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Properties

Label 900.4.a.g.1.1
Level $900$
Weight $4$
Character 900.1
Self dual yes
Analytic conductor $53.102$
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] = [900,4,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, 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("900.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{7} +O(q^{10})\) \(q-8.00000 q^{7} -36.0000 q^{11} +10.0000 q^{13} +18.0000 q^{17} -100.000 q^{19} +72.0000 q^{23} +234.000 q^{29} -16.0000 q^{31} +226.000 q^{37} -90.0000 q^{41} -452.000 q^{43} +432.000 q^{47} -279.000 q^{49} +414.000 q^{53} +684.000 q^{59} +422.000 q^{61} -332.000 q^{67} +360.000 q^{71} -26.0000 q^{73} +288.000 q^{77} +512.000 q^{79} -1188.00 q^{83} +630.000 q^{89} -80.0000 q^{91} +1054.00 q^{97} +O(q^{100})\)

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
\(7\) −8.00000 −0.431959 −0.215980 − 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 10.0000 0.213346 0.106673 − 0.994294i \(-0.465980\pi\)
0.106673 + 0.994294i \(0.465980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.0000 0.256802 0.128401 − 0.991722i \(-0.459015\pi\)
0.128401 + 0.991722i \(0.459015\pi\)
\(18\) 0 0
\(19\) −100.000 −1.20745 −0.603726 − 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 − 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 234.000 1.49837 0.749185 − 0.662361i \(-0.230446\pi\)
0.749185 + 0.662361i \(0.230446\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 − 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 226.000 1.00417 0.502083 − 0.864819i \(-0.332567\pi\)
0.502083 + 0.864819i \(0.332567\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −90.0000 −0.342820 −0.171410 − 0.985200i \(-0.554832\pi\)
−0.171410 + 0.985200i \(0.554832\pi\)
\(42\) 0 0
\(43\) −452.000 −1.60301 −0.801504 − 0.597989i \(-0.795967\pi\)
−0.801504 + 0.597989i \(0.795967\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 432.000 1.34072 0.670358 − 0.742038i \(-0.266140\pi\)
0.670358 + 0.742038i \(0.266140\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 414.000 1.07297 0.536484 − 0.843911i \(-0.319752\pi\)
0.536484 + 0.843911i \(0.319752\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 684.000 1.50931 0.754654 − 0.656123i \(-0.227805\pi\)
0.754654 + 0.656123i \(0.227805\pi\)
\(60\) 0 0
\(61\) 422.000 0.885763 0.442882 − 0.896580i \(-0.353956\pi\)
0.442882 + 0.896580i \(0.353956\pi\)
\(62\) 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
\(71\) 360.000 0.601748 0.300874 − 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) −26.0000 −0.0416859 −0.0208429 − 0.999783i \(-0.506635\pi\)
−0.0208429 + 0.999783i \(0.506635\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 288.000 0.426242
\(78\) 0 0
\(79\) 512.000 0.729171 0.364585 − 0.931170i \(-0.381211\pi\)
0.364585 + 0.931170i \(0.381211\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1188.00 −1.57108 −0.785542 − 0.618809i \(-0.787616\pi\)
−0.785542 + 0.618809i \(0.787616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 630.000 0.750336 0.375168 − 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1054.00 1.10327 0.551637 − 0.834085i \(-0.314004\pi\)
0.551637 + 0.834085i \(0.314004\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −558.000 −0.549733 −0.274867 − 0.961482i \(-0.588634\pi\)
−0.274867 + 0.961482i \(0.588634\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.00765304 −0.00382652 − 0.999993i \(-0.501218\pi\)
−0.00382652 + 0.999993i \(0.501218\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1764.00 1.59376 0.796880 − 0.604138i \(-0.206482\pi\)
0.796880 + 0.604138i \(0.206482\pi\)
\(108\) 0 0
\(109\) 1622.00 1.42532 0.712658 − 0.701512i \(-0.247491\pi\)
0.712658 + 0.701512i \(0.247491\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1134.00 −0.944051 −0.472025 − 0.881585i \(-0.656477\pi\)
−0.472025 + 0.881585i \(0.656477\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −144.000 −0.110928
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 592.000 0.413634 0.206817 − 0.978380i \(-0.433690\pi\)
0.206817 + 0.978380i \(0.433690\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1908.00 1.27254 0.636270 − 0.771466i \(-0.280476\pi\)
0.636270 + 0.771466i \(0.280476\pi\)
\(132\) 0 0
\(133\) 800.000 0.521570
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 954.000 0.594932 0.297466 − 0.954732i \(-0.403858\pi\)
0.297466 + 0.954732i \(0.403858\pi\)
\(138\) 0 0
\(139\) 2564.00 1.56457 0.782286 − 0.622919i \(-0.214053\pi\)
0.782286 + 0.622919i \(0.214053\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −360.000 −0.210522
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 738.000 0.405767 0.202884 − 0.979203i \(-0.434969\pi\)
0.202884 + 0.979203i \(0.434969\pi\)
\(150\) 0 0
\(151\) −2440.00 −1.31500 −0.657498 − 0.753456i \(-0.728385\pi\)
−0.657498 + 0.753456i \(0.728385\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2554.00 1.29829 0.649145 − 0.760665i \(-0.275127\pi\)
0.649145 + 0.760665i \(0.275127\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −576.000 −0.281958
\(162\) 0 0
\(163\) 820.000 0.394033 0.197016 − 0.980400i \(-0.436875\pi\)
0.197016 + 0.980400i \(0.436875\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1944.00 0.900786 0.450393 − 0.892830i \(-0.351284\pi\)
0.450393 + 0.892830i \(0.351284\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1242.00 −0.545824 −0.272912 − 0.962039i \(-0.587987\pi\)
−0.272912 + 0.962039i \(0.587987\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1116.00 −0.465999 −0.232999 − 0.972477i \(-0.574854\pi\)
−0.232999 + 0.972477i \(0.574854\pi\)
\(180\) 0 0
\(181\) 1070.00 0.439406 0.219703 − 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −648.000 −0.253403
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 576.000 0.218209 0.109104 − 0.994030i \(-0.465202\pi\)
0.109104 + 0.994030i \(0.465202\pi\)
\(192\) 0 0
\(193\) 1342.00 0.500514 0.250257 − 0.968179i \(-0.419485\pi\)
0.250257 + 0.968179i \(0.419485\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1422.00 0.514281 0.257140 − 0.966374i \(-0.417220\pi\)
0.257140 + 0.966374i \(0.417220\pi\)
\(198\) 0 0
\(199\) 872.000 0.310625 0.155313 − 0.987865i \(-0.450361\pi\)
0.155313 + 0.987865i \(0.450361\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1872.00 −0.647235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3600.00 1.19147
\(210\) 0 0
\(211\) 1340.00 0.437201 0.218600 − 0.975814i \(-0.429851\pi\)
0.218600 + 0.975814i \(0.429851\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 128.000 0.0400424
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 180.000 0.0547878
\(222\) 0 0
\(223\) −4880.00 −1.46542 −0.732711 − 0.680540i \(-0.761745\pi\)
−0.732711 + 0.680540i \(0.761745\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2700.00 0.789451 0.394725 − 0.918799i \(-0.370840\pi\)
0.394725 + 0.918799i \(0.370840\pi\)
\(228\) 0 0
\(229\) 254.000 0.0732960 0.0366480 − 0.999328i \(-0.488332\pi\)
0.0366480 + 0.999328i \(0.488332\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4410.00 1.23995 0.619976 − 0.784621i \(-0.287142\pi\)
0.619976 + 0.784621i \(0.287142\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3888.00 1.05228 0.526138 − 0.850399i \(-0.323640\pi\)
0.526138 + 0.850399i \(0.323640\pi\)
\(240\) 0 0
\(241\) 5138.00 1.37331 0.686655 − 0.726984i \(-0.259078\pi\)
0.686655 + 0.726984i \(0.259078\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1000.00 −0.257605
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 − 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5886.00 −1.42863 −0.714316 − 0.699823i \(-0.753262\pi\)
−0.714316 + 0.699823i \(0.753262\pi\)
\(258\) 0 0
\(259\) −1808.00 −0.433759
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2232.00 0.523312 0.261656 − 0.965161i \(-0.415731\pi\)
0.261656 + 0.965161i \(0.415731\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 666.000 0.150954 0.0754772 − 0.997148i \(-0.475952\pi\)
0.0754772 + 0.997148i \(0.475952\pi\)
\(270\) 0 0
\(271\) −5536.00 −1.24092 −0.620458 − 0.784240i \(-0.713053\pi\)
−0.620458 + 0.784240i \(0.713053\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2126.00 −0.461151 −0.230576 − 0.973054i \(-0.574061\pi\)
−0.230576 + 0.973054i \(0.574061\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2934.00 0.622875 0.311437 − 0.950267i \(-0.399190\pi\)
0.311437 + 0.950267i \(0.399190\pi\)
\(282\) 0 0
\(283\) −2036.00 −0.427659 −0.213830 − 0.976871i \(-0.568594\pi\)
−0.213830 + 0.976871i \(0.568594\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 720.000 0.148085
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2286.00 0.455800 0.227900 − 0.973684i \(-0.426814\pi\)
0.227900 + 0.973684i \(0.426814\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 720.000 0.139260
\(300\) 0 0
\(301\) 3616.00 0.692434
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1244.00 −0.231267 −0.115633 − 0.993292i \(-0.536890\pi\)
−0.115633 + 0.993292i \(0.536890\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1224.00 −0.223173 −0.111586 − 0.993755i \(-0.535593\pi\)
−0.111586 + 0.993755i \(0.535593\pi\)
\(312\) 0 0
\(313\) −1898.00 −0.342752 −0.171376 − 0.985206i \(-0.554821\pi\)
−0.171376 + 0.985206i \(0.554821\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9162.00 −1.62331 −0.811655 − 0.584137i \(-0.801433\pi\)
−0.811655 + 0.584137i \(0.801433\pi\)
\(318\) 0 0
\(319\) −8424.00 −1.47854
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1800.00 −0.310076
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3456.00 −0.579135
\(330\) 0 0
\(331\) −4348.00 −0.722017 −0.361009 − 0.932562i \(-0.617568\pi\)
−0.361009 + 0.932562i \(0.617568\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7154.00 −1.15639 −0.578195 − 0.815899i \(-0.696243\pi\)
−0.578195 + 0.815899i \(0.696243\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 576.000 0.0914726
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1836.00 −0.284039 −0.142020 − 0.989864i \(-0.545360\pi\)
−0.142020 + 0.989864i \(0.545360\pi\)
\(348\) 0 0
\(349\) 5894.00 0.904007 0.452004 − 0.892016i \(-0.350709\pi\)
0.452004 + 0.892016i \(0.350709\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11106.0 1.67454 0.837270 − 0.546789i \(-0.184150\pi\)
0.837270 + 0.546789i \(0.184150\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13176.0 −1.93705 −0.968527 − 0.248907i \(-0.919929\pi\)
−0.968527 + 0.248907i \(0.919929\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6112.00 0.869329 0.434665 − 0.900592i \(-0.356867\pi\)
0.434665 + 0.900592i \(0.356867\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3312.00 −0.463478
\(372\) 0 0
\(373\) 13618.0 1.89038 0.945192 − 0.326515i \(-0.105874\pi\)
0.945192 + 0.326515i \(0.105874\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2340.00 0.319671
\(378\) 0 0
\(379\) 692.000 0.0937880 0.0468940 − 0.998900i \(-0.485068\pi\)
0.0468940 + 0.998900i \(0.485068\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8064.00 −1.07585 −0.537926 − 0.842992i \(-0.680792\pi\)
−0.537926 + 0.842992i \(0.680792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12654.0 −1.64931 −0.824657 − 0.565633i \(-0.808632\pi\)
−0.824657 + 0.565633i \(0.808632\pi\)
\(390\) 0 0
\(391\) 1296.00 0.167625
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 106.000 0.0134005 0.00670024 − 0.999978i \(-0.497867\pi\)
0.00670024 + 0.999978i \(0.497867\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4014.00 0.499874 0.249937 − 0.968262i \(-0.419590\pi\)
0.249937 + 0.968262i \(0.419590\pi\)
\(402\) 0 0
\(403\) −160.000 −0.0197771
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8136.00 −0.990876
\(408\) 0 0
\(409\) 3914.00 0.473190 0.236595 − 0.971608i \(-0.423968\pi\)
0.236595 + 0.971608i \(0.423968\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5472.00 −0.651960
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4428.00 −0.516282 −0.258141 − 0.966107i \(-0.583110\pi\)
−0.258141 + 0.966107i \(0.583110\pi\)
\(420\) 0 0
\(421\) −15490.0 −1.79320 −0.896599 − 0.442843i \(-0.853970\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3376.00 −0.382614
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6768.00 −0.756388 −0.378194 − 0.925726i \(-0.623455\pi\)
−0.378194 + 0.925726i \(0.623455\pi\)
\(432\) 0 0
\(433\) −1298.00 −0.144060 −0.0720299 − 0.997402i \(-0.522948\pi\)
−0.0720299 + 0.997402i \(0.522948\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7200.00 −0.788153
\(438\) 0 0
\(439\) −2248.00 −0.244399 −0.122200 − 0.992506i \(-0.538995\pi\)
−0.122200 + 0.992506i \(0.538995\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9612.00 −1.03088 −0.515440 − 0.856926i \(-0.672372\pi\)
−0.515440 + 0.856926i \(0.672372\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −162.000 −0.0170273 −0.00851364 − 0.999964i \(-0.502710\pi\)
−0.00851364 + 0.999964i \(0.502710\pi\)
\(450\) 0 0
\(451\) 3240.00 0.338283
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1370.00 −0.140232 −0.0701159 − 0.997539i \(-0.522337\pi\)
−0.0701159 + 0.997539i \(0.522337\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15354.0 1.55121 0.775604 − 0.631220i \(-0.217445\pi\)
0.775604 + 0.631220i \(0.217445\pi\)
\(462\) 0 0
\(463\) 13024.0 1.30729 0.653646 − 0.756800i \(-0.273238\pi\)
0.653646 + 0.756800i \(0.273238\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14436.0 −1.43045 −0.715223 − 0.698896i \(-0.753675\pi\)
−0.715223 + 0.698896i \(0.753675\pi\)
\(468\) 0 0
\(469\) 2656.00 0.261498
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16272.0 1.58179
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12096.0 −1.15382 −0.576911 − 0.816807i \(-0.695742\pi\)
−0.576911 + 0.816807i \(0.695742\pi\)
\(480\) 0 0
\(481\) 2260.00 0.214235
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6056.00 −0.563498 −0.281749 − 0.959488i \(-0.590915\pi\)
−0.281749 + 0.959488i \(0.590915\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7524.00 −0.691555 −0.345777 − 0.938317i \(-0.612385\pi\)
−0.345777 + 0.938317i \(0.612385\pi\)
\(492\) 0 0
\(493\) 4212.00 0.384785
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2880.00 −0.259931
\(498\) 0 0
\(499\) 5276.00 0.473319 0.236660 − 0.971593i \(-0.423947\pi\)
0.236660 + 0.971593i \(0.423947\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4968.00 0.440382 0.220191 − 0.975457i \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10998.0 −0.957717 −0.478858 − 0.877892i \(-0.658949\pi\)
−0.478858 + 0.877892i \(0.658949\pi\)
\(510\) 0 0
\(511\) 208.000 0.0180066
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −15552.0 −1.32297
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8838.00 0.743186 0.371593 − 0.928396i \(-0.378812\pi\)
0.371593 + 0.928396i \(0.378812\pi\)
\(522\) 0 0
\(523\) −22436.0 −1.87583 −0.937914 − 0.346869i \(-0.887245\pi\)
−0.937914 + 0.346869i \(0.887245\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −288.000 −0.0238055
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −900.000 −0.0731395
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10044.0 0.802645
\(540\) 0 0
\(541\) −4762.00 −0.378437 −0.189218 − 0.981935i \(-0.560595\pi\)
−0.189218 + 0.981935i \(0.560595\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6004.00 0.469310 0.234655 − 0.972079i \(-0.424604\pi\)
0.234655 + 0.972079i \(0.424604\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23400.0 −1.80921
\(552\) 0 0
\(553\) −4096.00 −0.314972
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5274.00 −0.401197 −0.200598 − 0.979674i \(-0.564289\pi\)
−0.200598 + 0.979674i \(0.564289\pi\)
\(558\) 0 0
\(559\) −4520.00 −0.341996
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12420.0 −0.929735 −0.464867 − 0.885380i \(-0.653898\pi\)
−0.464867 + 0.885380i \(0.653898\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21366.0 1.57418 0.787091 − 0.616837i \(-0.211586\pi\)
0.787091 + 0.616837i \(0.211586\pi\)
\(570\) 0 0
\(571\) 21140.0 1.54935 0.774677 − 0.632357i \(-0.217912\pi\)
0.774677 + 0.632357i \(0.217912\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3266.00 −0.235642 −0.117821 − 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9504.00 0.678644
\(582\) 0 0
\(583\) −14904.0 −1.05877
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17028.0 1.19731 0.598655 − 0.801007i \(-0.295702\pi\)
0.598655 + 0.801007i \(0.295702\pi\)
\(588\) 0 0
\(589\) 1600.00 0.111930
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9522.00 0.659396 0.329698 − 0.944086i \(-0.393053\pi\)
0.329698 + 0.944086i \(0.393053\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10296.0 0.702309 0.351155 − 0.936318i \(-0.385789\pi\)
0.351155 + 0.936318i \(0.385789\pi\)
\(600\) 0 0
\(601\) −3382.00 −0.229542 −0.114771 − 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20656.0 1.38122 0.690611 − 0.723227i \(-0.257342\pi\)
0.690611 + 0.723227i \(0.257342\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4320.00 0.286037
\(612\) 0 0
\(613\) 22114.0 1.45706 0.728529 − 0.685015i \(-0.240205\pi\)
0.728529 + 0.685015i \(0.240205\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19962.0 1.30250 0.651248 − 0.758865i \(-0.274246\pi\)
0.651248 + 0.758865i \(0.274246\pi\)
\(618\) 0 0
\(619\) −604.000 −0.0392194 −0.0196097 − 0.999808i \(-0.506242\pi\)
−0.0196097 + 0.999808i \(0.506242\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5040.00 −0.324115
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4068.00 0.257872
\(630\) 0 0
\(631\) 152.000 0.00958958 0.00479479 − 0.999989i \(-0.498474\pi\)
0.00479479 + 0.999989i \(0.498474\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2790.00 −0.173538
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4194.00 −0.258429 −0.129215 − 0.991617i \(-0.541246\pi\)
−0.129215 + 0.991617i \(0.541246\pi\)
\(642\) 0 0
\(643\) 7252.00 0.444776 0.222388 − 0.974958i \(-0.428615\pi\)
0.222388 + 0.974958i \(0.428615\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6696.00 −0.406873 −0.203437 − 0.979088i \(-0.565211\pi\)
−0.203437 + 0.979088i \(0.565211\pi\)
\(648\) 0 0
\(649\) −24624.0 −1.48933
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28422.0 1.70328 0.851638 − 0.524131i \(-0.175610\pi\)
0.851638 + 0.524131i \(0.175610\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19908.0 1.17679 0.588396 − 0.808573i \(-0.299760\pi\)
0.588396 + 0.808573i \(0.299760\pi\)
\(660\) 0 0
\(661\) 14318.0 0.842520 0.421260 − 0.906940i \(-0.361588\pi\)
0.421260 + 0.906940i \(0.361588\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16848.0 0.978047
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15192.0 −0.874040
\(672\) 0 0
\(673\) −30050.0 −1.72116 −0.860581 − 0.509313i \(-0.829899\pi\)
−0.860581 + 0.509313i \(0.829899\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22158.0 1.25790 0.628952 − 0.777444i \(-0.283484\pi\)
0.628952 + 0.777444i \(0.283484\pi\)
\(678\) 0 0
\(679\) −8432.00 −0.476569
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3132.00 −0.175465 −0.0877325 − 0.996144i \(-0.527962\pi\)
−0.0877325 + 0.996144i \(0.527962\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4140.00 0.228914
\(690\) 0 0
\(691\) −20932.0 −1.15237 −0.576187 − 0.817318i \(-0.695460\pi\)
−0.576187 + 0.817318i \(0.695460\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1620.00 −0.0880371
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21834.0 1.17640 0.588202 − 0.808714i \(-0.299836\pi\)
0.588202 + 0.808714i \(0.299836\pi\)
\(702\) 0 0
\(703\) −22600.0 −1.21248
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4464.00 0.237463
\(708\) 0 0
\(709\) 12446.0 0.659266 0.329633 − 0.944109i \(-0.393075\pi\)
0.329633 + 0.944109i \(0.393075\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1152.00 −0.0605088
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12528.0 0.649813 0.324907 − 0.945746i \(-0.394667\pi\)
0.324907 + 0.945746i \(0.394667\pi\)
\(720\) 0 0
\(721\) 64.0000 0.00330580
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −11576.0 −0.590550 −0.295275 − 0.955412i \(-0.595411\pi\)
−0.295275 + 0.955412i \(0.595411\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8136.00 −0.411656
\(732\) 0 0
\(733\) 29338.0 1.47834 0.739170 − 0.673519i \(-0.235218\pi\)
0.739170 + 0.673519i \(0.235218\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11952.0 0.597364
\(738\) 0 0
\(739\) 2540.00 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18792.0 −0.927876 −0.463938 − 0.885868i \(-0.653564\pi\)
−0.463938 + 0.885868i \(0.653564\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14112.0 −0.688440
\(750\) 0 0
\(751\) 4832.00 0.234783 0.117392 − 0.993086i \(-0.462547\pi\)
0.117392 + 0.993086i \(0.462547\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20818.0 0.999529 0.499764 − 0.866161i \(-0.333420\pi\)
0.499764 + 0.866161i \(0.333420\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12042.0 −0.573617 −0.286808 − 0.957988i \(-0.592594\pi\)
−0.286808 + 0.957988i \(0.592594\pi\)
\(762\) 0 0
\(763\) −12976.0 −0.615679
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6840.00 0.322005
\(768\) 0 0
\(769\) 13058.0 0.612332 0.306166 − 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11826.0 −0.550261 −0.275130 − 0.961407i \(-0.588721\pi\)
−0.275130 + 0.961407i \(0.588721\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9000.00 0.413939
\(780\) 0 0
\(781\) −12960.0 −0.593784
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −11996.0 −0.543343 −0.271672 − 0.962390i \(-0.587576\pi\)
−0.271672 + 0.962390i \(0.587576\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9072.00 0.407792
\(792\) 0 0
\(793\) 4220.00 0.188974
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6966.00 0.309596 0.154798 − 0.987946i \(-0.450527\pi\)
0.154798 + 0.987946i \(0.450527\pi\)
\(798\) 0 0
\(799\) 7776.00 0.344299
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 936.000 0.0411342
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 40806.0 1.77338 0.886689 − 0.462367i \(-0.153000\pi\)
0.886689 + 0.462367i \(0.153000\pi\)
\(810\) 0 0
\(811\) −17980.0 −0.778500 −0.389250 − 0.921132i \(-0.627266\pi\)
−0.389250 + 0.921132i \(0.627266\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 45200.0 1.93555
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12834.0 0.545566 0.272783 − 0.962076i \(-0.412056\pi\)
0.272783 + 0.962076i \(0.412056\pi\)
\(822\) 0 0
\(823\) 37864.0 1.60371 0.801857 − 0.597516i \(-0.203846\pi\)
0.801857 + 0.597516i \(0.203846\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42516.0 1.78770 0.893849 − 0.448368i \(-0.147995\pi\)
0.893849 + 0.448368i \(0.147995\pi\)
\(828\) 0 0
\(829\) 45638.0 1.91203 0.956015 − 0.293317i \(-0.0947591\pi\)
0.956015 + 0.293317i \(0.0947591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5022.00 −0.208886
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −17496.0 −0.719939 −0.359970 − 0.932964i \(-0.617213\pi\)
−0.359970 + 0.932964i \(0.617213\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 280.000 0.0113588
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16272.0 0.655461
\(852\) 0 0
\(853\) −32174.0 −1.29146 −0.645731 − 0.763565i \(-0.723447\pi\)
−0.645731 + 0.763565i \(0.723447\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38934.0 −1.55188 −0.775939 − 0.630807i \(-0.782724\pi\)
−0.775939 + 0.630807i \(0.782724\pi\)
\(858\) 0 0
\(859\) 29780.0 1.18286 0.591432 − 0.806355i \(-0.298563\pi\)
0.591432 + 0.806355i \(0.298563\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48096.0 −1.89711 −0.948556 − 0.316611i \(-0.897455\pi\)
−0.948556 + 0.316611i \(0.897455\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18432.0 −0.719520
\(870\) 0 0
\(871\) −3320.00 −0.129155
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −21302.0 −0.820202 −0.410101 − 0.912040i \(-0.634507\pi\)
−0.410101 + 0.912040i \(0.634507\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7470.00 0.285665 0.142832 − 0.989747i \(-0.454379\pi\)
0.142832 + 0.989747i \(0.454379\pi\)
\(882\) 0 0
\(883\) −764.000 −0.0291174 −0.0145587 − 0.999894i \(-0.504634\pi\)
−0.0145587 + 0.999894i \(0.504634\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32328.0 1.22375 0.611876 − 0.790954i \(-0.290415\pi\)
0.611876 + 0.790954i \(0.290415\pi\)
\(888\) 0 0
\(889\) −4736.00 −0.178673
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −43200.0 −1.61885
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3744.00 −0.138898
\(900\) 0 0
\(901\) 7452.00 0.275541
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 36316.0 1.32950 0.664748 − 0.747068i \(-0.268539\pi\)
0.664748 + 0.747068i \(0.268539\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 13392.0 0.487044 0.243522 − 0.969895i \(-0.421697\pi\)
0.243522 + 0.969895i \(0.421697\pi\)
\(912\) 0 0
\(913\) 42768.0 1.55029
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15264.0 −0.549686
\(918\) 0 0
\(919\) 38072.0 1.36657 0.683286 − 0.730151i \(-0.260550\pi\)
0.683286 + 0.730151i \(0.260550\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3600.00 0.128381
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12798.0 0.451979 0.225990 − 0.974130i \(-0.427438\pi\)
0.225990 + 0.974130i \(0.427438\pi\)
\(930\) 0 0
\(931\) 27900.0 0.982154
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −34874.0 −1.21588 −0.607942 − 0.793981i \(-0.708005\pi\)
−0.607942 + 0.793981i \(0.708005\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17190.0 −0.595513 −0.297757 − 0.954642i \(-0.596238\pi\)
−0.297757 + 0.954642i \(0.596238\pi\)
\(942\) 0 0
\(943\) −6480.00 −0.223773
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40284.0 1.38232 0.691158 − 0.722703i \(-0.257101\pi\)
0.691158 + 0.722703i \(0.257101\pi\)
\(948\) 0 0
\(949\) −260.000 −0.00889353
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15498.0 0.526789 0.263394 − 0.964688i \(-0.415158\pi\)
0.263394 + 0.964688i \(0.415158\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7632.00 −0.256987
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −37160.0 −1.23577 −0.617883 − 0.786270i \(-0.712009\pi\)
−0.617883 + 0.786270i \(0.712009\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18468.0 −0.610367 −0.305183 − 0.952294i \(-0.598718\pi\)
−0.305183 + 0.952294i \(0.598718\pi\)
\(972\) 0 0
\(973\) −20512.0 −0.675832
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10386.0 0.340100 0.170050 − 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(978\) 0 0
\(979\) −22680.0 −0.740404
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 44136.0 1.43206 0.716032 − 0.698067i \(-0.245956\pi\)
0.716032 + 0.698067i \(0.245956\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32544.0 −1.04635
\(990\) 0 0
\(991\) −28432.0 −0.911375 −0.455687 − 0.890140i \(-0.650606\pi\)
−0.455687 + 0.890140i \(0.650606\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 39778.0 1.26357 0.631786 − 0.775143i \(-0.282322\pi\)
0.631786 + 0.775143i \(0.282322\pi\)
\(998\) 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 900.4.a.g.1.1 1
3.2 odd 2 300.4.a.b.1.1 1
5.2 odd 4 900.4.d.c.649.1 2
5.3 odd 4 900.4.d.c.649.2 2
5.4 even 2 36.4.a.a.1.1 1
12.11 even 2 1200.4.a.be.1.1 1
15.2 even 4 300.4.d.e.49.2 2
15.8 even 4 300.4.d.e.49.1 2
15.14 odd 2 12.4.a.a.1.1 ✓ 1
20.19 odd 2 144.4.a.g.1.1 1
35.4 even 6 1764.4.k.b.1549.1 2
35.9 even 6 1764.4.k.b.361.1 2
35.19 odd 6 1764.4.k.o.361.1 2
35.24 odd 6 1764.4.k.o.1549.1 2
35.34 odd 2 1764.4.a.b.1.1 1
40.19 odd 2 576.4.a.a.1.1 1
40.29 even 2 576.4.a.b.1.1 1
45.4 even 6 324.4.e.a.217.1 2
45.14 odd 6 324.4.e.h.217.1 2
45.29 odd 6 324.4.e.h.109.1 2
45.34 even 6 324.4.e.a.109.1 2
60.23 odd 4 1200.4.f.d.49.2 2
60.47 odd 4 1200.4.f.d.49.1 2
60.59 even 2 48.4.a.a.1.1 1
105.44 odd 6 588.4.i.d.361.1 2
105.59 even 6 588.4.i.e.373.1 2
105.74 odd 6 588.4.i.d.373.1 2
105.89 even 6 588.4.i.e.361.1 2
105.104 even 2 588.4.a.c.1.1 1
120.29 odd 2 192.4.a.f.1.1 1
120.59 even 2 192.4.a.l.1.1 1
165.164 even 2 1452.4.a.d.1.1 1
195.44 even 4 2028.4.b.c.337.2 2
195.164 even 4 2028.4.b.c.337.1 2
195.194 odd 2 2028.4.a.c.1.1 1
240.29 odd 4 768.4.d.g.385.2 2
240.59 even 4 768.4.d.j.385.2 2
240.149 odd 4 768.4.d.g.385.1 2
240.179 even 4 768.4.d.j.385.1 2
420.419 odd 2 2352.4.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 ✓ 1 15.14 odd 2
36.4.a.a.1.1 1 5.4 even 2
48.4.a.a.1.1 1 60.59 even 2
144.4.a.g.1.1 1 20.19 odd 2
192.4.a.f.1.1 1 120.29 odd 2
192.4.a.l.1.1 1 120.59 even 2
300.4.a.b.1.1 1 3.2 odd 2
300.4.d.e.49.1 2 15.8 even 4
300.4.d.e.49.2 2 15.2 even 4
324.4.e.a.109.1 2 45.34 even 6
324.4.e.a.217.1 2 45.4 even 6
324.4.e.h.109.1 2 45.29 odd 6
324.4.e.h.217.1 2 45.14 odd 6
576.4.a.a.1.1 1 40.19 odd 2
576.4.a.b.1.1 1 40.29 even 2
588.4.a.c.1.1 1 105.104 even 2
588.4.i.d.361.1 2 105.44 odd 6
588.4.i.d.373.1 2 105.74 odd 6
588.4.i.e.361.1 2 105.89 even 6
588.4.i.e.373.1 2 105.59 even 6
768.4.d.g.385.1 2 240.149 odd 4
768.4.d.g.385.2 2 240.29 odd 4
768.4.d.j.385.1 2 240.179 even 4
768.4.d.j.385.2 2 240.59 even 4
900.4.a.g.1.1 1 1.1 even 1 trivial
900.4.d.c.649.1 2 5.2 odd 4
900.4.d.c.649.2 2 5.3 odd 4
1200.4.a.be.1.1 1 12.11 even 2
1200.4.f.d.49.1 2 60.47 odd 4
1200.4.f.d.49.2 2 60.23 odd 4
1452.4.a.d.1.1 1 165.164 even 2
1764.4.a.b.1.1 1 35.34 odd 2
1764.4.k.b.361.1 2 35.9 even 6
1764.4.k.b.1549.1 2 35.4 even 6
1764.4.k.o.361.1 2 35.19 odd 6
1764.4.k.o.1549.1 2 35.24 odd 6
2028.4.a.c.1.1 1 195.194 odd 2
2028.4.b.c.337.1 2 195.164 even 4
2028.4.b.c.337.2 2 195.44 even 4
2352.4.a.bk.1.1 1 420.419 odd 2