Properties

Label 585.2.a.h.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.00000 q^{28} -2.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} -2.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.00000 q^{53} -2.00000 q^{55} +12.0000 q^{56} -2.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{70} -6.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} +6.00000 q^{76} +8.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +16.0000 q^{83} -2.00000 q^{85} +10.0000 q^{86} +6.00000 q^{88} -2.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -4.00000 q^{94} -6.00000 q^{95} -2.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 18.0000 1.45999
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −18.0000 −1.32698
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 40.0000 2.71538
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 30.0000 1.90500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −20.0000 −1.23560
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −40.0000 −2.30556
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) −10.0000 −0.567962
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) −16.0000 −0.878114
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −30.0000 −1.61749
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −10.0000 −0.533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) −27.0000 −1.36371
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) 40.0000 1.95413 0.977064 0.212946i \(-0.0683059\pi\)
0.977064 + 0.212946i \(0.0683059\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 40.0000 1.92006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 6.00000 0.286039
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) −28.0000 −1.32288
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 18.0000 0.828517
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 2.00000 0.0881305
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −14.0000 −0.610429
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) −24.0000 −1.04053
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) −10.0000 −0.432338
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 48.0000 2.04117
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 22.0000 0.927189 0.463595 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 18.0000 0.755263
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) −64.0000 −2.65517
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −40.0000 −1.63028
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 10.0000 0.401610
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 36.0000 1.43200
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −18.0000 −0.699590
\(663\) 0 0
\(664\) −48.0000 −1.86276
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 20.0000 0.765840
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 22.0000 0.835109
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −30.0000 −1.13552
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −72.0000 −2.70784
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −60.0000 −2.24702
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 10.0000 0.373197
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 22.0000 0.817624
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 6.00000 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 34.0000 1.24483
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 18.0000 0.652929
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −40.0000 −1.44810
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −10.0000 −0.353112
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) −54.0000 −1.89971
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) −60.0000 −2.09913
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) −46.0000 −1.60346 −0.801730 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 40.0000 1.38178
\(839\) −38.0000 −1.31191 −0.655953 0.754802i \(-0.727733\pi\)
−0.655953 + 0.754802i \(0.727733\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 30.0000 1.02538
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −10.0000 −0.340997
\(861\) 0 0
\(862\) −14.0000 −0.476842
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) −40.0000 −1.35769
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −30.0000 −1.01593
\(873\) 0 0
\(874\) −36.0000 −1.21772
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) −22.0000 −0.740359 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −14.0000 −0.470339
\(887\) −58.0000 −1.94745 −0.973725 0.227728i \(-0.926870\pi\)
−0.973725 + 0.227728i \(0.926870\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 80.0000 2.64183
\(918\) 0 0
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) −18.0000 −0.593442
\(921\) 0 0
\(922\) −10.0000 −0.329332
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −10.0000 −0.328266
\(929\) 38.0000 1.24674 0.623370 0.781927i \(-0.285763\pi\)
0.623370 + 0.781927i \(0.285763\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 21.0000 0.674966
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) −50.0000 −1.58750
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 10.0000 0.316544
\(999\) 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 585.2.a.h.1.1 1
3.2 odd 2 65.2.a.a.1.1 1
4.3 odd 2 9360.2.a.ca.1.1 1
5.2 odd 4 2925.2.c.h.2224.2 2
5.3 odd 4 2925.2.c.h.2224.1 2
5.4 even 2 2925.2.a.f.1.1 1
12.11 even 2 1040.2.a.f.1.1 1
13.12 even 2 7605.2.a.f.1.1 1
15.2 even 4 325.2.b.b.274.1 2
15.8 even 4 325.2.b.b.274.2 2
15.14 odd 2 325.2.a.d.1.1 1
21.20 even 2 3185.2.a.e.1.1 1
24.5 odd 2 4160.2.a.q.1.1 1
24.11 even 2 4160.2.a.f.1.1 1
33.32 even 2 7865.2.a.c.1.1 1
39.2 even 12 845.2.m.b.316.1 4
39.5 even 4 845.2.c.a.506.2 2
39.8 even 4 845.2.c.a.506.1 2
39.11 even 12 845.2.m.b.316.2 4
39.17 odd 6 845.2.e.a.146.1 2
39.20 even 12 845.2.m.b.361.1 4
39.23 odd 6 845.2.e.a.191.1 2
39.29 odd 6 845.2.e.b.191.1 2
39.32 even 12 845.2.m.b.361.2 4
39.35 odd 6 845.2.e.b.146.1 2
39.38 odd 2 845.2.a.a.1.1 1
60.59 even 2 5200.2.a.d.1.1 1
195.194 odd 2 4225.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.a.1.1 1 3.2 odd 2
325.2.a.d.1.1 1 15.14 odd 2
325.2.b.b.274.1 2 15.2 even 4
325.2.b.b.274.2 2 15.8 even 4
585.2.a.h.1.1 1 1.1 even 1 trivial
845.2.a.a.1.1 1 39.38 odd 2
845.2.c.a.506.1 2 39.8 even 4
845.2.c.a.506.2 2 39.5 even 4
845.2.e.a.146.1 2 39.17 odd 6
845.2.e.a.191.1 2 39.23 odd 6
845.2.e.b.146.1 2 39.35 odd 6
845.2.e.b.191.1 2 39.29 odd 6
845.2.m.b.316.1 4 39.2 even 12
845.2.m.b.316.2 4 39.11 even 12
845.2.m.b.361.1 4 39.20 even 12
845.2.m.b.361.2 4 39.32 even 12
1040.2.a.f.1.1 1 12.11 even 2
2925.2.a.f.1.1 1 5.4 even 2
2925.2.c.h.2224.1 2 5.3 odd 4
2925.2.c.h.2224.2 2 5.2 odd 4
3185.2.a.e.1.1 1 21.20 even 2
4160.2.a.f.1.1 1 24.11 even 2
4160.2.a.q.1.1 1 24.5 odd 2
4225.2.a.g.1.1 1 195.194 odd 2
5200.2.a.d.1.1 1 60.59 even 2
7605.2.a.f.1.1 1 13.12 even 2
7865.2.a.c.1.1 1 33.32 even 2
9360.2.a.ca.1.1 1 4.3 odd 2