Properties

Label 5040.2.a.bt.1.1
Level $5040$
Weight $2$
Character 5040.1
Self dual yes
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 5040.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +1.00000 q^{7} -1.56155 q^{11} +0.438447 q^{13} +0.438447 q^{17} +7.12311 q^{19} +3.12311 q^{23} +1.00000 q^{25} -6.68466 q^{29} -1.00000 q^{35} +6.00000 q^{37} -5.12311 q^{41} -0.876894 q^{43} -8.68466 q^{47} +1.00000 q^{49} +5.12311 q^{53} +1.56155 q^{55} -4.00000 q^{59} +15.3693 q^{61} -0.438447 q^{65} -10.2462 q^{67} +8.00000 q^{71} -12.2462 q^{73} -1.56155 q^{77} +2.43845 q^{79} +4.00000 q^{83} -0.438447 q^{85} +1.12311 q^{89} +0.438447 q^{91} -7.12311 q^{95} +5.80776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{7} + q^{11} + 5q^{13} + 5q^{17} + 6q^{19} - 2q^{23} + 2q^{25} - q^{29} - 2q^{35} + 12q^{37} - 2q^{41} - 10q^{43} - 5q^{47} + 2q^{49} + 2q^{53} - q^{55} - 8q^{59} + 6q^{61} - 5q^{65} - 4q^{67} + 16q^{71} - 8q^{73} + q^{77} + 9q^{79} + 8q^{83} - 5q^{85} - 6q^{89} + 5q^{91} - 6q^{95} - 9q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) 0.438447 0.121603 0.0608017 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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 \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 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 0
\(40\) 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 0
\(46\) 0 0
\(47\) −8.68466 −1.26679 −0.633394 0.773830i \(-0.718339\pi\)
−0.633394 + 0.773830i \(0.718339\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.12311 0.703713 0.351856 0.936054i \(-0.385551\pi\)
0.351856 + 0.936054i \(0.385551\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) 0 0
\(58\) 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 \(-0.0571605\pi\)
0.983920 + 0.178611i \(0.0571605\pi\)
\(62\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −12.2462 −1.43331 −0.716655 0.697428i \(-0.754328\pi\)
−0.716655 + 0.697428i \(0.754328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.56155 −0.177955
\(78\) 0 0
\(79\) 2.43845 0.274347 0.137173 0.990547i \(-0.456198\pi\)
0.137173 + 0.990547i \(0.456198\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) 1.12311 0.119049 0.0595245 0.998227i \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 0 0
\(91\) 0.438447 0.0459618
\(92\) 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 \(-0.404732\pi\)
0.294845 + 0.955545i \(0.404732\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 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 \(-0.588346\pi\)
−0.273998 + 0.961730i \(0.588346\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3693 1.29246 0.646230 0.763142i \(-0.276345\pi\)
0.646230 + 0.763142i \(0.276345\pi\)
\(108\) 0 0
\(109\) 5.31534 0.509117 0.254559 0.967057i \(-0.418070\pi\)
0.254559 + 0.967057i \(0.418070\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −3.12311 −0.291231
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.438447 0.0401924
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 0 0
\(131\) −0.876894 −0.0766146 −0.0383073 0.999266i \(-0.512197\pi\)
−0.0383073 + 0.999266i \(0.512197\pi\)
\(132\) 0 0
\(133\) 7.12311 0.617652
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.1231 1.46293 0.731463 0.681881i \(-0.238838\pi\)
0.731463 + 0.681881i \(0.238838\pi\)
\(138\) 0 0
\(139\) 15.1231 1.28273 0.641363 0.767238i \(-0.278369\pi\)
0.641363 + 0.767238i \(0.278369\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 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 \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 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 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2462 1.61582 0.807912 0.589303i \(-0.200598\pi\)
0.807912 + 0.589303i \(0.200598\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) 7.12311 0.557925 0.278962 0.960302i \(-0.410010\pi\)
0.278962 + 0.960302i \(0.410010\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 0 0
\(173\) 4.43845 0.337449 0.168724 0.985663i \(-0.446035\pi\)
0.168724 + 0.985663i \(0.446035\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −17.6155 −1.30935 −0.654676 0.755910i \(-0.727195\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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 \(-0.663237\pi\)
−0.490640 + 0.871363i \(0.663237\pi\)
\(192\) 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 0
\(196\) 0 0
\(197\) −1.12311 −0.0800180 −0.0400090 0.999199i \(-0.512739\pi\)
−0.0400090 + 0.999199i \(0.512739\pi\)
\(198\) 0 0
\(199\) 1.75379 0.124323 0.0621614 0.998066i \(-0.480201\pi\)
0.0621614 + 0.998066i \(0.480201\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.68466 −0.469171
\(204\) 0 0
\(205\) 5.12311 0.357813
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 0.876894 0.0598037
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.192236 0.0129312
\(222\) 0 0
\(223\) 2.43845 0.163291 0.0816453 0.996661i \(-0.473983\pi\)
0.0816453 + 0.996661i \(0.473983\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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 \(-0.382987\pi\)
0.359383 + 0.933190i \(0.382987\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.12311 −0.335626 −0.167813 0.985819i \(-0.553670\pi\)
−0.167813 + 0.985819i \(0.553670\pi\)
\(234\) 0 0
\(235\) 8.68466 0.566525
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.8078 1.28126 0.640629 0.767851i \(-0.278674\pi\)
0.640629 + 0.767851i \(0.278674\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 3.12311 0.198718
\(248\) 0 0
\(249\) 0 0
\(250\) 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 0
\(256\) 0 0
\(257\) 10.4924 0.654499 0.327250 0.944938i \(-0.393878\pi\)
0.327250 + 0.944938i \(0.393878\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.8769 −0.794023 −0.397012 0.917814i \(-0.629953\pi\)
−0.397012 + 0.917814i \(0.629953\pi\)
\(264\) 0 0
\(265\) −5.12311 −0.314710
\(266\) 0 0
\(267\) 0 0
\(268\) 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 \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 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 \(-0.620988\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(282\) 0 0
\(283\) 11.3153 0.672627 0.336314 0.941750i \(-0.390820\pi\)
0.336314 + 0.941750i \(0.390820\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.12311 −0.302407
\(288\) 0 0
\(289\) −16.8078 −0.988692
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) −0.876894 −0.0505434
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.3693 −0.880045
\(306\) 0 0
\(307\) 19.3153 1.10238 0.551192 0.834378i \(-0.314173\pi\)
0.551192 + 0.834378i \(0.314173\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 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 \(-0.717044\pi\)
−0.630241 + 0.776400i \(0.717044\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.4924 −0.589313 −0.294657 0.955603i \(-0.595205\pi\)
−0.294657 + 0.955603i \(0.595205\pi\)
\(318\) 0 0
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.12311 0.173774
\(324\) 0 0
\(325\) 0.438447 0.0243207
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.68466 −0.478801
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.12311 0.382388 0.191194 0.981552i \(-0.438764\pi\)
0.191194 + 0.981552i \(0.438764\pi\)
\(348\) 0 0
\(349\) 10.4924 0.561646 0.280823 0.959760i \(-0.409393\pi\)
0.280823 + 0.959760i \(0.409393\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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 \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.2462 0.640996
\(366\) 0 0
\(367\) 8.68466 0.453335 0.226668 0.973972i \(-0.427217\pi\)
0.226668 + 0.973972i \(0.427217\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.12311 0.265978
\(372\) 0 0
\(373\) 4.63068 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 0 0
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 1.56155 0.0795841
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.9309 1.26405 0.632023 0.774950i \(-0.282225\pi\)
0.632023 + 0.774950i \(0.282225\pi\)
\(390\) 0 0
\(391\) 1.36932 0.0692493
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.43845 −0.122692
\(396\) 0 0
\(397\) 27.5616 1.38327 0.691637 0.722245i \(-0.256890\pi\)
0.691637 + 0.722245i \(0.256890\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −31.5616 −1.57611 −0.788054 0.615606i \(-0.788911\pi\)
−0.788054 + 0.615606i \(0.788911\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.36932 −0.464420
\(408\) 0 0
\(409\) 6.49242 0.321030 0.160515 0.987033i \(-0.448685\pi\)
0.160515 + 0.987033i \(0.448685\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 0.438447 0.0212678
\(426\) 0 0
\(427\) 15.3693 0.743773
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.8078 −0.954106 −0.477053 0.878874i \(-0.658295\pi\)
−0.477053 + 0.878874i \(0.658295\pi\)
\(432\) 0 0
\(433\) 8.24621 0.396288 0.198144 0.980173i \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.2462 1.06418
\(438\) 0 0
\(439\) −9.36932 −0.447173 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.63068 −0.124988 −0.0624938 0.998045i \(-0.519905\pi\)
−0.0624938 + 0.998045i \(0.519905\pi\)
\(444\) 0 0
\(445\) −1.12311 −0.0532403
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.80776 0.0853137 0.0426568 0.999090i \(-0.486418\pi\)
0.0426568 + 0.999090i \(0.486418\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.438447 −0.0205547
\(456\) 0 0
\(457\) −17.1231 −0.800985 −0.400493 0.916300i \(-0.631161\pi\)
−0.400493 + 0.916300i \(0.631161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 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
\(469\) −10.2462 −0.473126
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.36932 0.0629613
\(474\) 0 0
\(475\) 7.12311 0.326831
\(476\) 0 0
\(477\) 0 0
\(478\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) −41.1771 −1.85830 −0.929148 0.369708i \(-0.879458\pi\)
−0.929148 + 0.369708i \(0.879458\pi\)
\(492\) 0 0
\(493\) −2.93087 −0.132000
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −41.1771 −1.84334 −0.921670 0.387976i \(-0.873174\pi\)
−0.921670 + 0.387976i \(0.873174\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 0 0
\(509\) 11.7538 0.520978 0.260489 0.965477i \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 0 0
\(511\) −12.2462 −0.541740
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.56155 0.245071
\(516\) 0 0
\(517\) 13.5616 0.596436
\(518\) 0 0
\(519\) 0 0
\(520\) 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 \(-0.846050\pi\)
−0.885305 + 0.465011i \(0.846050\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.24621 −0.0972942
\(534\) 0 0
\(535\) −13.3693 −0.578006
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.56155 −0.0672608
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) −47.6155 −2.02849
\(552\) 0 0
\(553\) 2.43845 0.103693
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.1231 0.556044 0.278022 0.960575i \(-0.410321\pi\)
0.278022 + 0.960575i \(0.410321\pi\)
\(558\) 0 0
\(559\) −0.384472 −0.0162614
\(560\) 0 0
\(561\) 0 0
\(562\) 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 \(-0.274988\pi\)
0.649476 + 0.760382i \(0.274988\pi\)
\(570\) 0 0
\(571\) −40.4924 −1.69456 −0.847278 0.531150i \(-0.821760\pi\)
−0.847278 + 0.531150i \(0.821760\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.12311 0.130243
\(576\) 0 0
\(577\) −24.0540 −1.00138 −0.500690 0.865627i \(-0.666920\pi\)
−0.500690 + 0.865627i \(0.666920\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 0 0
\(593\) 27.5616 1.13182 0.565909 0.824468i \(-0.308525\pi\)
0.565909 + 0.824468i \(0.308525\pi\)
\(594\) 0 0
\(595\) −0.438447 −0.0179746
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.8078 −0.482452 −0.241226 0.970469i \(-0.577550\pi\)
−0.241226 + 0.970469i \(0.577550\pi\)
\(600\) 0 0
\(601\) 6.49242 0.264831 0.132416 0.991194i \(-0.457727\pi\)
0.132416 + 0.991194i \(0.457727\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) 42.0540 1.70692 0.853459 0.521160i \(-0.174500\pi\)
0.853459 + 0.521160i \(0.174500\pi\)
\(608\) 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
\(615\) 0 0
\(616\) 0 0
\(617\) −32.2462 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(618\) 0 0
\(619\) −32.1080 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.12311 0.0449963
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) −6.24621 −0.247873
\(636\) 0 0
\(637\) 0.438447 0.0173719
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −1.56155 −0.0615816 −0.0307908 0.999526i \(-0.509803\pi\)
−0.0307908 + 0.999526i \(0.509803\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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 \(-0.274678\pi\)
0.650216 + 0.759750i \(0.274678\pi\)
\(654\) 0 0
\(655\) 0.876894 0.0342631
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.17708 0.357488 0.178744 0.983896i \(-0.442797\pi\)
0.178744 + 0.983896i \(0.442797\pi\)
\(660\) 0 0
\(661\) −5.12311 −0.199266 −0.0996329 0.995024i \(-0.531767\pi\)
−0.0996329 + 0.995024i \(0.531767\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.12311 −0.276222
\(666\) 0 0
\(667\) −20.8769 −0.808357
\(668\) 0 0
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 0 0
\(677\) −4.93087 −0.189509 −0.0947544 0.995501i \(-0.530207\pi\)
−0.0947544 + 0.995501i \(0.530207\pi\)
\(678\) 0 0
\(679\) 5.80776 0.222882
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.73863 −0.257847 −0.128923 0.991655i \(-0.541152\pi\)
−0.128923 + 0.991655i \(0.541152\pi\)
\(684\) 0 0
\(685\) −17.1231 −0.654240
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.24621 0.0855738
\(690\) 0 0
\(691\) 24.4924 0.931736 0.465868 0.884854i \(-0.345742\pi\)
0.465868 + 0.884854i \(0.345742\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.1231 −0.573652
\(696\) 0 0
\(697\) −2.24621 −0.0850813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.9309 −1.09270 −0.546352 0.837556i \(-0.683984\pi\)
−0.546352 + 0.837556i \(0.683984\pi\)
\(702\) 0 0
\(703\) 42.7386 1.61192
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.2462 0.611002
\(708\) 0 0
\(709\) 27.1771 1.02066 0.510328 0.859980i \(-0.329524\pi\)
0.510328 + 0.859980i \(0.329524\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.684658 0.0256048
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) −5.56155 −0.207123
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.68466 −0.248262
\(726\) 0 0
\(727\) −52.4924 −1.94684 −0.973418 0.229035i \(-0.926443\pi\)
−0.973418 + 0.229035i \(0.926443\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.384472 −0.0142202
\(732\) 0 0
\(733\) 6.68466 0.246903 0.123452 0.992351i \(-0.460604\pi\)
0.123452 + 0.992351i \(0.460604\pi\)
\(734\) 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
\(741\) 0 0
\(742\) 0 0
\(743\) −32.9848 −1.21010 −0.605048 0.796189i \(-0.706846\pi\)
−0.605048 + 0.796189i \(0.706846\pi\)
\(744\) 0 0
\(745\) 12.2462 0.448666
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.3693 0.488504
\(750\) 0 0
\(751\) −17.0691 −0.622861 −0.311431 0.950269i \(-0.600808\pi\)
−0.311431 + 0.950269i \(0.600808\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 0 0
\(763\) 5.31534 0.192428
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.75379 −0.0633256
\(768\) 0 0
\(769\) −42.4924 −1.53232 −0.766158 0.642652i \(-0.777834\pi\)
−0.766158 + 0.642652i \(0.777834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 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
\(783\) 0 0
\(784\) 0 0
\(785\) −20.2462 −0.722618
\(786\) 0 0
\(787\) 49.1771 1.75297 0.876487 0.481426i \(-0.159881\pi\)
0.876487 + 0.481426i \(0.159881\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 6.73863 0.239296
\(794\) 0 0
\(795\) 0 0
\(796\) 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 0
\(802\) 0 0
\(803\) 19.1231 0.674840
\(804\) 0 0
\(805\) −3.12311 −0.110075
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.5464 −0.581740 −0.290870 0.956763i \(-0.593945\pi\)
−0.290870 + 0.956763i \(0.593945\pi\)
\(810\) 0 0
\(811\) −19.6155 −0.688794 −0.344397 0.938824i \(-0.611917\pi\)
−0.344397 + 0.938824i \(0.611917\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 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 \(-0.378041\pi\)
0.373839 + 0.927494i \(0.378041\pi\)
\(822\) 0 0
\(823\) 36.4924 1.27205 0.636023 0.771670i \(-0.280578\pi\)
0.636023 + 0.771670i \(0.280578\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.36932 −0.186709 −0.0933547 0.995633i \(-0.529759\pi\)
−0.0933547 + 0.995633i \(0.529759\pi\)
\(828\) 0 0
\(829\) 34.8769 1.21132 0.605662 0.795722i \(-0.292908\pi\)
0.605662 + 0.795722i \(0.292908\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.438447 0.0151913
\(834\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) 12.8078 0.440600
\(846\) 0 0
\(847\) −8.56155 −0.294178
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 18.7386 0.642352
\(852\) 0 0
\(853\) −7.26137 −0.248624 −0.124312 0.992243i \(-0.539672\pi\)
−0.124312 + 0.992243i \(0.539672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 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 \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.7538 −0.876669 −0.438335 0.898812i \(-0.644431\pi\)
−0.438335 + 0.898812i \(0.644431\pi\)
\(864\) 0 0
\(865\) −4.43845 −0.150912
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) 0 0
\(871\) −4.49242 −0.152220
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) 0 0
\(887\) 20.4924 0.688068 0.344034 0.938957i \(-0.388206\pi\)
0.344034 + 0.938957i \(0.388206\pi\)
\(888\) 0 0
\(889\) 6.24621 0.209491
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −61.8617 −2.07012
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 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
\(909\) 0 0
\(910\) 0 0
\(911\) −28.4924 −0.943996 −0.471998 0.881600i \(-0.656467\pi\)
−0.471998 + 0.881600i \(0.656467\pi\)
\(912\) 0 0
\(913\) −6.24621 −0.206719
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.876894 −0.0289576
\(918\) 0 0
\(919\) −40.3002 −1.32938 −0.664690 0.747119i \(-0.731436\pi\)
−0.664690 + 0.747119i \(0.731436\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.50758 0.115453
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.1080 −0.725338 −0.362669 0.931918i \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(930\) 0 0
\(931\) 7.12311 0.233450
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.684658 0.0223907
\(936\) 0 0
\(937\) −55.6695 −1.81864 −0.909322 0.416094i \(-0.863399\pi\)
−0.909322 + 0.416094i \(0.863399\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 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 \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −5.36932 −0.174295
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 33.1231 1.07296 0.536481 0.843912i \(-0.319753\pi\)
0.536481 + 0.843912i \(0.319753\pi\)
\(954\) 0 0
\(955\) 13.5616 0.438842
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.1231 0.552934
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 0 0
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) 15.1231 0.484825
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.2311 −1.06316 −0.531578 0.847009i \(-0.678401\pi\)
−0.531578 + 0.847009i \(0.678401\pi\)
\(978\) 0 0
\(979\) −1.75379 −0.0560513
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) −1.75379 −0.0555988
\(996\) 0 0
\(997\) −2.68466 −0.0850240 −0.0425120 0.999096i \(-0.513536\pi\)
−0.0425120 + 0.999096i \(0.513536\pi\)
\(998\) 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 5040.2.a.bt.1.1 2
3.2 odd 2 560.2.a.i.1.1 2
4.3 odd 2 315.2.a.e.1.2 2
12.11 even 2 35.2.a.b.1.1 2
15.2 even 4 2800.2.g.t.449.3 4
15.8 even 4 2800.2.g.t.449.2 4
15.14 odd 2 2800.2.a.bi.1.2 2
20.3 even 4 1575.2.d.e.1324.1 4
20.7 even 4 1575.2.d.e.1324.4 4
20.19 odd 2 1575.2.a.p.1.1 2
21.20 even 2 3920.2.a.bs.1.2 2
24.5 odd 2 2240.2.a.bd.1.2 2
24.11 even 2 2240.2.a.bh.1.1 2
28.27 even 2 2205.2.a.x.1.2 2
60.23 odd 4 175.2.b.b.99.4 4
60.47 odd 4 175.2.b.b.99.1 4
60.59 even 2 175.2.a.f.1.2 2
84.11 even 6 245.2.e.i.226.2 4
84.23 even 6 245.2.e.i.116.2 4
84.47 odd 6 245.2.e.h.116.2 4
84.59 odd 6 245.2.e.h.226.2 4
84.83 odd 2 245.2.a.d.1.1 2
132.131 odd 2 4235.2.a.m.1.2 2
156.155 even 2 5915.2.a.l.1.2 2
420.83 even 4 1225.2.b.f.99.4 4
420.167 even 4 1225.2.b.f.99.1 4
420.419 odd 2 1225.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 12.11 even 2
175.2.a.f.1.2 2 60.59 even 2
175.2.b.b.99.1 4 60.47 odd 4
175.2.b.b.99.4 4 60.23 odd 4
245.2.a.d.1.1 2 84.83 odd 2
245.2.e.h.116.2 4 84.47 odd 6
245.2.e.h.226.2 4 84.59 odd 6
245.2.e.i.116.2 4 84.23 even 6
245.2.e.i.226.2 4 84.11 even 6
315.2.a.e.1.2 2 4.3 odd 2
560.2.a.i.1.1 2 3.2 odd 2
1225.2.a.s.1.2 2 420.419 odd 2
1225.2.b.f.99.1 4 420.167 even 4
1225.2.b.f.99.4 4 420.83 even 4
1575.2.a.p.1.1 2 20.19 odd 2
1575.2.d.e.1324.1 4 20.3 even 4
1575.2.d.e.1324.4 4 20.7 even 4
2205.2.a.x.1.2 2 28.27 even 2
2240.2.a.bd.1.2 2 24.5 odd 2
2240.2.a.bh.1.1 2 24.11 even 2
2800.2.a.bi.1.2 2 15.14 odd 2
2800.2.g.t.449.2 4 15.8 even 4
2800.2.g.t.449.3 4 15.2 even 4
3920.2.a.bs.1.2 2 21.20 even 2
4235.2.a.m.1.2 2 132.131 odd 2
5040.2.a.bt.1.1 2 1.1 even 1 trivial
5915.2.a.l.1.2 2 156.155 even 2