Properties

Label 490.2.a.e.1.1
Level 490
Weight 2
Character 490.1
Self dual yes
Analytic conductor 3.913
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 490.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 490.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} -3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} +3.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -9.00000 q^{27} +9.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +4.00000 q^{34} +6.00000 q^{36} -4.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +7.00000 q^{41} -5.00000 q^{43} -2.00000 q^{44} +6.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} -3.00000 q^{48} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{53} -9.00000 q^{54} -2.00000 q^{55} -18.0000 q^{57} +9.00000 q^{58} -10.0000 q^{59} -3.00000 q^{60} -1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -9.00000 q^{67} +4.00000 q^{68} -9.00000 q^{69} +2.00000 q^{71} +6.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -3.00000 q^{75} +6.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} +7.00000 q^{82} +7.00000 q^{83} +4.00000 q^{85} -5.00000 q^{86} -27.0000 q^{87} -2.00000 q^{88} -1.00000 q^{89} +6.00000 q^{90} +3.00000 q^{92} -12.0000 q^{93} -8.00000 q^{94} +6.00000 q^{95} -3.00000 q^{96} -14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −3.00000 −1.22474
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.00000 −0.866025
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 6.00000 1.41421
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) −3.00000 −0.547723
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −2.00000 −0.301511
\(45\) 6.00000 0.894427
\(46\) 3.00000 0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −3.00000 −0.433013
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −9.00000 −1.22474
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 9.00000 1.18176
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −3.00000 −0.387298
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 4.00000 0.485071
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 6.00000 0.707107
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −4.00000 −0.464991
\(75\) −3.00000 −0.346410
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) 7.00000 0.773021
\(83\) 7.00000 0.768350 0.384175 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −5.00000 −0.539164
\(87\) −27.0000 −2.89470
\(88\) −2.00000 −0.213201
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) −12.0000 −1.24434
\(94\) −8.00000 −0.825137
\(95\) 6.00000 0.615587
\(96\) −3.00000 −0.306186
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 1.00000 0.100000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) −12.0000 −1.18818
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −9.00000 −0.866025
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) −2.00000 −0.190693
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −18.0000 −1.68585
\(115\) 3.00000 0.279751
\(116\) 9.00000 0.835629
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) −7.00000 −0.636364
\(122\) −1.00000 −0.0905357
\(123\) −21.0000 −1.89351
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 15.0000 1.32068
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) −9.00000 −0.774597
\(136\) 4.00000 0.342997
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −9.00000 −0.766131
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 9.00000 0.747409
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) −3.00000 −0.244949
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 6.00000 0.486664
\(153\) 24.0000 1.94029
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 10.0000 0.795557
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 7.00000 0.546608
\(165\) 6.00000 0.467099
\(166\) 7.00000 0.543305
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 4.00000 0.306786
\(171\) 36.0000 2.75299
\(172\) −5.00000 −0.381246
\(173\) −8.00000 −0.608229 −0.304114 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) −27.0000 −2.04686
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 30.0000 2.25494
\(178\) −1.00000 −0.0749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 6.00000 0.447214
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) 3.00000 0.221163
\(185\) −4.00000 −0.294086
\(186\) −12.0000 −0.879883
\(187\) −8.00000 −0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −3.00000 −0.216506
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −12.0000 −0.852803
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 27.0000 1.90443
\(202\) −3.00000 −0.211079
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 7.00000 0.488901
\(206\) −1.00000 −0.0696733
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) −2.00000 −0.137361
\(213\) −6.00000 −0.411113
\(214\) 3.00000 0.205076
\(215\) −5.00000 −0.340997
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) −9.00000 −0.609557
\(219\) −12.0000 −0.810885
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −18.0000 −1.19208
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −10.0000 −0.650945
\(237\) −30.0000 −1.94871
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −3.00000 −0.193649
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) −21.0000 −1.33891
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −21.0000 −1.33082
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 16.0000 1.00393
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 15.0000 0.933859
\(259\) 0 0
\(260\) 0 0
\(261\) 54.0000 3.34252
\(262\) −8.00000 −0.494242
\(263\) 5.00000 0.308313 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(264\) 6.00000 0.369274
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 3.00000 0.183597
\(268\) −9.00000 −0.549762
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) −9.00000 −0.547723
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −2.00000 −0.120605
\(276\) −9.00000 −0.541736
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 14.0000 0.839664
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 24.0000 1.42918
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 2.00000 0.118678
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) 6.00000 0.353553
\(289\) −1.00000 −0.0588235
\(290\) 9.00000 0.528498
\(291\) 42.0000 2.46208
\(292\) 4.00000 0.234082
\(293\) −28.0000 −1.63578 −0.817889 0.575376i \(-0.804856\pi\)
−0.817889 + 0.575376i \(0.804856\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) −4.00000 −0.232495
\(297\) 18.0000 1.04447
\(298\) 3.00000 0.173785
\(299\) 0 0
\(300\) −3.00000 −0.173205
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 9.00000 0.517036
\(304\) 6.00000 0.344124
\(305\) −1.00000 −0.0572598
\(306\) 24.0000 1.37199
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 3.00000 0.170664
\(310\) 4.00000 0.227185
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −32.0000 −1.79730 −0.898650 0.438667i \(-0.855451\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(318\) 6.00000 0.336463
\(319\) −18.0000 −1.00781
\(320\) 1.00000 0.0559017
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 27.0000 1.49310
\(328\) 7.00000 0.386510
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 7.00000 0.384175
\(333\) −24.0000 −1.31519
\(334\) 21.0000 1.14907
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −13.0000 −0.707107
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) −8.00000 −0.433224
\(342\) 36.0000 1.94666
\(343\) 0 0
\(344\) −5.00000 −0.269582
\(345\) −9.00000 −0.484544
\(346\) −8.00000 −0.430083
\(347\) 19.0000 1.01997 0.509987 0.860182i \(-0.329650\pi\)
0.509987 + 0.860182i \(0.329650\pi\)
\(348\) −27.0000 −1.44735
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 30.0000 1.59448
\(355\) 2.00000 0.106149
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 6.00000 0.316228
\(361\) 17.0000 0.894737
\(362\) −7.00000 −0.367912
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 3.00000 0.156813
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) 3.00000 0.156386
\(369\) 42.0000 2.18643
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) −12.0000 −0.622171
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −8.00000 −0.413670
\(375\) −3.00000 −0.154919
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 6.00000 0.307794
\(381\) −48.0000 −2.45911
\(382\) −18.0000 −0.920960
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) −30.0000 −1.52499
\(388\) −14.0000 −0.710742
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 2.00000 0.100759
\(395\) 10.0000 0.503155
\(396\) −12.0000 −0.603023
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 31.0000 1.54807 0.774033 0.633145i \(-0.218236\pi\)
0.774033 + 0.633145i \(0.218236\pi\)
\(402\) 27.0000 1.34664
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −12.0000 −0.594089
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) 7.00000 0.345705
\(411\) −36.0000 −1.77575
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) 18.0000 0.884652
\(415\) 7.00000 0.343616
\(416\) 0 0
\(417\) −42.0000 −2.05675
\(418\) −12.0000 −0.586939
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) −26.0000 −1.26566
\(423\) −48.0000 −2.33384
\(424\) −2.00000 −0.0971286
\(425\) 4.00000 0.194029
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) −5.00000 −0.241121
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) −9.00000 −0.433013
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −27.0000 −1.29455
\(436\) −9.00000 −0.431022
\(437\) 18.0000 0.861057
\(438\) −12.0000 −0.573382
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 0 0
\(443\) 31.0000 1.47285 0.736427 0.676517i \(-0.236511\pi\)
0.736427 + 0.676517i \(0.236511\pi\)
\(444\) 12.0000 0.569495
\(445\) −1.00000 −0.0474045
\(446\) −28.0000 −1.32584
\(447\) −9.00000 −0.425685
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 6.00000 0.282843
\(451\) −14.0000 −0.659234
\(452\) 2.00000 0.0940721
\(453\) 48.0000 2.25524
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −22.0000 −1.02799
\(459\) −36.0000 −1.68034
\(460\) 3.00000 0.139876
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) 9.00000 0.417815
\(465\) −12.0000 −0.556487
\(466\) 24.0000 1.11178
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 30.0000 1.38233
\(472\) −10.0000 −0.460287
\(473\) 10.0000 0.459800
\(474\) −30.0000 −1.37795
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 16.0000 0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −3.00000 −0.136931
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −21.0000 −0.946753
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −21.0000 −0.941033
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 1.00000 0.0447214
\(501\) −63.0000 −2.81463
\(502\) 0 0
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) −6.00000 −0.266733
\(507\) 39.0000 1.73205
\(508\) 16.0000 0.709885
\(509\) −1.00000 −0.0443242 −0.0221621 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −54.0000 −2.38416
\(514\) −8.00000 −0.352865
\(515\) −1.00000 −0.0440653
\(516\) 15.0000 0.660338
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 54.0000 2.36352
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 5.00000 0.218010
\(527\) 16.0000 0.696971
\(528\) 6.00000 0.261116
\(529\) −14.0000 −0.608696
\(530\) −2.00000 −0.0868744
\(531\) −60.0000 −2.60378
\(532\) 0 0
\(533\) 0 0
\(534\) 3.00000 0.129823
\(535\) 3.00000 0.129701
\(536\) −9.00000 −0.388741
\(537\) −36.0000 −1.55351
\(538\) −3.00000 −0.129339
\(539\) 0 0
\(540\) −9.00000 −0.387298
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 6.00000 0.257722
\(543\) 21.0000 0.901196
\(544\) 4.00000 0.171499
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) −33.0000 −1.41098 −0.705489 0.708721i \(-0.749273\pi\)
−0.705489 + 0.708721i \(0.749273\pi\)
\(548\) 12.0000 0.512615
\(549\) −6.00000 −0.256074
\(550\) −2.00000 −0.0852803
\(551\) 54.0000 2.30048
\(552\) −9.00000 −0.383065
\(553\) 0 0
\(554\) 12.0000 0.509831
\(555\) 12.0000 0.509372
\(556\) 14.0000 0.593732
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 24.0000 1.01600
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 2.00000 0.0843649
\(563\) −17.0000 −0.716465 −0.358232 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) 24.0000 1.01058
\(565\) 2.00000 0.0841406
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) −18.0000 −0.753937
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) 54.0000 2.25588
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 6.00000 0.250000
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −78.0000 −3.24157
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) 42.0000 1.74096
\(583\) 4.00000 0.165663
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −28.0000 −1.15667
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −10.0000 −0.411693
\(591\) −6.00000 −0.246807
\(592\) −4.00000 −0.164399
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 18.0000 0.738549
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −3.00000 −0.122474
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) −54.0000 −2.19905
\(604\) −16.0000 −0.651031
\(605\) −7.00000 −0.284590
\(606\) 9.00000 0.365600
\(607\) −1.00000 −0.0405887 −0.0202944 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) 0 0
\(612\) 24.0000 0.970143
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) −7.00000 −0.282497
\(615\) −21.0000 −0.846802
\(616\) 0 0
\(617\) 44.0000 1.77137 0.885687 0.464283i \(-0.153688\pi\)
0.885687 + 0.464283i \(0.153688\pi\)
\(618\) 3.00000 0.120678
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 4.00000 0.160644
\(621\) −27.0000 −1.08347
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 36.0000 1.43770
\(628\) −10.0000 −0.399043
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 10.0000 0.397779
\(633\) 78.0000 3.10022
\(634\) −32.0000 −1.27088
\(635\) 16.0000 0.634941
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) 12.0000 0.474713
\(640\) 1.00000 0.0395285
\(641\) 5.00000 0.197488 0.0987441 0.995113i \(-0.468517\pi\)
0.0987441 + 0.995113i \(0.468517\pi\)
\(642\) −9.00000 −0.355202
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 15.0000 0.590624
\(646\) 24.0000 0.944267
\(647\) 11.0000 0.432455 0.216227 0.976343i \(-0.430625\pi\)
0.216227 + 0.976343i \(0.430625\pi\)
\(648\) 9.00000 0.353553
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 27.0000 1.05578
\(655\) −8.00000 −0.312586
\(656\) 7.00000 0.273304
\(657\) 24.0000 0.936329
\(658\) 0 0
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 6.00000 0.233550
\(661\) 11.0000 0.427850 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) −24.0000 −0.929981
\(667\) 27.0000 1.04544
\(668\) 21.0000 0.812514
\(669\) 84.0000 3.24763
\(670\) −9.00000 −0.347700
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −26.0000 −1.00148
\(675\) −9.00000 −0.346410
\(676\) −13.0000 −0.500000
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) −12.0000 −0.459841
\(682\) −8.00000 −0.306336
\(683\) −37.0000 −1.41577 −0.707883 0.706330i \(-0.750350\pi\)
−0.707883 + 0.706330i \(0.750350\pi\)
\(684\) 36.0000 1.37649
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 66.0000 2.51806
\(688\) −5.00000 −0.190623
\(689\) 0 0
\(690\) −9.00000 −0.342624
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) −8.00000 −0.304114
\(693\) 0 0
\(694\) 19.0000 0.721230
\(695\) 14.0000 0.531050
\(696\) −27.0000 −1.02343
\(697\) 28.0000 1.06058
\(698\) 35.0000 1.32477
\(699\) −72.0000 −2.72329
\(700\) 0 0
\(701\) −47.0000 −1.77517 −0.887583 0.460648i \(-0.847617\pi\)
−0.887583 + 0.460648i \(0.847617\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −2.00000 −0.0753778
\(705\) 24.0000 0.903892
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 30.0000 1.12747
\(709\) −11.0000 −0.413114 −0.206557 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(710\) 2.00000 0.0750587
\(711\) 60.0000 2.25018
\(712\) −1.00000 −0.0374766
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −48.0000 −1.79259
\(718\) −4.00000 −0.149279
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 6.00000 0.223607
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) 30.0000 1.11571
\(724\) −7.00000 −0.260153
\(725\) 9.00000 0.334252
\(726\) 21.0000 0.779383
\(727\) 21.0000 0.778847 0.389423 0.921059i \(-0.372674\pi\)
0.389423 + 0.921059i \(0.372674\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 4.00000 0.148047
\(731\) −20.0000 −0.739727
\(732\) 3.00000 0.110883
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 11.0000 0.406017
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 18.0000 0.663039
\(738\) 42.0000 1.54604
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) −12.0000 −0.439941
\(745\) 3.00000 0.109911
\(746\) −4.00000 −0.146450
\(747\) 42.0000 1.53670
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 30.0000 1.08965
\(759\) 18.0000 0.653359
\(760\) 6.00000 0.217643
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −48.0000 −1.73886
\(763\) 0 0
\(764\) −18.0000 −0.651217
\(765\) 24.0000 0.867722
\(766\) −15.0000 −0.541972
\(767\) 0 0
\(768\) −3.00000 −0.108253
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 26.0000 0.935760
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) −30.0000 −1.07833
\(775\) 4.00000 0.143684
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 12.0000 0.429119
\(783\) −81.0000 −2.89470
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 24.0000 0.856052
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 2.00000 0.0712470
\(789\) −15.0000 −0.534014
\(790\) 10.0000 0.355784
\(791\) 0 0
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) 4.00000 0.141776
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 31.0000 1.09465
\(803\) −8.00000 −0.282314
\(804\) 27.0000 0.952217
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) −3.00000 −0.105540
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 9.00000 0.316228
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) −12.0000 −0.420084
\(817\) −30.0000 −1.04957
\(818\) −3.00000 −0.104893
\(819\) 0 0
\(820\) 7.00000 0.244451
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −36.0000 −1.25564
\(823\) 19.0000 0.662298 0.331149 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −19.0000 −0.660695 −0.330347 0.943859i \(-0.607166\pi\)
−0.330347 + 0.943859i \(0.607166\pi\)
\(828\) 18.0000 0.625543
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 7.00000 0.242974
\(831\) −36.0000 −1.24883
\(832\) 0 0
\(833\) 0 0
\(834\) −42.0000 −1.45434
\(835\) 21.0000 0.726735
\(836\) −12.0000 −0.415029
\(837\) −36.0000 −1.24434
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −19.0000 −0.654783
\(843\) −6.00000 −0.206651
\(844\) −26.0000 −0.894957
\(845\) −13.0000 −0.447214
\(846\) −48.0000 −1.65027
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) −12.0000 −0.411839
\(850\) 4.00000 0.137199
\(851\) −12.0000 −0.411355
\(852\) −6.00000 −0.205557
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 36.0000 1.23117
\(856\) 3.00000 0.102538
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) −5.00000 −0.170499
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) −11.0000 −0.374444 −0.187222 0.982318i \(-0.559948\pi\)
−0.187222 + 0.982318i \(0.559948\pi\)
\(864\) −9.00000 −0.306186
\(865\) −8.00000 −0.272008
\(866\) −14.0000 −0.475739
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) −27.0000 −0.915386
\(871\) 0 0
\(872\) −9.00000 −0.304778
\(873\) −84.0000 −2.84297
\(874\) 18.0000 0.608859
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 20.0000 0.674967
\(879\) 84.0000 2.83325
\(880\) −2.00000 −0.0674200
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 30.0000 1.00844
\(886\) 31.0000 1.04147
\(887\) −29.0000 −0.973725 −0.486862 0.873479i \(-0.661859\pi\)
−0.486862 + 0.873479i \(0.661859\pi\)
\(888\) 12.0000 0.402694
\(889\) 0 0
\(890\) −1.00000 −0.0335201
\(891\) −18.0000 −0.603023
\(892\) −28.0000 −0.937509
\(893\) −48.0000 −1.60626
\(894\) −9.00000 −0.301005
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) −33.0000 −1.10122
\(899\) 36.0000 1.20067
\(900\) 6.00000 0.200000
\(901\) −8.00000 −0.266519
\(902\) −14.0000 −0.466149
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −7.00000 −0.232688
\(906\) 48.0000 1.59469
\(907\) 5.00000 0.166022 0.0830111 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 4.00000 0.132745
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −18.0000 −0.596040
\(913\) −14.0000 −0.463332
\(914\) −32.0000 −1.05847
\(915\) 3.00000 0.0991769
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) −36.0000 −1.18818
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 3.00000 0.0989071
\(921\) 21.0000 0.691974
\(922\) 14.0000 0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −19.0000 −0.624379
\(927\) −6.00000 −0.197066
\(928\) 9.00000 0.295439
\(929\) −43.0000 −1.41078 −0.705392 0.708817i \(-0.749229\pi\)
−0.705392 + 0.708817i \(0.749229\pi\)
\(930\) −12.0000 −0.393496
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) −54.0000 −1.76788
\(934\) 13.0000 0.425373
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) −8.00000 −0.260931
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 30.0000 0.977453
\(943\) 21.0000 0.683854
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) −25.0000 −0.812391 −0.406195 0.913786i \(-0.633145\pi\)
−0.406195 + 0.913786i \(0.633145\pi\)
\(948\) −30.0000 −0.974355
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) 96.0000 3.11301
\(952\) 0 0
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) −12.0000 −0.388514
\(955\) −18.0000 −0.582466
\(956\) 16.0000 0.517477
\(957\) 54.0000 1.74557
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) −3.00000 −0.0968246
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) −10.0000 −0.322078
\(965\) 26.0000 0.836970
\(966\) 0 0
\(967\) 37.0000 1.18984 0.594920 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) −7.00000 −0.224989
\(969\) −72.0000 −2.31297
\(970\) −14.0000 −0.449513
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 12.0000 0.383718
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) −54.0000 −1.72409
\(982\) −12.0000 −0.382935
\(983\) −17.0000 −0.542216 −0.271108 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(984\) −21.0000 −0.669456
\(985\) 2.00000 0.0637253
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) −12.0000 −0.381385
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) 96.0000 3.04647
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) −21.0000 −0.665410
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −18.0000 −0.569780
\(999\) 36.0000 1.13899
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 490.2.a.e.1.1 1
3.2 odd 2 4410.2.a.h.1.1 1
4.3 odd 2 3920.2.a.bk.1.1 1
5.2 odd 4 2450.2.c.a.99.2 2
5.3 odd 4 2450.2.c.a.99.1 2
5.4 even 2 2450.2.a.q.1.1 1
7.2 even 3 490.2.e.f.361.1 2
7.3 odd 6 70.2.e.a.51.1 yes 2
7.4 even 3 490.2.e.f.471.1 2
7.5 odd 6 70.2.e.a.11.1 2
7.6 odd 2 490.2.a.k.1.1 1
21.5 even 6 630.2.k.f.361.1 2
21.17 even 6 630.2.k.f.541.1 2
21.20 even 2 4410.2.a.r.1.1 1
28.3 even 6 560.2.q.i.401.1 2
28.19 even 6 560.2.q.i.81.1 2
28.27 even 2 3920.2.a.b.1.1 1
35.3 even 12 350.2.j.f.149.2 4
35.12 even 12 350.2.j.f.249.2 4
35.13 even 4 2450.2.c.s.99.1 2
35.17 even 12 350.2.j.f.149.1 4
35.19 odd 6 350.2.e.l.151.1 2
35.24 odd 6 350.2.e.l.51.1 2
35.27 even 4 2450.2.c.s.99.2 2
35.33 even 12 350.2.j.f.249.1 4
35.34 odd 2 2450.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.a.11.1 2 7.5 odd 6
70.2.e.a.51.1 yes 2 7.3 odd 6
350.2.e.l.51.1 2 35.24 odd 6
350.2.e.l.151.1 2 35.19 odd 6
350.2.j.f.149.1 4 35.17 even 12
350.2.j.f.149.2 4 35.3 even 12
350.2.j.f.249.1 4 35.33 even 12
350.2.j.f.249.2 4 35.12 even 12
490.2.a.e.1.1 1 1.1 even 1 trivial
490.2.a.k.1.1 1 7.6 odd 2
490.2.e.f.361.1 2 7.2 even 3
490.2.e.f.471.1 2 7.4 even 3
560.2.q.i.81.1 2 28.19 even 6
560.2.q.i.401.1 2 28.3 even 6
630.2.k.f.361.1 2 21.5 even 6
630.2.k.f.541.1 2 21.17 even 6
2450.2.a.b.1.1 1 35.34 odd 2
2450.2.a.q.1.1 1 5.4 even 2
2450.2.c.a.99.1 2 5.3 odd 4
2450.2.c.a.99.2 2 5.2 odd 4
2450.2.c.s.99.1 2 35.13 even 4
2450.2.c.s.99.2 2 35.27 even 4
3920.2.a.b.1.1 1 28.27 even 2
3920.2.a.bk.1.1 1 4.3 odd 2
4410.2.a.h.1.1 1 3.2 odd 2
4410.2.a.r.1.1 1 21.20 even 2