Properties

Label 2415.2.a.h.1.1
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
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 \(\chi\) \(=\) 2415.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +12.0000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} +4.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −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
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 5.00000 0.883883
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 3.00000 0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) −3.00000 −0.400892
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.00000 −0.353553
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 12.0000 1.27920
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 6.00000 0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.00000 −0.588348
\(105\) −1.00000 −0.0975900
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) −12.0000 −1.03664
\(135\) −1.00000 −0.0860663
\(136\) −18.0000 −1.54349
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 1.00000 0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) −1.00000 −0.0833333
\(145\) −6.00000 −0.498273
\(146\) 14.0000 1.15865
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 12.0000 0.973329
\(153\) 6.00000 0.485071
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) −2.00000 −0.160128
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 4.00000 0.318223
\(159\) −10.0000 −0.793052
\(160\) −5.00000 −0.395285
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 2.00000 0.148250
\(183\) 10.0000 0.739221
\(184\) −3.00000 −0.221163
\(185\) −6.00000 −0.441129
\(186\) 8.00000 0.586588
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 7.00000 0.505181
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −1.00000 −0.0714286
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) −4.00000 −0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −3.00000 −0.212132
\(201\) −12.0000 −0.846415
\(202\) 2.00000 0.140720
\(203\) 6.00000 0.421117
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) −16.0000 −1.11477
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) −1.00000 −0.0690066
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) −3.00000 −0.204124
\(217\) 8.00000 0.543075
\(218\) 6.00000 0.406371
\(219\) 14.0000 0.946032
\(220\) −4.00000 −0.269680
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 5.00000 0.334077
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 4.00000 0.264906
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −4.00000 −0.263181
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 6.00000 0.388922
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −4.00000 −0.251478
\(254\) 12.0000 0.752947
\(255\) −6.00000 −0.375735
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 4.00000 0.249029
\(259\) 6.00000 0.372822
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −8.00000 −0.494242
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 12.0000 0.738549
\(265\) 10.0000 0.614295
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −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
\(273\) 2.00000 0.121046
\(274\) 2.00000 0.120824
\(275\) −4.00000 −0.241209
\(276\) −1.00000 −0.0601929
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) 3.00000 0.179284
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 4.00000 0.236940
\(286\) −8.00000 −0.473050
\(287\) 6.00000 0.354169
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 2.00000 0.117242
\(292\) −14.0000 −0.819288
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −18.0000 −1.04623
\(297\) −4.00000 −0.232104
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 4.00000 0.227921
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −6.00000 −0.339683
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 22.0000 1.24153
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −10.0000 −0.560772
\(319\) −24.0000 −1.34374
\(320\) −7.00000 −0.391312
\(321\) −4.00000 −0.223258
\(322\) 1.00000 0.0557278
\(323\) −24.0000 −1.33540
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) −24.0000 −1.32924
\(327\) 6.00000 0.331801
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) 6.00000 0.325396
\(341\) −32.0000 −1.73290
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) −1.00000 −0.0538382
\(346\) −14.0000 −0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) −20.0000 −1.06600
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −6.00000 −0.317999
\(357\) 6.00000 0.317554
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −14.0000 −0.735824
\(363\) 5.00000 0.262432
\(364\) −2.00000 −0.104828
\(365\) −14.0000 −0.732793
\(366\) 10.0000 0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) −6.00000 −0.311925
\(371\) −10.0000 −0.519174
\(372\) −8.00000 −0.414781
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 1.00000 0.0514344
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −4.00000 −0.205196
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −3.00000 −0.153093
\(385\) 4.00000 0.203859
\(386\) 18.0000 0.916176
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) −2.00000 −0.101274
\(391\) 6.00000 0.303433
\(392\) −3.00000 −0.151523
\(393\) −8.00000 −0.403547
\(394\) 26.0000 1.30986
\(395\) −4.00000 −0.201262
\(396\) 4.00000 0.201008
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −12.0000 −0.598506
\(403\) 16.0000 0.797017
\(404\) −2.00000 −0.0995037
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) −24.0000 −1.18964
\(408\) −18.0000 −0.891133
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) 2.00000 0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 1.00000 0.0487950
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 30.0000 1.45693
\(425\) 6.00000 0.291043
\(426\) −8.00000 −0.387601
\(427\) 10.0000 0.483934
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) −4.00000 −0.192897
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 8.00000 0.384012
\(435\) −6.00000 −0.287678
\(436\) −6.00000 −0.287348
\(437\) −4.00000 −0.191346
\(438\) 14.0000 0.668946
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −12.0000 −0.572078
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −6.00000 −0.284747
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) 10.0000 0.472984
\(448\) 7.00000 0.330719
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) −24.0000 −1.13012
\(452\) −2.00000 −0.0940721
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 12.0000 0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −6.00000 −0.280362
\(459\) 6.00000 0.280056
\(460\) 1.00000 0.0466252
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) −4.00000 −0.186097
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −6.00000 −0.278543
\(465\) −8.00000 −0.370991
\(466\) 6.00000 0.277945
\(467\) −16.0000 −0.740392 −0.370196 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) −10.0000 −0.457869
\(478\) 8.00000 0.365911
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −5.00000 −0.228218
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) 1.00000 0.0455016
\(484\) −5.00000 −0.227273
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −30.0000 −1.35804
\(489\) −24.0000 −1.08532
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 36.0000 1.62136
\(494\) −8.00000 −0.359937
\(495\) 4.00000 0.179787
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) −3.00000 −0.133631
\(505\) −2.00000 −0.0889988
\(506\) −4.00000 −0.177822
\(507\) −9.00000 −0.399704
\(508\) −12.0000 −0.532414
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) −6.00000 −0.265684
\(511\) 14.0000 0.619324
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) −14.0000 −0.614532
\(520\) 6.00000 0.263117
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000 0.262613
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 8.00000 0.349482
\(525\) 1.00000 0.0436436
\(526\) −16.0000 −0.697633
\(527\) 48.0000 2.09091
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) 4.00000 0.172935
\(536\) 36.0000 1.55496
\(537\) 12.0000 0.517838
\(538\) −30.0000 −1.29339
\(539\) −4.00000 −0.172292
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 24.0000 1.03089
\(543\) −14.0000 −0.600798
\(544\) 30.0000 1.28624
\(545\) −6.00000 −0.257012
\(546\) 2.00000 0.0855921
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) −24.0000 −1.02243
\(552\) −3.00000 −0.127688
\(553\) 4.00000 0.170097
\(554\) 22.0000 0.934690
\(555\) −6.00000 −0.254686
\(556\) −12.0000 −0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) −24.0000 −1.01328
\(562\) −2.00000 −0.0843649
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) 24.0000 1.00702
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 4.00000 0.167542
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) 7.00000 0.291667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 19.0000 0.790296
\(579\) 18.0000 0.748054
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 40.0000 1.65663
\(584\) −42.0000 −1.73797
\(585\) −2.00000 −0.0826898
\(586\) −30.0000 −1.23929
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) −6.00000 −0.246598
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −4.00000 −0.164122
\(595\) −6.00000 −0.245976
\(596\) −10.0000 −0.409616
\(597\) 24.0000 0.982255
\(598\) 2.00000 0.0817861
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) −3.00000 −0.122474
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 4.00000 0.163028
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) −5.00000 −0.203279
\(606\) 2.00000 0.0812444
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −20.0000 −0.811107
\(609\) 6.00000 0.243132
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −28.0000 −1.12999
\(615\) −6.00000 −0.241943
\(616\) 12.0000 0.483494
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −16.0000 −0.643614
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) −12.0000 −0.481156
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 16.0000 0.638978
\(628\) −22.0000 −0.877896
\(629\) 36.0000 1.43541
\(630\) −1.00000 −0.0398410
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) −12.0000 −0.477334
\(633\) 4.00000 0.158986
\(634\) −22.0000 −0.873732
\(635\) −12.0000 −0.476205
\(636\) 10.0000 0.396526
\(637\) 2.00000 0.0792429
\(638\) −24.0000 −0.950169
\(639\) −8.00000 −0.316475
\(640\) 3.00000 0.118585
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −4.00000 −0.157867
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −4.00000 −0.157500
\(646\) −24.0000 −0.944267
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) 24.0000 0.939913
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 6.00000 0.234619
\(655\) 8.00000 0.312586
\(656\) −6.00000 −0.234261
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −4.00000 −0.155700
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 6.00000 0.232495
\(667\) 6.00000 0.232321
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 12.0000 0.463600
\(671\) −40.0000 −1.54418
\(672\) 5.00000 0.192879
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 2.00000 0.0768095
\(679\) 2.00000 0.0767530
\(680\) 18.0000 0.690268
\(681\) 0 0
\(682\) −32.0000 −1.22534
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) −4.00000 −0.152499
\(689\) −20.0000 −0.761939
\(690\) −1.00000 −0.0380693
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 14.0000 0.532200
\(693\) −4.00000 −0.151947
\(694\) 12.0000 0.455514
\(695\) −12.0000 −0.455186
\(696\) −18.0000 −0.682288
\(697\) 36.0000 1.36360
\(698\) 30.0000 1.13552
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 8.00000 0.300235
\(711\) 4.00000 0.150012
\(712\) −18.0000 −0.674579
\(713\) 8.00000 0.299602
\(714\) 6.00000 0.224544
\(715\) 8.00000 0.299183
\(716\) −12.0000 −0.448461
\(717\) 8.00000 0.298765
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) 14.0000 0.520306
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 8.00000 0.295285
\(735\) −1.00000 −0.0368856
\(736\) 5.00000 0.184302
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 6.00000 0.220564
\(741\) −8.00000 −0.293887
\(742\) −10.0000 −0.367112
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) −24.0000 −0.879883
\(745\) −10.0000 −0.366372
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) −4.00000 −0.146157
\(750\) −1.00000 −0.0365148
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 12.0000 0.437014
\(755\) 8.00000 0.291150
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 8.00000 0.290573
\(759\) −4.00000 −0.145191
\(760\) −12.0000 −0.435286
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 12.0000 0.434714
\(763\) 6.00000 0.217215
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) −17.0000 −0.613435
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 4.00000 0.144150
\(771\) −18.0000 −0.648254
\(772\) −18.0000 −0.647834
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) −6.00000 −0.215387
\(777\) 6.00000 0.215249
\(778\) −14.0000 −0.501924
\(779\) −24.0000 −0.859889
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) 6.00000 0.214560
\(783\) 6.00000 0.214423
\(784\) −1.00000 −0.0357143
\(785\) −22.0000 −0.785214
\(786\) −8.00000 −0.285351
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −26.0000 −0.926212
\(789\) −16.0000 −0.569615
\(790\) −4.00000 −0.142314
\(791\) 2.00000 0.0711118
\(792\) 12.0000 0.426401
\(793\) 20.0000 0.710221
\(794\) 10.0000 0.354887
\(795\) 10.0000 0.354663
\(796\) −24.0000 −0.850657
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 6.00000 0.212000
\(802\) −10.0000 −0.353112
\(803\) −56.0000 −1.97620
\(804\) 12.0000 0.423207
\(805\) −1.00000 −0.0352454
\(806\) 16.0000 0.563576
\(807\) −30.0000 −1.05605
\(808\) −6.00000 −0.211079
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) −6.00000 −0.210559
\(813\) 24.0000 0.841717
\(814\) −24.0000 −0.841200
\(815\) 24.0000 0.840683
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) −22.0000 −0.769212
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 2.00000 0.0697580
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 48.0000 1.67216
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 3.00000 0.103510
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) −2.00000 −0.0688837
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 6.00000 0.205798
\(851\) 6.00000 0.205677
\(852\) 8.00000 0.274075
\(853\) 50.0000 1.71197 0.855984 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 10.0000 0.342193
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) −8.00000 −0.273115
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 4.00000 0.136399
\(861\) 6.00000 0.204479
\(862\) 16.0000 0.544962
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.00000 0.170103
\(865\) 14.0000 0.476014
\(866\) −30.0000 −1.01944
\(867\) 19.0000 0.645274
\(868\) −8.00000 −0.271538
\(869\) −16.0000 −0.542763
\(870\) −6.00000 −0.203419
\(871\) −24.0000 −0.813209
\(872\) −18.0000 −0.609557
\(873\) 2.00000 0.0676897
\(874\) −4.00000 −0.135302
\(875\) −1.00000 −0.0338062
\(876\) −14.0000 −0.473016
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 24.0000 0.809961
\(879\) −30.0000 −1.01187
\(880\) −4.00000 −0.134840
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 1.00000 0.0336718
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) −18.0000 −0.604040
\(889\) 12.0000 0.402467
\(890\) −6.00000 −0.201120
\(891\) −4.00000 −0.134005
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) −12.0000 −0.401116
\(896\) −3.00000 −0.100223
\(897\) 2.00000 0.0667781
\(898\) 2.00000 0.0667409
\(899\) 48.0000 1.60089
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) −24.0000 −0.799113
\(903\) 4.00000 0.133112
\(904\) −6.00000 −0.199557
\(905\) 14.0000 0.465376
\(906\) −8.00000 −0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) −2.00000 −0.0662994
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −10.0000 −0.330590
\(916\) 6.00000 0.198246
\(917\) −8.00000 −0.264183
\(918\) 6.00000 0.198030
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 3.00000 0.0989071
\(921\) −28.0000 −0.922631
\(922\) 42.0000 1.38320
\(923\) −16.0000 −0.526646
\(924\) 4.00000 0.131590
\(925\) 6.00000 0.197279
\(926\) 12.0000 0.394344
\(927\) −16.0000 −0.525509
\(928\) 30.0000 0.984798
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −8.00000 −0.262330
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) −12.0000 −0.392862
\(934\) −16.0000 −0.523536
\(935\) 24.0000 0.784884
\(936\) −6.00000 −0.196116
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −12.0000 −0.391814
\(939\) 2.00000 0.0652675
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 22.0000 0.716799
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) −16.0000 −0.520205
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −4.00000 −0.129914
\(949\) 28.0000 0.908918
\(950\) −4.00000 −0.129777
\(951\) −22.0000 −0.713399
\(952\) −18.0000 −0.583383
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −24.0000 −0.775810
\(958\) −24.0000 −0.775405
\(959\) 2.00000 0.0645834
\(960\) −7.00000 −0.225924
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) −18.0000 −0.579441
\(966\) 1.00000 0.0321745
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) −15.0000 −0.482118
\(969\) −24.0000 −0.770991
\(970\) −2.00000 −0.0642161
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 12.0000 0.384702
\(974\) 28.0000 0.897178
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −24.0000 −0.767435
\(979\) −24.0000 −0.767043
\(980\) 1.00000 0.0319438
\(981\) 6.00000 0.191565
\(982\) −12.0000 −0.382935
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) −18.0000 −0.573819
\(985\) −26.0000 −0.828429
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 4.00000 0.127193
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 40.0000 1.27000
\(993\) −4.00000 −0.126936
\(994\) −8.00000 −0.253745
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −28.0000 −0.886325
\(999\) 6.00000 0.189832
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 2415.2.a.h.1.1 1
3.2 odd 2 7245.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.h.1.1 1 1.1 even 1 trivial
7245.2.a.h.1.1 1 3.2 odd 2