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Properties

Label 1800.4.a.b.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
Analytic rank $1$
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] = [1800,4,Mod(1,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, 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("1800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-34.0000 q^{7} +O(q^{10})\) \(q-34.0000 q^{7} +18.0000 q^{11} -12.0000 q^{13} +106.000 q^{17} -44.0000 q^{19} -56.0000 q^{23} +270.000 q^{29} +204.000 q^{31} -120.000 q^{37} +80.0000 q^{41} -536.000 q^{43} +536.000 q^{47} +813.000 q^{49} -542.000 q^{53} -174.000 q^{59} +186.000 q^{61} -332.000 q^{67} -132.000 q^{71} +602.000 q^{73} -612.000 q^{77} -548.000 q^{79} +492.000 q^{83} -1052.00 q^{89} +408.000 q^{91} -482.000 q^{97} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −34.0000 −1.83583 −0.917914 − 0.396780i \(-0.870128\pi\)
−0.917914 + 0.396780i \(0.870128\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.0000 0.493382 0.246691 − 0.969094i \(-0.420657\pi\)
0.246691 + 0.969094i \(0.420657\pi\)
\(12\) 0 0
\(13\) −12.0000 −0.256015 −0.128008 − 0.991773i \(-0.540858\pi\)
−0.128008 + 0.991773i \(0.540858\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 106.000 1.51228 0.756140 − 0.654409i \(-0.227083\pi\)
0.756140 + 0.654409i \(0.227083\pi\)
\(18\) 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
\(23\) −56.0000 −0.507687 −0.253844 − 0.967245i \(-0.581695\pi\)
−0.253844 + 0.967245i \(0.581695\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 270.000 1.72889 0.864444 − 0.502729i \(-0.167671\pi\)
0.864444 + 0.502729i \(0.167671\pi\)
\(30\) 0 0
\(31\) 204.000 1.18192 0.590959 − 0.806701i \(-0.298749\pi\)
0.590959 + 0.806701i \(0.298749\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −120.000 −0.533186 −0.266593 − 0.963809i \(-0.585898\pi\)
−0.266593 + 0.963809i \(0.585898\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 80.0000 0.304729 0.152365 − 0.988324i \(-0.451311\pi\)
0.152365 + 0.988324i \(0.451311\pi\)
\(42\) 0 0
\(43\) −536.000 −1.90091 −0.950456 − 0.310858i \(-0.899383\pi\)
−0.950456 + 0.310858i \(0.899383\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 536.000 1.66348 0.831741 − 0.555164i \(-0.187345\pi\)
0.831741 + 0.555164i \(0.187345\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −542.000 −1.40471 −0.702353 − 0.711829i \(-0.747867\pi\)
−0.702353 + 0.711829i \(0.747867\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −174.000 −0.383947 −0.191973 − 0.981400i \(-0.561489\pi\)
−0.191973 + 0.981400i \(0.561489\pi\)
\(60\) 0 0
\(61\) 186.000 0.390408 0.195204 − 0.980763i \(-0.437463\pi\)
0.195204 + 0.980763i \(0.437463\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −332.000 −0.605377 −0.302688 − 0.953090i \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −132.000 −0.220641 −0.110321 − 0.993896i \(-0.535188\pi\)
−0.110321 + 0.993896i \(0.535188\pi\)
\(72\) 0 0
\(73\) 602.000 0.965189 0.482594 − 0.875844i \(-0.339695\pi\)
0.482594 + 0.875844i \(0.339695\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −612.000 −0.905765
\(78\) 0 0
\(79\) −548.000 −0.780441 −0.390220 − 0.920721i \(-0.627601\pi\)
−0.390220 + 0.920721i \(0.627601\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 492.000 0.650651 0.325325 − 0.945602i \(-0.394526\pi\)
0.325325 + 0.945602i \(0.394526\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1052.00 −1.25294 −0.626471 − 0.779445i \(-0.715501\pi\)
−0.626471 + 0.779445i \(0.715501\pi\)
\(90\) 0 0
\(91\) 408.000 0.470000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −482.000 −0.504533 −0.252266 − 0.967658i \(-0.581176\pi\)
−0.252266 + 0.967658i \(0.581176\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1214.00 1.19601 0.598007 − 0.801491i \(-0.295959\pi\)
0.598007 + 0.801491i \(0.295959\pi\)
\(102\) 0 0
\(103\) −898.000 −0.859054 −0.429527 − 0.903054i \(-0.641320\pi\)
−0.429527 + 0.903054i \(0.641320\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1364.00 −1.23236 −0.616182 − 0.787604i \(-0.711321\pi\)
−0.616182 + 0.787604i \(0.711321\pi\)
\(108\) 0 0
\(109\) 218.000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1386.00 −1.15384 −0.576920 − 0.816801i \(-0.695746\pi\)
−0.576920 + 0.816801i \(0.695746\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3604.00 −2.77629
\(120\) 0 0
\(121\) −1007.00 −0.756574
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 814.000 0.568747 0.284373 − 0.958714i \(-0.408214\pi\)
0.284373 + 0.958714i \(0.408214\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1282.00 −0.855029 −0.427515 − 0.904008i \(-0.640611\pi\)
−0.427515 + 0.904008i \(0.640611\pi\)
\(132\) 0 0
\(133\) 1496.00 0.975336
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3066.00 −1.91202 −0.956008 − 0.293342i \(-0.905232\pi\)
−0.956008 + 0.293342i \(0.905232\pi\)
\(138\) 0 0
\(139\) −1332.00 −0.812797 −0.406398 − 0.913696i \(-0.633216\pi\)
−0.406398 + 0.913696i \(0.633216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −216.000 −0.126313
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1470.00 −0.808236 −0.404118 − 0.914707i \(-0.632421\pi\)
−0.404118 + 0.914707i \(0.632421\pi\)
\(150\) 0 0
\(151\) −2592.00 −1.39691 −0.698457 − 0.715652i \(-0.746130\pi\)
−0.698457 + 0.715652i \(0.746130\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3332.00 1.69377 0.846887 − 0.531773i \(-0.178474\pi\)
0.846887 + 0.531773i \(0.178474\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1904.00 0.932026
\(162\) 0 0
\(163\) 748.000 0.359435 0.179717 − 0.983718i \(-0.442482\pi\)
0.179717 + 0.983718i \(0.442482\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2560.00 1.18622 0.593110 − 0.805121i \(-0.297900\pi\)
0.593110 + 0.805121i \(0.297900\pi\)
\(168\) 0 0
\(169\) −2053.00 −0.934456
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1206.00 −0.530003 −0.265001 − 0.964248i \(-0.585372\pi\)
−0.265001 + 0.964248i \(0.585372\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1694.00 −0.707349 −0.353675 − 0.935369i \(-0.615068\pi\)
−0.353675 + 0.935369i \(0.615068\pi\)
\(180\) 0 0
\(181\) 3722.00 1.52848 0.764238 − 0.644935i \(-0.223115\pi\)
0.764238 + 0.644935i \(0.223115\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1908.00 0.746133
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2836.00 1.07438 0.537188 − 0.843463i \(-0.319487\pi\)
0.537188 + 0.843463i \(0.319487\pi\)
\(192\) 0 0
\(193\) 234.000 0.0872730 0.0436365 − 0.999047i \(-0.486106\pi\)
0.0436365 + 0.999047i \(0.486106\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3814.00 1.37937 0.689686 − 0.724109i \(-0.257749\pi\)
0.689686 + 0.724109i \(0.257749\pi\)
\(198\) 0 0
\(199\) 2352.00 0.837833 0.418917 − 0.908025i \(-0.362410\pi\)
0.418917 + 0.908025i \(0.362410\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9180.00 −3.17394
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −792.000 −0.262123
\(210\) 0 0
\(211\) −3660.00 −1.19415 −0.597073 − 0.802187i \(-0.703670\pi\)
−0.597073 + 0.802187i \(0.703670\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6936.00 −2.16980
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1272.00 −0.387167
\(222\) 0 0
\(223\) 2646.00 0.794571 0.397285 − 0.917695i \(-0.369952\pi\)
0.397285 + 0.917695i \(0.369952\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −240.000 −0.0701734 −0.0350867 − 0.999384i \(-0.511171\pi\)
−0.0350867 + 0.999384i \(0.511171\pi\)
\(228\) 0 0
\(229\) −4698.00 −1.35569 −0.677844 − 0.735206i \(-0.737086\pi\)
−0.677844 + 0.735206i \(0.737086\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3814.00 −1.07238 −0.536188 − 0.844099i \(-0.680136\pi\)
−0.536188 + 0.844099i \(0.680136\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2148.00 −0.581350 −0.290675 − 0.956822i \(-0.593880\pi\)
−0.290675 + 0.956822i \(0.593880\pi\)
\(240\) 0 0
\(241\) −3370.00 −0.900750 −0.450375 − 0.892839i \(-0.648710\pi\)
−0.450375 + 0.892839i \(0.648710\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 528.000 0.136016
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6134.00 1.54253 0.771264 − 0.636515i \(-0.219625\pi\)
0.771264 + 0.636515i \(0.219625\pi\)
\(252\) 0 0
\(253\) −1008.00 −0.250484
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4566.00 1.10825 0.554123 − 0.832435i \(-0.313054\pi\)
0.554123 + 0.832435i \(0.313054\pi\)
\(258\) 0 0
\(259\) 4080.00 0.978837
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1920.00 0.450161 0.225080 − 0.974340i \(-0.427736\pi\)
0.225080 + 0.974340i \(0.427736\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5802.00 −1.31507 −0.657536 − 0.753423i \(-0.728401\pi\)
−0.657536 + 0.753423i \(0.728401\pi\)
\(270\) 0 0
\(271\) 1640.00 0.367612 0.183806 − 0.982963i \(-0.441158\pi\)
0.183806 + 0.982963i \(0.441158\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2792.00 0.605614 0.302807 − 0.953052i \(-0.402076\pi\)
0.302807 + 0.953052i \(0.402076\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1108.00 0.235223 0.117612 − 0.993060i \(-0.462476\pi\)
0.117612 + 0.993060i \(0.462476\pi\)
\(282\) 0 0
\(283\) −6028.00 −1.26617 −0.633087 − 0.774080i \(-0.718213\pi\)
−0.633087 + 0.774080i \(0.718213\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2720.00 −0.559430
\(288\) 0 0
\(289\) 6323.00 1.28699
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7994.00 −1.59391 −0.796953 − 0.604041i \(-0.793556\pi\)
−0.796953 + 0.604041i \(0.793556\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 672.000 0.129976
\(300\) 0 0
\(301\) 18224.0 3.48975
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −736.000 −0.136827 −0.0684133 − 0.997657i \(-0.521794\pi\)
−0.0684133 + 0.997657i \(0.521794\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5380.00 −0.980938 −0.490469 − 0.871459i \(-0.663175\pi\)
−0.490469 + 0.871459i \(0.663175\pi\)
\(312\) 0 0
\(313\) 1370.00 0.247402 0.123701 − 0.992320i \(-0.460524\pi\)
0.123701 + 0.992320i \(0.460524\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5770.00 −1.02232 −0.511160 − 0.859486i \(-0.670784\pi\)
−0.511160 + 0.859486i \(0.670784\pi\)
\(318\) 0 0
\(319\) 4860.00 0.853002
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4664.00 −0.803442
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18224.0 −3.05387
\(330\) 0 0
\(331\) −4172.00 −0.692791 −0.346396 − 0.938089i \(-0.612594\pi\)
−0.346396 + 0.938089i \(0.612594\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8206.00 −1.32644 −0.663219 − 0.748426i \(-0.730810\pi\)
−0.663219 + 0.748426i \(0.730810\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3672.00 0.583138
\(342\) 0 0
\(343\) −15980.0 −2.51557
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10848.0 −1.67825 −0.839123 − 0.543942i \(-0.816931\pi\)
−0.839123 + 0.543942i \(0.816931\pi\)
\(348\) 0 0
\(349\) −1694.00 −0.259822 −0.129911 − 0.991526i \(-0.541469\pi\)
−0.129911 + 0.991526i \(0.541469\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6642.00 1.00147 0.500734 − 0.865601i \(-0.333064\pi\)
0.500734 + 0.865601i \(0.333064\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10376.0 −1.52542 −0.762708 − 0.646743i \(-0.776131\pi\)
−0.762708 + 0.646743i \(0.776131\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2198.00 0.312629 0.156314 − 0.987707i \(-0.450039\pi\)
0.156314 + 0.987707i \(0.450039\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18428.0 2.57880
\(372\) 0 0
\(373\) 12220.0 1.69632 0.848160 − 0.529740i \(-0.177710\pi\)
0.848160 + 0.529740i \(0.177710\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3240.00 −0.442622
\(378\) 0 0
\(379\) −10388.0 −1.40790 −0.703952 − 0.710247i \(-0.748583\pi\)
−0.703952 + 0.710247i \(0.748583\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10552.0 −1.40779 −0.703893 − 0.710306i \(-0.748557\pi\)
−0.703893 + 0.710306i \(0.748557\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8262.00 −1.07686 −0.538432 − 0.842669i \(-0.680983\pi\)
−0.538432 + 0.842669i \(0.680983\pi\)
\(390\) 0 0
\(391\) −5936.00 −0.767766
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2864.00 −0.362066 −0.181033 − 0.983477i \(-0.557944\pi\)
−0.181033 + 0.983477i \(0.557944\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12588.0 −1.56762 −0.783809 − 0.621002i \(-0.786726\pi\)
−0.783809 + 0.621002i \(0.786726\pi\)
\(402\) 0 0
\(403\) −2448.00 −0.302589
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2160.00 −0.263064
\(408\) 0 0
\(409\) 10330.0 1.24886 0.624432 − 0.781079i \(-0.285330\pi\)
0.624432 + 0.781079i \(0.285330\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5916.00 0.704860
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1250.00 −0.145743 −0.0728717 − 0.997341i \(-0.523216\pi\)
−0.0728717 + 0.997341i \(0.523216\pi\)
\(420\) 0 0
\(421\) 5670.00 0.656387 0.328193 − 0.944611i \(-0.393560\pi\)
0.328193 + 0.944611i \(0.393560\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6324.00 −0.716721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12976.0 −1.45019 −0.725095 − 0.688649i \(-0.758204\pi\)
−0.725095 + 0.688649i \(0.758204\pi\)
\(432\) 0 0
\(433\) 9050.00 1.00442 0.502212 − 0.864745i \(-0.332520\pi\)
0.502212 + 0.864745i \(0.332520\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2464.00 0.269723
\(438\) 0 0
\(439\) −17528.0 −1.90562 −0.952808 − 0.303572i \(-0.901821\pi\)
−0.952808 + 0.303572i \(0.901821\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2568.00 −0.275416 −0.137708 − 0.990473i \(-0.543974\pi\)
−0.137708 + 0.990473i \(0.543974\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12652.0 1.32981 0.664905 − 0.746928i \(-0.268472\pi\)
0.664905 + 0.746928i \(0.268472\pi\)
\(450\) 0 0
\(451\) 1440.00 0.150348
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6230.00 0.637696 0.318848 − 0.947806i \(-0.396704\pi\)
0.318848 + 0.947806i \(0.396704\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5290.00 0.534447 0.267223 − 0.963635i \(-0.413894\pi\)
0.267223 + 0.963635i \(0.413894\pi\)
\(462\) 0 0
\(463\) −8110.00 −0.814047 −0.407023 − 0.913418i \(-0.633433\pi\)
−0.407023 + 0.913418i \(0.633433\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2020.00 0.200159 0.100080 − 0.994979i \(-0.468090\pi\)
0.100080 + 0.994979i \(0.468090\pi\)
\(468\) 0 0
\(469\) 11288.0 1.11137
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9648.00 −0.937876
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9684.00 −0.923744 −0.461872 − 0.886947i \(-0.652822\pi\)
−0.461872 + 0.886947i \(0.652822\pi\)
\(480\) 0 0
\(481\) 1440.00 0.136504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18426.0 1.71450 0.857250 − 0.514900i \(-0.172171\pi\)
0.857250 + 0.514900i \(0.172171\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4558.00 0.418940 0.209470 − 0.977815i \(-0.432826\pi\)
0.209470 + 0.977815i \(0.432826\pi\)
\(492\) 0 0
\(493\) 28620.0 2.61456
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4488.00 0.405059
\(498\) 0 0
\(499\) −460.000 −0.0412674 −0.0206337 − 0.999787i \(-0.506568\pi\)
−0.0206337 + 0.999787i \(0.506568\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8568.00 0.759499 0.379750 − 0.925089i \(-0.376010\pi\)
0.379750 + 0.925089i \(0.376010\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16374.0 1.42586 0.712932 − 0.701233i \(-0.247367\pi\)
0.712932 + 0.701233i \(0.247367\pi\)
\(510\) 0 0
\(511\) −20468.0 −1.77192
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9648.00 0.820732
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21620.0 1.81802 0.909011 − 0.416772i \(-0.136839\pi\)
0.909011 + 0.416772i \(0.136839\pi\)
\(522\) 0 0
\(523\) −16524.0 −1.38154 −0.690769 − 0.723076i \(-0.742728\pi\)
−0.690769 + 0.723076i \(0.742728\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21624.0 1.78739
\(528\) 0 0
\(529\) −9031.00 −0.742254
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −960.000 −0.0780154
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14634.0 1.16945
\(540\) 0 0
\(541\) −4990.00 −0.396556 −0.198278 − 0.980146i \(-0.563535\pi\)
−0.198278 + 0.980146i \(0.563535\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15224.0 1.19000 0.595001 − 0.803725i \(-0.297152\pi\)
0.595001 + 0.803725i \(0.297152\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11880.0 −0.918521
\(552\) 0 0
\(553\) 18632.0 1.43275
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5698.00 0.433451 0.216725 − 0.976233i \(-0.430462\pi\)
0.216725 + 0.976233i \(0.430462\pi\)
\(558\) 0 0
\(559\) 6432.00 0.486663
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5976.00 −0.447351 −0.223675 − 0.974664i \(-0.571806\pi\)
−0.223675 + 0.974664i \(0.571806\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16460.0 1.21272 0.606361 − 0.795189i \(-0.292629\pi\)
0.606361 + 0.795189i \(0.292629\pi\)
\(570\) 0 0
\(571\) −18236.0 −1.33652 −0.668260 − 0.743928i \(-0.732961\pi\)
−0.668260 + 0.743928i \(0.732961\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −20842.0 −1.50375 −0.751875 − 0.659306i \(-0.770850\pi\)
−0.751875 + 0.659306i \(0.770850\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16728.0 −1.19448
\(582\) 0 0
\(583\) −9756.00 −0.693057
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11772.0 0.827738 0.413869 − 0.910336i \(-0.364177\pi\)
0.413869 + 0.910336i \(0.364177\pi\)
\(588\) 0 0
\(589\) −8976.00 −0.627928
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4514.00 −0.312593 −0.156297 − 0.987710i \(-0.549956\pi\)
−0.156297 + 0.987710i \(0.549956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25096.0 1.71184 0.855922 − 0.517105i \(-0.172990\pi\)
0.855922 + 0.517105i \(0.172990\pi\)
\(600\) 0 0
\(601\) 16262.0 1.10373 0.551864 − 0.833934i \(-0.313917\pi\)
0.551864 + 0.833934i \(0.313917\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2262.00 −0.151255 −0.0756275 − 0.997136i \(-0.524096\pi\)
−0.0756275 + 0.997136i \(0.524096\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6432.00 −0.425877
\(612\) 0 0
\(613\) −14216.0 −0.936670 −0.468335 − 0.883551i \(-0.655146\pi\)
−0.468335 + 0.883551i \(0.655146\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2558.00 −0.166906 −0.0834532 − 0.996512i \(-0.526595\pi\)
−0.0834532 + 0.996512i \(0.526595\pi\)
\(618\) 0 0
\(619\) −17044.0 −1.10671 −0.553357 − 0.832944i \(-0.686654\pi\)
−0.553357 + 0.832944i \(0.686654\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 35768.0 2.30018
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12720.0 −0.806327
\(630\) 0 0
\(631\) 20980.0 1.32361 0.661807 − 0.749674i \(-0.269790\pi\)
0.661807 + 0.749674i \(0.269790\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9756.00 −0.606824
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7176.00 0.442176 0.221088 − 0.975254i \(-0.429039\pi\)
0.221088 + 0.975254i \(0.429039\pi\)
\(642\) 0 0
\(643\) 2724.00 0.167067 0.0835335 − 0.996505i \(-0.473379\pi\)
0.0835335 + 0.996505i \(0.473379\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10392.0 0.631455 0.315728 − 0.948850i \(-0.397751\pi\)
0.315728 + 0.948850i \(0.397751\pi\)
\(648\) 0 0
\(649\) −3132.00 −0.189433
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11958.0 0.716620 0.358310 − 0.933603i \(-0.383353\pi\)
0.358310 + 0.933603i \(0.383353\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13366.0 0.790084 0.395042 − 0.918663i \(-0.370730\pi\)
0.395042 + 0.918663i \(0.370730\pi\)
\(660\) 0 0
\(661\) 14698.0 0.864880 0.432440 − 0.901663i \(-0.357653\pi\)
0.432440 + 0.901663i \(0.357653\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −15120.0 −0.877734
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3348.00 0.192620
\(672\) 0 0
\(673\) 7570.00 0.433584 0.216792 − 0.976218i \(-0.430441\pi\)
0.216792 + 0.976218i \(0.430441\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21378.0 −1.21362 −0.606812 − 0.794845i \(-0.707552\pi\)
−0.606812 + 0.794845i \(0.707552\pi\)
\(678\) 0 0
\(679\) 16388.0 0.926235
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15804.0 0.885393 0.442696 − 0.896672i \(-0.354022\pi\)
0.442696 + 0.896672i \(0.354022\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6504.00 0.359627
\(690\) 0 0
\(691\) −22028.0 −1.21271 −0.606356 − 0.795193i \(-0.707370\pi\)
−0.606356 + 0.795193i \(0.707370\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8480.00 0.460836
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1762.00 −0.0949356 −0.0474678 − 0.998873i \(-0.515115\pi\)
−0.0474678 + 0.998873i \(0.515115\pi\)
\(702\) 0 0
\(703\) 5280.00 0.283270
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −41276.0 −2.19568
\(708\) 0 0
\(709\) −2474.00 −0.131048 −0.0655240 − 0.997851i \(-0.520872\pi\)
−0.0655240 + 0.997851i \(0.520872\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −11424.0 −0.600045
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32040.0 −1.66188 −0.830939 − 0.556363i \(-0.812196\pi\)
−0.830939 + 0.556363i \(0.812196\pi\)
\(720\) 0 0
\(721\) 30532.0 1.57708
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12874.0 0.656768 0.328384 − 0.944544i \(-0.393496\pi\)
0.328384 + 0.944544i \(0.393496\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −56816.0 −2.87471
\(732\) 0 0
\(733\) −28208.0 −1.42140 −0.710700 − 0.703495i \(-0.751622\pi\)
−0.710700 + 0.703495i \(0.751622\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5976.00 −0.298682
\(738\) 0 0
\(739\) 29068.0 1.44693 0.723467 − 0.690359i \(-0.242548\pi\)
0.723467 + 0.690359i \(0.242548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28152.0 1.39004 0.695018 − 0.718992i \(-0.255396\pi\)
0.695018 + 0.718992i \(0.255396\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 46376.0 2.26241
\(750\) 0 0
\(751\) −29916.0 −1.45360 −0.726798 − 0.686851i \(-0.758992\pi\)
−0.726798 + 0.686851i \(0.758992\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −32904.0 −1.57981 −0.789905 − 0.613229i \(-0.789870\pi\)
−0.789905 + 0.613229i \(0.789870\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21764.0 −1.03672 −0.518360 − 0.855162i \(-0.673457\pi\)
−0.518360 + 0.855162i \(0.673457\pi\)
\(762\) 0 0
\(763\) −7412.00 −0.351681
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2088.00 0.0982964
\(768\) 0 0
\(769\) −3570.00 −0.167409 −0.0837045 − 0.996491i \(-0.526675\pi\)
−0.0837045 + 0.996491i \(0.526675\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 19486.0 0.906679 0.453339 − 0.891338i \(-0.350233\pi\)
0.453339 + 0.891338i \(0.350233\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3520.00 −0.161896
\(780\) 0 0
\(781\) −2376.00 −0.108860
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −19764.0 −0.895185 −0.447592 − 0.894238i \(-0.647718\pi\)
−0.447592 + 0.894238i \(0.647718\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 47124.0 2.11825
\(792\) 0 0
\(793\) −2232.00 −0.0999504
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −14390.0 −0.639548 −0.319774 − 0.947494i \(-0.603607\pi\)
−0.319774 + 0.947494i \(0.603607\pi\)
\(798\) 0 0
\(799\) 56816.0 2.51565
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10836.0 0.476207
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 28536.0 1.24014 0.620069 − 0.784547i \(-0.287104\pi\)
0.620069 + 0.784547i \(0.287104\pi\)
\(810\) 0 0
\(811\) 27732.0 1.20074 0.600371 − 0.799721i \(-0.295019\pi\)
0.600371 + 0.799721i \(0.295019\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 23584.0 1.00991
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8086.00 0.343731 0.171866 − 0.985120i \(-0.445021\pi\)
0.171866 + 0.985120i \(0.445021\pi\)
\(822\) 0 0
\(823\) −39854.0 −1.68800 −0.843999 − 0.536344i \(-0.819805\pi\)
−0.843999 + 0.536344i \(0.819805\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17752.0 0.746430 0.373215 − 0.927745i \(-0.378255\pi\)
0.373215 + 0.927745i \(0.378255\pi\)
\(828\) 0 0
\(829\) 23858.0 0.999545 0.499772 − 0.866157i \(-0.333417\pi\)
0.499772 + 0.866157i \(0.333417\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 86178.0 3.58450
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 13888.0 0.571474 0.285737 − 0.958308i \(-0.407762\pi\)
0.285737 + 0.958308i \(0.407762\pi\)
\(840\) 0 0
\(841\) 48511.0 1.98905
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 34238.0 1.38894
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6720.00 0.270692
\(852\) 0 0
\(853\) 16568.0 0.665038 0.332519 − 0.943097i \(-0.392101\pi\)
0.332519 + 0.943097i \(0.392101\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13034.0 0.519525 0.259763 − 0.965673i \(-0.416356\pi\)
0.259763 + 0.965673i \(0.416356\pi\)
\(858\) 0 0
\(859\) −34356.0 −1.36462 −0.682312 − 0.731061i \(-0.739025\pi\)
−0.682312 + 0.731061i \(0.739025\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16016.0 −0.631739 −0.315870 − 0.948803i \(-0.602296\pi\)
−0.315870 + 0.948803i \(0.602296\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9864.00 −0.385056
\(870\) 0 0
\(871\) 3984.00 0.154986
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19780.0 −0.761600 −0.380800 − 0.924657i \(-0.624351\pi\)
−0.380800 + 0.924657i \(0.624351\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −41036.0 −1.56928 −0.784641 − 0.619950i \(-0.787153\pi\)
−0.784641 + 0.619950i \(0.787153\pi\)
\(882\) 0 0
\(883\) 35108.0 1.33803 0.669014 − 0.743250i \(-0.266717\pi\)
0.669014 + 0.743250i \(0.266717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18648.0 −0.705906 −0.352953 − 0.935641i \(-0.614822\pi\)
−0.352953 + 0.935641i \(0.614822\pi\)
\(888\) 0 0
\(889\) −27676.0 −1.04412
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −23584.0 −0.883772
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 55080.0 2.04340
\(900\) 0 0
\(901\) −57452.0 −2.12431
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −21688.0 −0.793978 −0.396989 − 0.917823i \(-0.629945\pi\)
−0.396989 + 0.917823i \(0.629945\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −42064.0 −1.52979 −0.764897 − 0.644153i \(-0.777210\pi\)
−0.764897 + 0.644153i \(0.777210\pi\)
\(912\) 0 0
\(913\) 8856.00 0.321020
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43588.0 1.56969
\(918\) 0 0
\(919\) −44420.0 −1.59443 −0.797215 − 0.603696i \(-0.793694\pi\)
−0.797215 + 0.603696i \(0.793694\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1584.00 0.0564875
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17124.0 −0.604758 −0.302379 − 0.953188i \(-0.597781\pi\)
−0.302379 + 0.953188i \(0.597781\pi\)
\(930\) 0 0
\(931\) −35772.0 −1.25927
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11110.0 0.387351 0.193675 − 0.981066i \(-0.437959\pi\)
0.193675 + 0.981066i \(0.437959\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12962.0 0.449043 0.224521 − 0.974469i \(-0.427918\pi\)
0.224521 + 0.974469i \(0.427918\pi\)
\(942\) 0 0
\(943\) −4480.00 −0.154707
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25672.0 0.880916 0.440458 − 0.897773i \(-0.354816\pi\)
0.440458 + 0.897773i \(0.354816\pi\)
\(948\) 0 0
\(949\) −7224.00 −0.247103
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2082.00 −0.0707687 −0.0353844 − 0.999374i \(-0.511266\pi\)
−0.0353844 + 0.999374i \(0.511266\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 104244. 3.51013
\(960\) 0 0
\(961\) 11825.0 0.396932
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5666.00 −0.188424 −0.0942121 − 0.995552i \(-0.530033\pi\)
−0.0942121 + 0.995552i \(0.530033\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28622.0 0.945956 0.472978 − 0.881074i \(-0.343179\pi\)
0.472978 + 0.881074i \(0.343179\pi\)
\(972\) 0 0
\(973\) 45288.0 1.49215
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24586.0 0.805093 0.402546 − 0.915400i \(-0.368125\pi\)
0.402546 + 0.915400i \(0.368125\pi\)
\(978\) 0 0
\(979\) −18936.0 −0.618179
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −40632.0 −1.31837 −0.659186 − 0.751980i \(-0.729099\pi\)
−0.659186 + 0.751980i \(0.729099\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30016.0 0.965069
\(990\) 0 0
\(991\) 8768.00 0.281054 0.140527 − 0.990077i \(-0.455120\pi\)
0.140527 + 0.990077i \(0.455120\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −37212.0 −1.18206 −0.591031 − 0.806649i \(-0.701279\pi\)
−0.591031 + 0.806649i \(0.701279\pi\)
\(998\) 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 1800.4.a.b.1.1 1
3.2 odd 2 1800.4.a.a.1.1 1
5.2 odd 4 1800.4.f.o.649.1 2
5.3 odd 4 1800.4.f.o.649.2 2
5.4 even 2 360.4.a.g.1.1 ✓ 1
15.2 even 4 1800.4.f.i.649.1 2
15.8 even 4 1800.4.f.i.649.2 2
15.14 odd 2 360.4.a.o.1.1 yes 1
20.19 odd 2 720.4.a.a.1.1 1
60.59 even 2 720.4.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.4.a.g.1.1 ✓ 1 5.4 even 2
360.4.a.o.1.1 yes 1 15.14 odd 2
720.4.a.a.1.1 1 20.19 odd 2
720.4.a.p.1.1 1 60.59 even 2
1800.4.a.a.1.1 1 3.2 odd 2
1800.4.a.b.1.1 1 1.1 even 1 trivial
1800.4.f.i.649.1 2 15.2 even 4
1800.4.f.i.649.2 2 15.8 even 4
1800.4.f.o.649.1 2 5.2 odd 4
1800.4.f.o.649.2 2 5.3 odd 4