Properties

Label 690.2.a.j.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +4.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -8.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} +4.00000 q^{58} +8.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} -6.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} +10.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} +6.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -2.00000 q^{88} -12.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} -12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.00000 −1.29777
\(39\) 4.00000 0.640513
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 4.00000 0.525226
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 1.00000 0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −2.00000 −0.246183
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 2.00000 0.215666
\(87\) 4.00000 0.428845
\(88\) −2.00000 −0.213201
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −7.00000 −0.707107
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −2.00000 −0.190693
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −8.00000 −0.749269
\(115\) −1.00000 −0.0932505
\(116\) 4.00000 0.371391
\(117\) 4.00000 0.369800
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 4.00000 0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 10.0000 0.839181
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −2.00000 −0.165521
\(147\) −7.00000 −0.577350
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 1.00000 0.0816497
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −8.00000 −0.648886
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −2.00000 −0.156174
\(165\) −2.00000 −0.155700
\(166\) 8.00000 0.620920
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) −8.00000 −0.611775
\(172\) 2.00000 0.152499
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 8.00000 0.601317
\(178\) −12.0000 −0.899438
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −16.0000 −1.14873
\(195\) 4.00000 0.286446
\(196\) −7.00000 −0.500000
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −2.00000 −0.142134
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.00000 −0.423207
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 4.00000 0.277350
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 10.0000 0.685189
\(214\) −12.0000 −0.820303
\(215\) 2.00000 0.136399
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −2.00000 −0.135147
\(220\) −2.00000 −0.134840
\(221\) 24.0000 1.61441
\(222\) −2.00000 −0.134231
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −8.00000 −0.529813
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) −12.0000 −0.782794
\(236\) 8.00000 0.520756
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 1.00000 0.0645497
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −7.00000 −0.447214
\(246\) −2.00000 −0.127515
\(247\) −32.0000 −2.03611
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 1.00000 0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 2.00000 0.125491
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 4.00000 0.247594
\(262\) −4.00000 −0.247121
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −2.00000 −0.123091
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) −6.00000 −0.366508
\(269\) 8.00000 0.487769 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) 1.00000 0.0608581
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −12.0000 −0.714590
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 10.0000 0.593391
\(285\) −8.00000 −0.473879
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) −16.0000 −0.937937
\(292\) −2.00000 −0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −7.00000 −0.408248
\(295\) 8.00000 0.465778
\(296\) −2.00000 −0.116248
\(297\) −2.00000 −0.116052
\(298\) 14.0000 0.810998
\(299\) −4.00000 −0.231326
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) 12.0000 0.689382
\(304\) −8.00000 −0.458831
\(305\) 2.00000 0.114520
\(306\) 6.00000 0.342997
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.0000 1.47432 0.737162 0.675716i \(-0.236165\pi\)
0.737162 + 0.675716i \(0.236165\pi\)
\(312\) 4.00000 0.226455
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −48.0000 −2.67079
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −16.0000 −0.886158
\(327\) −10.0000 −0.553001
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 8.00000 0.439057
\(333\) −2.00000 −0.109599
\(334\) −16.0000 −0.875481
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 3.00000 0.163178
\(339\) 18.0000 0.977626
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) −1.00000 −0.0538382
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 4.00000 0.214423
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) −2.00000 −0.106600
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 8.00000 0.425195
\(355\) 10.0000 0.530745
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 2.00000 0.104542
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −12.0000 −0.620505
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −8.00000 −0.410391
\(381\) 2.00000 0.102463
\(382\) 24.0000 1.22795
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) 2.00000 0.101666
\(388\) −16.0000 −0.812277
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 4.00000 0.202548
\(391\) −6.00000 −0.303433
\(392\) −7.00000 −0.353553
\(393\) −4.00000 −0.201773
\(394\) −18.0000 −0.906827
\(395\) −8.00000 −0.402524
\(396\) −2.00000 −0.100504
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) −6.00000 −0.299253
\(403\) 0 0
\(404\) 12.0000 0.597022
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 6.00000 0.297044
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 8.00000 0.392705
\(416\) 4.00000 0.196116
\(417\) 4.00000 0.195881
\(418\) 16.0000 0.782586
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 4.00000 0.194717
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) 2.00000 0.0964486
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −10.0000 −0.478913
\(437\) 8.00000 0.382692
\(438\) −2.00000 −0.0955637
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −7.00000 −0.333333
\(442\) 24.0000 1.14156
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −12.0000 −0.568855
\(446\) −2.00000 −0.0947027
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 1.00000 0.0471405
\(451\) 4.00000 0.188353
\(452\) 18.0000 0.846649
\(453\) −4.00000 −0.187936
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −10.0000 −0.467269
\(459\) 6.00000 0.280056
\(460\) −1.00000 −0.0466252
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) −10.0000 −0.460776
\(472\) 8.00000 0.368230
\(473\) −4.00000 −0.183920
\(474\) −8.00000 −0.367452
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −14.0000 −0.640345
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 1.00000 0.0456435
\(481\) −8.00000 −0.364769
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −16.0000 −0.726523
\(486\) 1.00000 0.0453609
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 2.00000 0.0905357
\(489\) −16.0000 −0.723545
\(490\) −7.00000 −0.316228
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 24.0000 1.08091
\(494\) −32.0000 −1.43975
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0000 −0.714827
\(502\) 2.00000 0.0892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 2.00000 0.0889108
\(507\) 3.00000 0.133235
\(508\) 2.00000 0.0887357
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 4.00000 0.175412
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 4.00000 0.175075
\(523\) 30.0000 1.31181 0.655904 0.754844i \(-0.272288\pi\)
0.655904 + 0.754844i \(0.272288\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) −12.0000 −0.519291
\(535\) −12.0000 −0.518805
\(536\) −6.00000 −0.259161
\(537\) 16.0000 0.690451
\(538\) 8.00000 0.344904
\(539\) 14.0000 0.603023
\(540\) 1.00000 0.0430331
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 24.0000 1.03089
\(543\) 10.0000 0.429141
\(544\) 6.00000 0.257248
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 2.00000 0.0854358
\(549\) 2.00000 0.0853579
\(550\) −2.00000 −0.0852803
\(551\) −32.0000 −1.36325
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) −2.00000 −0.0848953
\(556\) 4.00000 0.169638
\(557\) 38.0000 1.61011 0.805056 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 12.0000 0.506189
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −12.0000 −0.505291
\(565\) 18.0000 0.757266
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) −8.00000 −0.335083
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −8.00000 −0.334497
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 19.0000 0.790296
\(579\) 26.0000 1.08052
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 12.0000 0.496989
\(584\) −2.00000 −0.0827606
\(585\) 4.00000 0.165380
\(586\) −14.0000 −0.578335
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −7.00000 −0.288675
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) −18.0000 −0.740421
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) −16.0000 −0.654836
\(598\) −4.00000 −0.163572
\(599\) −34.0000 −1.38920 −0.694601 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) −4.00000 −0.162758
\(605\) −7.00000 −0.284590
\(606\) 12.0000 0.487467
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −48.0000 −1.94187
\(612\) 6.00000 0.242536
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 28.0000 1.12999
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 26.0000 1.04251
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 28.0000 1.11911
\(627\) 16.0000 0.638978
\(628\) −10.0000 −0.399043
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) 18.0000 0.714871
\(635\) 2.00000 0.0793676
\(636\) −6.00000 −0.237915
\(637\) −28.0000 −1.10940
\(638\) −8.00000 −0.316723
\(639\) 10.0000 0.395594
\(640\) 1.00000 0.0395285
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) −12.0000 −0.473602
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) −48.0000 −1.88853
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) −10.0000 −0.391031
\(655\) −4.00000 −0.156293
\(656\) −2.00000 −0.0780869
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 28.0000 1.08825
\(663\) 24.0000 0.932083
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −4.00000 −0.154881
\(668\) −16.0000 −0.619059
\(669\) −2.00000 −0.0773245
\(670\) −6.00000 −0.231800
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −28.0000 −1.07852
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 18.0000 0.691286
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −8.00000 −0.305888
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 2.00000 0.0762493
\(689\) −24.0000 −0.914327
\(690\) −1.00000 −0.0380693
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 4.00000 0.151729
\(696\) 4.00000 0.151620
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 4.00000 0.150970
\(703\) 16.0000 0.603451
\(704\) −2.00000 −0.0753778
\(705\) −12.0000 −0.451946
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 10.0000 0.375293
\(711\) −8.00000 −0.300023
\(712\) −12.0000 −0.449719
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 16.0000 0.597948
\(717\) −14.0000 −0.522840
\(718\) −16.0000 −0.597115
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) −6.00000 −0.223142
\(724\) 10.0000 0.371647
\(725\) 4.00000 0.148556
\(726\) −7.00000 −0.259794
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 16.0000 0.590571
\(735\) −7.00000 −0.258199
\(736\) −1.00000 −0.0368605
\(737\) 12.0000 0.442026
\(738\) −2.00000 −0.0736210
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 6.00000 0.219676
\(747\) 8.00000 0.292705
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −12.0000 −0.437595
\(753\) 2.00000 0.0728841
\(754\) 16.0000 0.582686
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 12.0000 0.435860
\(759\) 2.00000 0.0725954
\(760\) −8.00000 −0.290191
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 2.00000 0.0724524
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) 6.00000 0.216930
\(766\) −32.0000 −1.15621
\(767\) 32.0000 1.15545
\(768\) 1.00000 0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 26.0000 0.935760
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 16.0000 0.573259
\(780\) 4.00000 0.143223
\(781\) −20.0000 −0.715656
\(782\) −6.00000 −0.214560
\(783\) 4.00000 0.142948
\(784\) −7.00000 −0.250000
\(785\) −10.0000 −0.356915
\(786\) −4.00000 −0.142675
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 8.00000 0.284088
\(794\) 28.0000 0.993683
\(795\) −6.00000 −0.212798
\(796\) −16.0000 −0.567105
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) −8.00000 −0.282490
\(803\) 4.00000 0.141157
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) 0 0
\(807\) 8.00000 0.281613
\(808\) 12.0000 0.422159
\(809\) −46.0000 −1.61727 −0.808637 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 1.00000 0.0351364
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 4.00000 0.140200
\(815\) −16.0000 −0.560456
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 2.00000 0.0697580
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 8.00000 0.277684
\(831\) −28.0000 −0.971309
\(832\) 4.00000 0.138675
\(833\) −42.0000 −1.45521
\(834\) 4.00000 0.138509
\(835\) −16.0000 −0.553703
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −14.0000 −0.482472
\(843\) 12.0000 0.413302
\(844\) 4.00000 0.137686
\(845\) 3.00000 0.103203
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 14.0000 0.480479
\(850\) 6.00000 0.205798
\(851\) 2.00000 0.0685591
\(852\) 10.0000 0.342594
\(853\) 36.0000 1.23262 0.616308 0.787505i \(-0.288628\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −8.00000 −0.273115
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 1.00000 0.0340207
\(865\) −10.0000 −0.340010
\(866\) 16.0000 0.543702
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 4.00000 0.135613
\(871\) −24.0000 −0.813209
\(872\) −10.0000 −0.338643
\(873\) −16.0000 −0.541518
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 28.0000 0.944954
\(879\) −14.0000 −0.472208
\(880\) −2.00000 −0.0674200
\(881\) 4.00000 0.134763 0.0673817 0.997727i \(-0.478535\pi\)
0.0673817 + 0.997727i \(0.478535\pi\)
\(882\) −7.00000 −0.235702
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 24.0000 0.807207
\(885\) 8.00000 0.268917
\(886\) −4.00000 −0.134383
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) −2.00000 −0.0670025
\(892\) −2.00000 −0.0669650
\(893\) 96.0000 3.21252
\(894\) 14.0000 0.468230
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 10.0000 0.332411
\(906\) −4.00000 −0.132891
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 28.0000 0.929213
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) −8.00000 −0.264906
\(913\) −16.0000 −0.529523
\(914\) −8.00000 −0.264616
\(915\) 2.00000 0.0661180
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 28.0000 0.922631
\(922\) −12.0000 −0.395199
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −26.0000 −0.854413
\(927\) 0 0
\(928\) 4.00000 0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 56.0000 1.83533
\(932\) 6.00000 0.196537
\(933\) 26.0000 0.851202
\(934\) −24.0000 −0.785304
\(935\) −12.0000 −0.392442
\(936\) 4.00000 0.130744
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) −12.0000 −0.391397
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −10.0000 −0.325818
\(943\) 2.00000 0.0651290
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −8.00000 −0.259828
\(949\) −8.00000 −0.259691
\(950\) −8.00000 −0.259554
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) −6.00000 −0.194257
\(955\) 24.0000 0.776622
\(956\) −14.0000 −0.452792
\(957\) −8.00000 −0.258603
\(958\) −16.0000 −0.516937
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) −8.00000 −0.257930
\(963\) −12.0000 −0.386695
\(964\) −6.00000 −0.193247
\(965\) 26.0000 0.836970
\(966\) 0 0
\(967\) −42.0000 −1.35063 −0.675314 0.737530i \(-0.735992\pi\)
−0.675314 + 0.737530i \(0.735992\pi\)
\(968\) −7.00000 −0.224989
\(969\) −48.0000 −1.54198
\(970\) −16.0000 −0.513729
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 4.00000 0.128103
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 24.0000 0.767043
\(980\) −7.00000 −0.223607
\(981\) −10.0000 −0.319275
\(982\) 20.0000 0.638226
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −18.0000 −0.573528
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) −32.0000 −1.01806
\(989\) −2.00000 −0.0635963
\(990\) −2.00000 −0.0635642
\(991\) 12.0000 0.381193 0.190596 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 8.00000 0.253490
\(997\) −4.00000 −0.126681 −0.0633406 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) −4.00000 −0.126618
\(999\) −2.00000 −0.0632772
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 690.2.a.j.1.1 1
3.2 odd 2 2070.2.a.c.1.1 1
4.3 odd 2 5520.2.a.l.1.1 1
5.2 odd 4 3450.2.d.m.2899.2 2
5.3 odd 4 3450.2.d.m.2899.1 2
5.4 even 2 3450.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.j.1.1 1 1.1 even 1 trivial
2070.2.a.c.1.1 1 3.2 odd 2
3450.2.a.b.1.1 1 5.4 even 2
3450.2.d.m.2899.1 2 5.3 odd 4
3450.2.d.m.2899.2 2 5.2 odd 4
5520.2.a.l.1.1 1 4.3 odd 2