Properties

Label 1950.2.a.d.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} +6.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -10.0000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} +4.00000 q^{42} -6.00000 q^{44} -6.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -2.00000 q^{57} +10.0000 q^{58} -6.00000 q^{59} -6.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +12.0000 q^{67} +4.00000 q^{68} -6.00000 q^{69} -1.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +2.00000 q^{76} -24.0000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} -4.00000 q^{84} +10.0000 q^{87} +6.00000 q^{88} +14.0000 q^{89} -4.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} -9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 6.00000 1.27920
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −2.00000 −0.324443
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000 0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −2.00000 −0.264906
\(58\) 10.0000 1.31306
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 4.00000 0.485071
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −24.0000 −2.73505
\(78\) −1.00000 −0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 6.00000 0.639602
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −9.00000 −0.909137
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 4.00000 0.396059
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 4.00000 0.377964
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) −1.00000 −0.0924500
\(118\) 6.00000 0.552345
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 6.00000 0.522233
\(133\) 8.00000 0.693688
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 6.00000 0.510754
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) 6.00000 0.493197
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −2.00000 −0.162221
\(153\) 4.00000 0.323381
\(154\) 24.0000 1.93398
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 4.00000 0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 6.00000 0.450988
\(178\) −14.0000 −1.04934
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 4.00000 0.296500
\(183\) 6.00000 0.443533
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −24.0000 −1.75505
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 6.00000 0.426401
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) −10.0000 −0.703598
\(203\) −40.0000 −2.80745
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −4.00000 −0.270914
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 6.00000 0.402694
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −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
\(231\) 24.0000 1.57908
\(232\) 10.0000 0.656532
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 8.00000 0.519656
\(238\) −16.0000 −1.03713
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) −2.00000 −0.127257
\(248\) −4.00000 −0.254000
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 4.00000 0.251976
\(253\) −36.0000 −2.26330
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.0000 −1.74659 −0.873296 0.487190i \(-0.838022\pi\)
−0.873296 + 0.487190i \(0.838022\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 8.00000 0.494242
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) −14.0000 −0.856786
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 4.00000 0.242536
\(273\) 4.00000 0.242091
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −16.0000 −0.959616
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −8.00000 −0.476393
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 40.0000 2.36113
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) −2.00000 −0.117041
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 6.00000 0.348155
\(298\) −16.0000 −0.926855
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −24.0000 −1.36753
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 6.00000 0.336463
\(319\) 60.0000 3.35936
\(320\) 0 0
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −4.00000 −0.221201
\(328\) −10.0000 −0.552158
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) −2.00000 −0.108148
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 10.0000 0.536056
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 6.00000 0.319801
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −16.0000 −0.846810
\(358\) 24.0000 1.26844
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −6.00000 −0.315353
\(363\) −25.0000 −1.31216
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 6.00000 0.312772
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) −4.00000 −0.207390
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 10.0000 0.515026
\(378\) 4.00000 0.205738
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −10.0000 −0.512316
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −9.00000 −0.454569
\(393\) 8.00000 0.403547
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 8.00000 0.401004
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 12.0000 0.598506
\(403\) −4.00000 −0.199254
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 40.0000 1.98517
\(407\) −36.0000 −1.78445
\(408\) 4.00000 0.198030
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 2.00000 0.0985329
\(413\) −24.0000 −1.18096
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −16.0000 −0.783523
\(418\) 12.0000 0.586939
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −4.00000 −0.194717
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 12.0000 0.574038
\(438\) −2.00000 −0.0955637
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) −16.0000 −0.756774
\(448\) 4.00000 0.188982
\(449\) 38.0000 1.79333 0.896665 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) 0 0
\(451\) −60.0000 −2.82529
\(452\) 4.00000 0.188144
\(453\) −16.0000 −0.751746
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −8.00000 −0.373815
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) −24.0000 −1.11658
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 6.00000 0.274721
\(478\) 4.00000 0.182956
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −18.0000 −0.819878
\(483\) −24.0000 −1.09204
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 6.00000 0.271607
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −10.0000 −0.450835
\(493\) −40.0000 −1.80151
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 20.0000 0.892644
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 36.0000 1.60040
\(507\) −1.00000 −0.0444116
\(508\) 10.0000 0.443678
\(509\) 16.0000 0.709188 0.354594 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −2.00000 −0.0883022
\(514\) 28.0000 1.23503
\(515\) 0 0
\(516\) 0 0
\(517\) 48.0000 2.11104
\(518\) −24.0000 −1.05450
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 10.0000 0.437688
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) 16.0000 0.696971
\(528\) 6.00000 0.261116
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 8.00000 0.346844
\(533\) −10.0000 −0.433148
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 24.0000 1.03568
\(538\) −10.0000 −0.431131
\(539\) −54.0000 −2.32594
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 8.00000 0.343629
\(543\) −6.00000 −0.257485
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 14.0000 0.598050
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −20.0000 −0.852029
\(552\) 6.00000 0.255377
\(553\) −32.0000 −1.36078
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 18.0000 0.759284
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) −40.0000 −1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 1.00000 0.0415945
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) −14.0000 −0.580319
\(583\) −36.0000 −1.49097
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −9.00000 −0.371154
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 6.00000 0.246598
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 16.0000 0.655386
\(597\) 8.00000 0.327418
\(598\) 6.00000 0.245358
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −42.0000 −1.70473 −0.852364 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 40.0000 1.62088
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 4.00000 0.161690
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 24.0000 0.966988
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 2.00000 0.0804518
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 8.00000 0.320771
\(623\) 56.0000 2.24359
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 12.0000 0.479234
\(628\) −6.00000 −0.239426
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −9.00000 −0.356593
\(638\) −60.0000 −2.37542
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 4.00000 0.156652
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −2.00000 −0.0780274
\(658\) 32.0000 1.24749
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) −2.00000 −0.0777322
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −60.0000 −2.32321
\(668\) 16.0000 0.619059
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 4.00000 0.154303
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 4.00000 0.153619
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 24.0000 0.919007
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) −8.00000 −0.305219
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 18.0000 0.684752 0.342376 0.939563i \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) 22.0000 0.836315
\(693\) −24.0000 −0.911685
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 40.0000 1.51511
\(698\) 4.00000 0.151402
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 12.0000 0.452589
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 40.0000 1.50435
\(708\) 6.00000 0.225494
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 24.0000 0.898807
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 4.00000 0.149383
\(718\) 20.0000 0.746393
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 15.0000 0.558242
\(723\) −18.0000 −0.669427
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 6.00000 0.221766
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −72.0000 −2.65215
\(738\) −10.0000 −0.368105
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) −24.0000 −0.881068
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) 4.00000 0.146352
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −8.00000 −0.291730
\(753\) 20.0000 0.728841
\(754\) −10.0000 −0.364179
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −10.0000 −0.363216
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 10.0000 0.362262
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 6.00000 0.216647
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 2.00000 0.0719816
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) −24.0000 −0.860995
\(778\) −30.0000 −1.07555
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) 10.0000 0.357371
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 6.00000 0.213741
\(789\) −10.0000 −0.356009
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 6.00000 0.213201
\(793\) 6.00000 0.213066
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 8.00000 0.283197
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) −30.0000 −1.05934
\(803\) 12.0000 0.423471
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −10.0000 −0.352017
\(808\) −10.0000 −0.351799
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) −40.0000 −1.40372
\(813\) 8.00000 0.280572
\(814\) 36.0000 1.26180
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −28.0000 −0.977207 −0.488603 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 14.0000 0.488306
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 6.00000 0.208514
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) −1.00000 −0.0346688
\(833\) 36.0000 1.24733
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) −4.00000 −0.138260
\(838\) −8.00000 −0.276355
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −4.00000 −0.137849
\(843\) 18.0000 0.619953
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 100.000 3.43604
\(848\) 6.00000 0.206041
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 6.00000 0.204837
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 16.0000 0.544962
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 1.00000 0.0339618
\(868\) 16.0000 0.543075
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −4.00000 −0.135457
\(873\) −14.0000 −0.473828
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 32.0000 1.07995
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −9.00000 −0.303046
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 6.00000 0.201347
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −8.00000 −0.267860
\(893\) −16.0000 −0.535420
\(894\) 16.0000 0.535120
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 6.00000 0.200334
\(898\) −38.0000 −1.26808
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 60.0000 1.99778
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 4.00000 0.132745
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −24.0000 −0.794284
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) −32.0000 −1.05673
\(918\) 4.00000 0.132020
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 4.00000 0.131733
\(923\) 0 0
\(924\) 24.0000 0.789542
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 2.00000 0.0656886
\(928\) 10.0000 0.328266
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 12.0000 0.393073
\(933\) 8.00000 0.261908
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −48.0000 −1.56809 −0.784046 0.620703i \(-0.786847\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) −48.0000 −1.56726
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) −6.00000 −0.195491
\(943\) 60.0000 1.95387
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 8.00000 0.259828
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) −16.0000 −0.518563
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) −60.0000 −1.93952
\(958\) 20.0000 0.646171
\(959\) 56.0000 1.80833
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.00000 0.193448
\(963\) 0 0
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −25.0000 −0.803530
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 64.0000 2.05175
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −58.0000 −1.85558 −0.927792 0.373097i \(-0.878296\pi\)
−0.927792 + 0.373097i \(0.878296\pi\)
\(978\) 4.00000 0.127906
\(979\) −84.0000 −2.68465
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 28.0000 0.893061 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 40.0000 1.27386
\(987\) 32.0000 1.01857
\(988\) −2.00000 −0.0636285
\(989\) 0 0
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −4.00000 −0.127000
\(993\) −2.00000 −0.0634681
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 18.0000 0.569780
\(999\) −6.00000 −0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.a.d.1.1 1
3.2 odd 2 5850.2.a.cc.1.1 1
5.2 odd 4 390.2.e.d.79.1 2
5.3 odd 4 390.2.e.d.79.2 yes 2
5.4 even 2 1950.2.a.v.1.1 1
15.2 even 4 1170.2.e.c.469.2 2
15.8 even 4 1170.2.e.c.469.1 2
15.14 odd 2 5850.2.a.e.1.1 1
20.3 even 4 3120.2.l.i.1249.2 2
20.7 even 4 3120.2.l.i.1249.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.d.79.1 2 5.2 odd 4
390.2.e.d.79.2 yes 2 5.3 odd 4
1170.2.e.c.469.1 2 15.8 even 4
1170.2.e.c.469.2 2 15.2 even 4
1950.2.a.d.1.1 1 1.1 even 1 trivial
1950.2.a.v.1.1 1 5.4 even 2
3120.2.l.i.1249.1 2 20.7 even 4
3120.2.l.i.1249.2 2 20.3 even 4
5850.2.a.e.1.1 1 15.14 odd 2
5850.2.a.cc.1.1 1 3.2 odd 2