Properties

Label 1110.2.a.n.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +3.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +9.00000 q^{41} -1.00000 q^{42} +11.0000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} +2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -1.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} -6.00000 q^{59} -1.00000 q^{60} -1.00000 q^{61} -1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +3.00000 q^{66} +8.00000 q^{67} +3.00000 q^{68} +1.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -3.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} -6.00000 q^{83} -1.00000 q^{84} -3.00000 q^{85} +11.0000 q^{86} -3.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -1.00000 q^{93} -2.00000 q^{95} +1.00000 q^{96} -13.0000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 3.00000 0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) −1.00000 −0.154303
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 3.00000 0.420084
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) −3.00000 −0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −1.00000 −0.129099
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −1.00000 −0.127000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 3.00000 0.369274
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −3.00000 −0.341882
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) −3.00000 −0.325396
\(86\) 11.0000 1.18616
\(87\) −3.00000 −0.321634
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −6.00000 −0.606092
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 3.00000 0.297044
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) −9.00000 −0.874157
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) −3.00000 −0.286039
\(111\) 1.00000 0.0949158
\(112\) −1.00000 −0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 2.00000 0.184900
\(118\) −6.00000 −0.552345
\(119\) −3.00000 −0.275010
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) 9.00000 0.811503
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.0000 0.968496
\(130\) −2.00000 −0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 3.00000 0.261116
\(133\) −2.00000 −0.173422
\(134\) 8.00000 0.691095
\(135\) −1.00000 −0.0860663
\(136\) 3.00000 0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 8.00000 0.662085
\(147\) −6.00000 −0.494872
\(148\) 1.00000 0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 2.00000 0.162221
\(153\) 3.00000 0.242536
\(154\) −3.00000 −0.241747
\(155\) 1.00000 0.0803219
\(156\) 2.00000 0.160128
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −4.00000 −0.318223
\(159\) −9.00000 −0.713746
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 9.00000 0.702782
\(165\) −3.00000 −0.233550
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −3.00000 −0.230089
\(171\) 2.00000 0.152944
\(172\) 11.0000 0.838742
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) −3.00000 −0.227429
\(175\) −1.00000 −0.0755929
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −2.00000 −0.148250
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) −1.00000 −0.0733236
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) −2.00000 −0.145095
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −13.0000 −0.933346
\(195\) −2.00000 −0.143223
\(196\) −6.00000 −0.428571
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 3.00000 0.213201
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 3.00000 0.210042
\(205\) −9.00000 −0.628587
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 1.00000 0.0690066
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) −9.00000 −0.618123
\(213\) −12.0000 −0.822226
\(214\) −18.0000 −1.23045
\(215\) −11.0000 −0.750194
\(216\) 1.00000 0.0680414
\(217\) 1.00000 0.0678844
\(218\) −1.00000 −0.0677285
\(219\) 8.00000 0.540590
\(220\) −3.00000 −0.202260
\(221\) 6.00000 0.403604
\(222\) 1.00000 0.0671156
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 9.00000 0.598671
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 2.00000 0.132453
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) −3.00000 −0.196960
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −4.00000 −0.259828
\(238\) −3.00000 −0.194461
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 6.00000 0.383326
\(246\) 9.00000 0.573819
\(247\) 4.00000 0.254514
\(248\) −1.00000 −0.0635001
\(249\) −6.00000 −0.380235
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −3.00000 −0.187867
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 11.0000 0.684830
\(259\) −1.00000 −0.0621370
\(260\) −2.00000 −0.124035
\(261\) −3.00000 −0.185695
\(262\) −6.00000 −0.370681
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 3.00000 0.184637
\(265\) 9.00000 0.552866
\(266\) −2.00000 −0.122628
\(267\) −6.00000 −0.367194
\(268\) 8.00000 0.488678
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 3.00000 0.181902
\(273\) −2.00000 −0.121046
\(274\) −18.0000 −1.08742
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 5.00000 0.299880
\(279\) −1.00000 −0.0598684
\(280\) 1.00000 0.0597614
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −12.0000 −0.712069
\(285\) −2.00000 −0.118470
\(286\) 6.00000 0.354787
\(287\) −9.00000 −0.531253
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 3.00000 0.176166
\(291\) −13.0000 −0.762073
\(292\) 8.00000 0.468165
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) −6.00000 −0.349927
\(295\) 6.00000 0.349334
\(296\) 1.00000 0.0581238
\(297\) 3.00000 0.174078
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −11.0000 −0.634029
\(302\) 2.00000 0.115087
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 1.00000 0.0572598
\(306\) 3.00000 0.171499
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −3.00000 −0.170941
\(309\) −4.00000 −0.227552
\(310\) 1.00000 0.0567962
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 2.00000 0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −7.00000 −0.395033
\(315\) 1.00000 0.0563436
\(316\) −4.00000 −0.225018
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) −9.00000 −0.504695
\(319\) −9.00000 −0.503903
\(320\) −1.00000 −0.0559017
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 11.0000 0.609234
\(327\) −1.00000 −0.0553001
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −6.00000 −0.329293
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 9.00000 0.488813
\(340\) −3.00000 −0.162698
\(341\) −3.00000 −0.162459
\(342\) 2.00000 0.108148
\(343\) 13.0000 0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −3.00000 −0.160817
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 3.00000 0.159901
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) −6.00000 −0.318896
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) −3.00000 −0.158777
\(358\) −12.0000 −0.634220
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) −2.00000 −0.104973
\(364\) −2.00000 −0.104828
\(365\) −8.00000 −0.418739
\(366\) −1.00000 −0.0522708
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 0 0
\(369\) 9.00000 0.468521
\(370\) −1.00000 −0.0519875
\(371\) 9.00000 0.467257
\(372\) −1.00000 −0.0518476
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 9.00000 0.465379
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) −1.00000 −0.0514344
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −2.00000 −0.102598
\(381\) 8.00000 0.409852
\(382\) 15.0000 0.767467
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 3.00000 0.152894
\(386\) 2.00000 0.101797
\(387\) 11.0000 0.559161
\(388\) −13.0000 −0.659975
\(389\) −33.0000 −1.67317 −0.836583 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) −6.00000 −0.302660
\(394\) −6.00000 −0.302276
\(395\) 4.00000 0.201262
\(396\) 3.00000 0.150756
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −4.00000 −0.200502
\(399\) −2.00000 −0.100125
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 8.00000 0.399004
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 3.00000 0.148888
\(407\) 3.00000 0.148704
\(408\) 3.00000 0.148522
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −9.00000 −0.444478
\(411\) −18.0000 −0.887875
\(412\) −4.00000 −0.197066
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 2.00000 0.0980581
\(417\) 5.00000 0.244851
\(418\) 6.00000 0.293470
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 1.00000 0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 11.0000 0.535472
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 3.00000 0.145521
\(426\) −12.0000 −0.581402
\(427\) 1.00000 0.0483934
\(428\) −18.0000 −0.870063
\(429\) 6.00000 0.289683
\(430\) −11.0000 −0.530467
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 1.00000 0.0481125
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 1.00000 0.0480015
\(435\) 3.00000 0.143839
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) −3.00000 −0.143019
\(441\) −6.00000 −0.285714
\(442\) 6.00000 0.285391
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 1.00000 0.0474579
\(445\) 6.00000 0.284427
\(446\) 5.00000 0.236757
\(447\) −6.00000 −0.283790
\(448\) −1.00000 −0.0472456
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 1.00000 0.0471405
\(451\) 27.0000 1.27138
\(452\) 9.00000 0.423324
\(453\) 2.00000 0.0939682
\(454\) 3.00000 0.140797
\(455\) 2.00000 0.0937614
\(456\) 2.00000 0.0936586
\(457\) 35.0000 1.63723 0.818615 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) 8.00000 0.373815
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 39.0000 1.81641 0.908206 0.418524i \(-0.137453\pi\)
0.908206 + 0.418524i \(0.137453\pi\)
\(462\) −3.00000 −0.139573
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −3.00000 −0.139272
\(465\) 1.00000 0.0463739
\(466\) −12.0000 −0.555889
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) 2.00000 0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −7.00000 −0.322543
\(472\) −6.00000 −0.276172
\(473\) 33.0000 1.51734
\(474\) −4.00000 −0.183726
\(475\) 2.00000 0.0917663
\(476\) −3.00000 −0.137505
\(477\) −9.00000 −0.412082
\(478\) −9.00000 −0.411650
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 13.0000 0.590300
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 11.0000 0.497437
\(490\) 6.00000 0.271052
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 9.00000 0.405751
\(493\) −9.00000 −0.405340
\(494\) 4.00000 0.179969
\(495\) −3.00000 −0.134840
\(496\) −1.00000 −0.0449013
\(497\) 12.0000 0.538274
\(498\) −6.00000 −0.268866
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) −3.00000 −0.132842
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −6.00000 −0.264649
\(515\) 4.00000 0.176261
\(516\) 11.0000 0.484248
\(517\) 0 0
\(518\) −1.00000 −0.0439375
\(519\) 15.0000 0.658427
\(520\) −2.00000 −0.0877058
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) −3.00000 −0.131306
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −6.00000 −0.262111
\(525\) −1.00000 −0.0436436
\(526\) −9.00000 −0.392419
\(527\) −3.00000 −0.130682
\(528\) 3.00000 0.130558
\(529\) −23.0000 −1.00000
\(530\) 9.00000 0.390935
\(531\) −6.00000 −0.260378
\(532\) −2.00000 −0.0867110
\(533\) 18.0000 0.779667
\(534\) −6.00000 −0.259645
\(535\) 18.0000 0.778208
\(536\) 8.00000 0.345547
\(537\) −12.0000 −0.517838
\(538\) −24.0000 −1.03471
\(539\) −18.0000 −0.775315
\(540\) −1.00000 −0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −4.00000 −0.171815
\(543\) 2.00000 0.0858282
\(544\) 3.00000 0.128624
\(545\) 1.00000 0.0428353
\(546\) −2.00000 −0.0855921
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) −18.0000 −0.768922
\(549\) −1.00000 −0.0426790
\(550\) 3.00000 0.127920
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 8.00000 0.339887
\(555\) −1.00000 −0.0424476
\(556\) 5.00000 0.212047
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 22.0000 0.930501
\(560\) 1.00000 0.0422577
\(561\) 9.00000 0.379980
\(562\) 6.00000 0.253095
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 20.0000 0.840663
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 6.00000 0.250873
\(573\) 15.0000 0.626634
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −8.00000 −0.332756
\(579\) 2.00000 0.0831172
\(580\) 3.00000 0.124568
\(581\) 6.00000 0.248922
\(582\) −13.0000 −0.538867
\(583\) −27.0000 −1.11823
\(584\) 8.00000 0.331042
\(585\) −2.00000 −0.0826898
\(586\) 21.0000 0.867502
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) −6.00000 −0.247436
\(589\) −2.00000 −0.0824086
\(590\) 6.00000 0.247016
\(591\) −6.00000 −0.246807
\(592\) 1.00000 0.0410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 3.00000 0.123091
\(595\) 3.00000 0.122988
\(596\) −6.00000 −0.245770
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 1.00000 0.0408248
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) −11.0000 −0.448327
\(603\) 8.00000 0.325785
\(604\) 2.00000 0.0813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 2.00000 0.0811107
\(609\) 3.00000 0.121566
\(610\) 1.00000 0.0404888
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −25.0000 −1.00974 −0.504870 0.863195i \(-0.668460\pi\)
−0.504870 + 0.863195i \(0.668460\pi\)
\(614\) 2.00000 0.0807134
\(615\) −9.00000 −0.362915
\(616\) −3.00000 −0.120873
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −4.00000 −0.160904
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 3.00000 0.120289
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 6.00000 0.239617
\(628\) −7.00000 −0.279330
\(629\) 3.00000 0.119618
\(630\) 1.00000 0.0398410
\(631\) 47.0000 1.87104 0.935520 0.353273i \(-0.114931\pi\)
0.935520 + 0.353273i \(0.114931\pi\)
\(632\) −4.00000 −0.159111
\(633\) 11.0000 0.437211
\(634\) −33.0000 −1.31060
\(635\) −8.00000 −0.317470
\(636\) −9.00000 −0.356873
\(637\) −12.0000 −0.475457
\(638\) −9.00000 −0.356313
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) −18.0000 −0.710403
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) 0 0
\(645\) −11.0000 −0.433125
\(646\) 6.00000 0.236067
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) −18.0000 −0.706562
\(650\) 2.00000 0.0784465
\(651\) 1.00000 0.0391931
\(652\) 11.0000 0.430793
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −1.00000 −0.0391031
\(655\) 6.00000 0.234439
\(656\) 9.00000 0.351391
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −3.00000 −0.116775
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 20.0000 0.777322
\(663\) 6.00000 0.233021
\(664\) −6.00000 −0.232845
\(665\) 2.00000 0.0775567
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 0 0
\(669\) 5.00000 0.193311
\(670\) −8.00000 −0.309067
\(671\) −3.00000 −0.115814
\(672\) −1.00000 −0.0385758
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 9.00000 0.345643
\(679\) 13.0000 0.498894
\(680\) −3.00000 −0.115045
\(681\) 3.00000 0.114960
\(682\) −3.00000 −0.114876
\(683\) 27.0000 1.03313 0.516563 0.856249i \(-0.327211\pi\)
0.516563 + 0.856249i \(0.327211\pi\)
\(684\) 2.00000 0.0764719
\(685\) 18.0000 0.687745
\(686\) 13.0000 0.496342
\(687\) 8.00000 0.305219
\(688\) 11.0000 0.419371
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −25.0000 −0.951045 −0.475522 0.879704i \(-0.657741\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(692\) 15.0000 0.570214
\(693\) −3.00000 −0.113961
\(694\) −12.0000 −0.455514
\(695\) −5.00000 −0.189661
\(696\) −3.00000 −0.113715
\(697\) 27.0000 1.02270
\(698\) −28.0000 −1.05982
\(699\) −12.0000 −0.453882
\(700\) −1.00000 −0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.00000 0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 12.0000 0.450352
\(711\) −4.00000 −0.150012
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −3.00000 −0.112272
\(715\) −6.00000 −0.224387
\(716\) −12.0000 −0.448461
\(717\) −9.00000 −0.336111
\(718\) 6.00000 0.223918
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 4.00000 0.148968
\(722\) −15.0000 −0.558242
\(723\) −10.0000 −0.371904
\(724\) 2.00000 0.0743294
\(725\) −3.00000 −0.111417
\(726\) −2.00000 −0.0742270
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) 33.0000 1.22055
\(732\) −1.00000 −0.0369611
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) 23.0000 0.848945
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 9.00000 0.331295
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 4.00000 0.146944
\(742\) 9.00000 0.330400
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 6.00000 0.219823
\(746\) −10.0000 −0.366126
\(747\) −6.00000 −0.219529
\(748\) 9.00000 0.329073
\(749\) 18.0000 0.657706
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) −2.00000 −0.0727875
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 8.00000 0.289809
\(763\) 1.00000 0.0362024
\(764\) 15.0000 0.542681
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 3.00000 0.108112
\(771\) −6.00000 −0.216085
\(772\) 2.00000 0.0719816
\(773\) −51.0000 −1.83434 −0.917171 0.398493i \(-0.869533\pi\)
−0.917171 + 0.398493i \(0.869533\pi\)
\(774\) 11.0000 0.395387
\(775\) −1.00000 −0.0359211
\(776\) −13.0000 −0.466673
\(777\) −1.00000 −0.0358748
\(778\) −33.0000 −1.18311
\(779\) 18.0000 0.644917
\(780\) −2.00000 −0.0716115
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) −6.00000 −0.214286
\(785\) 7.00000 0.249841
\(786\) −6.00000 −0.214013
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −6.00000 −0.213741
\(789\) −9.00000 −0.320408
\(790\) 4.00000 0.142314
\(791\) −9.00000 −0.320003
\(792\) 3.00000 0.106600
\(793\) −2.00000 −0.0710221
\(794\) 14.0000 0.496841
\(795\) 9.00000 0.319197
\(796\) −4.00000 −0.141776
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) 24.0000 0.846942
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 3.00000 0.105279
\(813\) −4.00000 −0.140286
\(814\) 3.00000 0.105150
\(815\) −11.0000 −0.385313
\(816\) 3.00000 0.105021
\(817\) 22.0000 0.769683
\(818\) −22.0000 −0.769212
\(819\) −2.00000 −0.0698857
\(820\) −9.00000 −0.314294
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −18.0000 −0.627822
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −4.00000 −0.139347
\(825\) 3.00000 0.104447
\(826\) 6.00000 0.208767
\(827\) −9.00000 −0.312961 −0.156480 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) 6.00000 0.208263
\(831\) 8.00000 0.277517
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) −1.00000 −0.0345651
\(838\) 36.0000 1.24360
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 1.00000 0.0345033
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) 6.00000 0.206651
\(844\) 11.0000 0.378636
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −9.00000 −0.309061
\(849\) 20.0000 0.686398
\(850\) 3.00000 0.102899
\(851\) 0 0
\(852\) −12.0000 −0.411113
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 1.00000 0.0342193
\(855\) −2.00000 −0.0683986
\(856\) −18.0000 −0.615227
\(857\) 45.0000 1.53717 0.768585 0.639747i \(-0.220961\pi\)
0.768585 + 0.639747i \(0.220961\pi\)
\(858\) 6.00000 0.204837
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −11.0000 −0.375097
\(861\) −9.00000 −0.306719
\(862\) 3.00000 0.102180
\(863\) 9.00000 0.306364 0.153182 0.988198i \(-0.451048\pi\)
0.153182 + 0.988198i \(0.451048\pi\)
\(864\) 1.00000 0.0340207
\(865\) −15.0000 −0.510015
\(866\) 20.0000 0.679628
\(867\) −8.00000 −0.271694
\(868\) 1.00000 0.0339422
\(869\) −12.0000 −0.407072
\(870\) 3.00000 0.101710
\(871\) 16.0000 0.542139
\(872\) −1.00000 −0.0338643
\(873\) −13.0000 −0.439983
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 8.00000 0.270295
\(877\) 41.0000 1.38447 0.692236 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) 17.0000 0.573722
\(879\) 21.0000 0.708312
\(880\) −3.00000 −0.101130
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) −6.00000 −0.202031
\(883\) −43.0000 −1.44707 −0.723533 0.690290i \(-0.757483\pi\)
−0.723533 + 0.690290i \(0.757483\pi\)
\(884\) 6.00000 0.201802
\(885\) 6.00000 0.201688
\(886\) 6.00000 0.201574
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) 1.00000 0.0335578
\(889\) −8.00000 −0.268311
\(890\) 6.00000 0.201120
\(891\) 3.00000 0.100504
\(892\) 5.00000 0.167412
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 1.00000 0.0333333
\(901\) −27.0000 −0.899500
\(902\) 27.0000 0.899002
\(903\) −11.0000 −0.366057
\(904\) 9.00000 0.299336
\(905\) −2.00000 −0.0664822
\(906\) 2.00000 0.0664455
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 3.00000 0.0995585
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 2.00000 0.0662266
\(913\) −18.0000 −0.595713
\(914\) 35.0000 1.15770
\(915\) 1.00000 0.0330590
\(916\) 8.00000 0.264327
\(917\) 6.00000 0.198137
\(918\) 3.00000 0.0990148
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 39.0000 1.28440
\(923\) −24.0000 −0.789970
\(924\) −3.00000 −0.0986928
\(925\) 1.00000 0.0328798
\(926\) 26.0000 0.854413
\(927\) −4.00000 −0.131377
\(928\) −3.00000 −0.0984798
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 1.00000 0.0327913
\(931\) −12.0000 −0.393284
\(932\) −12.0000 −0.393073
\(933\) 3.00000 0.0982156
\(934\) −15.0000 −0.490815
\(935\) −9.00000 −0.294331
\(936\) 2.00000 0.0653720
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −8.00000 −0.261209
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −7.00000 −0.228072
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 1.00000 0.0325300
\(946\) 33.0000 1.07292
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) −4.00000 −0.129914
\(949\) 16.0000 0.519382
\(950\) 2.00000 0.0648886
\(951\) −33.0000 −1.07010
\(952\) −3.00000 −0.0972306
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −9.00000 −0.291386
\(955\) −15.0000 −0.485389
\(956\) −9.00000 −0.291081
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −30.0000 −0.967742
\(962\) 2.00000 0.0644826
\(963\) −18.0000 −0.580042
\(964\) −10.0000 −0.322078
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 6.00000 0.192748
\(970\) 13.0000 0.417405
\(971\) −51.0000 −1.63667 −0.818334 0.574743i \(-0.805102\pi\)
−0.818334 + 0.574743i \(0.805102\pi\)
\(972\) 1.00000 0.0320750
\(973\) −5.00000 −0.160293
\(974\) 2.00000 0.0640841
\(975\) 2.00000 0.0640513
\(976\) −1.00000 −0.0320092
\(977\) 15.0000 0.479893 0.239946 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) 11.0000 0.351741
\(979\) −18.0000 −0.575282
\(980\) 6.00000 0.191663
\(981\) −1.00000 −0.0319275
\(982\) −12.0000 −0.382935
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 9.00000 0.286910
\(985\) 6.00000 0.191176
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −3.00000 −0.0953463
\(991\) 29.0000 0.921215 0.460608 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 20.0000 0.634681
\(994\) 12.0000 0.380617
\(995\) 4.00000 0.126809
\(996\) −6.00000 −0.190117
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 32.0000 1.01294
\(999\) 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.n.1.1 1
3.2 odd 2 3330.2.a.h.1.1 1
4.3 odd 2 8880.2.a.g.1.1 1
5.4 even 2 5550.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.n.1.1 1 1.1 even 1 trivial
3330.2.a.h.1.1 1 3.2 odd 2
5550.2.a.g.1.1 1 5.4 even 2
8880.2.a.g.1.1 1 4.3 odd 2