Properties

Label 4560.2.a.bv.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +2.52444 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +2.52444 q^{7} +1.00000 q^{9} +3.10278 q^{11} +6.72999 q^{13} +1.00000 q^{15} -2.57834 q^{17} +1.00000 q^{19} +2.52444 q^{21} +4.57834 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.10278 q^{29} -7.83276 q^{31} +3.10278 q^{33} +2.52444 q^{35} -4.52444 q^{37} +6.72999 q^{39} +6.15165 q^{41} +7.68111 q^{43} +1.00000 q^{45} +3.42166 q^{47} -0.627213 q^{49} -2.57834 q^{51} +2.57834 q^{53} +3.10278 q^{55} +1.00000 q^{57} +10.2056 q^{59} -14.8816 q^{61} +2.52444 q^{63} +6.72999 q^{65} -8.41110 q^{67} +4.57834 q^{69} -9.45998 q^{71} -9.25443 q^{73} +1.00000 q^{75} +7.83276 q^{77} +1.00000 q^{81} +4.47054 q^{83} -2.57834 q^{85} -1.10278 q^{87} -14.5628 q^{89} +16.9894 q^{91} -7.83276 q^{93} +1.00000 q^{95} -12.9355 q^{97} +3.10278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{15} - 6 q^{17} + 3 q^{19} + 2 q^{21} + 12 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 14 q^{43} + 3 q^{45} + 12 q^{47} + 11 q^{49} - 6 q^{51} + 6 q^{53} + 2 q^{55} + 3 q^{57} + 16 q^{59} - 6 q^{61} + 2 q^{63} + 4 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} + 3 q^{75} - 4 q^{77} + 3 q^{81} + 4 q^{83} - 6 q^{85} + 4 q^{87} + 4 q^{89} + 20 q^{91} + 4 q^{93} + 3 q^{95} - 4 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.52444 0.954148 0.477074 0.878863i \(-0.341697\pi\)
0.477074 + 0.878863i \(0.341697\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.10278 0.935522 0.467761 0.883855i \(-0.345061\pi\)
0.467761 + 0.883855i \(0.345061\pi\)
\(12\) 0 0
\(13\) 6.72999 1.86656 0.933281 0.359146i \(-0.116932\pi\)
0.933281 + 0.359146i \(0.116932\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.57834 −0.625339 −0.312669 0.949862i \(-0.601223\pi\)
−0.312669 + 0.949862i \(0.601223\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.52444 0.550878
\(22\) 0 0
\(23\) 4.57834 0.954649 0.477325 0.878727i \(-0.341607\pi\)
0.477325 + 0.878727i \(0.341607\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.10278 −0.204780 −0.102390 0.994744i \(-0.532649\pi\)
−0.102390 + 0.994744i \(0.532649\pi\)
\(30\) 0 0
\(31\) −7.83276 −1.40681 −0.703403 0.710791i \(-0.748337\pi\)
−0.703403 + 0.710791i \(0.748337\pi\)
\(32\) 0 0
\(33\) 3.10278 0.540124
\(34\) 0 0
\(35\) 2.52444 0.426708
\(36\) 0 0
\(37\) −4.52444 −0.743813 −0.371907 0.928270i \(-0.621296\pi\)
−0.371907 + 0.928270i \(0.621296\pi\)
\(38\) 0 0
\(39\) 6.72999 1.07766
\(40\) 0 0
\(41\) 6.15165 0.960726 0.480363 0.877070i \(-0.340505\pi\)
0.480363 + 0.877070i \(0.340505\pi\)
\(42\) 0 0
\(43\) 7.68111 1.17136 0.585679 0.810543i \(-0.300828\pi\)
0.585679 + 0.810543i \(0.300828\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 3.42166 0.499101 0.249550 0.968362i \(-0.419717\pi\)
0.249550 + 0.968362i \(0.419717\pi\)
\(48\) 0 0
\(49\) −0.627213 −0.0896019
\(50\) 0 0
\(51\) −2.57834 −0.361039
\(52\) 0 0
\(53\) 2.57834 0.354162 0.177081 0.984196i \(-0.443335\pi\)
0.177081 + 0.984196i \(0.443335\pi\)
\(54\) 0 0
\(55\) 3.10278 0.418378
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 10.2056 1.32865 0.664325 0.747444i \(-0.268719\pi\)
0.664325 + 0.747444i \(0.268719\pi\)
\(60\) 0 0
\(61\) −14.8816 −1.90540 −0.952699 0.303914i \(-0.901706\pi\)
−0.952699 + 0.303914i \(0.901706\pi\)
\(62\) 0 0
\(63\) 2.52444 0.318049
\(64\) 0 0
\(65\) 6.72999 0.834752
\(66\) 0 0
\(67\) −8.41110 −1.02758 −0.513790 0.857916i \(-0.671759\pi\)
−0.513790 + 0.857916i \(0.671759\pi\)
\(68\) 0 0
\(69\) 4.57834 0.551167
\(70\) 0 0
\(71\) −9.45998 −1.12269 −0.561346 0.827581i \(-0.689716\pi\)
−0.561346 + 0.827581i \(0.689716\pi\)
\(72\) 0 0
\(73\) −9.25443 −1.08315 −0.541574 0.840653i \(-0.682172\pi\)
−0.541574 + 0.840653i \(0.682172\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 7.83276 0.892626
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.47054 0.490705 0.245353 0.969434i \(-0.421096\pi\)
0.245353 + 0.969434i \(0.421096\pi\)
\(84\) 0 0
\(85\) −2.57834 −0.279660
\(86\) 0 0
\(87\) −1.10278 −0.118230
\(88\) 0 0
\(89\) −14.5628 −1.54365 −0.771824 0.635836i \(-0.780656\pi\)
−0.771824 + 0.635836i \(0.780656\pi\)
\(90\) 0 0
\(91\) 16.9894 1.78098
\(92\) 0 0
\(93\) −7.83276 −0.812220
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −12.9355 −1.31340 −0.656702 0.754150i \(-0.728049\pi\)
−0.656702 + 0.754150i \(0.728049\pi\)
\(98\) 0 0
\(99\) 3.10278 0.311841
\(100\) 0 0
\(101\) −11.4600 −1.14031 −0.570155 0.821537i \(-0.693117\pi\)
−0.570155 + 0.821537i \(0.693117\pi\)
\(102\) 0 0
\(103\) −14.5089 −1.42960 −0.714800 0.699329i \(-0.753482\pi\)
−0.714800 + 0.699329i \(0.753482\pi\)
\(104\) 0 0
\(105\) 2.52444 0.246360
\(106\) 0 0
\(107\) 15.2544 1.47470 0.737351 0.675510i \(-0.236077\pi\)
0.737351 + 0.675510i \(0.236077\pi\)
\(108\) 0 0
\(109\) 16.6167 1.59159 0.795793 0.605568i \(-0.207054\pi\)
0.795793 + 0.605568i \(0.207054\pi\)
\(110\) 0 0
\(111\) −4.52444 −0.429441
\(112\) 0 0
\(113\) −4.37279 −0.411357 −0.205679 0.978620i \(-0.565940\pi\)
−0.205679 + 0.978620i \(0.565940\pi\)
\(114\) 0 0
\(115\) 4.57834 0.426932
\(116\) 0 0
\(117\) 6.72999 0.622188
\(118\) 0 0
\(119\) −6.50885 −0.596665
\(120\) 0 0
\(121\) −1.37279 −0.124799
\(122\) 0 0
\(123\) 6.15165 0.554676
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.2544 1.70855 0.854277 0.519818i \(-0.174000\pi\)
0.854277 + 0.519818i \(0.174000\pi\)
\(128\) 0 0
\(129\) 7.68111 0.676284
\(130\) 0 0
\(131\) −11.4061 −0.996554 −0.498277 0.867018i \(-0.666034\pi\)
−0.498277 + 0.867018i \(0.666034\pi\)
\(132\) 0 0
\(133\) 2.52444 0.218897
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.74557 −0.234570 −0.117285 0.993098i \(-0.537419\pi\)
−0.117285 + 0.993098i \(0.537419\pi\)
\(138\) 0 0
\(139\) −5.79445 −0.491479 −0.245739 0.969336i \(-0.579031\pi\)
−0.245739 + 0.969336i \(0.579031\pi\)
\(140\) 0 0
\(141\) 3.42166 0.288156
\(142\) 0 0
\(143\) 20.8816 1.74621
\(144\) 0 0
\(145\) −1.10278 −0.0915805
\(146\) 0 0
\(147\) −0.627213 −0.0517317
\(148\) 0 0
\(149\) −4.09775 −0.335701 −0.167850 0.985812i \(-0.553683\pi\)
−0.167850 + 0.985812i \(0.553683\pi\)
\(150\) 0 0
\(151\) 6.67609 0.543292 0.271646 0.962397i \(-0.412432\pi\)
0.271646 + 0.962397i \(0.412432\pi\)
\(152\) 0 0
\(153\) −2.57834 −0.208446
\(154\) 0 0
\(155\) −7.83276 −0.629143
\(156\) 0 0
\(157\) −19.4600 −1.55308 −0.776538 0.630071i \(-0.783026\pi\)
−0.776538 + 0.630071i \(0.783026\pi\)
\(158\) 0 0
\(159\) 2.57834 0.204475
\(160\) 0 0
\(161\) 11.5577 0.910877
\(162\) 0 0
\(163\) −12.7300 −0.997090 −0.498545 0.866864i \(-0.666132\pi\)
−0.498545 + 0.866864i \(0.666132\pi\)
\(164\) 0 0
\(165\) 3.10278 0.241551
\(166\) 0 0
\(167\) 20.8816 1.61587 0.807935 0.589272i \(-0.200585\pi\)
0.807935 + 0.589272i \(0.200585\pi\)
\(168\) 0 0
\(169\) 32.2927 2.48406
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −0.372787 −0.0283425 −0.0141712 0.999900i \(-0.504511\pi\)
−0.0141712 + 0.999900i \(0.504511\pi\)
\(174\) 0 0
\(175\) 2.52444 0.190830
\(176\) 0 0
\(177\) 10.2056 0.767096
\(178\) 0 0
\(179\) −1.04888 −0.0783967 −0.0391983 0.999231i \(-0.512480\pi\)
−0.0391983 + 0.999231i \(0.512480\pi\)
\(180\) 0 0
\(181\) 5.36222 0.398571 0.199285 0.979941i \(-0.436138\pi\)
0.199285 + 0.979941i \(0.436138\pi\)
\(182\) 0 0
\(183\) −14.8816 −1.10008
\(184\) 0 0
\(185\) −4.52444 −0.332643
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 2.52444 0.183626
\(190\) 0 0
\(191\) 25.7194 1.86099 0.930496 0.366302i \(-0.119376\pi\)
0.930496 + 0.366302i \(0.119376\pi\)
\(192\) 0 0
\(193\) 7.77886 0.559935 0.279967 0.960009i \(-0.409676\pi\)
0.279967 + 0.960009i \(0.409676\pi\)
\(194\) 0 0
\(195\) 6.72999 0.481944
\(196\) 0 0
\(197\) −5.42166 −0.386277 −0.193139 0.981171i \(-0.561867\pi\)
−0.193139 + 0.981171i \(0.561867\pi\)
\(198\) 0 0
\(199\) −14.2056 −1.00700 −0.503502 0.863994i \(-0.667955\pi\)
−0.503502 + 0.863994i \(0.667955\pi\)
\(200\) 0 0
\(201\) −8.41110 −0.593273
\(202\) 0 0
\(203\) −2.78389 −0.195391
\(204\) 0 0
\(205\) 6.15165 0.429650
\(206\) 0 0
\(207\) 4.57834 0.318216
\(208\) 0 0
\(209\) 3.10278 0.214623
\(210\) 0 0
\(211\) −6.09775 −0.419787 −0.209893 0.977724i \(-0.567312\pi\)
−0.209893 + 0.977724i \(0.567312\pi\)
\(212\) 0 0
\(213\) −9.45998 −0.648187
\(214\) 0 0
\(215\) 7.68111 0.523848
\(216\) 0 0
\(217\) −19.7733 −1.34230
\(218\) 0 0
\(219\) −9.25443 −0.625356
\(220\) 0 0
\(221\) −17.3522 −1.16723
\(222\) 0 0
\(223\) 24.8222 1.66222 0.831109 0.556110i \(-0.187707\pi\)
0.831109 + 0.556110i \(0.187707\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.7839 1.51222 0.756110 0.654445i \(-0.227098\pi\)
0.756110 + 0.654445i \(0.227098\pi\)
\(228\) 0 0
\(229\) 6.47054 0.427585 0.213793 0.976879i \(-0.431418\pi\)
0.213793 + 0.976879i \(0.431418\pi\)
\(230\) 0 0
\(231\) 7.83276 0.515358
\(232\) 0 0
\(233\) −19.3522 −1.26780 −0.633902 0.773414i \(-0.718548\pi\)
−0.633902 + 0.773414i \(0.718548\pi\)
\(234\) 0 0
\(235\) 3.42166 0.223205
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.9250 −1.54758 −0.773789 0.633443i \(-0.781641\pi\)
−0.773789 + 0.633443i \(0.781641\pi\)
\(240\) 0 0
\(241\) −8.09775 −0.521622 −0.260811 0.965390i \(-0.583990\pi\)
−0.260811 + 0.965390i \(0.583990\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.627213 −0.0400712
\(246\) 0 0
\(247\) 6.72999 0.428219
\(248\) 0 0
\(249\) 4.47054 0.283309
\(250\) 0 0
\(251\) −1.00502 −0.0634365 −0.0317183 0.999497i \(-0.510098\pi\)
−0.0317183 + 0.999497i \(0.510098\pi\)
\(252\) 0 0
\(253\) 14.2056 0.893095
\(254\) 0 0
\(255\) −2.57834 −0.161462
\(256\) 0 0
\(257\) −16.7839 −1.04695 −0.523475 0.852041i \(-0.675365\pi\)
−0.523475 + 0.852041i \(0.675365\pi\)
\(258\) 0 0
\(259\) −11.4217 −0.709708
\(260\) 0 0
\(261\) −1.10278 −0.0682601
\(262\) 0 0
\(263\) 22.0383 1.35894 0.679470 0.733703i \(-0.262210\pi\)
0.679470 + 0.733703i \(0.262210\pi\)
\(264\) 0 0
\(265\) 2.57834 0.158386
\(266\) 0 0
\(267\) −14.5628 −0.891226
\(268\) 0 0
\(269\) 32.3572 1.97285 0.986427 0.164202i \(-0.0525050\pi\)
0.986427 + 0.164202i \(0.0525050\pi\)
\(270\) 0 0
\(271\) 23.1466 1.40606 0.703029 0.711161i \(-0.251831\pi\)
0.703029 + 0.711161i \(0.251831\pi\)
\(272\) 0 0
\(273\) 16.9894 1.02825
\(274\) 0 0
\(275\) 3.10278 0.187104
\(276\) 0 0
\(277\) 30.8222 1.85193 0.925963 0.377614i \(-0.123255\pi\)
0.925963 + 0.377614i \(0.123255\pi\)
\(278\) 0 0
\(279\) −7.83276 −0.468935
\(280\) 0 0
\(281\) −3.94610 −0.235405 −0.117702 0.993049i \(-0.537553\pi\)
−0.117702 + 0.993049i \(0.537553\pi\)
\(282\) 0 0
\(283\) −17.4756 −1.03881 −0.519407 0.854527i \(-0.673847\pi\)
−0.519407 + 0.854527i \(0.673847\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 15.5295 0.916675
\(288\) 0 0
\(289\) −10.3522 −0.608952
\(290\) 0 0
\(291\) −12.9355 −0.758295
\(292\) 0 0
\(293\) 2.05944 0.120314 0.0601568 0.998189i \(-0.480840\pi\)
0.0601568 + 0.998189i \(0.480840\pi\)
\(294\) 0 0
\(295\) 10.2056 0.594190
\(296\) 0 0
\(297\) 3.10278 0.180041
\(298\) 0 0
\(299\) 30.8122 1.78191
\(300\) 0 0
\(301\) 19.3905 1.11765
\(302\) 0 0
\(303\) −11.4600 −0.658358
\(304\) 0 0
\(305\) −14.8816 −0.852120
\(306\) 0 0
\(307\) −10.2056 −0.582462 −0.291231 0.956653i \(-0.594065\pi\)
−0.291231 + 0.956653i \(0.594065\pi\)
\(308\) 0 0
\(309\) −14.5089 −0.825380
\(310\) 0 0
\(311\) 1.20053 0.0680756 0.0340378 0.999421i \(-0.489163\pi\)
0.0340378 + 0.999421i \(0.489163\pi\)
\(312\) 0 0
\(313\) −7.45998 −0.421663 −0.210831 0.977522i \(-0.567617\pi\)
−0.210831 + 0.977522i \(0.567617\pi\)
\(314\) 0 0
\(315\) 2.52444 0.142236
\(316\) 0 0
\(317\) −4.48059 −0.251655 −0.125827 0.992052i \(-0.540159\pi\)
−0.125827 + 0.992052i \(0.540159\pi\)
\(318\) 0 0
\(319\) −3.42166 −0.191576
\(320\) 0 0
\(321\) 15.2544 0.851419
\(322\) 0 0
\(323\) −2.57834 −0.143463
\(324\) 0 0
\(325\) 6.72999 0.373313
\(326\) 0 0
\(327\) 16.6167 0.918903
\(328\) 0 0
\(329\) 8.63778 0.476216
\(330\) 0 0
\(331\) −24.2439 −1.33256 −0.666282 0.745700i \(-0.732115\pi\)
−0.666282 + 0.745700i \(0.732115\pi\)
\(332\) 0 0
\(333\) −4.52444 −0.247938
\(334\) 0 0
\(335\) −8.41110 −0.459547
\(336\) 0 0
\(337\) 23.4444 1.27710 0.638549 0.769581i \(-0.279535\pi\)
0.638549 + 0.769581i \(0.279535\pi\)
\(338\) 0 0
\(339\) −4.37279 −0.237497
\(340\) 0 0
\(341\) −24.3033 −1.31610
\(342\) 0 0
\(343\) −19.2544 −1.03964
\(344\) 0 0
\(345\) 4.57834 0.246489
\(346\) 0 0
\(347\) −0.881639 −0.0473289 −0.0236644 0.999720i \(-0.507533\pi\)
−0.0236644 + 0.999720i \(0.507533\pi\)
\(348\) 0 0
\(349\) −1.66553 −0.0891536 −0.0445768 0.999006i \(-0.514194\pi\)
−0.0445768 + 0.999006i \(0.514194\pi\)
\(350\) 0 0
\(351\) 6.72999 0.359220
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −9.45998 −0.502083
\(356\) 0 0
\(357\) −6.50885 −0.344485
\(358\) 0 0
\(359\) 17.7194 0.935196 0.467598 0.883941i \(-0.345120\pi\)
0.467598 + 0.883941i \(0.345120\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.37279 −0.0720526
\(364\) 0 0
\(365\) −9.25443 −0.484399
\(366\) 0 0
\(367\) 6.93554 0.362032 0.181016 0.983480i \(-0.442061\pi\)
0.181016 + 0.983480i \(0.442061\pi\)
\(368\) 0 0
\(369\) 6.15165 0.320242
\(370\) 0 0
\(371\) 6.50885 0.337923
\(372\) 0 0
\(373\) 3.25997 0.168795 0.0843973 0.996432i \(-0.473104\pi\)
0.0843973 + 0.996432i \(0.473104\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −7.42166 −0.382235
\(378\) 0 0
\(379\) −14.1461 −0.726637 −0.363318 0.931665i \(-0.618356\pi\)
−0.363318 + 0.931665i \(0.618356\pi\)
\(380\) 0 0
\(381\) 19.2544 0.986434
\(382\) 0 0
\(383\) −28.0766 −1.43465 −0.717324 0.696739i \(-0.754633\pi\)
−0.717324 + 0.696739i \(0.754633\pi\)
\(384\) 0 0
\(385\) 7.83276 0.399195
\(386\) 0 0
\(387\) 7.68111 0.390453
\(388\) 0 0
\(389\) 35.3522 1.79243 0.896213 0.443623i \(-0.146307\pi\)
0.896213 + 0.443623i \(0.146307\pi\)
\(390\) 0 0
\(391\) −11.8045 −0.596979
\(392\) 0 0
\(393\) −11.4061 −0.575360
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −21.2544 −1.06673 −0.533365 0.845885i \(-0.679073\pi\)
−0.533365 + 0.845885i \(0.679073\pi\)
\(398\) 0 0
\(399\) 2.52444 0.126380
\(400\) 0 0
\(401\) 23.8272 1.18987 0.594937 0.803772i \(-0.297177\pi\)
0.594937 + 0.803772i \(0.297177\pi\)
\(402\) 0 0
\(403\) −52.7144 −2.62589
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −14.0383 −0.695853
\(408\) 0 0
\(409\) 15.9900 0.790652 0.395326 0.918541i \(-0.370632\pi\)
0.395326 + 0.918541i \(0.370632\pi\)
\(410\) 0 0
\(411\) −2.74557 −0.135429
\(412\) 0 0
\(413\) 25.7633 1.26773
\(414\) 0 0
\(415\) 4.47054 0.219450
\(416\) 0 0
\(417\) −5.79445 −0.283755
\(418\) 0 0
\(419\) −1.30833 −0.0639159 −0.0319579 0.999489i \(-0.510174\pi\)
−0.0319579 + 0.999489i \(0.510174\pi\)
\(420\) 0 0
\(421\) −39.2333 −1.91211 −0.956057 0.293181i \(-0.905286\pi\)
−0.956057 + 0.293181i \(0.905286\pi\)
\(422\) 0 0
\(423\) 3.42166 0.166367
\(424\) 0 0
\(425\) −2.57834 −0.125068
\(426\) 0 0
\(427\) −37.5678 −1.81803
\(428\) 0 0
\(429\) 20.8816 1.00818
\(430\) 0 0
\(431\) −27.5577 −1.32741 −0.663705 0.747995i \(-0.731017\pi\)
−0.663705 + 0.747995i \(0.731017\pi\)
\(432\) 0 0
\(433\) 3.58336 0.172205 0.0861027 0.996286i \(-0.472559\pi\)
0.0861027 + 0.996286i \(0.472559\pi\)
\(434\) 0 0
\(435\) −1.10278 −0.0528740
\(436\) 0 0
\(437\) 4.57834 0.219012
\(438\) 0 0
\(439\) −5.68665 −0.271409 −0.135705 0.990749i \(-0.543330\pi\)
−0.135705 + 0.990749i \(0.543330\pi\)
\(440\) 0 0
\(441\) −0.627213 −0.0298673
\(442\) 0 0
\(443\) −0.362741 −0.0172343 −0.00861716 0.999963i \(-0.502743\pi\)
−0.00861716 + 0.999963i \(0.502743\pi\)
\(444\) 0 0
\(445\) −14.5628 −0.690341
\(446\) 0 0
\(447\) −4.09775 −0.193817
\(448\) 0 0
\(449\) 0.799473 0.0377295 0.0188647 0.999822i \(-0.493995\pi\)
0.0188647 + 0.999822i \(0.493995\pi\)
\(450\) 0 0
\(451\) 19.0872 0.898781
\(452\) 0 0
\(453\) 6.67609 0.313670
\(454\) 0 0
\(455\) 16.9894 0.796477
\(456\) 0 0
\(457\) 24.2056 1.13229 0.566144 0.824306i \(-0.308435\pi\)
0.566144 + 0.824306i \(0.308435\pi\)
\(458\) 0 0
\(459\) −2.57834 −0.120346
\(460\) 0 0
\(461\) 23.7633 1.10677 0.553383 0.832927i \(-0.313337\pi\)
0.553383 + 0.832927i \(0.313337\pi\)
\(462\) 0 0
\(463\) 4.62219 0.214811 0.107406 0.994215i \(-0.465746\pi\)
0.107406 + 0.994215i \(0.465746\pi\)
\(464\) 0 0
\(465\) −7.83276 −0.363236
\(466\) 0 0
\(467\) 4.98944 0.230884 0.115442 0.993314i \(-0.463172\pi\)
0.115442 + 0.993314i \(0.463172\pi\)
\(468\) 0 0
\(469\) −21.2333 −0.980463
\(470\) 0 0
\(471\) −19.4600 −0.896668
\(472\) 0 0
\(473\) 23.8328 1.09583
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 2.57834 0.118054
\(478\) 0 0
\(479\) −12.9739 −0.592790 −0.296395 0.955065i \(-0.595785\pi\)
−0.296395 + 0.955065i \(0.595785\pi\)
\(480\) 0 0
\(481\) −30.4494 −1.38837
\(482\) 0 0
\(483\) 11.5577 0.525895
\(484\) 0 0
\(485\) −12.9355 −0.587373
\(486\) 0 0
\(487\) −7.14663 −0.323845 −0.161922 0.986804i \(-0.551769\pi\)
−0.161922 + 0.986804i \(0.551769\pi\)
\(488\) 0 0
\(489\) −12.7300 −0.575670
\(490\) 0 0
\(491\) −22.1416 −0.999237 −0.499618 0.866246i \(-0.666526\pi\)
−0.499618 + 0.866246i \(0.666526\pi\)
\(492\) 0 0
\(493\) 2.84333 0.128057
\(494\) 0 0
\(495\) 3.10278 0.139459
\(496\) 0 0
\(497\) −23.8811 −1.07121
\(498\) 0 0
\(499\) 4.51890 0.202294 0.101147 0.994872i \(-0.467749\pi\)
0.101147 + 0.994872i \(0.467749\pi\)
\(500\) 0 0
\(501\) 20.8816 0.932923
\(502\) 0 0
\(503\) 8.35166 0.372382 0.186191 0.982514i \(-0.440386\pi\)
0.186191 + 0.982514i \(0.440386\pi\)
\(504\) 0 0
\(505\) −11.4600 −0.509962
\(506\) 0 0
\(507\) 32.2927 1.43417
\(508\) 0 0
\(509\) 4.28057 0.189733 0.0948666 0.995490i \(-0.469758\pi\)
0.0948666 + 0.995490i \(0.469758\pi\)
\(510\) 0 0
\(511\) −23.3622 −1.03348
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −14.5089 −0.639336
\(516\) 0 0
\(517\) 10.6167 0.466920
\(518\) 0 0
\(519\) −0.372787 −0.0163635
\(520\) 0 0
\(521\) 40.1305 1.75815 0.879075 0.476683i \(-0.158161\pi\)
0.879075 + 0.476683i \(0.158161\pi\)
\(522\) 0 0
\(523\) −20.5189 −0.897229 −0.448614 0.893725i \(-0.648082\pi\)
−0.448614 + 0.893725i \(0.648082\pi\)
\(524\) 0 0
\(525\) 2.52444 0.110176
\(526\) 0 0
\(527\) 20.1955 0.879730
\(528\) 0 0
\(529\) −2.03883 −0.0886448
\(530\) 0 0
\(531\) 10.2056 0.442883
\(532\) 0 0
\(533\) 41.4005 1.79326
\(534\) 0 0
\(535\) 15.2544 0.659506
\(536\) 0 0
\(537\) −1.04888 −0.0452623
\(538\) 0 0
\(539\) −1.94610 −0.0838245
\(540\) 0 0
\(541\) −15.6867 −0.674422 −0.337211 0.941429i \(-0.609484\pi\)
−0.337211 + 0.941429i \(0.609484\pi\)
\(542\) 0 0
\(543\) 5.36222 0.230115
\(544\) 0 0
\(545\) 16.6167 0.711779
\(546\) 0 0
\(547\) −23.8922 −1.02156 −0.510778 0.859712i \(-0.670643\pi\)
−0.510778 + 0.859712i \(0.670643\pi\)
\(548\) 0 0
\(549\) −14.8816 −0.635133
\(550\) 0 0
\(551\) −1.10278 −0.0469798
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.52444 −0.192052
\(556\) 0 0
\(557\) 22.8222 0.967008 0.483504 0.875342i \(-0.339364\pi\)
0.483504 + 0.875342i \(0.339364\pi\)
\(558\) 0 0
\(559\) 51.6938 2.18641
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 15.6061 0.657718 0.328859 0.944379i \(-0.393336\pi\)
0.328859 + 0.944379i \(0.393336\pi\)
\(564\) 0 0
\(565\) −4.37279 −0.183965
\(566\) 0 0
\(567\) 2.52444 0.106016
\(568\) 0 0
\(569\) −0.691675 −0.0289965 −0.0144983 0.999895i \(-0.504615\pi\)
−0.0144983 + 0.999895i \(0.504615\pi\)
\(570\) 0 0
\(571\) 44.7910 1.87445 0.937223 0.348730i \(-0.113387\pi\)
0.937223 + 0.348730i \(0.113387\pi\)
\(572\) 0 0
\(573\) 25.7194 1.07444
\(574\) 0 0
\(575\) 4.57834 0.190930
\(576\) 0 0
\(577\) 3.45998 0.144041 0.0720203 0.997403i \(-0.477055\pi\)
0.0720203 + 0.997403i \(0.477055\pi\)
\(578\) 0 0
\(579\) 7.77886 0.323279
\(580\) 0 0
\(581\) 11.2856 0.468205
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 6.72999 0.278251
\(586\) 0 0
\(587\) −27.8328 −1.14878 −0.574391 0.818581i \(-0.694761\pi\)
−0.574391 + 0.818581i \(0.694761\pi\)
\(588\) 0 0
\(589\) −7.83276 −0.322743
\(590\) 0 0
\(591\) −5.42166 −0.223017
\(592\) 0 0
\(593\) −17.5889 −0.722290 −0.361145 0.932510i \(-0.617614\pi\)
−0.361145 + 0.932510i \(0.617614\pi\)
\(594\) 0 0
\(595\) −6.50885 −0.266837
\(596\) 0 0
\(597\) −14.2056 −0.581394
\(598\) 0 0
\(599\) 19.4700 0.795524 0.397762 0.917489i \(-0.369787\pi\)
0.397762 + 0.917489i \(0.369787\pi\)
\(600\) 0 0
\(601\) 40.4777 1.65112 0.825560 0.564315i \(-0.190860\pi\)
0.825560 + 0.564315i \(0.190860\pi\)
\(602\) 0 0
\(603\) −8.41110 −0.342526
\(604\) 0 0
\(605\) −1.37279 −0.0558117
\(606\) 0 0
\(607\) −4.71440 −0.191352 −0.0956758 0.995413i \(-0.530501\pi\)
−0.0956758 + 0.995413i \(0.530501\pi\)
\(608\) 0 0
\(609\) −2.78389 −0.112809
\(610\) 0 0
\(611\) 23.0278 0.931603
\(612\) 0 0
\(613\) −9.48110 −0.382938 −0.191469 0.981499i \(-0.561325\pi\)
−0.191469 + 0.981499i \(0.561325\pi\)
\(614\) 0 0
\(615\) 6.15165 0.248059
\(616\) 0 0
\(617\) 7.73501 0.311400 0.155700 0.987804i \(-0.450237\pi\)
0.155700 + 0.987804i \(0.450237\pi\)
\(618\) 0 0
\(619\) −6.12892 −0.246342 −0.123171 0.992385i \(-0.539306\pi\)
−0.123171 + 0.992385i \(0.539306\pi\)
\(620\) 0 0
\(621\) 4.57834 0.183722
\(622\) 0 0
\(623\) −36.7628 −1.47287
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.10278 0.123913
\(628\) 0 0
\(629\) 11.6655 0.465135
\(630\) 0 0
\(631\) −4.19550 −0.167020 −0.0835102 0.996507i \(-0.526613\pi\)
−0.0835102 + 0.996507i \(0.526613\pi\)
\(632\) 0 0
\(633\) −6.09775 −0.242364
\(634\) 0 0
\(635\) 19.2544 0.764089
\(636\) 0 0
\(637\) −4.22114 −0.167247
\(638\) 0 0
\(639\) −9.45998 −0.374231
\(640\) 0 0
\(641\) −47.3850 −1.87159 −0.935797 0.352541i \(-0.885318\pi\)
−0.935797 + 0.352541i \(0.885318\pi\)
\(642\) 0 0
\(643\) −21.3678 −0.842662 −0.421331 0.906907i \(-0.638437\pi\)
−0.421331 + 0.906907i \(0.638437\pi\)
\(644\) 0 0
\(645\) 7.68111 0.302443
\(646\) 0 0
\(647\) −9.29274 −0.365335 −0.182668 0.983175i \(-0.558473\pi\)
−0.182668 + 0.983175i \(0.558473\pi\)
\(648\) 0 0
\(649\) 31.6655 1.24298
\(650\) 0 0
\(651\) −19.7733 −0.774978
\(652\) 0 0
\(653\) −21.9094 −0.857381 −0.428690 0.903451i \(-0.641025\pi\)
−0.428690 + 0.903451i \(0.641025\pi\)
\(654\) 0 0
\(655\) −11.4061 −0.445672
\(656\) 0 0
\(657\) −9.25443 −0.361050
\(658\) 0 0
\(659\) 8.52998 0.332281 0.166140 0.986102i \(-0.446870\pi\)
0.166140 + 0.986102i \(0.446870\pi\)
\(660\) 0 0
\(661\) 5.36222 0.208566 0.104283 0.994548i \(-0.466745\pi\)
0.104283 + 0.994548i \(0.466745\pi\)
\(662\) 0 0
\(663\) −17.3522 −0.673903
\(664\) 0 0
\(665\) 2.52444 0.0978935
\(666\) 0 0
\(667\) −5.04888 −0.195493
\(668\) 0 0
\(669\) 24.8222 0.959682
\(670\) 0 0
\(671\) −46.1744 −1.78254
\(672\) 0 0
\(673\) −2.31889 −0.0893866 −0.0446933 0.999001i \(-0.514231\pi\)
−0.0446933 + 0.999001i \(0.514231\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −35.9305 −1.38092 −0.690461 0.723370i \(-0.742592\pi\)
−0.690461 + 0.723370i \(0.742592\pi\)
\(678\) 0 0
\(679\) −32.6550 −1.25318
\(680\) 0 0
\(681\) 22.7839 0.873080
\(682\) 0 0
\(683\) 5.15667 0.197315 0.0986573 0.995121i \(-0.468545\pi\)
0.0986573 + 0.995121i \(0.468545\pi\)
\(684\) 0 0
\(685\) −2.74557 −0.104903
\(686\) 0 0
\(687\) 6.47054 0.246866
\(688\) 0 0
\(689\) 17.3522 0.661065
\(690\) 0 0
\(691\) −19.9688 −0.759650 −0.379825 0.925058i \(-0.624016\pi\)
−0.379825 + 0.925058i \(0.624016\pi\)
\(692\) 0 0
\(693\) 7.83276 0.297542
\(694\) 0 0
\(695\) −5.79445 −0.219796
\(696\) 0 0
\(697\) −15.8610 −0.600779
\(698\) 0 0
\(699\) −19.3522 −0.731967
\(700\) 0 0
\(701\) −39.2333 −1.48182 −0.740911 0.671604i \(-0.765606\pi\)
−0.740911 + 0.671604i \(0.765606\pi\)
\(702\) 0 0
\(703\) −4.52444 −0.170642
\(704\) 0 0
\(705\) 3.42166 0.128867
\(706\) 0 0
\(707\) −28.9300 −1.08802
\(708\) 0 0
\(709\) 36.8605 1.38433 0.692163 0.721741i \(-0.256658\pi\)
0.692163 + 0.721741i \(0.256658\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −35.8610 −1.34301
\(714\) 0 0
\(715\) 20.8816 0.780929
\(716\) 0 0
\(717\) −23.9250 −0.893495
\(718\) 0 0
\(719\) 17.8383 0.665256 0.332628 0.943058i \(-0.392065\pi\)
0.332628 + 0.943058i \(0.392065\pi\)
\(720\) 0 0
\(721\) −36.6267 −1.36405
\(722\) 0 0
\(723\) −8.09775 −0.301159
\(724\) 0 0
\(725\) −1.10278 −0.0409560
\(726\) 0 0
\(727\) 2.00554 0.0743813 0.0371907 0.999308i \(-0.488159\pi\)
0.0371907 + 0.999308i \(0.488159\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −19.8045 −0.732496
\(732\) 0 0
\(733\) −43.4288 −1.60408 −0.802040 0.597271i \(-0.796252\pi\)
−0.802040 + 0.597271i \(0.796252\pi\)
\(734\) 0 0
\(735\) −0.627213 −0.0231351
\(736\) 0 0
\(737\) −26.0978 −0.961323
\(738\) 0 0
\(739\) 20.6066 0.758026 0.379013 0.925391i \(-0.376264\pi\)
0.379013 + 0.925391i \(0.376264\pi\)
\(740\) 0 0
\(741\) 6.72999 0.247232
\(742\) 0 0
\(743\) 47.4499 1.74077 0.870385 0.492373i \(-0.163870\pi\)
0.870385 + 0.492373i \(0.163870\pi\)
\(744\) 0 0
\(745\) −4.09775 −0.150130
\(746\) 0 0
\(747\) 4.47054 0.163568
\(748\) 0 0
\(749\) 38.5089 1.40708
\(750\) 0 0
\(751\) 7.83276 0.285822 0.142911 0.989736i \(-0.454354\pi\)
0.142911 + 0.989736i \(0.454354\pi\)
\(752\) 0 0
\(753\) −1.00502 −0.0366251
\(754\) 0 0
\(755\) 6.67609 0.242968
\(756\) 0 0
\(757\) −43.4600 −1.57958 −0.789790 0.613378i \(-0.789810\pi\)
−0.789790 + 0.613378i \(0.789810\pi\)
\(758\) 0 0
\(759\) 14.2056 0.515629
\(760\) 0 0
\(761\) −2.41110 −0.0874023 −0.0437012 0.999045i \(-0.513915\pi\)
−0.0437012 + 0.999045i \(0.513915\pi\)
\(762\) 0 0
\(763\) 41.9477 1.51861
\(764\) 0 0
\(765\) −2.57834 −0.0932200
\(766\) 0 0
\(767\) 68.6832 2.48001
\(768\) 0 0
\(769\) −8.09775 −0.292012 −0.146006 0.989284i \(-0.546642\pi\)
−0.146006 + 0.989284i \(0.546642\pi\)
\(770\) 0 0
\(771\) −16.7839 −0.604457
\(772\) 0 0
\(773\) 13.6172 0.489775 0.244888 0.969551i \(-0.421249\pi\)
0.244888 + 0.969551i \(0.421249\pi\)
\(774\) 0 0
\(775\) −7.83276 −0.281361
\(776\) 0 0
\(777\) −11.4217 −0.409750
\(778\) 0 0
\(779\) 6.15165 0.220406
\(780\) 0 0
\(781\) −29.3522 −1.05030
\(782\) 0 0
\(783\) −1.10278 −0.0394100
\(784\) 0 0
\(785\) −19.4600 −0.694556
\(786\) 0 0
\(787\) −36.9099 −1.31570 −0.657848 0.753151i \(-0.728533\pi\)
−0.657848 + 0.753151i \(0.728533\pi\)
\(788\) 0 0
\(789\) 22.0383 0.784585
\(790\) 0 0
\(791\) −11.0388 −0.392496
\(792\) 0 0
\(793\) −100.153 −3.55655
\(794\) 0 0
\(795\) 2.57834 0.0914442
\(796\) 0 0
\(797\) −26.4705 −0.937635 −0.468817 0.883295i \(-0.655320\pi\)
−0.468817 + 0.883295i \(0.655320\pi\)
\(798\) 0 0
\(799\) −8.82220 −0.312107
\(800\) 0 0
\(801\) −14.5628 −0.514550
\(802\) 0 0
\(803\) −28.7144 −1.01331
\(804\) 0 0
\(805\) 11.5577 0.407356
\(806\) 0 0
\(807\) 32.3572 1.13903
\(808\) 0 0
\(809\) 29.2544 1.02853 0.514265 0.857631i \(-0.328065\pi\)
0.514265 + 0.857631i \(0.328065\pi\)
\(810\) 0 0
\(811\) −38.9200 −1.36666 −0.683332 0.730108i \(-0.739470\pi\)
−0.683332 + 0.730108i \(0.739470\pi\)
\(812\) 0 0
\(813\) 23.1466 0.811788
\(814\) 0 0
\(815\) −12.7300 −0.445912
\(816\) 0 0
\(817\) 7.68111 0.268728
\(818\) 0 0
\(819\) 16.9894 0.593659
\(820\) 0 0
\(821\) −36.6167 −1.27793 −0.638965 0.769236i \(-0.720637\pi\)
−0.638965 + 0.769236i \(0.720637\pi\)
\(822\) 0 0
\(823\) −41.5522 −1.44842 −0.724209 0.689580i \(-0.757795\pi\)
−0.724209 + 0.689580i \(0.757795\pi\)
\(824\) 0 0
\(825\) 3.10278 0.108025
\(826\) 0 0
\(827\) −25.0972 −0.872716 −0.436358 0.899773i \(-0.643732\pi\)
−0.436358 + 0.899773i \(0.643732\pi\)
\(828\) 0 0
\(829\) 16.9511 0.588737 0.294368 0.955692i \(-0.404891\pi\)
0.294368 + 0.955692i \(0.404891\pi\)
\(830\) 0 0
\(831\) 30.8222 1.06921
\(832\) 0 0
\(833\) 1.61717 0.0560315
\(834\) 0 0
\(835\) 20.8816 0.722639
\(836\) 0 0
\(837\) −7.83276 −0.270740
\(838\) 0 0
\(839\) 29.6555 1.02382 0.511910 0.859039i \(-0.328938\pi\)
0.511910 + 0.859039i \(0.328938\pi\)
\(840\) 0 0
\(841\) −27.7839 −0.958065
\(842\) 0 0
\(843\) −3.94610 −0.135911
\(844\) 0 0
\(845\) 32.2927 1.11090
\(846\) 0 0
\(847\) −3.46552 −0.119077
\(848\) 0 0
\(849\) −17.4756 −0.599760
\(850\) 0 0
\(851\) −20.7144 −0.710081
\(852\) 0 0
\(853\) 36.4777 1.24897 0.624486 0.781036i \(-0.285308\pi\)
0.624486 + 0.781036i \(0.285308\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −24.8716 −0.849597 −0.424799 0.905288i \(-0.639655\pi\)
−0.424799 + 0.905288i \(0.639655\pi\)
\(858\) 0 0
\(859\) −15.0388 −0.513118 −0.256559 0.966529i \(-0.582589\pi\)
−0.256559 + 0.966529i \(0.582589\pi\)
\(860\) 0 0
\(861\) 15.5295 0.529243
\(862\) 0 0
\(863\) 40.4877 1.37822 0.689109 0.724658i \(-0.258002\pi\)
0.689109 + 0.724658i \(0.258002\pi\)
\(864\) 0 0
\(865\) −0.372787 −0.0126751
\(866\) 0 0
\(867\) −10.3522 −0.351578
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −56.6066 −1.91804
\(872\) 0 0
\(873\) −12.9355 −0.437802
\(874\) 0 0
\(875\) 2.52444 0.0853416
\(876\) 0 0
\(877\) −40.0822 −1.35348 −0.676739 0.736223i \(-0.736608\pi\)
−0.676739 + 0.736223i \(0.736608\pi\)
\(878\) 0 0
\(879\) 2.05944 0.0694631
\(880\) 0 0
\(881\) −21.0278 −0.708443 −0.354221 0.935162i \(-0.615254\pi\)
−0.354221 + 0.935162i \(0.615254\pi\)
\(882\) 0 0
\(883\) 0.287716 0.00968241 0.00484121 0.999988i \(-0.498459\pi\)
0.00484121 + 0.999988i \(0.498459\pi\)
\(884\) 0 0
\(885\) 10.2056 0.343056
\(886\) 0 0
\(887\) 20.4111 0.685338 0.342669 0.939456i \(-0.388669\pi\)
0.342669 + 0.939456i \(0.388669\pi\)
\(888\) 0 0
\(889\) 48.6066 1.63021
\(890\) 0 0
\(891\) 3.10278 0.103947
\(892\) 0 0
\(893\) 3.42166 0.114502
\(894\) 0 0
\(895\) −1.04888 −0.0350601
\(896\) 0 0
\(897\) 30.8122 1.02879
\(898\) 0 0
\(899\) 8.63778 0.288086
\(900\) 0 0
\(901\) −6.64782 −0.221471
\(902\) 0 0
\(903\) 19.3905 0.645275
\(904\) 0 0
\(905\) 5.36222 0.178246
\(906\) 0 0
\(907\) 16.3799 0.543887 0.271943 0.962313i \(-0.412334\pi\)
0.271943 + 0.962313i \(0.412334\pi\)
\(908\) 0 0
\(909\) −11.4600 −0.380103
\(910\) 0 0
\(911\) 50.5855 1.67597 0.837986 0.545692i \(-0.183733\pi\)
0.837986 + 0.545692i \(0.183733\pi\)
\(912\) 0 0
\(913\) 13.8711 0.459066
\(914\) 0 0
\(915\) −14.8816 −0.491972
\(916\) 0 0
\(917\) −28.7939 −0.950859
\(918\) 0 0
\(919\) 28.7456 0.948229 0.474114 0.880463i \(-0.342768\pi\)
0.474114 + 0.880463i \(0.342768\pi\)
\(920\) 0 0
\(921\) −10.2056 −0.336284
\(922\) 0 0
\(923\) −63.6655 −2.09558
\(924\) 0 0
\(925\) −4.52444 −0.148763
\(926\) 0 0
\(927\) −14.5089 −0.476533
\(928\) 0 0
\(929\) −49.7733 −1.63301 −0.816505 0.577339i \(-0.804091\pi\)
−0.816505 + 0.577339i \(0.804091\pi\)
\(930\) 0 0
\(931\) −0.627213 −0.0205561
\(932\) 0 0
\(933\) 1.20053 0.0393035
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 34.3033 1.12064 0.560320 0.828276i \(-0.310678\pi\)
0.560320 + 0.828276i \(0.310678\pi\)
\(938\) 0 0
\(939\) −7.45998 −0.243447
\(940\) 0 0
\(941\) 34.6705 1.13023 0.565114 0.825013i \(-0.308832\pi\)
0.565114 + 0.825013i \(0.308832\pi\)
\(942\) 0 0
\(943\) 28.1643 0.917157
\(944\) 0 0
\(945\) 2.52444 0.0821200
\(946\) 0 0
\(947\) −5.96169 −0.193729 −0.0968644 0.995298i \(-0.530881\pi\)
−0.0968644 + 0.995298i \(0.530881\pi\)
\(948\) 0 0
\(949\) −62.2822 −2.02177
\(950\) 0 0
\(951\) −4.48059 −0.145293
\(952\) 0 0
\(953\) 9.52946 0.308690 0.154345 0.988017i \(-0.450673\pi\)
0.154345 + 0.988017i \(0.450673\pi\)
\(954\) 0 0
\(955\) 25.7194 0.832261
\(956\) 0 0
\(957\) −3.42166 −0.110607
\(958\) 0 0
\(959\) −6.93103 −0.223815
\(960\) 0 0
\(961\) 30.3522 0.979103
\(962\) 0 0
\(963\) 15.2544 0.491567
\(964\) 0 0
\(965\) 7.77886 0.250410
\(966\) 0 0
\(967\) 19.9844 0.642655 0.321328 0.946968i \(-0.395871\pi\)
0.321328 + 0.946968i \(0.395871\pi\)
\(968\) 0 0
\(969\) −2.57834 −0.0828281
\(970\) 0 0
\(971\) −18.3900 −0.590162 −0.295081 0.955472i \(-0.595347\pi\)
−0.295081 + 0.955472i \(0.595347\pi\)
\(972\) 0 0
\(973\) −14.6277 −0.468943
\(974\) 0 0
\(975\) 6.72999 0.215532
\(976\) 0 0
\(977\) −38.4394 −1.22978 −0.614892 0.788611i \(-0.710800\pi\)
−0.614892 + 0.788611i \(0.710800\pi\)
\(978\) 0 0
\(979\) −45.1849 −1.44412
\(980\) 0 0
\(981\) 16.6167 0.530529
\(982\) 0 0
\(983\) −54.5260 −1.73911 −0.869555 0.493836i \(-0.835594\pi\)
−0.869555 + 0.493836i \(0.835594\pi\)
\(984\) 0 0
\(985\) −5.42166 −0.172749
\(986\) 0 0
\(987\) 8.63778 0.274943
\(988\) 0 0
\(989\) 35.1667 1.11824
\(990\) 0 0
\(991\) −2.91995 −0.0927553 −0.0463777 0.998924i \(-0.514768\pi\)
−0.0463777 + 0.998924i \(0.514768\pi\)
\(992\) 0 0
\(993\) −24.2439 −0.769356
\(994\) 0 0
\(995\) −14.2056 −0.450346
\(996\) 0 0
\(997\) −36.2822 −1.14907 −0.574534 0.818481i \(-0.694817\pi\)
−0.574534 + 0.818481i \(0.694817\pi\)
\(998\) 0 0
\(999\) −4.52444 −0.143147
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 4560.2.a.bv.1.2 3
4.3 odd 2 2280.2.a.s.1.2 3
12.11 even 2 6840.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.s.1.2 3 4.3 odd 2
4560.2.a.bv.1.2 3 1.1 even 1 trivial
6840.2.a.bf.1.2 3 12.11 even 2