Properties

Label 7056.2.b.w.1567.5
Level $7056$
Weight $2$
Character 7056.1567
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
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: no (minimal twist has level 2352)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.5
Root \(0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 7056.1567
Dual form 7056.2.b.w.1567.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.601731i q^{5} +O(q^{10})\) \(q+0.601731i q^{5} -0.163636i q^{11} +3.37849i q^{13} +2.13246i q^{17} +0.953669 q^{19} -6.99598i q^{23} +4.63792 q^{25} +1.28897 q^{29} +3.35449 q^{31} +5.16373 q^{37} -6.67896i q^{41} -6.09172i q^{43} +3.69764 q^{47} -13.8806 q^{53} +0.0984649 q^{55} -8.05188 q^{59} +0.181465i q^{61} -2.03294 q^{65} +9.09299i q^{67} -3.36066i q^{71} -12.1442i q^{73} +6.98840i q^{79} +12.9788 q^{83} -1.28317 q^{85} +14.8853i q^{89} +0.573852i q^{95} +9.08274i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} - 16q^{29} - 32q^{31} + 16q^{47} - 16q^{53} - 64q^{55} - 48q^{59} + 16q^{65} - 64q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.601731i 0.269102i 0.990907 + 0.134551i \(0.0429592\pi\)
−0.990907 + 0.134551i \(0.957041\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.163636i − 0.0493382i −0.999696 0.0246691i \(-0.992147\pi\)
0.999696 0.0246691i \(-0.00785321\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i 0.883457 + 0.468513i \(0.155210\pi\)
−0.883457 + 0.468513i \(0.844790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.13246i 0.517199i 0.965985 + 0.258599i \(0.0832609\pi\)
−0.965985 + 0.258599i \(0.916739\pi\)
\(18\) 0 0
\(19\) 0.953669 0.218787 0.109393 0.993999i \(-0.465109\pi\)
0.109393 + 0.993999i \(0.465109\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.99598i − 1.45876i −0.684107 0.729382i \(-0.739808\pi\)
0.684107 0.729382i \(-0.260192\pi\)
\(24\) 0 0
\(25\) 4.63792 0.927584
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.28897 0.239356 0.119678 0.992813i \(-0.461814\pi\)
0.119678 + 0.992813i \(0.461814\pi\)
\(30\) 0 0
\(31\) 3.35449 0.602485 0.301242 0.953548i \(-0.402599\pi\)
0.301242 + 0.953548i \(0.402599\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.16373 0.848912 0.424456 0.905449i \(-0.360465\pi\)
0.424456 + 0.905449i \(0.360465\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.67896i − 1.04308i −0.853227 0.521539i \(-0.825358\pi\)
0.853227 0.521539i \(-0.174642\pi\)
\(42\) 0 0
\(43\) − 6.09172i − 0.928978i −0.885579 0.464489i \(-0.846238\pi\)
0.885579 0.464489i \(-0.153762\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.69764 0.539356 0.269678 0.962951i \(-0.413083\pi\)
0.269678 + 0.962951i \(0.413083\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.8806 −1.90664 −0.953321 0.301959i \(-0.902359\pi\)
−0.953321 + 0.301959i \(0.902359\pi\)
\(54\) 0 0
\(55\) 0.0984649 0.0132770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.05188 −1.04827 −0.524133 0.851637i \(-0.675610\pi\)
−0.524133 + 0.851637i \(0.675610\pi\)
\(60\) 0 0
\(61\) 0.181465i 0.0232342i 0.999933 + 0.0116171i \(0.00369792\pi\)
−0.999933 + 0.0116171i \(0.996302\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.03294 −0.252156
\(66\) 0 0
\(67\) 9.09299i 1.11089i 0.831555 + 0.555443i \(0.187451\pi\)
−0.831555 + 0.555443i \(0.812549\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.36066i − 0.398837i −0.979914 0.199419i \(-0.936095\pi\)
0.979914 0.199419i \(-0.0639054\pi\)
\(72\) 0 0
\(73\) − 12.1442i − 1.42137i −0.703509 0.710686i \(-0.748385\pi\)
0.703509 0.710686i \(-0.251615\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.98840i 0.786257i 0.919484 + 0.393128i \(0.128607\pi\)
−0.919484 + 0.393128i \(0.871393\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.9788 1.42460 0.712302 0.701873i \(-0.247652\pi\)
0.712302 + 0.701873i \(0.247652\pi\)
\(84\) 0 0
\(85\) −1.28317 −0.139179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.8853i 1.57784i 0.614496 + 0.788920i \(0.289359\pi\)
−0.614496 + 0.788920i \(0.710641\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.573852i 0.0588760i
\(96\) 0 0
\(97\) 9.08274i 0.922213i 0.887345 + 0.461106i \(0.152547\pi\)
−0.887345 + 0.461106i \(0.847453\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.51647i 0.847420i 0.905798 + 0.423710i \(0.139273\pi\)
−0.905798 + 0.423710i \(0.860727\pi\)
\(102\) 0 0
\(103\) 16.7609 1.65150 0.825749 0.564038i \(-0.190753\pi\)
0.825749 + 0.564038i \(0.190753\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.39735i − 0.425108i −0.977149 0.212554i \(-0.931822\pi\)
0.977149 0.212554i \(-0.0681781\pi\)
\(108\) 0 0
\(109\) 11.5453 1.10584 0.552918 0.833236i \(-0.313514\pi\)
0.552918 + 0.833236i \(0.313514\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.09821 0.103311 0.0516554 0.998665i \(-0.483550\pi\)
0.0516554 + 0.998665i \(0.483550\pi\)
\(114\) 0 0
\(115\) 4.20970 0.392556
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.9732 0.997566
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.79943i 0.518717i
\(126\) 0 0
\(127\) 5.22625i 0.463755i 0.972745 + 0.231877i \(0.0744868\pi\)
−0.972745 + 0.231877i \(0.925513\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 22.5319 1.96862 0.984309 0.176452i \(-0.0564620\pi\)
0.984309 + 0.176452i \(0.0564620\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.5916 −1.24664 −0.623322 0.781965i \(-0.714217\pi\)
−0.623322 + 0.781965i \(0.714217\pi\)
\(138\) 0 0
\(139\) −8.30816 −0.704689 −0.352345 0.935870i \(-0.614615\pi\)
−0.352345 + 0.935870i \(0.614615\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.552844 0.0462311
\(144\) 0 0
\(145\) 0.775614i 0.0644112i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.8017 −0.884906 −0.442453 0.896792i \(-0.645892\pi\)
−0.442453 + 0.896792i \(0.645892\pi\)
\(150\) 0 0
\(151\) 14.1587i 1.15222i 0.817371 + 0.576111i \(0.195430\pi\)
−0.817371 + 0.576111i \(0.804570\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.01850i 0.162130i
\(156\) 0 0
\(157\) 1.09264i 0.0872021i 0.999049 + 0.0436011i \(0.0138830\pi\)
−0.999049 + 0.0436011i \(0.986117\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 18.0857i − 1.41658i −0.705922 0.708290i \(-0.749467\pi\)
0.705922 0.708290i \(-0.250533\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.0196 1.47178 0.735889 0.677103i \(-0.236765\pi\)
0.735889 + 0.677103i \(0.236765\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.25540i 0.627646i 0.949481 + 0.313823i \(0.101610\pi\)
−0.949481 + 0.313823i \(0.898390\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.49647i 0.410825i 0.978675 + 0.205413i \(0.0658536\pi\)
−0.978675 + 0.205413i \(0.934146\pi\)
\(180\) 0 0
\(181\) 11.8519i 0.880946i 0.897766 + 0.440473i \(0.145189\pi\)
−0.897766 + 0.440473i \(0.854811\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.10717i 0.228444i
\(186\) 0 0
\(187\) 0.348948 0.0255176
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 23.1483i − 1.67495i −0.546474 0.837476i \(-0.684030\pi\)
0.546474 0.837476i \(-0.315970\pi\)
\(192\) 0 0
\(193\) −0.512058 −0.0368588 −0.0184294 0.999830i \(-0.505867\pi\)
−0.0184294 + 0.999830i \(0.505867\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.7934 0.911495 0.455748 0.890109i \(-0.349372\pi\)
0.455748 + 0.890109i \(0.349372\pi\)
\(198\) 0 0
\(199\) −15.8072 −1.12054 −0.560271 0.828309i \(-0.689303\pi\)
−0.560271 + 0.828309i \(0.689303\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.01893 0.280695
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 0.156055i − 0.0107945i
\(210\) 0 0
\(211\) 0.191712i 0.0131980i 0.999978 + 0.00659899i \(0.00210054\pi\)
−0.999978 + 0.00659899i \(0.997899\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.66557 0.249990
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.20452 −0.484628
\(222\) 0 0
\(223\) 10.0111 0.670392 0.335196 0.942148i \(-0.391197\pi\)
0.335196 + 0.942148i \(0.391197\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.13633 −0.407283 −0.203641 0.979046i \(-0.565278\pi\)
−0.203641 + 0.979046i \(0.565278\pi\)
\(228\) 0 0
\(229\) 9.44500i 0.624143i 0.950059 + 0.312072i \(0.101023\pi\)
−0.950059 + 0.312072i \(0.898977\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.2217 1.45579 0.727895 0.685689i \(-0.240499\pi\)
0.727895 + 0.685689i \(0.240499\pi\)
\(234\) 0 0
\(235\) 2.22498i 0.145142i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 20.7962i − 1.34520i −0.740008 0.672598i \(-0.765178\pi\)
0.740008 0.672598i \(-0.234822\pi\)
\(240\) 0 0
\(241\) − 13.0554i − 0.840971i −0.907299 0.420486i \(-0.861860\pi\)
0.907299 0.420486i \(-0.138140\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.22196i 0.205009i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.8975 1.12968 0.564839 0.825201i \(-0.308938\pi\)
0.564839 + 0.825201i \(0.308938\pi\)
\(252\) 0 0
\(253\) −1.14480 −0.0719727
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.67896i 0.416622i 0.978063 + 0.208311i \(0.0667966\pi\)
−0.978063 + 0.208311i \(0.933203\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.4164i 1.50558i 0.658261 + 0.752790i \(0.271292\pi\)
−0.658261 + 0.752790i \(0.728708\pi\)
\(264\) 0 0
\(265\) − 8.35236i − 0.513081i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.14593i 0.0698685i 0.999390 + 0.0349342i \(0.0111222\pi\)
−0.999390 + 0.0349342i \(0.988878\pi\)
\(270\) 0 0
\(271\) 5.65157 0.343308 0.171654 0.985157i \(-0.445089\pi\)
0.171654 + 0.985157i \(0.445089\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 0.758932i − 0.0457653i
\(276\) 0 0
\(277\) −19.2758 −1.15817 −0.579087 0.815266i \(-0.696591\pi\)
−0.579087 + 0.815266i \(0.696591\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.80369 −0.525184 −0.262592 0.964907i \(-0.584577\pi\)
−0.262592 + 0.964907i \(0.584577\pi\)
\(282\) 0 0
\(283\) 26.1451 1.55416 0.777081 0.629401i \(-0.216700\pi\)
0.777081 + 0.629401i \(0.216700\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.4526 0.732506
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.2640i 1.06699i 0.845802 + 0.533497i \(0.179123\pi\)
−0.845802 + 0.533497i \(0.820877\pi\)
\(294\) 0 0
\(295\) − 4.84506i − 0.282090i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.6359 1.36690
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.109193 −0.00625237
\(306\) 0 0
\(307\) 21.4472 1.22405 0.612027 0.790837i \(-0.290354\pi\)
0.612027 + 0.790837i \(0.290354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.4800 0.821085 0.410543 0.911841i \(-0.365339\pi\)
0.410543 + 0.911841i \(0.365339\pi\)
\(312\) 0 0
\(313\) 1.48483i 0.0839275i 0.999119 + 0.0419638i \(0.0133614\pi\)
−0.999119 + 0.0419638i \(0.986639\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.4908 0.645389 0.322695 0.946503i \(-0.395411\pi\)
0.322695 + 0.946503i \(0.395411\pi\)
\(318\) 0 0
\(319\) − 0.210922i − 0.0118094i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.03366i 0.113156i
\(324\) 0 0
\(325\) 15.6692i 0.869170i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 21.0967i − 1.15958i −0.814766 0.579790i \(-0.803134\pi\)
0.814766 0.579790i \(-0.196866\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.47153 −0.298942
\(336\) 0 0
\(337\) −5.62683 −0.306513 −0.153256 0.988186i \(-0.548976\pi\)
−0.153256 + 0.988186i \(0.548976\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 0.548917i − 0.0297255i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.3767i 1.30861i 0.756231 + 0.654305i \(0.227039\pi\)
−0.756231 + 0.654305i \(0.772961\pi\)
\(348\) 0 0
\(349\) − 2.84502i − 0.152290i −0.997097 0.0761451i \(-0.975739\pi\)
0.997097 0.0761451i \(-0.0242612\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 24.6581i − 1.31242i −0.754580 0.656208i \(-0.772159\pi\)
0.754580 0.656208i \(-0.227841\pi\)
\(354\) 0 0
\(355\) 2.02221 0.107328
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 35.6193i − 1.87991i −0.341292 0.939957i \(-0.610864\pi\)
0.341292 0.939957i \(-0.389136\pi\)
\(360\) 0 0
\(361\) −18.0905 −0.952132
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.30754 0.382494
\(366\) 0 0
\(367\) −23.2676 −1.21456 −0.607280 0.794488i \(-0.707740\pi\)
−0.607280 + 0.794488i \(0.707740\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21.9732 −1.13773 −0.568865 0.822431i \(-0.692617\pi\)
−0.568865 + 0.822431i \(0.692617\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.35478i 0.224283i
\(378\) 0 0
\(379\) − 19.7876i − 1.01642i −0.861233 0.508211i \(-0.830307\pi\)
0.861233 0.508211i \(-0.169693\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.6569 0.697833 0.348916 0.937154i \(-0.386550\pi\)
0.348916 + 0.937154i \(0.386550\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.0501 0.661666 0.330833 0.943689i \(-0.392670\pi\)
0.330833 + 0.943689i \(0.392670\pi\)
\(390\) 0 0
\(391\) 14.9187 0.754470
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.20514 −0.211583
\(396\) 0 0
\(397\) 6.52604i 0.327533i 0.986499 + 0.163766i \(0.0523643\pi\)
−0.986499 + 0.163766i \(0.947636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.4311 1.56959 0.784797 0.619753i \(-0.212767\pi\)
0.784797 + 0.619753i \(0.212767\pi\)
\(402\) 0 0
\(403\) 11.3331i 0.564544i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 0.844973i − 0.0418838i
\(408\) 0 0
\(409\) 34.8221i 1.72184i 0.508739 + 0.860921i \(0.330112\pi\)
−0.508739 + 0.860921i \(0.669888\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.80972i 0.383364i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.9597 −0.584271 −0.292135 0.956377i \(-0.594366\pi\)
−0.292135 + 0.956377i \(0.594366\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.89020i 0.479745i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.1460i 0.922230i 0.887340 + 0.461115i \(0.152550\pi\)
−0.887340 + 0.461115i \(0.847450\pi\)
\(432\) 0 0
\(433\) − 1.66205i − 0.0798730i −0.999202 0.0399365i \(-0.987284\pi\)
0.999202 0.0399365i \(-0.0127156\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.67185i − 0.319158i
\(438\) 0 0
\(439\) 16.5586 0.790301 0.395151 0.918616i \(-0.370692\pi\)
0.395151 + 0.918616i \(0.370692\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.3344i 0.871096i 0.900166 + 0.435548i \(0.143445\pi\)
−0.900166 + 0.435548i \(0.856555\pi\)
\(444\) 0 0
\(445\) −8.95695 −0.424600
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.9899 −0.896187 −0.448093 0.893987i \(-0.647897\pi\)
−0.448093 + 0.893987i \(0.647897\pi\)
\(450\) 0 0
\(451\) −1.09292 −0.0514636
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.8395 −0.787719 −0.393860 0.919171i \(-0.628860\pi\)
−0.393860 + 0.919171i \(0.628860\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 36.8939i 1.71832i 0.511708 + 0.859160i \(0.329013\pi\)
−0.511708 + 0.859160i \(0.670987\pi\)
\(462\) 0 0
\(463\) 35.2394i 1.63771i 0.573997 + 0.818857i \(0.305392\pi\)
−0.573997 + 0.818857i \(0.694608\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 35.8246 1.65776 0.828882 0.559423i \(-0.188977\pi\)
0.828882 + 0.559423i \(0.188977\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.996826 −0.0458341
\(474\) 0 0
\(475\) 4.42304 0.202943
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.10594 0.278988 0.139494 0.990223i \(-0.455452\pi\)
0.139494 + 0.990223i \(0.455452\pi\)
\(480\) 0 0
\(481\) 17.4456i 0.795452i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.46536 −0.248169
\(486\) 0 0
\(487\) − 17.0071i − 0.770663i −0.922778 0.385332i \(-0.874087\pi\)
0.922778 0.385332i \(-0.125913\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.2897i 1.41209i 0.708169 + 0.706043i \(0.249521\pi\)
−0.708169 + 0.706043i \(0.750479\pi\)
\(492\) 0 0
\(493\) 2.74869i 0.123795i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 21.2515i − 0.951347i −0.879622 0.475674i \(-0.842204\pi\)
0.879622 0.475674i \(-0.157796\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.728285 0.0324726 0.0162363 0.999868i \(-0.494832\pi\)
0.0162363 + 0.999868i \(0.494832\pi\)
\(504\) 0 0
\(505\) −5.12462 −0.228043
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.6424i 1.40253i 0.712903 + 0.701263i \(0.247380\pi\)
−0.712903 + 0.701263i \(0.752620\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.0855i 0.444421i
\(516\) 0 0
\(517\) − 0.605068i − 0.0266108i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 19.0336i − 0.833878i −0.908935 0.416939i \(-0.863103\pi\)
0.908935 0.416939i \(-0.136897\pi\)
\(522\) 0 0
\(523\) −4.91288 −0.214825 −0.107413 0.994215i \(-0.534257\pi\)
−0.107413 + 0.994215i \(0.534257\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.15334i 0.311604i
\(528\) 0 0
\(529\) −25.9438 −1.12799
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 22.5648 0.977391
\(534\) 0 0
\(535\) 2.64602 0.114397
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −28.0886 −1.20762 −0.603811 0.797127i \(-0.706352\pi\)
−0.603811 + 0.797127i \(0.706352\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.94714i 0.297583i
\(546\) 0 0
\(547\) − 18.5886i − 0.794792i −0.917647 0.397396i \(-0.869914\pi\)
0.917647 0.397396i \(-0.130086\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.22925 0.0523679
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.9877 1.52485 0.762424 0.647078i \(-0.224009\pi\)
0.762424 + 0.647078i \(0.224009\pi\)
\(558\) 0 0
\(559\) 20.5808 0.870476
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.3463 −1.06822 −0.534109 0.845415i \(-0.679353\pi\)
−0.534109 + 0.845415i \(0.679353\pi\)
\(564\) 0 0
\(565\) 0.660825i 0.0278011i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.6432 0.739642 0.369821 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(570\) 0 0
\(571\) − 29.9226i − 1.25222i −0.779733 0.626112i \(-0.784645\pi\)
0.779733 0.626112i \(-0.215355\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 32.4468i − 1.35313i
\(576\) 0 0
\(577\) − 8.49884i − 0.353811i −0.984228 0.176906i \(-0.943391\pi\)
0.984228 0.176906i \(-0.0566087\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.27136i 0.0940702i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.1156 0.871535 0.435767 0.900059i \(-0.356477\pi\)
0.435767 + 0.900059i \(0.356477\pi\)
\(588\) 0 0
\(589\) 3.19908 0.131816
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 35.8055i − 1.47035i −0.677875 0.735177i \(-0.737099\pi\)
0.677875 0.735177i \(-0.262901\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.0909i 0.412302i 0.978520 + 0.206151i \(0.0660938\pi\)
−0.978520 + 0.206151i \(0.933906\pi\)
\(600\) 0 0
\(601\) − 31.0913i − 1.26824i −0.773234 0.634120i \(-0.781362\pi\)
0.773234 0.634120i \(-0.218638\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.60293i 0.268447i
\(606\) 0 0
\(607\) 21.7145 0.881366 0.440683 0.897663i \(-0.354736\pi\)
0.440683 + 0.897663i \(0.354736\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.4924i 0.505390i
\(612\) 0 0
\(613\) −20.2810 −0.819143 −0.409571 0.912278i \(-0.634322\pi\)
−0.409571 + 0.912278i \(0.634322\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.1442 −0.569423 −0.284712 0.958613i \(-0.591898\pi\)
−0.284712 + 0.958613i \(0.591898\pi\)
\(618\) 0 0
\(619\) −42.5162 −1.70887 −0.854435 0.519559i \(-0.826096\pi\)
−0.854435 + 0.519559i \(0.826096\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.6999 0.787996
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.0115i 0.439056i
\(630\) 0 0
\(631\) − 11.2426i − 0.447561i −0.974640 0.223781i \(-0.928160\pi\)
0.974640 0.223781i \(-0.0718399\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.14480 −0.124797
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.471636 −0.0186285 −0.00931425 0.999957i \(-0.502965\pi\)
−0.00931425 + 0.999957i \(0.502965\pi\)
\(642\) 0 0
\(643\) 1.10040 0.0433955 0.0216977 0.999765i \(-0.493093\pi\)
0.0216977 + 0.999765i \(0.493093\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.7314 1.24749 0.623746 0.781627i \(-0.285610\pi\)
0.623746 + 0.781627i \(0.285610\pi\)
\(648\) 0 0
\(649\) 1.31758i 0.0517195i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.98421 0.351579 0.175790 0.984428i \(-0.443752\pi\)
0.175790 + 0.984428i \(0.443752\pi\)
\(654\) 0 0
\(655\) 13.5581i 0.529759i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 26.0679i − 1.01546i −0.861516 0.507731i \(-0.830484\pi\)
0.861516 0.507731i \(-0.169516\pi\)
\(660\) 0 0
\(661\) 1.36376i 0.0530441i 0.999648 + 0.0265221i \(0.00844323\pi\)
−0.999648 + 0.0265221i \(0.991557\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.01763i − 0.349164i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.0296942 0.00114633
\(672\) 0 0
\(673\) −17.6874 −0.681799 −0.340899 0.940100i \(-0.610732\pi\)
−0.340899 + 0.940100i \(0.610732\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 7.95083i − 0.305575i −0.988259 0.152788i \(-0.951175\pi\)
0.988259 0.152788i \(-0.0488250\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0286i 0.689846i 0.938631 + 0.344923i \(0.112095\pi\)
−0.938631 + 0.344923i \(0.887905\pi\)
\(684\) 0 0
\(685\) − 8.78021i − 0.335474i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 46.8954i − 1.78657i
\(690\) 0 0
\(691\) −44.4976 −1.69277 −0.846384 0.532572i \(-0.821225\pi\)
−0.846384 + 0.532572i \(0.821225\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 4.99928i − 0.189633i
\(696\) 0 0
\(697\) 14.2426 0.539478
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11.8785 0.448646 0.224323 0.974515i \(-0.427983\pi\)
0.224323 + 0.974515i \(0.427983\pi\)
\(702\) 0 0
\(703\) 4.92449 0.185731
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 24.8643 0.933798 0.466899 0.884311i \(-0.345371\pi\)
0.466899 + 0.884311i \(0.345371\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 23.4680i − 0.878883i
\(714\) 0 0
\(715\) 0.332663i 0.0124409i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.604720 −0.0225523 −0.0112761 0.999936i \(-0.503589\pi\)
−0.0112761 + 0.999936i \(0.503589\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.97815 0.222023
\(726\) 0 0
\(727\) −17.7628 −0.658786 −0.329393 0.944193i \(-0.606844\pi\)
−0.329393 + 0.944193i \(0.606844\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.9904 0.480466
\(732\) 0 0
\(733\) 15.5269i 0.573501i 0.958005 + 0.286750i \(0.0925750\pi\)
−0.958005 + 0.286750i \(0.907425\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.48794 0.0548090
\(738\) 0 0
\(739\) 17.2042i 0.632865i 0.948615 + 0.316433i \(0.102485\pi\)
−0.948615 + 0.316433i \(0.897515\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.1996i 0.484247i 0.970245 + 0.242123i \(0.0778439\pi\)
−0.970245 + 0.242123i \(0.922156\pi\)
\(744\) 0 0
\(745\) − 6.49968i − 0.238130i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 20.2946i − 0.740561i −0.928920 0.370280i \(-0.879262\pi\)
0.928920 0.370280i \(-0.120738\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.51975 −0.310065
\(756\) 0 0
\(757\) 24.7011 0.897778 0.448889 0.893587i \(-0.351820\pi\)
0.448889 + 0.893587i \(0.351820\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.77524i 0.173102i 0.996247 + 0.0865512i \(0.0275846\pi\)
−0.996247 + 0.0865512i \(0.972415\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 27.2032i − 0.982251i
\(768\) 0 0
\(769\) − 14.7721i − 0.532696i −0.963877 0.266348i \(-0.914183\pi\)
0.963877 0.266348i \(-0.0858170\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.9175i 1.32783i 0.747807 + 0.663916i \(0.231107\pi\)
−0.747807 + 0.663916i \(0.768893\pi\)
\(774\) 0 0
\(775\) 15.5579 0.558855
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 6.36951i − 0.228212i
\(780\) 0 0
\(781\) −0.549926 −0.0196779
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.657475 −0.0234663
\(786\) 0 0
\(787\) 37.0398 1.32033 0.660164 0.751122i \(-0.270487\pi\)
0.660164 + 0.751122i \(0.270487\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.613077 −0.0217710
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.9752i 1.30973i 0.755746 + 0.654865i \(0.227275\pi\)
−0.755746 + 0.654865i \(0.772725\pi\)
\(798\) 0 0
\(799\) 7.88509i 0.278954i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.98723 −0.0701279
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.34100 −0.187780 −0.0938898 0.995583i \(-0.529930\pi\)
−0.0938898 + 0.995583i \(0.529930\pi\)
\(810\) 0 0
\(811\) −37.4180 −1.31392 −0.656961 0.753924i \(-0.728159\pi\)
−0.656961 + 0.753924i \(0.728159\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.8827 0.381205
\(816\) 0 0
\(817\) − 5.80948i − 0.203248i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.3460 −0.710081 −0.355041 0.934851i \(-0.615533\pi\)
−0.355041 + 0.934851i \(0.615533\pi\)
\(822\) 0 0
\(823\) − 46.3367i − 1.61520i −0.589734 0.807598i \(-0.700767\pi\)
0.589734 0.807598i \(-0.299233\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.20358i − 0.0766260i −0.999266 0.0383130i \(-0.987802\pi\)
0.999266 0.0383130i \(-0.0121984\pi\)
\(828\) 0 0
\(829\) 17.2639i 0.599599i 0.954002 + 0.299800i \(0.0969199\pi\)
−0.954002 + 0.299800i \(0.903080\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 11.4446i 0.396058i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −17.4781 −0.603410 −0.301705 0.953401i \(-0.597556\pi\)
−0.301705 + 0.953401i \(0.597556\pi\)
\(840\) 0 0
\(841\) −27.3386 −0.942709
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.954216i 0.0328260i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 36.1254i − 1.23836i
\(852\) 0 0
\(853\) 30.7781i 1.05382i 0.849920 + 0.526912i \(0.176650\pi\)
−0.849920 + 0.526912i \(0.823350\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.4865i 0.802283i 0.916016 + 0.401142i \(0.131386\pi\)
−0.916016 + 0.401142i \(0.868614\pi\)
\(858\) 0 0
\(859\) 42.2249 1.44070 0.720348 0.693613i \(-0.243982\pi\)
0.720348 + 0.693613i \(0.243982\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 6.37615i − 0.217047i −0.994094 0.108523i \(-0.965388\pi\)
0.994094 0.108523i \(-0.0346122\pi\)
\(864\) 0 0
\(865\) −4.96753 −0.168901
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.14356 0.0387925
\(870\) 0 0
\(871\) −30.7206 −1.04093
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.2199 −0.412638 −0.206319 0.978485i \(-0.566148\pi\)
−0.206319 + 0.978485i \(0.566148\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 45.0545i − 1.51793i −0.651134 0.758963i \(-0.725706\pi\)
0.651134 0.758963i \(-0.274294\pi\)
\(882\) 0 0
\(883\) 21.8481i 0.735246i 0.929975 + 0.367623i \(0.119828\pi\)
−0.929975 + 0.367623i \(0.880172\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.2604 −1.08320 −0.541599 0.840637i \(-0.682181\pi\)
−0.541599 + 0.840637i \(0.682181\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.52632 0.118004
\(894\) 0 0
\(895\) −3.30739 −0.110554
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.32385 0.144208
\(900\) 0 0
\(901\) − 29.5998i − 0.986112i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.13166 −0.237065
\(906\) 0 0
\(907\) 17.5626i 0.583158i 0.956547 + 0.291579i \(0.0941806\pi\)
−0.956547 + 0.291579i \(0.905819\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 0.475746i − 0.0157622i −0.999969 0.00788108i \(-0.997491\pi\)
0.999969 0.00788108i \(-0.00250865\pi\)
\(912\) 0 0
\(913\) − 2.12380i − 0.0702874i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.8870i 1.08484i 0.840107 + 0.542420i \(0.182492\pi\)
−0.840107 + 0.542420i \(0.817508\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11.3540 0.373721
\(924\) 0 0
\(925\) 23.9490 0.787437
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.7798i 0.419293i 0.977777 + 0.209646i \(0.0672313\pi\)
−0.977777 + 0.209646i \(0.932769\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.209973i 0.00686685i
\(936\) 0 0
\(937\) − 4.03304i − 0.131754i −0.997828 0.0658768i \(-0.979016\pi\)
0.997828 0.0658768i \(-0.0209844\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 58.7586i 1.91548i 0.287640 + 0.957739i \(0.407129\pi\)
−0.287640 + 0.957739i \(0.592871\pi\)
\(942\) 0 0
\(943\) −46.7259 −1.52160
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.4052i 0.435611i 0.975992 + 0.217805i \(0.0698898\pi\)
−0.975992 + 0.217805i \(0.930110\pi\)
\(948\) 0 0
\(949\) 41.0291 1.33186
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.89697 0.0614490 0.0307245 0.999528i \(-0.490219\pi\)
0.0307245 + 0.999528i \(0.490219\pi\)
\(954\) 0 0
\(955\) 13.9290 0.450733
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.7474 −0.637012
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 0.308121i − 0.00991877i
\(966\) 0 0
\(967\) − 4.84855i − 0.155919i −0.996957 0.0779595i \(-0.975160\pi\)
0.996957 0.0779595i \(-0.0248405\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −37.7418 −1.21119 −0.605596 0.795772i \(-0.707065\pi\)
−0.605596 + 0.795772i \(0.707065\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.6686 −0.789217 −0.394609 0.918849i \(-0.629120\pi\)
−0.394609 + 0.918849i \(0.629120\pi\)
\(978\) 0 0
\(979\) 2.43578 0.0778477
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −37.7500 −1.20404 −0.602019 0.798481i \(-0.705637\pi\)
−0.602019 + 0.798481i \(0.705637\pi\)
\(984\) 0 0
\(985\) 7.69821i 0.245285i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −42.6176 −1.35516
\(990\) 0 0
\(991\) 55.5501i 1.76461i 0.470682 + 0.882303i \(0.344008\pi\)
−0.470682 + 0.882303i \(0.655992\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 9.51167i − 0.301540i
\(996\) 0 0
\(997\) − 24.1419i − 0.764583i −0.924042 0.382291i \(-0.875135\pi\)
0.924042 0.382291i \(-0.124865\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7056.2.b.w.1567.5 8
3.2 odd 2 2352.2.b.l.1567.4 yes 8
4.3 odd 2 7056.2.b.x.1567.5 8
7.6 odd 2 7056.2.b.x.1567.4 8
12.11 even 2 2352.2.b.k.1567.4 8
21.2 odd 6 2352.2.bl.q.31.2 8
21.5 even 6 2352.2.bl.t.31.3 8
21.11 odd 6 2352.2.bl.o.607.3 8
21.17 even 6 2352.2.bl.r.607.2 8
21.20 even 2 2352.2.b.k.1567.5 yes 8
28.27 even 2 inner 7056.2.b.w.1567.4 8
84.11 even 6 2352.2.bl.t.607.3 8
84.23 even 6 2352.2.bl.r.31.2 8
84.47 odd 6 2352.2.bl.o.31.3 8
84.59 odd 6 2352.2.bl.q.607.2 8
84.83 odd 2 2352.2.b.l.1567.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.4 8 12.11 even 2
2352.2.b.k.1567.5 yes 8 21.20 even 2
2352.2.b.l.1567.4 yes 8 3.2 odd 2
2352.2.b.l.1567.5 yes 8 84.83 odd 2
2352.2.bl.o.31.3 8 84.47 odd 6
2352.2.bl.o.607.3 8 21.11 odd 6
2352.2.bl.q.31.2 8 21.2 odd 6
2352.2.bl.q.607.2 8 84.59 odd 6
2352.2.bl.r.31.2 8 84.23 even 6
2352.2.bl.r.607.2 8 21.17 even 6
2352.2.bl.t.31.3 8 21.5 even 6
2352.2.bl.t.607.3 8 84.11 even 6
7056.2.b.w.1567.4 8 28.27 even 2 inner
7056.2.b.w.1567.5 8 1.1 even 1 trivial
7056.2.b.x.1567.4 8 7.6 odd 2
7056.2.b.x.1567.5 8 4.3 odd 2