Properties

Label 4650.2.a.bv.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} +3.00000 q^{21} +3.00000 q^{22} -5.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -3.00000 q^{38} +2.00000 q^{39} +4.00000 q^{41} +3.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} -5.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{51} +2.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} +4.00000 q^{58} +6.00000 q^{59} -2.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -2.00000 q^{67} +4.00000 q^{68} -5.00000 q^{69} +7.00000 q^{71} +1.00000 q^{72} -5.00000 q^{73} -3.00000 q^{76} +9.00000 q^{77} +2.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -12.0000 q^{83} +3.00000 q^{84} -1.00000 q^{86} +4.00000 q^{87} +3.00000 q^{88} +1.00000 q^{89} +6.00000 q^{91} -5.00000 q^{92} +1.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} +10.0000 q^{97} +2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 3.00000 0.639602
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −3.00000 −0.486664
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 3.00000 0.462910
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −3.00000 −0.397360
\(58\) 4.00000 0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000 0.127000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 4.00000 0.485071
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 1.00000 0.117851
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 9.00000 1.02565
\(78\) 2.00000 0.226455
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 4.00000 0.428845
\(88\) 3.00000 0.319801
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −5.00000 −0.521286
\(93\) 1.00000 0.103695
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 2.00000 0.202031
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 4.00000 0.396059
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 1.00000 0.0962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 2.00000 0.184900
\(118\) 6.00000 0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 3.00000 0.261116
\(133\) −9.00000 −0.780399
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −5.00000 −0.425628
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 7.00000 0.587427
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −5.00000 −0.413803
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −3.00000 −0.243332
\(153\) 4.00000 0.323381
\(154\) 9.00000 0.725241
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) 1.00000 0.0785674
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −1.00000 −0.0762493
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 6.00000 0.450988
\(178\) 1.00000 0.0749532
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 6.00000 0.444750
\(183\) −2.00000 −0.147844
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 12.0000 0.877527
\(188\) −10.0000 −0.729325
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 3.00000 0.213201
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −1.00000 −0.0703598
\(203\) 12.0000 0.842235
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −5.00000 −0.347524
\(208\) 2.00000 0.138675
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) −3.00000 −0.206041
\(213\) 7.00000 0.479632
\(214\) −9.00000 −0.615227
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 3.00000 0.203653
\(218\) 20.0000 1.35457
\(219\) −5.00000 −0.337869
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) −3.00000 −0.198680
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) 4.00000 0.262613
\(233\) −15.0000 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −1.00000 −0.0649570
\(238\) 12.0000 0.777844
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) −6.00000 −0.381771
\(248\) 1.00000 0.0635001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 3.00000 0.188982
\(253\) −15.0000 −0.943042
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.0000 1.43470 0.717350 0.696713i \(-0.245355\pi\)
0.717350 + 0.696713i \(0.245355\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 6.00000 0.370681
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −9.00000 −0.551825
\(267\) 1.00000 0.0611990
\(268\) −2.00000 −0.122169
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 4.00000 0.242536
\(273\) 6.00000 0.363137
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −14.0000 −0.839664
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) −10.0000 −0.595491
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −5.00000 −0.292603
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) −11.0000 −0.637213
\(299\) −10.0000 −0.578315
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) −1.00000 −0.0574485
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 9.00000 0.512823
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 5.00000 0.282166
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) −3.00000 −0.168232
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) −15.0000 −0.835917
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 20.0000 1.10600
\(328\) 4.00000 0.220863
\(329\) −30.0000 −1.65395
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 19.0000 1.03963
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) −3.00000 −0.162221
\(343\) −15.0000 −0.809924
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 4.00000 0.214423
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 1.00000 0.0529999
\(357\) 12.0000 0.635107
\(358\) 4.00000 0.211407
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 5.00000 0.262794
\(363\) −2.00000 −0.104973
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −5.00000 −0.260643
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 1.00000 0.0518476
\(373\) 9.00000 0.466002 0.233001 0.972476i \(-0.425145\pi\)
0.233001 + 0.972476i \(0.425145\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 8.00000 0.412021
\(378\) 3.00000 0.154303
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 16.0000 0.818631
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −1.00000 −0.0508329
\(388\) 10.0000 0.507673
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 2.00000 0.101015
\(393\) 6.00000 0.302660
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −35.0000 −1.75660 −0.878300 0.478110i \(-0.841322\pi\)
−0.878300 + 0.478110i \(0.841322\pi\)
\(398\) 7.00000 0.350878
\(399\) −9.00000 −0.450564
\(400\) 0 0
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 2.00000 0.0996271
\(404\) −1.00000 −0.0497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 4.00000 0.198030
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −16.0000 −0.788263
\(413\) 18.0000 0.885722
\(414\) −5.00000 −0.245737
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −14.0000 −0.685583
\(418\) −9.00000 −0.440204
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −19.0000 −0.924906
\(423\) −10.0000 −0.486217
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 7.00000 0.339151
\(427\) −6.00000 −0.290360
\(428\) −9.00000 −0.435031
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 15.0000 0.717547
\(438\) −5.00000 −0.238909
\(439\) 18.0000 0.859093 0.429547 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 8.00000 0.380521
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.00000 0.284108
\(447\) −11.0000 −0.520282
\(448\) 3.00000 0.141737
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) 27.0000 1.26717
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 13.0000 0.607450
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 9.00000 0.418718
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −15.0000 −0.694862
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 2.00000 0.0924500
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) 5.00000 0.230388
\(472\) 6.00000 0.276172
\(473\) −3.00000 −0.137940
\(474\) −1.00000 −0.0459315
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) −3.00000 −0.137361
\(478\) 12.0000 0.548867
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) −15.0000 −0.682524
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) −37.0000 −1.66979 −0.834893 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) 4.00000 0.180334
\(493\) 16.0000 0.720604
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 21.0000 0.941979
\(498\) −12.0000 −0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 19.0000 0.848857
\(502\) −12.0000 −0.535586
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −15.0000 −0.666831
\(507\) −9.00000 −0.399704
\(508\) −8.00000 −0.354943
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) 1.00000 0.0441942
\(513\) −3.00000 −0.132453
\(514\) 23.0000 1.01449
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 4.00000 0.175075
\(523\) 43.0000 1.88026 0.940129 0.340818i \(-0.110704\pi\)
0.940129 + 0.340818i \(0.110704\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 4.00000 0.174243
\(528\) 3.00000 0.130558
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −9.00000 −0.390199
\(533\) 8.00000 0.346518
\(534\) 1.00000 0.0432742
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 4.00000 0.172613
\(538\) 2.00000 0.0862261
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 13.0000 0.558398
\(543\) 5.00000 0.214571
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) −5.00000 −0.212814
\(553\) −3.00000 −0.127573
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 1.00000 0.0423714 0.0211857 0.999776i \(-0.493256\pi\)
0.0211857 + 0.999776i \(0.493256\pi\)
\(558\) 1.00000 0.0423334
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 16.0000 0.674919
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −10.0000 −0.421076
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) 3.00000 0.125988
\(568\) 7.00000 0.293713
\(569\) −23.0000 −0.964210 −0.482105 0.876113i \(-0.660128\pi\)
−0.482105 + 0.876113i \(0.660128\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 6.00000 0.250873
\(573\) 16.0000 0.668410
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 10.0000 0.414513
\(583\) −9.00000 −0.372742
\(584\) −5.00000 −0.206901
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 2.00000 0.0824786
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −11.0000 −0.450578
\(597\) 7.00000 0.286491
\(598\) −10.0000 −0.408930
\(599\) −27.0000 −1.10319 −0.551595 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −3.00000 −0.122271
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 0 0
\(606\) −1.00000 −0.0406222
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) −3.00000 −0.121666
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −20.0000 −0.809113
\(612\) 4.00000 0.161690
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 9.00000 0.362620
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) −16.0000 −0.643614
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) −9.00000 −0.359425
\(628\) 5.00000 0.199522
\(629\) 0 0
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −19.0000 −0.755182
\(634\) 8.00000 0.317721
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 4.00000 0.158486
\(638\) 12.0000 0.475085
\(639\) 7.00000 0.276916
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −9.00000 −0.355202
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) −15.0000 −0.591083
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) −14.0000 −0.548282
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 20.0000 0.782062
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) −5.00000 −0.195069
\(658\) −30.0000 −1.16952
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 12.0000 0.466393
\(663\) 8.00000 0.310694
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) 19.0000 0.735132
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 3.00000 0.115728
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) −9.00000 −0.345643
\(679\) 30.0000 1.15129
\(680\) 0 0
\(681\) 27.0000 1.03464
\(682\) 3.00000 0.114876
\(683\) 19.0000 0.727015 0.363507 0.931591i \(-0.381579\pi\)
0.363507 + 0.931591i \(0.381579\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 13.0000 0.495981
\(688\) −1.00000 −0.0381246
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) 22.0000 0.836315
\(693\) 9.00000 0.341882
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 16.0000 0.606043
\(698\) −24.0000 −0.908413
\(699\) −15.0000 −0.567352
\(700\) 0 0
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −3.00000 −0.112827
\(708\) 6.00000 0.225494
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 1.00000 0.0374766
\(713\) −5.00000 −0.187251
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 12.0000 0.448148
\(718\) 15.0000 0.559795
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −10.0000 −0.372161
\(723\) −18.0000 −0.669427
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −45.0000 −1.66896 −0.834479 0.551040i \(-0.814231\pi\)
−0.834479 + 0.551040i \(0.814231\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) −2.00000 −0.0739221
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −6.00000 −0.221013
\(738\) 4.00000 0.147242
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) −9.00000 −0.330400
\(743\) 37.0000 1.35740 0.678699 0.734416i \(-0.262544\pi\)
0.678699 + 0.734416i \(0.262544\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 9.00000 0.329513
\(747\) −12.0000 −0.439057
\(748\) 12.0000 0.438763
\(749\) −27.0000 −0.986559
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) −10.0000 −0.364662
\(753\) −12.0000 −0.437304
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 15.0000 0.544825
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) −8.00000 −0.289809
\(763\) 60.0000 2.17215
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 12.0000 0.433295
\(768\) 1.00000 0.0360844
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 0 0
\(771\) 23.0000 0.828325
\(772\) 6.00000 0.215945
\(773\) 31.0000 1.11499 0.557496 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 21.0000 0.751439
\(782\) −20.0000 −0.715199
\(783\) 4.00000 0.142948
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 35.0000 1.24762 0.623808 0.781578i \(-0.285585\pi\)
0.623808 + 0.781578i \(0.285585\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −27.0000 −0.960009
\(792\) 3.00000 0.106600
\(793\) −4.00000 −0.142044
\(794\) −35.0000 −1.24210
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −9.00000 −0.318597
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 1.00000 0.0353333
\(802\) 25.0000 0.882781
\(803\) −15.0000 −0.529339
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 2.00000 0.0704033
\(808\) −1.00000 −0.0351799
\(809\) 17.0000 0.597688 0.298844 0.954302i \(-0.403399\pi\)
0.298844 + 0.954302i \(0.403399\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 12.0000 0.421117
\(813\) 13.0000 0.455930
\(814\) 0 0
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 3.00000 0.104957
\(818\) 8.00000 0.279713
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 40.0000 1.39601 0.698005 0.716093i \(-0.254071\pi\)
0.698005 + 0.716093i \(0.254071\pi\)
\(822\) 10.0000 0.348790
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) −5.00000 −0.173762
\(829\) −35.0000 −1.21560 −0.607800 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 2.00000 0.0693375
\(833\) 8.00000 0.277184
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) −9.00000 −0.311272
\(837\) 1.00000 0.0345651
\(838\) 12.0000 0.414533
\(839\) 51.0000 1.76072 0.880358 0.474310i \(-0.157302\pi\)
0.880358 + 0.474310i \(0.157302\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −22.0000 −0.758170
\(843\) 16.0000 0.551069
\(844\) −19.0000 −0.654007
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) −6.00000 −0.206162
\(848\) −3.00000 −0.103020
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 7.00000 0.239816
\(853\) 41.0000 1.40381 0.701907 0.712269i \(-0.252332\pi\)
0.701907 + 0.712269i \(0.252332\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 6.00000 0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 16.0000 0.544962
\(863\) −43.0000 −1.46374 −0.731869 0.681446i \(-0.761351\pi\)
−0.731869 + 0.681446i \(0.761351\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 1.00000 0.0339814
\(867\) −1.00000 −0.0339618
\(868\) 3.00000 0.101827
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 20.0000 0.677285
\(873\) 10.0000 0.338449
\(874\) 15.0000 0.507383
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 18.0000 0.607471
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 2.00000 0.0673435
\(883\) −11.0000 −0.370179 −0.185090 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −15.0000 −0.503935
\(887\) 46.0000 1.54453 0.772264 0.635301i \(-0.219124\pi\)
0.772264 + 0.635301i \(0.219124\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 6.00000 0.200895
\(893\) 30.0000 1.00391
\(894\) −11.0000 −0.367895
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) −10.0000 −0.333890
\(898\) 2.00000 0.0667409
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 12.0000 0.399556
\(903\) −3.00000 −0.0998337
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 27.0000 0.896026
\(909\) −1.00000 −0.0331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) −3.00000 −0.0993399
\(913\) −36.0000 −1.19143
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 13.0000 0.429532
\(917\) 18.0000 0.594412
\(918\) 4.00000 0.132020
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 9.00000 0.296078
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) −16.0000 −0.525509
\(928\) 4.00000 0.131306
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −15.0000 −0.491341
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −6.00000 −0.195907
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 5.00000 0.162909
\(943\) −20.0000 −0.651290
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −1.00000 −0.0324785
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 8.00000 0.259418
\(952\) 12.0000 0.388922
\(953\) 52.0000 1.68445 0.842223 0.539130i \(-0.181247\pi\)
0.842223 + 0.539130i \(0.181247\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 12.0000 0.387905
\(958\) 3.00000 0.0969256
\(959\) 30.0000 0.968751
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −9.00000 −0.290021
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) −15.0000 −0.482617
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 1.00000 0.0320750
\(973\) −42.0000 −1.34646
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −14.0000 −0.447671
\(979\) 3.00000 0.0958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) −37.0000 −1.18072
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) −30.0000 −0.954911
\(988\) −6.00000 −0.190885
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 1.00000 0.0317500
\(993\) 12.0000 0.380808
\(994\) 21.0000 0.666080
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 4650.2.a.bv.1.1 1
5.2 odd 4 4650.2.d.y.3349.2 2
5.3 odd 4 4650.2.d.y.3349.1 2
5.4 even 2 930.2.a.a.1.1 1
15.14 odd 2 2790.2.a.y.1.1 1
20.19 odd 2 7440.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.a.1.1 1 5.4 even 2
2790.2.a.y.1.1 1 15.14 odd 2
4650.2.a.bv.1.1 1 1.1 even 1 trivial
4650.2.d.y.3349.1 2 5.3 odd 4
4650.2.d.y.3349.2 2 5.2 odd 4
7440.2.a.v.1.1 1 20.19 odd 2