Properties

Label 2352.2.b.l.1567.3
Level $2352$
Weight $2$
Character 2352.1567
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1567
Dual form 2352.2.b.l.1567.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.68412i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.68412i q^{5} +1.00000 q^{9} -3.53188i q^{11} -2.93015i q^{13} -1.68412i q^{15} +2.01140i q^{17} +1.69764 q^{19} +1.59851i q^{23} +2.16373 q^{25} +1.00000 q^{27} +7.94028 q^{29} -4.95367 q^{31} -3.53188i q^{33} -10.4663 q^{37} -2.93015i q^{39} -2.86237i q^{41} -11.7518i q^{43} -1.68412i q^{45} -6.70319 q^{47} +2.01140i q^{51} +2.92109 q^{53} -5.94812 q^{55} +1.69764 q^{57} -7.75532 q^{59} -12.5925i q^{61} -4.93473 q^{65} -1.70195i q^{67} +1.59851i q^{69} +6.13052i q^{71} -2.43166i q^{73} +2.16373 q^{75} -0.865467i q^{79} +1.00000 q^{81} +14.5319 q^{83} +3.38744 q^{85} +7.94028 q^{87} -15.9387i q^{89} -4.95367 q^{93} -2.85903i q^{95} +9.82270i q^{97} -3.53188i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 8q^{9} - 24q^{25} + 8q^{27} + 16q^{29} - 32q^{31} - 16q^{47} + 16q^{53} - 64q^{55} + 48q^{59} - 16q^{65} - 24q^{75} + 8q^{81} - 64q^{85} + 16q^{87} - 32q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) − 1.68412i − 0.753163i −0.926384 0.376581i \(-0.877100\pi\)
0.926384 0.376581i \(-0.122900\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 3.53188i − 1.06490i −0.846461 0.532451i \(-0.821271\pi\)
0.846461 0.532451i \(-0.178729\pi\)
\(12\) 0 0
\(13\) − 2.93015i − 0.812678i −0.913722 0.406339i \(-0.866805\pi\)
0.913722 0.406339i \(-0.133195\pi\)
\(14\) 0 0
\(15\) − 1.68412i − 0.434839i
\(16\) 0 0
\(17\) 2.01140i 0.487835i 0.969796 + 0.243918i \(0.0784326\pi\)
−0.969796 + 0.243918i \(0.921567\pi\)
\(18\) 0 0
\(19\) 1.69764 0.389465 0.194733 0.980856i \(-0.437616\pi\)
0.194733 + 0.980856i \(0.437616\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.59851i 0.333313i 0.986015 + 0.166657i \(0.0532971\pi\)
−0.986015 + 0.166657i \(0.946703\pi\)
\(24\) 0 0
\(25\) 2.16373 0.432746
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.94028 1.47447 0.737237 0.675635i \(-0.236130\pi\)
0.737237 + 0.675635i \(0.236130\pi\)
\(30\) 0 0
\(31\) −4.95367 −0.889705 −0.444853 0.895604i \(-0.646744\pi\)
−0.444853 + 0.895604i \(0.646744\pi\)
\(32\) 0 0
\(33\) − 3.53188i − 0.614822i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.4663 −1.72066 −0.860328 0.509740i \(-0.829742\pi\)
−0.860328 + 0.509740i \(0.829742\pi\)
\(38\) 0 0
\(39\) − 2.93015i − 0.469200i
\(40\) 0 0
\(41\) − 2.86237i − 0.447027i −0.974701 0.223514i \(-0.928247\pi\)
0.974701 0.223514i \(-0.0717527\pi\)
\(42\) 0 0
\(43\) − 11.7518i − 1.79214i −0.443917 0.896068i \(-0.646411\pi\)
0.443917 0.896068i \(-0.353589\pi\)
\(44\) 0 0
\(45\) − 1.68412i − 0.251054i
\(46\) 0 0
\(47\) −6.70319 −0.977760 −0.488880 0.872351i \(-0.662594\pi\)
−0.488880 + 0.872351i \(0.662594\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.01140i 0.281652i
\(52\) 0 0
\(53\) 2.92109 0.401242 0.200621 0.979669i \(-0.435704\pi\)
0.200621 + 0.979669i \(0.435704\pi\)
\(54\) 0 0
\(55\) −5.94812 −0.802045
\(56\) 0 0
\(57\) 1.69764 0.224858
\(58\) 0 0
\(59\) −7.75532 −1.00966 −0.504828 0.863220i \(-0.668444\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(60\) 0 0
\(61\) − 12.5925i − 1.61231i −0.591704 0.806155i \(-0.701545\pi\)
0.591704 0.806155i \(-0.298455\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.93473 −0.612079
\(66\) 0 0
\(67\) − 1.70195i − 0.207926i −0.994581 0.103963i \(-0.966848\pi\)
0.994581 0.103963i \(-0.0331524\pi\)
\(68\) 0 0
\(69\) 1.59851i 0.192438i
\(70\) 0 0
\(71\) 6.13052i 0.727558i 0.931485 + 0.363779i \(0.118514\pi\)
−0.931485 + 0.363779i \(0.881486\pi\)
\(72\) 0 0
\(73\) − 2.43166i − 0.284604i −0.989823 0.142302i \(-0.954550\pi\)
0.989823 0.142302i \(-0.0454505\pi\)
\(74\) 0 0
\(75\) 2.16373 0.249846
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 0.865467i − 0.0973727i −0.998814 0.0486863i \(-0.984497\pi\)
0.998814 0.0486863i \(-0.0155035\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.5319 1.59508 0.797540 0.603266i \(-0.206134\pi\)
0.797540 + 0.603266i \(0.206134\pi\)
\(84\) 0 0
\(85\) 3.38744 0.367419
\(86\) 0 0
\(87\) 7.94028 0.851287
\(88\) 0 0
\(89\) − 15.9387i − 1.68950i −0.535160 0.844751i \(-0.679749\pi\)
0.535160 0.844751i \(-0.320251\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.95367 −0.513672
\(94\) 0 0
\(95\) − 2.85903i − 0.293331i
\(96\) 0 0
\(97\) 9.82270i 0.997344i 0.866791 + 0.498672i \(0.166179\pi\)
−0.866791 + 0.498672i \(0.833821\pi\)
\(98\) 0 0
\(99\) − 3.53188i − 0.354967i
\(100\) 0 0
\(101\) − 9.42764i − 0.938086i −0.883175 0.469043i \(-0.844599\pi\)
0.883175 0.469043i \(-0.155401\pi\)
\(102\) 0 0
\(103\) −15.6627 −1.54329 −0.771644 0.636055i \(-0.780565\pi\)
−0.771644 + 0.636055i \(0.780565\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.9792i 1.83479i 0.397977 + 0.917395i \(0.369712\pi\)
−0.397977 + 0.917395i \(0.630288\pi\)
\(108\) 0 0
\(109\) 10.5590 1.01137 0.505685 0.862718i \(-0.331240\pi\)
0.505685 + 0.862718i \(0.331240\pi\)
\(110\) 0 0
\(111\) −10.4663 −0.993422
\(112\) 0 0
\(113\) 15.4530 1.45369 0.726846 0.686800i \(-0.240985\pi\)
0.726846 + 0.686800i \(0.240985\pi\)
\(114\) 0 0
\(115\) 2.69209 0.251039
\(116\) 0 0
\(117\) − 2.93015i − 0.270893i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.47419 −0.134017
\(122\) 0 0
\(123\) − 2.86237i − 0.258091i
\(124\) 0 0
\(125\) − 12.0646i − 1.07909i
\(126\) 0 0
\(127\) − 2.16478i − 0.192094i −0.995377 0.0960468i \(-0.969380\pi\)
0.995377 0.0960468i \(-0.0306198\pi\)
\(128\) 0 0
\(129\) − 11.7518i − 1.03469i
\(130\) 0 0
\(131\) −15.3219 −1.33868 −0.669341 0.742955i \(-0.733423\pi\)
−0.669341 + 0.742955i \(0.733423\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 1.68412i − 0.144946i
\(136\) 0 0
\(137\) 12.8614 1.09882 0.549410 0.835553i \(-0.314852\pi\)
0.549410 + 0.835553i \(0.314852\pi\)
\(138\) 0 0
\(139\) −0.743971 −0.0631028 −0.0315514 0.999502i \(-0.510045\pi\)
−0.0315514 + 0.999502i \(0.510045\pi\)
\(140\) 0 0
\(141\) −6.70319 −0.564510
\(142\) 0 0
\(143\) −10.3489 −0.865423
\(144\) 0 0
\(145\) − 13.3724i − 1.11052i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.30262 −0.598254 −0.299127 0.954213i \(-0.596695\pi\)
−0.299127 + 0.954213i \(0.596695\pi\)
\(150\) 0 0
\(151\) − 6.28343i − 0.511338i −0.966764 0.255669i \(-0.917704\pi\)
0.966764 0.255669i \(-0.0822958\pi\)
\(152\) 0 0
\(153\) 2.01140i 0.162612i
\(154\) 0 0
\(155\) 8.34259i 0.670093i
\(156\) 0 0
\(157\) − 8.91152i − 0.711217i −0.934635 0.355608i \(-0.884274\pi\)
0.934635 0.355608i \(-0.115726\pi\)
\(158\) 0 0
\(159\) 2.92109 0.231657
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 0.242138i − 0.0189657i −0.999955 0.00948287i \(-0.996981\pi\)
0.999955 0.00948287i \(-0.00301854\pi\)
\(164\) 0 0
\(165\) −5.94812 −0.463061
\(166\) 0 0
\(167\) −5.82817 −0.450997 −0.225499 0.974243i \(-0.572401\pi\)
−0.225499 + 0.974243i \(0.572401\pi\)
\(168\) 0 0
\(169\) 4.41421 0.339555
\(170\) 0 0
\(171\) 1.69764 0.129822
\(172\) 0 0
\(173\) 16.7935i 1.27678i 0.769712 + 0.638392i \(0.220400\pi\)
−0.769712 + 0.638392i \(0.779600\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.75532 −0.582925
\(178\) 0 0
\(179\) 21.2399i 1.58754i 0.608217 + 0.793771i \(0.291885\pi\)
−0.608217 + 0.793771i \(0.708115\pi\)
\(180\) 0 0
\(181\) 2.74444i 0.203993i 0.994785 + 0.101996i \(0.0325230\pi\)
−0.994785 + 0.101996i \(0.967477\pi\)
\(182\) 0 0
\(183\) − 12.5925i − 0.930868i
\(184\) 0 0
\(185\) 17.6266i 1.29593i
\(186\) 0 0
\(187\) 7.10401 0.519497
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.60924i 0.116440i 0.998304 + 0.0582201i \(0.0185425\pi\)
−0.998304 + 0.0582201i \(0.981457\pi\)
\(192\) 0 0
\(193\) 4.01109 0.288725 0.144362 0.989525i \(-0.453887\pi\)
0.144362 + 0.989525i \(0.453887\pi\)
\(194\) 0 0
\(195\) −4.93473 −0.353384
\(196\) 0 0
\(197\) 11.7413 0.836534 0.418267 0.908324i \(-0.362638\pi\)
0.418267 + 0.908324i \(0.362638\pi\)
\(198\) 0 0
\(199\) 17.3603 1.23064 0.615319 0.788278i \(-0.289027\pi\)
0.615319 + 0.788278i \(0.289027\pi\)
\(200\) 0 0
\(201\) − 1.70195i − 0.120046i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.82058 −0.336684
\(206\) 0 0
\(207\) 1.59851i 0.111104i
\(208\) 0 0
\(209\) − 5.99586i − 0.414743i
\(210\) 0 0
\(211\) − 24.1172i − 1.66030i −0.557543 0.830148i \(-0.688256\pi\)
0.557543 0.830148i \(-0.311744\pi\)
\(212\) 0 0
\(213\) 6.13052i 0.420056i
\(214\) 0 0
\(215\) −19.7915 −1.34977
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 2.43166i − 0.164316i
\(220\) 0 0
\(221\) 5.89369 0.396453
\(222\) 0 0
\(223\) −20.1154 −1.34702 −0.673512 0.739176i \(-0.735215\pi\)
−0.673512 + 0.739176i \(0.735215\pi\)
\(224\) 0 0
\(225\) 2.16373 0.144249
\(226\) 0 0
\(227\) −17.5893 −1.16744 −0.583721 0.811954i \(-0.698404\pi\)
−0.583721 + 0.811954i \(0.698404\pi\)
\(228\) 0 0
\(229\) − 3.99205i − 0.263802i −0.991263 0.131901i \(-0.957892\pi\)
0.991263 0.131901i \(-0.0421081\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.26967 0.214203 0.107102 0.994248i \(-0.465843\pi\)
0.107102 + 0.994248i \(0.465843\pi\)
\(234\) 0 0
\(235\) 11.2890i 0.736412i
\(236\) 0 0
\(237\) − 0.865467i − 0.0562181i
\(238\) 0 0
\(239\) 28.9127i 1.87021i 0.354371 + 0.935105i \(0.384695\pi\)
−0.354371 + 0.935105i \(0.615305\pi\)
\(240\) 0 0
\(241\) − 6.11269i − 0.393753i −0.980428 0.196876i \(-0.936920\pi\)
0.980428 0.196876i \(-0.0630798\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.97434i − 0.316510i
\(248\) 0 0
\(249\) 14.5319 0.920920
\(250\) 0 0
\(251\) 27.7471 1.75138 0.875691 0.482872i \(-0.160406\pi\)
0.875691 + 0.482872i \(0.160406\pi\)
\(252\) 0 0
\(253\) 5.64576 0.354946
\(254\) 0 0
\(255\) 3.38744 0.212130
\(256\) 0 0
\(257\) 2.86237i 0.178550i 0.996007 + 0.0892749i \(0.0284550\pi\)
−0.996007 + 0.0892749i \(0.971545\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 7.94028 0.491491
\(262\) 0 0
\(263\) − 19.4528i − 1.19951i −0.800184 0.599755i \(-0.795265\pi\)
0.800184 0.599755i \(-0.204735\pi\)
\(264\) 0 0
\(265\) − 4.91947i − 0.302201i
\(266\) 0 0
\(267\) − 15.9387i − 0.975434i
\(268\) 0 0
\(269\) 16.6831i 1.01719i 0.861007 + 0.508594i \(0.169835\pi\)
−0.861007 + 0.508594i \(0.830165\pi\)
\(270\) 0 0
\(271\) 19.9057 1.20918 0.604591 0.796536i \(-0.293336\pi\)
0.604591 + 0.796536i \(0.293336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 7.64204i − 0.460832i
\(276\) 0 0
\(277\) −14.3275 −0.860854 −0.430427 0.902625i \(-0.641637\pi\)
−0.430427 + 0.902625i \(0.641637\pi\)
\(278\) 0 0
\(279\) −4.95367 −0.296568
\(280\) 0 0
\(281\) 16.5450 0.986992 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(282\) 0 0
\(283\) 29.8591 1.77494 0.887469 0.460868i \(-0.152462\pi\)
0.887469 + 0.460868i \(0.152462\pi\)
\(284\) 0 0
\(285\) − 2.85903i − 0.169355i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.9543 0.762017
\(290\) 0 0
\(291\) 9.82270i 0.575817i
\(292\) 0 0
\(293\) 6.40183i 0.373999i 0.982360 + 0.187000i \(0.0598763\pi\)
−0.982360 + 0.187000i \(0.940124\pi\)
\(294\) 0 0
\(295\) 13.0609i 0.760436i
\(296\) 0 0
\(297\) − 3.53188i − 0.204941i
\(298\) 0 0
\(299\) 4.68389 0.270876
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 9.42764i − 0.541604i
\(304\) 0 0
\(305\) −21.2074 −1.21433
\(306\) 0 0
\(307\) 11.6511 0.664961 0.332480 0.943110i \(-0.392114\pi\)
0.332480 + 0.943110i \(0.392114\pi\)
\(308\) 0 0
\(309\) −15.6627 −0.891017
\(310\) 0 0
\(311\) −23.0772 −1.30859 −0.654295 0.756239i \(-0.727034\pi\)
−0.654295 + 0.756239i \(0.727034\pi\)
\(312\) 0 0
\(313\) 25.9505i 1.46681i 0.679794 + 0.733404i \(0.262069\pi\)
−0.679794 + 0.733404i \(0.737931\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.5430 1.15381 0.576904 0.816812i \(-0.304261\pi\)
0.576904 + 0.816812i \(0.304261\pi\)
\(318\) 0 0
\(319\) − 28.0441i − 1.57017i
\(320\) 0 0
\(321\) 18.9792i 1.05932i
\(322\) 0 0
\(323\) 3.41462i 0.189995i
\(324\) 0 0
\(325\) − 6.34006i − 0.351683i
\(326\) 0 0
\(327\) 10.5590 0.583915
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 32.7763i 1.80155i 0.434286 + 0.900775i \(0.357001\pi\)
−0.434286 + 0.900775i \(0.642999\pi\)
\(332\) 0 0
\(333\) −10.4663 −0.573552
\(334\) 0 0
\(335\) −2.86630 −0.156602
\(336\) 0 0
\(337\) −33.2791 −1.81283 −0.906414 0.422391i \(-0.861191\pi\)
−0.906414 + 0.422391i \(0.861191\pi\)
\(338\) 0 0
\(339\) 15.4530 0.839290
\(340\) 0 0
\(341\) 17.4958i 0.947449i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.69209 0.144937
\(346\) 0 0
\(347\) 9.65926i 0.518536i 0.965805 + 0.259268i \(0.0834813\pi\)
−0.965805 + 0.259268i \(0.916519\pi\)
\(348\) 0 0
\(349\) − 11.9525i − 0.639802i −0.947451 0.319901i \(-0.896350\pi\)
0.947451 0.319901i \(-0.103650\pi\)
\(350\) 0 0
\(351\) − 2.93015i − 0.156400i
\(352\) 0 0
\(353\) 12.9230i 0.687820i 0.939003 + 0.343910i \(0.111751\pi\)
−0.939003 + 0.343910i \(0.888249\pi\)
\(354\) 0 0
\(355\) 10.3245 0.547970
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 11.0629i − 0.583879i −0.956437 0.291939i \(-0.905699\pi\)
0.956437 0.291939i \(-0.0943005\pi\)
\(360\) 0 0
\(361\) −16.1180 −0.848317
\(362\) 0 0
\(363\) −1.47419 −0.0773749
\(364\) 0 0
\(365\) −4.09522 −0.214353
\(366\) 0 0
\(367\) −11.8888 −0.620589 −0.310294 0.950641i \(-0.600428\pi\)
−0.310294 + 0.950641i \(0.600428\pi\)
\(368\) 0 0
\(369\) − 2.86237i − 0.149009i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.52581 −0.493228 −0.246614 0.969114i \(-0.579318\pi\)
−0.246614 + 0.969114i \(0.579318\pi\)
\(374\) 0 0
\(375\) − 12.0646i − 0.623013i
\(376\) 0 0
\(377\) − 23.2662i − 1.19827i
\(378\) 0 0
\(379\) 4.52128i 0.232243i 0.993235 + 0.116121i \(0.0370461\pi\)
−0.993235 + 0.116121i \(0.962954\pi\)
\(380\) 0 0
\(381\) − 2.16478i − 0.110905i
\(382\) 0 0
\(383\) −2.34315 −0.119729 −0.0598646 0.998207i \(-0.519067\pi\)
−0.0598646 + 0.998207i \(0.519067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 11.7518i − 0.597379i
\(388\) 0 0
\(389\) 18.0981 0.917610 0.458805 0.888537i \(-0.348278\pi\)
0.458805 + 0.888537i \(0.348278\pi\)
\(390\) 0 0
\(391\) −3.21524 −0.162602
\(392\) 0 0
\(393\) −15.3219 −0.772888
\(394\) 0 0
\(395\) −1.45755 −0.0733375
\(396\) 0 0
\(397\) − 2.81508i − 0.141285i −0.997502 0.0706425i \(-0.977495\pi\)
0.997502 0.0706425i \(-0.0225050\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.08242 0.303741 0.151871 0.988400i \(-0.451470\pi\)
0.151871 + 0.988400i \(0.451470\pi\)
\(402\) 0 0
\(403\) 14.5150i 0.723044i
\(404\) 0 0
\(405\) − 1.68412i − 0.0836847i
\(406\) 0 0
\(407\) 36.9659i 1.83233i
\(408\) 0 0
\(409\) − 8.41275i − 0.415984i −0.978131 0.207992i \(-0.933307\pi\)
0.978131 0.207992i \(-0.0666928\pi\)
\(410\) 0 0
\(411\) 12.8614 0.634405
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 24.4735i − 1.20135i
\(416\) 0 0
\(417\) −0.743971 −0.0364324
\(418\) 0 0
\(419\) 18.6496 0.911094 0.455547 0.890212i \(-0.349444\pi\)
0.455547 + 0.890212i \(0.349444\pi\)
\(420\) 0 0
\(421\) −22.6274 −1.10279 −0.551396 0.834243i \(-0.685905\pi\)
−0.551396 + 0.834243i \(0.685905\pi\)
\(422\) 0 0
\(423\) −6.70319 −0.325920
\(424\) 0 0
\(425\) 4.35212i 0.211109i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −10.3489 −0.499652
\(430\) 0 0
\(431\) 15.9070i 0.766215i 0.923704 + 0.383107i \(0.125146\pi\)
−0.923704 + 0.383107i \(0.874854\pi\)
\(432\) 0 0
\(433\) 14.4650i 0.695146i 0.937653 + 0.347573i \(0.112994\pi\)
−0.937653 + 0.347573i \(0.887006\pi\)
\(434\) 0 0
\(435\) − 13.3724i − 0.641158i
\(436\) 0 0
\(437\) 2.71370i 0.129814i
\(438\) 0 0
\(439\) 21.7961 1.04027 0.520136 0.854084i \(-0.325881\pi\)
0.520136 + 0.854084i \(0.325881\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.8637i 0.563664i 0.959464 + 0.281832i \(0.0909420\pi\)
−0.959464 + 0.281832i \(0.909058\pi\)
\(444\) 0 0
\(445\) −26.8428 −1.27247
\(446\) 0 0
\(447\) −7.30262 −0.345402
\(448\) 0 0
\(449\) −38.6472 −1.82388 −0.911938 0.410329i \(-0.865414\pi\)
−0.911938 + 0.410329i \(0.865414\pi\)
\(450\) 0 0
\(451\) −10.1096 −0.476040
\(452\) 0 0
\(453\) − 6.28343i − 0.295221i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.9438 0.886153 0.443076 0.896484i \(-0.353887\pi\)
0.443076 + 0.896484i \(0.353887\pi\)
\(458\) 0 0
\(459\) 2.01140i 0.0938839i
\(460\) 0 0
\(461\) 24.5269i 1.14233i 0.820834 + 0.571167i \(0.193509\pi\)
−0.820834 + 0.571167i \(0.806491\pi\)
\(462\) 0 0
\(463\) − 7.59791i − 0.353105i −0.984291 0.176552i \(-0.943505\pi\)
0.984291 0.176552i \(-0.0564945\pi\)
\(464\) 0 0
\(465\) 8.34259i 0.386878i
\(466\) 0 0
\(467\) 11.0188 0.509891 0.254945 0.966955i \(-0.417943\pi\)
0.254945 + 0.966955i \(0.417943\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 8.91152i − 0.410621i
\(472\) 0 0
\(473\) −41.5060 −1.90845
\(474\) 0 0
\(475\) 3.67323 0.168540
\(476\) 0 0
\(477\) 2.92109 0.133747
\(478\) 0 0
\(479\) −36.2027 −1.65415 −0.827073 0.562095i \(-0.809995\pi\)
−0.827073 + 0.562095i \(0.809995\pi\)
\(480\) 0 0
\(481\) 30.6680i 1.39834i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.5426 0.751162
\(486\) 0 0
\(487\) − 17.4119i − 0.789010i −0.918894 0.394505i \(-0.870916\pi\)
0.918894 0.394505i \(-0.129084\pi\)
\(488\) 0 0
\(489\) − 0.242138i − 0.0109499i
\(490\) 0 0
\(491\) 21.5154i 0.970977i 0.874243 + 0.485489i \(0.161358\pi\)
−0.874243 + 0.485489i \(0.838642\pi\)
\(492\) 0 0
\(493\) 15.9710i 0.719300i
\(494\) 0 0
\(495\) −5.94812 −0.267348
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 36.4384i − 1.63121i −0.578610 0.815604i \(-0.696405\pi\)
0.578610 0.815604i \(-0.303595\pi\)
\(500\) 0 0
\(501\) −5.82817 −0.260383
\(502\) 0 0
\(503\) 39.5840 1.76496 0.882482 0.470347i \(-0.155871\pi\)
0.882482 + 0.470347i \(0.155871\pi\)
\(504\) 0 0
\(505\) −15.8773 −0.706531
\(506\) 0 0
\(507\) 4.41421 0.196042
\(508\) 0 0
\(509\) 35.1759i 1.55914i 0.626313 + 0.779572i \(0.284563\pi\)
−0.626313 + 0.779572i \(0.715437\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.69764 0.0749526
\(514\) 0 0
\(515\) 26.3778i 1.16235i
\(516\) 0 0
\(517\) 23.6749i 1.04122i
\(518\) 0 0
\(519\) 16.7935i 0.737151i
\(520\) 0 0
\(521\) 19.5734i 0.857525i 0.903417 + 0.428762i \(0.141050\pi\)
−0.903417 + 0.428762i \(0.858950\pi\)
\(522\) 0 0
\(523\) 8.66240 0.378780 0.189390 0.981902i \(-0.439349\pi\)
0.189390 + 0.981902i \(0.439349\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 9.96379i − 0.434029i
\(528\) 0 0
\(529\) 20.4448 0.888902
\(530\) 0 0
\(531\) −7.75532 −0.336552
\(532\) 0 0
\(533\) −8.38718 −0.363289
\(534\) 0 0
\(535\) 31.9633 1.38190
\(536\) 0 0
\(537\) 21.2399i 0.916568i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 25.0905 1.07873 0.539363 0.842074i \(-0.318665\pi\)
0.539363 + 0.842074i \(0.318665\pi\)
\(542\) 0 0
\(543\) 2.74444i 0.117775i
\(544\) 0 0
\(545\) − 17.7827i − 0.761726i
\(546\) 0 0
\(547\) − 0.523032i − 0.0223632i −0.999937 0.0111816i \(-0.996441\pi\)
0.999937 0.0111816i \(-0.00355929\pi\)
\(548\) 0 0
\(549\) − 12.5925i − 0.537437i
\(550\) 0 0
\(551\) 13.4797 0.574256
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 17.6266i 0.748208i
\(556\) 0 0
\(557\) 22.1339 0.937845 0.468922 0.883239i \(-0.344642\pi\)
0.468922 + 0.883239i \(0.344642\pi\)
\(558\) 0 0
\(559\) −34.4346 −1.45643
\(560\) 0 0
\(561\) 7.10401 0.299932
\(562\) 0 0
\(563\) 10.6076 0.447059 0.223529 0.974697i \(-0.428242\pi\)
0.223529 + 0.974697i \(0.428242\pi\)
\(564\) 0 0
\(565\) − 26.0247i − 1.09487i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.3988 0.435940 0.217970 0.975955i \(-0.430057\pi\)
0.217970 + 0.975955i \(0.430057\pi\)
\(570\) 0 0
\(571\) 21.3518i 0.893544i 0.894648 + 0.446772i \(0.147427\pi\)
−0.894648 + 0.446772i \(0.852573\pi\)
\(572\) 0 0
\(573\) 1.60924i 0.0672268i
\(574\) 0 0
\(575\) 3.45875i 0.144240i
\(576\) 0 0
\(577\) 0.922093i 0.0383872i 0.999816 + 0.0191936i \(0.00610989\pi\)
−0.999816 + 0.0191936i \(0.993890\pi\)
\(578\) 0 0
\(579\) 4.01109 0.166695
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 10.3169i − 0.427284i
\(584\) 0 0
\(585\) −4.93473 −0.204026
\(586\) 0 0
\(587\) 9.11149 0.376071 0.188036 0.982162i \(-0.439788\pi\)
0.188036 + 0.982162i \(0.439788\pi\)
\(588\) 0 0
\(589\) −8.40955 −0.346509
\(590\) 0 0
\(591\) 11.7413 0.482973
\(592\) 0 0
\(593\) 12.2075i 0.501304i 0.968077 + 0.250652i \(0.0806450\pi\)
−0.968077 + 0.250652i \(0.919355\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.3603 0.710510
\(598\) 0 0
\(599\) − 35.2167i − 1.43891i −0.694537 0.719457i \(-0.744391\pi\)
0.694537 0.719457i \(-0.255609\pi\)
\(600\) 0 0
\(601\) 30.6430i 1.24995i 0.780644 + 0.624976i \(0.214891\pi\)
−0.780644 + 0.624976i \(0.785109\pi\)
\(602\) 0 0
\(603\) − 1.70195i − 0.0693088i
\(604\) 0 0
\(605\) 2.48272i 0.100937i
\(606\) 0 0
\(607\) −9.96502 −0.404468 −0.202234 0.979337i \(-0.564820\pi\)
−0.202234 + 0.979337i \(0.564820\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.6413i 0.794604i
\(612\) 0 0
\(613\) −15.1259 −0.610928 −0.305464 0.952204i \(-0.598812\pi\)
−0.305464 + 0.952204i \(0.598812\pi\)
\(614\) 0 0
\(615\) −4.82058 −0.194385
\(616\) 0 0
\(617\) 11.7055 0.471245 0.235622 0.971845i \(-0.424287\pi\)
0.235622 + 0.971845i \(0.424287\pi\)
\(618\) 0 0
\(619\) 7.26763 0.292111 0.146055 0.989276i \(-0.453342\pi\)
0.146055 + 0.989276i \(0.453342\pi\)
\(620\) 0 0
\(621\) 1.59851i 0.0641462i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.49962 −0.379985
\(626\) 0 0
\(627\) − 5.99586i − 0.239452i
\(628\) 0 0
\(629\) − 21.0520i − 0.839397i
\(630\) 0 0
\(631\) 1.40366i 0.0558789i 0.999610 + 0.0279395i \(0.00889456\pi\)
−0.999610 + 0.0279395i \(0.991105\pi\)
\(632\) 0 0
\(633\) − 24.1172i − 0.958573i
\(634\) 0 0
\(635\) −3.64576 −0.144678
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.13052i 0.242519i
\(640\) 0 0
\(641\) −33.2272 −1.31240 −0.656198 0.754589i \(-0.727836\pi\)
−0.656198 + 0.754589i \(0.727836\pi\)
\(642\) 0 0
\(643\) 46.2604 1.82433 0.912166 0.409820i \(-0.134409\pi\)
0.912166 + 0.409820i \(0.134409\pi\)
\(644\) 0 0
\(645\) −19.7915 −0.779290
\(646\) 0 0
\(647\) 34.6332 1.36157 0.680786 0.732482i \(-0.261638\pi\)
0.680786 + 0.732482i \(0.261638\pi\)
\(648\) 0 0
\(649\) 27.3909i 1.07519i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.22863 0.322011 0.161006 0.986953i \(-0.448526\pi\)
0.161006 + 0.986953i \(0.448526\pi\)
\(654\) 0 0
\(655\) 25.8040i 1.00825i
\(656\) 0 0
\(657\) − 2.43166i − 0.0948681i
\(658\) 0 0
\(659\) − 10.9381i − 0.426087i −0.977043 0.213044i \(-0.931662\pi\)
0.977043 0.213044i \(-0.0683376\pi\)
\(660\) 0 0
\(661\) 25.1953i 0.979985i 0.871727 + 0.489993i \(0.163001\pi\)
−0.871727 + 0.489993i \(0.836999\pi\)
\(662\) 0 0
\(663\) 5.89369 0.228892
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.6926i 0.491461i
\(668\) 0 0
\(669\) −20.1154 −0.777705
\(670\) 0 0
\(671\) −44.4754 −1.71695
\(672\) 0 0
\(673\) 11.5831 0.446497 0.223248 0.974762i \(-0.428334\pi\)
0.223248 + 0.974762i \(0.428334\pi\)
\(674\) 0 0
\(675\) 2.16373 0.0832820
\(676\) 0 0
\(677\) 34.7641i 1.33609i 0.744120 + 0.668046i \(0.232869\pi\)
−0.744120 + 0.668046i \(0.767131\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −17.5893 −0.674023
\(682\) 0 0
\(683\) 11.7692i 0.450335i 0.974320 + 0.225167i \(0.0722929\pi\)
−0.974320 + 0.225167i \(0.927707\pi\)
\(684\) 0 0
\(685\) − 21.6601i − 0.827591i
\(686\) 0 0
\(687\) − 3.99205i − 0.152306i
\(688\) 0 0
\(689\) − 8.55923i − 0.326081i
\(690\) 0 0
\(691\) 11.3031 0.429991 0.214996 0.976615i \(-0.431026\pi\)
0.214996 + 0.976615i \(0.431026\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.25294i 0.0475267i
\(696\) 0 0
\(697\) 5.75736 0.218076
\(698\) 0 0
\(699\) 3.26967 0.123670
\(700\) 0 0
\(701\) 24.9265 0.941462 0.470731 0.882277i \(-0.343990\pi\)
0.470731 + 0.882277i \(0.343990\pi\)
\(702\) 0 0
\(703\) −17.7681 −0.670136
\(704\) 0 0
\(705\) 11.2890i 0.425168i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 44.2380 1.66139 0.830697 0.556724i \(-0.187942\pi\)
0.830697 + 0.556724i \(0.187942\pi\)
\(710\) 0 0
\(711\) − 0.865467i − 0.0324576i
\(712\) 0 0
\(713\) − 7.91851i − 0.296551i
\(714\) 0 0
\(715\) 17.4289i 0.651804i
\(716\) 0 0
\(717\) 28.9127i 1.07977i
\(718\) 0 0
\(719\) −5.40637 −0.201624 −0.100812 0.994906i \(-0.532144\pi\)
−0.100812 + 0.994906i \(0.532144\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 6.11269i − 0.227333i
\(724\) 0 0
\(725\) 17.1806 0.638072
\(726\) 0 0
\(727\) −36.5459 −1.35541 −0.677706 0.735333i \(-0.737026\pi\)
−0.677706 + 0.735333i \(0.737026\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 23.6376 0.874267
\(732\) 0 0
\(733\) 27.3245i 1.00925i 0.863338 + 0.504626i \(0.168370\pi\)
−0.863338 + 0.504626i \(0.831630\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.01109 −0.221421
\(738\) 0 0
\(739\) 17.7874i 0.654320i 0.944969 + 0.327160i \(0.106092\pi\)
−0.944969 + 0.327160i \(0.893908\pi\)
\(740\) 0 0
\(741\) − 4.97434i − 0.182737i
\(742\) 0 0
\(743\) 30.1701i 1.10683i 0.832904 + 0.553417i \(0.186676\pi\)
−0.832904 + 0.553417i \(0.813324\pi\)
\(744\) 0 0
\(745\) 12.2985i 0.450582i
\(746\) 0 0
\(747\) 14.5319 0.531693
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 42.9930i − 1.56884i −0.620232 0.784418i \(-0.712962\pi\)
0.620232 0.784418i \(-0.287038\pi\)
\(752\) 0 0
\(753\) 27.7471 1.01116
\(754\) 0 0
\(755\) −10.5821 −0.385121
\(756\) 0 0
\(757\) −13.2021 −0.479839 −0.239919 0.970793i \(-0.577121\pi\)
−0.239919 + 0.970793i \(0.577121\pi\)
\(758\) 0 0
\(759\) 5.64576 0.204928
\(760\) 0 0
\(761\) − 18.1325i − 0.657302i −0.944451 0.328651i \(-0.893406\pi\)
0.944451 0.328651i \(-0.106594\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.38744 0.122473
\(766\) 0 0
\(767\) 22.7243i 0.820525i
\(768\) 0 0
\(769\) − 45.0980i − 1.62628i −0.582070 0.813139i \(-0.697757\pi\)
0.582070 0.813139i \(-0.302243\pi\)
\(770\) 0 0
\(771\) 2.86237i 0.103086i
\(772\) 0 0
\(773\) − 36.0470i − 1.29652i −0.761420 0.648259i \(-0.775497\pi\)
0.761420 0.648259i \(-0.224503\pi\)
\(774\) 0 0
\(775\) −10.7184 −0.385016
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 4.85927i − 0.174102i
\(780\) 0 0
\(781\) 21.6523 0.774779
\(782\) 0 0
\(783\) 7.94028 0.283762
\(784\) 0 0
\(785\) −15.0081 −0.535662
\(786\) 0 0
\(787\) −26.3844 −0.940503 −0.470251 0.882533i \(-0.655837\pi\)
−0.470251 + 0.882533i \(0.655837\pi\)
\(788\) 0 0
\(789\) − 19.4528i − 0.692537i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −36.8981 −1.31029
\(794\) 0 0
\(795\) − 4.91947i − 0.174476i
\(796\) 0 0
\(797\) 52.8839i 1.87324i 0.350344 + 0.936621i \(0.386065\pi\)
−0.350344 + 0.936621i \(0.613935\pi\)
\(798\) 0 0
\(799\) − 13.4828i − 0.476986i
\(800\) 0 0
\(801\) − 15.9387i − 0.563167i
\(802\) 0 0
\(803\) −8.58834 −0.303076
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.6831i 0.587274i
\(808\) 0 0
\(809\) 16.1702 0.568512 0.284256 0.958748i \(-0.408253\pi\)
0.284256 + 0.958748i \(0.408253\pi\)
\(810\) 0 0
\(811\) −4.18532 −0.146967 −0.0734833 0.997296i \(-0.523412\pi\)
−0.0734833 + 0.997296i \(0.523412\pi\)
\(812\) 0 0
\(813\) 19.9057 0.698122
\(814\) 0 0
\(815\) −0.407791 −0.0142843
\(816\) 0 0
\(817\) − 19.9504i − 0.697975i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.89720 −0.170913 −0.0854567 0.996342i \(-0.527235\pi\)
−0.0854567 + 0.996342i \(0.527235\pi\)
\(822\) 0 0
\(823\) 6.49031i 0.226238i 0.993581 + 0.113119i \(0.0360841\pi\)
−0.993581 + 0.113119i \(0.963916\pi\)
\(824\) 0 0
\(825\) − 7.64204i − 0.266062i
\(826\) 0 0
\(827\) − 18.8437i − 0.655258i −0.944806 0.327629i \(-0.893750\pi\)
0.944806 0.327629i \(-0.106250\pi\)
\(828\) 0 0
\(829\) 15.8101i 0.549106i 0.961572 + 0.274553i \(0.0885300\pi\)
−0.961572 + 0.274553i \(0.911470\pi\)
\(830\) 0 0
\(831\) −14.3275 −0.497014
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.81535i 0.339674i
\(836\) 0 0
\(837\) −4.95367 −0.171224
\(838\) 0 0
\(839\) −25.1313 −0.867629 −0.433814 0.901002i \(-0.642833\pi\)
−0.433814 + 0.901002i \(0.642833\pi\)
\(840\) 0 0
\(841\) 34.0481 1.17407
\(842\) 0 0
\(843\) 16.5450 0.569840
\(844\) 0 0
\(845\) − 7.43408i − 0.255740i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 29.8591 1.02476
\(850\) 0 0
\(851\) − 16.7306i − 0.573518i
\(852\) 0 0
\(853\) 31.5399i 1.07991i 0.841695 + 0.539953i \(0.181558\pi\)
−0.841695 + 0.539953i \(0.818442\pi\)
\(854\) 0 0
\(855\) − 2.85903i − 0.0977769i
\(856\) 0 0
\(857\) 28.3495i 0.968401i 0.874957 + 0.484201i \(0.160890\pi\)
−0.874957 + 0.484201i \(0.839110\pi\)
\(858\) 0 0
\(859\) −34.6798 −1.18326 −0.591630 0.806210i \(-0.701515\pi\)
−0.591630 + 0.806210i \(0.701515\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 18.4754i − 0.628910i −0.949272 0.314455i \(-0.898178\pi\)
0.949272 0.314455i \(-0.101822\pi\)
\(864\) 0 0
\(865\) 28.2823 0.961626
\(866\) 0 0
\(867\) 12.9543 0.439951
\(868\) 0 0
\(869\) −3.05673 −0.103692
\(870\) 0 0
\(871\) −4.98697 −0.168977
\(872\) 0 0
\(873\) 9.82270i 0.332448i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −42.9784 −1.45128 −0.725639 0.688076i \(-0.758456\pi\)
−0.725639 + 0.688076i \(0.758456\pi\)
\(878\) 0 0
\(879\) 6.40183i 0.215929i
\(880\) 0 0
\(881\) 4.50973i 0.151937i 0.997110 + 0.0759684i \(0.0242048\pi\)
−0.997110 + 0.0759684i \(0.975795\pi\)
\(882\) 0 0
\(883\) − 42.8824i − 1.44311i −0.692359 0.721553i \(-0.743429\pi\)
0.692359 0.721553i \(-0.256571\pi\)
\(884\) 0 0
\(885\) 13.0609i 0.439038i
\(886\) 0 0
\(887\) −4.75725 −0.159733 −0.0798665 0.996806i \(-0.525449\pi\)
−0.0798665 + 0.996806i \(0.525449\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 3.53188i − 0.118322i
\(892\) 0 0
\(893\) −11.3796 −0.380804
\(894\) 0 0
\(895\) 35.7705 1.19568
\(896\) 0 0
\(897\) 4.68389 0.156390
\(898\) 0 0
\(899\) −39.3335 −1.31185
\(900\) 0 0
\(901\) 5.87547i 0.195740i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.62198 0.153640
\(906\) 0 0
\(907\) − 15.8103i − 0.524971i −0.964936 0.262486i \(-0.915458\pi\)
0.964936 0.262486i \(-0.0845422\pi\)
\(908\) 0 0
\(909\) − 9.42764i − 0.312695i
\(910\) 0 0
\(911\) − 15.5236i − 0.514320i −0.966369 0.257160i \(-0.917213\pi\)
0.966369 0.257160i \(-0.0827868\pi\)
\(912\) 0 0
\(913\) − 51.3248i − 1.69860i
\(914\) 0 0
\(915\) −21.2074 −0.701095
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 58.5264i 1.93061i 0.261129 + 0.965304i \(0.415905\pi\)
−0.261129 + 0.965304i \(0.584095\pi\)
\(920\) 0 0
\(921\) 11.6511 0.383915
\(922\) 0 0
\(923\) 17.9633 0.591271
\(924\) 0 0
\(925\) −22.6464 −0.744607
\(926\) 0 0
\(927\) −15.6627 −0.514429
\(928\) 0 0
\(929\) − 53.1650i − 1.74429i −0.489249 0.872144i \(-0.662729\pi\)
0.489249 0.872144i \(-0.337271\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −23.0772 −0.755515
\(934\) 0 0
\(935\) − 11.9640i − 0.391266i
\(936\) 0 0
\(937\) 17.0577i 0.557250i 0.960400 + 0.278625i \(0.0898787\pi\)
−0.960400 + 0.278625i \(0.910121\pi\)
\(938\) 0 0
\(939\) 25.9505i 0.846861i
\(940\) 0 0
\(941\) − 16.4321i − 0.535671i −0.963465 0.267836i \(-0.913692\pi\)
0.963465 0.267836i \(-0.0863084\pi\)
\(942\) 0 0
\(943\) 4.57554 0.149000
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 25.8513i − 0.840054i −0.907512 0.420027i \(-0.862021\pi\)
0.907512 0.420027i \(-0.137979\pi\)
\(948\) 0 0
\(949\) −7.12514 −0.231292
\(950\) 0 0
\(951\) 20.5430 0.666151
\(952\) 0 0
\(953\) −3.79848 −0.123045 −0.0615224 0.998106i \(-0.519596\pi\)
−0.0615224 + 0.998106i \(0.519596\pi\)
\(954\) 0 0
\(955\) 2.71015 0.0876984
\(956\) 0 0
\(957\) − 28.0441i − 0.906538i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.46117 −0.208425
\(962\) 0 0
\(963\) 18.9792i 0.611597i
\(964\) 0 0
\(965\) − 6.75517i − 0.217457i
\(966\) 0 0
\(967\) 20.7285i 0.666582i 0.942824 + 0.333291i \(0.108159\pi\)
−0.942824 + 0.333291i \(0.891841\pi\)
\(968\) 0 0
\(969\) 3.41462i 0.109694i
\(970\) 0 0
\(971\) 39.5188 1.26822 0.634110 0.773243i \(-0.281367\pi\)
0.634110 + 0.773243i \(0.281367\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 6.34006i − 0.203044i
\(976\) 0 0
\(977\) −21.1235 −0.675799 −0.337900 0.941182i \(-0.609716\pi\)
−0.337900 + 0.941182i \(0.609716\pi\)
\(978\) 0 0
\(979\) −56.2937 −1.79915
\(980\) 0 0
\(981\) 10.5590 0.337123
\(982\) 0 0
\(983\) 45.9575 1.46582 0.732909 0.680327i \(-0.238162\pi\)
0.732909 + 0.680327i \(0.238162\pi\)
\(984\) 0 0
\(985\) − 19.7738i − 0.630046i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.7855 0.597343
\(990\) 0 0
\(991\) 3.93322i 0.124943i 0.998047 + 0.0624714i \(0.0198982\pi\)
−0.998047 + 0.0624714i \(0.980102\pi\)
\(992\) 0 0
\(993\) 32.7763i 1.04013i
\(994\) 0 0
\(995\) − 29.2369i − 0.926871i
\(996\) 0 0
\(997\) − 10.7049i − 0.339027i −0.985528 0.169514i \(-0.945780\pi\)
0.985528 0.169514i \(-0.0542197\pi\)
\(998\) 0 0
\(999\) −10.4663 −0.331141
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 2352.2.b.l.1567.3 yes 8
3.2 odd 2 7056.2.b.w.1567.6 8
4.3 odd 2 2352.2.b.k.1567.3 8
7.2 even 3 2352.2.bl.o.31.2 8
7.3 odd 6 2352.2.bl.t.607.2 8
7.4 even 3 2352.2.bl.q.607.3 8
7.5 odd 6 2352.2.bl.r.31.3 8
7.6 odd 2 2352.2.b.k.1567.6 yes 8
12.11 even 2 7056.2.b.x.1567.6 8
21.20 even 2 7056.2.b.x.1567.3 8
28.3 even 6 2352.2.bl.o.607.2 8
28.11 odd 6 2352.2.bl.r.607.3 8
28.19 even 6 2352.2.bl.q.31.3 8
28.23 odd 6 2352.2.bl.t.31.2 8
28.27 even 2 inner 2352.2.b.l.1567.6 yes 8
84.83 odd 2 7056.2.b.w.1567.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.3 8 4.3 odd 2
2352.2.b.k.1567.6 yes 8 7.6 odd 2
2352.2.b.l.1567.3 yes 8 1.1 even 1 trivial
2352.2.b.l.1567.6 yes 8 28.27 even 2 inner
2352.2.bl.o.31.2 8 7.2 even 3
2352.2.bl.o.607.2 8 28.3 even 6
2352.2.bl.q.31.3 8 28.19 even 6
2352.2.bl.q.607.3 8 7.4 even 3
2352.2.bl.r.31.3 8 7.5 odd 6
2352.2.bl.r.607.3 8 28.11 odd 6
2352.2.bl.t.31.2 8 28.23 odd 6
2352.2.bl.t.607.2 8 7.3 odd 6
7056.2.b.w.1567.3 8 84.83 odd 2
7056.2.b.w.1567.6 8 3.2 odd 2
7056.2.b.x.1567.3 8 21.20 even 2
7056.2.b.x.1567.6 8 12.11 even 2