Properties

Label 7800.2.a.bk.1.3
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
Defining polynomial: \( x^{3} - 15x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.41883\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.41883 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.41883 q^{7} +1.00000 q^{9} -4.41883 q^{11} -1.00000 q^{13} -1.31158 q^{17} +0.311583 q^{19} -4.41883 q^{21} +1.37683 q^{23} -1.00000 q^{27} +1.00000 q^{29} -4.41883 q^{31} +4.41883 q^{33} -3.68842 q^{37} +1.00000 q^{39} +1.68842 q^{41} +2.00000 q^{43} +2.73042 q^{47} +12.5261 q^{49} +1.31158 q^{51} -2.37683 q^{53} -0.311583 q^{57} -1.04200 q^{59} -4.06525 q^{61} +4.41883 q^{63} +6.10725 q^{67} -1.37683 q^{69} +3.68842 q^{71} +12.8377 q^{73} -19.5261 q^{77} +14.5261 q^{79} +1.00000 q^{81} +5.04200 q^{83} -1.00000 q^{87} +3.37683 q^{89} -4.41883 q^{91} +4.41883 q^{93} -5.37683 q^{97} -4.41883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.41883 1.67016 0.835081 0.550127i \(-0.185421\pi\)
0.835081 + 0.550127i \(0.185421\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.41883 −1.33233 −0.666164 0.745805i \(-0.732065\pi\)
−0.666164 + 0.745805i \(0.732065\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.31158 −0.318106 −0.159053 0.987270i \(-0.550844\pi\)
−0.159053 + 0.987270i \(0.550844\pi\)
\(18\) 0 0
\(19\) 0.311583 0.0714821 0.0357410 0.999361i \(-0.488621\pi\)
0.0357410 + 0.999361i \(0.488621\pi\)
\(20\) 0 0
\(21\) −4.41883 −0.964268
\(22\) 0 0
\(23\) 1.37683 0.287090 0.143545 0.989644i \(-0.454150\pi\)
0.143545 + 0.989644i \(0.454150\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −4.41883 −0.793646 −0.396823 0.917895i \(-0.629887\pi\)
−0.396823 + 0.917895i \(0.629887\pi\)
\(32\) 0 0
\(33\) 4.41883 0.769220
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.68842 −0.606372 −0.303186 0.952931i \(-0.598050\pi\)
−0.303186 + 0.952931i \(0.598050\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 1.68842 0.263686 0.131843 0.991271i \(-0.457910\pi\)
0.131843 + 0.991271i \(0.457910\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.73042 0.398272 0.199136 0.979972i \(-0.436186\pi\)
0.199136 + 0.979972i \(0.436186\pi\)
\(48\) 0 0
\(49\) 12.5261 1.78944
\(50\) 0 0
\(51\) 1.31158 0.183658
\(52\) 0 0
\(53\) −2.37683 −0.326483 −0.163242 0.986586i \(-0.552195\pi\)
−0.163242 + 0.986586i \(0.552195\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.311583 −0.0412702
\(58\) 0 0
\(59\) −1.04200 −0.135657 −0.0678284 0.997697i \(-0.521607\pi\)
−0.0678284 + 0.997697i \(0.521607\pi\)
\(60\) 0 0
\(61\) −4.06525 −0.520502 −0.260251 0.965541i \(-0.583805\pi\)
−0.260251 + 0.965541i \(0.583805\pi\)
\(62\) 0 0
\(63\) 4.41883 0.556721
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.10725 0.746119 0.373060 0.927807i \(-0.378309\pi\)
0.373060 + 0.927807i \(0.378309\pi\)
\(68\) 0 0
\(69\) −1.37683 −0.165751
\(70\) 0 0
\(71\) 3.68842 0.437735 0.218867 0.975755i \(-0.429764\pi\)
0.218867 + 0.975755i \(0.429764\pi\)
\(72\) 0 0
\(73\) 12.8377 1.50254 0.751268 0.659998i \(-0.229443\pi\)
0.751268 + 0.659998i \(0.229443\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19.5261 −2.22520
\(78\) 0 0
\(79\) 14.5261 1.63431 0.817156 0.576417i \(-0.195549\pi\)
0.817156 + 0.576417i \(0.195549\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.04200 0.553431 0.276716 0.960952i \(-0.410754\pi\)
0.276716 + 0.960952i \(0.410754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 3.37683 0.357944 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(90\) 0 0
\(91\) −4.41883 −0.463220
\(92\) 0 0
\(93\) 4.41883 0.458212
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.37683 −0.545935 −0.272967 0.962023i \(-0.588005\pi\)
−0.272967 + 0.962023i \(0.588005\pi\)
\(98\) 0 0
\(99\) −4.41883 −0.444109
\(100\) 0 0
\(101\) 14.9029 1.48290 0.741448 0.671011i \(-0.234139\pi\)
0.741448 + 0.671011i \(0.234139\pi\)
\(102\) 0 0
\(103\) −6.21450 −0.612333 −0.306166 0.951978i \(-0.599046\pi\)
−0.306166 + 0.951978i \(0.599046\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.311583 0.0301219 0.0150609 0.999887i \(-0.495206\pi\)
0.0150609 + 0.999887i \(0.495206\pi\)
\(108\) 0 0
\(109\) 19.9869 1.91440 0.957200 0.289429i \(-0.0934653\pi\)
0.957200 + 0.289429i \(0.0934653\pi\)
\(110\) 0 0
\(111\) 3.68842 0.350089
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −5.79567 −0.531288
\(120\) 0 0
\(121\) 8.52608 0.775098
\(122\) 0 0
\(123\) −1.68842 −0.152239
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.14925 0.634393 0.317197 0.948360i \(-0.397258\pi\)
0.317197 + 0.948360i \(0.397258\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 6.31158 0.551446 0.275723 0.961237i \(-0.411083\pi\)
0.275723 + 0.961237i \(0.411083\pi\)
\(132\) 0 0
\(133\) 1.37683 0.119387
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.7724 −1.51840 −0.759200 0.650857i \(-0.774410\pi\)
−0.759200 + 0.650857i \(0.774410\pi\)
\(138\) 0 0
\(139\) −4.83767 −0.410325 −0.205163 0.978728i \(-0.565772\pi\)
−0.205163 + 0.978728i \(0.565772\pi\)
\(140\) 0 0
\(141\) −2.73042 −0.229942
\(142\) 0 0
\(143\) 4.41883 0.369521
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −12.5261 −1.03313
\(148\) 0 0
\(149\) −19.0522 −1.56081 −0.780407 0.625272i \(-0.784988\pi\)
−0.780407 + 0.625272i \(0.784988\pi\)
\(150\) 0 0
\(151\) 15.1725 1.23472 0.617360 0.786681i \(-0.288202\pi\)
0.617360 + 0.786681i \(0.288202\pi\)
\(152\) 0 0
\(153\) −1.31158 −0.106035
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.93475 −0.154410 −0.0772049 0.997015i \(-0.524600\pi\)
−0.0772049 + 0.997015i \(0.524600\pi\)
\(158\) 0 0
\(159\) 2.37683 0.188495
\(160\) 0 0
\(161\) 6.08400 0.479486
\(162\) 0 0
\(163\) −0.837665 −0.0656110 −0.0328055 0.999462i \(-0.510444\pi\)
−0.0328055 + 0.999462i \(0.510444\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.934749 −0.0723331 −0.0361665 0.999346i \(-0.511515\pi\)
−0.0361665 + 0.999346i \(0.511515\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.311583 0.0238274
\(172\) 0 0
\(173\) −15.9217 −1.21050 −0.605251 0.796035i \(-0.706927\pi\)
−0.605251 + 0.796035i \(0.706927\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.04200 0.0783214
\(178\) 0 0
\(179\) −10.2145 −0.763468 −0.381734 0.924272i \(-0.624673\pi\)
−0.381734 + 0.924272i \(0.624673\pi\)
\(180\) 0 0
\(181\) 18.9869 1.41129 0.705643 0.708567i \(-0.250658\pi\)
0.705643 + 0.708567i \(0.250658\pi\)
\(182\) 0 0
\(183\) 4.06525 0.300512
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.79567 0.423821
\(188\) 0 0
\(189\) −4.41883 −0.321423
\(190\) 0 0
\(191\) −4.21450 −0.304950 −0.152475 0.988307i \(-0.548724\pi\)
−0.152475 + 0.988307i \(0.548724\pi\)
\(192\) 0 0
\(193\) 15.4608 1.11290 0.556448 0.830883i \(-0.312164\pi\)
0.556448 + 0.830883i \(0.312164\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.2145 −1.58272 −0.791359 0.611352i \(-0.790626\pi\)
−0.791359 + 0.611352i \(0.790626\pi\)
\(198\) 0 0
\(199\) 26.6101 1.88634 0.943169 0.332313i \(-0.107829\pi\)
0.943169 + 0.332313i \(0.107829\pi\)
\(200\) 0 0
\(201\) −6.10725 −0.430772
\(202\) 0 0
\(203\) 4.41883 0.310141
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.37683 0.0956966
\(208\) 0 0
\(209\) −1.37683 −0.0952376
\(210\) 0 0
\(211\) 2.62317 0.180586 0.0902931 0.995915i \(-0.471220\pi\)
0.0902931 + 0.995915i \(0.471220\pi\)
\(212\) 0 0
\(213\) −3.68842 −0.252726
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −19.5261 −1.32552
\(218\) 0 0
\(219\) −12.8377 −0.867489
\(220\) 0 0
\(221\) 1.31158 0.0882266
\(222\) 0 0
\(223\) 6.53917 0.437895 0.218948 0.975737i \(-0.429738\pi\)
0.218948 + 0.975737i \(0.429738\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.87966 −0.124758 −0.0623789 0.998053i \(-0.519869\pi\)
−0.0623789 + 0.998053i \(0.519869\pi\)
\(228\) 0 0
\(229\) 21.2797 1.40621 0.703103 0.711088i \(-0.251797\pi\)
0.703103 + 0.711088i \(0.251797\pi\)
\(230\) 0 0
\(231\) 19.5261 1.28472
\(232\) 0 0
\(233\) −15.0522 −0.986100 −0.493050 0.870001i \(-0.664118\pi\)
−0.493050 + 0.870001i \(0.664118\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.5261 −0.943570
\(238\) 0 0
\(239\) 23.4710 1.51821 0.759106 0.650967i \(-0.225636\pi\)
0.759106 + 0.650967i \(0.225636\pi\)
\(240\) 0 0
\(241\) −8.70716 −0.560878 −0.280439 0.959872i \(-0.590480\pi\)
−0.280439 + 0.959872i \(0.590480\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.311583 −0.0198256
\(248\) 0 0
\(249\) −5.04200 −0.319524
\(250\) 0 0
\(251\) 28.6101 1.80585 0.902926 0.429796i \(-0.141414\pi\)
0.902926 + 0.429796i \(0.141414\pi\)
\(252\) 0 0
\(253\) −6.08400 −0.382498
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.0653 −1.00212 −0.501061 0.865412i \(-0.667057\pi\)
−0.501061 + 0.865412i \(0.667057\pi\)
\(258\) 0 0
\(259\) −16.2985 −1.01274
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −17.5913 −1.08473 −0.542364 0.840144i \(-0.682471\pi\)
−0.542364 + 0.840144i \(0.682471\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.37683 −0.206659
\(268\) 0 0
\(269\) −8.67533 −0.528944 −0.264472 0.964393i \(-0.585198\pi\)
−0.264472 + 0.964393i \(0.585198\pi\)
\(270\) 0 0
\(271\) 16.4188 0.997373 0.498687 0.866782i \(-0.333816\pi\)
0.498687 + 0.866782i \(0.333816\pi\)
\(272\) 0 0
\(273\) 4.41883 0.267440
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.67533 0.461166 0.230583 0.973053i \(-0.425937\pi\)
0.230583 + 0.973053i \(0.425937\pi\)
\(278\) 0 0
\(279\) −4.41883 −0.264549
\(280\) 0 0
\(281\) 1.47392 0.0879266 0.0439633 0.999033i \(-0.486002\pi\)
0.0439633 + 0.999033i \(0.486002\pi\)
\(282\) 0 0
\(283\) −18.5130 −1.10048 −0.550242 0.835005i \(-0.685464\pi\)
−0.550242 + 0.835005i \(0.685464\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.46083 0.440399
\(288\) 0 0
\(289\) −15.2797 −0.898809
\(290\) 0 0
\(291\) 5.37683 0.315196
\(292\) 0 0
\(293\) 19.0522 1.11304 0.556520 0.830834i \(-0.312136\pi\)
0.556520 + 0.830834i \(0.312136\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.41883 0.256407
\(298\) 0 0
\(299\) −1.37683 −0.0796244
\(300\) 0 0
\(301\) 8.83767 0.509395
\(302\) 0 0
\(303\) −14.9029 −0.856150
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.1174 −0.919869 −0.459935 0.887953i \(-0.652127\pi\)
−0.459935 + 0.887953i \(0.652127\pi\)
\(308\) 0 0
\(309\) 6.21450 0.353531
\(310\) 0 0
\(311\) 3.78550 0.214656 0.107328 0.994224i \(-0.465770\pi\)
0.107328 + 0.994224i \(0.465770\pi\)
\(312\) 0 0
\(313\) 15.0000 0.847850 0.423925 0.905697i \(-0.360652\pi\)
0.423925 + 0.905697i \(0.360652\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.5130 1.26446 0.632228 0.774782i \(-0.282141\pi\)
0.632228 + 0.774782i \(0.282141\pi\)
\(318\) 0 0
\(319\) −4.41883 −0.247407
\(320\) 0 0
\(321\) −0.311583 −0.0173909
\(322\) 0 0
\(323\) −0.408667 −0.0227389
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −19.9869 −1.10528
\(328\) 0 0
\(329\) 12.0653 0.665179
\(330\) 0 0
\(331\) 12.8377 0.705622 0.352811 0.935695i \(-0.385226\pi\)
0.352811 + 0.935695i \(0.385226\pi\)
\(332\) 0 0
\(333\) −3.68842 −0.202124
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.279750 0.0152389 0.00761947 0.999971i \(-0.497575\pi\)
0.00761947 + 0.999971i \(0.497575\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 19.5261 1.05740
\(342\) 0 0
\(343\) 24.4188 1.31849
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.7406 −0.683950 −0.341975 0.939709i \(-0.611096\pi\)
−0.341975 + 0.939709i \(0.611096\pi\)
\(348\) 0 0
\(349\) −8.21450 −0.439712 −0.219856 0.975532i \(-0.570559\pi\)
−0.219856 + 0.975532i \(0.570559\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −3.36375 −0.179034 −0.0895171 0.995985i \(-0.528532\pi\)
−0.0895171 + 0.995985i \(0.528532\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.79567 0.306739
\(358\) 0 0
\(359\) 24.4057 1.28809 0.644043 0.764989i \(-0.277256\pi\)
0.644043 + 0.764989i \(0.277256\pi\)
\(360\) 0 0
\(361\) −18.9029 −0.994890
\(362\) 0 0
\(363\) −8.52608 −0.447503
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.7724 0.718914 0.359457 0.933162i \(-0.382962\pi\)
0.359457 + 0.933162i \(0.382962\pi\)
\(368\) 0 0
\(369\) 1.68842 0.0878955
\(370\) 0 0
\(371\) −10.5028 −0.545280
\(372\) 0 0
\(373\) 12.5579 0.650224 0.325112 0.945675i \(-0.394598\pi\)
0.325112 + 0.945675i \(0.394598\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) −8.71733 −0.447779 −0.223890 0.974615i \(-0.571875\pi\)
−0.223890 + 0.974615i \(0.571875\pi\)
\(380\) 0 0
\(381\) −7.14925 −0.366267
\(382\) 0 0
\(383\) 20.7406 1.05979 0.529897 0.848062i \(-0.322231\pi\)
0.529897 + 0.848062i \(0.322231\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) −33.4477 −1.69587 −0.847934 0.530102i \(-0.822154\pi\)
−0.847934 + 0.530102i \(0.822154\pi\)
\(390\) 0 0
\(391\) −1.80583 −0.0913248
\(392\) 0 0
\(393\) −6.31158 −0.318377
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.1492 −0.760319 −0.380159 0.924921i \(-0.624131\pi\)
−0.380159 + 0.924921i \(0.624131\pi\)
\(398\) 0 0
\(399\) −1.37683 −0.0689279
\(400\) 0 0
\(401\) −4.21450 −0.210462 −0.105231 0.994448i \(-0.533558\pi\)
−0.105231 + 0.994448i \(0.533558\pi\)
\(402\) 0 0
\(403\) 4.41883 0.220118
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.2985 0.807887
\(408\) 0 0
\(409\) 12.5392 0.620022 0.310011 0.950733i \(-0.399667\pi\)
0.310011 + 0.950733i \(0.399667\pi\)
\(410\) 0 0
\(411\) 17.7724 0.876649
\(412\) 0 0
\(413\) −4.60442 −0.226569
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.83767 0.236901
\(418\) 0 0
\(419\) 11.3637 0.555155 0.277578 0.960703i \(-0.410468\pi\)
0.277578 + 0.960703i \(0.410468\pi\)
\(420\) 0 0
\(421\) −12.6232 −0.615215 −0.307608 0.951513i \(-0.599528\pi\)
−0.307608 + 0.951513i \(0.599528\pi\)
\(422\) 0 0
\(423\) 2.73042 0.132757
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17.9637 −0.869323
\(428\) 0 0
\(429\) −4.41883 −0.213343
\(430\) 0 0
\(431\) 13.8189 0.665634 0.332817 0.942991i \(-0.392001\pi\)
0.332817 + 0.942991i \(0.392001\pi\)
\(432\) 0 0
\(433\) −15.5782 −0.748643 −0.374321 0.927299i \(-0.622124\pi\)
−0.374321 + 0.927299i \(0.622124\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.428998 0.0205218
\(438\) 0 0
\(439\) 13.9029 0.663550 0.331775 0.943359i \(-0.392353\pi\)
0.331775 + 0.943359i \(0.392353\pi\)
\(440\) 0 0
\(441\) 12.5261 0.596480
\(442\) 0 0
\(443\) 33.4477 1.58915 0.794575 0.607166i \(-0.207694\pi\)
0.794575 + 0.607166i \(0.207694\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 19.0522 0.901136
\(448\) 0 0
\(449\) 17.7724 0.838732 0.419366 0.907817i \(-0.362252\pi\)
0.419366 + 0.907817i \(0.362252\pi\)
\(450\) 0 0
\(451\) −7.46083 −0.351317
\(452\) 0 0
\(453\) −15.1725 −0.712866
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.2145 1.22626 0.613131 0.789981i \(-0.289910\pi\)
0.613131 + 0.789981i \(0.289910\pi\)
\(458\) 0 0
\(459\) 1.31158 0.0612195
\(460\) 0 0
\(461\) −24.7275 −1.15167 −0.575837 0.817564i \(-0.695324\pi\)
−0.575837 + 0.817564i \(0.695324\pi\)
\(462\) 0 0
\(463\) 31.7957 1.47767 0.738835 0.673886i \(-0.235376\pi\)
0.738835 + 0.673886i \(0.235376\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.85075 0.317015 0.158507 0.987358i \(-0.449332\pi\)
0.158507 + 0.987358i \(0.449332\pi\)
\(468\) 0 0
\(469\) 26.9869 1.24614
\(470\) 0 0
\(471\) 1.93475 0.0891485
\(472\) 0 0
\(473\) −8.83767 −0.406356
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.37683 −0.108828
\(478\) 0 0
\(479\) −13.8928 −0.634776 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(480\) 0 0
\(481\) 3.68842 0.168177
\(482\) 0 0
\(483\) −6.08400 −0.276831
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −5.47100 −0.247915 −0.123957 0.992288i \(-0.539559\pi\)
−0.123957 + 0.992288i \(0.539559\pi\)
\(488\) 0 0
\(489\) 0.837665 0.0378805
\(490\) 0 0
\(491\) 1.91600 0.0864680 0.0432340 0.999065i \(-0.486234\pi\)
0.0432340 + 0.999065i \(0.486234\pi\)
\(492\) 0 0
\(493\) −1.31158 −0.0590707
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.2985 0.731088
\(498\) 0 0
\(499\) 38.0811 1.70474 0.852372 0.522937i \(-0.175164\pi\)
0.852372 + 0.522937i \(0.175164\pi\)
\(500\) 0 0
\(501\) 0.934749 0.0417615
\(502\) 0 0
\(503\) 35.8898 1.60025 0.800124 0.599834i \(-0.204767\pi\)
0.800124 + 0.599834i \(0.204767\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 33.1362 1.46873 0.734367 0.678752i \(-0.237479\pi\)
0.734367 + 0.678752i \(0.237479\pi\)
\(510\) 0 0
\(511\) 56.7275 2.50948
\(512\) 0 0
\(513\) −0.311583 −0.0137567
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0653 −0.530629
\(518\) 0 0
\(519\) 15.9217 0.698883
\(520\) 0 0
\(521\) 37.8058 1.65630 0.828152 0.560504i \(-0.189393\pi\)
0.828152 + 0.560504i \(0.189393\pi\)
\(522\) 0 0
\(523\) 12.2145 0.534103 0.267051 0.963682i \(-0.413951\pi\)
0.267051 + 0.963682i \(0.413951\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.79567 0.252463
\(528\) 0 0
\(529\) −21.1043 −0.917580
\(530\) 0 0
\(531\) −1.04200 −0.0452189
\(532\) 0 0
\(533\) −1.68842 −0.0731335
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.2145 0.440788
\(538\) 0 0
\(539\) −55.3507 −2.38412
\(540\) 0 0
\(541\) −39.5652 −1.70104 −0.850520 0.525943i \(-0.823712\pi\)
−0.850520 + 0.525943i \(0.823712\pi\)
\(542\) 0 0
\(543\) −18.9869 −0.814806
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.8377 0.805440 0.402720 0.915323i \(-0.368065\pi\)
0.402720 + 0.915323i \(0.368065\pi\)
\(548\) 0 0
\(549\) −4.06525 −0.173501
\(550\) 0 0
\(551\) 0.311583 0.0132739
\(552\) 0 0
\(553\) 64.1883 2.72957
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21.5913 −0.914854 −0.457427 0.889247i \(-0.651229\pi\)
−0.457427 + 0.889247i \(0.651229\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) −5.79567 −0.244693
\(562\) 0 0
\(563\) −5.01875 −0.211515 −0.105757 0.994392i \(-0.533727\pi\)
−0.105757 + 0.994392i \(0.533727\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.41883 0.185574
\(568\) 0 0
\(569\) 43.2854 1.81462 0.907309 0.420464i \(-0.138133\pi\)
0.907309 + 0.420464i \(0.138133\pi\)
\(570\) 0 0
\(571\) 12.1305 0.507646 0.253823 0.967251i \(-0.418312\pi\)
0.253823 + 0.967251i \(0.418312\pi\)
\(572\) 0 0
\(573\) 4.21450 0.176063
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.86950 0.327612 0.163806 0.986493i \(-0.447623\pi\)
0.163806 + 0.986493i \(0.447623\pi\)
\(578\) 0 0
\(579\) −15.4608 −0.642530
\(580\) 0 0
\(581\) 22.2797 0.924320
\(582\) 0 0
\(583\) 10.5028 0.434983
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.4188 0.842775 0.421388 0.906881i \(-0.361543\pi\)
0.421388 + 0.906881i \(0.361543\pi\)
\(588\) 0 0
\(589\) −1.37683 −0.0567314
\(590\) 0 0
\(591\) 22.2145 0.913782
\(592\) 0 0
\(593\) 24.0334 0.986934 0.493467 0.869764i \(-0.335729\pi\)
0.493467 + 0.869764i \(0.335729\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.6101 −1.08908
\(598\) 0 0
\(599\) 11.8898 0.485805 0.242903 0.970051i \(-0.421900\pi\)
0.242903 + 0.970051i \(0.421900\pi\)
\(600\) 0 0
\(601\) −16.3768 −0.668025 −0.334012 0.942569i \(-0.608403\pi\)
−0.334012 + 0.942569i \(0.608403\pi\)
\(602\) 0 0
\(603\) 6.10725 0.248706
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.0653 0.611480 0.305740 0.952115i \(-0.401096\pi\)
0.305740 + 0.952115i \(0.401096\pi\)
\(608\) 0 0
\(609\) −4.41883 −0.179060
\(610\) 0 0
\(611\) −2.73042 −0.110461
\(612\) 0 0
\(613\) 31.6753 1.27935 0.639677 0.768644i \(-0.279068\pi\)
0.639677 + 0.768644i \(0.279068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −45.4477 −1.82966 −0.914829 0.403842i \(-0.867674\pi\)
−0.914829 + 0.403842i \(0.867674\pi\)
\(618\) 0 0
\(619\) −29.2667 −1.17633 −0.588163 0.808742i \(-0.700149\pi\)
−0.588163 + 0.808742i \(0.700149\pi\)
\(620\) 0 0
\(621\) −1.37683 −0.0552504
\(622\) 0 0
\(623\) 14.9217 0.597824
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.37683 0.0549854
\(628\) 0 0
\(629\) 4.83767 0.192890
\(630\) 0 0
\(631\) −25.0987 −0.999162 −0.499581 0.866267i \(-0.666513\pi\)
−0.499581 + 0.866267i \(0.666513\pi\)
\(632\) 0 0
\(633\) −2.62317 −0.104261
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.5261 −0.496301
\(638\) 0 0
\(639\) 3.68842 0.145912
\(640\) 0 0
\(641\) −34.7927 −1.37423 −0.687115 0.726548i \(-0.741123\pi\)
−0.687115 + 0.726548i \(0.741123\pi\)
\(642\) 0 0
\(643\) 41.5317 1.63785 0.818926 0.573899i \(-0.194570\pi\)
0.818926 + 0.573899i \(0.194570\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.50733 0.373772 0.186886 0.982382i \(-0.440161\pi\)
0.186886 + 0.982382i \(0.440161\pi\)
\(648\) 0 0
\(649\) 4.60442 0.180739
\(650\) 0 0
\(651\) 19.5261 0.765287
\(652\) 0 0
\(653\) −36.1492 −1.41463 −0.707315 0.706899i \(-0.750094\pi\)
−0.707315 + 0.706899i \(0.750094\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.8377 0.500845
\(658\) 0 0
\(659\) −7.77241 −0.302770 −0.151385 0.988475i \(-0.548373\pi\)
−0.151385 + 0.988475i \(0.548373\pi\)
\(660\) 0 0
\(661\) −19.3637 −0.753162 −0.376581 0.926384i \(-0.622900\pi\)
−0.376581 + 0.926384i \(0.622900\pi\)
\(662\) 0 0
\(663\) −1.31158 −0.0509377
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.37683 0.0533112
\(668\) 0 0
\(669\) −6.53917 −0.252819
\(670\) 0 0
\(671\) 17.9637 0.693479
\(672\) 0 0
\(673\) 22.6753 0.874070 0.437035 0.899445i \(-0.356029\pi\)
0.437035 + 0.899445i \(0.356029\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.92166 0.189155 0.0945774 0.995518i \(-0.469850\pi\)
0.0945774 + 0.995518i \(0.469850\pi\)
\(678\) 0 0
\(679\) −23.7593 −0.911799
\(680\) 0 0
\(681\) 1.87966 0.0720289
\(682\) 0 0
\(683\) 6.50283 0.248824 0.124412 0.992231i \(-0.460296\pi\)
0.124412 + 0.992231i \(0.460296\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −21.2797 −0.811873
\(688\) 0 0
\(689\) 2.37683 0.0905502
\(690\) 0 0
\(691\) 17.0289 0.647810 0.323905 0.946090i \(-0.395004\pi\)
0.323905 + 0.946090i \(0.395004\pi\)
\(692\) 0 0
\(693\) −19.5261 −0.741735
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.21450 −0.0838801
\(698\) 0 0
\(699\) 15.0522 0.569325
\(700\) 0 0
\(701\) 49.8246 1.88185 0.940924 0.338617i \(-0.109959\pi\)
0.940924 + 0.338617i \(0.109959\pi\)
\(702\) 0 0
\(703\) −1.14925 −0.0433447
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 65.8535 2.47668
\(708\) 0 0
\(709\) 39.2667 1.47469 0.737345 0.675516i \(-0.236079\pi\)
0.737345 + 0.675516i \(0.236079\pi\)
\(710\) 0 0
\(711\) 14.5261 0.544771
\(712\) 0 0
\(713\) −6.08400 −0.227848
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −23.4710 −0.876540
\(718\) 0 0
\(719\) 37.5652 1.40094 0.700472 0.713680i \(-0.252973\pi\)
0.700472 + 0.713680i \(0.252973\pi\)
\(720\) 0 0
\(721\) −27.4608 −1.02269
\(722\) 0 0
\(723\) 8.70716 0.323823
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 19.6288 0.727993 0.363996 0.931400i \(-0.381412\pi\)
0.363996 + 0.931400i \(0.381412\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.62317 −0.0970213
\(732\) 0 0
\(733\) −46.3694 −1.71269 −0.856347 0.516402i \(-0.827271\pi\)
−0.856347 + 0.516402i \(0.827271\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −26.9869 −0.994076
\(738\) 0 0
\(739\) −32.4057 −1.19206 −0.596032 0.802960i \(-0.703257\pi\)
−0.596032 + 0.802960i \(0.703257\pi\)
\(740\) 0 0
\(741\) 0.311583 0.0114463
\(742\) 0 0
\(743\) −46.9652 −1.72299 −0.861494 0.507768i \(-0.830471\pi\)
−0.861494 + 0.507768i \(0.830471\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.04200 0.184477
\(748\) 0 0
\(749\) 1.37683 0.0503084
\(750\) 0 0
\(751\) 50.0709 1.82711 0.913557 0.406711i \(-0.133324\pi\)
0.913557 + 0.406711i \(0.133324\pi\)
\(752\) 0 0
\(753\) −28.6101 −1.04261
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −9.47958 −0.344541 −0.172271 0.985050i \(-0.555110\pi\)
−0.172271 + 0.985050i \(0.555110\pi\)
\(758\) 0 0
\(759\) 6.08400 0.220835
\(760\) 0 0
\(761\) 4.31158 0.156295 0.0781474 0.996942i \(-0.475100\pi\)
0.0781474 + 0.996942i \(0.475100\pi\)
\(762\) 0 0
\(763\) 88.3188 3.19736
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.04200 0.0376244
\(768\) 0 0
\(769\) −37.0260 −1.33519 −0.667596 0.744524i \(-0.732677\pi\)
−0.667596 + 0.744524i \(0.732677\pi\)
\(770\) 0 0
\(771\) 16.0653 0.578576
\(772\) 0 0
\(773\) 39.6492 1.42608 0.713041 0.701123i \(-0.247318\pi\)
0.713041 + 0.701123i \(0.247318\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 16.2985 0.584705
\(778\) 0 0
\(779\) 0.526082 0.0188489
\(780\) 0 0
\(781\) −16.2985 −0.583206
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −44.7377 −1.59473 −0.797363 0.603500i \(-0.793772\pi\)
−0.797363 + 0.603500i \(0.793772\pi\)
\(788\) 0 0
\(789\) 17.5913 0.626268
\(790\) 0 0
\(791\) 26.5130 0.942694
\(792\) 0 0
\(793\) 4.06525 0.144361
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.7724 −0.948328 −0.474164 0.880437i \(-0.657250\pi\)
−0.474164 + 0.880437i \(0.657250\pi\)
\(798\) 0 0
\(799\) −3.58117 −0.126693
\(800\) 0 0
\(801\) 3.37683 0.119315
\(802\) 0 0
\(803\) −56.7275 −2.00187
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.67533 0.305386
\(808\) 0 0
\(809\) 47.0260 1.65335 0.826673 0.562683i \(-0.190231\pi\)
0.826673 + 0.562683i \(0.190231\pi\)
\(810\) 0 0
\(811\) −37.9637 −1.33308 −0.666542 0.745467i \(-0.732226\pi\)
−0.666542 + 0.745467i \(0.732226\pi\)
\(812\) 0 0
\(813\) −16.4188 −0.575834
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.623166 0.0218018
\(818\) 0 0
\(819\) −4.41883 −0.154407
\(820\) 0 0
\(821\) 23.1362 0.807458 0.403729 0.914879i \(-0.367714\pi\)
0.403729 + 0.914879i \(0.367714\pi\)
\(822\) 0 0
\(823\) 37.1492 1.29494 0.647471 0.762090i \(-0.275827\pi\)
0.647471 + 0.762090i \(0.275827\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.91150 −0.240336 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(828\) 0 0
\(829\) 15.6101 0.542160 0.271080 0.962557i \(-0.412619\pi\)
0.271080 + 0.962557i \(0.412619\pi\)
\(830\) 0 0
\(831\) −7.67533 −0.266254
\(832\) 0 0
\(833\) −16.4290 −0.569231
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.41883 0.152737
\(838\) 0 0
\(839\) −52.1883 −1.80174 −0.900871 0.434088i \(-0.857071\pi\)
−0.900871 + 0.434088i \(0.857071\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −1.47392 −0.0507644
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 37.6753 1.29454
\(848\) 0 0
\(849\) 18.5130 0.635364
\(850\) 0 0
\(851\) −5.07834 −0.174083
\(852\) 0 0
\(853\) 3.01875 0.103360 0.0516800 0.998664i \(-0.483542\pi\)
0.0516800 + 0.998664i \(0.483542\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −44.1043 −1.50658 −0.753288 0.657691i \(-0.771533\pi\)
−0.753288 + 0.657691i \(0.771533\pi\)
\(858\) 0 0
\(859\) 18.1680 0.619884 0.309942 0.950755i \(-0.399690\pi\)
0.309942 + 0.950755i \(0.399690\pi\)
\(860\) 0 0
\(861\) −7.46083 −0.254264
\(862\) 0 0
\(863\) 16.6464 0.566651 0.283325 0.959024i \(-0.408562\pi\)
0.283325 + 0.959024i \(0.408562\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 15.2797 0.518928
\(868\) 0 0
\(869\) −64.1883 −2.17744
\(870\) 0 0
\(871\) −6.10725 −0.206936
\(872\) 0 0
\(873\) −5.37683 −0.181978
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.7724 0.465061 0.232531 0.972589i \(-0.425299\pi\)
0.232531 + 0.972589i \(0.425299\pi\)
\(878\) 0 0
\(879\) −19.0522 −0.642614
\(880\) 0 0
\(881\) −8.06525 −0.271725 −0.135863 0.990728i \(-0.543381\pi\)
−0.135863 + 0.990728i \(0.543381\pi\)
\(882\) 0 0
\(883\) 33.9363 1.14205 0.571024 0.820933i \(-0.306546\pi\)
0.571024 + 0.820933i \(0.306546\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.2985 −1.08448 −0.542239 0.840224i \(-0.682423\pi\)
−0.542239 + 0.840224i \(0.682423\pi\)
\(888\) 0 0
\(889\) 31.5913 1.05954
\(890\) 0 0
\(891\) −4.41883 −0.148036
\(892\) 0 0
\(893\) 0.850752 0.0284693
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.37683 0.0459711
\(898\) 0 0
\(899\) −4.41883 −0.147376
\(900\) 0 0
\(901\) 3.11742 0.103856
\(902\) 0 0
\(903\) −8.83767 −0.294099
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 23.8058 0.790460 0.395230 0.918582i \(-0.370665\pi\)
0.395230 + 0.918582i \(0.370665\pi\)
\(908\) 0 0
\(909\) 14.9029 0.494299
\(910\) 0 0
\(911\) −18.5970 −0.616146 −0.308073 0.951363i \(-0.599684\pi\)
−0.308073 + 0.951363i \(0.599684\pi\)
\(912\) 0 0
\(913\) −22.2797 −0.737352
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.8898 0.921003
\(918\) 0 0
\(919\) 14.4421 0.476400 0.238200 0.971216i \(-0.423443\pi\)
0.238200 + 0.971216i \(0.423443\pi\)
\(920\) 0 0
\(921\) 16.1174 0.531087
\(922\) 0 0
\(923\) −3.68842 −0.121406
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6.21450 −0.204111
\(928\) 0 0
\(929\) −16.8246 −0.551997 −0.275998 0.961158i \(-0.589008\pi\)
−0.275998 + 0.961158i \(0.589008\pi\)
\(930\) 0 0
\(931\) 3.90292 0.127913
\(932\) 0 0
\(933\) −3.78550 −0.123932
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.1231 −1.24543 −0.622713 0.782450i \(-0.713970\pi\)
−0.622713 + 0.782450i \(0.713970\pi\)
\(938\) 0 0
\(939\) −15.0000 −0.489506
\(940\) 0 0
\(941\) −0.408667 −0.0133222 −0.00666108 0.999978i \(-0.502120\pi\)
−0.00666108 + 0.999978i \(0.502120\pi\)
\(942\) 0 0
\(943\) 2.32467 0.0757016
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.5028 0.991209 0.495604 0.868548i \(-0.334947\pi\)
0.495604 + 0.868548i \(0.334947\pi\)
\(948\) 0 0
\(949\) −12.8377 −0.416728
\(950\) 0 0
\(951\) −22.5130 −0.730034
\(952\) 0 0
\(953\) −24.4942 −0.793447 −0.396723 0.917938i \(-0.629853\pi\)
−0.396723 + 0.917938i \(0.629853\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.41883 0.142841
\(958\) 0 0
\(959\) −78.5333 −2.53597
\(960\) 0 0
\(961\) −11.4739 −0.370126
\(962\) 0 0
\(963\) 0.311583 0.0100406
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.5868 −0.533396 −0.266698 0.963780i \(-0.585933\pi\)
−0.266698 + 0.963780i \(0.585933\pi\)
\(968\) 0 0
\(969\) 0.408667 0.0131283
\(970\) 0 0
\(971\) −54.8711 −1.76090 −0.880448 0.474142i \(-0.842758\pi\)
−0.880448 + 0.474142i \(0.842758\pi\)
\(972\) 0 0
\(973\) −21.3768 −0.685310
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.5448 −0.625294 −0.312647 0.949869i \(-0.601216\pi\)
−0.312647 + 0.949869i \(0.601216\pi\)
\(978\) 0 0
\(979\) −14.9217 −0.476898
\(980\) 0 0
\(981\) 19.9869 0.638133
\(982\) 0 0
\(983\) −58.8217 −1.87612 −0.938060 0.346473i \(-0.887379\pi\)
−0.938060 + 0.346473i \(0.887379\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −12.0653 −0.384041
\(988\) 0 0
\(989\) 2.75367 0.0875615
\(990\) 0 0
\(991\) −10.2651 −0.326081 −0.163041 0.986619i \(-0.552130\pi\)
−0.163041 + 0.986619i \(0.552130\pi\)
\(992\) 0 0
\(993\) −12.8377 −0.407391
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −23.6101 −0.747739 −0.373869 0.927481i \(-0.621969\pi\)
−0.373869 + 0.927481i \(0.621969\pi\)
\(998\) 0 0
\(999\) 3.68842 0.116696
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 7800.2.a.bk.1.3 3
5.4 even 2 7800.2.a.bq.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bk.1.3 3 1.1 even 1 trivial
7800.2.a.bq.1.1 yes 3 5.4 even 2