Properties

Label 3920.2.a.bs.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{3} -1.00000 q^{5} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} -1.00000 q^{5} -0.561553 q^{9} +1.56155 q^{11} -0.438447 q^{13} -1.56155 q^{15} +0.438447 q^{17} -7.12311 q^{19} -3.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +6.68466 q^{29} +2.43845 q^{33} +6.00000 q^{37} -0.684658 q^{39} -5.12311 q^{41} -0.876894 q^{43} +0.561553 q^{45} -8.68466 q^{47} +0.684658 q^{51} -5.12311 q^{53} -1.56155 q^{55} -11.1231 q^{57} -4.00000 q^{59} -15.3693 q^{61} +0.438447 q^{65} -10.2462 q^{67} -4.87689 q^{69} -8.00000 q^{71} +12.2462 q^{73} +1.56155 q^{75} +2.43845 q^{79} -7.00000 q^{81} +4.00000 q^{83} -0.438447 q^{85} +10.4384 q^{87} +1.12311 q^{89} +7.12311 q^{95} -5.80776 q^{97} -0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 2q^{5} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} - 2q^{5} + 3q^{9} - q^{11} - 5q^{13} + q^{15} + 5q^{17} - 6q^{19} + 2q^{23} + 2q^{25} - 7q^{27} + q^{29} + 9q^{33} + 12q^{37} + 11q^{39} - 2q^{41} - 10q^{43} - 3q^{45} - 5q^{47} - 11q^{51} - 2q^{53} + q^{55} - 14q^{57} - 8q^{59} - 6q^{61} + 5q^{65} - 4q^{67} - 18q^{69} - 16q^{71} + 8q^{73} - q^{75} + 9q^{79} - 14q^{81} + 8q^{83} - 5q^{85} + 25q^{87} - 6q^{89} + 6q^{95} + 9q^{97} - 10q^{99} + O(q^{100}) \)

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
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 0.438447 0.106339 0.0531695 0.998586i \(-0.483068\pi\)
0.0531695 + 0.998586i \(0.483068\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.12311 −0.651213 −0.325606 0.945505i \(-0.605568\pi\)
−0.325606 + 0.945505i \(0.605568\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.43845 0.424479
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −0.684658 −0.109633
\(40\) 0 0
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) −0.876894 −0.133725 −0.0668626 0.997762i \(-0.521299\pi\)
−0.0668626 + 0.997762i \(0.521299\pi\)
\(44\) 0 0
\(45\) 0.561553 0.0837114
\(46\) 0 0
\(47\) −8.68466 −1.26679 −0.633394 0.773830i \(-0.718339\pi\)
−0.633394 + 0.773830i \(0.718339\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.684658 0.0958714
\(52\) 0 0
\(53\) −5.12311 −0.703713 −0.351856 0.936054i \(-0.614449\pi\)
−0.351856 + 0.936054i \(0.614449\pi\)
\(54\) 0 0
\(55\) −1.56155 −0.210560
\(56\) 0 0
\(57\) −11.1231 −1.47329
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −15.3693 −1.96784 −0.983920 0.178611i \(-0.942839\pi\)
−0.983920 + 0.178611i \(0.942839\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.438447 0.0543827
\(66\) 0 0
\(67\) −10.2462 −1.25177 −0.625887 0.779914i \(-0.715263\pi\)
−0.625887 + 0.779914i \(0.715263\pi\)
\(68\) 0 0
\(69\) −4.87689 −0.587109
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.43845 0.274347 0.137173 0.990547i \(-0.456198\pi\)
0.137173 + 0.990547i \(0.456198\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −0.438447 −0.0475563
\(86\) 0 0
\(87\) 10.4384 1.11912
\(88\) 0 0
\(89\) 1.12311 0.119049 0.0595245 0.998227i \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) −5.80776 −0.589689 −0.294845 0.955545i \(-0.595268\pi\)
−0.294845 + 0.955545i \(0.595268\pi\)
\(98\) 0 0
\(99\) −0.876894 −0.0881312
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) 5.56155 0.547996 0.273998 0.961730i \(-0.411654\pi\)
0.273998 + 0.961730i \(0.411654\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.3693 −1.29246 −0.646230 0.763142i \(-0.723655\pi\)
−0.646230 + 0.763142i \(0.723655\pi\)
\(108\) 0 0
\(109\) 5.31534 0.509117 0.254559 0.967057i \(-0.418070\pi\)
0.254559 + 0.967057i \(0.418070\pi\)
\(110\) 0 0
\(111\) 9.36932 0.889296
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 3.12311 0.291231
\(116\) 0 0
\(117\) 0.246211 0.0227622
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.24621 0.554262 0.277131 0.960832i \(-0.410616\pi\)
0.277131 + 0.960832i \(0.410616\pi\)
\(128\) 0 0
\(129\) −1.36932 −0.120562
\(130\) 0 0
\(131\) −0.876894 −0.0766146 −0.0383073 0.999266i \(-0.512197\pi\)
−0.0383073 + 0.999266i \(0.512197\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.56155 0.478662
\(136\) 0 0
\(137\) −17.1231 −1.46293 −0.731463 0.681881i \(-0.761162\pi\)
−0.731463 + 0.681881i \(0.761162\pi\)
\(138\) 0 0
\(139\) −15.1231 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) 0 0
\(143\) −0.684658 −0.0572540
\(144\) 0 0
\(145\) −6.68466 −0.555131
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 0 0
\(151\) 6.93087 0.564026 0.282013 0.959411i \(-0.408998\pi\)
0.282013 + 0.959411i \(0.408998\pi\)
\(152\) 0 0
\(153\) −0.246211 −0.0199050
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.2462 −1.61582 −0.807912 0.589303i \(-0.799402\pi\)
−0.807912 + 0.589303i \(0.799402\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.12311 0.557925 0.278962 0.960302i \(-0.410010\pi\)
0.278962 + 0.960302i \(0.410010\pi\)
\(164\) 0 0
\(165\) −2.43845 −0.189833
\(166\) 0 0
\(167\) −6.93087 −0.536327 −0.268163 0.963373i \(-0.586417\pi\)
−0.268163 + 0.963373i \(0.586417\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 4.43845 0.337449 0.168724 0.985663i \(-0.446035\pi\)
0.168724 + 0.985663i \(0.446035\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.24621 −0.469494
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 17.6155 1.30935 0.654676 0.755910i \(-0.272805\pi\)
0.654676 + 0.755910i \(0.272805\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0.684658 0.0500672
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.5616 0.981280 0.490640 0.871363i \(-0.336763\pi\)
0.490640 + 0.871363i \(0.336763\pi\)
\(192\) 0 0
\(193\) 19.3693 1.39423 0.697117 0.716957i \(-0.254466\pi\)
0.697117 + 0.716957i \(0.254466\pi\)
\(194\) 0 0
\(195\) 0.684658 0.0490294
\(196\) 0 0
\(197\) 1.12311 0.0800180 0.0400090 0.999199i \(-0.487261\pi\)
0.0400090 + 0.999199i \(0.487261\pi\)
\(198\) 0 0
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.12311 0.357813
\(206\) 0 0
\(207\) 1.75379 0.121897
\(208\) 0 0
\(209\) −11.1231 −0.769401
\(210\) 0 0
\(211\) −14.0540 −0.967516 −0.483758 0.875202i \(-0.660728\pi\)
−0.483758 + 0.875202i \(0.660728\pi\)
\(212\) 0 0
\(213\) −12.4924 −0.855967
\(214\) 0 0
\(215\) 0.876894 0.0598037
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 19.1231 1.29222
\(220\) 0 0
\(221\) −0.192236 −0.0129312
\(222\) 0 0
\(223\) −2.43845 −0.163291 −0.0816453 0.996661i \(-0.526017\pi\)
−0.0816453 + 0.996661i \(0.526017\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) 11.3153 0.751026 0.375513 0.926817i \(-0.377467\pi\)
0.375513 + 0.926817i \(0.377467\pi\)
\(228\) 0 0
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.12311 0.335626 0.167813 0.985819i \(-0.446330\pi\)
0.167813 + 0.985819i \(0.446330\pi\)
\(234\) 0 0
\(235\) 8.68466 0.566525
\(236\) 0 0
\(237\) 3.80776 0.247341
\(238\) 0 0
\(239\) −19.8078 −1.28126 −0.640629 0.767851i \(-0.721326\pi\)
−0.640629 + 0.767851i \(0.721326\pi\)
\(240\) 0 0
\(241\) 4.24621 0.273523 0.136761 0.990604i \(-0.456331\pi\)
0.136761 + 0.990604i \(0.456331\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.12311 0.198718
\(248\) 0 0
\(249\) 6.24621 0.395838
\(250\) 0 0
\(251\) −8.87689 −0.560305 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(252\) 0 0
\(253\) −4.87689 −0.306608
\(254\) 0 0
\(255\) −0.684658 −0.0428750
\(256\) 0 0
\(257\) 10.4924 0.654499 0.327250 0.944938i \(-0.393878\pi\)
0.327250 + 0.944938i \(0.393878\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) 0 0
\(263\) 12.8769 0.794023 0.397012 0.917814i \(-0.370047\pi\)
0.397012 + 0.917814i \(0.370047\pi\)
\(264\) 0 0
\(265\) 5.12311 0.314710
\(266\) 0 0
\(267\) 1.75379 0.107330
\(268\) 0 0
\(269\) 20.7386 1.26446 0.632228 0.774782i \(-0.282140\pi\)
0.632228 + 0.774782i \(0.282140\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.56155 0.0941652
\(276\) 0 0
\(277\) −0.246211 −0.0147934 −0.00739670 0.999973i \(-0.502354\pi\)
−0.00739670 + 0.999973i \(0.502354\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.4384 0.742016 0.371008 0.928630i \(-0.379012\pi\)
0.371008 + 0.928630i \(0.379012\pi\)
\(282\) 0 0
\(283\) −11.3153 −0.672627 −0.336314 0.941750i \(-0.609180\pi\)
−0.336314 + 0.941750i \(0.609180\pi\)
\(284\) 0 0
\(285\) 11.1231 0.658876
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8078 −0.988692
\(290\) 0 0
\(291\) −9.06913 −0.531642
\(292\) 0 0
\(293\) 2.68466 0.156839 0.0784197 0.996920i \(-0.475013\pi\)
0.0784197 + 0.996920i \(0.475013\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) −8.68466 −0.503935
\(298\) 0 0
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 25.3693 1.45743
\(304\) 0 0
\(305\) 15.3693 0.880045
\(306\) 0 0
\(307\) −19.3153 −1.10238 −0.551192 0.834378i \(-0.685827\pi\)
−0.551192 + 0.834378i \(0.685827\pi\)
\(308\) 0 0
\(309\) 8.68466 0.494053
\(310\) 0 0
\(311\) 31.6155 1.79275 0.896376 0.443294i \(-0.146190\pi\)
0.896376 + 0.443294i \(0.146190\pi\)
\(312\) 0 0
\(313\) 22.3002 1.26048 0.630241 0.776400i \(-0.282956\pi\)
0.630241 + 0.776400i \(0.282956\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.4924 0.589313 0.294657 0.955603i \(-0.404795\pi\)
0.294657 + 0.955603i \(0.404795\pi\)
\(318\) 0 0
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) −20.8769 −1.16523
\(322\) 0 0
\(323\) −3.12311 −0.173774
\(324\) 0 0
\(325\) −0.438447 −0.0243207
\(326\) 0 0
\(327\) 8.30019 0.459001
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) −3.36932 −0.184637
\(334\) 0 0
\(335\) 10.2462 0.559810
\(336\) 0 0
\(337\) −1.50758 −0.0821230 −0.0410615 0.999157i \(-0.513074\pi\)
−0.0410615 + 0.999157i \(0.513074\pi\)
\(338\) 0 0
\(339\) −21.8617 −1.18737
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.87689 0.262563
\(346\) 0 0
\(347\) −7.12311 −0.382388 −0.191194 0.981552i \(-0.561236\pi\)
−0.191194 + 0.981552i \(0.561236\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 2.43845 0.130155
\(352\) 0 0
\(353\) −5.80776 −0.309116 −0.154558 0.987984i \(-0.549395\pi\)
−0.154558 + 0.987984i \(0.549395\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) −13.3693 −0.701707
\(364\) 0 0
\(365\) −12.2462 −0.640996
\(366\) 0 0
\(367\) −8.68466 −0.453335 −0.226668 0.973972i \(-0.572783\pi\)
−0.226668 + 0.973972i \(0.572783\pi\)
\(368\) 0 0
\(369\) 2.87689 0.149765
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.63068 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(374\) 0 0
\(375\) −1.56155 −0.0806382
\(376\) 0 0
\(377\) −2.93087 −0.150947
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) 0 0
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.492423 0.0250312
\(388\) 0 0
\(389\) −24.9309 −1.26405 −0.632023 0.774950i \(-0.717775\pi\)
−0.632023 + 0.774950i \(0.717775\pi\)
\(390\) 0 0
\(391\) −1.36932 −0.0692493
\(392\) 0 0
\(393\) −1.36932 −0.0690729
\(394\) 0 0
\(395\) −2.43845 −0.122692
\(396\) 0 0
\(397\) −27.5616 −1.38327 −0.691637 0.722245i \(-0.743110\pi\)
−0.691637 + 0.722245i \(0.743110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.5616 1.57611 0.788054 0.615606i \(-0.211089\pi\)
0.788054 + 0.615606i \(0.211089\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 9.36932 0.464420
\(408\) 0 0
\(409\) −6.49242 −0.321030 −0.160515 0.987033i \(-0.551315\pi\)
−0.160515 + 0.987033i \(0.551315\pi\)
\(410\) 0 0
\(411\) −26.7386 −1.31892
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −23.6155 −1.15646
\(418\) 0 0
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) 0 0
\(421\) −2.68466 −0.130842 −0.0654211 0.997858i \(-0.520839\pi\)
−0.0654211 + 0.997858i \(0.520839\pi\)
\(422\) 0 0
\(423\) 4.87689 0.237123
\(424\) 0 0
\(425\) 0.438447 0.0212678
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.06913 −0.0516181
\(430\) 0 0
\(431\) 19.8078 0.954106 0.477053 0.878874i \(-0.341705\pi\)
0.477053 + 0.878874i \(0.341705\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) −10.4384 −0.500485
\(436\) 0 0
\(437\) 22.2462 1.06418
\(438\) 0 0
\(439\) 9.36932 0.447173 0.223587 0.974684i \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.63068 0.124988 0.0624938 0.998045i \(-0.480095\pi\)
0.0624938 + 0.998045i \(0.480095\pi\)
\(444\) 0 0
\(445\) −1.12311 −0.0532403
\(446\) 0 0
\(447\) 19.1231 0.904492
\(448\) 0 0
\(449\) −1.80776 −0.0853137 −0.0426568 0.999090i \(-0.513582\pi\)
−0.0426568 + 0.999090i \(0.513582\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 10.8229 0.508505
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.1231 −0.800985 −0.400493 0.916300i \(-0.631161\pi\)
−0.400493 + 0.916300i \(0.631161\pi\)
\(458\) 0 0
\(459\) −2.43845 −0.113817
\(460\) 0 0
\(461\) 13.1231 0.611204 0.305602 0.952159i \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 0 0
\(463\) −12.4924 −0.580572 −0.290286 0.956940i \(-0.593750\pi\)
−0.290286 + 0.956940i \(0.593750\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.4384 1.03833 0.519164 0.854675i \(-0.326243\pi\)
0.519164 + 0.854675i \(0.326243\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −31.6155 −1.45677
\(472\) 0 0
\(473\) −1.36932 −0.0629613
\(474\) 0 0
\(475\) −7.12311 −0.326831
\(476\) 0 0
\(477\) 2.87689 0.131724
\(478\) 0 0
\(479\) 4.87689 0.222831 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(480\) 0 0
\(481\) −2.63068 −0.119949
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.80776 0.263717
\(486\) 0 0
\(487\) 3.12311 0.141521 0.0707607 0.997493i \(-0.477457\pi\)
0.0707607 + 0.997493i \(0.477457\pi\)
\(488\) 0 0
\(489\) 11.1231 0.503004
\(490\) 0 0
\(491\) 41.1771 1.85830 0.929148 0.369708i \(-0.120542\pi\)
0.929148 + 0.369708i \(0.120542\pi\)
\(492\) 0 0
\(493\) 2.93087 0.132000
\(494\) 0 0
\(495\) 0.876894 0.0394135
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −41.1771 −1.84334 −0.921670 0.387976i \(-0.873174\pi\)
−0.921670 + 0.387976i \(0.873174\pi\)
\(500\) 0 0
\(501\) −10.8229 −0.483532
\(502\) 0 0
\(503\) 38.9309 1.73584 0.867921 0.496703i \(-0.165456\pi\)
0.867921 + 0.496703i \(0.165456\pi\)
\(504\) 0 0
\(505\) −16.2462 −0.722947
\(506\) 0 0
\(507\) −20.0000 −0.888231
\(508\) 0 0
\(509\) 11.7538 0.520978 0.260489 0.965477i \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 39.6155 1.74907
\(514\) 0 0
\(515\) −5.56155 −0.245071
\(516\) 0 0
\(517\) −13.5616 −0.596436
\(518\) 0 0
\(519\) 6.93087 0.304231
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 40.4924 1.77061 0.885305 0.465011i \(-0.153950\pi\)
0.885305 + 0.465011i \(0.153950\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) 2.24621 0.0974773
\(532\) 0 0
\(533\) 2.24621 0.0972942
\(534\) 0 0
\(535\) 13.3693 0.578006
\(536\) 0 0
\(537\) −31.2311 −1.34772
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) 0 0
\(543\) 27.5076 1.18046
\(544\) 0 0
\(545\) −5.31534 −0.227684
\(546\) 0 0
\(547\) 2.24621 0.0960411 0.0480205 0.998846i \(-0.484709\pi\)
0.0480205 + 0.998846i \(0.484709\pi\)
\(548\) 0 0
\(549\) 8.63068 0.368349
\(550\) 0 0
\(551\) −47.6155 −2.02849
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −9.36932 −0.397705
\(556\) 0 0
\(557\) −13.1231 −0.556044 −0.278022 0.960575i \(-0.589679\pi\)
−0.278022 + 0.960575i \(0.589679\pi\)
\(558\) 0 0
\(559\) 0.384472 0.0162614
\(560\) 0 0
\(561\) 1.06913 0.0451387
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.9848 −1.29895 −0.649476 0.760382i \(-0.725012\pi\)
−0.649476 + 0.760382i \(0.725012\pi\)
\(570\) 0 0
\(571\) −40.4924 −1.69456 −0.847278 0.531150i \(-0.821760\pi\)
−0.847278 + 0.531150i \(0.821760\pi\)
\(572\) 0 0
\(573\) 21.1771 0.884685
\(574\) 0 0
\(575\) −3.12311 −0.130243
\(576\) 0 0
\(577\) 24.0540 1.00138 0.500690 0.865627i \(-0.333080\pi\)
0.500690 + 0.865627i \(0.333080\pi\)
\(578\) 0 0
\(579\) 30.2462 1.25699
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) −0.246211 −0.0101796
\(586\) 0 0
\(587\) −26.2462 −1.08330 −0.541649 0.840605i \(-0.682200\pi\)
−0.541649 + 0.840605i \(0.682200\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 1.75379 0.0721412
\(592\) 0 0
\(593\) 27.5616 1.13182 0.565909 0.824468i \(-0.308525\pi\)
0.565909 + 0.824468i \(0.308525\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.73863 −0.112085
\(598\) 0 0
\(599\) 11.8078 0.482452 0.241226 0.970469i \(-0.422450\pi\)
0.241226 + 0.970469i \(0.422450\pi\)
\(600\) 0 0
\(601\) −6.49242 −0.264831 −0.132416 0.991194i \(-0.542273\pi\)
−0.132416 + 0.991194i \(0.542273\pi\)
\(602\) 0 0
\(603\) 5.75379 0.234312
\(604\) 0 0
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) −42.0540 −1.70692 −0.853459 0.521160i \(-0.825500\pi\)
−0.853459 + 0.521160i \(0.825500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.80776 0.154046
\(612\) 0 0
\(613\) 40.7386 1.64542 0.822709 0.568463i \(-0.192462\pi\)
0.822709 + 0.568463i \(0.192462\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 32.2462 1.29818 0.649092 0.760710i \(-0.275149\pi\)
0.649092 + 0.760710i \(0.275149\pi\)
\(618\) 0 0
\(619\) 32.1080 1.29053 0.645264 0.763960i \(-0.276747\pi\)
0.645264 + 0.763960i \(0.276747\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.3693 −0.693664
\(628\) 0 0
\(629\) 2.63068 0.104892
\(630\) 0 0
\(631\) 11.8078 0.470060 0.235030 0.971988i \(-0.424481\pi\)
0.235030 + 0.971988i \(0.424481\pi\)
\(632\) 0 0
\(633\) −21.9460 −0.872276
\(634\) 0 0
\(635\) −6.24621 −0.247873
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.49242 0.177717
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 1.56155 0.0615816 0.0307908 0.999526i \(-0.490197\pi\)
0.0307908 + 0.999526i \(0.490197\pi\)
\(644\) 0 0
\(645\) 1.36932 0.0539168
\(646\) 0 0
\(647\) 36.4924 1.43467 0.717333 0.696731i \(-0.245363\pi\)
0.717333 + 0.696731i \(0.245363\pi\)
\(648\) 0 0
\(649\) −6.24621 −0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −33.2311 −1.30043 −0.650216 0.759750i \(-0.725322\pi\)
−0.650216 + 0.759750i \(0.725322\pi\)
\(654\) 0 0
\(655\) 0.876894 0.0342631
\(656\) 0 0
\(657\) −6.87689 −0.268293
\(658\) 0 0
\(659\) −9.17708 −0.357488 −0.178744 0.983896i \(-0.557203\pi\)
−0.178744 + 0.983896i \(0.557203\pi\)
\(660\) 0 0
\(661\) 5.12311 0.199266 0.0996329 0.995024i \(-0.468233\pi\)
0.0996329 + 0.995024i \(0.468233\pi\)
\(662\) 0 0
\(663\) −0.300187 −0.0116583
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.8769 −0.808357
\(668\) 0 0
\(669\) −3.80776 −0.147217
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 31.8617 1.22818 0.614090 0.789236i \(-0.289523\pi\)
0.614090 + 0.789236i \(0.289523\pi\)
\(674\) 0 0
\(675\) −5.56155 −0.214064
\(676\) 0 0
\(677\) −4.93087 −0.189509 −0.0947544 0.995501i \(-0.530207\pi\)
−0.0947544 + 0.995501i \(0.530207\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) 6.73863 0.257847 0.128923 0.991655i \(-0.458848\pi\)
0.128923 + 0.991655i \(0.458848\pi\)
\(684\) 0 0
\(685\) 17.1231 0.654240
\(686\) 0 0
\(687\) −16.9848 −0.648012
\(688\) 0 0
\(689\) 2.24621 0.0855738
\(690\) 0 0
\(691\) −24.4924 −0.931736 −0.465868 0.884854i \(-0.654258\pi\)
−0.465868 + 0.884854i \(0.654258\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.1231 0.573652
\(696\) 0 0
\(697\) −2.24621 −0.0850813
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 28.9309 1.09270 0.546352 0.837556i \(-0.316016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(702\) 0 0
\(703\) −42.7386 −1.61192
\(704\) 0 0
\(705\) 13.5616 0.510758
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 27.1771 1.02066 0.510328 0.859980i \(-0.329524\pi\)
0.510328 + 0.859980i \(0.329524\pi\)
\(710\) 0 0
\(711\) −1.36932 −0.0513534
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.684658 0.0256048
\(716\) 0 0
\(717\) −30.9309 −1.15513
\(718\) 0 0
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 6.63068 0.246598
\(724\) 0 0
\(725\) 6.68466 0.248262
\(726\) 0 0
\(727\) 52.4924 1.94684 0.973418 0.229035i \(-0.0735572\pi\)
0.973418 + 0.229035i \(0.0735572\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −0.384472 −0.0142202
\(732\) 0 0
\(733\) −6.68466 −0.246903 −0.123452 0.992351i \(-0.539396\pi\)
−0.123452 + 0.992351i \(0.539396\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −34.9309 −1.28495 −0.642476 0.766305i \(-0.722093\pi\)
−0.642476 + 0.766305i \(0.722093\pi\)
\(740\) 0 0
\(741\) 4.87689 0.179157
\(742\) 0 0
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) −12.2462 −0.448666
\(746\) 0 0
\(747\) −2.24621 −0.0821846
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −17.0691 −0.622861 −0.311431 0.950269i \(-0.600808\pi\)
−0.311431 + 0.950269i \(0.600808\pi\)
\(752\) 0 0
\(753\) −13.8617 −0.505150
\(754\) 0 0
\(755\) −6.93087 −0.252240
\(756\) 0 0
\(757\) 39.3693 1.43090 0.715451 0.698663i \(-0.246221\pi\)
0.715451 + 0.698663i \(0.246221\pi\)
\(758\) 0 0
\(759\) −7.61553 −0.276426
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.246211 0.00890179
\(766\) 0 0
\(767\) 1.75379 0.0633256
\(768\) 0 0
\(769\) 42.4924 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(770\) 0 0
\(771\) 16.3845 0.590072
\(772\) 0 0
\(773\) −36.9309 −1.32831 −0.664156 0.747594i \(-0.731209\pi\)
−0.664156 + 0.747594i \(0.731209\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.4924 1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) 0 0
\(783\) −37.1771 −1.32860
\(784\) 0 0
\(785\) 20.2462 0.722618
\(786\) 0 0
\(787\) −49.1771 −1.75297 −0.876487 0.481426i \(-0.840119\pi\)
−0.876487 + 0.481426i \(0.840119\pi\)
\(788\) 0 0
\(789\) 20.1080 0.715862
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.73863 0.239296
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) −24.0540 −0.852036 −0.426018 0.904715i \(-0.640084\pi\)
−0.426018 + 0.904715i \(0.640084\pi\)
\(798\) 0 0
\(799\) −3.80776 −0.134709
\(800\) 0 0
\(801\) −0.630683 −0.0222841
\(802\) 0 0
\(803\) 19.1231 0.674840
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 32.3845 1.13999
\(808\) 0 0
\(809\) 16.5464 0.581740 0.290870 0.956763i \(-0.406055\pi\)
0.290870 + 0.956763i \(0.406055\pi\)
\(810\) 0 0
\(811\) 19.6155 0.688794 0.344397 0.938824i \(-0.388083\pi\)
0.344397 + 0.938824i \(0.388083\pi\)
\(812\) 0 0
\(813\) −24.9848 −0.876257
\(814\) 0 0
\(815\) −7.12311 −0.249512
\(816\) 0 0
\(817\) 6.24621 0.218527
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.4233 −0.747678 −0.373839 0.927494i \(-0.621959\pi\)
−0.373839 + 0.927494i \(0.621959\pi\)
\(822\) 0 0
\(823\) 36.4924 1.27205 0.636023 0.771670i \(-0.280578\pi\)
0.636023 + 0.771670i \(0.280578\pi\)
\(824\) 0 0
\(825\) 2.43845 0.0848958
\(826\) 0 0
\(827\) 5.36932 0.186709 0.0933547 0.995633i \(-0.470241\pi\)
0.0933547 + 0.995633i \(0.470241\pi\)
\(828\) 0 0
\(829\) −34.8769 −1.21132 −0.605662 0.795722i \(-0.707092\pi\)
−0.605662 + 0.795722i \(0.707092\pi\)
\(830\) 0 0
\(831\) −0.384472 −0.0133372
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 6.93087 0.239853
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.8769 −0.996941 −0.498471 0.866907i \(-0.666105\pi\)
−0.498471 + 0.866907i \(0.666105\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) 19.4233 0.668974
\(844\) 0 0
\(845\) 12.8078 0.440600
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −17.6695 −0.606416
\(850\) 0 0
\(851\) −18.7386 −0.642352
\(852\) 0 0
\(853\) 7.26137 0.248624 0.124312 0.992243i \(-0.460328\pi\)
0.124312 + 0.992243i \(0.460328\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 15.7538 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.7538 0.876669 0.438335 0.898812i \(-0.355569\pi\)
0.438335 + 0.898812i \(0.355569\pi\)
\(864\) 0 0
\(865\) −4.43845 −0.150912
\(866\) 0 0
\(867\) −26.2462 −0.891368
\(868\) 0 0
\(869\) 3.80776 0.129170
\(870\) 0 0
\(871\) 4.49242 0.152220
\(872\) 0 0
\(873\) 3.26137 0.110381
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) 0 0
\(879\) 4.19224 0.141401
\(880\) 0 0
\(881\) 11.8617 0.399632 0.199816 0.979833i \(-0.435966\pi\)
0.199816 + 0.979833i \(0.435966\pi\)
\(882\) 0 0
\(883\) 8.49242 0.285793 0.142896 0.989738i \(-0.454358\pi\)
0.142896 + 0.989738i \(0.454358\pi\)
\(884\) 0 0
\(885\) 6.24621 0.209964
\(886\) 0 0
\(887\) 20.4924 0.688068 0.344034 0.938957i \(-0.388206\pi\)
0.344034 + 0.938957i \(0.388206\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −10.9309 −0.366198
\(892\) 0 0
\(893\) 61.8617 2.07012
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 2.13826 0.0713944
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.24621 −0.0748321
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −17.6155 −0.585560
\(906\) 0 0
\(907\) 24.1080 0.800491 0.400246 0.916408i \(-0.368925\pi\)
0.400246 + 0.916408i \(0.368925\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) 28.4924 0.943996 0.471998 0.881600i \(-0.343533\pi\)
0.471998 + 0.881600i \(0.343533\pi\)
\(912\) 0 0
\(913\) 6.24621 0.206719
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −40.3002 −1.32938 −0.664690 0.747119i \(-0.731436\pi\)
−0.664690 + 0.747119i \(0.731436\pi\)
\(920\) 0 0
\(921\) −30.1619 −0.993869
\(922\) 0 0
\(923\) 3.50758 0.115453
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −3.12311 −0.102576
\(928\) 0 0
\(929\) −22.1080 −0.725338 −0.362669 0.931918i \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 49.3693 1.61628
\(934\) 0 0
\(935\) −0.684658 −0.0223907
\(936\) 0 0
\(937\) 55.6695 1.81864 0.909322 0.416094i \(-0.136601\pi\)
0.909322 + 0.416094i \(0.136601\pi\)
\(938\) 0 0
\(939\) 34.8229 1.13640
\(940\) 0 0
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) −5.36932 −0.174295
\(950\) 0 0
\(951\) 16.3845 0.531303
\(952\) 0 0
\(953\) −33.1231 −1.07296 −0.536481 0.843912i \(-0.680247\pi\)
−0.536481 + 0.843912i \(0.680247\pi\)
\(954\) 0 0
\(955\) −13.5616 −0.438842
\(956\) 0 0
\(957\) 16.3002 0.526910
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 7.50758 0.241928
\(964\) 0 0
\(965\) −19.3693 −0.623520
\(966\) 0 0
\(967\) 35.1231 1.12948 0.564741 0.825268i \(-0.308976\pi\)
0.564741 + 0.825268i \(0.308976\pi\)
\(968\) 0 0
\(969\) −4.87689 −0.156668
\(970\) 0 0
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.684658 −0.0219266
\(976\) 0 0
\(977\) 33.2311 1.06316 0.531578 0.847009i \(-0.321599\pi\)
0.531578 + 0.847009i \(0.321599\pi\)
\(978\) 0 0
\(979\) 1.75379 0.0560513
\(980\) 0 0
\(981\) −2.98485 −0.0952988
\(982\) 0 0
\(983\) 51.4233 1.64015 0.820074 0.572257i \(-0.193932\pi\)
0.820074 + 0.572257i \(0.193932\pi\)
\(984\) 0 0
\(985\) −1.12311 −0.0357851
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.73863 0.0870835
\(990\) 0 0
\(991\) −12.4924 −0.396835 −0.198417 0.980118i \(-0.563580\pi\)
−0.198417 + 0.980118i \(0.563580\pi\)
\(992\) 0 0
\(993\) −18.7386 −0.594653
\(994\) 0 0
\(995\) 1.75379 0.0555988
\(996\) 0 0
\(997\) 2.68466 0.0850240 0.0425120 0.999096i \(-0.486464\pi\)
0.0425120 + 0.999096i \(0.486464\pi\)
\(998\) 0 0
\(999\) −33.3693 −1.05576
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 3920.2.a.bs.1.2 2
4.3 odd 2 245.2.a.d.1.1 2
7.6 odd 2 560.2.a.i.1.1 2
12.11 even 2 2205.2.a.x.1.2 2
20.3 even 4 1225.2.b.f.99.4 4
20.7 even 4 1225.2.b.f.99.1 4
20.19 odd 2 1225.2.a.s.1.2 2
21.20 even 2 5040.2.a.bt.1.1 2
28.3 even 6 245.2.e.i.226.2 4
28.11 odd 6 245.2.e.h.226.2 4
28.19 even 6 245.2.e.i.116.2 4
28.23 odd 6 245.2.e.h.116.2 4
28.27 even 2 35.2.a.b.1.1 2
35.13 even 4 2800.2.g.t.449.2 4
35.27 even 4 2800.2.g.t.449.3 4
35.34 odd 2 2800.2.a.bi.1.2 2
56.13 odd 2 2240.2.a.bd.1.2 2
56.27 even 2 2240.2.a.bh.1.1 2
84.83 odd 2 315.2.a.e.1.2 2
140.27 odd 4 175.2.b.b.99.1 4
140.83 odd 4 175.2.b.b.99.4 4
140.139 even 2 175.2.a.f.1.2 2
308.307 odd 2 4235.2.a.m.1.2 2
364.363 even 2 5915.2.a.l.1.2 2
420.83 even 4 1575.2.d.e.1324.1 4
420.167 even 4 1575.2.d.e.1324.4 4
420.419 odd 2 1575.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 28.27 even 2
175.2.a.f.1.2 2 140.139 even 2
175.2.b.b.99.1 4 140.27 odd 4
175.2.b.b.99.4 4 140.83 odd 4
245.2.a.d.1.1 2 4.3 odd 2
245.2.e.h.116.2 4 28.23 odd 6
245.2.e.h.226.2 4 28.11 odd 6
245.2.e.i.116.2 4 28.19 even 6
245.2.e.i.226.2 4 28.3 even 6
315.2.a.e.1.2 2 84.83 odd 2
560.2.a.i.1.1 2 7.6 odd 2
1225.2.a.s.1.2 2 20.19 odd 2
1225.2.b.f.99.1 4 20.7 even 4
1225.2.b.f.99.4 4 20.3 even 4
1575.2.a.p.1.1 2 420.419 odd 2
1575.2.d.e.1324.1 4 420.83 even 4
1575.2.d.e.1324.4 4 420.167 even 4
2205.2.a.x.1.2 2 12.11 even 2
2240.2.a.bd.1.2 2 56.13 odd 2
2240.2.a.bh.1.1 2 56.27 even 2
2800.2.a.bi.1.2 2 35.34 odd 2
2800.2.g.t.449.2 4 35.13 even 4
2800.2.g.t.449.3 4 35.27 even 4
3920.2.a.bs.1.2 2 1.1 even 1 trivial
4235.2.a.m.1.2 2 308.307 odd 2
5040.2.a.bt.1.1 2 21.20 even 2
5915.2.a.l.1.2 2 364.363 even 2