Properties

Label 1600.4.a.t.1.1
Level $1600$
Weight $4$
Character 1600.1
Self dual yes
Analytic conductor $94.403$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,4,Mod(1,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, 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("1600.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.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: \(94.4030560092\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} -28.0000 q^{11} +12.0000 q^{13} +64.0000 q^{17} -60.0000 q^{19} -52.0000 q^{21} -58.0000 q^{23} +100.000 q^{27} -90.0000 q^{29} +128.000 q^{31} +56.0000 q^{33} +236.000 q^{37} -24.0000 q^{39} +242.000 q^{41} -362.000 q^{43} +226.000 q^{47} +333.000 q^{49} -128.000 q^{51} -108.000 q^{53} +120.000 q^{57} -20.0000 q^{59} -542.000 q^{61} -598.000 q^{63} +434.000 q^{67} +116.000 q^{69} +1128.00 q^{71} -632.000 q^{73} -728.000 q^{77} +720.000 q^{79} +421.000 q^{81} +478.000 q^{83} +180.000 q^{87} -490.000 q^{89} +312.000 q^{91} -256.000 q^{93} -1456.00 q^{97} +644.000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 26.0000 1.40387 0.701934 0.712242i \(-0.252320\pi\)
0.701934 + 0.712242i \(0.252320\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −28.0000 −0.767483 −0.383742 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(12\) 0 0
\(13\) 12.0000 0.256015 0.128008 0.991773i \(-0.459142\pi\)
0.128008 + 0.991773i \(0.459142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 64.0000 0.913075 0.456538 0.889704i \(-0.349089\pi\)
0.456538 + 0.889704i \(0.349089\pi\)
\(18\) 0 0
\(19\) −60.0000 −0.724471 −0.362235 0.932087i \(-0.617986\pi\)
−0.362235 + 0.932087i \(0.617986\pi\)
\(20\) 0 0
\(21\) −52.0000 −0.540349
\(22\) 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
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) −90.0000 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) 0 0
\(31\) 128.000 0.741596 0.370798 0.928714i \(-0.379084\pi\)
0.370798 + 0.928714i \(0.379084\pi\)
\(32\) 0 0
\(33\) 56.0000 0.295405
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 236.000 1.04860 0.524299 0.851534i \(-0.324327\pi\)
0.524299 + 0.851534i \(0.324327\pi\)
\(38\) 0 0
\(39\) −24.0000 −0.0985404
\(40\) 0 0
\(41\) 242.000 0.921806 0.460903 0.887450i \(-0.347526\pi\)
0.460903 + 0.887450i \(0.347526\pi\)
\(42\) 0 0
\(43\) −362.000 −1.28383 −0.641913 0.766778i \(-0.721859\pi\)
−0.641913 + 0.766778i \(0.721859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 226.000 0.701393 0.350697 0.936489i \(-0.385945\pi\)
0.350697 + 0.936489i \(0.385945\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) −128.000 −0.351443
\(52\) 0 0
\(53\) −108.000 −0.279905 −0.139952 0.990158i \(-0.544695\pi\)
−0.139952 + 0.990158i \(0.544695\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 120.000 0.278849
\(58\) 0 0
\(59\) −20.0000 −0.0441318 −0.0220659 0.999757i \(-0.507024\pi\)
−0.0220659 + 0.999757i \(0.507024\pi\)
\(60\) 0 0
\(61\) −542.000 −1.13764 −0.568820 0.822462i \(-0.692600\pi\)
−0.568820 + 0.822462i \(0.692600\pi\)
\(62\) 0 0
\(63\) −598.000 −1.19589
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 434.000 0.791366 0.395683 0.918387i \(-0.370508\pi\)
0.395683 + 0.918387i \(0.370508\pi\)
\(68\) 0 0
\(69\) 116.000 0.202388
\(70\) 0 0
\(71\) 1128.00 1.88548 0.942739 0.333531i \(-0.108240\pi\)
0.942739 + 0.333531i \(0.108240\pi\)
\(72\) 0 0
\(73\) −632.000 −1.01329 −0.506644 0.862155i \(-0.669114\pi\)
−0.506644 + 0.862155i \(0.669114\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −728.000 −1.07745
\(78\) 0 0
\(79\) 720.000 1.02540 0.512698 0.858569i \(-0.328646\pi\)
0.512698 + 0.858569i \(0.328646\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 478.000 0.632136 0.316068 0.948736i \(-0.397637\pi\)
0.316068 + 0.948736i \(0.397637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 180.000 0.221816
\(88\) 0 0
\(89\) −490.000 −0.583594 −0.291797 0.956480i \(-0.594253\pi\)
−0.291797 + 0.956480i \(0.594253\pi\)
\(90\) 0 0
\(91\) 312.000 0.359412
\(92\) 0 0
\(93\) −256.000 −0.285440
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1456.00 −1.52407 −0.762033 0.647538i \(-0.775799\pi\)
−0.762033 + 0.647538i \(0.775799\pi\)
\(98\) 0 0
\(99\) 644.000 0.653782
\(100\) 0 0
\(101\) 578.000 0.569437 0.284719 0.958611i \(-0.408100\pi\)
0.284719 + 0.958611i \(0.408100\pi\)
\(102\) 0 0
\(103\) 1462.00 1.39859 0.699297 0.714831i \(-0.253497\pi\)
0.699297 + 0.714831i \(0.253497\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −966.000 −0.872773 −0.436387 0.899759i \(-0.643742\pi\)
−0.436387 + 0.899759i \(0.643742\pi\)
\(108\) 0 0
\(109\) −370.000 −0.325134 −0.162567 0.986698i \(-0.551977\pi\)
−0.162567 + 0.986698i \(0.551977\pi\)
\(110\) 0 0
\(111\) −472.000 −0.403606
\(112\) 0 0
\(113\) 528.000 0.439558 0.219779 0.975550i \(-0.429466\pi\)
0.219779 + 0.975550i \(0.429466\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −276.000 −0.218087
\(118\) 0 0
\(119\) 1664.00 1.28184
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) −484.000 −0.354803
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1534.00 −1.07181 −0.535907 0.844277i \(-0.680030\pi\)
−0.535907 + 0.844277i \(0.680030\pi\)
\(128\) 0 0
\(129\) 724.000 0.494145
\(130\) 0 0
\(131\) 12.0000 0.00800340 0.00400170 0.999992i \(-0.498726\pi\)
0.00400170 + 0.999992i \(0.498726\pi\)
\(132\) 0 0
\(133\) −1560.00 −1.01706
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1224.00 0.763309 0.381655 0.924305i \(-0.375354\pi\)
0.381655 + 0.924305i \(0.375354\pi\)
\(138\) 0 0
\(139\) 3100.00 1.89164 0.945822 0.324685i \(-0.105258\pi\)
0.945822 + 0.324685i \(0.105258\pi\)
\(140\) 0 0
\(141\) −452.000 −0.269966
\(142\) 0 0
\(143\) −336.000 −0.196488
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −666.000 −0.373679
\(148\) 0 0
\(149\) −250.000 −0.137455 −0.0687275 0.997635i \(-0.521894\pi\)
−0.0687275 + 0.997635i \(0.521894\pi\)
\(150\) 0 0
\(151\) −2152.00 −1.15978 −0.579892 0.814694i \(-0.696905\pi\)
−0.579892 + 0.814694i \(0.696905\pi\)
\(152\) 0 0
\(153\) −1472.00 −0.777805
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −524.000 −0.266368 −0.133184 0.991091i \(-0.542520\pi\)
−0.133184 + 0.991091i \(0.542520\pi\)
\(158\) 0 0
\(159\) 216.000 0.107735
\(160\) 0 0
\(161\) −1508.00 −0.738180
\(162\) 0 0
\(163\) 3518.00 1.69050 0.845249 0.534373i \(-0.179452\pi\)
0.845249 + 0.534373i \(0.179452\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −534.000 −0.247438 −0.123719 0.992317i \(-0.539482\pi\)
−0.123719 + 0.992317i \(0.539482\pi\)
\(168\) 0 0
\(169\) −2053.00 −0.934456
\(170\) 0 0
\(171\) 1380.00 0.617142
\(172\) 0 0
\(173\) 4252.00 1.86863 0.934317 0.356444i \(-0.116011\pi\)
0.934317 + 0.356444i \(0.116011\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 40.0000 0.0169864
\(178\) 0 0
\(179\) 2500.00 1.04390 0.521952 0.852975i \(-0.325204\pi\)
0.521952 + 0.852975i \(0.325204\pi\)
\(180\) 0 0
\(181\) 2578.00 1.05868 0.529340 0.848410i \(-0.322439\pi\)
0.529340 + 0.848410i \(0.322439\pi\)
\(182\) 0 0
\(183\) 1084.00 0.437878
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1792.00 −0.700770
\(188\) 0 0
\(189\) 2600.00 1.00065
\(190\) 0 0
\(191\) 768.000 0.290945 0.145473 0.989362i \(-0.453530\pi\)
0.145473 + 0.989362i \(0.453530\pi\)
\(192\) 0 0
\(193\) 2608.00 0.972684 0.486342 0.873769i \(-0.338331\pi\)
0.486342 + 0.873769i \(0.338331\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5116.00 1.85025 0.925127 0.379659i \(-0.123959\pi\)
0.925127 + 0.379659i \(0.123959\pi\)
\(198\) 0 0
\(199\) 3480.00 1.23965 0.619826 0.784739i \(-0.287203\pi\)
0.619826 + 0.784739i \(0.287203\pi\)
\(200\) 0 0
\(201\) −868.000 −0.304597
\(202\) 0 0
\(203\) −2340.00 −0.809043
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1334.00 0.447920
\(208\) 0 0
\(209\) 1680.00 0.556019
\(210\) 0 0
\(211\) 3132.00 1.02188 0.510938 0.859618i \(-0.329298\pi\)
0.510938 + 0.859618i \(0.329298\pi\)
\(212\) 0 0
\(213\) −2256.00 −0.725721
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3328.00 1.04110
\(218\) 0 0
\(219\) 1264.00 0.390015
\(220\) 0 0
\(221\) 768.000 0.233761
\(222\) 0 0
\(223\) 62.0000 0.0186181 0.00930903 0.999957i \(-0.497037\pi\)
0.00930903 + 0.999957i \(0.497037\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5314.00 1.55376 0.776878 0.629651i \(-0.216802\pi\)
0.776878 + 0.629651i \(0.216802\pi\)
\(228\) 0 0
\(229\) 190.000 0.0548277 0.0274139 0.999624i \(-0.491273\pi\)
0.0274139 + 0.999624i \(0.491273\pi\)
\(230\) 0 0
\(231\) 1456.00 0.414709
\(232\) 0 0
\(233\) 2408.00 0.677053 0.338526 0.940957i \(-0.390072\pi\)
0.338526 + 0.940957i \(0.390072\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1440.00 −0.394675
\(238\) 0 0
\(239\) 5680.00 1.53727 0.768637 0.639685i \(-0.220935\pi\)
0.768637 + 0.639685i \(0.220935\pi\)
\(240\) 0 0
\(241\) −278.000 −0.0743052 −0.0371526 0.999310i \(-0.511829\pi\)
−0.0371526 + 0.999310i \(0.511829\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −720.000 −0.185476
\(248\) 0 0
\(249\) −956.000 −0.243309
\(250\) 0 0
\(251\) 3252.00 0.817787 0.408893 0.912582i \(-0.365915\pi\)
0.408893 + 0.912582i \(0.365915\pi\)
\(252\) 0 0
\(253\) 1624.00 0.403557
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1536.00 −0.372813 −0.186407 0.982473i \(-0.559684\pi\)
−0.186407 + 0.982473i \(0.559684\pi\)
\(258\) 0 0
\(259\) 6136.00 1.47209
\(260\) 0 0
\(261\) 2070.00 0.490919
\(262\) 0 0
\(263\) −4858.00 −1.13900 −0.569500 0.821991i \(-0.692863\pi\)
−0.569500 + 0.821991i \(0.692863\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 980.000 0.224626
\(268\) 0 0
\(269\) −2610.00 −0.591578 −0.295789 0.955253i \(-0.595583\pi\)
−0.295789 + 0.955253i \(0.595583\pi\)
\(270\) 0 0
\(271\) 5168.00 1.15843 0.579213 0.815176i \(-0.303360\pi\)
0.579213 + 0.815176i \(0.303360\pi\)
\(272\) 0 0
\(273\) −624.000 −0.138338
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1924.00 −0.417336 −0.208668 0.977987i \(-0.566913\pi\)
−0.208668 + 0.977987i \(0.566913\pi\)
\(278\) 0 0
\(279\) −2944.00 −0.631730
\(280\) 0 0
\(281\) 3042.00 0.645803 0.322901 0.946433i \(-0.395342\pi\)
0.322901 + 0.946433i \(0.395342\pi\)
\(282\) 0 0
\(283\) 1718.00 0.360864 0.180432 0.983587i \(-0.442250\pi\)
0.180432 + 0.983587i \(0.442250\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6292.00 1.29409
\(288\) 0 0
\(289\) −817.000 −0.166294
\(290\) 0 0
\(291\) 2912.00 0.586613
\(292\) 0 0
\(293\) 2292.00 0.456997 0.228498 0.973544i \(-0.426618\pi\)
0.228498 + 0.973544i \(0.426618\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2800.00 −0.547045
\(298\) 0 0
\(299\) −696.000 −0.134618
\(300\) 0 0
\(301\) −9412.00 −1.80232
\(302\) 0 0
\(303\) −1156.00 −0.219176
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5406.00 −1.00501 −0.502503 0.864576i \(-0.667587\pi\)
−0.502503 + 0.864576i \(0.667587\pi\)
\(308\) 0 0
\(309\) −2924.00 −0.538319
\(310\) 0 0
\(311\) 5688.00 1.03710 0.518548 0.855048i \(-0.326473\pi\)
0.518548 + 0.855048i \(0.326473\pi\)
\(312\) 0 0
\(313\) −7352.00 −1.32767 −0.663833 0.747881i \(-0.731072\pi\)
−0.663833 + 0.747881i \(0.731072\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3484.00 −0.617290 −0.308645 0.951177i \(-0.599876\pi\)
−0.308645 + 0.951177i \(0.599876\pi\)
\(318\) 0 0
\(319\) 2520.00 0.442298
\(320\) 0 0
\(321\) 1932.00 0.335931
\(322\) 0 0
\(323\) −3840.00 −0.661496
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 740.000 0.125144
\(328\) 0 0
\(329\) 5876.00 0.984664
\(330\) 0 0
\(331\) −7868.00 −1.30654 −0.653269 0.757125i \(-0.726603\pi\)
−0.653269 + 0.757125i \(0.726603\pi\)
\(332\) 0 0
\(333\) −5428.00 −0.893251
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −656.000 −0.106037 −0.0530187 0.998594i \(-0.516884\pi\)
−0.0530187 + 0.998594i \(0.516884\pi\)
\(338\) 0 0
\(339\) −1056.00 −0.169186
\(340\) 0 0
\(341\) −3584.00 −0.569163
\(342\) 0 0
\(343\) −260.000 −0.0409291
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5754.00 0.890176 0.445088 0.895487i \(-0.353172\pi\)
0.445088 + 0.895487i \(0.353172\pi\)
\(348\) 0 0
\(349\) 3110.00 0.477004 0.238502 0.971142i \(-0.423344\pi\)
0.238502 + 0.971142i \(0.423344\pi\)
\(350\) 0 0
\(351\) 1200.00 0.182482
\(352\) 0 0
\(353\) 7808.00 1.17727 0.588637 0.808397i \(-0.299665\pi\)
0.588637 + 0.808397i \(0.299665\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3328.00 −0.493379
\(358\) 0 0
\(359\) 9240.00 1.35841 0.679204 0.733949i \(-0.262325\pi\)
0.679204 + 0.733949i \(0.262325\pi\)
\(360\) 0 0
\(361\) −3259.00 −0.475142
\(362\) 0 0
\(363\) 1094.00 0.158182
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3214.00 −0.457137 −0.228569 0.973528i \(-0.573405\pi\)
−0.228569 + 0.973528i \(0.573405\pi\)
\(368\) 0 0
\(369\) −5566.00 −0.785242
\(370\) 0 0
\(371\) −2808.00 −0.392949
\(372\) 0 0
\(373\) −348.000 −0.0483077 −0.0241538 0.999708i \(-0.507689\pi\)
−0.0241538 + 0.999708i \(0.507689\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1080.00 −0.147541
\(378\) 0 0
\(379\) 4940.00 0.669527 0.334764 0.942302i \(-0.391344\pi\)
0.334764 + 0.942302i \(0.391344\pi\)
\(380\) 0 0
\(381\) 3068.00 0.412542
\(382\) 0 0
\(383\) 6142.00 0.819430 0.409715 0.912214i \(-0.365628\pi\)
0.409715 + 0.912214i \(0.365628\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8326.00 1.09363
\(388\) 0 0
\(389\) −3050.00 −0.397535 −0.198768 0.980047i \(-0.563694\pi\)
−0.198768 + 0.980047i \(0.563694\pi\)
\(390\) 0 0
\(391\) −3712.00 −0.480112
\(392\) 0 0
\(393\) −24.0000 −0.00308051
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5396.00 0.682160 0.341080 0.940034i \(-0.389207\pi\)
0.341080 + 0.940034i \(0.389207\pi\)
\(398\) 0 0
\(399\) 3120.00 0.391467
\(400\) 0 0
\(401\) 14482.0 1.80348 0.901741 0.432276i \(-0.142289\pi\)
0.901741 + 0.432276i \(0.142289\pi\)
\(402\) 0 0
\(403\) 1536.00 0.189860
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6608.00 −0.804782
\(408\) 0 0
\(409\) −1090.00 −0.131778 −0.0658888 0.997827i \(-0.520988\pi\)
−0.0658888 + 0.997827i \(0.520988\pi\)
\(410\) 0 0
\(411\) −2448.00 −0.293798
\(412\) 0 0
\(413\) −520.000 −0.0619553
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6200.00 −0.728094
\(418\) 0 0
\(419\) −7180.00 −0.837150 −0.418575 0.908182i \(-0.637470\pi\)
−0.418575 + 0.908182i \(0.637470\pi\)
\(420\) 0 0
\(421\) 8138.00 0.942095 0.471047 0.882108i \(-0.343876\pi\)
0.471047 + 0.882108i \(0.343876\pi\)
\(422\) 0 0
\(423\) −5198.00 −0.597483
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −14092.0 −1.59710
\(428\) 0 0
\(429\) 672.000 0.0756281
\(430\) 0 0
\(431\) 208.000 0.0232460 0.0116230 0.999932i \(-0.496300\pi\)
0.0116230 + 0.999932i \(0.496300\pi\)
\(432\) 0 0
\(433\) −12992.0 −1.44193 −0.720965 0.692971i \(-0.756301\pi\)
−0.720965 + 0.692971i \(0.756301\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3480.00 0.380940
\(438\) 0 0
\(439\) −1080.00 −0.117416 −0.0587080 0.998275i \(-0.518698\pi\)
−0.0587080 + 0.998275i \(0.518698\pi\)
\(440\) 0 0
\(441\) −7659.00 −0.827017
\(442\) 0 0
\(443\) 9078.00 0.973609 0.486805 0.873511i \(-0.338162\pi\)
0.486805 + 0.873511i \(0.338162\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 500.000 0.0529065
\(448\) 0 0
\(449\) 14310.0 1.50408 0.752039 0.659119i \(-0.229071\pi\)
0.752039 + 0.659119i \(0.229071\pi\)
\(450\) 0 0
\(451\) −6776.00 −0.707471
\(452\) 0 0
\(453\) 4304.00 0.446401
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2344.00 0.239929 0.119965 0.992778i \(-0.461722\pi\)
0.119965 + 0.992778i \(0.461722\pi\)
\(458\) 0 0
\(459\) 6400.00 0.650820
\(460\) 0 0
\(461\) −11382.0 −1.14992 −0.574959 0.818182i \(-0.694982\pi\)
−0.574959 + 0.818182i \(0.694982\pi\)
\(462\) 0 0
\(463\) 16062.0 1.61223 0.806117 0.591756i \(-0.201565\pi\)
0.806117 + 0.591756i \(0.201565\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17166.0 −1.70096 −0.850479 0.526008i \(-0.823688\pi\)
−0.850479 + 0.526008i \(0.823688\pi\)
\(468\) 0 0
\(469\) 11284.0 1.11097
\(470\) 0 0
\(471\) 1048.00 0.102525
\(472\) 0 0
\(473\) 10136.0 0.985315
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2484.00 0.238437
\(478\) 0 0
\(479\) −7520.00 −0.717323 −0.358661 0.933468i \(-0.616767\pi\)
−0.358661 + 0.933468i \(0.616767\pi\)
\(480\) 0 0
\(481\) 2832.00 0.268458
\(482\) 0 0
\(483\) 3016.00 0.284126
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −11814.0 −1.09927 −0.549634 0.835406i \(-0.685233\pi\)
−0.549634 + 0.835406i \(0.685233\pi\)
\(488\) 0 0
\(489\) −7036.00 −0.650673
\(490\) 0 0
\(491\) 14052.0 1.29156 0.645782 0.763522i \(-0.276532\pi\)
0.645782 + 0.763522i \(0.276532\pi\)
\(492\) 0 0
\(493\) −5760.00 −0.526202
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29328.0 2.64696
\(498\) 0 0
\(499\) 7620.00 0.683603 0.341802 0.939772i \(-0.388963\pi\)
0.341802 + 0.939772i \(0.388963\pi\)
\(500\) 0 0
\(501\) 1068.00 0.0952390
\(502\) 0 0
\(503\) −1818.00 −0.161154 −0.0805772 0.996748i \(-0.525676\pi\)
−0.0805772 + 0.996748i \(0.525676\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4106.00 0.359672
\(508\) 0 0
\(509\) −17850.0 −1.55440 −0.777198 0.629256i \(-0.783360\pi\)
−0.777198 + 0.629256i \(0.783360\pi\)
\(510\) 0 0
\(511\) −16432.0 −1.42252
\(512\) 0 0
\(513\) −6000.00 −0.516387
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6328.00 −0.538308
\(518\) 0 0
\(519\) −8504.00 −0.719237
\(520\) 0 0
\(521\) −19238.0 −1.61772 −0.808860 0.588001i \(-0.799915\pi\)
−0.808860 + 0.588001i \(0.799915\pi\)
\(522\) 0 0
\(523\) 6278.00 0.524891 0.262445 0.964947i \(-0.415471\pi\)
0.262445 + 0.964947i \(0.415471\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8192.00 0.677133
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) 0 0
\(531\) 460.000 0.0375938
\(532\) 0 0
\(533\) 2904.00 0.235997
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5000.00 −0.401799
\(538\) 0 0
\(539\) −9324.00 −0.745108
\(540\) 0 0
\(541\) 9818.00 0.780238 0.390119 0.920764i \(-0.372434\pi\)
0.390119 + 0.920764i \(0.372434\pi\)
\(542\) 0 0
\(543\) −5156.00 −0.407486
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12514.0 0.978172 0.489086 0.872236i \(-0.337330\pi\)
0.489086 + 0.872236i \(0.337330\pi\)
\(548\) 0 0
\(549\) 12466.0 0.969100
\(550\) 0 0
\(551\) 5400.00 0.417509
\(552\) 0 0
\(553\) 18720.0 1.43952
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10596.0 0.806045 0.403022 0.915190i \(-0.367960\pi\)
0.403022 + 0.915190i \(0.367960\pi\)
\(558\) 0 0
\(559\) −4344.00 −0.328679
\(560\) 0 0
\(561\) 3584.00 0.269727
\(562\) 0 0
\(563\) −14002.0 −1.04816 −0.524080 0.851669i \(-0.675591\pi\)
−0.524080 + 0.851669i \(0.675591\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 10946.0 0.810739
\(568\) 0 0
\(569\) −7330.00 −0.540052 −0.270026 0.962853i \(-0.587032\pi\)
−0.270026 + 0.962853i \(0.587032\pi\)
\(570\) 0 0
\(571\) 5812.00 0.425963 0.212981 0.977056i \(-0.431683\pi\)
0.212981 + 0.977056i \(0.431683\pi\)
\(572\) 0 0
\(573\) −1536.00 −0.111985
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −16736.0 −1.20750 −0.603751 0.797173i \(-0.706328\pi\)
−0.603751 + 0.797173i \(0.706328\pi\)
\(578\) 0 0
\(579\) −5216.00 −0.374386
\(580\) 0 0
\(581\) 12428.0 0.887436
\(582\) 0 0
\(583\) 3024.00 0.214822
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7434.00 0.522716 0.261358 0.965242i \(-0.415830\pi\)
0.261358 + 0.965242i \(0.415830\pi\)
\(588\) 0 0
\(589\) −7680.00 −0.537265
\(590\) 0 0
\(591\) −10232.0 −0.712163
\(592\) 0 0
\(593\) −25872.0 −1.79163 −0.895814 0.444429i \(-0.853407\pi\)
−0.895814 + 0.444429i \(0.853407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6960.00 −0.477142
\(598\) 0 0
\(599\) 3720.00 0.253748 0.126874 0.991919i \(-0.459506\pi\)
0.126874 + 0.991919i \(0.459506\pi\)
\(600\) 0 0
\(601\) −12958.0 −0.879481 −0.439740 0.898125i \(-0.644930\pi\)
−0.439740 + 0.898125i \(0.644930\pi\)
\(602\) 0 0
\(603\) −9982.00 −0.674127
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −7214.00 −0.482384 −0.241192 0.970477i \(-0.577538\pi\)
−0.241192 + 0.970477i \(0.577538\pi\)
\(608\) 0 0
\(609\) 4680.00 0.311401
\(610\) 0 0
\(611\) 2712.00 0.179568
\(612\) 0 0
\(613\) −4828.00 −0.318109 −0.159055 0.987270i \(-0.550845\pi\)
−0.159055 + 0.987270i \(0.550845\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27656.0 −1.80452 −0.902260 0.431193i \(-0.858093\pi\)
−0.902260 + 0.431193i \(0.858093\pi\)
\(618\) 0 0
\(619\) −21220.0 −1.37787 −0.688937 0.724821i \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(620\) 0 0
\(621\) −5800.00 −0.374792
\(622\) 0 0
\(623\) −12740.0 −0.819289
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3360.00 −0.214012
\(628\) 0 0
\(629\) 15104.0 0.957450
\(630\) 0 0
\(631\) −17672.0 −1.11491 −0.557457 0.830206i \(-0.688223\pi\)
−0.557457 + 0.830206i \(0.688223\pi\)
\(632\) 0 0
\(633\) −6264.00 −0.393320
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3996.00 0.248551
\(638\) 0 0
\(639\) −25944.0 −1.60615
\(640\) 0 0
\(641\) 7322.00 0.451173 0.225586 0.974223i \(-0.427570\pi\)
0.225586 + 0.974223i \(0.427570\pi\)
\(642\) 0 0
\(643\) 8238.00 0.505249 0.252624 0.967564i \(-0.418706\pi\)
0.252624 + 0.967564i \(0.418706\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6426.00 0.390467 0.195233 0.980757i \(-0.437454\pi\)
0.195233 + 0.980757i \(0.437454\pi\)
\(648\) 0 0
\(649\) 560.000 0.0338705
\(650\) 0 0
\(651\) −6656.00 −0.400721
\(652\) 0 0
\(653\) −5908.00 −0.354055 −0.177027 0.984206i \(-0.556648\pi\)
−0.177027 + 0.984206i \(0.556648\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14536.0 0.863171
\(658\) 0 0
\(659\) −26780.0 −1.58301 −0.791503 0.611166i \(-0.790701\pi\)
−0.791503 + 0.611166i \(0.790701\pi\)
\(660\) 0 0
\(661\) 24538.0 1.44390 0.721950 0.691945i \(-0.243246\pi\)
0.721950 + 0.691945i \(0.243246\pi\)
\(662\) 0 0
\(663\) −1536.00 −0.0899748
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5220.00 0.303027
\(668\) 0 0
\(669\) −124.000 −0.00716609
\(670\) 0 0
\(671\) 15176.0 0.873119
\(672\) 0 0
\(673\) 28848.0 1.65232 0.826158 0.563439i \(-0.190522\pi\)
0.826158 + 0.563439i \(0.190522\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26884.0 −1.52620 −0.763099 0.646282i \(-0.776323\pi\)
−0.763099 + 0.646282i \(0.776323\pi\)
\(678\) 0 0
\(679\) −37856.0 −2.13959
\(680\) 0 0
\(681\) −10628.0 −0.598041
\(682\) 0 0
\(683\) −14282.0 −0.800125 −0.400063 0.916488i \(-0.631012\pi\)
−0.400063 + 0.916488i \(0.631012\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −380.000 −0.0211032
\(688\) 0 0
\(689\) −1296.00 −0.0716599
\(690\) 0 0
\(691\) −3428.00 −0.188723 −0.0943613 0.995538i \(-0.530081\pi\)
−0.0943613 + 0.995538i \(0.530081\pi\)
\(692\) 0 0
\(693\) 16744.0 0.917824
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15488.0 0.841678
\(698\) 0 0
\(699\) −4816.00 −0.260598
\(700\) 0 0
\(701\) −26942.0 −1.45162 −0.725810 0.687895i \(-0.758535\pi\)
−0.725810 + 0.687895i \(0.758535\pi\)
\(702\) 0 0
\(703\) −14160.0 −0.759679
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15028.0 0.799415
\(708\) 0 0
\(709\) 1950.00 0.103292 0.0516458 0.998665i \(-0.483553\pi\)
0.0516458 + 0.998665i \(0.483553\pi\)
\(710\) 0 0
\(711\) −16560.0 −0.873486
\(712\) 0 0
\(713\) −7424.00 −0.389945
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −11360.0 −0.591697
\(718\) 0 0
\(719\) −12080.0 −0.626576 −0.313288 0.949658i \(-0.601430\pi\)
−0.313288 + 0.949658i \(0.601430\pi\)
\(720\) 0 0
\(721\) 38012.0 1.96344
\(722\) 0 0
\(723\) 556.000 0.0286001
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17226.0 0.878785 0.439393 0.898295i \(-0.355194\pi\)
0.439393 + 0.898295i \(0.355194\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) −23168.0 −1.17223
\(732\) 0 0
\(733\) −788.000 −0.0397073 −0.0198536 0.999803i \(-0.506320\pi\)
−0.0198536 + 0.999803i \(0.506320\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12152.0 −0.607360
\(738\) 0 0
\(739\) −2060.00 −0.102542 −0.0512709 0.998685i \(-0.516327\pi\)
−0.0512709 + 0.998685i \(0.516327\pi\)
\(740\) 0 0
\(741\) 1440.00 0.0713896
\(742\) 0 0
\(743\) −3258.00 −0.160867 −0.0804337 0.996760i \(-0.525631\pi\)
−0.0804337 + 0.996760i \(0.525631\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −10994.0 −0.538487
\(748\) 0 0
\(749\) −25116.0 −1.22526
\(750\) 0 0
\(751\) 4528.00 0.220012 0.110006 0.993931i \(-0.464913\pi\)
0.110006 + 0.993931i \(0.464913\pi\)
\(752\) 0 0
\(753\) −6504.00 −0.314766
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 18236.0 0.875560 0.437780 0.899082i \(-0.355765\pi\)
0.437780 + 0.899082i \(0.355765\pi\)
\(758\) 0 0
\(759\) −3248.00 −0.155329
\(760\) 0 0
\(761\) −18678.0 −0.889720 −0.444860 0.895600i \(-0.646747\pi\)
−0.444860 + 0.895600i \(0.646747\pi\)
\(762\) 0 0
\(763\) −9620.00 −0.456445
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −240.000 −0.0112984
\(768\) 0 0
\(769\) 27390.0 1.28441 0.642203 0.766534i \(-0.278020\pi\)
0.642203 + 0.766534i \(0.278020\pi\)
\(770\) 0 0
\(771\) 3072.00 0.143496
\(772\) 0 0
\(773\) 9252.00 0.430493 0.215247 0.976560i \(-0.430944\pi\)
0.215247 + 0.976560i \(0.430944\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −12272.0 −0.566609
\(778\) 0 0
\(779\) −14520.0 −0.667822
\(780\) 0 0
\(781\) −31584.0 −1.44707
\(782\) 0 0
\(783\) −9000.00 −0.410771
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −5726.00 −0.259352 −0.129676 0.991556i \(-0.541394\pi\)
−0.129676 + 0.991556i \(0.541394\pi\)
\(788\) 0 0
\(789\) 9716.00 0.438401
\(790\) 0 0
\(791\) 13728.0 0.617082
\(792\) 0 0
\(793\) −6504.00 −0.291253
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27236.0 1.21048 0.605238 0.796045i \(-0.293078\pi\)
0.605238 + 0.796045i \(0.293078\pi\)
\(798\) 0 0
\(799\) 14464.0 0.640425
\(800\) 0 0
\(801\) 11270.0 0.497136
\(802\) 0 0
\(803\) 17696.0 0.777682
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5220.00 0.227699
\(808\) 0 0
\(809\) 10950.0 0.475873 0.237937 0.971281i \(-0.423529\pi\)
0.237937 + 0.971281i \(0.423529\pi\)
\(810\) 0 0
\(811\) −8828.00 −0.382236 −0.191118 0.981567i \(-0.561211\pi\)
−0.191118 + 0.981567i \(0.561211\pi\)
\(812\) 0 0
\(813\) −10336.0 −0.445879
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21720.0 0.930094
\(818\) 0 0
\(819\) −7176.00 −0.306166
\(820\) 0 0
\(821\) 16058.0 0.682616 0.341308 0.939951i \(-0.389130\pi\)
0.341308 + 0.939951i \(0.389130\pi\)
\(822\) 0 0
\(823\) 41862.0 1.77305 0.886523 0.462684i \(-0.153113\pi\)
0.886523 + 0.462684i \(0.153113\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12154.0 0.511047 0.255524 0.966803i \(-0.417752\pi\)
0.255524 + 0.966803i \(0.417752\pi\)
\(828\) 0 0
\(829\) 15390.0 0.644773 0.322386 0.946608i \(-0.395515\pi\)
0.322386 + 0.946608i \(0.395515\pi\)
\(830\) 0 0
\(831\) 3848.00 0.160633
\(832\) 0 0
\(833\) 21312.0 0.886455
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 12800.0 0.528593
\(838\) 0 0
\(839\) 4280.00 0.176117 0.0880584 0.996115i \(-0.471934\pi\)
0.0880584 + 0.996115i \(0.471934\pi\)
\(840\) 0 0
\(841\) −16289.0 −0.667883
\(842\) 0 0
\(843\) −6084.00 −0.248570
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14222.0 −0.576947
\(848\) 0 0
\(849\) −3436.00 −0.138897
\(850\) 0 0
\(851\) −13688.0 −0.551373
\(852\) 0 0
\(853\) 14452.0 0.580102 0.290051 0.957011i \(-0.406328\pi\)
0.290051 + 0.957011i \(0.406328\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22584.0 0.900181 0.450090 0.892983i \(-0.351392\pi\)
0.450090 + 0.892983i \(0.351392\pi\)
\(858\) 0 0
\(859\) −26740.0 −1.06212 −0.531058 0.847336i \(-0.678205\pi\)
−0.531058 + 0.847336i \(0.678205\pi\)
\(860\) 0 0
\(861\) −12584.0 −0.498097
\(862\) 0 0
\(863\) −498.000 −0.0196432 −0.00982162 0.999952i \(-0.503126\pi\)
−0.00982162 + 0.999952i \(0.503126\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1634.00 0.0640064
\(868\) 0 0
\(869\) −20160.0 −0.786975
\(870\) 0 0
\(871\) 5208.00 0.202602
\(872\) 0 0
\(873\) 33488.0 1.29828
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13244.0 −0.509941 −0.254970 0.966949i \(-0.582066\pi\)
−0.254970 + 0.966949i \(0.582066\pi\)
\(878\) 0 0
\(879\) −4584.00 −0.175898
\(880\) 0 0
\(881\) 40842.0 1.56186 0.780932 0.624616i \(-0.214745\pi\)
0.780932 + 0.624616i \(0.214745\pi\)
\(882\) 0 0
\(883\) 12078.0 0.460314 0.230157 0.973154i \(-0.426076\pi\)
0.230157 + 0.973154i \(0.426076\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18294.0 −0.692506 −0.346253 0.938141i \(-0.612546\pi\)
−0.346253 + 0.938141i \(0.612546\pi\)
\(888\) 0 0
\(889\) −39884.0 −1.50469
\(890\) 0 0
\(891\) −11788.0 −0.443224
\(892\) 0 0
\(893\) −13560.0 −0.508139
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1392.00 0.0518144
\(898\) 0 0
\(899\) −11520.0 −0.427379
\(900\) 0 0
\(901\) −6912.00 −0.255574
\(902\) 0 0
\(903\) 18824.0 0.693714
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −22566.0 −0.826121 −0.413060 0.910704i \(-0.635540\pi\)
−0.413060 + 0.910704i \(0.635540\pi\)
\(908\) 0 0
\(909\) −13294.0 −0.485076
\(910\) 0 0
\(911\) 6768.00 0.246140 0.123070 0.992398i \(-0.460726\pi\)
0.123070 + 0.992398i \(0.460726\pi\)
\(912\) 0 0
\(913\) −13384.0 −0.485154
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 312.000 0.0112357
\(918\) 0 0
\(919\) −22200.0 −0.796856 −0.398428 0.917200i \(-0.630444\pi\)
−0.398428 + 0.917200i \(0.630444\pi\)
\(920\) 0 0
\(921\) 10812.0 0.386827
\(922\) 0 0
\(923\) 13536.0 0.482712
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −33626.0 −1.19139
\(928\) 0 0
\(929\) −6330.00 −0.223553 −0.111776 0.993733i \(-0.535654\pi\)
−0.111776 + 0.993733i \(0.535654\pi\)
\(930\) 0 0
\(931\) −19980.0 −0.703349
\(932\) 0 0
\(933\) −11376.0 −0.399178
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19544.0 0.681403 0.340702 0.940172i \(-0.389335\pi\)
0.340702 + 0.940172i \(0.389335\pi\)
\(938\) 0 0
\(939\) 14704.0 0.511019
\(940\) 0 0
\(941\) 9898.00 0.342896 0.171448 0.985193i \(-0.445155\pi\)
0.171448 + 0.985193i \(0.445155\pi\)
\(942\) 0 0
\(943\) −14036.0 −0.484703
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41406.0 −1.42082 −0.710409 0.703789i \(-0.751490\pi\)
−0.710409 + 0.703789i \(0.751490\pi\)
\(948\) 0 0
\(949\) −7584.00 −0.259417
\(950\) 0 0
\(951\) 6968.00 0.237595
\(952\) 0 0
\(953\) −25432.0 −0.864453 −0.432226 0.901765i \(-0.642272\pi\)
−0.432226 + 0.901765i \(0.642272\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −5040.00 −0.170240
\(958\) 0 0
\(959\) 31824.0 1.07159
\(960\) 0 0
\(961\) −13407.0 −0.450035
\(962\) 0 0
\(963\) 22218.0 0.743474
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 12106.0 0.402588 0.201294 0.979531i \(-0.435485\pi\)
0.201294 + 0.979531i \(0.435485\pi\)
\(968\) 0 0
\(969\) 7680.00 0.254610
\(970\) 0 0
\(971\) 7812.00 0.258186 0.129093 0.991632i \(-0.458793\pi\)
0.129093 + 0.991632i \(0.458793\pi\)
\(972\) 0 0
\(973\) 80600.0 2.65562
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12576.0 −0.411814 −0.205907 0.978572i \(-0.566014\pi\)
−0.205907 + 0.978572i \(0.566014\pi\)
\(978\) 0 0
\(979\) 13720.0 0.447899
\(980\) 0 0
\(981\) 8510.00 0.276966
\(982\) 0 0
\(983\) 4342.00 0.140883 0.0704417 0.997516i \(-0.477559\pi\)
0.0704417 + 0.997516i \(0.477559\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −11752.0 −0.378997
\(988\) 0 0
\(989\) 20996.0 0.675060
\(990\) 0 0
\(991\) −26272.0 −0.842137 −0.421068 0.907029i \(-0.638345\pi\)
−0.421068 + 0.907029i \(0.638345\pi\)
\(992\) 0 0
\(993\) 15736.0 0.502887
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 44796.0 1.42297 0.711486 0.702700i \(-0.248022\pi\)
0.711486 + 0.702700i \(0.248022\pi\)
\(998\) 0 0
\(999\) 23600.0 0.747418
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 1600.4.a.t.1.1 1
4.3 odd 2 1600.4.a.bh.1.1 1
5.2 odd 4 320.4.c.c.129.2 2
5.3 odd 4 320.4.c.c.129.1 2
5.4 even 2 1600.4.a.bg.1.1 1
8.3 odd 2 50.4.a.b.1.1 1
8.5 even 2 400.4.a.n.1.1 1
20.3 even 4 320.4.c.d.129.2 2
20.7 even 4 320.4.c.d.129.1 2
20.19 odd 2 1600.4.a.u.1.1 1
24.11 even 2 450.4.a.k.1.1 1
40.3 even 4 10.4.b.a.9.2 yes 2
40.13 odd 4 80.4.c.a.49.2 2
40.19 odd 2 50.4.a.d.1.1 1
40.27 even 4 10.4.b.a.9.1 2
40.29 even 2 400.4.a.h.1.1 1
40.37 odd 4 80.4.c.a.49.1 2
56.27 even 2 2450.4.a.o.1.1 1
120.53 even 4 720.4.f.f.289.2 2
120.59 even 2 450.4.a.j.1.1 1
120.77 even 4 720.4.f.f.289.1 2
120.83 odd 4 90.4.c.b.19.1 2
120.107 odd 4 90.4.c.b.19.2 2
280.27 odd 4 490.4.c.b.99.1 2
280.83 odd 4 490.4.c.b.99.2 2
280.139 even 2 2450.4.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.b.a.9.1 2 40.27 even 4
10.4.b.a.9.2 yes 2 40.3 even 4
50.4.a.b.1.1 1 8.3 odd 2
50.4.a.d.1.1 1 40.19 odd 2
80.4.c.a.49.1 2 40.37 odd 4
80.4.c.a.49.2 2 40.13 odd 4
90.4.c.b.19.1 2 120.83 odd 4
90.4.c.b.19.2 2 120.107 odd 4
320.4.c.c.129.1 2 5.3 odd 4
320.4.c.c.129.2 2 5.2 odd 4
320.4.c.d.129.1 2 20.7 even 4
320.4.c.d.129.2 2 20.3 even 4
400.4.a.h.1.1 1 40.29 even 2
400.4.a.n.1.1 1 8.5 even 2
450.4.a.j.1.1 1 120.59 even 2
450.4.a.k.1.1 1 24.11 even 2
490.4.c.b.99.1 2 280.27 odd 4
490.4.c.b.99.2 2 280.83 odd 4
720.4.f.f.289.1 2 120.77 even 4
720.4.f.f.289.2 2 120.53 even 4
1600.4.a.t.1.1 1 1.1 even 1 trivial
1600.4.a.u.1.1 1 20.19 odd 2
1600.4.a.bg.1.1 1 5.4 even 2
1600.4.a.bh.1.1 1 4.3 odd 2
2450.4.a.o.1.1 1 56.27 even 2
2450.4.a.bb.1.1 1 280.139 even 2