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Properties

Label 1800.4.a.p.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{7} -39.0000 q^{11} -84.0000 q^{13} -61.0000 q^{17} +151.000 q^{19} -58.0000 q^{23} -192.000 q^{29} -18.0000 q^{31} +138.000 q^{37} -229.000 q^{41} +164.000 q^{43} -212.000 q^{47} -339.000 q^{49} +578.000 q^{53} +336.000 q^{59} +858.000 q^{61} +209.000 q^{67} +780.000 q^{71} +403.000 q^{73} +78.0000 q^{77} -230.000 q^{79} -1293.00 q^{83} +1369.00 q^{89} +168.000 q^{91} -382.000 q^{97} +O(q^{100})\)

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
\(7\) −2.00000 −0.107990 −0.0539949 − 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −39.0000 −1.06899 −0.534497 − 0.845170i \(-0.679499\pi\)
−0.534497 + 0.845170i \(0.679499\pi\)
\(12\) 0 0
\(13\) −84.0000 −1.79211 −0.896054 − 0.443945i \(-0.853579\pi\)
−0.896054 + 0.443945i \(0.853579\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −61.0000 −0.870275 −0.435137 − 0.900364i \(-0.643300\pi\)
−0.435137 + 0.900364i \(0.643300\pi\)
\(18\) 0 0
\(19\) 151.000 1.82325 0.911626 − 0.411021i \(-0.134828\pi\)
0.911626 + 0.411021i \(0.134828\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −58.0000 −0.525819 −0.262909 − 0.964821i \(-0.584682\pi\)
−0.262909 + 0.964821i \(0.584682\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −192.000 −1.22943 −0.614716 − 0.788749i \(-0.710729\pi\)
−0.614716 + 0.788749i \(0.710729\pi\)
\(30\) 0 0
\(31\) −18.0000 −0.104287 −0.0521435 − 0.998640i \(-0.516605\pi\)
−0.0521435 + 0.998640i \(0.516605\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 138.000 0.613164 0.306582 − 0.951844i \(-0.400815\pi\)
0.306582 + 0.951844i \(0.400815\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −229.000 −0.872288 −0.436144 − 0.899877i \(-0.643656\pi\)
−0.436144 + 0.899877i \(0.643656\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 − 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −212.000 −0.657944 −0.328972 − 0.944340i \(-0.606702\pi\)
−0.328972 + 0.944340i \(0.606702\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 578.000 1.49801 0.749004 − 0.662566i \(-0.230532\pi\)
0.749004 + 0.662566i \(0.230532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 336.000 0.741415 0.370707 − 0.928750i \(-0.379115\pi\)
0.370707 + 0.928750i \(0.379115\pi\)
\(60\) 0 0
\(61\) 858.000 1.80091 0.900456 − 0.434947i \(-0.143233\pi\)
0.900456 + 0.434947i \(0.143233\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 209.000 0.381096 0.190548 − 0.981678i \(-0.438974\pi\)
0.190548 + 0.981678i \(0.438974\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 780.000 1.30379 0.651894 − 0.758310i \(-0.273975\pi\)
0.651894 + 0.758310i \(0.273975\pi\)
\(72\) 0 0
\(73\) 403.000 0.646131 0.323066 − 0.946377i \(-0.395287\pi\)
0.323066 + 0.946377i \(0.395287\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 78.0000 0.115441
\(78\) 0 0
\(79\) −230.000 −0.327557 −0.163779 − 0.986497i \(-0.552368\pi\)
−0.163779 + 0.986497i \(0.552368\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1293.00 −1.70994 −0.854971 − 0.518676i \(-0.826425\pi\)
−0.854971 + 0.518676i \(0.826425\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1369.00 1.63049 0.815246 − 0.579115i \(-0.196602\pi\)
0.815246 + 0.579115i \(0.196602\pi\)
\(90\) 0 0
\(91\) 168.000 0.193530
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −382.000 −0.399858 −0.199929 − 0.979810i \(-0.564071\pi\)
−0.199929 + 0.979810i \(0.564071\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 794.000 0.782237 0.391119 − 0.920340i \(-0.372088\pi\)
0.391119 + 0.920340i \(0.372088\pi\)
\(102\) 0 0
\(103\) 1348.00 1.28954 0.644769 − 0.764378i \(-0.276954\pi\)
0.644769 + 0.764378i \(0.276954\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −775.000 −0.700206 −0.350103 − 0.936711i \(-0.613853\pi\)
−0.350103 + 0.936711i \(0.613853\pi\)
\(108\) 0 0
\(109\) 446.000 0.391918 0.195959 − 0.980612i \(-0.437218\pi\)
0.195959 + 0.980612i \(0.437218\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −231.000 −0.192307 −0.0961533 − 0.995367i \(-0.530654\pi\)
−0.0961533 + 0.995367i \(0.530654\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 122.000 0.0939809
\(120\) 0 0
\(121\) 190.000 0.142750
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2386.00 −1.66711 −0.833556 − 0.552435i \(-0.813699\pi\)
−0.833556 + 0.552435i \(0.813699\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2452.00 −1.63536 −0.817680 − 0.575673i \(-0.804740\pi\)
−0.817680 + 0.575673i \(0.804740\pi\)
\(132\) 0 0
\(133\) −302.000 −0.196893
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1125.00 −0.701571 −0.350786 − 0.936456i \(-0.614085\pi\)
−0.350786 + 0.936456i \(0.614085\pi\)
\(138\) 0 0
\(139\) −1377.00 −0.840256 −0.420128 − 0.907465i \(-0.638015\pi\)
−0.420128 + 0.907465i \(0.638015\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3276.00 1.91575
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1920.00 −1.05565 −0.527827 − 0.849352i \(-0.676993\pi\)
−0.527827 + 0.849352i \(0.676993\pi\)
\(150\) 0 0
\(151\) 1854.00 0.999181 0.499591 − 0.866262i \(-0.333484\pi\)
0.499591 + 0.866262i \(0.333484\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 634.000 0.322285 0.161142 − 0.986931i \(-0.448482\pi\)
0.161142 + 0.986931i \(0.448482\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 116.000 0.0567831
\(162\) 0 0
\(163\) −103.000 −0.0494944 −0.0247472 − 0.999694i \(-0.507878\pi\)
−0.0247472 + 0.999694i \(0.507878\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 44.0000 0.0203882 0.0101941 − 0.999948i \(-0.496755\pi\)
0.0101941 + 0.999948i \(0.496755\pi\)
\(168\) 0 0
\(169\) 4859.00 2.21165
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1128.00 −0.495724 −0.247862 − 0.968795i \(-0.579728\pi\)
−0.247862 + 0.968795i \(0.579728\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2245.00 0.937426 0.468713 − 0.883351i \(-0.344718\pi\)
0.468713 + 0.883351i \(0.344718\pi\)
\(180\) 0 0
\(181\) 3050.00 1.25251 0.626256 − 0.779617i \(-0.284586\pi\)
0.626256 + 0.779617i \(0.284586\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2379.00 0.930319
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4222.00 1.59944 0.799720 − 0.600373i \(-0.204981\pi\)
0.799720 + 0.600373i \(0.204981\pi\)
\(192\) 0 0
\(193\) 3357.00 1.25203 0.626016 − 0.779810i \(-0.284684\pi\)
0.626016 + 0.779810i \(0.284684\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −166.000 −0.0600356 −0.0300178 − 0.999549i \(-0.509556\pi\)
−0.0300178 + 0.999549i \(0.509556\pi\)
\(198\) 0 0
\(199\) 3372.00 1.20118 0.600590 − 0.799557i \(-0.294932\pi\)
0.600590 + 0.799557i \(0.294932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 384.000 0.132766
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5889.00 −1.94905
\(210\) 0 0
\(211\) 5601.00 1.82743 0.913717 − 0.406350i \(-0.133199\pi\)
0.913717 + 0.406350i \(0.133199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 36.0000 0.0112619
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5124.00 1.55963
\(222\) 0 0
\(223\) −828.000 −0.248641 −0.124321 − 0.992242i \(-0.539675\pi\)
−0.124321 + 0.992242i \(0.539675\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2388.00 −0.698225 −0.349113 − 0.937081i \(-0.613517\pi\)
−0.349113 + 0.937081i \(0.613517\pi\)
\(228\) 0 0
\(229\) −2844.00 −0.820685 −0.410342 − 0.911932i \(-0.634591\pi\)
−0.410342 + 0.911932i \(0.634591\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5962.00 1.67632 0.838162 − 0.545421i \(-0.183630\pi\)
0.838162 + 0.545421i \(0.183630\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4320.00 1.16919 0.584597 − 0.811324i \(-0.301252\pi\)
0.584597 + 0.811324i \(0.301252\pi\)
\(240\) 0 0
\(241\) 3857.00 1.03092 0.515459 − 0.856914i \(-0.327621\pi\)
0.515459 + 0.856914i \(0.327621\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12684.0 −3.26746
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 287.000 0.0721724 0.0360862 − 0.999349i \(-0.488511\pi\)
0.0360862 + 0.999349i \(0.488511\pi\)
\(252\) 0 0
\(253\) 2262.00 0.562098
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2130.00 0.516987 0.258494 − 0.966013i \(-0.416774\pi\)
0.258494 + 0.966013i \(0.416774\pi\)
\(258\) 0 0
\(259\) −276.000 −0.0662155
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3066.00 −0.718850 −0.359425 − 0.933174i \(-0.617027\pi\)
−0.359425 + 0.933174i \(0.617027\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3744.00 0.848609 0.424304 − 0.905520i \(-0.360519\pi\)
0.424304 + 0.905520i \(0.360519\pi\)
\(270\) 0 0
\(271\) −3346.00 −0.750019 −0.375009 − 0.927021i \(-0.622360\pi\)
−0.375009 + 0.927021i \(0.622360\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7040.00 −1.52705 −0.763525 − 0.645779i \(-0.776533\pi\)
−0.763525 + 0.645779i \(0.776533\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3010.00 0.639009 0.319505 − 0.947585i \(-0.396484\pi\)
0.319505 + 0.947585i \(0.396484\pi\)
\(282\) 0 0
\(283\) 6001.00 1.26050 0.630252 − 0.776391i \(-0.282952\pi\)
0.630252 + 0.776391i \(0.282952\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 458.000 0.0941982
\(288\) 0 0
\(289\) −1192.00 −0.242622
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4802.00 0.957460 0.478730 − 0.877962i \(-0.341097\pi\)
0.478730 + 0.877962i \(0.341097\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4872.00 0.942325
\(300\) 0 0
\(301\) −328.000 −0.0628093
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6149.00 −1.14313 −0.571567 − 0.820556i \(-0.693664\pi\)
−0.571567 + 0.820556i \(0.693664\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 878.000 0.160086 0.0800431 − 0.996791i \(-0.474494\pi\)
0.0800431 + 0.996791i \(0.474494\pi\)
\(312\) 0 0
\(313\) 4042.00 0.729928 0.364964 − 0.931022i \(-0.381081\pi\)
0.364964 + 0.931022i \(0.381081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3844.00 0.681074 0.340537 − 0.940231i \(-0.389391\pi\)
0.340537 + 0.940231i \(0.389391\pi\)
\(318\) 0 0
\(319\) 7488.00 1.31426
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9211.00 −1.58673
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 424.000 0.0710513
\(330\) 0 0
\(331\) −2717.00 −0.451178 −0.225589 − 0.974223i \(-0.572431\pi\)
−0.225589 + 0.974223i \(0.572431\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1603.00 0.259113 0.129556 − 0.991572i \(-0.458645\pi\)
0.129556 + 0.991572i \(0.458645\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 702.000 0.111482
\(342\) 0 0
\(343\) 1364.00 0.214720
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −11607.0 −1.79567 −0.897833 − 0.440335i \(-0.854860\pi\)
−0.897833 + 0.440335i \(0.854860\pi\)
\(348\) 0 0
\(349\) 4030.00 0.618112 0.309056 − 0.951044i \(-0.399987\pi\)
0.309056 + 0.951044i \(0.399987\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2106.00 0.317538 0.158769 − 0.987316i \(-0.449247\pi\)
0.158769 + 0.987316i \(0.449247\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7394.00 −1.08702 −0.543510 − 0.839402i \(-0.682905\pi\)
−0.543510 + 0.839402i \(0.682905\pi\)
\(360\) 0 0
\(361\) 15942.0 2.32425
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6940.00 0.987098 0.493549 − 0.869718i \(-0.335699\pi\)
0.493549 + 0.869718i \(0.335699\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1156.00 −0.161770
\(372\) 0 0
\(373\) −7486.00 −1.03917 −0.519585 − 0.854419i \(-0.673913\pi\)
−0.519585 + 0.854419i \(0.673913\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16128.0 2.20327
\(378\) 0 0
\(379\) 1285.00 0.174158 0.0870792 − 0.996201i \(-0.472247\pi\)
0.0870792 + 0.996201i \(0.472247\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9622.00 1.28371 0.641855 − 0.766826i \(-0.278165\pi\)
0.641855 + 0.766826i \(0.278165\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1974.00 −0.257290 −0.128645 − 0.991691i \(-0.541063\pi\)
−0.128645 + 0.991691i \(0.541063\pi\)
\(390\) 0 0
\(391\) 3538.00 0.457607
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8084.00 1.02198 0.510988 − 0.859588i \(-0.329280\pi\)
0.510988 + 0.859588i \(0.329280\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5667.00 0.705727 0.352863 − 0.935675i \(-0.385208\pi\)
0.352863 + 0.935675i \(0.385208\pi\)
\(402\) 0 0
\(403\) 1512.00 0.186894
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5382.00 −0.655469
\(408\) 0 0
\(409\) −4835.00 −0.584536 −0.292268 − 0.956336i \(-0.594410\pi\)
−0.292268 + 0.956336i \(0.594410\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −672.000 −0.0800653
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4619.00 −0.538551 −0.269276 − 0.963063i \(-0.586784\pi\)
−0.269276 + 0.963063i \(0.586784\pi\)
\(420\) 0 0
\(421\) 7476.00 0.865458 0.432729 − 0.901524i \(-0.357551\pi\)
0.432729 + 0.901524i \(0.357551\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1716.00 −0.194480
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7810.00 −0.872841 −0.436420 − 0.899743i \(-0.643754\pi\)
−0.436420 + 0.899743i \(0.643754\pi\)
\(432\) 0 0
\(433\) 2029.00 0.225191 0.112595 − 0.993641i \(-0.464084\pi\)
0.112595 + 0.993641i \(0.464084\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8758.00 −0.958700
\(438\) 0 0
\(439\) 3208.00 0.348769 0.174384 − 0.984678i \(-0.444206\pi\)
0.174384 + 0.984678i \(0.444206\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13227.0 1.41859 0.709293 − 0.704914i \(-0.249014\pi\)
0.709293 + 0.704914i \(0.249014\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3617.00 −0.380171 −0.190086 − 0.981768i \(-0.560877\pi\)
−0.190086 + 0.981768i \(0.560877\pi\)
\(450\) 0 0
\(451\) 8931.00 0.932471
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6215.00 −0.636161 −0.318080 − 0.948064i \(-0.603038\pi\)
−0.318080 + 0.948064i \(0.603038\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7108.00 0.718118 0.359059 − 0.933315i \(-0.383098\pi\)
0.359059 + 0.933315i \(0.383098\pi\)
\(462\) 0 0
\(463\) 3364.00 0.337664 0.168832 − 0.985645i \(-0.446000\pi\)
0.168832 + 0.985645i \(0.446000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18964.0 −1.87912 −0.939560 − 0.342384i \(-0.888766\pi\)
−0.939560 + 0.342384i \(0.888766\pi\)
\(468\) 0 0
\(469\) −418.000 −0.0411545
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6396.00 −0.621751
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10926.0 1.04222 0.521108 − 0.853491i \(-0.325519\pi\)
0.521108 + 0.853491i \(0.325519\pi\)
\(480\) 0 0
\(481\) −11592.0 −1.09886
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4350.00 0.404758 0.202379 − 0.979307i \(-0.435133\pi\)
0.202379 + 0.979307i \(0.435133\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1324.00 0.121693 0.0608465 − 0.998147i \(-0.480620\pi\)
0.0608465 + 0.998147i \(0.480620\pi\)
\(492\) 0 0
\(493\) 11712.0 1.06994
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1560.00 −0.140796
\(498\) 0 0
\(499\) 9068.00 0.813506 0.406753 − 0.913538i \(-0.366661\pi\)
0.406753 + 0.913538i \(0.366661\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19836.0 −1.75834 −0.879169 − 0.476511i \(-0.841901\pi\)
−0.879169 + 0.476511i \(0.841901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2682.00 −0.233551 −0.116776 − 0.993158i \(-0.537256\pi\)
−0.116776 + 0.993158i \(0.537256\pi\)
\(510\) 0 0
\(511\) −806.000 −0.0697756
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8268.00 0.703339
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3035.00 0.255213 0.127606 − 0.991825i \(-0.459271\pi\)
0.127606 + 0.991825i \(0.459271\pi\)
\(522\) 0 0
\(523\) −7701.00 −0.643865 −0.321932 − 0.946763i \(-0.604332\pi\)
−0.321932 + 0.946763i \(0.604332\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1098.00 0.0907583
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 19236.0 1.56323
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13221.0 1.05653
\(540\) 0 0
\(541\) −18112.0 −1.43936 −0.719682 − 0.694304i \(-0.755712\pi\)
−0.719682 + 0.694304i \(0.755712\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19541.0 −1.52745 −0.763723 − 0.645544i \(-0.776631\pi\)
−0.763723 + 0.645544i \(0.776631\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −28992.0 −2.24156
\(552\) 0 0
\(553\) 460.000 0.0353729
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13508.0 1.02756 0.513781 − 0.857921i \(-0.328244\pi\)
0.513781 + 0.857921i \(0.328244\pi\)
\(558\) 0 0
\(559\) −13776.0 −1.04233
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8712.00 −0.652162 −0.326081 − 0.945342i \(-0.605728\pi\)
−0.326081 + 0.945342i \(0.605728\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9623.00 0.708993 0.354497 − 0.935057i \(-0.384652\pi\)
0.354497 + 0.935057i \(0.384652\pi\)
\(570\) 0 0
\(571\) 604.000 0.0442673 0.0221336 − 0.999755i \(-0.492954\pi\)
0.0221336 + 0.999755i \(0.492954\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3629.00 −0.261832 −0.130916 − 0.991393i \(-0.541792\pi\)
−0.130916 + 0.991393i \(0.541792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2586.00 0.184656
\(582\) 0 0
\(583\) −22542.0 −1.60136
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9219.00 0.648226 0.324113 − 0.946018i \(-0.394934\pi\)
0.324113 + 0.946018i \(0.394934\pi\)
\(588\) 0 0
\(589\) −2718.00 −0.190141
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −19111.0 −1.32343 −0.661716 − 0.749755i \(-0.730171\pi\)
−0.661716 + 0.749755i \(0.730171\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 17086.0 1.16547 0.582734 − 0.812663i \(-0.301983\pi\)
0.582734 + 0.812663i \(0.301983\pi\)
\(600\) 0 0
\(601\) 9035.00 0.613220 0.306610 − 0.951835i \(-0.400805\pi\)
0.306610 + 0.951835i \(0.400805\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 14784.0 0.988573 0.494287 − 0.869299i \(-0.335429\pi\)
0.494287 + 0.869299i \(0.335429\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17808.0 1.17911
\(612\) 0 0
\(613\) 17846.0 1.17585 0.587923 − 0.808917i \(-0.299946\pi\)
0.587923 + 0.808917i \(0.299946\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11618.0 0.758060 0.379030 − 0.925384i \(-0.376258\pi\)
0.379030 + 0.925384i \(0.376258\pi\)
\(618\) 0 0
\(619\) −9556.00 −0.620498 −0.310249 − 0.950655i \(-0.600412\pi\)
−0.310249 + 0.950655i \(0.600412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2738.00 −0.176076
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8418.00 −0.533621
\(630\) 0 0
\(631\) −19394.0 −1.22355 −0.611777 − 0.791030i \(-0.709545\pi\)
−0.611777 + 0.791030i \(0.709545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 28476.0 1.77121
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12138.0 −0.747929 −0.373964 − 0.927443i \(-0.622002\pi\)
−0.373964 + 0.927443i \(0.622002\pi\)
\(642\) 0 0
\(643\) −27036.0 −1.65816 −0.829079 − 0.559131i \(-0.811135\pi\)
−0.829079 + 0.559131i \(0.811135\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17556.0 −1.06677 −0.533383 − 0.845874i \(-0.679080\pi\)
−0.533383 + 0.845874i \(0.679080\pi\)
\(648\) 0 0
\(649\) −13104.0 −0.792569
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17262.0 −1.03448 −0.517239 − 0.855841i \(-0.673040\pi\)
−0.517239 + 0.855841i \(0.673040\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10517.0 −0.621675 −0.310838 − 0.950463i \(-0.600610\pi\)
−0.310838 + 0.950463i \(0.600610\pi\)
\(660\) 0 0
\(661\) 1408.00 0.0828515 0.0414258 − 0.999142i \(-0.486810\pi\)
0.0414258 + 0.999142i \(0.486810\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11136.0 0.646458
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33462.0 −1.92517
\(672\) 0 0
\(673\) 9626.00 0.551345 0.275672 − 0.961252i \(-0.411100\pi\)
0.275672 + 0.961252i \(0.411100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28464.0 1.61589 0.807947 − 0.589255i \(-0.200579\pi\)
0.807947 + 0.589255i \(0.200579\pi\)
\(678\) 0 0
\(679\) 764.000 0.0431806
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3963.00 −0.222020 −0.111010 − 0.993819i \(-0.535409\pi\)
−0.111010 + 0.993819i \(0.535409\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −48552.0 −2.68459
\(690\) 0 0
\(691\) −31781.0 −1.74965 −0.874824 − 0.484442i \(-0.839023\pi\)
−0.874824 + 0.484442i \(0.839023\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13969.0 0.759130
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28004.0 1.50884 0.754420 − 0.656392i \(-0.227918\pi\)
0.754420 + 0.656392i \(0.227918\pi\)
\(702\) 0 0
\(703\) 20838.0 1.11795
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1588.00 −0.0844737
\(708\) 0 0
\(709\) −35228.0 −1.86603 −0.933015 − 0.359837i \(-0.882832\pi\)
−0.933015 + 0.359837i \(0.882832\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1044.00 0.0548361
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8658.00 0.449081 0.224540 − 0.974465i \(-0.427912\pi\)
0.224540 + 0.974465i \(0.427912\pi\)
\(720\) 0 0
\(721\) −2696.00 −0.139257
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −5728.00 −0.292214 −0.146107 − 0.989269i \(-0.546674\pi\)
−0.146107 + 0.989269i \(0.546674\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10004.0 −0.506171
\(732\) 0 0
\(733\) −21460.0 −1.08137 −0.540684 − 0.841226i \(-0.681835\pi\)
−0.540684 + 0.841226i \(0.681835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8151.00 −0.407389
\(738\) 0 0
\(739\) 29164.0 1.45171 0.725856 − 0.687847i \(-0.241444\pi\)
0.725856 + 0.687847i \(0.241444\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29478.0 1.45551 0.727754 − 0.685838i \(-0.240564\pi\)
0.727754 + 0.685838i \(0.240564\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1550.00 0.0756152
\(750\) 0 0
\(751\) 576.000 0.0279874 0.0139937 − 0.999902i \(-0.495546\pi\)
0.0139937 + 0.999902i \(0.495546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2880.00 −0.138277 −0.0691383 − 0.997607i \(-0.522025\pi\)
−0.0691383 + 0.997607i \(0.522025\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20789.0 −0.990277 −0.495138 − 0.868814i \(-0.664883\pi\)
−0.495138 + 0.868814i \(0.664883\pi\)
\(762\) 0 0
\(763\) −892.000 −0.0423232
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28224.0 −1.32870
\(768\) 0 0
\(769\) 26421.0 1.23897 0.619484 − 0.785010i \(-0.287342\pi\)
0.619484 + 0.785010i \(0.287342\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32504.0 1.51240 0.756202 − 0.654339i \(-0.227053\pi\)
0.756202 + 0.654339i \(0.227053\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −34579.0 −1.59040
\(780\) 0 0
\(781\) −30420.0 −1.39374
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −996.000 −0.0451125 −0.0225563 − 0.999746i \(-0.507180\pi\)
−0.0225563 + 0.999746i \(0.507180\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 462.000 0.0207672
\(792\) 0 0
\(793\) −72072.0 −3.22743
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15134.0 0.672615 0.336307 − 0.941752i \(-0.390822\pi\)
0.336307 + 0.941752i \(0.390822\pi\)
\(798\) 0 0
\(799\) 12932.0 0.572592
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −15717.0 −0.690711
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36942.0 1.60545 0.802727 − 0.596347i \(-0.203382\pi\)
0.802727 + 0.596347i \(0.203382\pi\)
\(810\) 0 0
\(811\) −11748.0 −0.508666 −0.254333 − 0.967117i \(-0.581856\pi\)
−0.254333 + 0.967117i \(0.581856\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24764.0 1.06044
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1198.00 0.0509263 0.0254631 − 0.999676i \(-0.491894\pi\)
0.0254631 + 0.999676i \(0.491894\pi\)
\(822\) 0 0
\(823\) 6788.00 0.287503 0.143751 − 0.989614i \(-0.454083\pi\)
0.143751 + 0.989614i \(0.454083\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33011.0 1.38803 0.694017 − 0.719958i \(-0.255839\pi\)
0.694017 + 0.719958i \(0.255839\pi\)
\(828\) 0 0
\(829\) 17732.0 0.742892 0.371446 − 0.928454i \(-0.378862\pi\)
0.371446 + 0.928454i \(0.378862\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20679.0 0.860126
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8480.00 −0.348942 −0.174471 − 0.984662i \(-0.555821\pi\)
−0.174471 + 0.984662i \(0.555821\pi\)
\(840\) 0 0
\(841\) 12475.0 0.511501
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −380.000 −0.0154155
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8004.00 −0.322413
\(852\) 0 0
\(853\) −30014.0 −1.20476 −0.602380 − 0.798210i \(-0.705781\pi\)
−0.602380 + 0.798210i \(0.705781\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21643.0 0.862673 0.431337 − 0.902191i \(-0.358042\pi\)
0.431337 + 0.902191i \(0.358042\pi\)
\(858\) 0 0
\(859\) 2799.00 0.111177 0.0555883 − 0.998454i \(-0.482297\pi\)
0.0555883 + 0.998454i \(0.482297\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19384.0 −0.764588 −0.382294 − 0.924041i \(-0.624866\pi\)
−0.382294 + 0.924041i \(0.624866\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8970.00 0.350157
\(870\) 0 0
\(871\) −17556.0 −0.682965
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5132.00 −0.197600 −0.0988001 − 0.995107i \(-0.531500\pi\)
−0.0988001 + 0.995107i \(0.531500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4430.00 −0.169410 −0.0847052 − 0.996406i \(-0.526995\pi\)
−0.0847052 + 0.996406i \(0.526995\pi\)
\(882\) 0 0
\(883\) −24317.0 −0.926764 −0.463382 − 0.886159i \(-0.653364\pi\)
−0.463382 + 0.886159i \(0.653364\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26100.0 −0.987996 −0.493998 − 0.869463i \(-0.664465\pi\)
−0.493998 + 0.869463i \(0.664465\pi\)
\(888\) 0 0
\(889\) 4772.00 0.180031
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32012.0 −1.19960
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3456.00 0.128214
\(900\) 0 0
\(901\) −35258.0 −1.30368
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −24356.0 −0.891651 −0.445826 − 0.895120i \(-0.647090\pi\)
−0.445826 + 0.895120i \(0.647090\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 29900.0 1.08741 0.543705 − 0.839276i \(-0.317021\pi\)
0.543705 + 0.839276i \(0.317021\pi\)
\(912\) 0 0
\(913\) 50427.0 1.82792
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4904.00 0.176602
\(918\) 0 0
\(919\) −34838.0 −1.25049 −0.625245 − 0.780429i \(-0.715001\pi\)
−0.625245 + 0.780429i \(0.715001\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −65520.0 −2.33653
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26334.0 −0.930022 −0.465011 − 0.885305i \(-0.653950\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(930\) 0 0
\(931\) −51189.0 −1.80199
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −30949.0 −1.07904 −0.539520 − 0.841973i \(-0.681394\pi\)
−0.539520 + 0.841973i \(0.681394\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25276.0 −0.875637 −0.437818 − 0.899063i \(-0.644249\pi\)
−0.437818 + 0.899063i \(0.644249\pi\)
\(942\) 0 0
\(943\) 13282.0 0.458665
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1216.00 −0.0417262 −0.0208631 − 0.999782i \(-0.506641\pi\)
−0.0208631 + 0.999782i \(0.506641\pi\)
\(948\) 0 0
\(949\) −33852.0 −1.15794
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6033.00 −0.205066 −0.102533 − 0.994730i \(-0.532695\pi\)
−0.102533 + 0.994730i \(0.532695\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2250.00 0.0757626
\(960\) 0 0
\(961\) −29467.0 −0.989124
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 41792.0 1.38980 0.694902 − 0.719105i \(-0.255448\pi\)
0.694902 + 0.719105i \(0.255448\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2105.00 0.0695702 0.0347851 − 0.999395i \(-0.488925\pi\)
0.0347851 + 0.999395i \(0.488925\pi\)
\(972\) 0 0
\(973\) 2754.00 0.0907391
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30119.0 0.986277 0.493138 − 0.869951i \(-0.335850\pi\)
0.493138 + 0.869951i \(0.335850\pi\)
\(978\) 0 0
\(979\) −53391.0 −1.74299
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18438.0 0.598251 0.299126 − 0.954214i \(-0.403305\pi\)
0.299126 + 0.954214i \(0.403305\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9512.00 −0.305828
\(990\) 0 0
\(991\) −2230.00 −0.0714816 −0.0357408 − 0.999361i \(-0.511379\pi\)
−0.0357408 + 0.999361i \(0.511379\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6804.00 0.216133 0.108067 − 0.994144i \(-0.465534\pi\)
0.108067 + 0.994144i \(0.465534\pi\)
\(998\) 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.p.1.1 1
3.2 odd 2 200.4.a.c.1.1 ✓ 1
5.2 odd 4 1800.4.f.c.649.1 2
5.3 odd 4 1800.4.f.c.649.2 2
5.4 even 2 1800.4.a.t.1.1 1
12.11 even 2 400.4.a.q.1.1 1
15.2 even 4 200.4.c.d.49.2 2
15.8 even 4 200.4.c.d.49.1 2
15.14 odd 2 200.4.a.h.1.1 yes 1
24.5 odd 2 1600.4.a.bn.1.1 1
24.11 even 2 1600.4.a.n.1.1 1
60.23 odd 4 400.4.c.g.49.2 2
60.47 odd 4 400.4.c.g.49.1 2
60.59 even 2 400.4.a.f.1.1 1
120.29 odd 2 1600.4.a.m.1.1 1
120.59 even 2 1600.4.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.c.1.1 ✓ 1 3.2 odd 2
200.4.a.h.1.1 yes 1 15.14 odd 2
200.4.c.d.49.1 2 15.8 even 4
200.4.c.d.49.2 2 15.2 even 4
400.4.a.f.1.1 1 60.59 even 2
400.4.a.q.1.1 1 12.11 even 2
400.4.c.g.49.1 2 60.47 odd 4
400.4.c.g.49.2 2 60.23 odd 4
1600.4.a.m.1.1 1 120.29 odd 2
1600.4.a.n.1.1 1 24.11 even 2
1600.4.a.bn.1.1 1 24.5 odd 2
1600.4.a.bo.1.1 1 120.59 even 2
1800.4.a.p.1.1 1 1.1 even 1 trivial
1800.4.a.t.1.1 1 5.4 even 2
1800.4.f.c.649.1 2 5.2 odd 4
1800.4.f.c.649.2 2 5.3 odd 4