Properties

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

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Newspace parameters

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

Newform invariants

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

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +2.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -6.00000 q^{55} +2.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +6.00000 q^{66} +1.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -12.0000 q^{77} -2.00000 q^{78} -14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -6.00000 q^{83} +2.00000 q^{84} +8.00000 q^{86} +2.00000 q^{87} -6.00000 q^{88} -16.0000 q^{89} +1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −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
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −2.00000 −0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 6.00000 0.904534
\(45\) −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
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.00000 −0.809040
\(56\) 2.00000 0.267261
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 8.00000 1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 6.00000 0.738549
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −12.0000 −1.36753
\(78\) −2.00000 −0.226455
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 2.00000 0.214423
\(88\) −6.00000 −0.639602
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) 6.00000 0.603023
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 2.00000 0.196116
\(105\) −2.00000 −0.195180
\(106\) 2.00000 0.194257
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 6.00000 0.572078
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) −2.00000 −0.175412
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −6.00000 −0.522233
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 20.0000 1.70872 0.854358 0.519685i \(-0.173951\pi\)
0.854358 + 0.519685i \(0.173951\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −6.00000 −0.496564
\(147\) 3.00000 0.247436
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 14.0000 1.11378
\(159\) 2.00000 0.158610
\(160\) 1.00000 0.0790569
\(161\) 2.00000 0.157622
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 2.00000 0.156174
\(165\) 6.00000 0.467099
\(166\) 6.00000 0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −2.00000 −0.151620
\(175\) −2.00000 −0.151186
\(176\) 6.00000 0.452267
\(177\) 4.00000 0.300658
\(178\) 16.0000 1.19925
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 4.00000 0.297318 0.148659 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 4.00000 0.294086
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −2.00000 −0.143592
\(195\) −2.00000 −0.143223
\(196\) −3.00000 −0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −6.00000 −0.426401
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 10.0000 0.696733
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) −24.0000 −1.66011
\(210\) 2.00000 0.138013
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) −2.00000 −0.137361
\(213\) 8.00000 0.548151
\(214\) 6.00000 0.410152
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −12.0000 −0.812743
\(219\) −6.00000 −0.405442
\(220\) −6.00000 −0.404520
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −16.0000 −1.06430
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 4.00000 0.264906
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 12.0000 0.789542
\(232\) 2.00000 0.131306
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 2.00000 0.127515
\(247\) 8.00000 0.509028
\(248\) 8.00000 0.508001
\(249\) 6.00000 0.380235
\(250\) 1.00000 0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −2.00000 −0.125988
\(253\) −6.00000 −0.377217
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −8.00000 −0.498058
\(259\) 8.00000 0.497096
\(260\) 2.00000 0.124035
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 6.00000 0.369274
\(265\) 2.00000 0.122859
\(266\) −8.00000 −0.490511
\(267\) 16.0000 0.979184
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) −20.0000 −1.20824
\(275\) 6.00000 0.361814
\(276\) 1.00000 0.0601929
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −2.00000 −0.119523
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) −4.00000 −0.236940
\(286\) 12.0000 0.709575
\(287\) −4.00000 −0.236113
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −2.00000 −0.117444
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) −3.00000 −0.174964
\(295\) 4.00000 0.232889
\(296\) 4.00000 0.232495
\(297\) −6.00000 −0.348155
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 16.0000 0.922225
\(302\) −20.0000 −1.15087
\(303\) −2.00000 −0.114897
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −12.0000 −0.683763
\(309\) 10.0000 0.568880
\(310\) −8.00000 −0.454369
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) −2.00000 −0.113228
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 4.00000 0.225733
\(315\) 2.00000 0.112687
\(316\) −14.0000 −0.787562
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −2.00000 −0.112154
\(319\) −12.0000 −0.671871
\(320\) −1.00000 −0.0559017
\(321\) 6.00000 0.334887
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) −12.0000 −0.663602
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000 0.489535
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 4.00000 0.216295
\(343\) 20.0000 1.07990
\(344\) 8.00000 0.431331
\(345\) −1.00000 −0.0538382
\(346\) −6.00000 −0.322562
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 2.00000 0.107211
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 2.00000 0.106904
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −4.00000 −0.212598
\(355\) 8.00000 0.424596
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −4.00000 −0.210235
\(363\) −25.0000 −1.31216
\(364\) 4.00000 0.209657
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) 4.00000 0.207670
\(372\) 8.00000 0.414781
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) −2.00000 −0.102869
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 4.00000 0.205196
\(381\) 12.0000 0.614779
\(382\) −8.00000 −0.409316
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 1.00000 0.0510310
\(385\) 12.0000 0.611577
\(386\) 18.0000 0.916176
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −8.00000 −0.403547
\(394\) 22.0000 1.10834
\(395\) 14.0000 0.704416
\(396\) 6.00000 0.301511
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 22.0000 1.10276
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 2.00000 0.0995037
\(405\) −1.00000 −0.0496904
\(406\) −4.00000 −0.198517
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 2.00000 0.0987730
\(411\) −20.0000 −0.986527
\(412\) −10.0000 −0.492665
\(413\) 8.00000 0.393654
\(414\) 1.00000 0.0491473
\(415\) 6.00000 0.294528
\(416\) 2.00000 0.0980581
\(417\) 4.00000 0.195881
\(418\) 24.0000 1.17388
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 12.0000 0.579365
\(430\) −8.00000 −0.385794
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −16.0000 −0.768025
\(435\) −2.00000 −0.0958927
\(436\) 12.0000 0.574696
\(437\) 4.00000 0.191346
\(438\) 6.00000 0.286691
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 6.00000 0.286039
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 4.00000 0.189832
\(445\) 16.0000 0.758473
\(446\) 28.0000 1.32584
\(447\) −10.0000 −0.472984
\(448\) −2.00000 −0.0944911
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 16.0000 0.752577
\(453\) −20.0000 −0.939682
\(454\) −10.0000 −0.469323
\(455\) −4.00000 −0.187523
\(456\) −4.00000 −0.187317
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) −12.0000 −0.558291
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −8.00000 −0.370991
\(466\) −2.00000 −0.0926482
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 4.00000 0.184115
\(473\) −48.0000 −2.20704
\(474\) −14.0000 −0.643041
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 8.00000 0.364769
\(482\) −18.0000 −0.819878
\(483\) −2.00000 −0.0910032
\(484\) 25.0000 1.13636
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) −3.00000 −0.135526
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) −6.00000 −0.269680
\(496\) −8.00000 −0.359211
\(497\) 16.0000 0.717698
\(498\) −6.00000 −0.268866
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 2.00000 0.0890871
\(505\) −2.00000 −0.0889988
\(506\) 6.00000 0.266733
\(507\) 9.00000 0.399704
\(508\) −12.0000 −0.532414
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 6.00000 0.264649
\(515\) 10.0000 0.440653
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) −2.00000 −0.0877058
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 2.00000 0.0875376
\(523\) 40.0000 1.74908 0.874539 0.484955i \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) 8.00000 0.349482
\(525\) 2.00000 0.0872872
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 1.00000 0.0434783
\(530\) −2.00000 −0.0868744
\(531\) −4.00000 −0.173585
\(532\) 8.00000 0.346844
\(533\) −4.00000 −0.173259
\(534\) −16.0000 −0.692388
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −18.0000 −0.775315
\(540\) 1.00000 0.0430331
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 4.00000 0.171815
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 4.00000 0.171184
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 20.0000 0.854358
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) 28.0000 1.19068
\(554\) −10.0000 −0.424859
\(555\) −4.00000 −0.169791
\(556\) −4.00000 −0.169638
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) −28.0000 −1.17693
\(567\) −2.00000 −0.0839921
\(568\) 8.00000 0.335673
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 4.00000 0.167542
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −12.0000 −0.501745
\(573\) −8.00000 −0.334205
\(574\) 4.00000 0.166957
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 17.0000 0.707107
\(579\) 18.0000 0.748054
\(580\) 2.00000 0.0830455
\(581\) 12.0000 0.497844
\(582\) 2.00000 0.0829027
\(583\) −12.0000 −0.496989
\(584\) −6.00000 −0.248282
\(585\) 2.00000 0.0826898
\(586\) −22.0000 −0.908812
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 3.00000 0.123718
\(589\) 32.0000 1.31854
\(590\) −4.00000 −0.164677
\(591\) 22.0000 0.904959
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 22.0000 0.900400
\(598\) −2.00000 −0.0817861
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) −25.0000 −1.01639
\(606\) 2.00000 0.0812444
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 4.00000 0.162221
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 12.0000 0.484281
\(615\) 2.00000 0.0806478
\(616\) 12.0000 0.483494
\(617\) −40.0000 −1.61034 −0.805170 0.593045i \(-0.797926\pi\)
−0.805170 + 0.593045i \(0.797926\pi\)
\(618\) −10.0000 −0.402259
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) 32.0000 1.28308
\(623\) 32.0000 1.28205
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 24.0000 0.958468
\(628\) −4.00000 −0.159617
\(629\) 0 0
\(630\) −2.00000 −0.0796819
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 14.0000 0.556890
\(633\) −28.0000 −1.11290
\(634\) −30.0000 −1.19145
\(635\) 12.0000 0.476205
\(636\) 2.00000 0.0793052
\(637\) 6.00000 0.237729
\(638\) 12.0000 0.475085
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) −6.00000 −0.236801
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 2.00000 0.0788110
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) 2.00000 0.0784465
\(651\) −16.0000 −0.627089
\(652\) 12.0000 0.469956
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 12.0000 0.469237
\(655\) −8.00000 −0.312586
\(656\) 2.00000 0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 6.00000 0.233550
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −8.00000 −0.310227
\(666\) 4.00000 0.154997
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −22.0000 −0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 16.0000 0.614476
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 48.0000 1.83801
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −4.00000 −0.152944
\(685\) −20.0000 −0.764161
\(686\) −20.0000 −0.763604
\(687\) −4.00000 −0.152610
\(688\) −8.00000 −0.304997
\(689\) 4.00000 0.152388
\(690\) 1.00000 0.0380693
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000 0.228086
\(693\) −12.0000 −0.455842
\(694\) 32.0000 1.21470
\(695\) 4.00000 0.151729
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) −10.0000 −0.378506
\(699\) −2.00000 −0.0756469
\(700\) −2.00000 −0.0755929
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 16.0000 0.603451
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −4.00000 −0.150435
\(708\) 4.00000 0.150329
\(709\) 24.0000 0.901339 0.450669 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) −8.00000 −0.300235
\(711\) −14.0000 −0.525041
\(712\) 16.0000 0.599625
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 20.0000 0.744839
\(722\) 3.00000 0.111648
\(723\) −18.0000 −0.669427
\(724\) 4.00000 0.148659
\(725\) −2.00000 −0.0742781
\(726\) 25.0000 0.927837
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) 0 0
\(732\) 0 0
\(733\) −48.0000 −1.77292 −0.886460 0.462805i \(-0.846843\pi\)
−0.886460 + 0.462805i \(0.846843\pi\)
\(734\) −2.00000 −0.0738213
\(735\) −3.00000 −0.110657
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −2.00000 −0.0736210
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 4.00000 0.147043
\(741\) −8.00000 −0.293887
\(742\) −4.00000 −0.146845
\(743\) 52.0000 1.90769 0.953847 0.300291i \(-0.0970839\pi\)
0.953847 + 0.300291i \(0.0970839\pi\)
\(744\) −8.00000 −0.293294
\(745\) −10.0000 −0.366372
\(746\) 4.00000 0.146450
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) −4.00000 −0.145671
\(755\) −20.0000 −0.727875
\(756\) 2.00000 0.0727393
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 36.0000 1.30758
\(759\) 6.00000 0.217786
\(760\) −4.00000 −0.145095
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −12.0000 −0.434714
\(763\) −24.0000 −0.868858
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) −12.0000 −0.432450
\(771\) 6.00000 0.216085
\(772\) −18.0000 −0.647834
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 8.00000 0.287554
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) −8.00000 −0.286998
\(778\) −10.0000 −0.358517
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 4.00000 0.142766
\(786\) 8.00000 0.285351
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −22.0000 −0.783718
\(789\) −4.00000 −0.142404
\(790\) −14.0000 −0.498098
\(791\) −32.0000 −1.13779
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −2.00000 −0.0709327
\(796\) −22.0000 −0.779769
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −16.0000 −0.565332
\(802\) −8.00000 −0.282490
\(803\) 36.0000 1.27041
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) −16.0000 −0.563576
\(807\) 30.0000 1.05605
\(808\) −2.00000 −0.0703598
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 1.00000 0.0351364
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 4.00000 0.140372
\(813\) 4.00000 0.140286
\(814\) 24.0000 0.841200
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 10.0000 0.349642
\(819\) 4.00000 0.139771
\(820\) −2.00000 −0.0698430
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 20.0000 0.697580
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 10.0000 0.348367
\(825\) −6.00000 −0.208893
\(826\) −8.00000 −0.278356
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −6.00000 −0.208263
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 8.00000 0.276520
\(838\) −26.0000 −0.898155
\(839\) −44.0000 −1.51905 −0.759524 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 2.00000 0.0690066
\(841\) −25.0000 −0.862069
\(842\) −24.0000 −0.827095
\(843\) 16.0000 0.551069
\(844\) 28.0000 0.963800
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −50.0000 −1.71802
\(848\) −2.00000 −0.0686803
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 8.00000 0.274075
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 6.00000 0.205076
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −12.0000 −0.409673
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 8.00000 0.272798
\(861\) 4.00000 0.136320
\(862\) 36.0000 1.22616
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) 26.0000 0.883516
\(867\) 17.0000 0.577350
\(868\) 16.0000 0.543075
\(869\) −84.0000 −2.84950
\(870\) 2.00000 0.0678064
\(871\) 0 0
\(872\) −12.0000 −0.406371
\(873\) 2.00000 0.0676897
\(874\) −4.00000 −0.135302
\(875\) 2.00000 0.0676123
\(876\) −6.00000 −0.202721
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) −22.0000 −0.742042
\(880\) −6.00000 −0.202260
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 3.00000 0.101015
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) −4.00000 −0.134383
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) −4.00000 −0.134231
\(889\) 24.0000 0.804934
\(890\) −16.0000 −0.536321
\(891\) 6.00000 0.201008
\(892\) −28.0000 −0.937509
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) −2.00000 −0.0667781
\(898\) −10.0000 −0.333704
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −12.0000 −0.399556
\(903\) −16.0000 −0.532447
\(904\) −16.0000 −0.532152
\(905\) −4.00000 −0.132964
\(906\) 20.0000 0.664455
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 10.0000 0.331862
\(909\) 2.00000 0.0663358
\(910\) 4.00000 0.132599
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 4.00000 0.132453
\(913\) −36.0000 −1.19143
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 12.0000 0.395413
\(922\) 14.0000 0.461065
\(923\) 16.0000 0.526646
\(924\) 12.0000 0.394771
\(925\) −4.00000 −0.131519
\(926\) −32.0000 −1.05159
\(927\) −10.0000 −0.328443
\(928\) 2.00000 0.0656532
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 8.00000 0.262330
\(931\) 12.0000 0.393284
\(932\) 2.00000 0.0655122
\(933\) 32.0000 1.04763
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −4.00000 −0.130327
\(943\) −2.00000 −0.0651290
\(944\) −4.00000 −0.130189
\(945\) −2.00000 −0.0650600
\(946\) 48.0000 1.56061
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 14.0000 0.454699
\(949\) −12.0000 −0.389536
\(950\) 4.00000 0.129777
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 2.00000 0.0647524
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 40.0000 1.29234
\(959\) −40.0000 −1.29167
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −8.00000 −0.257930
\(963\) −6.00000 −0.193347
\(964\) 18.0000 0.579741
\(965\) 18.0000 0.579441
\(966\) 2.00000 0.0643489
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 32.0000 1.02535
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 12.0000 0.383718
\(979\) −96.0000 −3.06817
\(980\) 3.00000 0.0958315
\(981\) 12.0000 0.383131
\(982\) 32.0000 1.02116
\(983\) 28.0000 0.893061 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) 2.00000 0.0637577
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 8.00000 0.254385
\(990\) 6.00000 0.190693
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) 20.0000 0.634681
\(994\) −16.0000 −0.507489
\(995\) 22.0000 0.697447
\(996\) 6.00000 0.190117
\(997\) 34.0000 1.07679 0.538395 0.842692i \(-0.319031\pi\)
0.538395 + 0.842692i \(0.319031\pi\)
\(998\) −36.0000 −1.13956
\(999\) 4.00000 0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 690.2.a.a.1.1 1
3.2 odd 2 2070.2.a.q.1.1 1
4.3 odd 2 5520.2.a.y.1.1 1
5.2 odd 4 3450.2.d.u.2899.1 2
5.3 odd 4 3450.2.d.u.2899.2 2
5.4 even 2 3450.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.a.1.1 1 1.1 even 1 trivial
2070.2.a.q.1.1 1 3.2 odd 2
3450.2.a.ba.1.1 1 5.4 even 2
3450.2.d.u.2899.1 2 5.2 odd 4
3450.2.d.u.2899.2 2 5.3 odd 4
5520.2.a.y.1.1 1 4.3 odd 2