.274951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4842.00 1.36142 0.680708 − 0.732555i \(-0.261672\pi\)
0.680708 + 0.732555i \(0.261672\pi\)
\(234\) 0 0
\(235\) −1422.00 −0.394728
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5334.00 1.44363 0.721815 − 0.692086i \(-0.243308\pi\)
0.721815 + 0.692086i \(0.243308\pi\)
\(240\) 0 0
\(241\) −3994.00 −1.06754 −0.533768 − 0.845631i \(-0.679224\pi\)
−0.533768 + 0.845631i \(0.679224\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1494.00 0.389584
\(246\) 0 0
\(247\) 2120.00 0.546123
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1008.00 0.253484 0.126742 − 0.991936i \(-0.459548\pi\)
0.126742 + 0.991936i \(0.459548\pi\)
\(252\) 0 0
\(253\) −9918.00 −2.46458
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −924.000 −0.224271 −0.112135 − 0.993693i \(-0.535769\pi\)
−0.112135 + 0.993693i \(0.535769\pi\)
\(258\) 0 0
\(259\) −58.0000 −0.0139148
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1014.00 −0.237741 −0.118871 − 0.992910i \(-0.537927\pi\)
−0.118871 + 0.992910i \(0.537927\pi\)
\(264\) 0 0
\(265\) 243.000 0.0563297
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2970.00 −0.673175 −0.336588 − 0.941652i \(-0.609273\pi\)
−0.336588 + 0.941652i \(0.609273\pi\)
\(270\) 0 0
\(271\) −245.000 −0.0549177 −0.0274588 − 0.999623i \(-0.508742\pi\)
−0.0274588 + 0.999623i \(0.508742\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6612.00 −1.44989
\(276\) 0 0
\(277\) 4376.00 0.949200 0.474600 − 0.880202i \(-0.342593\pi\)
0.474600 + 0.880202i \(0.342593\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 240.000 0.0509509 0.0254754 − 0.999675i \(-0.491890\pi\)
0.0254754 + 0.999675i \(0.491890\pi\)
\(282\) 0 0
\(283\) 6838.00 1.43631 0.718157 − 0.695881i \(-0.244986\pi\)
0.718157 + 0.695881i \(0.244986\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 174.000 0.0357871
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5118.00 −1.02047 −0.510233 − 0.860036i \(-0.670441\pi\)
−0.510233 + 0.860036i \(0.670441\pi\)
\(294\) 0 0
\(295\) −252.000 −0.0497356
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3480.00 −0.673089
\(300\) 0 0
\(301\) 6322.00 1.21061
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 168.000 0.0315398
\(306\) 0 0
\(307\) 5560.00 1.03364 0.516818 − 0.856096i \(-0.327117\pi\)
0.516818 + 0.856096i \(0.327117\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7662.00 −1.39702 −0.698508 − 0.715602i \(-0.746152\pi\)
−0.698508 + 0.715602i \(0.746152\pi\)
\(312\) 0 0
\(313\) 3485.00 0.629341 0.314671 − 0.949201i \(-0.398106\pi\)
0.314671 + 0.949201i \(0.398106\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7059.00 −1.25070 −0.625352 − 0.780343i \(-0.715044\pi\)
−0.625352 + 0.780343i \(0.715044\pi\)
\(318\) 0 0
\(319\) −11970.0 −2.10091
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7632.00 −1.31472
\(324\) 0 0
\(325\) −2320.00 −0.395971
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13746.0 2.30347
\(330\) 0 0
\(331\) −9290.00 −1.54267 −0.771336 − 0.636428i \(-0.780411\pi\)
−0.771336 + 0.636428i \(0.780411\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 426.000 0.0694772
\(336\) 0 0
\(337\) −3814.00 −0.616504 −0.308252 − 0.951305i \(-0.599744\pi\)
−0.308252 + 0.951305i \(0.599744\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2679.00 −0.425443
\(342\) 0 0
\(343\) −4495.00 −0.707601
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1929.00 −0.298427 −0.149213 − 0.988805i \(-0.547674\pi\)
−0.149213 + 0.988805i \(0.547674\pi\)
\(348\) 0 0
\(349\) −6586.00 −1.01014 −0.505072 − 0.863077i \(-0.668534\pi\)
−0.505072 + 0.863077i \(0.668534\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6042.00 −0.911001 −0.455500 − 0.890236i \(-0.650540\pi\)
−0.455500 + 0.890236i \(0.650540\pi\)
\(354\) 0 0
\(355\) −1080.00 −0.161466
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3762.00 0.553066 0.276533 − 0.961004i \(-0.410814\pi\)
0.276533 + 0.961004i \(0.410814\pi\)
\(360\) 0 0
\(361\) 4377.00 0.638140
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3477.00 −0.498615
\(366\) 0 0
\(367\) 7261.00 1.03276 0.516378 − 0.856361i \(-0.327280\pi\)
0.516378 + 0.856361i \(0.327280\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2349.00 −0.328717
\(372\) 0 0
\(373\) 1640.00 0.227657 0.113828 − 0.993500i \(-0.463689\pi\)
0.113828 + 0.993500i \(0.463689\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4200.00 −0.573769
\(378\) 0 0
\(379\) 7396.00 1.00239 0.501197 − 0.865333i \(-0.332893\pi\)
0.501197 + 0.865333i \(0.332893\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4992.00 0.666003 0.333002 − 0.942926i \(-0.391939\pi\)
0.333002 + 0.942926i \(0.391939\pi\)
\(384\) 0 0
\(385\) −4959.00 −0.656452
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9453.00 −1.23210 −0.616049 − 0.787708i \(-0.711268\pi\)
−0.616049 + 0.787708i \(0.711268\pi\)
\(390\) 0 0
\(391\) 12528.0 1.62038
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 480.000 0.0611428
\(396\) 0 0
\(397\) 8588.00 1.08569 0.542846 − 0.839833i \(-0.317347\pi\)
0.542846 + 0.839833i \(0.317347\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1716.00 −0.213698 −0.106849 − 0.994275i \(-0.534076\pi\)
−0.106849 + 0.994275i \(0.534076\pi\)
\(402\) 0 0
\(403\) −940.000 −0.116190
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 114.000 0.0138840
\(408\) 0 0
\(409\) −9889.00 −1.19555 −0.597775 − 0.801664i \(-0.703948\pi\)
−0.597775 + 0.801664i \(0.703948\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2436.00 0.290237
\(414\) 0 0
\(415\) −2205.00 −0.260817
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5556.00 −0.647800 −0.323900 − 0.946091i \(-0.604994\pi\)
−0.323900 + 0.946091i \(0.604994\pi\)
\(420\) 0 0
\(421\) −2104.00 −0.243569 −0.121785 − 0.992557i \(-0.538862\pi\)
−0.121785 + 0.992557i \(0.538862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8352.00 0.953251
\(426\) 0 0
\(427\) −1624.00 −0.184054
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7614.00 −0.850936 −0.425468 − 0.904973i \(-0.639891\pi\)
−0.425468 + 0.904973i \(0.639891\pi\)
\(432\) 0 0
\(433\) 7805.00 0.866246 0.433123 − 0.901335i \(-0.357412\pi\)
0.433123 + 0.901335i \(0.357412\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18444.0 −2.01898
\(438\) 0 0
\(439\) 5209.00 0.566314 0.283157 − 0.959074i \(-0.408618\pi\)
0.283157 + 0.959074i \(0.408618\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4236.00 −0.454308 −0.227154 − 0.973859i \(-0.572942\pi\)
−0.227154 + 0.973859i \(0.572942\pi\)
\(444\) 0 0
\(445\) −2862.00 −0.304880
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16002.0 −1.68192 −0.840959 − 0.541099i \(-0.818008\pi\)
−0.840959 + 0.541099i \(0.818008\pi\)
\(450\) 0 0
\(451\) −342.000 −0.0357077
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1740.00 −0.179280
\(456\) 0 0
\(457\) 7319.00 0.749165 0.374582 − 0.927194i \(-0.377786\pi\)
0.374582 + 0.927194i \(0.377786\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9483.00 0.958064 0.479032 − 0.877798i \(-0.340988\pi\)
0.479032 + 0.877798i \(0.340988\pi\)
\(462\) 0 0
\(463\) −10793.0 −1.08335 −0.541677 − 0.840586i \(-0.682211\pi\)
−0.541677 + 0.840586i \(0.682211\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2583.00 −0.255946 −0.127973 − 0.991778i \(-0.540847\pi\)
−0.127973 + 0.991778i \(0.540847\pi\)
\(468\) 0 0
\(469\) −4118.00 −0.405440
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12426.0 −1.20792
\(474\) 0 0
\(475\) −12296.0 −1.18775
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1254.00 0.119617 0.0598087 − 0.998210i \(-0.480951\pi\)
0.0598087 + 0.998210i \(0.480951\pi\)
\(480\) 0 0
\(481\) 40.0000 0.00379177
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 573.000 0.0536466
\(486\) 0 0
\(487\) −17336.0 −1.61308 −0.806539 − 0.591181i \(-0.798662\pi\)
−0.806539 + 0.591181i \(0.798662\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15171.0 1.39441 0.697207 − 0.716869i \(-0.254426\pi\)
0.697207 + 0.716869i \(0.254426\pi\)
\(492\) 0 0
\(493\) 15120.0 1.38128
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10440.0 0.942249
\(498\) 0 0
\(499\) −8930.00 −0.801126 −0.400563 − 0.916269i \(-0.631185\pi\)
−0.400563 + 0.916269i \(0.631185\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15210.0 −1.34827 −0.674136 − 0.738608i \(-0.735484\pi\)
−0.674136 + 0.738608i \(0.735484\pi\)
\(504\) 0 0
\(505\) −1089.00 −0.0959601
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19641.0 1.71036 0.855179 − 0.518333i \(-0.173447\pi\)
0.855179 + 0.518333i \(0.173447\pi\)
\(510\) 0 0
\(511\) 33611.0 2.90971
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1884.00 0.161202
\(516\) 0 0
\(517\) −27018.0 −2.29836
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22428.0 1.88597 0.942983 − 0.332840i \(-0.108007\pi\)
0.942983 + 0.332840i \(0.108007\pi\)
\(522\) 0 0
\(523\) 8152.00 0.681572 0.340786 − 0.940141i \(-0.389307\pi\)
0.340786 + 0.940141i \(0.389307\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3384.00 0.279714
\(528\) 0 0
\(529\) 18109.0 1.48837
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −120.000 −0.00975193
\(534\) 0 0
\(535\) −2025.00 −0.163642
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 28386.0 2.26841
\(540\) 0 0
\(541\) −2860.00 −0.227285 −0.113642 − 0.993522i \(-0.536252\pi\)
−0.113642 + 0.993522i \(0.536252\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5190.00 0.407918
\(546\) 0 0
\(547\) 9664.00 0.755398 0.377699 − 0.925928i \(-0.376715\pi\)
0.377699 + 0.925928i \(0.376715\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −22260.0 −1.72107
\(552\) 0 0
\(553\) −4640.00 −0.356804
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14859.0 1.13033 0.565167 − 0.824977i \(-0.308812\pi\)
0.565167 + 0.824977i \(0.308812\pi\)
\(558\) 0 0
\(559\) −4360.00 −0.329890
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8193.00 −0.613310 −0.306655 − 0.951821i \(-0.599210\pi\)
−0.306655 + 0.951821i \(0.599210\pi\)
\(564\) 0 0
\(565\) 5598.00 0.416831
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16572.0 −1.22097 −0.610487 − 0.792026i \(-0.709026\pi\)
−0.610487 + 0.792026i \(0.709026\pi\)
\(570\) 0 0
\(571\) 6244.00 0.457624 0.228812 − 0.973471i \(-0.426516\pi\)
0.228812 + 0.973471i \(0.426516\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20184.0 1.46388
\(576\) 0 0
\(577\) −14794.0 −1.06739 −0.533693 − 0.845678i \(-0.679196\pi\)
−0.533693 + 0.845678i \(0.679196\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 21315.0 1.52202
\(582\) 0 0
\(583\) 4617.00 0.327987
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26769.0 1.88224 0.941120 − 0.338073i \(-0.109775\pi\)
0.941120 + 0.338073i \(0.109775\pi\)
\(588\) 0 0
\(589\) −4982.00 −0.348522
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3078.00 −0.213151 −0.106575 − 0.994305i \(-0.533989\pi\)
−0.106575 + 0.994305i \(0.533989\pi\)
\(594\) 0 0
\(595\) 6264.00 0.431595
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1002.00 −0.0683483 −0.0341741 − 0.999416i \(-0.510880\pi\)
−0.0341741 + 0.999416i \(0.510880\pi\)
\(600\) 0 0
\(601\) −20653.0 −1.40175 −0.700876 − 0.713283i \(-0.747208\pi\)
−0.700876 + 0.713283i \(0.747208\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5754.00 0.386667
\(606\) 0 0
\(607\) −27128.0 −1.81399 −0.906995 − 0.421142i \(-0.861629\pi\)
−0.906995 + 0.421142i \(0.861629\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9480.00 −0.627692
\(612\) 0 0
\(613\) 24518.0 1.61545 0.807727 − 0.589557i \(-0.200698\pi\)
0.807727 + 0.589557i \(0.200698\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −474.000 −0.0309279 −0.0154640 − 0.999880i \(-0.504923\pi\)
−0.0154640 + 0.999880i \(0.504923\pi\)
\(618\) 0 0
\(619\) 1132.00 0.0735039 0.0367520 − 0.999324i \(-0.488299\pi\)
0.0367520 + 0.999324i \(0.488299\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27666.0 1.77916
\(624\) 0 0
\(625\) 12331.0 0.789184
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −144.000 −0.00912823
\(630\) 0 0
\(631\) −6725.00 −0.424276 −0.212138 − 0.977240i \(-0.568043\pi\)
−0.212138 + 0.977240i \(0.568043\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4137.00 −0.258538
\(636\) 0 0
\(637\) 9960.00 0.619513
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21126.0 −1.30176 −0.650879 − 0.759182i \(-0.725599\pi\)
−0.650879 + 0.759182i \(0.725599\pi\)
\(642\) 0 0
\(643\) −19460.0 −1.19351 −0.596755 − 0.802423i \(-0.703544\pi\)
−0.596755 + 0.802423i \(0.703544\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11664.0 0.708747 0.354373 − 0.935104i \(-0.384694\pi\)
0.354373 + 0.935104i \(0.384694\pi\)
\(648\) 0 0
\(649\) −4788.00 −0.289592
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3345.00 −0.200459 −0.100230 − 0.994964i \(-0.531958\pi\)
−0.100230 + 0.994964i \(0.531958\pi\)
\(654\) 0 0
\(655\) −1737.00 −0.103619
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9393.00 −0.555234 −0.277617 − 0.960692i \(-0.589545\pi\)
−0.277617 + 0.960692i \(0.589545\pi\)
\(660\) 0 0
\(661\) −1762.00 −0.103682 −0.0518410 − 0.998655i \(-0.516509\pi\)
−0.0518410 + 0.998655i \(0.516509\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9222.00 −0.537765
\(666\) 0 0
\(667\) 36540.0 2.12119
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3192.00 0.183645
\(672\) 0 0
\(673\) 25517.0 1.46153 0.730764 − 0.682630i \(-0.239164\pi\)
0.730764 + 0.682630i \(0.239164\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26898.0 1.52699 0.763496 − 0.645812i \(-0.223481\pi\)
0.763496 + 0.645812i \(0.223481\pi\)
\(678\) 0 0
\(679\) −5539.00 −0.313059
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23940.0 −1.34120 −0.670599 − 0.741820i \(-0.733963\pi\)
−0.670599 + 0.741820i \(0.733963\pi\)
\(684\) 0 0
\(685\) 1962.00 0.109437
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1620.00 0.0895749
\(690\) 0 0
\(691\) −23060.0 −1.26953 −0.634764 − 0.772706i \(-0.718903\pi\)
−0.634764 + 0.772706i \(0.718903\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9012.00 0.491863
\(696\) 0 0
\(697\) 432.000 0.0234766
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14175.0 −0.763741 −0.381870 − 0.924216i \(-0.624720\pi\)
−0.381870 + 0.924216i \(0.624720\pi\)
\(702\) 0 0
\(703\) 212.000 0.0113737
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10527.0 0.559984
\(708\) 0 0
\(709\) −8692.00 −0.460416 −0.230208 − 0.973141i \(-0.573941\pi\)
−0.230208 + 0.973141i \(0.573941\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8178.00 0.429549
\(714\) 0 0
\(715\) 3420.00 0.178882
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29556.0 1.53304 0.766518 − 0.642223i \(-0.221988\pi\)
0.766518 + 0.642223i \(0.221988\pi\)
\(720\) 0 0
\(721\) −18212.0 −0.940708
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24360.0 1.24787
\(726\) 0 0
\(727\) 36691.0 1.87179 0.935897 − 0.352274i \(-0.114592\pi\)
0.935897 + 0.352274i \(0.114592\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15696.0 0.794169
\(732\) 0 0
\(733\) −19798.0 −0.997620 −0.498810 − 0.866711i \(-0.666230\pi\)
−0.498810 + 0.866711i \(0.666230\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8094.00 0.404540
\(738\) 0 0
\(739\) 21976.0 1.09391 0.546955 − 0.837162i \(-0.315787\pi\)
0.546955 + 0.837162i \(0.315787\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13236.0 0.653542 0.326771 − 0.945104i \(-0.394039\pi\)
0.326771 + 0.945104i \(0.394039\pi\)
\(744\) 0 0
\(745\) 5409.00 0.266001
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19575.0 0.954947
\(750\) 0 0
\(751\) 6325.00 0.307327 0.153663 − 0.988123i \(-0.450893\pi\)
0.153663 + 0.988123i \(0.450893\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7377.00 −0.355598
\(756\) 0 0
\(757\) −3238.00 −0.155465 −0.0777326 − 0.996974i \(-0.524768\pi\)
−0.0777326 + 0.996974i \(0.524768\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40416.0 −1.92520 −0.962601 − 0.270923i \(-0.912671\pi\)
−0.962601 + 0.270923i \(0.912671\pi\)
\(762\) 0 0
\(763\) −50170.0 −2.38044
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1680.00 −0.0790890
\(768\) 0 0
\(769\) −4759.00 −0.223165 −0.111583 − 0.993755i \(-0.535592\pi\)
−0.111583 + 0.993755i \(0.535592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27414.0 1.27557 0.637783 − 0.770216i \(-0.279852\pi\)
0.637783 + 0.770216i \(0.279852\pi\)
\(774\) 0 0
\(775\) 5452.00 0.252699
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −636.000 −0.0292517
\(780\) 0 0
\(781\) −20520.0 −0.940158
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −588.000 −0.0267345
\(786\) 0 0
\(787\) −6176.00 −0.279734 −0.139867 − 0.990170i \(-0.544667\pi\)
−0.139867 + 0.990170i \(0.544667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −54114.0 −2.43246
\(792\) 0 0
\(793\) 1120.00 0.0501543
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6879.00 0.305730 0.152865 − 0.988247i \(-0.451150\pi\)
0.152865 + 0.988247i \(0.451150\pi\)
\(798\) 0 0
\(799\) 34128.0 1.51109
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −66063.0 −2.90325
\(804\) 0 0
\(805\) 15138.0 0.662788
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16902.0 −0.734540 −0.367270 − 0.930114i \(-0.619707\pi\)
−0.367270 + 0.930114i \(0.619707\pi\)
\(810\) 0 0
\(811\) −24086.0 −1.04288 −0.521439 − 0.853289i \(-0.674605\pi\)
−0.521439 + 0.853289i \(0.674605\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4692.00 0.201661
\(816\) 0 0
\(817\) −23108.0 −0.989531
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7854.00 0.333869 0.166935 − 0.985968i \(-0.446613\pi\)
0.166935 + 0.985968i \(0.446613\pi\)
\(822\) 0 0
\(823\) −5771.00 −0.244428 −0.122214 − 0.992504i \(-0.538999\pi\)
−0.122214 + 0.992504i \(0.538999\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17568.0 0.738693 0.369347 − 0.929292i \(-0.379582\pi\)
0.369347 + 0.929292i \(0.379582\pi\)
\(828\) 0 0
\(829\) 31322.0 1.31225 0.656127 − 0.754651i \(-0.272194\pi\)
0.656127 + 0.754651i \(0.272194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −35856.0 −1.49140
\(834\) 0 0
\(835\) −5922.00 −0.245436
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −41856.0 −1.72232 −0.861162 − 0.508331i \(-0.830263\pi\)
−0.861162 + 0.508331i \(0.830263\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5391.00 −0.219475
\(846\) 0 0
\(847\) −55622.0 −2.25643
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −348.000 −0.0140180
\(852\) 0 0
\(853\) 15662.0 0.628671 0.314336 − 0.949312i \(-0.398218\pi\)
0.314336 + 0.949312i \(0.398218\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39864.0 −1.58895 −0.794474 − 0.607298i \(-0.792253\pi\)
−0.794474 + 0.607298i \(0.792253\pi\)
\(858\) 0 0
\(859\) 9160.00 0.363836 0.181918 − 0.983314i \(-0.441769\pi\)
0.181918 + 0.983314i \(0.441769\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5076.00 −0.200219 −0.100110 − 0.994976i \(-0.531919\pi\)
−0.100110 + 0.994976i \(0.531919\pi\)
\(864\) 0 0
\(865\) −6651.00 −0.261434
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9120.00 0.356012
\(870\) 0 0
\(871\) 2840.00 0.110482
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 20967.0 0.810073
\(876\) 0 0
\(877\) 14978.0 0.576706 0.288353 − 0.957524i \(-0.406892\pi\)
0.288353 + 0.957524i \(0.406892\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −22860.0 −0.874203 −0.437102 − 0.899412i \(-0.643995\pi\)
−0.437102 + 0.899412i \(0.643995\pi\)
\(882\) 0 0
\(883\) 32506.0 1.23886 0.619430 − 0.785052i \(-0.287364\pi\)
0.619430 + 0.785052i \(0.287364\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −35868.0 −1.35776 −0.678878 − 0.734251i \(-0.737533\pi\)
−0.678878 + 0.734251i \(0.737533\pi\)
\(888\) 0 0
\(889\) 39991.0 1.50872
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −50244.0 −1.88281
\(894\) 0 0
\(895\) 7425.00 0.277308
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9870.00 0.366166
\(900\) 0 0
\(901\) −5832.00 −0.215640
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4704.00 0.172780
\(906\) 0 0
\(907\) 33586.0 1.22955 0.614777 − 0.788701i \(-0.289246\pi\)
0.614777 + 0.788701i \(0.289246\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28902.0 1.05112 0.525558 − 0.850758i \(-0.323857\pi\)
0.525558 + 0.850758i \(0.323857\pi\)
\(912\) 0 0
\(913\) −41895.0 −1.51864
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16791.0 0.604676
\(918\) 0 0
\(919\) −28271.0 −1.01477 −0.507385 − 0.861719i \(-0.669388\pi\)
−0.507385 + 0.861719i \(0.669388\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7200.00 −0.256762
\(924\) 0 0
\(925\) −232.000 −0.00824661
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 19140.0 0.675956 0.337978 − 0.941154i \(-0.390257\pi\)
0.337978 + 0.941154i \(0.390257\pi\)
\(930\) 0 0
\(931\) 52788.0 1.85828
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12312.0 −0.430637
\(936\) 0 0
\(937\) 31619.0 1.10240 0.551199 − 0.834374i \(-0.314170\pi\)
0.551199 + 0.834374i \(0.314170\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20913.0 −0.724489 −0.362245 − 0.932083i \(-0.617989\pi\)
−0.362245 + 0.932083i \(0.617989\pi\)
\(942\) 0 0
\(943\) 1044.00 0.0360523
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17529.0 −0.601495 −0.300748 − 0.953704i \(-0.597236\pi\)
−0.300748 + 0.953704i \(0.597236\pi\)
\(948\) 0 0
\(949\) −23180.0 −0.792892
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −53604.0 −1.82204 −0.911020 − 0.412362i \(-0.864704\pi\)
−0.911020 + 0.412362i \(0.864704\pi\)
\(954\) 0 0
\(955\) −3420.00 −0.115883
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18966.0 −0.638628
\(960\) 0 0
\(961\) −27582.0 −0.925850
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6135.00 0.204656
\(966\) 0 0
\(967\) −11117.0 −0.369699 −0.184849 − 0.982767i \(-0.559180\pi\)
−0.184849 + 0.982767i \(0.559180\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27297.0 0.902165 0.451083 − 0.892482i \(-0.351038\pi\)
0.451083 + 0.892482i \(0.351038\pi\)
\(972\) 0 0
\(973\) −87116.0 −2.87031
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25086.0 0.821466 0.410733 − 0.911756i \(-0.365273\pi\)
0.410733 + 0.911756i \(0.365273\pi\)
\(978\) 0 0
\(979\) −54378.0 −1.77521
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20982.0 0.680795 0.340398 − 0.940282i \(-0.389438\pi\)
0.340398 + 0.940282i \(0.389438\pi\)
\(984\) 0 0
\(985\) −11205.0 −0.362458
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 37932.0 1.21958
\(990\) 0 0
\(991\) −11477.0 −0.367890 −0.183945 − 0.982937i \(-0.558887\pi\)
−0.183945 + 0.982937i \(0.558887\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3489.00 −0.111165
\(996\) 0 0
\(997\) 8588.00 0.272803 0.136402 − 0.990654i \(-0.456446\pi\)
0.136402 + 0.990654i \(0.456446\pi\)
\(998\) 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 432.4.a.j.1.1 1
3.2 odd 2 432.4.a.e.1.1 1
4.3 odd 2 54.4.a.c.1.1 yes 1
8.3 odd 2 1728.4.a.l.1.1 1
8.5 even 2 1728.4.a.k.1.1 1
12.11 even 2 54.4.a.b.1.1 ✓ 1
20.3 even 4 1350.4.c.b.649.1 2
20.7 even 4 1350.4.c.b.649.2 2
20.19 odd 2 1350.4.a.a.1.1 1
24.5 odd 2 1728.4.a.u.1.1 1
24.11 even 2 1728.4.a.v.1.1 1
36.7 odd 6 162.4.c.b.109.1 2
36.11 even 6 162.4.c.g.109.1 2
36.23 even 6 162.4.c.g.55.1 2
36.31 odd 6 162.4.c.b.55.1 2
60.23 odd 4 1350.4.c.s.649.2 2
60.47 odd 4 1350.4.c.s.649.1 2
60.59 even 2 1350.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.4.a.b.1.1 ✓ 1 12.11 even 2
54.4.a.c.1.1 yes 1 4.3 odd 2
162.4.c.b.55.1 2 36.31 odd 6
162.4.c.b.109.1 2 36.7 odd 6
162.4.c.g.55.1 2 36.23 even 6
162.4.c.g.109.1 2 36.11 even 6
432.4.a.e.1.1 1 3.2 odd 2
432.4.a.j.1.1 1 1.1 even 1 trivial
1350.4.a.a.1.1 1 20.19 odd 2
1350.4.a.o.1.1 1 60.59 even 2
1350.4.c.b.649.1 2 20.3 even 4
1350.4.c.b.649.2 2 20.7 even 4
1350.4.c.s.649.1 2 60.47 odd 4
1350.4.c.s.649.2 2 60.23 odd 4
1728.4.a.k.1.1 1 8.5 even 2
1728.4.a.l.1.1 1 8.3 odd 2
1728.4.a.u.1.1 1 24.5 odd 2
1728.4.a.v.1.1 1 24.11 even 2