Properties

Label 5400.2.f.b.649.2
Level $5400$
Weight $2$
Character 5400.649
Analytic conductor $43.119$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5400.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(43.1192170915\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5400.649
Dual form 5400.2.f.b.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{7} +O(q^{10})\) \(q+3.00000i q^{7} -5.00000 q^{11} -4.00000i q^{13} +8.00000i q^{17} -2.00000 q^{19} +2.00000i q^{23} +6.00000 q^{29} -7.00000 q^{31} -6.00000i q^{37} +6.00000 q^{41} +2.00000i q^{43} -6.00000i q^{47} -2.00000 q^{49} +5.00000i q^{53} -4.00000 q^{59} -8.00000 q^{61} -10.0000i q^{67} +8.00000 q^{71} -1.00000i q^{73} -15.0000i q^{77} -16.0000 q^{79} -11.0000i q^{83} +6.00000 q^{89} +12.0000 q^{91} -1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 10q^{11} - 4q^{19} + 12q^{29} - 14q^{31} + 12q^{41} - 4q^{49} - 8q^{59} - 16q^{61} + 16q^{71} - 32q^{79} + 12q^{89} + 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\) \(2701\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000i 0.686803i 0.939189 + 0.343401i \(0.111579\pi\)
−0.939189 + 0.343401i \(0.888421\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 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
\(73\) − 1.00000i − 0.117041i −0.998286 0.0585206i \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15.0000i − 1.70941i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 11.0000i − 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.00000i − 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 11.0000i − 0.976092i −0.872818 0.488046i \(-0.837710\pi\)
0.872818 0.488046i \(-0.162290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.0000i 1.67248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.00000 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 20.0000i − 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000i 0.464294i 0.972681 + 0.232147i \(0.0745750\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 21.0000i − 1.59660i −0.602260 0.798300i \(-0.705733\pi\)
0.602260 0.798300i \(-0.294267\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 40.0000i − 2.92509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 19.0000i 1.36765i 0.729646 + 0.683825i \(0.239685\pi\)
−0.729646 + 0.683825i \(0.760315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.0000i 1.26335i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 21.0000i − 1.42557i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 32.0000 2.15255
\(222\) 0 0
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 22.0000i − 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) − 10.0000i − 0.628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4.00000i − 0.249513i −0.992187 0.124757i \(-0.960185\pi\)
0.992187 0.124757i \(-0.0398150\pi\)
\(258\) 0 0
\(259\) 18.0000 1.11847
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.0000i 1.60323i 0.597841 + 0.801614i \(0.296025\pi\)
−0.597841 + 0.801614i \(0.703975\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0000i 1.06251i
\(288\) 0 0
\(289\) −47.0000 −2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 24.0000i 1.36975i 0.728659 + 0.684876i \(0.240144\pi\)
−0.728659 + 0.684876i \(0.759856\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) − 21.0000i − 1.18699i −0.804838 0.593495i \(-0.797748\pi\)
0.804838 0.593495i \(-0.202252\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 9.00000i − 0.505490i −0.967533 0.252745i \(-0.918667\pi\)
0.967533 0.252745i \(-0.0813334\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 16.0000i − 0.890264i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.00000i − 0.326841i −0.986557 0.163420i \(-0.947747\pi\)
0.986557 0.163420i \(-0.0522527\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 35.0000 1.89536
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.0000i − 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.0000i 1.17094i 0.810693 + 0.585471i \(0.199090\pi\)
−0.810693 + 0.585471i \(0.800910\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 11.0000i − 0.574195i −0.957901 0.287098i \(-0.907310\pi\)
0.957901 0.287098i \(-0.0926904\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) − 8.00000i − 0.414224i −0.978317 0.207112i \(-0.933593\pi\)
0.978317 0.207112i \(-0.0664065\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.00000 −0.0507020 −0.0253510 0.999679i \(-0.508070\pi\)
−0.0253510 + 0.999679i \(0.508070\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 4.00000i − 0.200754i −0.994949 0.100377i \(-0.967995\pi\)
0.994949 0.100377i \(-0.0320049\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 28.0000i 1.39478i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.0000i 1.48704i
\(408\) 0 0
\(409\) 9.00000 0.445021 0.222511 0.974930i \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 24.0000i − 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −34.0000 −1.63772 −0.818861 0.573992i \(-0.805394\pi\)
−0.818861 + 0.573992i \(0.805394\pi\)
\(432\) 0 0
\(433\) − 13.0000i − 0.624740i −0.949960 0.312370i \(-0.898877\pi\)
0.949960 0.312370i \(-0.101123\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −9.00000 −0.429547 −0.214773 0.976664i \(-0.568901\pi\)
−0.214773 + 0.976664i \(0.568901\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.00000i − 0.0467780i −0.999726 0.0233890i \(-0.992554\pi\)
0.999726 0.0233890i \(-0.00744563\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00000 0.0465746 0.0232873 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) 1.00000i 0.0464739i 0.999730 + 0.0232370i \(0.00739722\pi\)
−0.999730 + 0.0232370i \(0.992603\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.00000i 0.138823i 0.997588 + 0.0694117i \(0.0221122\pi\)
−0.997588 + 0.0694117i \(0.977888\pi\)
\(468\) 0 0
\(469\) 30.0000 1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 10.0000i − 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 48.0000i 2.16181i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000i 1.31940i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 56.0000i − 2.43940i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 36.0000 1.54776 0.773880 0.633332i \(-0.218313\pi\)
0.773880 + 0.633332i \(0.218313\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) − 48.0000i − 2.04117i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000i 0.0423714i 0.999776 + 0.0211857i \(0.00674412\pi\)
−0.999776 + 0.0211857i \(0.993256\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.0000i 0.463595i 0.972764 + 0.231797i \(0.0744606\pi\)
−0.972764 + 0.231797i \(0.925539\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 10.0000i − 0.416305i −0.978096 0.208153i \(-0.933255\pi\)
0.978096 0.208153i \(-0.0667451\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.0000 1.36907
\(582\) 0 0
\(583\) − 25.0000i − 1.03539i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.0000i 0.784214i 0.919920 + 0.392107i \(0.128254\pi\)
−0.919920 + 0.392107i \(0.871746\pi\)
\(588\) 0 0
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.0000i 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) 43.0000 1.71180 0.855901 0.517139i \(-0.173003\pi\)
0.855901 + 0.517139i \(0.173003\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.00000i 0.316972i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) − 12.0000i − 0.473234i −0.971603 0.236617i \(-0.923961\pi\)
0.971603 0.236617i \(-0.0760386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 16.0000i − 0.629025i −0.949253 0.314512i \(-0.898159\pi\)
0.949253 0.314512i \(-0.101841\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000i 0.117399i 0.998276 + 0.0586995i \(0.0186954\pi\)
−0.998276 + 0.0586995i \(0.981305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) − 13.0000i − 0.501113i −0.968102 0.250557i \(-0.919386\pi\)
0.968102 0.250557i \(-0.0806136\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 0 0
\(679\) 3.00000 0.115129
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000i 1.81813i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) 12.0000i 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 27.0000i − 1.01544i
\(708\) 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
\(713\) − 14.0000i − 0.524304i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000i 0.111264i 0.998451 + 0.0556319i \(0.0177173\pi\)
−0.998451 + 0.0556319i \(0.982283\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 50.0000i 1.84177i
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.0000i 0.733729i 0.930274 + 0.366864i \(0.119569\pi\)
−0.930274 + 0.366864i \(0.880431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.0000 0.986559
\(750\) 0 0
\(751\) 45.0000 1.64207 0.821037 0.570875i \(-0.193396\pi\)
0.821037 + 0.570875i \(0.193396\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 46.0000i − 1.67190i −0.548807 0.835949i \(-0.684918\pi\)
0.548807 0.835949i \(-0.315082\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.0000 1.16000 0.580000 0.814617i \(-0.303053\pi\)
0.580000 + 0.814617i \(0.303053\pi\)
\(762\) 0 0
\(763\) − 30.0000i − 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000i 0.577727i
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 32.0000i 1.13635i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 35.0000i − 1.23976i −0.784695 0.619882i \(-0.787181\pi\)
0.784695 0.619882i \(-0.212819\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.00000i 0.176446i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.00000i − 0.139942i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) 43.0000i 1.49889i 0.662069 + 0.749443i \(0.269679\pi\)
−0.662069 + 0.749443i \(0.730321\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.0000i − 0.554367i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 42.0000i 1.44314i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000i 0.819824i 0.912125 + 0.409912i \(0.134441\pi\)
−0.912125 + 0.409912i \(0.865559\pi\)
\(858\) 0 0
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 52.0000i − 1.77010i −0.465495 0.885050i \(-0.654124\pi\)
0.465495 0.885050i \(-0.345876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 80.0000 2.71381
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.0000i 1.41824i 0.705088 + 0.709120i \(0.250907\pi\)
−0.705088 + 0.709120i \(0.749093\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) 46.0000i 1.54802i 0.633171 + 0.774012i \(0.281753\pi\)
−0.633171 + 0.774012i \(0.718247\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 4.00000i − 0.134307i −0.997743 0.0671534i \(-0.978608\pi\)
0.997743 0.0671534i \(-0.0213917\pi\)
\(888\) 0 0
\(889\) 33.0000 1.10678
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −42.0000 −1.40078
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 58.0000i 1.92586i 0.269754 + 0.962929i \(0.413058\pi\)
−0.269754 + 0.962929i \(0.586942\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38.0000 −1.25900 −0.629498 0.777002i \(-0.716739\pi\)
−0.629498 + 0.777002i \(0.716739\pi\)
\(912\) 0 0
\(913\) 55.0000i 1.82023i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3.00000i − 0.0990687i
\(918\) 0 0
\(919\) 47.0000 1.55039 0.775193 0.631724i \(-0.217652\pi\)
0.775193 + 0.631724i \(0.217652\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 32.0000i − 1.05329i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 21.0000i − 0.686040i −0.939328 0.343020i \(-0.888550\pi\)
0.939328 0.343020i \(-0.111450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 45.0000 1.46696 0.733479 0.679712i \(-0.237895\pi\)
0.733479 + 0.679712i \(0.237895\pi\)
\(942\) 0 0
\(943\) 12.0000i 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.00000i − 0.0974869i −0.998811 0.0487435i \(-0.984478\pi\)
0.998811 0.0487435i \(-0.0155217\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 36.0000i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −54.0000 −1.74375
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 35.0000i 1.12552i 0.826619 + 0.562762i \(0.190261\pi\)
−0.826619 + 0.562762i \(0.809739\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) − 36.0000i − 1.15411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.00000i 0.191370i 0.995412 + 0.0956851i \(0.0305042\pi\)
−0.995412 + 0.0956851i \(0.969496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0000i 0.886769i 0.896332 + 0.443384i \(0.146222\pi\)
−0.896332 + 0.443384i \(0.853778\pi\)
\(998\) 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 5400.2.f.b.649.2 2
3.2 odd 2 5400.2.f.z.649.2 2
5.2 odd 4 5400.2.a.e.1.1 1
5.3 odd 4 216.2.a.c.1.1 yes 1
5.4 even 2 inner 5400.2.f.b.649.1 2
15.2 even 4 5400.2.a.h.1.1 1
15.8 even 4 216.2.a.b.1.1 1
15.14 odd 2 5400.2.f.z.649.1 2
20.3 even 4 432.2.a.f.1.1 1
40.3 even 4 1728.2.a.i.1.1 1
40.13 odd 4 1728.2.a.l.1.1 1
45.13 odd 12 648.2.i.c.217.1 2
45.23 even 12 648.2.i.e.217.1 2
45.38 even 12 648.2.i.e.433.1 2
45.43 odd 12 648.2.i.c.433.1 2
60.23 odd 4 432.2.a.c.1.1 1
120.53 even 4 1728.2.a.s.1.1 1
120.83 odd 4 1728.2.a.r.1.1 1
180.23 odd 12 1296.2.i.l.865.1 2
180.43 even 12 1296.2.i.g.433.1 2
180.83 odd 12 1296.2.i.l.433.1 2
180.103 even 12 1296.2.i.g.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.a.b.1.1 1 15.8 even 4
216.2.a.c.1.1 yes 1 5.3 odd 4
432.2.a.c.1.1 1 60.23 odd 4
432.2.a.f.1.1 1 20.3 even 4
648.2.i.c.217.1 2 45.13 odd 12
648.2.i.c.433.1 2 45.43 odd 12
648.2.i.e.217.1 2 45.23 even 12
648.2.i.e.433.1 2 45.38 even 12
1296.2.i.g.433.1 2 180.43 even 12
1296.2.i.g.865.1 2 180.103 even 12
1296.2.i.l.433.1 2 180.83 odd 12
1296.2.i.l.865.1 2 180.23 odd 12
1728.2.a.i.1.1 1 40.3 even 4
1728.2.a.l.1.1 1 40.13 odd 4
1728.2.a.r.1.1 1 120.83 odd 4
1728.2.a.s.1.1 1 120.53 even 4
5400.2.a.e.1.1 1 5.2 odd 4
5400.2.a.h.1.1 1 15.2 even 4
5400.2.f.b.649.1 2 5.4 even 2 inner
5400.2.f.b.649.2 2 1.1 even 1 trivial
5400.2.f.z.649.1 2 15.14 odd 2
5400.2.f.z.649.2 2 3.2 odd 2