Properties

Label 5175.2.a.a.1.1
Level $5175$
Weight $2$
Character 5175.1
Self dual yes
Analytic conductor $41.323$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3225830460\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5175.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{7} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +5.00000 q^{17} +8.00000 q^{19} +1.00000 q^{23} -4.00000 q^{26} +2.00000 q^{28} +5.00000 q^{29} -5.00000 q^{31} +8.00000 q^{32} -10.0000 q^{34} +7.00000 q^{37} -16.0000 q^{38} +7.00000 q^{41} +4.00000 q^{43} -2.00000 q^{46} +2.00000 q^{47} -6.00000 q^{49} +4.00000 q^{52} +1.00000 q^{53} -10.0000 q^{58} -3.00000 q^{59} -6.00000 q^{61} +10.0000 q^{62} -8.00000 q^{64} +13.0000 q^{67} +10.0000 q^{68} -13.0000 q^{71} +8.00000 q^{73} -14.0000 q^{74} +16.0000 q^{76} -14.0000 q^{79} -14.0000 q^{82} +3.00000 q^{83} -8.00000 q^{86} +14.0000 q^{89} +2.00000 q^{91} +2.00000 q^{92} -4.00000 q^{94} +14.0000 q^{97} +12.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −16.0000 −2.59554
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 10.0000 1.21268
\(69\) 0 0
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −14.0000 −1.62747
\(75\) 0 0
\(76\) 16.0000 1.83533
\(77\) 0 0
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.0000 −1.54604
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 12.0000 1.21218
\(99\) 0 0
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 12.0000 1.08643
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) −26.0000 −2.24606
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 26.0000 2.18187
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 14.0000 1.15079
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 28.0000 2.22756
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 14.0000 1.09322
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −28.0000 −2.09869
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) −28.0000 −2.01028
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 30.0000 2.11079
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) −36.0000 −2.43823
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −10.0000 −0.648204
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 22.0000 1.41421
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 40.0000 2.50982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.0000 0.801614 0.400807 0.916162i \(-0.368730\pi\)
0.400807 + 0.916162i \(0.368730\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) 26.0000 1.58820
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −20.0000 −1.21268
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −18.0000 −1.07957
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) −26.0000 −1.54282
\(285\) 0 0
\(286\) 0 0
\(287\) 7.00000 0.413197
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −29.0000 −1.69420 −0.847099 0.531435i \(-0.821653\pi\)
−0.847099 + 0.531435i \(0.821653\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −32.0000 −1.83533
\(305\) 0 0
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −28.0000 −1.57512
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 40.0000 2.22566
\(324\) 0 0
\(325\) 0 0
\(326\) −48.0000 −2.65847
\(327\) 0 0
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 32.0000 1.75096
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) 23.0000 1.23116 0.615581 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 28.0000 1.48400
\(357\) 0 0
\(358\) −8.00000 −0.422813
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 28.0000 1.47165
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −3.00000 −0.153293 −0.0766464 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 0 0
\(388\) 28.0000 1.42148
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) −30.0000 −1.49256
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) 0 0
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 18.0000 0.876226
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 36.0000 1.72409
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) −8.00000 −0.373815
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 13.0000 0.600284
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 10.0000 0.458349
\(477\) 0 0
\(478\) −2.00000 −0.0914779
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) −12.0000 −0.546585
\(483\) 0 0
\(484\) −22.0000 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.0000 1.39901 0.699505 0.714628i \(-0.253404\pi\)
0.699505 + 0.714628i \(0.253404\pi\)
\(492\) 0 0
\(493\) 25.0000 1.12594
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) −13.0000 −0.583130
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 36.0000 1.60676
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −40.0000 −1.77471
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 44.0000 1.94076
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −14.0000 −0.615125
\(519\) 0 0
\(520\) 0 0
\(521\) 20.0000 0.876216 0.438108 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) −25.0000 −1.08902
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) 14.0000 0.606407
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 30.0000 1.28861
\(543\) 0 0
\(544\) 40.0000 1.71499
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) 52.0000 2.20927
\(555\) 0 0
\(556\) 18.0000 0.763370
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 25.0000 1.05362 0.526812 0.849982i \(-0.323387\pi\)
0.526812 + 0.849982i \(0.323387\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −14.0000 −0.584349
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 58.0000 2.39596
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 64.0000 2.59554
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 47.0000 1.89215 0.946074 0.323949i \(-0.105011\pi\)
0.946074 + 0.323949i \(0.105011\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 0 0
\(626\) −42.0000 −1.67866
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 35.0000 1.39554
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −48.0000 −1.90632
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 37.0000 1.45914 0.729569 0.683907i \(-0.239721\pi\)
0.729569 + 0.683907i \(0.239721\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) −80.0000 −3.14756
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 48.0000 1.87983
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −28.0000 −1.09322
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) −32.0000 −1.23812
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −52.0000 −2.00297
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −46.0000 −1.76014 −0.880071 0.474843i \(-0.842505\pi\)
−0.880071 + 0.474843i \(0.842505\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.0000 0.992685
\(687\) 0 0
\(688\) −16.0000 −0.609994
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) 35.0000 1.32572
\(698\) −46.0000 −1.74113
\(699\) 0 0
\(700\) 0 0
\(701\) 28.0000 1.05755 0.528773 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) 0 0
\(703\) 56.0000 2.11208
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 8.00000 0.298974
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −90.0000 −3.34945
\(723\) 0 0
\(724\) −28.0000 −1.04061
\(725\) 0 0
\(726\) 0 0
\(727\) 27.0000 1.00137 0.500687 0.865628i \(-0.333081\pi\)
0.500687 + 0.865628i \(0.333081\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 0 0
\(739\) 7.00000 0.257499 0.128750 0.991677i \(-0.458904\pi\)
0.128750 + 0.991677i \(0.458904\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) 0 0
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) −35.0000 −1.27210 −0.636048 0.771649i \(-0.719432\pi\)
−0.636048 + 0.771649i \(0.719432\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) −25.0000 −0.906249 −0.453125 0.891447i \(-0.649691\pi\)
−0.453125 + 0.891447i \(0.649691\pi\)
\(762\) 0 0
\(763\) 18.0000 0.651644
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.0000 −0.863779
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 44.0000 1.57748
\(779\) 56.0000 2.00641
\(780\) 0 0
\(781\) 0 0
\(782\) −10.0000 −0.357599
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) 0 0
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −32.0000 −1.13564
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 15.0000 0.531327 0.265664 0.964066i \(-0.414409\pi\)
0.265664 + 0.964066i \(0.414409\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) 0 0
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 0 0
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) 0 0
\(811\) 31.0000 1.08856 0.544279 0.838905i \(-0.316803\pi\)
0.544279 + 0.838905i \(0.316803\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) 37.0000 1.28506 0.642532 0.766259i \(-0.277884\pi\)
0.642532 + 0.766259i \(0.277884\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16.0000 −0.554700
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) −18.0000 −0.619586
\(845\) 0 0
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 12.0000 0.410632
\(855\) 0 0
\(856\) 0 0
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 43.0000 1.46714 0.733571 0.679613i \(-0.237852\pi\)
0.733571 + 0.679613i \(0.237852\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) 0 0
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 26.0000 0.880976
\(872\) 0 0
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 56.0000 1.88991
\(879\) 0 0
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 20.0000 0.672673
\(885\) 0 0
\(886\) −48.0000 −1.61259
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) −28.0000 −0.937509
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −25.0000 −0.833797
\(900\) 0 0
\(901\) 5.00000 0.166574
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.00000 0.298840 0.149420 0.988774i \(-0.452259\pi\)
0.149420 + 0.988774i \(0.452259\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 0 0
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −36.0000 −1.18560
\(923\) −26.0000 −0.855800
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) 40.0000 1.31306
\(929\) −35.0000 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) 0 0
\(931\) −48.0000 −1.57314
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) −54.0000 −1.76693
\(935\) 0 0
\(936\) 0 0
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) −26.0000 −0.848930
\(939\) 0 0
\(940\) 0 0
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 0 0
\(943\) 7.00000 0.227951
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 −0.454939 −0.227469 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) 0 0
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2.00000 0.0646846
\(957\) 0 0
\(958\) −56.0000 −1.80928
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −28.0000 −0.902756
\(963\) 0 0
\(964\) 12.0000 0.386494
\(965\) 0 0
\(966\) 0 0
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) 9.00000 0.288527
\(974\) 64.0000 2.05069
\(975\) 0 0
\(976\) 24.0000 0.768221
\(977\) 39.0000 1.24772 0.623860 0.781536i \(-0.285563\pi\)
0.623860 + 0.781536i \(0.285563\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −62.0000 −1.97850
\(983\) −21.0000 −0.669796 −0.334898 0.942254i \(-0.608702\pi\)
−0.334898 + 0.942254i \(0.608702\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −50.0000 −1.59232
\(987\) 0 0
\(988\) 32.0000 1.01806
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) −40.0000 −1.27000
\(993\) 0 0
\(994\) 26.0000 0.824670
\(995\) 0 0
\(996\) 0 0
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 22.0000 0.696398
\(999\) 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 5175.2.a.a.1.1 1
3.2 odd 2 575.2.a.e.1.1 1
5.2 odd 4 1035.2.b.a.829.1 2
5.3 odd 4 1035.2.b.a.829.2 2
5.4 even 2 5175.2.a.z.1.1 1
12.11 even 2 9200.2.a.g.1.1 1
15.2 even 4 115.2.b.a.24.2 yes 2
15.8 even 4 115.2.b.a.24.1 2
15.14 odd 2 575.2.a.a.1.1 1
60.23 odd 4 1840.2.e.b.369.1 2
60.47 odd 4 1840.2.e.b.369.2 2
60.59 even 2 9200.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.b.a.24.1 2 15.8 even 4
115.2.b.a.24.2 yes 2 15.2 even 4
575.2.a.a.1.1 1 15.14 odd 2
575.2.a.e.1.1 1 3.2 odd 2
1035.2.b.a.829.1 2 5.2 odd 4
1035.2.b.a.829.2 2 5.3 odd 4
1840.2.e.b.369.1 2 60.23 odd 4
1840.2.e.b.369.2 2 60.47 odd 4
5175.2.a.a.1.1 1 1.1 even 1 trivial
5175.2.a.z.1.1 1 5.4 even 2
9200.2.a.g.1.1 1 12.11 even 2
9200.2.a.bg.1.1 1 60.59 even 2