Properties

Label 4608.2.a.z.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{5} +1.41421 q^{7} +O(q^{10})\) \(q-2.44949 q^{5} +1.41421 q^{7} +3.46410 q^{11} -4.89898 q^{13} +4.00000 q^{17} +6.92820 q^{19} +5.65685 q^{23} +1.00000 q^{25} -2.44949 q^{29} -1.41421 q^{31} -3.46410 q^{35} -4.89898 q^{37} +4.00000 q^{41} -6.92820 q^{43} -5.65685 q^{47} -5.00000 q^{49} +7.34847 q^{53} -8.48528 q^{55} -13.8564 q^{59} -4.89898 q^{61} +12.0000 q^{65} +11.3137 q^{71} -4.00000 q^{73} +4.89898 q^{77} +7.07107 q^{79} +10.3923 q^{83} -9.79796 q^{85} +16.0000 q^{89} -6.92820 q^{91} -16.9706 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{17} + 4 q^{25} + 16 q^{41} - 20 q^{49} + 48 q^{65} - 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.44949 −1.09545 −0.547723 0.836660i \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 −0.454859 −0.227429 0.973795i \(-0.573032\pi\)
−0.227429 + 0.973795i \(0.573032\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) −4.89898 −0.805387 −0.402694 0.915335i \(-0.631926\pi\)
−0.402694 + 0.915335i \(0.631926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.34847 1.00939 0.504695 0.863298i \(-0.331605\pi\)
0.504695 + 0.863298i \(0.331605\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) −4.89898 −0.627250 −0.313625 0.949547i \(-0.601543\pi\)
−0.313625 + 0.949547i \(0.601543\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.89898 0.558291
\(78\) 0 0
\(79\) 7.07107 0.795557 0.397779 0.917481i \(-0.369781\pi\)
0.397779 + 0.917481i \(0.369781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923 1.14070 0.570352 0.821401i \(-0.306807\pi\)
0.570352 + 0.821401i \(0.306807\pi\)
\(84\) 0 0
\(85\) −9.79796 −1.06274
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) −6.92820 −0.726273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.9706 −1.74114
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.44949 0.243733 0.121867 0.992546i \(-0.461112\pi\)
0.121867 + 0.992546i \(0.461112\pi\)
\(102\) 0 0
\(103\) 18.3848 1.81151 0.905753 0.423806i \(-0.139306\pi\)
0.905753 + 0.423806i \(0.139306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.8564 1.33955 0.669775 0.742564i \(-0.266391\pi\)
0.669775 + 0.742564i \(0.266391\pi\)
\(108\) 0 0
\(109\) 14.6969 1.40771 0.703856 0.710343i \(-0.251460\pi\)
0.703856 + 0.710343i \(0.251460\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −13.8564 −1.29212
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796 0.876356
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 9.79796 0.849591
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.0000 1.70872 0.854358 0.519685i \(-0.173951\pi\)
0.854358 + 0.519685i \(0.173951\pi\)
\(138\) 0 0
\(139\) 13.8564 1.17529 0.587643 0.809121i \(-0.300056\pi\)
0.587643 + 0.809121i \(0.300056\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −16.9706 −1.41915
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.0454 −1.80603 −0.903015 0.429609i \(-0.858651\pi\)
−0.903015 + 0.429609i \(0.858651\pi\)
\(150\) 0 0
\(151\) −18.3848 −1.49613 −0.748066 0.663624i \(-0.769017\pi\)
−0.748066 + 0.663624i \(0.769017\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.46410 0.278243
\(156\) 0 0
\(157\) 4.89898 0.390981 0.195491 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 6.92820 0.542659 0.271329 0.962487i \(-0.412537\pi\)
0.271329 + 0.962487i \(0.412537\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.9706 1.31322 0.656611 0.754230i \(-0.271989\pi\)
0.656611 + 0.754230i \(0.271989\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −17.1464 −1.30362 −0.651809 0.758383i \(-0.725990\pi\)
−0.651809 + 0.758383i \(0.725990\pi\)
\(174\) 0 0
\(175\) 1.41421 0.106904
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.8564 −1.03568 −0.517838 0.855479i \(-0.673263\pi\)
−0.517838 + 0.855479i \(0.673263\pi\)
\(180\) 0 0
\(181\) 24.4949 1.82069 0.910346 0.413849i \(-0.135816\pi\)
0.910346 + 0.413849i \(0.135816\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 13.8564 1.01328
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.34847 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(198\) 0 0
\(199\) 18.3848 1.30326 0.651631 0.758536i \(-0.274085\pi\)
0.651631 + 0.758536i \(0.274085\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.46410 −0.243132
\(204\) 0 0
\(205\) −9.79796 −0.684319
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −13.8564 −0.953914 −0.476957 0.878927i \(-0.658260\pi\)
−0.476957 + 0.878927i \(0.658260\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.9706 1.15738
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −19.5959 −1.31816
\(222\) 0 0
\(223\) −7.07107 −0.473514 −0.236757 0.971569i \(-0.576084\pi\)
−0.236757 + 0.971569i \(0.576084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.3923 −0.689761 −0.344881 0.938647i \(-0.612081\pi\)
−0.344881 + 0.938647i \(0.612081\pi\)
\(228\) 0 0
\(229\) 4.89898 0.323734 0.161867 0.986813i \(-0.448248\pi\)
0.161867 + 0.986813i \(0.448248\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 13.8564 0.903892
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.2474 0.782461
\(246\) 0 0
\(247\) −33.9411 −2.15962
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) 0 0
\(253\) 19.5959 1.23198
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) −6.92820 −0.430498
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.2474 0.746740 0.373370 0.927682i \(-0.378202\pi\)
0.373370 + 0.927682i \(0.378202\pi\)
\(270\) 0 0
\(271\) 26.8701 1.63224 0.816120 0.577883i \(-0.196121\pi\)
0.816120 + 0.577883i \(0.196121\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410 0.208893
\(276\) 0 0
\(277\) 4.89898 0.294351 0.147176 0.989110i \(-0.452982\pi\)
0.147176 + 0.989110i \(0.452982\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 13.8564 0.823678 0.411839 0.911257i \(-0.364887\pi\)
0.411839 + 0.911257i \(0.364887\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.34847 0.429302 0.214651 0.976691i \(-0.431139\pi\)
0.214651 + 0.976691i \(0.431139\pi\)
\(294\) 0 0
\(295\) 33.9411 1.97613
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −27.7128 −1.60267
\(300\) 0 0
\(301\) −9.79796 −0.564745
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 27.7128 1.58165 0.790827 0.612040i \(-0.209651\pi\)
0.790827 + 0.612040i \(0.209651\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 33.9411 1.92462 0.962312 0.271947i \(-0.0876674\pi\)
0.962312 + 0.271947i \(0.0876674\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.8434 1.78850 0.894251 0.447566i \(-0.147709\pi\)
0.894251 + 0.447566i \(0.147709\pi\)
\(318\) 0 0
\(319\) −8.48528 −0.475085
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.7128 1.54198
\(324\) 0 0
\(325\) −4.89898 −0.271746
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −27.7128 −1.52323 −0.761617 0.648027i \(-0.775594\pi\)
−0.761617 + 0.648027i \(0.775594\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.89898 −0.265295
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.2487 1.30174 0.650870 0.759190i \(-0.274404\pi\)
0.650870 + 0.759190i \(0.274404\pi\)
\(348\) 0 0
\(349\) −4.89898 −0.262236 −0.131118 0.991367i \(-0.541857\pi\)
−0.131118 + 0.991367i \(0.541857\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.0000 1.70319 0.851594 0.524202i \(-0.175636\pi\)
0.851594 + 0.524202i \(0.175636\pi\)
\(354\) 0 0
\(355\) −27.7128 −1.47084
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.79796 0.512849
\(366\) 0 0
\(367\) −1.41421 −0.0738213 −0.0369107 0.999319i \(-0.511752\pi\)
−0.0369107 + 0.999319i \(0.511752\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.3923 0.539542
\(372\) 0 0
\(373\) −34.2929 −1.77562 −0.887808 0.460213i \(-0.847773\pi\)
−0.887808 + 0.460213i \(0.847773\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −34.6410 −1.77939 −0.889695 0.456556i \(-0.849083\pi\)
−0.889695 + 0.456556i \(0.849083\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.9706 −0.867155 −0.433578 0.901116i \(-0.642749\pi\)
−0.433578 + 0.901116i \(0.642749\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.9444 −1.36613 −0.683067 0.730355i \(-0.739354\pi\)
−0.683067 + 0.730355i \(0.739354\pi\)
\(390\) 0 0
\(391\) 22.6274 1.14432
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −17.3205 −0.871489
\(396\) 0 0
\(397\) 24.4949 1.22936 0.614682 0.788775i \(-0.289284\pi\)
0.614682 + 0.788775i \(0.289284\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) 6.92820 0.345118
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.9706 −0.841200
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −19.5959 −0.964252
\(414\) 0 0
\(415\) −25.4558 −1.24958
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3205 0.846162 0.423081 0.906092i \(-0.360949\pi\)
0.423081 + 0.906092i \(0.360949\pi\)
\(420\) 0 0
\(421\) −14.6969 −0.716285 −0.358142 0.933667i \(-0.616590\pi\)
−0.358142 + 0.933667i \(0.616590\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −6.92820 −0.335279
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685 0.272481 0.136241 0.990676i \(-0.456498\pi\)
0.136241 + 0.990676i \(0.456498\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 39.1918 1.87480
\(438\) 0 0
\(439\) −26.8701 −1.28244 −0.641219 0.767358i \(-0.721571\pi\)
−0.641219 + 0.767358i \(0.721571\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.2487 1.15209 0.576046 0.817418i \(-0.304595\pi\)
0.576046 + 0.817418i \(0.304595\pi\)
\(444\) 0 0
\(445\) −39.1918 −1.85787
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) 0 0
\(451\) 13.8564 0.652473
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.9706 0.795592
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.44949 0.114084 0.0570421 0.998372i \(-0.481833\pi\)
0.0570421 + 0.998372i \(0.481833\pi\)
\(462\) 0 0
\(463\) 1.41421 0.0657241 0.0328620 0.999460i \(-0.489538\pi\)
0.0328620 + 0.999460i \(0.489538\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −38.1051 −1.76329 −0.881647 0.471909i \(-0.843565\pi\)
−0.881647 + 0.471909i \(0.843565\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 6.92820 0.317888
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.6969 −0.667354
\(486\) 0 0
\(487\) 24.0416 1.08943 0.544715 0.838621i \(-0.316638\pi\)
0.544715 + 0.838621i \(0.316638\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −9.79796 −0.441278
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 13.8564 0.620298 0.310149 0.950688i \(-0.399621\pi\)
0.310149 + 0.950688i \(0.399621\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −22.6274 −1.00891 −0.504453 0.863439i \(-0.668306\pi\)
−0.504453 + 0.863439i \(0.668306\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.0454 −0.977146 −0.488573 0.872523i \(-0.662482\pi\)
−0.488573 + 0.872523i \(0.662482\pi\)
\(510\) 0 0
\(511\) −5.65685 −0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −45.0333 −1.98441
\(516\) 0 0
\(517\) −19.5959 −0.861827
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) −20.7846 −0.908848 −0.454424 0.890786i \(-0.650155\pi\)
−0.454424 + 0.890786i \(0.650155\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.65685 −0.246416
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −19.5959 −0.848793
\(534\) 0 0
\(535\) −33.9411 −1.46740
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.3205 −0.746047
\(540\) 0 0
\(541\) −14.6969 −0.631871 −0.315935 0.948781i \(-0.602318\pi\)
−0.315935 + 0.948781i \(0.602318\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) 34.6410 1.48114 0.740571 0.671978i \(-0.234555\pi\)
0.740571 + 0.671978i \(0.234555\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −16.9706 −0.722970
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.8434 −1.34925 −0.674623 0.738162i \(-0.735694\pi\)
−0.674623 + 0.738162i \(0.735694\pi\)
\(558\) 0 0
\(559\) 33.9411 1.43556
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) 0 0
\(565\) −19.5959 −0.824406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.65685 0.235907
\(576\) 0 0
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.6969 0.609732
\(582\) 0 0
\(583\) 25.4558 1.05427
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −9.79796 −0.403718
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8.00000 −0.328521 −0.164260 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) 0 0
\(595\) −13.8564 −0.568057
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.44949 −0.0995859
\(606\) 0 0
\(607\) 24.0416 0.975820 0.487910 0.872894i \(-0.337759\pi\)
0.487910 + 0.872894i \(0.337759\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.7128 1.12114
\(612\) 0 0
\(613\) 44.0908 1.78081 0.890406 0.455168i \(-0.150421\pi\)
0.890406 + 0.455168i \(0.150421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16.0000 −0.644136 −0.322068 0.946717i \(-0.604378\pi\)
−0.322068 + 0.946717i \(0.604378\pi\)
\(618\) 0 0
\(619\) 27.7128 1.11387 0.556936 0.830555i \(-0.311977\pi\)
0.556936 + 0.830555i \(0.311977\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.6274 0.906548
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −19.5959 −0.781340
\(630\) 0 0
\(631\) −9.89949 −0.394093 −0.197046 0.980394i \(-0.563135\pi\)
−0.197046 + 0.980394i \(0.563135\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.2487 0.962281
\(636\) 0 0
\(637\) 24.4949 0.970523
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.0000 −1.10593 −0.552967 0.833203i \(-0.686504\pi\)
−0.552967 + 0.833203i \(0.686504\pi\)
\(642\) 0 0
\(643\) 20.7846 0.819665 0.409832 0.912161i \(-0.365587\pi\)
0.409832 + 0.912161i \(0.365587\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.2843 −1.11197 −0.555985 0.831193i \(-0.687659\pi\)
−0.555985 + 0.831193i \(0.687659\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.0454 0.862703 0.431352 0.902184i \(-0.358037\pi\)
0.431352 + 0.902184i \(0.358037\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.7128 1.07954 0.539769 0.841813i \(-0.318512\pi\)
0.539769 + 0.841813i \(0.318512\pi\)
\(660\) 0 0
\(661\) 44.0908 1.71493 0.857467 0.514539i \(-0.172037\pi\)
0.857467 + 0.514539i \(0.172037\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) −13.8564 −0.536522
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16.9706 −0.655141
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.1464 0.658991 0.329495 0.944157i \(-0.393121\pi\)
0.329495 + 0.944157i \(0.393121\pi\)
\(678\) 0 0
\(679\) 8.48528 0.325635
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.46410 0.132550 0.0662751 0.997801i \(-0.478889\pi\)
0.0662751 + 0.997801i \(0.478889\pi\)
\(684\) 0 0
\(685\) −48.9898 −1.87180
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −20.7846 −0.790684 −0.395342 0.918534i \(-0.629374\pi\)
−0.395342 + 0.918534i \(0.629374\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.9411 −1.28746
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.8434 1.20271 0.601354 0.798983i \(-0.294628\pi\)
0.601354 + 0.798983i \(0.294628\pi\)
\(702\) 0 0
\(703\) −33.9411 −1.28011
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.46410 0.130281
\(708\) 0 0
\(709\) −14.6969 −0.551955 −0.275978 0.961164i \(-0.589002\pi\)
−0.275978 + 0.961164i \(0.589002\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 41.5692 1.55460
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.6274 0.843860 0.421930 0.906628i \(-0.361353\pi\)
0.421930 + 0.906628i \(0.361353\pi\)
\(720\) 0 0
\(721\) 26.0000 0.968291
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.44949 −0.0909718
\(726\) 0 0
\(727\) −7.07107 −0.262251 −0.131126 0.991366i \(-0.541859\pi\)
−0.131126 + 0.991366i \(0.541859\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −27.7128 −1.02500
\(732\) 0 0
\(733\) 24.4949 0.904740 0.452370 0.891830i \(-0.350579\pi\)
0.452370 + 0.891830i \(0.350579\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.9706 −0.622590 −0.311295 0.950313i \(-0.600763\pi\)
−0.311295 + 0.950313i \(0.600763\pi\)
\(744\) 0 0
\(745\) 54.0000 1.97841
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.5959 0.716019
\(750\) 0 0
\(751\) −15.5563 −0.567659 −0.283830 0.958875i \(-0.591605\pi\)
−0.283830 + 0.958875i \(0.591605\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 45.0333 1.63893
\(756\) 0 0
\(757\) 4.89898 0.178056 0.0890282 0.996029i \(-0.471624\pi\)
0.0890282 + 0.996029i \(0.471624\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −52.0000 −1.88500 −0.942499 0.334208i \(-0.891531\pi\)
−0.942499 + 0.334208i \(0.891531\pi\)
\(762\) 0 0
\(763\) 20.7846 0.752453
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 67.8823 2.45109
\(768\) 0 0
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.44949 0.0881020 0.0440510 0.999029i \(-0.485974\pi\)
0.0440510 + 0.999029i \(0.485974\pi\)
\(774\) 0 0
\(775\) −1.41421 −0.0508001
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.7128 0.992915
\(780\) 0 0
\(781\) 39.1918 1.40239
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) −20.7846 −0.740891 −0.370446 0.928854i \(-0.620795\pi\)
−0.370446 + 0.928854i \(0.620795\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.3137 0.402269
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.5403 1.64854 0.824271 0.566195i \(-0.191585\pi\)
0.824271 + 0.566195i \(0.191585\pi\)
\(798\) 0 0
\(799\) −22.6274 −0.800500
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −13.8564 −0.488982
\(804\) 0 0
\(805\) −19.5959 −0.690665
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −20.7846 −0.729846 −0.364923 0.931038i \(-0.618905\pi\)
−0.364923 + 0.931038i \(0.618905\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.9706 −0.594453
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.44949 0.0854878 0.0427439 0.999086i \(-0.486390\pi\)
0.0427439 + 0.999086i \(0.486390\pi\)
\(822\) 0 0
\(823\) 1.41421 0.0492964 0.0246482 0.999696i \(-0.492153\pi\)
0.0246482 + 0.999696i \(0.492153\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.5692 −1.44550 −0.722752 0.691108i \(-0.757123\pi\)
−0.722752 + 0.691108i \(0.757123\pi\)
\(828\) 0 0
\(829\) 34.2929 1.19104 0.595520 0.803340i \(-0.296946\pi\)
0.595520 + 0.803340i \(0.296946\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −20.0000 −0.692959
\(834\) 0 0
\(835\) −41.5692 −1.43856
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −45.2548 −1.56237 −0.781185 0.624299i \(-0.785385\pi\)
−0.781185 + 0.624299i \(0.785385\pi\)
\(840\) 0 0
\(841\) −23.0000 −0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.9444 −0.926915
\(846\) 0 0
\(847\) 1.41421 0.0485930
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.7128 −0.949983
\(852\) 0 0
\(853\) 4.89898 0.167738 0.0838689 0.996477i \(-0.473272\pi\)
0.0838689 + 0.996477i \(0.473272\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.0000 0.956462 0.478231 0.878234i \(-0.341278\pi\)
0.478231 + 0.878234i \(0.341278\pi\)
\(858\) 0 0
\(859\) −6.92820 −0.236387 −0.118194 0.992991i \(-0.537710\pi\)
−0.118194 + 0.992991i \(0.537710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.9706 0.577685 0.288842 0.957377i \(-0.406730\pi\)
0.288842 + 0.957377i \(0.406730\pi\)
\(864\) 0 0
\(865\) 42.0000 1.42804
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.4949 0.830932
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 13.8564 0.468432
\(876\) 0 0
\(877\) 34.2929 1.15799 0.578994 0.815332i \(-0.303446\pi\)
0.578994 + 0.815332i \(0.303446\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) −34.6410 −1.16576 −0.582882 0.812557i \(-0.698075\pi\)
−0.582882 + 0.812557i \(0.698075\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −39.1918 −1.31150
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.46410 0.115534
\(900\) 0 0
\(901\) 29.3939 0.979252
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −60.0000 −1.99447
\(906\) 0 0
\(907\) 20.7846 0.690142 0.345071 0.938577i \(-0.387855\pi\)
0.345071 + 0.938577i \(0.387855\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 50.9117 1.68678 0.843390 0.537302i \(-0.180557\pi\)
0.843390 + 0.537302i \(0.180557\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.41421 0.0466506 0.0233253 0.999728i \(-0.492575\pi\)
0.0233253 + 0.999728i \(0.492575\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −55.4256 −1.82436
\(924\) 0 0
\(925\) −4.89898 −0.161077
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) −34.6410 −1.13531
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −33.9411 −1.10999
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.1464 −0.558958 −0.279479 0.960152i \(-0.590162\pi\)
−0.279479 + 0.960152i \(0.590162\pi\)
\(942\) 0 0
\(943\) 22.6274 0.736850
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.8564 0.450273 0.225136 0.974327i \(-0.427717\pi\)
0.225136 + 0.974327i \(0.427717\pi\)
\(948\) 0 0
\(949\) 19.5959 0.636110
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) 41.5692 1.34515
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 28.2843 0.913347
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 29.3939 0.946222
\(966\) 0 0
\(967\) 32.5269 1.04599 0.522997 0.852334i \(-0.324814\pi\)
0.522997 + 0.852334i \(0.324814\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 31.1769 1.00051 0.500257 0.865877i \(-0.333239\pi\)
0.500257 + 0.865877i \(0.333239\pi\)
\(972\) 0 0
\(973\) 19.5959 0.628216
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) 55.4256 1.77141
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −33.9411 −1.08255 −0.541277 0.840844i \(-0.682059\pi\)
−0.541277 + 0.840844i \(0.682059\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −39.1918 −1.24623
\(990\) 0 0
\(991\) −15.5563 −0.494164 −0.247082 0.968995i \(-0.579472\pi\)
−0.247082 + 0.968995i \(0.579472\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −45.0333 −1.42765
\(996\) 0 0
\(997\) 14.6969 0.465457 0.232728 0.972542i \(-0.425235\pi\)
0.232728 + 0.972542i \(0.425235\pi\)
\(998\) 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 4608.2.a.z.1.2 yes 4
3.2 odd 2 4608.2.a.v.1.4 yes 4
4.3 odd 2 inner 4608.2.a.z.1.1 yes 4
8.3 odd 2 inner 4608.2.a.z.1.3 yes 4
8.5 even 2 inner 4608.2.a.z.1.4 yes 4
12.11 even 2 4608.2.a.v.1.3 yes 4
16.3 odd 4 4608.2.d.m.2305.4 4
16.5 even 4 4608.2.d.m.2305.1 4
16.11 odd 4 4608.2.d.m.2305.2 4
16.13 even 4 4608.2.d.m.2305.3 4
24.5 odd 2 4608.2.a.v.1.2 yes 4
24.11 even 2 4608.2.a.v.1.1 4
48.5 odd 4 4608.2.d.e.2305.3 4
48.11 even 4 4608.2.d.e.2305.4 4
48.29 odd 4 4608.2.d.e.2305.1 4
48.35 even 4 4608.2.d.e.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.v.1.1 4 24.11 even 2
4608.2.a.v.1.2 yes 4 24.5 odd 2
4608.2.a.v.1.3 yes 4 12.11 even 2
4608.2.a.v.1.4 yes 4 3.2 odd 2
4608.2.a.z.1.1 yes 4 4.3 odd 2 inner
4608.2.a.z.1.2 yes 4 1.1 even 1 trivial
4608.2.a.z.1.3 yes 4 8.3 odd 2 inner
4608.2.a.z.1.4 yes 4 8.5 even 2 inner
4608.2.d.e.2305.1 4 48.29 odd 4
4608.2.d.e.2305.2 4 48.35 even 4
4608.2.d.e.2305.3 4 48.5 odd 4
4608.2.d.e.2305.4 4 48.11 even 4
4608.2.d.m.2305.1 4 16.5 even 4
4608.2.d.m.2305.2 4 16.11 odd 4
4608.2.d.m.2305.3 4 16.13 even 4
4608.2.d.m.2305.4 4 16.3 odd 4