Properties

Label 462.2.a.a.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
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 \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +1.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{84} +12.0000 q^{85} -2.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +2.00000 q^{91} -4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) −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
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 4.00000 0.481543
\(70\) 2.00000 0.239046
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 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
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) 2.00000 0.209657
\(92\) −4.00000 −0.417029
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −6.00000 −0.594089
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 2.00000 0.195180
\(106\) 14.0000 1.35980
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) −6.00000 −0.550019
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 6.00000 0.514496
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) −4.00000 −0.340503
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −2.00000 −0.169031
\(141\) 8.00000 0.673722
\(142\) −12.0000 −1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −6.00000 −0.496564
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 14.0000 1.11027
\(160\) 2.00000 0.158114
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −6.00000 −0.468521
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 2.00000 0.151620
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) 4.00000 0.294884
\(185\) 4.00000 0.294086
\(186\) −4.00000 −0.293294
\(187\) −6.00000 −0.438763
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 14.0000 1.00514
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) 6.00000 0.420084
\(205\) 12.0000 0.838116
\(206\) 4.00000 0.278693
\(207\) −4.00000 −0.278019
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −2.00000 −0.138013
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −14.0000 −0.961524
\(213\) −12.0000 −0.822226
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) −6.00000 −0.405442
\(220\) −2.00000 −0.134840
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −8.00000 −0.527504
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) −2.00000 −0.127775
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) −16.0000 −1.00393
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −4.00000 −0.248069
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 1.00000 0.0615457
\(265\) 28.0000 1.72003
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) −2.00000 −0.121716
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) 22.0000 1.32907
\(275\) −1.00000 −0.0603023
\(276\) 4.00000 0.240772
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −12.0000 −0.719712
\(279\) −4.00000 −0.239474
\(280\) 2.00000 0.119523
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) −8.00000 −0.473879
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 14.0000 0.820695
\(292\) 6.00000 0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) −10.0000 −0.579284
\(299\) −8.00000 −0.462652
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) 28.0000 1.60328
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 1.00000 0.0569803
\(309\) 4.00000 0.227552
\(310\) −8.00000 −0.454369
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −2.00000 −0.112867
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −14.0000 −0.785081
\(319\) 2.00000 0.111979
\(320\) −2.00000 −0.111803
\(321\) −12.0000 −0.669775
\(322\) 4.00000 0.222911
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) −10.0000 −0.553001
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) −2.00000 −0.110096
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 12.0000 0.650791
\(341\) −4.00000 −0.216612
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) −10.0000 −0.537603
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −2.00000 −0.107211
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 1.00000 0.0534522
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 12.0000 0.637793
\(355\) −24.0000 −1.27379
\(356\) −6.00000 −0.317999
\(357\) 6.00000 0.317554
\(358\) 4.00000 0.211407
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) −1.00000 −0.0524864
\(364\) 2.00000 0.104828
\(365\) −12.0000 −0.628109
\(366\) −14.0000 −0.731792
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) −4.00000 −0.207950
\(371\) −14.0000 −0.726844
\(372\) 4.00000 0.207390
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 6.00000 0.310253
\(375\) −12.0000 −0.619677
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 1.00000 0.0514344
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 8.00000 0.410391
\(381\) −16.0000 −0.819705
\(382\) −4.00000 −0.204658
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.00000 −0.101929
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −4.00000 −0.202548
\(391\) 24.0000 1.21373
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 12.0000 0.601506
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 4.00000 0.199502
\(403\) −8.00000 −0.398508
\(404\) 10.0000 0.497519
\(405\) −2.00000 −0.0993808
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) −6.00000 −0.297044
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) −12.0000 −0.592638
\(411\) 22.0000 1.08518
\(412\) −4.00000 −0.197066
\(413\) 12.0000 0.590481
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 4.00000 0.195646
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 14.0000 0.679900
\(425\) 6.00000 0.291043
\(426\) 12.0000 0.581402
\(427\) −14.0000 −0.677507
\(428\) 12.0000 0.580042
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 4.00000 0.192006
\(435\) 4.00000 0.191785
\(436\) 10.0000 0.478913
\(437\) 16.0000 0.765384
\(438\) 6.00000 0.286691
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 2.00000 0.0953463
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) 12.0000 0.568855
\(446\) 12.0000 0.568216
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) −4.00000 −0.187317
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 8.00000 0.373002
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 1.00000 0.0465242
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 2.00000 0.0928477
\(465\) −8.00000 −0.370991
\(466\) 2.00000 0.0926482
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 2.00000 0.0924500
\(469\) 4.00000 0.184703
\(470\) −16.0000 −0.738025
\(471\) −2.00000 −0.0921551
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −6.00000 −0.275010
\(477\) −14.0000 −0.641016
\(478\) −24.0000 −1.09773
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −4.00000 −0.182384
\(482\) 26.0000 1.18427
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) 28.0000 1.27141
\(486\) 1.00000 0.0453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 14.0000 0.633750
\(489\) 20.0000 0.904431
\(490\) 2.00000 0.0903508
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 6.00000 0.270501
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) −2.00000 −0.0898933
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −20.0000 −0.889988
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 12.0000 0.531369
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −18.0000 −0.793946
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 2.00000 0.0878750
\(519\) −10.0000 −0.438951
\(520\) 4.00000 0.175412
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) −28.0000 −1.21624
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) −6.00000 −0.259645
\(535\) −24.0000 −1.03761
\(536\) −4.00000 −0.172774
\(537\) 4.00000 0.172613
\(538\) −30.0000 −1.29339
\(539\) 1.00000 0.0430730
\(540\) 2.00000 0.0860663
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 6.00000 0.257248
\(545\) −20.0000 −0.856706
\(546\) 2.00000 0.0855921
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) −22.0000 −0.939793
\(549\) −14.0000 −0.597505
\(550\) 1.00000 0.0426401
\(551\) −8.00000 −0.340811
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) −4.00000 −0.169791
\(556\) 12.0000 0.508913
\(557\) −46.0000 −1.94908 −0.974541 0.224208i \(-0.928020\pi\)
−0.974541 + 0.224208i \(0.928020\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 6.00000 0.253320
\(562\) 18.0000 0.759284
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 8.00000 0.336861
\(565\) 12.0000 0.504844
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 8.00000 0.335083
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 2.00000 0.0836242
\(573\) −4.00000 −0.167102
\(574\) 6.00000 0.250435
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) −19.0000 −0.790296
\(579\) 6.00000 0.249351
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) −14.0000 −0.579821
\(584\) −6.00000 −0.248282
\(585\) −4.00000 −0.165380
\(586\) 14.0000 0.578335
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 16.0000 0.659269
\(590\) 24.0000 0.988064
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 1.00000 0.0410305
\(595\) 12.0000 0.491952
\(596\) 10.0000 0.409616
\(597\) 12.0000 0.491127
\(598\) 8.00000 0.327144
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 10.0000 0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.00000 0.162221
\(609\) −2.00000 −0.0810441
\(610\) −28.0000 −1.13369
\(611\) −16.0000 −0.647291
\(612\) −6.00000 −0.242536
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 12.0000 0.484281
\(615\) −12.0000 −0.483887
\(616\) −1.00000 −0.0402911
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −4.00000 −0.160904
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 8.00000 0.321288
\(621\) 4.00000 0.160514
\(622\) −8.00000 −0.320771
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 14.0000 0.559553
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) 2.00000 0.0796819
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −10.0000 −0.397151
\(635\) −32.0000 −1.26988
\(636\) 14.0000 0.555136
\(637\) 2.00000 0.0792429
\(638\) −2.00000 −0.0791808
\(639\) 12.0000 0.474713
\(640\) 2.00000 0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 12.0000 0.473602
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 12.0000 0.471041
\(650\) 2.00000 0.0784465
\(651\) 4.00000 0.156772
\(652\) −20.0000 −0.783260
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) 8.00000 0.311872
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 2.00000 0.0778499
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 20.0000 0.777322
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 2.00000 0.0774984
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 8.00000 0.309067
\(671\) −14.0000 −0.540464
\(672\) 1.00000 0.0385758
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 14.0000 0.539260
\(675\) 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 \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) −14.0000 −0.537271
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) −4.00000 −0.152944
\(685\) 44.0000 1.68115
\(686\) −1.00000 −0.0381802
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −28.0000 −1.06672
\(690\) 8.00000 0.304555
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 10.0000 0.380143
\(693\) 1.00000 0.0379869
\(694\) 28.0000 1.06287
\(695\) −24.0000 −0.910372
\(696\) 2.00000 0.0758098
\(697\) 36.0000 1.36360
\(698\) −34.0000 −1.28692
\(699\) 2.00000 0.0756469
\(700\) −1.00000 −0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) 1.00000 0.0376889
\(705\) −16.0000 −0.602595
\(706\) 14.0000 0.526897
\(707\) 10.0000 0.376089
\(708\) −12.0000 −0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 16.0000 0.599205
\(714\) −6.00000 −0.224544
\(715\) −4.00000 −0.149592
\(716\) −4.00000 −0.149487
\(717\) −24.0000 −0.896296
\(718\) −32.0000 −1.19423
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −4.00000 −0.148968
\(722\) 3.00000 0.111648
\(723\) 26.0000 0.966950
\(724\) −14.0000 −0.520306
\(725\) −2.00000 −0.0742781
\(726\) 1.00000 0.0371135
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 28.0000 1.03350
\(735\) 2.00000 0.0737711
\(736\) 4.00000 0.147442
\(737\) 4.00000 0.147342
\(738\) 6.00000 0.220863
\(739\) −48.0000 −1.76571 −0.882854 0.469647i \(-0.844381\pi\)
−0.882854 + 0.469647i \(0.844381\pi\)
\(740\) 4.00000 0.147043
\(741\) 8.00000 0.293887
\(742\) 14.0000 0.513956
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) −4.00000 −0.146647
\(745\) −20.0000 −0.732743
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) 12.0000 0.438470
\(750\) 12.0000 0.438178
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) −20.0000 −0.728841
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −36.0000 −1.30758
\(759\) 4.00000 0.145191
\(760\) −8.00000 −0.290191
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 16.0000 0.579619
\(763\) 10.0000 0.362024
\(764\) 4.00000 0.144715
\(765\) 12.0000 0.433861
\(766\) −16.0000 −0.578103
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 2.00000 0.0720750
\(771\) −18.0000 −0.648254
\(772\) −6.00000 −0.215945
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 14.0000 0.502571
\(777\) 2.00000 0.0717496
\(778\) −18.0000 −0.645331
\(779\) 24.0000 0.859889
\(780\) 4.00000 0.143223
\(781\) 12.0000 0.429394
\(782\) −24.0000 −0.858238
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −1.00000 −0.0355335
\(793\) −28.0000 −0.994309
\(794\) −18.0000 −0.638796
\(795\) −28.0000 −0.993058
\(796\) −12.0000 −0.425329
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −4.00000 −0.141598
\(799\) 48.0000 1.69812
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 14.0000 0.494357
\(803\) 6.00000 0.211735
\(804\) −4.00000 −0.141069
\(805\) 8.00000 0.281963
\(806\) 8.00000 0.281788
\(807\) −30.0000 −1.05605
\(808\) −10.0000 −0.351799
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 2.00000 0.0702728
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 40.0000 1.40114
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) −38.0000 −1.32864
\(819\) 2.00000 0.0698857
\(820\) 12.0000 0.419058
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −22.0000 −0.767338
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 4.00000 0.139347
\(825\) 1.00000 0.0348155
\(826\) −12.0000 −0.417533
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −4.00000 −0.139010
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 4.00000 0.138260
\(838\) 12.0000 0.414533
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 18.0000 0.619219
\(846\) 8.00000 0.275046
\(847\) 1.00000 0.0343604
\(848\) −14.0000 −0.480762
\(849\) 4.00000 0.137280
\(850\) −6.00000 −0.205798
\(851\) 8.00000 0.274236
\(852\) −12.0000 −0.411113
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 14.0000 0.479070
\(855\) 8.00000 0.273594
\(856\) −12.0000 −0.410152
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 2.00000 0.0682789
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 8.00000 0.272481
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) −20.0000 −0.680020
\(866\) −34.0000 −1.15537
\(867\) −19.0000 −0.645274
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) −14.0000 −0.473828
\(874\) −16.0000 −0.541208
\(875\) 12.0000 0.405674
\(876\) −6.00000 −0.202721
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −16.0000 −0.539974
\(879\) 14.0000 0.472208
\(880\) −2.00000 −0.0674200
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −12.0000 −0.403604
\(885\) 24.0000 0.806751
\(886\) −12.0000 −0.403148
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 16.0000 0.536623
\(890\) −12.0000 −0.402241
\(891\) 1.00000 0.0335013
\(892\) −12.0000 −0.401790
\(893\) 32.0000 1.07084
\(894\) 10.0000 0.334450
\(895\) 8.00000 0.267411
\(896\) −1.00000 −0.0334077
\(897\) 8.00000 0.267112
\(898\) 22.0000 0.734150
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 84.0000 2.79845
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 4.00000 0.132599
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) −28.0000 −0.925651
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −8.00000 −0.263752
\(921\) 12.0000 0.395413
\(922\) 30.0000 0.987997
\(923\) 24.0000 0.789970
\(924\) −1.00000 −0.0328976
\(925\) 2.00000 0.0657596
\(926\) −40.0000 −1.31448
\(927\) −4.00000 −0.131377
\(928\) −2.00000 −0.0656532
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 8.00000 0.262330
\(931\) −4.00000 −0.131095
\(932\) −2.00000 −0.0655122
\(933\) −8.00000 −0.261908
\(934\) −20.0000 −0.654420
\(935\) 12.0000 0.392442
\(936\) −2.00000 −0.0653720
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −4.00000 −0.130605
\(939\) 14.0000 0.456873
\(940\) 16.0000 0.521862
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 2.00000 0.0651635
\(943\) 24.0000 0.781548
\(944\) 12.0000 0.390567
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) −4.00000 −0.129777
\(951\) −10.0000 −0.324272
\(952\) 6.00000 0.194461
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 14.0000 0.453267
\(955\) −8.00000 −0.258874
\(956\) 24.0000 0.776215
\(957\) −2.00000 −0.0646508
\(958\) 24.0000 0.775405
\(959\) −22.0000 −0.710417
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 12.0000 0.386695
\(964\) −26.0000 −0.837404
\(965\) 12.0000 0.386294
\(966\) −4.00000 −0.128698
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −24.0000 −0.770991
\(970\) −28.0000 −0.899026
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) −14.0000 −0.448129
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −20.0000 −0.639529
\(979\) −6.00000 −0.191761
\(980\) −2.00000 −0.0638877
\(981\) 10.0000 0.319275
\(982\) 28.0000 0.893516
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −6.00000 −0.191273
\(985\) 12.0000 0.382352
\(986\) 12.0000 0.382158
\(987\) 8.00000 0.254643
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 4.00000 0.127000
\(993\) 20.0000 0.634681
\(994\) −12.0000 −0.380617
\(995\) 24.0000 0.760851
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −12.0000 −0.379853
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 462.2.a.a.1.1 1
3.2 odd 2 1386.2.a.k.1.1 1
4.3 odd 2 3696.2.a.s.1.1 1
7.6 odd 2 3234.2.a.n.1.1 1
11.10 odd 2 5082.2.a.q.1.1 1
21.20 even 2 9702.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.a.1.1 1 1.1 even 1 trivial
1386.2.a.k.1.1 1 3.2 odd 2
3234.2.a.n.1.1 1 7.6 odd 2
3696.2.a.s.1.1 1 4.3 odd 2
5082.2.a.q.1.1 1 11.10 odd 2
9702.2.a.bf.1.1 1 21.20 even 2