Properties

Label 390.2.a.b.1.1
Level $390$
Weight $2$
Character 390.1
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.11416567883\)
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 \(\chi\) \(=\) 390.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -4.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -8.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -6.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -8.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -8.00000 q^{87} -4.00000 q^{88} -10.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +2.00000 q^{92} -4.00000 q^{93} -2.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −4.00000 −0.852803
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −4.00000 −0.685994
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 2.00000 0.324443
\(39\) 1.00000 0.160128
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) 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
\(46\) −2.00000 −0.294884
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −4.00000 −0.560112
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 2.00000 0.267261
\(57\) 2.00000 0.264906
\(58\) −8.00000 −1.05045
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 4.00000 0.492366
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 4.00000 0.485071
\(69\) −2.00000 −0.240772
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) −8.00000 −0.911685
\(78\) −1.00000 −0.113228
\(79\) −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
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) −8.00000 −0.857690
\(88\) −4.00000 −0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) 2.00000 0.208514
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 3.00000 0.303046
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 4.00000 0.396059
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −4.00000 −0.381385
\(111\) −6.00000 −0.569495
\(112\) −2.00000 −0.188982
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −2.00000 −0.187317
\(115\) 2.00000 0.186501
\(116\) 8.00000 0.742781
\(117\) −1.00000 −0.0924500
\(118\) 12.0000 1.10469
\(119\) −8.00000 −0.733359
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 1.00000 0.0877058
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) −4.00000 −0.348155
\(133\) 4.00000 0.346844
\(134\) 8.00000 0.691095
\(135\) −1.00000 −0.0860663
\(136\) −4.00000 −0.342997
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 2.00000 0.170251
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 6.00000 0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 1.00000 0.0816497
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 2.00000 0.162221
\(153\) 4.00000 0.323381
\(154\) 8.00000 0.644658
\(155\) 4.00000 0.321288
\(156\) 1.00000 0.0800641
\(157\) −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
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) −4.00000 −0.311400
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −2.00000 −0.154303
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 8.00000 0.606478
\(175\) −2.00000 −0.151186
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 1.00000 0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −2.00000 −0.148250
\(183\) 2.00000 0.147844
\(184\) −2.00000 −0.147442
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 2.00000 0.145095
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 8.00000 0.574367
\(195\) 1.00000 0.0716115
\(196\) −3.00000 −0.214286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −4.00000 −0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −20.0000 −1.40720
\(203\) −16.0000 −1.12298
\(204\) −4.00000 −0.280056
\(205\) 10.0000 0.698430
\(206\) −4.00000 −0.278693
\(207\) 2.00000 0.139010
\(208\) −1.00000 −0.0693375
\(209\) −8.00000 −0.553372
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) 6.00000 0.402694
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 16.0000 1.06430
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 2.00000 0.132453
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) −2.00000 −0.131876
\(231\) 8.00000 0.526361
\(232\) −8.00000 −0.525226
\(233\) 28.0000 1.83434 0.917170 0.398495i \(-0.130467\pi\)
0.917170 + 0.398495i \(0.130467\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 8.00000 0.518563
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −3.00000 −0.191663
\(246\) 10.0000 0.637577
\(247\) 2.00000 0.127257
\(248\) −4.00000 −0.254000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) −2.00000 −0.125988
\(253\) 8.00000 0.502956
\(254\) −20.0000 −1.25491
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 4.00000 0.249029
\(259\) −12.0000 −0.745644
\(260\) −1.00000 −0.0620174
\(261\) 8.00000 0.495188
\(262\) −14.0000 −0.864923
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 10.0000 0.611990
\(268\) −8.00000 −0.488678
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 1.00000 0.0608581
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 4.00000 0.242536
\(273\) −2.00000 −0.121046
\(274\) 2.00000 0.120824
\(275\) 4.00000 0.241209
\(276\) −2.00000 −0.120386
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 16.0000 0.959616
\(279\) 4.00000 0.239474
\(280\) 2.00000 0.119523
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 4.00000 0.236525
\(287\) −20.0000 −1.18056
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −8.00000 −0.469776
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −3.00000 −0.174964
\(295\) −12.0000 −0.698667
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) 22.0000 1.27443
\(299\) −2.00000 −0.115663
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) 4.00000 0.230174
\(303\) −20.0000 −1.14897
\(304\) −2.00000 −0.114708
\(305\) −2.00000 −0.114520
\(306\) −4.00000 −0.228665
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −8.00000 −0.455842
\(309\) −4.00000 −0.227552
\(310\) −4.00000 −0.227185
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 10.0000 0.564333
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 6.00000 0.336463
\(319\) 32.0000 1.79166
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) 4.00000 0.222911
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −12.0000 −0.664619
\(327\) −4.00000 −0.221201
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) 2.00000 0.109109
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 16.0000 0.869001
\(340\) 4.00000 0.216930
\(341\) 16.0000 0.866449
\(342\) 2.00000 0.108148
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) −2.00000 −0.107676
\(346\) −14.0000 −0.752645
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −8.00000 −0.428845
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 2.00000 0.106904
\(351\) 1.00000 0.0533761
\(352\) −4.00000 −0.213201
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 8.00000 0.423405
\(358\) −2.00000 −0.105703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 22.0000 1.15629
\(363\) −5.00000 −0.262432
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 2.00000 0.104257
\(369\) 10.0000 0.520579
\(370\) −6.00000 −0.311925
\(371\) −12.0000 −0.623009
\(372\) −4.00000 −0.207390
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −16.0000 −0.827340
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) −2.00000 −0.102869
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) −2.00000 −0.102598
\(381\) −20.0000 −1.02463
\(382\) 8.00000 0.409316
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) −12.0000 −0.610784
\(387\) 4.00000 0.203331
\(388\) −8.00000 −0.406138
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 8.00000 0.404577
\(392\) 3.00000 0.151523
\(393\) −14.0000 −0.706207
\(394\) −10.0000 −0.503793
\(395\) −8.00000 −0.402524
\(396\) 4.00000 0.201008
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −24.0000 −1.20301
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −8.00000 −0.399004
\(403\) −4.00000 −0.199254
\(404\) 20.0000 0.995037
\(405\) 1.00000 0.0496904
\(406\) 16.0000 0.794067
\(407\) 24.0000 1.18964
\(408\) 4.00000 0.198030
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −10.0000 −0.493865
\(411\) 2.00000 0.0986527
\(412\) 4.00000 0.197066
\(413\) 24.0000 1.18096
\(414\) −2.00000 −0.0982946
\(415\) −12.0000 −0.589057
\(416\) 1.00000 0.0490290
\(417\) 16.0000 0.783523
\(418\) 8.00000 0.391293
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 2.00000 0.0975900
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 4.00000 0.193347
\(429\) 4.00000 0.193122
\(430\) −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
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 8.00000 0.384012
\(435\) −8.00000 −0.383571
\(436\) 4.00000 0.191565
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −4.00000 −0.190693
\(441\) −3.00000 −0.142857
\(442\) 4.00000 0.190261
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) −6.00000 −0.284747
\(445\) −10.0000 −0.474045
\(446\) 22.0000 1.04173
\(447\) 22.0000 1.04056
\(448\) −2.00000 −0.0944911
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 40.0000 1.88353
\(452\) −16.0000 −0.752577
\(453\) 4.00000 0.187936
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) −2.00000 −0.0936586
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) −28.0000 −1.30835
\(459\) −4.00000 −0.186704
\(460\) 2.00000 0.0932505
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) −8.00000 −0.372194
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 8.00000 0.371391
\(465\) −4.00000 −0.185496
\(466\) −28.0000 −1.29707
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 12.0000 0.552345
\(473\) 16.0000 0.735681
\(474\) −8.00000 −0.367452
\(475\) −2.00000 −0.0917663
\(476\) −8.00000 −0.366679
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) −6.00000 −0.273576
\(482\) 22.0000 1.00207
\(483\) 4.00000 0.182006
\(484\) 5.00000 0.227273
\(485\) −8.00000 −0.363261
\(486\) 1.00000 0.0453609
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 2.00000 0.0905357
\(489\) −12.0000 −0.542659
\(490\) 3.00000 0.135526
\(491\) 38.0000 1.71492 0.857458 0.514554i \(-0.172042\pi\)
0.857458 + 0.514554i \(0.172042\pi\)
\(492\) −10.0000 −0.450835
\(493\) 32.0000 1.44121
\(494\) −2.00000 −0.0899843
\(495\) 4.00000 0.179787
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.0000 0.536120
\(502\) −10.0000 −0.446322
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 2.00000 0.0890871
\(505\) 20.0000 0.889988
\(506\) −8.00000 −0.355643
\(507\) −1.00000 −0.0444116
\(508\) 20.0000 0.887357
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 8.00000 0.352865
\(515\) 4.00000 0.176261
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) −14.0000 −0.614532
\(520\) 1.00000 0.0438529
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −8.00000 −0.350150
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 14.0000 0.611593
\(525\) 2.00000 0.0872872
\(526\) 18.0000 0.784837
\(527\) 16.0000 0.696971
\(528\) −4.00000 −0.174078
\(529\) −19.0000 −0.826087
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 4.00000 0.173422
\(533\) −10.0000 −0.433148
\(534\) −10.0000 −0.432742
\(535\) 4.00000 0.172935
\(536\) 8.00000 0.345547
\(537\) −2.00000 −0.0863064
\(538\) 4.00000 0.172452
\(539\) −12.0000 −0.516877
\(540\) −1.00000 −0.0430331
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 28.0000 1.20270
\(543\) 22.0000 0.944110
\(544\) −4.00000 −0.171499
\(545\) 4.00000 0.171341
\(546\) 2.00000 0.0855921
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) −16.0000 −0.681623
\(552\) 2.00000 0.0851257
\(553\) 16.0000 0.680389
\(554\) −22.0000 −0.934690
\(555\) −6.00000 −0.254686
\(556\) −16.0000 −0.678551
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) −4.00000 −0.169334
\(559\) −4.00000 −0.169182
\(560\) −2.00000 −0.0845154
\(561\) −16.0000 −0.675521
\(562\) 10.0000 0.421825
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) −28.0000 −1.17693
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −4.00000 −0.167248
\(573\) 8.00000 0.334205
\(574\) 20.0000 0.834784
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 1.00000 0.0415945
\(579\) −12.0000 −0.498703
\(580\) 8.00000 0.332182
\(581\) 24.0000 0.995688
\(582\) −8.00000 −0.331611
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 3.00000 0.123718
\(589\) −8.00000 −0.329634
\(590\) 12.0000 0.494032
\(591\) −10.0000 −0.411345
\(592\) 6.00000 0.246598
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 4.00000 0.164122
\(595\) −8.00000 −0.327968
\(596\) −22.0000 −0.901155
\(597\) −24.0000 −0.982255
\(598\) 2.00000 0.0817861
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000 0.0408248
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 8.00000 0.326056
\(603\) −8.00000 −0.325785
\(604\) −4.00000 −0.162758
\(605\) 5.00000 0.203279
\(606\) 20.0000 0.812444
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 2.00000 0.0811107
\(609\) 16.0000 0.648353
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) 8.00000 0.322329
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 4.00000 0.160904
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 4.00000 0.160644
\(621\) −2.00000 −0.0802572
\(622\) 16.0000 0.641542
\(623\) 20.0000 0.801283
\(624\) 1.00000 0.0400320
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 8.00000 0.319489
\(628\) −10.0000 −0.399043
\(629\) 24.0000 0.956943
\(630\) 2.00000 0.0796819
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) 2.00000 0.0794301
\(635\) 20.0000 0.793676
\(636\) −6.00000 −0.237915
\(637\) 3.00000 0.118864
\(638\) −32.0000 −1.26689
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −4.00000 −0.157622
\(645\) −4.00000 −0.157500
\(646\) 8.00000 0.314756
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −48.0000 −1.88416
\(650\) 1.00000 0.0392232
\(651\) 8.00000 0.313545
\(652\) 12.0000 0.469956
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 4.00000 0.156412
\(655\) 14.0000 0.547025
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) −4.00000 −0.155700
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 10.0000 0.388661
\(663\) 4.00000 0.155347
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) −6.00000 −0.232495
\(667\) 16.0000 0.619522
\(668\) −12.0000 −0.464294
\(669\) 22.0000 0.850569
\(670\) 8.00000 0.309067
\(671\) −8.00000 −0.308837
\(672\) −2.00000 −0.0771517
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 26.0000 1.00148
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −16.0000 −0.614476
\(679\) 16.0000 0.614024
\(680\) −4.00000 −0.153393
\(681\) −12.0000 −0.459841
\(682\) −16.0000 −0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −2.00000 −0.0764161
\(686\) −20.0000 −0.763604
\(687\) −28.0000 −1.06827
\(688\) 4.00000 0.152499
\(689\) −6.00000 −0.228582
\(690\) 2.00000 0.0761387
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 14.0000 0.532200
\(693\) −8.00000 −0.303895
\(694\) −16.0000 −0.607352
\(695\) −16.0000 −0.606915
\(696\) 8.00000 0.303239
\(697\) 40.0000 1.51511
\(698\) 28.0000 1.05982
\(699\) −28.0000 −1.05906
\(700\) −2.00000 −0.0755929
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −12.0000 −0.452589
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −40.0000 −1.50435
\(708\) 12.0000 0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) 8.00000 0.299602
\(714\) −8.00000 −0.299392
\(715\) −4.00000 −0.149592
\(716\) 2.00000 0.0747435
\(717\) 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
\(721\) −8.00000 −0.297936
\(722\) 15.0000 0.558242
\(723\) 22.0000 0.818189
\(724\) −22.0000 −0.817624
\(725\) 8.00000 0.297113
\(726\) 5.00000 0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 2.00000 0.0739221
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 4.00000 0.147643
\(735\) 3.00000 0.110657
\(736\) −2.00000 −0.0737210
\(737\) −32.0000 −1.17874
\(738\) −10.0000 −0.368105
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) 6.00000 0.220564
\(741\) −2.00000 −0.0734718
\(742\) 12.0000 0.440534
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 4.00000 0.146647
\(745\) −22.0000 −0.806018
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) 16.0000 0.585018
\(749\) −8.00000 −0.292314
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 8.00000 0.291343
\(755\) −4.00000 −0.145575
\(756\) 2.00000 0.0727393
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 14.0000 0.508503
\(759\) −8.00000 −0.290382
\(760\) 2.00000 0.0725476
\(761\) −46.0000 −1.66750 −0.833749 0.552143i \(-0.813810\pi\)
−0.833749 + 0.552143i \(0.813810\pi\)
\(762\) 20.0000 0.724524
\(763\) −8.00000 −0.289619
\(764\) −8.00000 −0.289430
\(765\) 4.00000 0.144620
\(766\) 20.0000 0.722629
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 8.00000 0.288300
\(771\) 8.00000 0.288113
\(772\) 12.0000 0.431889
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) 8.00000 0.287183
\(777\) 12.0000 0.430498
\(778\) −20.0000 −0.717035
\(779\) −20.0000 −0.716574
\(780\) 1.00000 0.0358057
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) −8.00000 −0.285897
\(784\) −3.00000 −0.107143
\(785\) −10.0000 −0.356915
\(786\) 14.0000 0.499363
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 10.0000 0.356235
\(789\) 18.0000 0.640817
\(790\) 8.00000 0.284627
\(791\) 32.0000 1.13779
\(792\) −4.00000 −0.142134
\(793\) 2.00000 0.0710221
\(794\) −6.00000 −0.212932
\(795\) −6.00000 −0.212798
\(796\) 24.0000 0.850657
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −10.0000 −0.353333
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) −4.00000 −0.140981
\(806\) 4.00000 0.140894
\(807\) 4.00000 0.140807
\(808\) −20.0000 −0.703598
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) −16.0000 −0.561490
\(813\) 28.0000 0.982003
\(814\) −24.0000 −0.841200
\(815\) 12.0000 0.420342
\(816\) −4.00000 −0.140028
\(817\) −8.00000 −0.279885
\(818\) 18.0000 0.629355
\(819\) 2.00000 0.0698857
\(820\) 10.0000 0.349215
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −4.00000 −0.139347
\(825\) −4.00000 −0.139262
\(826\) −24.0000 −0.835067
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 2.00000 0.0695048
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 12.0000 0.416526
\(831\) −22.0000 −0.763172
\(832\) −1.00000 −0.0346688
\(833\) −12.0000 −0.415775
\(834\) −16.0000 −0.554035
\(835\) −12.0000 −0.415277
\(836\) −8.00000 −0.276686
\(837\) −4.00000 −0.138260
\(838\) −14.0000 −0.483622
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 35.0000 1.20690
\(842\) 16.0000 0.551396
\(843\) 10.0000 0.344418
\(844\) −4.00000 −0.137686
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 6.00000 0.206041
\(849\) −28.0000 −0.960958
\(850\) −4.00000 −0.137199
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −4.00000 −0.136877
\(855\) −2.00000 −0.0683986
\(856\) −4.00000 −0.136717
\(857\) −16.0000 −0.546550 −0.273275 0.961936i \(-0.588107\pi\)
−0.273275 + 0.961936i \(0.588107\pi\)
\(858\) −4.00000 −0.136558
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 4.00000 0.136399
\(861\) 20.0000 0.681598
\(862\) −16.0000 −0.544962
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 26.0000 0.883516
\(867\) 1.00000 0.0339618
\(868\) −8.00000 −0.271538
\(869\) −32.0000 −1.08553
\(870\) 8.00000 0.271225
\(871\) 8.00000 0.271070
\(872\) −4.00000 −0.135457
\(873\) −8.00000 −0.270759
\(874\) 4.00000 0.135302
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −8.00000 −0.269987
\(879\) 6.00000 0.202375
\(880\) 4.00000 0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000 0.101015
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −4.00000 −0.134535
\(885\) 12.0000 0.403376
\(886\) 16.0000 0.537531
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 6.00000 0.201347
\(889\) −40.0000 −1.34156
\(890\) 10.0000 0.335201
\(891\) 4.00000 0.134005
\(892\) −22.0000 −0.736614
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 2.00000 0.0668526
\(896\) 2.00000 0.0668153
\(897\) 2.00000 0.0667781
\(898\) 18.0000 0.600668
\(899\) 32.0000 1.06726
\(900\) 1.00000 0.0333333
\(901\) 24.0000 0.799556
\(902\) −40.0000 −1.33185
\(903\) 8.00000 0.266223
\(904\) 16.0000 0.532152
\(905\) −22.0000 −0.731305
\(906\) −4.00000 −0.132891
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 12.0000 0.398234
\(909\) 20.0000 0.663358
\(910\) −2.00000 −0.0662994
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 2.00000 0.0662266
\(913\) −48.0000 −1.58857
\(914\) 16.0000 0.529233
\(915\) 2.00000 0.0661180
\(916\) 28.0000 0.925146
\(917\) −28.0000 −0.924641
\(918\) 4.00000 0.132020
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 6.00000 0.197279
\(926\) −2.00000 −0.0657241
\(927\) 4.00000 0.131377
\(928\) −8.00000 −0.262613
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 4.00000 0.131165
\(931\) 6.00000 0.196642
\(932\) 28.0000 0.917170
\(933\) 16.0000 0.523816
\(934\) −12.0000 −0.392652
\(935\) 16.0000 0.523256
\(936\) 1.00000 0.0326860
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) −16.0000 −0.522419
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) −10.0000 −0.325818
\(943\) 20.0000 0.651290
\(944\) −12.0000 −0.390567
\(945\) 2.00000 0.0650600
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) 2.00000 0.0648544
\(952\) 8.00000 0.259281
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −6.00000 −0.194257
\(955\) −8.00000 −0.258874
\(956\) −8.00000 −0.258738
\(957\) −32.0000 −1.03441
\(958\) −24.0000 −0.775405
\(959\) 4.00000 0.129167
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 6.00000 0.193448
\(963\) 4.00000 0.128898
\(964\) −22.0000 −0.708572
\(965\) 12.0000 0.386294
\(966\) −4.00000 −0.128698
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) −5.00000 −0.160706
\(969\) 8.00000 0.256997
\(970\) 8.00000 0.256865
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 32.0000 1.02587
\(974\) −38.0000 −1.21760
\(975\) 1.00000 0.0320256
\(976\) −2.00000 −0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 12.0000 0.383718
\(979\) −40.0000 −1.27841
\(980\) −3.00000 −0.0958315
\(981\) 4.00000 0.127710
\(982\) −38.0000 −1.21263
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 10.0000 0.318788
\(985\) 10.0000 0.318626
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 8.00000 0.254385
\(990\) −4.00000 −0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −4.00000 −0.127000
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 12.0000 0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 14.0000 0.443162
\(999\) −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 390.2.a.b.1.1 1
3.2 odd 2 1170.2.a.j.1.1 1
4.3 odd 2 3120.2.a.y.1.1 1
5.2 odd 4 1950.2.e.m.1249.1 2
5.3 odd 4 1950.2.e.m.1249.2 2
5.4 even 2 1950.2.a.ba.1.1 1
12.11 even 2 9360.2.a.v.1.1 1
13.5 odd 4 5070.2.b.f.1351.2 2
13.8 odd 4 5070.2.b.f.1351.1 2
13.12 even 2 5070.2.a.n.1.1 1
15.2 even 4 5850.2.e.h.5149.2 2
15.8 even 4 5850.2.e.h.5149.1 2
15.14 odd 2 5850.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 1.1 even 1 trivial
1170.2.a.j.1.1 1 3.2 odd 2
1950.2.a.ba.1.1 1 5.4 even 2
1950.2.e.m.1249.1 2 5.2 odd 4
1950.2.e.m.1249.2 2 5.3 odd 4
3120.2.a.y.1.1 1 4.3 odd 2
5070.2.a.n.1.1 1 13.12 even 2
5070.2.b.f.1351.1 2 13.8 odd 4
5070.2.b.f.1351.2 2 13.5 odd 4
5850.2.a.s.1.1 1 15.14 odd 2
5850.2.e.h.5149.1 2 15.8 even 4
5850.2.e.h.5149.2 2 15.2 even 4
9360.2.a.v.1.1 1 12.11 even 2