Properties

Label 2400.2.d.e.49.5
Level 2400
Weight 2
Character 2400.49
Analytic conductor 19.164
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.5
Root \(1.40680 - 0.144584i\) of defining polynomial
Character \(\chi\) \(=\) 2400.49
Dual form 2400.2.d.e.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.62721i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.62721i q^{7} +1.00000 q^{9} +6.20555i q^{11} +0.578337 q^{13} +1.42166i q^{17} +5.62721i q^{19} -3.62721i q^{21} -5.62721i q^{23} -1.00000 q^{27} +2.00000i q^{29} +2.57834 q^{31} -6.20555i q^{33} -7.83276 q^{37} -0.578337 q^{39} +5.25443 q^{41} -7.25443 q^{43} -6.78389i q^{47} -6.15667 q^{49} -1.42166i q^{51} +2.00000 q^{53} -5.62721i q^{57} +2.20555i q^{59} -12.4111i q^{61} +3.62721i q^{63} -4.00000 q^{67} +5.62721i q^{69} -8.41110 q^{71} +6.00000i q^{73} -22.5089 q^{77} +5.42166 q^{79} +1.00000 q^{81} +3.25443 q^{83} -2.00000i q^{87} +13.2544 q^{89} +2.09775i q^{91} -2.57834 q^{93} +4.84333i q^{97} +6.20555i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 6q^{9} - 6q^{27} + 12q^{31} + 8q^{37} - 20q^{41} + 8q^{43} - 30q^{49} + 12q^{53} - 24q^{67} + 8q^{71} - 32q^{77} + 36q^{79} + 6q^{81} - 32q^{83} + 28q^{89} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.62721i 1.37096i 0.728093 + 0.685479i \(0.240407\pi\)
−0.728093 + 0.685479i \(0.759593\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.20555i 1.87104i 0.353269 + 0.935522i \(0.385070\pi\)
−0.353269 + 0.935522i \(0.614930\pi\)
\(12\) 0 0
\(13\) 0.578337 0.160402 0.0802009 0.996779i \(-0.474444\pi\)
0.0802009 + 0.996779i \(0.474444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.42166i 0.344804i 0.985027 + 0.172402i \(0.0551528\pi\)
−0.985027 + 0.172402i \(0.944847\pi\)
\(18\) 0 0
\(19\) 5.62721i 1.29097i 0.763772 + 0.645486i \(0.223345\pi\)
−0.763772 + 0.645486i \(0.776655\pi\)
\(20\) 0 0
\(21\) − 3.62721i − 0.791523i
\(22\) 0 0
\(23\) − 5.62721i − 1.17336i −0.809821 0.586678i \(-0.800436\pi\)
0.809821 0.586678i \(-0.199564\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 2.57834 0.463083 0.231542 0.972825i \(-0.425623\pi\)
0.231542 + 0.972825i \(0.425623\pi\)
\(32\) 0 0
\(33\) − 6.20555i − 1.08025i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.83276 −1.28770 −0.643849 0.765152i \(-0.722664\pi\)
−0.643849 + 0.765152i \(0.722664\pi\)
\(38\) 0 0
\(39\) −0.578337 −0.0926081
\(40\) 0 0
\(41\) 5.25443 0.820603 0.410302 0.911950i \(-0.365423\pi\)
0.410302 + 0.911950i \(0.365423\pi\)
\(42\) 0 0
\(43\) −7.25443 −1.10629 −0.553145 0.833085i \(-0.686572\pi\)
−0.553145 + 0.833085i \(0.686572\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.78389i − 0.989532i −0.869026 0.494766i \(-0.835254\pi\)
0.869026 0.494766i \(-0.164746\pi\)
\(48\) 0 0
\(49\) −6.15667 −0.879525
\(50\) 0 0
\(51\) − 1.42166i − 0.199073i
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.62721i − 0.745343i
\(58\) 0 0
\(59\) 2.20555i 0.287138i 0.989640 + 0.143569i \(0.0458579\pi\)
−0.989640 + 0.143569i \(0.954142\pi\)
\(60\) 0 0
\(61\) − 12.4111i − 1.58908i −0.607213 0.794539i \(-0.707712\pi\)
0.607213 0.794539i \(-0.292288\pi\)
\(62\) 0 0
\(63\) 3.62721i 0.456986i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 5.62721i 0.677437i
\(70\) 0 0
\(71\) −8.41110 −0.998214 −0.499107 0.866540i \(-0.666339\pi\)
−0.499107 + 0.866540i \(0.666339\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22.5089 −2.56512
\(78\) 0 0
\(79\) 5.42166 0.609985 0.304992 0.952355i \(-0.401346\pi\)
0.304992 + 0.952355i \(0.401346\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.25443 0.357220 0.178610 0.983920i \(-0.442840\pi\)
0.178610 + 0.983920i \(0.442840\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.00000i − 0.214423i
\(88\) 0 0
\(89\) 13.2544 1.40497 0.702483 0.711700i \(-0.252075\pi\)
0.702483 + 0.711700i \(0.252075\pi\)
\(90\) 0 0
\(91\) 2.09775i 0.219904i
\(92\) 0 0
\(93\) −2.57834 −0.267361
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.84333i 0.491765i 0.969300 + 0.245883i \(0.0790778\pi\)
−0.969300 + 0.245883i \(0.920922\pi\)
\(98\) 0 0
\(99\) 6.20555i 0.623681i
\(100\) 0 0
\(101\) 2.00000i 0.199007i 0.995037 + 0.0995037i \(0.0317255\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) 2.47054i 0.243429i 0.992565 + 0.121715i \(0.0388393\pi\)
−0.992565 + 0.121715i \(0.961161\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0978 −1.36288 −0.681441 0.731873i \(-0.738646\pi\)
−0.681441 + 0.731873i \(0.738646\pi\)
\(108\) 0 0
\(109\) 7.25443i 0.694848i 0.937708 + 0.347424i \(0.112944\pi\)
−0.937708 + 0.347424i \(0.887056\pi\)
\(110\) 0 0
\(111\) 7.83276 0.743453
\(112\) 0 0
\(113\) 9.08719i 0.854851i 0.904051 + 0.427425i \(0.140579\pi\)
−0.904051 + 0.427425i \(0.859421\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.578337 0.0534673
\(118\) 0 0
\(119\) −5.15667 −0.472712
\(120\) 0 0
\(121\) −27.5089 −2.50080
\(122\) 0 0
\(123\) −5.25443 −0.473776
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.4705i 0.929110i 0.885544 + 0.464555i \(0.153786\pi\)
−0.885544 + 0.464555i \(0.846214\pi\)
\(128\) 0 0
\(129\) 7.25443 0.638717
\(130\) 0 0
\(131\) − 13.4600i − 1.17600i −0.808860 0.588002i \(-0.799915\pi\)
0.808860 0.588002i \(-0.200085\pi\)
\(132\) 0 0
\(133\) −20.4111 −1.76987
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.5783i − 0.903768i −0.892077 0.451884i \(-0.850752\pi\)
0.892077 0.451884i \(-0.149248\pi\)
\(138\) 0 0
\(139\) 12.4705i 1.05774i 0.848704 + 0.528869i \(0.177384\pi\)
−0.848704 + 0.528869i \(0.822616\pi\)
\(140\) 0 0
\(141\) 6.78389i 0.571306i
\(142\) 0 0
\(143\) 3.58890i 0.300119i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.15667 0.507794
\(148\) 0 0
\(149\) 2.00000i 0.163846i 0.996639 + 0.0819232i \(0.0261062\pi\)
−0.996639 + 0.0819232i \(0.973894\pi\)
\(150\) 0 0
\(151\) −12.6761 −1.03157 −0.515783 0.856719i \(-0.672499\pi\)
−0.515783 + 0.856719i \(0.672499\pi\)
\(152\) 0 0
\(153\) 1.42166i 0.114935i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.32391 −0.105660 −0.0528298 0.998604i \(-0.516824\pi\)
−0.0528298 + 0.998604i \(0.516824\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 20.4111 1.60862
\(162\) 0 0
\(163\) 15.2544 1.19482 0.597409 0.801936i \(-0.296197\pi\)
0.597409 + 0.801936i \(0.296197\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 10.7839i − 0.834482i −0.908796 0.417241i \(-0.862997\pi\)
0.908796 0.417241i \(-0.137003\pi\)
\(168\) 0 0
\(169\) −12.6655 −0.974271
\(170\) 0 0
\(171\) 5.62721i 0.430324i
\(172\) 0 0
\(173\) −13.6655 −1.03897 −0.519485 0.854479i \(-0.673876\pi\)
−0.519485 + 0.854479i \(0.673876\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2.20555i − 0.165779i
\(178\) 0 0
\(179\) 9.04888i 0.676345i 0.941084 + 0.338172i \(0.109809\pi\)
−0.941084 + 0.338172i \(0.890191\pi\)
\(180\) 0 0
\(181\) 23.2544i 1.72849i 0.503073 + 0.864244i \(0.332203\pi\)
−0.503073 + 0.864244i \(0.667797\pi\)
\(182\) 0 0
\(183\) 12.4111i 0.917455i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.82220 −0.645143
\(188\) 0 0
\(189\) − 3.62721i − 0.263841i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) − 25.6655i − 1.84745i −0.383062 0.923723i \(-0.625131\pi\)
0.383062 0.923723i \(-0.374869\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.1567 1.07987 0.539934 0.841707i \(-0.318449\pi\)
0.539934 + 0.841707i \(0.318449\pi\)
\(198\) 0 0
\(199\) −20.6761 −1.46569 −0.732845 0.680396i \(-0.761808\pi\)
−0.732845 + 0.680396i \(0.761808\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −7.25443 −0.509161
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 5.62721i − 0.391118i
\(208\) 0 0
\(209\) −34.9200 −2.41546
\(210\) 0 0
\(211\) − 2.03831i − 0.140323i −0.997536 0.0701616i \(-0.977648\pi\)
0.997536 0.0701616i \(-0.0223515\pi\)
\(212\) 0 0
\(213\) 8.41110 0.576319
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.35218i 0.634867i
\(218\) 0 0
\(219\) − 6.00000i − 0.405442i
\(220\) 0 0
\(221\) 0.822200i 0.0553072i
\(222\) 0 0
\(223\) − 7.21611i − 0.483227i −0.970373 0.241613i \(-0.922323\pi\)
0.970373 0.241613i \(-0.0776766\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.15667 −0.0767712 −0.0383856 0.999263i \(-0.512222\pi\)
−0.0383856 + 0.999263i \(0.512222\pi\)
\(228\) 0 0
\(229\) 14.0978i 0.931606i 0.884889 + 0.465803i \(0.154234\pi\)
−0.884889 + 0.465803i \(0.845766\pi\)
\(230\) 0 0
\(231\) 22.5089 1.48097
\(232\) 0 0
\(233\) 14.5783i 0.955059i 0.878616 + 0.477529i \(0.158468\pi\)
−0.878616 + 0.477529i \(0.841532\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.42166 −0.352175
\(238\) 0 0
\(239\) 19.2544 1.24547 0.622733 0.782435i \(-0.286022\pi\)
0.622733 + 0.782435i \(0.286022\pi\)
\(240\) 0 0
\(241\) −13.6655 −0.880274 −0.440137 0.897931i \(-0.645070\pi\)
−0.440137 + 0.897931i \(0.645070\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.25443i 0.207074i
\(248\) 0 0
\(249\) −3.25443 −0.206241
\(250\) 0 0
\(251\) − 7.14663i − 0.451091i −0.974233 0.225546i \(-0.927584\pi\)
0.974233 0.225546i \(-0.0724165\pi\)
\(252\) 0 0
\(253\) 34.9200 2.19540
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.73501i 0.482497i 0.970463 + 0.241248i \(0.0775568\pi\)
−0.970463 + 0.241248i \(0.922443\pi\)
\(258\) 0 0
\(259\) − 28.4111i − 1.76538i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) − 18.7839i − 1.15826i −0.815234 0.579132i \(-0.803392\pi\)
0.815234 0.579132i \(-0.196608\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −13.2544 −0.811158
\(268\) 0 0
\(269\) 8.50885i 0.518794i 0.965771 + 0.259397i \(0.0835238\pi\)
−0.965771 + 0.259397i \(0.916476\pi\)
\(270\) 0 0
\(271\) −30.9894 −1.88247 −0.941237 0.337746i \(-0.890335\pi\)
−0.941237 + 0.337746i \(0.890335\pi\)
\(272\) 0 0
\(273\) − 2.09775i − 0.126962i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.51941 0.571966 0.285983 0.958235i \(-0.407680\pi\)
0.285983 + 0.958235i \(0.407680\pi\)
\(278\) 0 0
\(279\) 2.57834 0.154361
\(280\) 0 0
\(281\) 13.6655 0.815217 0.407608 0.913157i \(-0.366363\pi\)
0.407608 + 0.913157i \(0.366363\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.0589i 1.12501i
\(288\) 0 0
\(289\) 14.9789 0.881110
\(290\) 0 0
\(291\) − 4.84333i − 0.283921i
\(292\) 0 0
\(293\) 4.31335 0.251989 0.125994 0.992031i \(-0.459788\pi\)
0.125994 + 0.992031i \(0.459788\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 6.20555i − 0.360083i
\(298\) 0 0
\(299\) − 3.25443i − 0.188208i
\(300\) 0 0
\(301\) − 26.3133i − 1.51668i
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −25.5678 −1.45923 −0.729615 0.683858i \(-0.760301\pi\)
−0.729615 + 0.683858i \(0.760301\pi\)
\(308\) 0 0
\(309\) − 2.47054i − 0.140544i
\(310\) 0 0
\(311\) 20.0766 1.13844 0.569221 0.822185i \(-0.307245\pi\)
0.569221 + 0.822185i \(0.307245\pi\)
\(312\) 0 0
\(313\) − 7.15667i − 0.404519i −0.979332 0.202260i \(-0.935172\pi\)
0.979332 0.202260i \(-0.0648285\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.1744 1.35777 0.678884 0.734245i \(-0.262464\pi\)
0.678884 + 0.734245i \(0.262464\pi\)
\(318\) 0 0
\(319\) −12.4111 −0.694888
\(320\) 0 0
\(321\) 14.0978 0.786860
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.25443i − 0.401171i
\(328\) 0 0
\(329\) 24.6066 1.35661
\(330\) 0 0
\(331\) − 27.1950i − 1.49477i −0.664390 0.747386i \(-0.731309\pi\)
0.664390 0.747386i \(-0.268691\pi\)
\(332\) 0 0
\(333\) −7.83276 −0.429233
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.8222i 1.24320i 0.783333 + 0.621602i \(0.213518\pi\)
−0.783333 + 0.621602i \(0.786482\pi\)
\(338\) 0 0
\(339\) − 9.08719i − 0.493548i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 3.05892i 0.165166i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.6655 1.27043 0.635216 0.772335i \(-0.280911\pi\)
0.635216 + 0.772335i \(0.280911\pi\)
\(348\) 0 0
\(349\) − 34.9200i − 1.86922i −0.355671 0.934611i \(-0.615748\pi\)
0.355671 0.934611i \(-0.384252\pi\)
\(350\) 0 0
\(351\) −0.578337 −0.0308694
\(352\) 0 0
\(353\) 15.9305i 0.847896i 0.905687 + 0.423948i \(0.139356\pi\)
−0.905687 + 0.423948i \(0.860644\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.15667 0.272920
\(358\) 0 0
\(359\) 8.41110 0.443921 0.221960 0.975056i \(-0.428754\pi\)
0.221960 + 0.975056i \(0.428754\pi\)
\(360\) 0 0
\(361\) −12.6655 −0.666607
\(362\) 0 0
\(363\) 27.5089 1.44384
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 24.4494i 1.27625i 0.769933 + 0.638124i \(0.220289\pi\)
−0.769933 + 0.638124i \(0.779711\pi\)
\(368\) 0 0
\(369\) 5.25443 0.273534
\(370\) 0 0
\(371\) 7.25443i 0.376631i
\(372\) 0 0
\(373\) 0.167237 0.00865920 0.00432960 0.999991i \(-0.498622\pi\)
0.00432960 + 0.999991i \(0.498622\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.15667i 0.0595718i
\(378\) 0 0
\(379\) − 7.72496i − 0.396805i −0.980121 0.198402i \(-0.936425\pi\)
0.980121 0.198402i \(-0.0635753\pi\)
\(380\) 0 0
\(381\) − 10.4705i − 0.536422i
\(382\) 0 0
\(383\) − 1.62721i − 0.0831467i −0.999135 0.0415734i \(-0.986763\pi\)
0.999135 0.0415734i \(-0.0132370\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.25443 −0.368763
\(388\) 0 0
\(389\) 12.3133i 0.624312i 0.950031 + 0.312156i \(0.101051\pi\)
−0.950031 + 0.312156i \(0.898949\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 13.4600i 0.678966i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −19.0872 −0.957959 −0.478979 0.877826i \(-0.658993\pi\)
−0.478979 + 0.877826i \(0.658993\pi\)
\(398\) 0 0
\(399\) 20.4111 1.02183
\(400\) 0 0
\(401\) −14.4111 −0.719656 −0.359828 0.933019i \(-0.617165\pi\)
−0.359828 + 0.933019i \(0.617165\pi\)
\(402\) 0 0
\(403\) 1.49115 0.0742794
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 48.6066i − 2.40934i
\(408\) 0 0
\(409\) 8.31335 0.411069 0.205534 0.978650i \(-0.434107\pi\)
0.205534 + 0.978650i \(0.434107\pi\)
\(410\) 0 0
\(411\) 10.5783i 0.521791i
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.4705i − 0.610685i
\(418\) 0 0
\(419\) 7.36222i 0.359668i 0.983697 + 0.179834i \(0.0575561\pi\)
−0.983697 + 0.179834i \(0.942444\pi\)
\(420\) 0 0
\(421\) 30.0978i 1.46687i 0.679757 + 0.733437i \(0.262085\pi\)
−0.679757 + 0.733437i \(0.737915\pi\)
\(422\) 0 0
\(423\) − 6.78389i − 0.329844i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 45.0177 2.17856
\(428\) 0 0
\(429\) − 3.58890i − 0.173274i
\(430\) 0 0
\(431\) −8.41110 −0.405148 −0.202574 0.979267i \(-0.564931\pi\)
−0.202574 + 0.979267i \(0.564931\pi\)
\(432\) 0 0
\(433\) 4.31335i 0.207286i 0.994615 + 0.103643i \(0.0330500\pi\)
−0.994615 + 0.103643i \(0.966950\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.6655 1.51477
\(438\) 0 0
\(439\) 9.83276 0.469292 0.234646 0.972081i \(-0.424607\pi\)
0.234646 + 0.972081i \(0.424607\pi\)
\(440\) 0 0
\(441\) −6.15667 −0.293175
\(442\) 0 0
\(443\) 21.3522 1.01447 0.507236 0.861807i \(-0.330667\pi\)
0.507236 + 0.861807i \(0.330667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.00000i − 0.0945968i
\(448\) 0 0
\(449\) 20.3133 0.958646 0.479323 0.877639i \(-0.340882\pi\)
0.479323 + 0.877639i \(0.340882\pi\)
\(450\) 0 0
\(451\) 32.6066i 1.53539i
\(452\) 0 0
\(453\) 12.6761 0.595575
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.35218i − 0.156808i −0.996922 0.0784041i \(-0.975018\pi\)
0.996922 0.0784041i \(-0.0249825\pi\)
\(458\) 0 0
\(459\) − 1.42166i − 0.0663575i
\(460\) 0 0
\(461\) − 28.5089i − 1.32779i −0.747826 0.663895i \(-0.768902\pi\)
0.747826 0.663895i \(-0.231098\pi\)
\(462\) 0 0
\(463\) 23.6272i 1.09805i 0.835806 + 0.549025i \(0.185001\pi\)
−0.835806 + 0.549025i \(0.814999\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.5678 1.36823 0.684117 0.729373i \(-0.260188\pi\)
0.684117 + 0.729373i \(0.260188\pi\)
\(468\) 0 0
\(469\) − 14.5089i − 0.669957i
\(470\) 0 0
\(471\) 1.32391 0.0610026
\(472\) 0 0
\(473\) − 45.0177i − 2.06992i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −22.0978 −1.00967 −0.504836 0.863215i \(-0.668447\pi\)
−0.504836 + 0.863215i \(0.668447\pi\)
\(480\) 0 0
\(481\) −4.52998 −0.206549
\(482\) 0 0
\(483\) −20.4111 −0.928737
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.03831i − 0.182993i −0.995805 0.0914967i \(-0.970835\pi\)
0.995805 0.0914967i \(-0.0291651\pi\)
\(488\) 0 0
\(489\) −15.2544 −0.689829
\(490\) 0 0
\(491\) 18.2056i 0.821605i 0.911724 + 0.410802i \(0.134751\pi\)
−0.911724 + 0.410802i \(0.865249\pi\)
\(492\) 0 0
\(493\) −2.84333 −0.128057
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 30.5089i − 1.36851i
\(498\) 0 0
\(499\) − 0.0594386i − 0.00266084i −0.999999 0.00133042i \(-0.999577\pi\)
0.999999 0.00133042i \(-0.000423486\pi\)
\(500\) 0 0
\(501\) 10.7839i 0.481789i
\(502\) 0 0
\(503\) 2.03831i 0.0908839i 0.998967 + 0.0454419i \(0.0144696\pi\)
−0.998967 + 0.0454419i \(0.985530\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.6655 0.562496
\(508\) 0 0
\(509\) − 40.7044i − 1.80419i −0.431539 0.902094i \(-0.642029\pi\)
0.431539 0.902094i \(-0.357971\pi\)
\(510\) 0 0
\(511\) −21.7633 −0.962751
\(512\) 0 0
\(513\) − 5.62721i − 0.248448i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 42.0978 1.85146
\(518\) 0 0
\(519\) 13.6655 0.599850
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 35.3311 1.54492 0.772460 0.635064i \(-0.219026\pi\)
0.772460 + 0.635064i \(0.219026\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.66553i 0.159673i
\(528\) 0 0
\(529\) −8.66553 −0.376762
\(530\) 0 0
\(531\) 2.20555i 0.0957127i
\(532\) 0 0
\(533\) 3.03883 0.131626
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.04888i − 0.390488i
\(538\) 0 0
\(539\) − 38.2056i − 1.64563i
\(540\) 0 0
\(541\) − 3.05892i − 0.131513i −0.997836 0.0657567i \(-0.979054\pi\)
0.997836 0.0657567i \(-0.0209461\pi\)
\(542\) 0 0
\(543\) − 23.2544i − 0.997943i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0766 1.37150 0.685749 0.727838i \(-0.259475\pi\)
0.685749 + 0.727838i \(0.259475\pi\)
\(548\) 0 0
\(549\) − 12.4111i − 0.529693i
\(550\) 0 0
\(551\) −11.2544 −0.479455
\(552\) 0 0
\(553\) 19.6655i 0.836263i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −33.6655 −1.42645 −0.713227 0.700933i \(-0.752767\pi\)
−0.713227 + 0.700933i \(0.752767\pi\)
\(558\) 0 0
\(559\) −4.19550 −0.177451
\(560\) 0 0
\(561\) 8.82220 0.372474
\(562\) 0 0
\(563\) −5.35218 −0.225567 −0.112784 0.993620i \(-0.535977\pi\)
−0.112784 + 0.993620i \(0.535977\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.62721i 0.152329i
\(568\) 0 0
\(569\) −5.58890 −0.234299 −0.117149 0.993114i \(-0.537376\pi\)
−0.117149 + 0.993114i \(0.537376\pi\)
\(570\) 0 0
\(571\) − 10.3728i − 0.434088i −0.976162 0.217044i \(-0.930359\pi\)
0.976162 0.217044i \(-0.0696414\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 21.6655i 0.901948i 0.892537 + 0.450974i \(0.148923\pi\)
−0.892537 + 0.450974i \(0.851077\pi\)
\(578\) 0 0
\(579\) 25.6655i 1.06662i
\(580\) 0 0
\(581\) 11.8045i 0.489733i
\(582\) 0 0
\(583\) 12.4111i 0.514015i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.90225 −0.0785142 −0.0392571 0.999229i \(-0.512499\pi\)
−0.0392571 + 0.999229i \(0.512499\pi\)
\(588\) 0 0
\(589\) 14.5089i 0.597827i
\(590\) 0 0
\(591\) −15.1567 −0.623462
\(592\) 0 0
\(593\) 2.57834i 0.105880i 0.998598 + 0.0529398i \(0.0168591\pi\)
−0.998598 + 0.0529398i \(0.983141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.6761 0.846216
\(598\) 0 0
\(599\) 26.7244 1.09193 0.545966 0.837808i \(-0.316163\pi\)
0.545966 + 0.837808i \(0.316163\pi\)
\(600\) 0 0
\(601\) 33.3311 1.35960 0.679801 0.733397i \(-0.262066\pi\)
0.679801 + 0.733397i \(0.262066\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 21.9406i 0.890540i 0.895396 + 0.445270i \(0.146892\pi\)
−0.895396 + 0.445270i \(0.853108\pi\)
\(608\) 0 0
\(609\) 7.25443 0.293964
\(610\) 0 0
\(611\) − 3.92337i − 0.158723i
\(612\) 0 0
\(613\) 3.42166 0.138200 0.0690998 0.997610i \(-0.477987\pi\)
0.0690998 + 0.997610i \(0.477987\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.7350i 0.794502i 0.917710 + 0.397251i \(0.130036\pi\)
−0.917710 + 0.397251i \(0.869964\pi\)
\(618\) 0 0
\(619\) 20.4705i 0.822780i 0.911459 + 0.411390i \(0.134957\pi\)
−0.911459 + 0.411390i \(0.865043\pi\)
\(620\) 0 0
\(621\) 5.62721i 0.225812i
\(622\) 0 0
\(623\) 48.0766i 1.92615i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 34.9200 1.39457
\(628\) 0 0
\(629\) − 11.1355i − 0.444003i
\(630\) 0 0
\(631\) −1.08719 −0.0432803 −0.0216402 0.999766i \(-0.506889\pi\)
−0.0216402 + 0.999766i \(0.506889\pi\)
\(632\) 0 0
\(633\) 2.03831i 0.0810157i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.56063 −0.141077
\(638\) 0 0
\(639\) −8.41110 −0.332738
\(640\) 0 0
\(641\) −27.9789 −1.10510 −0.552550 0.833480i \(-0.686345\pi\)
−0.552550 + 0.833480i \(0.686345\pi\)
\(642\) 0 0
\(643\) −4.94108 −0.194857 −0.0974285 0.995243i \(-0.531062\pi\)
−0.0974285 + 0.995243i \(0.531062\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 49.3694i 1.94091i 0.241282 + 0.970455i \(0.422432\pi\)
−0.241282 + 0.970455i \(0.577568\pi\)
\(648\) 0 0
\(649\) −13.6867 −0.537248
\(650\) 0 0
\(651\) − 9.35218i − 0.366541i
\(652\) 0 0
\(653\) 40.1744 1.57214 0.786072 0.618134i \(-0.212111\pi\)
0.786072 + 0.618134i \(0.212111\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) − 21.1255i − 0.822933i −0.911425 0.411466i \(-0.865017\pi\)
0.911425 0.411466i \(-0.134983\pi\)
\(660\) 0 0
\(661\) − 10.9200i − 0.424737i −0.977190 0.212368i \(-0.931882\pi\)
0.977190 0.212368i \(-0.0681177\pi\)
\(662\) 0 0
\(663\) − 0.822200i − 0.0319316i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.2544 0.435773
\(668\) 0 0
\(669\) 7.21611i 0.278991i
\(670\) 0 0
\(671\) 77.0177 2.97324
\(672\) 0 0
\(673\) 18.0000i 0.693849i 0.937893 + 0.346925i \(0.112774\pi\)
−0.937893 + 0.346925i \(0.887226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.4877 −1.17174 −0.585869 0.810406i \(-0.699247\pi\)
−0.585869 + 0.810406i \(0.699247\pi\)
\(678\) 0 0
\(679\) −17.5678 −0.674189
\(680\) 0 0
\(681\) 1.15667 0.0443239
\(682\) 0 0
\(683\) 35.2544 1.34897 0.674487 0.738287i \(-0.264365\pi\)
0.674487 + 0.738287i \(0.264365\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.0978i − 0.537863i
\(688\) 0 0
\(689\) 1.15667 0.0440658
\(690\) 0 0
\(691\) 28.1361i 1.07035i 0.844742 + 0.535173i \(0.179754\pi\)
−0.844742 + 0.535173i \(0.820246\pi\)
\(692\) 0 0
\(693\) −22.5089 −0.855041
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.47002i 0.282947i
\(698\) 0 0
\(699\) − 14.5783i − 0.551403i
\(700\) 0 0
\(701\) − 34.8222i − 1.31522i −0.753360 0.657608i \(-0.771568\pi\)
0.753360 0.657608i \(-0.228432\pi\)
\(702\) 0 0
\(703\) − 44.0766i − 1.66238i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.25443 −0.272831
\(708\) 0 0
\(709\) 7.58890i 0.285007i 0.989794 + 0.142504i \(0.0455152\pi\)
−0.989794 + 0.142504i \(0.954485\pi\)
\(710\) 0 0
\(711\) 5.42166 0.203328
\(712\) 0 0
\(713\) − 14.5089i − 0.543361i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −19.2544 −0.719070
\(718\) 0 0
\(719\) −3.66553 −0.136701 −0.0683505 0.997661i \(-0.521774\pi\)
−0.0683505 + 0.997661i \(0.521774\pi\)
\(720\) 0 0
\(721\) −8.96117 −0.333731
\(722\) 0 0
\(723\) 13.6655 0.508226
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 36.1149i 1.33943i 0.742619 + 0.669714i \(0.233584\pi\)
−0.742619 + 0.669714i \(0.766416\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) − 10.3133i − 0.381453i
\(732\) 0 0
\(733\) −34.0071 −1.25608 −0.628041 0.778180i \(-0.716143\pi\)
−0.628041 + 0.778180i \(0.716143\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 24.8222i − 0.914338i
\(738\) 0 0
\(739\) − 52.0172i − 1.91348i −0.290939 0.956742i \(-0.593968\pi\)
0.290939 0.956742i \(-0.406032\pi\)
\(740\) 0 0
\(741\) − 3.25443i − 0.119554i
\(742\) 0 0
\(743\) − 23.3139i − 0.855303i −0.903944 0.427651i \(-0.859341\pi\)
0.903944 0.427651i \(-0.140659\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.25443 0.119073
\(748\) 0 0
\(749\) − 51.1355i − 1.86845i
\(750\) 0 0
\(751\) −11.1083 −0.405348 −0.202674 0.979246i \(-0.564963\pi\)
−0.202674 + 0.979246i \(0.564963\pi\)
\(752\) 0 0
\(753\) 7.14663i 0.260438i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.3239 0.484266 0.242133 0.970243i \(-0.422153\pi\)
0.242133 + 0.970243i \(0.422153\pi\)
\(758\) 0 0
\(759\) −34.9200 −1.26751
\(760\) 0 0
\(761\) −17.1355 −0.621163 −0.310582 0.950547i \(-0.600524\pi\)
−0.310582 + 0.950547i \(0.600524\pi\)
\(762\) 0 0
\(763\) −26.3133 −0.952607
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.27555i 0.0460575i
\(768\) 0 0
\(769\) −5.47002 −0.197254 −0.0986270 0.995124i \(-0.531445\pi\)
−0.0986270 + 0.995124i \(0.531445\pi\)
\(770\) 0 0
\(771\) − 7.73501i − 0.278570i
\(772\) 0 0
\(773\) 3.15667 0.113538 0.0567688 0.998387i \(-0.481920\pi\)
0.0567688 + 0.998387i \(0.481920\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 28.4111i 1.01924i
\(778\) 0 0
\(779\) 29.5678i 1.05938i
\(780\) 0 0
\(781\) − 52.1955i − 1.86770i
\(782\) 0 0
\(783\) − 2.00000i − 0.0714742i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −31.1355 −1.10986 −0.554931 0.831896i \(-0.687255\pi\)
−0.554931 + 0.831896i \(0.687255\pi\)
\(788\) 0 0
\(789\) 18.7839i 0.668724i
\(790\) 0 0
\(791\) −32.9612 −1.17196
\(792\) 0 0
\(793\) − 7.17780i − 0.254891i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 9.64440 0.341194
\(800\) 0 0
\(801\) 13.2544 0.468322
\(802\) 0 0
\(803\) −37.2333 −1.31393
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.50885i − 0.299526i
\(808\) 0 0
\(809\) −29.0388 −1.02095 −0.510475 0.859892i \(-0.670531\pi\)
−0.510475 + 0.859892i \(0.670531\pi\)
\(810\) 0 0
\(811\) 2.58838i 0.0908904i 0.998967 + 0.0454452i \(0.0144706\pi\)
−0.998967 + 0.0454452i \(0.985529\pi\)
\(812\) 0 0
\(813\) 30.9894 1.08685
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 40.8222i − 1.42819i
\(818\) 0 0
\(819\) 2.09775i 0.0733014i
\(820\) 0 0
\(821\) 38.1955i 1.33303i 0.745491 + 0.666516i \(0.232215\pi\)
−0.745491 + 0.666516i \(0.767785\pi\)
\(822\) 0 0
\(823\) − 18.3517i − 0.639699i −0.947468 0.319849i \(-0.896368\pi\)
0.947468 0.319849i \(-0.103632\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) − 24.7456i − 0.859449i −0.902960 0.429725i \(-0.858611\pi\)
0.902960 0.429725i \(-0.141389\pi\)
\(830\) 0 0
\(831\) −9.51941 −0.330225
\(832\) 0 0
\(833\) − 8.75272i − 0.303264i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.57834 −0.0891204
\(838\) 0 0
\(839\) −53.4288 −1.84457 −0.922284 0.386514i \(-0.873679\pi\)
−0.922284 + 0.386514i \(0.873679\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) −13.6655 −0.470666
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 99.7805i − 3.42850i
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 44.0766i 1.51093i
\(852\) 0 0
\(853\) 29.0661 0.995203 0.497602 0.867406i \(-0.334214\pi\)
0.497602 + 0.867406i \(0.334214\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 10.2439i − 0.349924i −0.984575 0.174962i \(-0.944020\pi\)
0.984575 0.174962i \(-0.0559802\pi\)
\(858\) 0 0
\(859\) 10.9794i 0.374612i 0.982302 + 0.187306i \(0.0599756\pi\)
−0.982302 + 0.187306i \(0.940024\pi\)
\(860\) 0 0
\(861\) − 19.0589i − 0.649526i
\(862\) 0 0
\(863\) 38.8605i 1.32283i 0.750021 + 0.661414i \(0.230043\pi\)
−0.750021 + 0.661414i \(0.769957\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −14.9789 −0.508709
\(868\) 0 0
\(869\) 33.6444i 1.14131i
\(870\) 0 0
\(871\) −2.31335 −0.0783848
\(872\) 0 0
\(873\) 4.84333i 0.163922i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.3416 −0.754423 −0.377211 0.926127i \(-0.623117\pi\)
−0.377211 + 0.926127i \(0.623117\pi\)
\(878\) 0 0
\(879\) −4.31335 −0.145486
\(880\) 0 0
\(881\) 9.88112 0.332903 0.166452 0.986050i \(-0.446769\pi\)
0.166452 + 0.986050i \(0.446769\pi\)
\(882\) 0 0
\(883\) 10.6277 0.357652 0.178826 0.983881i \(-0.442770\pi\)
0.178826 + 0.983881i \(0.442770\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 11.6061i − 0.389694i −0.980834 0.194847i \(-0.937579\pi\)
0.980834 0.194847i \(-0.0624211\pi\)
\(888\) 0 0
\(889\) −37.9789 −1.27377
\(890\) 0 0
\(891\) 6.20555i 0.207894i
\(892\) 0 0
\(893\) 38.1744 1.27746
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.25443i 0.108662i
\(898\) 0 0
\(899\) 5.15667i 0.171985i
\(900\) 0 0
\(901\) 2.84333i 0.0947249i
\(902\) 0 0
\(903\) 26.3133i 0.875653i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.195504 −0.00649159 −0.00324580 0.999995i \(-0.501033\pi\)
−0.00324580 + 0.999995i \(0.501033\pi\)
\(908\) 0 0
\(909\) 2.00000i 0.0663358i
\(910\) 0 0
\(911\) 7.88112 0.261113 0.130557 0.991441i \(-0.458324\pi\)
0.130557 + 0.991441i \(0.458324\pi\)
\(912\) 0 0
\(913\) 20.1955i 0.668374i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 48.8222 1.61225
\(918\) 0 0
\(919\) 9.75614 0.321825 0.160913 0.986969i \(-0.448556\pi\)
0.160913 + 0.986969i \(0.448556\pi\)
\(920\) 0 0
\(921\) 25.5678 0.842487
\(922\) 0 0
\(923\) −4.86445 −0.160115
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.47054i 0.0811431i
\(928\) 0 0
\(929\) −6.82220 −0.223829 −0.111915 0.993718i \(-0.535698\pi\)
−0.111915 + 0.993718i \(0.535698\pi\)
\(930\) 0 0
\(931\) − 34.6449i − 1.13544i
\(932\) 0 0
\(933\) −20.0766 −0.657279
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 57.5266i 1.87931i 0.342123 + 0.939655i \(0.388854\pi\)
−0.342123 + 0.939655i \(0.611146\pi\)
\(938\) 0 0
\(939\) 7.15667i 0.233549i
\(940\) 0 0
\(941\) 0.508852i 0.0165881i 0.999966 + 0.00829405i \(0.00264011\pi\)
−0.999966 + 0.00829405i \(0.997360\pi\)
\(942\) 0 0
\(943\) − 29.5678i − 0.962859i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.68665 −0.0548088 −0.0274044 0.999624i \(-0.508724\pi\)
−0.0274044 + 0.999624i \(0.508724\pi\)
\(948\) 0 0
\(949\) 3.47002i 0.112642i
\(950\) 0 0
\(951\) −24.1744 −0.783908
\(952\) 0 0
\(953\) − 9.22616i − 0.298865i −0.988772 0.149432i \(-0.952255\pi\)
0.988772 0.149432i \(-0.0477446\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.4111 0.401194
\(958\) 0 0
\(959\) 38.3699 1.23903
\(960\) 0 0
\(961\) −24.3522 −0.785554
\(962\) 0 0
\(963\) −14.0978 −0.454294
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 12.2338i − 0.393413i −0.980462 0.196707i \(-0.936975\pi\)
0.980462 0.196707i \(-0.0630246\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 33.2444i 1.06686i 0.845843 + 0.533431i \(0.179098\pi\)
−0.845843 + 0.533431i \(0.820902\pi\)
\(972\) 0 0
\(973\) −45.2333 −1.45011
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.93051i 0.253720i 0.991921 + 0.126860i \(0.0404898\pi\)
−0.991921 + 0.126860i \(0.959510\pi\)
\(978\) 0 0
\(979\) 82.2510i 2.62875i
\(980\) 0 0
\(981\) 7.25443i 0.231616i
\(982\) 0 0
\(983\) 41.8993i 1.33638i 0.743990 + 0.668191i \(0.232931\pi\)
−0.743990 + 0.668191i \(0.767069\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −24.6066 −0.783237
\(988\) 0 0
\(989\) 40.8222i 1.29807i
\(990\) 0 0
\(991\) 35.1849 1.11769 0.558843 0.829273i \(-0.311245\pi\)
0.558843 + 0.829273i \(0.311245\pi\)
\(992\) 0 0
\(993\) 27.1950i 0.863007i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.04836 0.254894 0.127447 0.991845i \(-0.459322\pi\)
0.127447 + 0.991845i \(0.459322\pi\)
\(998\) 0 0
\(999\) 7.83276 0.247818
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 2400.2.d.e.49.5 6
3.2 odd 2 7200.2.d.r.2449.5 6
4.3 odd 2 600.2.d.f.349.4 6
5.2 odd 4 2400.2.k.c.1201.4 6
5.3 odd 4 480.2.k.b.241.3 6
5.4 even 2 2400.2.d.f.49.2 6
8.3 odd 2 600.2.d.e.349.4 6
8.5 even 2 2400.2.d.f.49.5 6
12.11 even 2 1800.2.d.r.1549.3 6
15.2 even 4 7200.2.k.p.3601.1 6
15.8 even 4 1440.2.k.f.721.3 6
15.14 odd 2 7200.2.d.q.2449.2 6
20.3 even 4 120.2.k.b.61.5 6
20.7 even 4 600.2.k.c.301.2 6
20.19 odd 2 600.2.d.e.349.3 6
24.5 odd 2 7200.2.d.q.2449.5 6
24.11 even 2 1800.2.d.q.1549.3 6
40.3 even 4 120.2.k.b.61.6 yes 6
40.13 odd 4 480.2.k.b.241.6 6
40.19 odd 2 600.2.d.f.349.3 6
40.27 even 4 600.2.k.c.301.1 6
40.29 even 2 inner 2400.2.d.e.49.2 6
40.37 odd 4 2400.2.k.c.1201.1 6
60.23 odd 4 360.2.k.f.181.2 6
60.47 odd 4 1800.2.k.p.901.5 6
60.59 even 2 1800.2.d.q.1549.4 6
80.3 even 4 3840.2.a.bp.1.3 3
80.13 odd 4 3840.2.a.br.1.1 3
80.43 even 4 3840.2.a.bq.1.3 3
80.53 odd 4 3840.2.a.bo.1.1 3
120.29 odd 2 7200.2.d.r.2449.2 6
120.53 even 4 1440.2.k.f.721.6 6
120.59 even 2 1800.2.d.r.1549.4 6
120.77 even 4 7200.2.k.p.3601.2 6
120.83 odd 4 360.2.k.f.181.1 6
120.107 odd 4 1800.2.k.p.901.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.5 6 20.3 even 4
120.2.k.b.61.6 yes 6 40.3 even 4
360.2.k.f.181.1 6 120.83 odd 4
360.2.k.f.181.2 6 60.23 odd 4
480.2.k.b.241.3 6 5.3 odd 4
480.2.k.b.241.6 6 40.13 odd 4
600.2.d.e.349.3 6 20.19 odd 2
600.2.d.e.349.4 6 8.3 odd 2
600.2.d.f.349.3 6 40.19 odd 2
600.2.d.f.349.4 6 4.3 odd 2
600.2.k.c.301.1 6 40.27 even 4
600.2.k.c.301.2 6 20.7 even 4
1440.2.k.f.721.3 6 15.8 even 4
1440.2.k.f.721.6 6 120.53 even 4
1800.2.d.q.1549.3 6 24.11 even 2
1800.2.d.q.1549.4 6 60.59 even 2
1800.2.d.r.1549.3 6 12.11 even 2
1800.2.d.r.1549.4 6 120.59 even 2
1800.2.k.p.901.5 6 60.47 odd 4
1800.2.k.p.901.6 6 120.107 odd 4
2400.2.d.e.49.2 6 40.29 even 2 inner
2400.2.d.e.49.5 6 1.1 even 1 trivial
2400.2.d.f.49.2 6 5.4 even 2
2400.2.d.f.49.5 6 8.5 even 2
2400.2.k.c.1201.1 6 40.37 odd 4
2400.2.k.c.1201.4 6 5.2 odd 4
3840.2.a.bo.1.1 3 80.53 odd 4
3840.2.a.bp.1.3 3 80.3 even 4
3840.2.a.bq.1.3 3 80.43 even 4
3840.2.a.br.1.1 3 80.13 odd 4
7200.2.d.q.2449.2 6 15.14 odd 2
7200.2.d.q.2449.5 6 24.5 odd 2
7200.2.d.r.2449.2 6 120.29 odd 2
7200.2.d.r.2449.5 6 3.2 odd 2
7200.2.k.p.3601.1 6 15.2 even 4
7200.2.k.p.3601.2 6 120.77 even 4