Properties

Label 1110.2.a.i.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -6.00000 q^{57} +2.00000 q^{58} +1.00000 q^{60} -4.00000 q^{61} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} -1.00000 q^{74} -1.00000 q^{75} +6.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -6.00000 q^{85} -2.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +8.00000 q^{92} +6.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 6.00000 0.973329
\(39\) 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 \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 8.00000 1.17954
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.00000 −0.115470
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −6.00000 −0.615587
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −7.00000 −0.707107
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 2.00000 0.190693
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −6.00000 −0.561951
\(115\) −8.00000 −0.746004
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −4.00000 −0.362143
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −8.00000 −0.681005
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 6.00000 0.496564
\(147\) 7.00000 0.577350
\(148\) −1.00000 −0.0821995
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.00000 0.486664
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 2.00000 0.156174
\(165\) −2.00000 −0.155700
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 6.00000 0.458831
\(172\) 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
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 8.00000 0.589768
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 8.00000 0.574367
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −2.00000 −0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) −2.00000 −0.139686
\(206\) 10.0000 0.696733
\(207\) 8.00000 0.556038
\(208\) −2.00000 −0.138675
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) −6.00000 −0.405442
\(220\) 2.00000 0.134840
\(221\) −12.0000 −0.807207
\(222\) 1.00000 0.0671156
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −6.00000 −0.397360
\(229\) −30.0000 −1.98246 −0.991228 0.132164i \(-0.957808\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −2.00000 −0.130744
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 1.00000 0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 7.00000 0.447214
\(246\) −2.00000 −0.127515
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) −16.0000 −1.00393
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) −8.00000 −0.494242
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 2.00000 0.123091
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 8.00000 0.487769 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) −2.00000 −0.120605
\(276\) −8.00000 −0.481543
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −6.00000 −0.357295
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) −8.00000 −0.468968
\(292\) 6.00000 0.351123
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 2.00000 0.116052
\(298\) −12.0000 −0.695141
\(299\) −16.0000 −0.925304
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 4.00000 0.229039
\(306\) 6.00000 0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 2.00000 0.113228
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −2.00000 −0.112154
\(319\) −4.00000 −0.223957
\(320\) −1.00000 −0.0559017
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 36.0000 2.00309
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 8.00000 0.443079
\(327\) 16.0000 0.884802
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 12.0000 0.658586
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 10.0000 0.537603
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) −22.0000 −1.17094 −0.585471 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) −2.00000 −0.105118
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 4.00000 0.209083
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 8.00000 0.417029
\(369\) 2.00000 0.104116
\(370\) 1.00000 0.0519875
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −12.0000 −0.620505
\(375\) 1.00000 0.0516398
\(376\) 6.00000 0.309426
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −6.00000 −0.307794
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) 48.0000 2.42746
\(392\) −7.00000 −0.353553
\(393\) 8.00000 0.403547
\(394\) −2.00000 −0.100759
\(395\) −8.00000 −0.402524
\(396\) −2.00000 −0.100504
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) −6.00000 −0.297044
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 16.0000 0.789222
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) −12.0000 −0.586939
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) −8.00000 −0.389434
\(423\) 6.00000 0.291730
\(424\) 2.00000 0.0971286
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −16.0000 −0.766261
\(437\) 48.0000 2.29615
\(438\) −6.00000 −0.286691
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 2.00000 0.0953463
\(441\) −7.00000 −0.333333
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 1.00000 0.0474579
\(445\) −6.00000 −0.284427
\(446\) 16.0000 0.757622
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 1.00000 0.0471405
\(451\) −4.00000 −0.188353
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −30.0000 −1.40181
\(459\) −6.00000 −0.280056
\(460\) −8.00000 −0.373002
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −6.00000 −0.276759
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 8.00000 0.365911
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 1.00000 0.0456435
\(481\) 2.00000 0.0911922
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) −42.0000 −1.90320 −0.951601 0.307337i \(-0.900562\pi\)
−0.951601 + 0.307337i \(0.900562\pi\)
\(488\) −4.00000 −0.181071
\(489\) −8.00000 −0.361773
\(490\) 7.00000 0.316228
\(491\) −14.0000 −0.631811 −0.315906 0.948791i \(-0.602308\pi\)
−0.315906 + 0.948791i \(0.602308\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 12.0000 0.540453
\(494\) −12.0000 −0.539906
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −6.00000 −0.264906
\(514\) 2.00000 0.0882162
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 2.00000 0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 2.00000 0.0875376
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) −6.00000 −0.259645
\(535\) −8.00000 −0.345870
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 8.00000 0.344904
\(539\) 14.0000 0.603023
\(540\) 1.00000 0.0430331
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) −16.0000 −0.687259
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −16.0000 −0.683486
\(549\) −4.00000 −0.170716
\(550\) −2.00000 −0.0852803
\(551\) 12.0000 0.511217
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) −1.00000 −0.0424476
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) −22.0000 −0.928014
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −6.00000 −0.252646
\(565\) 6.00000 0.252422
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 6.00000 0.251312
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 19.0000 0.790296
\(579\) −24.0000 −0.997406
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) −4.00000 −0.165663
\(584\) 6.00000 0.248282
\(585\) 2.00000 0.0826898
\(586\) −2.00000 −0.0826192
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −1.00000 −0.0410997
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 4.00000 0.163709
\(598\) −16.0000 −0.654289
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −46.0000 −1.86708 −0.933541 0.358470i \(-0.883298\pi\)
−0.933541 + 0.358470i \(0.883298\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −12.0000 −0.485468
\(612\) 6.00000 0.242536
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 4.00000 0.161427
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 40.0000 1.61034 0.805170 0.593045i \(-0.202074\pi\)
0.805170 + 0.593045i \(0.202074\pi\)
\(618\) −10.0000 −0.402259
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 32.0000 1.28308
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −24.0000 −0.959233
\(627\) 12.0000 0.479234
\(628\) −22.0000 −0.877896
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 8.00000 0.318223
\(633\) 8.00000 0.317971
\(634\) 18.0000 0.714871
\(635\) 16.0000 0.634941
\(636\) −2.00000 −0.0793052
\(637\) 14.0000 0.554700
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −8.00000 −0.315735
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −50.0000 −1.95665 −0.978326 0.207072i \(-0.933606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 16.0000 0.625650
\(655\) 8.00000 0.312586
\(656\) 2.00000 0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 10.0000 0.388661
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 16.0000 0.619522
\(668\) −12.0000 −0.464294
\(669\) −16.0000 −0.618596
\(670\) 4.00000 0.154533
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −18.0000 −0.693334
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 6.00000 0.229416
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 30.0000 1.14457
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 8.00000 0.304555
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) −2.00000 −0.0758098
\(697\) 12.0000 0.454532
\(698\) 10.0000 0.378506
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) −6.00000 −0.226294
\(704\) −2.00000 −0.0753778
\(705\) 6.00000 0.225973
\(706\) −22.0000 −0.827981
\(707\) 0 0
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 24.0000 0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −26.0000 −0.966950
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) 7.00000 0.259794
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 0 0
\(732\) 4.00000 0.147844
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −24.0000 −0.885856
\(735\) −7.00000 −0.258199
\(736\) 8.00000 0.294884
\(737\) 8.00000 0.294684
\(738\) 2.00000 0.0736210
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 1.00000 0.0367607
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −10.0000 −0.366126
\(747\) 12.0000 0.439057
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 6.00000 0.218797
\(753\) −12.0000 −0.437304
\(754\) −4.00000 −0.145671
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −30.0000 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(758\) −28.0000 −1.01701
\(759\) 16.0000 0.580763
\(760\) −6.00000 −0.217643
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 4.00000 0.144526
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 24.0000 0.863779
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 12.0000 0.429945
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 48.0000 1.71648
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) 22.0000 0.785214
\(786\) 8.00000 0.285351
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −2.00000 −0.0712019
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 8.00000 0.284088
\(794\) 30.0000 1.06466
\(795\) 2.00000 0.0709327
\(796\) −4.00000 −0.141776
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) −38.0000 −1.34183
\(803\) −12.0000 −0.423471
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −8.00000 −0.281613
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 2.00000 0.0701000
\(815\) −8.00000 −0.280228
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 16.0000 0.558064
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 10.0000 0.348367
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 8.00000 0.278019
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) −12.0000 −0.416526
\(831\) −2.00000 −0.0693792
\(832\) −2.00000 −0.0693375
\(833\) −42.0000 −1.45521
\(834\) 4.00000 0.138509
\(835\) 12.0000 0.415277
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 6.00000 0.207267
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −28.0000 −0.964944
\(843\) 22.0000 0.757720
\(844\) −8.00000 −0.275371
\(845\) 9.00000 0.309609
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 8.00000 0.274559
\(850\) 6.00000 0.205798
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 8.00000 0.273434
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −4.00000 −0.136558
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −10.0000 −0.340010
\(866\) −2.00000 −0.0679628
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 2.00000 0.0678064
\(871\) 8.00000 0.271070
\(872\) −16.0000 −0.541828
\(873\) 8.00000 0.270759
\(874\) 48.0000 1.62362
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 4.00000 0.134993
\(879\) 2.00000 0.0674583
\(880\) 2.00000 0.0674200
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −7.00000 −0.235702
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) −2.00000 −0.0670025
\(892\) 16.0000 0.535720
\(893\) 36.0000 1.20469
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 2.00000 0.0664822
\(906\) 8.00000 0.265782
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −6.00000 −0.198680
\(913\) −24.0000 −0.794284
\(914\) −8.00000 −0.264616
\(915\) −4.00000 −0.132236
\(916\) −30.0000 −0.991228
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) −8.00000 −0.263752
\(921\) −4.00000 −0.131804
\(922\) −42.0000 −1.38320
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 34.0000 1.11731
\(927\) 10.0000 0.328443
\(928\) 2.00000 0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 12.0000 0.393073
\(933\) −32.0000 −1.04763
\(934\) 36.0000 1.17796
\(935\) 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 \(0.426559\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) −6.00000 −0.195698
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 22.0000 0.716799
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) −8.00000 −0.259828
\(949\) −12.0000 −0.389536
\(950\) 6.00000 0.194666
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 4.00000 0.129302
\(958\) 8.00000 0.258468
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 2.00000 0.0644826
\(963\) 8.00000 0.257796
\(964\) 26.0000 0.837404
\(965\) −24.0000 −0.772587
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) −7.00000 −0.224989
\(969\) −36.0000 −1.15649
\(970\) −8.00000 −0.256865
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −42.0000 −1.34577
\(975\) 2.00000 0.0640513
\(976\) −4.00000 −0.128037
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) −8.00000 −0.255812
\(979\) −12.0000 −0.383522
\(980\) 7.00000 0.223607
\(981\) −16.0000 −0.510841
\(982\) −14.0000 −0.446758
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 2.00000 0.0637253
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) −12.0000 −0.380235
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −6.00000 −0.189927
\(999\) 1.00000 0.0316386
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 1110.2.a.i.1.1 1
3.2 odd 2 3330.2.a.j.1.1 1
4.3 odd 2 8880.2.a.v.1.1 1
5.4 even 2 5550.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.i.1.1 1 1.1 even 1 trivial
3330.2.a.j.1.1 1 3.2 odd 2
5550.2.a.q.1.1 1 5.4 even 2
8880.2.a.v.1.1 1 4.3 odd 2