Properties

Label 2352.2.b.l.1567.4
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 1567.4
Root \(0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1567
Dual form 2352.2.b.l.1567.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.601731i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.601731i q^{5} +1.00000 q^{9} +0.163636i q^{11} +3.37849i q^{13} -0.601731i q^{15} -2.13246i q^{17} +0.953669 q^{19} +6.99598i q^{23} +4.63792 q^{25} +1.00000 q^{27} -1.28897 q^{29} +3.35449 q^{31} +0.163636i q^{33} +5.16373 q^{37} +3.37849i q^{39} +6.67896i q^{41} -6.09172i q^{43} -0.601731i q^{45} -3.69764 q^{47} -2.13246i q^{51} +13.8806 q^{53} +0.0984649 q^{55} +0.953669 q^{57} +8.05188 q^{59} +0.181465i q^{61} +2.03294 q^{65} +9.09299i q^{67} +6.99598i q^{69} +3.36066i q^{71} -12.1442i q^{73} +4.63792 q^{75} +6.98840i q^{79} +1.00000 q^{81} -12.9788 q^{83} -1.28317 q^{85} -1.28897 q^{87} -14.8853i q^{89} +3.35449 q^{93} -0.573852i q^{95} +9.08274i q^{97} +0.163636i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 8q^{9} - 24q^{25} + 8q^{27} + 16q^{29} - 32q^{31} - 16q^{47} + 16q^{53} - 64q^{55} + 48q^{59} - 16q^{65} - 24q^{75} + 8q^{81} - 64q^{85} + 16q^{87} - 32q^{93} + O(q^{100}) \)

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
\(5\) − 0.601731i − 0.269102i −0.990907 0.134551i \(-0.957041\pi\)
0.990907 0.134551i \(-0.0429592\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.163636i 0.0493382i 0.999696 + 0.0246691i \(0.00785321\pi\)
−0.999696 + 0.0246691i \(0.992147\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i 0.883457 + 0.468513i \(0.155210\pi\)
−0.883457 + 0.468513i \(0.844790\pi\)
\(14\) 0 0
\(15\) − 0.601731i − 0.155366i
\(16\) 0 0
\(17\) − 2.13246i − 0.517199i −0.965985 0.258599i \(-0.916739\pi\)
0.965985 0.258599i \(-0.0832609\pi\)
\(18\) 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 \(0.260192\pi\)
−0.684107 + 0.729382i \(0.739808\pi\)
\(24\) 0 0
\(25\) 4.63792 0.927584
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.28897 −0.239356 −0.119678 0.992813i \(-0.538186\pi\)
−0.119678 + 0.992813i \(0.538186\pi\)
\(30\) 0 0
\(31\) 3.35449 0.602485 0.301242 0.953548i \(-0.402599\pi\)
0.301242 + 0.953548i \(0.402599\pi\)
\(32\) 0 0
\(33\) 0.163636i 0.0284854i
\(34\) 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
\(39\) 3.37849i 0.540992i
\(40\) 0 0
\(41\) 6.67896i 1.04308i 0.853227 + 0.521539i \(0.174642\pi\)
−0.853227 + 0.521539i \(0.825358\pi\)
\(42\) 0 0
\(43\) − 6.09172i − 0.928978i −0.885579 0.464489i \(-0.846238\pi\)
0.885579 0.464489i \(-0.153762\pi\)
\(44\) 0 0
\(45\) − 0.601731i − 0.0897007i
\(46\) 0 0
\(47\) −3.69764 −0.539356 −0.269678 0.962951i \(-0.586917\pi\)
−0.269678 + 0.962951i \(0.586917\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 2.13246i − 0.298605i
\(52\) 0 0
\(53\) 13.8806 1.90664 0.953321 0.301959i \(-0.0976407\pi\)
0.953321 + 0.301959i \(0.0976407\pi\)
\(54\) 0 0
\(55\) 0.0984649 0.0132770
\(56\) 0 0
\(57\) 0.953669 0.126317
\(58\) 0 0
\(59\) 8.05188 1.04827 0.524133 0.851637i \(-0.324390\pi\)
0.524133 + 0.851637i \(0.324390\pi\)
\(60\) 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
\(69\) 6.99598i 0.842217i
\(70\) 0 0
\(71\) 3.36066i 0.398837i 0.979914 + 0.199419i \(0.0639054\pi\)
−0.979914 + 0.199419i \(0.936095\pi\)
\(72\) 0 0
\(73\) − 12.1442i − 1.42137i −0.703509 0.710686i \(-0.748385\pi\)
0.703509 0.710686i \(-0.251615\pi\)
\(74\) 0 0
\(75\) 4.63792 0.535541
\(76\) 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
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.9788 −1.42460 −0.712302 0.701873i \(-0.752348\pi\)
−0.712302 + 0.701873i \(0.752348\pi\)
\(84\) 0 0
\(85\) −1.28317 −0.139179
\(86\) 0 0
\(87\) −1.28897 −0.138192
\(88\) 0 0
\(89\) − 14.8853i − 1.57784i −0.614496 0.788920i \(-0.710641\pi\)
0.614496 0.788920i \(-0.289359\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.35449 0.347845
\(94\) 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
\(99\) 0.163636i 0.0164461i
\(100\) 0 0
\(101\) − 8.51647i − 0.847420i −0.905798 0.423710i \(-0.860727\pi\)
0.905798 0.423710i \(-0.139273\pi\)
\(102\) 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 \(0.0681781\pi\)
−0.977149 + 0.212554i \(0.931822\pi\)
\(108\) 0 0
\(109\) 11.5453 1.10584 0.552918 0.833236i \(-0.313514\pi\)
0.552918 + 0.833236i \(0.313514\pi\)
\(110\) 0 0
\(111\) 5.16373 0.490120
\(112\) 0 0
\(113\) −1.09821 −0.103311 −0.0516554 0.998665i \(-0.516450\pi\)
−0.0516554 + 0.998665i \(0.516450\pi\)
\(114\) 0 0
\(115\) 4.20970 0.392556
\(116\) 0 0
\(117\) 3.37849i 0.312342i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.9732 0.997566
\(122\) 0 0
\(123\) 6.67896i 0.602221i
\(124\) 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
\(129\) − 6.09172i − 0.536346i
\(130\) 0 0
\(131\) −22.5319 −1.96862 −0.984309 0.176452i \(-0.943538\pi\)
−0.984309 + 0.176452i \(0.943538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 0.601731i − 0.0517887i
\(136\) 0 0
\(137\) 14.5916 1.24664 0.623322 0.781965i \(-0.285783\pi\)
0.623322 + 0.781965i \(0.285783\pi\)
\(138\) 0 0
\(139\) −8.30816 −0.704689 −0.352345 0.935870i \(-0.614615\pi\)
−0.352345 + 0.935870i \(0.614615\pi\)
\(140\) 0 0
\(141\) −3.69764 −0.311397
\(142\) 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 \(-0.354108\pi\)
0.442453 + 0.896792i \(0.354108\pi\)
\(150\) 0 0
\(151\) 14.1587i 1.15222i 0.817371 + 0.576111i \(0.195430\pi\)
−0.817371 + 0.576111i \(0.804570\pi\)
\(152\) 0 0
\(153\) − 2.13246i − 0.172400i
\(154\) 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
\(159\) 13.8806 1.10080
\(160\) 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
\(165\) 0.0984649 0.00766548
\(166\) 0 0
\(167\) −19.0196 −1.47178 −0.735889 0.677103i \(-0.763235\pi\)
−0.735889 + 0.677103i \(0.763235\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 0.953669 0.0729289
\(172\) 0 0
\(173\) − 8.25540i − 0.627646i −0.949481 0.313823i \(-0.898390\pi\)
0.949481 0.313823i \(-0.101610\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.05188 0.605216
\(178\) 0 0
\(179\) − 5.49647i − 0.410825i −0.978675 0.205413i \(-0.934146\pi\)
0.978675 0.205413i \(-0.0658536\pi\)
\(180\) 0 0
\(181\) 11.8519i 0.880946i 0.897766 + 0.440473i \(0.145189\pi\)
−0.897766 + 0.440473i \(0.854811\pi\)
\(182\) 0 0
\(183\) 0.181465i 0.0134143i
\(184\) 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 \(0.315970\pi\)
−0.546474 + 0.837476i \(0.684030\pi\)
\(192\) 0 0
\(193\) −0.512058 −0.0368588 −0.0184294 0.999830i \(-0.505867\pi\)
−0.0184294 + 0.999830i \(0.505867\pi\)
\(194\) 0 0
\(195\) 2.03294 0.145582
\(196\) 0 0
\(197\) −12.7934 −0.911495 −0.455748 0.890109i \(-0.650628\pi\)
−0.455748 + 0.890109i \(0.650628\pi\)
\(198\) 0 0
\(199\) −15.8072 −1.12054 −0.560271 0.828309i \(-0.689303\pi\)
−0.560271 + 0.828309i \(0.689303\pi\)
\(200\) 0 0
\(201\) 9.09299i 0.641370i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.01893 0.280695
\(206\) 0 0
\(207\) 6.99598i 0.486254i
\(208\) 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
\(213\) 3.36066i 0.230269i
\(214\) 0 0
\(215\) −3.66557 −0.249990
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 12.1442i − 0.820630i
\(220\) 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
\(225\) 4.63792 0.309195
\(226\) 0 0
\(227\) 6.13633 0.407283 0.203641 0.979046i \(-0.434722\pi\)
0.203641 + 0.979046i \(0.434722\pi\)
\(228\) 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 \(-0.759501\pi\)
−0.727895 + 0.685689i \(0.759501\pi\)
\(234\) 0 0
\(235\) 2.22498i 0.145142i
\(236\) 0 0
\(237\) 6.98840i 0.453945i
\(238\) 0 0
\(239\) 20.7962i 1.34520i 0.740008 + 0.672598i \(0.234822\pi\)
−0.740008 + 0.672598i \(0.765178\pi\)
\(240\) 0 0
\(241\) − 13.0554i − 0.840971i −0.907299 0.420486i \(-0.861860\pi\)
0.907299 0.420486i \(-0.138140\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.22196i 0.205009i
\(248\) 0 0
\(249\) −12.9788 −0.822496
\(250\) 0 0
\(251\) −17.8975 −1.12968 −0.564839 0.825201i \(-0.691062\pi\)
−0.564839 + 0.825201i \(0.691062\pi\)
\(252\) 0 0
\(253\) −1.14480 −0.0719727
\(254\) 0 0
\(255\) −1.28317 −0.0803552
\(256\) 0 0
\(257\) − 6.67896i − 0.416622i −0.978063 0.208311i \(-0.933203\pi\)
0.978063 0.208311i \(-0.0667966\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.28897 −0.0797854
\(262\) 0 0
\(263\) − 24.4164i − 1.50558i −0.658261 0.752790i \(-0.728708\pi\)
0.658261 0.752790i \(-0.271292\pi\)
\(264\) 0 0
\(265\) − 8.35236i − 0.513081i
\(266\) 0 0
\(267\) − 14.8853i − 0.910966i
\(268\) 0 0
\(269\) − 1.14593i − 0.0698685i −0.999390 0.0349342i \(-0.988878\pi\)
0.999390 0.0349342i \(-0.0111222\pi\)
\(270\) 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
\(279\) 3.35449 0.200828
\(280\) 0 0
\(281\) 8.80369 0.525184 0.262592 0.964907i \(-0.415423\pi\)
0.262592 + 0.964907i \(0.415423\pi\)
\(282\) 0 0
\(283\) 26.1451 1.55416 0.777081 0.629401i \(-0.216700\pi\)
0.777081 + 0.629401i \(0.216700\pi\)
\(284\) 0 0
\(285\) − 0.573852i − 0.0339920i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.4526 0.732506
\(290\) 0 0
\(291\) 9.08274i 0.532440i
\(292\) 0 0
\(293\) − 18.2640i − 1.06699i −0.845802 0.533497i \(-0.820877\pi\)
0.845802 0.533497i \(-0.179123\pi\)
\(294\) 0 0
\(295\) − 4.84506i − 0.282090i
\(296\) 0 0
\(297\) 0.163636i 0.00949513i
\(298\) 0 0
\(299\) −23.6359 −1.36690
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 8.51647i − 0.489258i
\(304\) 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
\(309\) 16.7609 0.953492
\(310\) 0 0
\(311\) −14.4800 −0.821085 −0.410543 0.911841i \(-0.634661\pi\)
−0.410543 + 0.911841i \(0.634661\pi\)
\(312\) 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 \(-0.604589\pi\)
−0.322695 + 0.946503i \(0.604589\pi\)
\(318\) 0 0
\(319\) − 0.210922i − 0.0118094i
\(320\) 0 0
\(321\) 4.39735i 0.245436i
\(322\) 0 0
\(323\) − 2.03366i − 0.113156i
\(324\) 0 0
\(325\) 15.6692i 0.869170i
\(326\) 0 0
\(327\) 11.5453 0.638454
\(328\) 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
\(333\) 5.16373 0.282971
\(334\) 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
\(339\) −1.09821 −0.0596465
\(340\) 0 0
\(341\) 0.548917i 0.0297255i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.20970 0.226643
\(346\) 0 0
\(347\) − 24.3767i − 1.30861i −0.756231 0.654305i \(-0.772961\pi\)
0.756231 0.654305i \(-0.227039\pi\)
\(348\) 0 0
\(349\) − 2.84502i − 0.152290i −0.997097 0.0761451i \(-0.975739\pi\)
0.997097 0.0761451i \(-0.0242612\pi\)
\(350\) 0 0
\(351\) 3.37849i 0.180331i
\(352\) 0 0
\(353\) 24.6581i 1.31242i 0.754580 + 0.656208i \(0.227841\pi\)
−0.754580 + 0.656208i \(0.772159\pi\)
\(354\) 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 \(0.389136\pi\)
−0.341292 + 0.939957i \(0.610864\pi\)
\(360\) 0 0
\(361\) −18.0905 −0.952132
\(362\) 0 0
\(363\) 10.9732 0.575945
\(364\) 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
\(369\) 6.67896i 0.347693i
\(370\) 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
\(375\) − 5.79943i − 0.299481i
\(376\) 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
\(381\) 5.22625i 0.267749i
\(382\) 0 0
\(383\) −13.6569 −0.697833 −0.348916 0.937154i \(-0.613450\pi\)
−0.348916 + 0.937154i \(0.613450\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 6.09172i − 0.309659i
\(388\) 0 0
\(389\) −13.0501 −0.661666 −0.330833 0.943689i \(-0.607330\pi\)
−0.330833 + 0.943689i \(0.607330\pi\)
\(390\) 0 0
\(391\) 14.9187 0.754470
\(392\) 0 0
\(393\) −22.5319 −1.13658
\(394\) 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 \(-0.787233\pi\)
−0.784797 + 0.619753i \(0.787233\pi\)
\(402\) 0 0
\(403\) 11.3331i 0.564544i
\(404\) 0 0
\(405\) − 0.601731i − 0.0299002i
\(406\) 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
\(411\) 14.5916 0.719750
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.80972i 0.383364i
\(416\) 0 0
\(417\) −8.30816 −0.406852
\(418\) 0 0
\(419\) 11.9597 0.584271 0.292135 0.956377i \(-0.405634\pi\)
0.292135 + 0.956377i \(0.405634\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) −3.69764 −0.179785
\(424\) 0 0
\(425\) − 9.89020i − 0.479745i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.552844 −0.0266915
\(430\) 0 0
\(431\) − 19.1460i − 0.922230i −0.887340 0.461115i \(-0.847450\pi\)
0.887340 0.461115i \(-0.152550\pi\)
\(432\) 0 0
\(433\) − 1.66205i − 0.0798730i −0.999202 0.0399365i \(-0.987284\pi\)
0.999202 0.0399365i \(-0.0127156\pi\)
\(434\) 0 0
\(435\) 0.775614i 0.0371878i
\(436\) 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 \(-0.856555\pi\)
0.900166 0.435548i \(-0.143445\pi\)
\(444\) 0 0
\(445\) −8.95695 −0.424600
\(446\) 0 0
\(447\) 10.8017 0.510901
\(448\) 0 0
\(449\) 18.9899 0.896187 0.448093 0.893987i \(-0.352103\pi\)
0.448093 + 0.893987i \(0.352103\pi\)
\(450\) 0 0
\(451\) −1.09292 −0.0514636
\(452\) 0 0
\(453\) 14.1587i 0.665236i
\(454\) 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
\(459\) − 2.13246i − 0.0995349i
\(460\) 0 0
\(461\) − 36.8939i − 1.71832i −0.511708 0.859160i \(-0.670987\pi\)
0.511708 0.859160i \(-0.329013\pi\)
\(462\) 0 0
\(463\) 35.2394i 1.63771i 0.573997 + 0.818857i \(0.305392\pi\)
−0.573997 + 0.818857i \(0.694608\pi\)
\(464\) 0 0
\(465\) − 2.01850i − 0.0936058i
\(466\) 0 0
\(467\) −35.8246 −1.65776 −0.828882 0.559423i \(-0.811023\pi\)
−0.828882 + 0.559423i \(0.811023\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.09264i 0.0503462i
\(472\) 0 0
\(473\) 0.996826 0.0458341
\(474\) 0 0
\(475\) 4.42304 0.202943
\(476\) 0 0
\(477\) 13.8806 0.635547
\(478\) 0 0
\(479\) −6.10594 −0.278988 −0.139494 0.990223i \(-0.544548\pi\)
−0.139494 + 0.990223i \(0.544548\pi\)
\(480\) 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
\(489\) − 18.0857i − 0.817863i
\(490\) 0 0
\(491\) − 31.2897i − 1.41209i −0.708169 0.706043i \(-0.750479\pi\)
0.708169 0.706043i \(-0.249521\pi\)
\(492\) 0 0
\(493\) 2.74869i 0.123795i
\(494\) 0 0
\(495\) 0.0984649 0.00442567
\(496\) 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
\(501\) −19.0196 −0.849731
\(502\) 0 0
\(503\) −0.728285 −0.0324726 −0.0162363 0.999868i \(-0.505168\pi\)
−0.0162363 + 0.999868i \(0.505168\pi\)
\(504\) 0 0
\(505\) −5.12462 −0.228043
\(506\) 0 0
\(507\) 1.58579 0.0704272
\(508\) 0 0
\(509\) − 31.6424i − 1.40253i −0.712903 0.701263i \(-0.752620\pi\)
0.712903 0.701263i \(-0.247380\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.953669 0.0421055
\(514\) 0 0
\(515\) − 10.0855i − 0.444421i
\(516\) 0 0
\(517\) − 0.605068i − 0.0266108i
\(518\) 0 0
\(519\) − 8.25540i − 0.362372i
\(520\) 0 0
\(521\) 19.0336i 0.833878i 0.908935 + 0.416939i \(0.136897\pi\)
−0.908935 + 0.416939i \(0.863103\pi\)
\(522\) 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
\(531\) 8.05188 0.349422
\(532\) 0 0
\(533\) −22.5648 −0.977391
\(534\) 0 0
\(535\) 2.64602 0.114397
\(536\) 0 0
\(537\) − 5.49647i − 0.237190i
\(538\) 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
\(543\) 11.8519i 0.508615i
\(544\) 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
\(549\) 0.181465i 0.00774472i
\(550\) 0 0
\(551\) −1.22925 −0.0523679
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 3.10717i − 0.131892i
\(556\) 0 0
\(557\) −35.9877 −1.52485 −0.762424 0.647078i \(-0.775991\pi\)
−0.762424 + 0.647078i \(0.775991\pi\)
\(558\) 0 0
\(559\) 20.5808 0.870476
\(560\) 0 0
\(561\) 0.348948 0.0147326
\(562\) 0 0
\(563\) 25.3463 1.06822 0.534109 0.845415i \(-0.320647\pi\)
0.534109 + 0.845415i \(0.320647\pi\)
\(564\) 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 \(-0.620581\pi\)
−0.369821 + 0.929103i \(0.620581\pi\)
\(570\) 0 0
\(571\) − 29.9226i − 1.25222i −0.779733 0.626112i \(-0.784645\pi\)
0.779733 0.626112i \(-0.215355\pi\)
\(572\) 0 0
\(573\) 23.1483i 0.967034i
\(574\) 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
\(579\) −0.512058 −0.0212804
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.27136i 0.0940702i
\(584\) 0 0
\(585\) 2.03294 0.0840518
\(586\) 0 0
\(587\) −21.1156 −0.871535 −0.435767 0.900059i \(-0.643523\pi\)
−0.435767 + 0.900059i \(0.643523\pi\)
\(588\) 0 0
\(589\) 3.19908 0.131816
\(590\) 0 0
\(591\) −12.7934 −0.526252
\(592\) 0 0
\(593\) 35.8055i 1.47035i 0.677875 + 0.735177i \(0.262901\pi\)
−0.677875 + 0.735177i \(0.737099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15.8072 −0.646945
\(598\) 0 0
\(599\) − 10.0909i − 0.412302i −0.978520 0.206151i \(-0.933906\pi\)
0.978520 0.206151i \(-0.0660938\pi\)
\(600\) 0 0
\(601\) − 31.0913i − 1.26824i −0.773234 0.634120i \(-0.781362\pi\)
0.773234 0.634120i \(-0.218638\pi\)
\(602\) 0 0
\(603\) 9.09299i 0.370295i
\(604\) 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
\(615\) 4.01893 0.162059
\(616\) 0 0
\(617\) 14.1442 0.569423 0.284712 0.958613i \(-0.408102\pi\)
0.284712 + 0.958613i \(0.408102\pi\)
\(618\) 0 0
\(619\) −42.5162 −1.70887 −0.854435 0.519559i \(-0.826096\pi\)
−0.854435 + 0.519559i \(0.826096\pi\)
\(620\) 0 0
\(621\) 6.99598i 0.280739i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.6999 0.787996
\(626\) 0 0
\(627\) 0.156055i 0.00623223i
\(628\) 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
\(633\) 0.191712i 0.00761986i
\(634\) 0 0
\(635\) 3.14480 0.124797
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.36066i 0.132946i
\(640\) 0 0
\(641\) 0.471636 0.0186285 0.00931425 0.999957i \(-0.497035\pi\)
0.00931425 + 0.999957i \(0.497035\pi\)
\(642\) 0 0
\(643\) 1.10040 0.0433955 0.0216977 0.999765i \(-0.493093\pi\)
0.0216977 + 0.999765i \(0.493093\pi\)
\(644\) 0 0
\(645\) −3.66557 −0.144332
\(646\) 0 0
\(647\) −31.7314 −1.24749 −0.623746 0.781627i \(-0.714390\pi\)
−0.623746 + 0.781627i \(0.714390\pi\)
\(648\) 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 \(-0.556248\pi\)
−0.175790 + 0.984428i \(0.556248\pi\)
\(654\) 0 0
\(655\) 13.5581i 0.529759i
\(656\) 0 0
\(657\) − 12.1442i − 0.473791i
\(658\) 0 0
\(659\) 26.0679i 1.01546i 0.861516 + 0.507731i \(0.169516\pi\)
−0.861516 + 0.507731i \(0.830484\pi\)
\(660\) 0 0
\(661\) 1.36376i 0.0530441i 0.999648 + 0.0265221i \(0.00844323\pi\)
−0.999648 + 0.0265221i \(0.991557\pi\)
\(662\) 0 0
\(663\) 7.20452 0.279800
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.01763i − 0.349164i
\(668\) 0 0
\(669\) 10.0111 0.387051
\(670\) 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
\(675\) 4.63792 0.178514
\(676\) 0 0
\(677\) 7.95083i 0.305575i 0.988259 + 0.152788i \(0.0488250\pi\)
−0.988259 + 0.152788i \(0.951175\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6.13633 0.235145
\(682\) 0 0
\(683\) − 18.0286i − 0.689846i −0.938631 0.344923i \(-0.887905\pi\)
0.938631 0.344923i \(-0.112095\pi\)
\(684\) 0 0
\(685\) − 8.78021i − 0.335474i
\(686\) 0 0
\(687\) 9.44500i 0.360349i
\(688\) 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
\(699\) −22.2217 −0.840501
\(700\) 0 0
\(701\) −11.8785 −0.448646 −0.224323 0.974515i \(-0.572017\pi\)
−0.224323 + 0.974515i \(0.572017\pi\)
\(702\) 0 0
\(703\) 4.92449 0.185731
\(704\) 0 0
\(705\) 2.22498i 0.0837977i
\(706\) 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
\(711\) 6.98840i 0.262086i
\(712\) 0 0
\(713\) 23.4680i 0.878883i
\(714\) 0 0
\(715\) 0.332663i 0.0124409i
\(716\) 0 0
\(717\) 20.7962i 0.776650i
\(718\) 0 0
\(719\) 0.604720 0.0225523 0.0112761 0.999936i \(-0.496411\pi\)
0.0112761 + 0.999936i \(0.496411\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 13.0554i − 0.485535i
\(724\) 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
\(729\) 1.00000 0.0370370
\(730\) 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
\(741\) 3.22196i 0.118362i
\(742\) 0 0
\(743\) − 13.1996i − 0.484247i −0.970245 0.242123i \(-0.922156\pi\)
0.970245 0.242123i \(-0.0778439\pi\)
\(744\) 0 0
\(745\) − 6.49968i − 0.238130i
\(746\) 0 0
\(747\) −12.9788 −0.474868
\(748\) 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
\(753\) −17.8975 −0.652220
\(754\) 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
\(759\) −1.14480 −0.0415535
\(760\) 0 0
\(761\) − 4.77524i − 0.173102i −0.996247 0.0865512i \(-0.972415\pi\)
0.996247 0.0865512i \(-0.0275846\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.28317 −0.0463931
\(766\) 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
\(771\) − 6.67896i − 0.240537i
\(772\) 0 0
\(773\) − 36.9175i − 1.32783i −0.747807 0.663916i \(-0.768893\pi\)
0.747807 0.663916i \(-0.231107\pi\)
\(774\) 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
\(783\) −1.28897 −0.0460641
\(784\) 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
\(789\) − 24.4164i − 0.869247i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.613077 −0.0217710
\(794\) 0 0
\(795\) − 8.35236i − 0.296228i
\(796\) 0 0
\(797\) − 36.9752i − 1.30973i −0.755746 0.654865i \(-0.772725\pi\)
0.755746 0.654865i \(-0.227275\pi\)
\(798\) 0 0
\(799\) 7.88509i 0.278954i
\(800\) 0 0
\(801\) − 14.8853i − 0.525947i
\(802\) 0 0
\(803\) 1.98723 0.0701279
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.14593i − 0.0403386i
\(808\) 0 0
\(809\) 5.34100 0.187780 0.0938898 0.995583i \(-0.470070\pi\)
0.0938898 + 0.995583i \(0.470070\pi\)
\(810\) 0 0
\(811\) −37.4180 −1.31392 −0.656961 0.753924i \(-0.728159\pi\)
−0.656961 + 0.753924i \(0.728159\pi\)
\(812\) 0 0
\(813\) 5.65157 0.198209
\(814\) 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 \(-0.384467\pi\)
0.355041 + 0.934851i \(0.384467\pi\)
\(822\) 0 0
\(823\) − 46.3367i − 1.61520i −0.589734 0.807598i \(-0.700767\pi\)
0.589734 0.807598i \(-0.299233\pi\)
\(824\) 0 0
\(825\) 0.758932i 0.0264226i
\(826\) 0 0
\(827\) 2.20358i 0.0766260i 0.999266 + 0.0383130i \(0.0121984\pi\)
−0.999266 + 0.0383130i \(0.987802\pi\)
\(828\) 0 0
\(829\) 17.2639i 0.599599i 0.954002 + 0.299800i \(0.0969199\pi\)
−0.954002 + 0.299800i \(0.903080\pi\)
\(830\) 0 0
\(831\) −19.2758 −0.668671
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 11.4446i 0.396058i
\(836\) 0 0
\(837\) 3.35449 0.115948
\(838\) 0 0
\(839\) 17.4781 0.603410 0.301705 0.953401i \(-0.402444\pi\)
0.301705 + 0.953401i \(0.402444\pi\)
\(840\) 0 0
\(841\) −27.3386 −0.942709
\(842\) 0 0
\(843\) 8.80369 0.303215
\(844\) 0 0
\(845\) − 0.954216i − 0.0328260i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 26.1451 0.897296
\(850\) 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
\(855\) − 0.573852i − 0.0196253i
\(856\) 0 0
\(857\) − 23.4865i − 0.802283i −0.916016 0.401142i \(-0.868614\pi\)
0.916016 0.401142i \(-0.131386\pi\)
\(858\) 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 \(0.0346122\pi\)
−0.994094 + 0.108523i \(0.965388\pi\)
\(864\) 0 0
\(865\) −4.96753 −0.168901
\(866\) 0 0
\(867\) 12.4526 0.422912
\(868\) 0 0
\(869\) −1.14356 −0.0387925
\(870\) 0 0
\(871\) −30.7206 −1.04093
\(872\) 0 0
\(873\) 9.08274i 0.307404i
\(874\) 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
\(879\) − 18.2640i − 0.616030i
\(880\) 0 0
\(881\) 45.0545i 1.51793i 0.651134 + 0.758963i \(0.274294\pi\)
−0.651134 + 0.758963i \(0.725706\pi\)
\(882\) 0 0
\(883\) 21.8481i 0.735246i 0.929975 + 0.367623i \(0.119828\pi\)
−0.929975 + 0.367623i \(0.880172\pi\)
\(884\) 0 0
\(885\) − 4.84506i − 0.162865i
\(886\) 0 0
\(887\) 32.2604 1.08320 0.541599 0.840637i \(-0.317819\pi\)
0.541599 + 0.840637i \(0.317819\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.163636i 0.00548202i
\(892\) 0 0
\(893\) −3.52632 −0.118004
\(894\) 0 0
\(895\) −3.30739 −0.110554
\(896\) 0 0
\(897\) −23.6359 −0.789179
\(898\) 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
\(909\) − 8.51647i − 0.282473i
\(910\) 0 0
\(911\) 0.475746i 0.0157622i 0.999969 + 0.00788108i \(0.00250865\pi\)
−0.999969 + 0.00788108i \(0.997491\pi\)
\(912\) 0 0
\(913\) − 2.12380i − 0.0702874i
\(914\) 0 0
\(915\) 0.109193 0.00360980
\(916\) 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
\(921\) 21.4472 0.706708
\(922\) 0 0
\(923\) −11.3540 −0.373721
\(924\) 0 0
\(925\) 23.9490 0.787437
\(926\) 0 0
\(927\) 16.7609 0.550499
\(928\) 0 0
\(929\) − 12.7798i − 0.419293i −0.977777 0.209646i \(-0.932769\pi\)
0.977777 0.209646i \(-0.0672313\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −14.4800 −0.474054
\(934\) 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
\(939\) 1.48483i 0.0484556i
\(940\) 0 0
\(941\) − 58.7586i − 1.91548i −0.287640 0.957739i \(-0.592871\pi\)
0.287640 0.957739i \(-0.407129\pi\)
\(942\) 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 \(-0.930110\pi\)
0.975992 0.217805i \(-0.0698898\pi\)
\(948\) 0 0
\(949\) 41.0291 1.33186
\(950\) 0 0
\(951\) −11.4908 −0.372616
\(952\) 0 0
\(953\) −1.89697 −0.0614490 −0.0307245 0.999528i \(-0.509781\pi\)
−0.0307245 + 0.999528i \(0.509781\pi\)
\(954\) 0 0
\(955\) 13.9290 0.450733
\(956\) 0 0
\(957\) − 0.210922i − 0.00681815i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.7474 −0.637012
\(962\) 0 0
\(963\) 4.39735i 0.141703i
\(964\) 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
\(969\) − 2.03366i − 0.0653307i
\(970\) 0 0
\(971\) 37.7418 1.21119 0.605596 0.795772i \(-0.292935\pi\)
0.605596 + 0.795772i \(0.292935\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 15.6692i 0.501815i
\(976\) 0 0
\(977\) 24.6686 0.789217 0.394609 0.918849i \(-0.370880\pi\)
0.394609 + 0.918849i \(0.370880\pi\)
\(978\) 0 0
\(979\) 2.43578 0.0778477
\(980\) 0 0
\(981\) 11.5453 0.368612
\(982\) 0 0
\(983\) 37.7500 1.20404 0.602019 0.798481i \(-0.294363\pi\)
0.602019 + 0.798481i \(0.294363\pi\)
\(984\) 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
\(993\) − 21.0967i − 0.669484i
\(994\) 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
\(999\) 5.16373 0.163373
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.1567.4 yes 8
3.2 odd 2 7056.2.b.w.1567.5 8
4.3 odd 2 2352.2.b.k.1567.4 8
7.2 even 3 2352.2.bl.q.31.2 8
7.3 odd 6 2352.2.bl.r.607.2 8
7.4 even 3 2352.2.bl.o.607.3 8
7.5 odd 6 2352.2.bl.t.31.3 8
7.6 odd 2 2352.2.b.k.1567.5 yes 8
12.11 even 2 7056.2.b.x.1567.5 8
21.20 even 2 7056.2.b.x.1567.4 8
28.3 even 6 2352.2.bl.q.607.2 8
28.11 odd 6 2352.2.bl.t.607.3 8
28.19 even 6 2352.2.bl.o.31.3 8
28.23 odd 6 2352.2.bl.r.31.2 8
28.27 even 2 inner 2352.2.b.l.1567.5 yes 8
84.83 odd 2 7056.2.b.w.1567.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.4 8 4.3 odd 2
2352.2.b.k.1567.5 yes 8 7.6 odd 2
2352.2.b.l.1567.4 yes 8 1.1 even 1 trivial
2352.2.b.l.1567.5 yes 8 28.27 even 2 inner
2352.2.bl.o.31.3 8 28.19 even 6
2352.2.bl.o.607.3 8 7.4 even 3
2352.2.bl.q.31.2 8 7.2 even 3
2352.2.bl.q.607.2 8 28.3 even 6
2352.2.bl.r.31.2 8 28.23 odd 6
2352.2.bl.r.607.2 8 7.3 odd 6
2352.2.bl.t.31.3 8 7.5 odd 6
2352.2.bl.t.607.3 8 28.11 odd 6
7056.2.b.w.1567.4 8 84.83 odd 2
7056.2.b.w.1567.5 8 3.2 odd 2
7056.2.b.x.1567.4 8 21.20 even 2
7056.2.b.x.1567.5 8 12.11 even 2