Properties

Label 150.2.a.c.1.1
Level $150$
Weight $2$
Character 150.1
Self dual yes
Analytic conductor $1.198$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +10.0000 q^{59} +2.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} +4.00000 q^{73} -2.00000 q^{74} -4.00000 q^{77} -6.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +4.00000 q^{86} +2.00000 q^{88} -10.0000 q^{89} +12.0000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) −6.00000 −0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −2.00000 −0.198030
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 10.0000 0.920575
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 4.00000 0.340503
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 12.0000 1.00702
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −3.00000 −0.247436
\(148\) −2.00000 −0.164399
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −2.00000 −0.154303
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 10.0000 0.751646
\(178\) −10.0000 −0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 12.0000 0.889499
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −4.00000 −0.292509
\(188\) 8.00000 0.583460
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 2.00000 0.142134
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −8.00000 −0.562878
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 4.00000 0.278019
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) 10.0000 0.677285
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −2.00000 −0.125988
\(253\) 8.00000 0.502956
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 12.0000 0.726273
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) −8.00000 −0.478947
\(280\) 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
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 4.00000 0.234082
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 2.00000 0.116052
\(298\) −20.0000 −1.15857
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −8.00000 −0.460348
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −4.00000 −0.227921
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −6.00000 −0.339683
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 10.0000 0.553001
\(328\) 2.00000 0.110432
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 23.0000 1.25104
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 2.00000 0.106600
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 4.00000 0.211702
\(358\) 10.0000 0.528516
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) −7.00000 −0.367405
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 12.0000 0.613973
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 4.00000 0.203331
\(388\) 8.00000 0.406138
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −3.00000 −0.151523
\(393\) −18.0000 −0.907980
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 8.00000 0.399004
\(403\) 48.0000 2.39105
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) −2.00000 −0.0990148
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 14.0000 0.689730
\(413\) −20.0000 −0.984136
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 12.0000 0.584151
\(423\) 8.00000 0.388973
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) −20.0000 −0.945968
\(448\) −2.00000 −0.0944911
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) −4.00000 −0.186097
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −6.00000 −0.277350
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 10.0000 0.460287
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −6.00000 −0.274721
\(478\) 20.0000 0.914779
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 22.0000 1.00207
\(483\) −8.00000 −0.364013
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 2.00000 0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −24.0000 −1.07655
\(498\) 4.00000 0.179244
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) −18.0000 −0.803379
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 23.0000 1.02147
\(508\) −2.00000 −0.0887357
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 16.0000 0.703679
\(518\) 4.00000 0.175750
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 16.0000 0.696971
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −8.00000 −0.343629
\(543\) 2.00000 0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −18.0000 −0.759284
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) −2.00000 −0.0839921
\(568\) 12.0000 0.503509
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) −12.0000 −0.501745
\(573\) 12.0000 0.501307
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 8.00000 0.331611
\(583\) −12.0000 −0.496989
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) −2.00000 −0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −8.00000 −0.326056
\(603\) 8.00000 0.325785
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −8.00000 −0.324978
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) −2.00000 −0.0808452
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 14.0000 0.563163
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 12.0000 0.481156
\(623\) 20.0000 0.801283
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −12.0000 −0.473602
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −16.0000 −0.626608
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 4.00000 0.156055
\(658\) −16.0000 −0.623745
\(659\) −50.0000 −1.94772 −0.973862 0.227142i \(-0.927062\pi\)
−0.973862 + 0.227142i \(0.927062\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −8.00000 −0.310929
\(663\) 12.0000 0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) −2.00000 −0.0771517
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) −6.00000 −0.230429
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) −16.0000 −0.612672
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 14.0000 0.532200
\(693\) −4.00000 −0.151947
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 10.0000 0.378506
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) −6.00000 −0.226455
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 16.0000 0.601742
\(708\) 10.0000 0.375823
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) −32.0000 −1.19841
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) −19.0000 −0.707107
\(723\) 22.0000 0.818189
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 12.0000 0.444750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 2.00000 0.0739221
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 16.0000 0.589368
\(738\) 2.00000 0.0736210
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) −4.00000 −0.146254
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 20.0000 0.726433
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −2.00000 −0.0724524
\(763\) −20.0000 −0.724049
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −60.0000 −2.16647
\(768\) 1.00000 0.0360844
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 4.00000 0.143963
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 4.00000 0.143499
\(778\) 20.0000 0.717035
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −22.0000 −0.783718
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 2.00000 0.0710669
\(793\) −12.0000 −0.426132
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 22.0000 0.776847
\(803\) 8.00000 0.282314
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 18.0000 0.627822
\(823\) −6.00000 −0.209147 −0.104573 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −20.0000 −0.695889
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 4.00000 0.139010
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −6.00000 −0.208013
\(833\) 6.00000 0.207888
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −10.0000 −0.345444
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 22.0000 0.758170
\(843\) −18.0000 −0.619953
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 14.0000 0.481046
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 12.0000 0.411113
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −12.0000 −0.409673
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 32.0000 1.08992
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 10.0000 0.338643
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −3.00000 −0.101015
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −26.0000 −0.870544
\(893\) 0 0
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −24.0000 −0.801337
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 4.00000 0.133185
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 28.0000 0.929213
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) −32.0000 −1.05847
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 36.0000 1.18882
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 12.0000 0.395199
\(923\) −72.0000 −2.36991
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 12.0000 0.392862
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) −16.0000 −0.522419
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) −22.0000 −0.716799
\(943\) 8.00000 0.260516
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 4.00000 0.129641
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 20.0000 0.646171
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) −12.0000 −0.386695
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) 38.0000 1.22200 0.610999 0.791632i \(-0.290768\pi\)
0.610999 + 0.791632i \(0.290768\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 1.00000 0.0320750
\(973\) 40.0000 1.28234
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −16.0000 −0.511624
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) −18.0000 −0.574403
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −8.00000 −0.254000
\(993\) −8.00000 −0.253872
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 150.2.a.c.1.1 1
3.2 odd 2 450.2.a.b.1.1 1
4.3 odd 2 1200.2.a.g.1.1 1
5.2 odd 4 30.2.c.a.19.2 yes 2
5.3 odd 4 30.2.c.a.19.1 2
5.4 even 2 150.2.a.a.1.1 1
7.6 odd 2 7350.2.a.cc.1.1 1
8.3 odd 2 4800.2.a.cj.1.1 1
8.5 even 2 4800.2.a.l.1.1 1
12.11 even 2 3600.2.a.bg.1.1 1
15.2 even 4 90.2.c.a.19.1 2
15.8 even 4 90.2.c.a.19.2 2
15.14 odd 2 450.2.a.f.1.1 1
20.3 even 4 240.2.f.a.49.1 2
20.7 even 4 240.2.f.a.49.2 2
20.19 odd 2 1200.2.a.m.1.1 1
35.2 odd 12 1470.2.n.h.949.2 4
35.3 even 12 1470.2.n.a.79.2 4
35.12 even 12 1470.2.n.a.949.2 4
35.13 even 4 1470.2.g.g.589.1 2
35.17 even 12 1470.2.n.a.79.1 4
35.18 odd 12 1470.2.n.h.79.2 4
35.23 odd 12 1470.2.n.h.949.1 4
35.27 even 4 1470.2.g.g.589.2 2
35.32 odd 12 1470.2.n.h.79.1 4
35.33 even 12 1470.2.n.a.949.1 4
35.34 odd 2 7350.2.a.bg.1.1 1
40.3 even 4 960.2.f.i.769.2 2
40.13 odd 4 960.2.f.h.769.1 2
40.19 odd 2 4800.2.a.m.1.1 1
40.27 even 4 960.2.f.i.769.1 2
40.29 even 2 4800.2.a.cg.1.1 1
40.37 odd 4 960.2.f.h.769.2 2
45.2 even 12 810.2.i.b.109.1 4
45.7 odd 12 810.2.i.e.109.2 4
45.13 odd 12 810.2.i.e.379.2 4
45.22 odd 12 810.2.i.e.379.1 4
45.23 even 12 810.2.i.b.379.1 4
45.32 even 12 810.2.i.b.379.2 4
45.38 even 12 810.2.i.b.109.2 4
45.43 odd 12 810.2.i.e.109.1 4
60.23 odd 4 720.2.f.f.289.2 2
60.47 odd 4 720.2.f.f.289.1 2
60.59 even 2 3600.2.a.o.1.1 1
80.3 even 4 3840.2.d.x.2689.2 2
80.13 odd 4 3840.2.d.g.2689.2 2
80.27 even 4 3840.2.d.x.2689.1 2
80.37 odd 4 3840.2.d.g.2689.1 2
80.43 even 4 3840.2.d.j.2689.1 2
80.53 odd 4 3840.2.d.y.2689.1 2
80.67 even 4 3840.2.d.j.2689.2 2
80.77 odd 4 3840.2.d.y.2689.2 2
120.53 even 4 2880.2.f.e.1729.1 2
120.77 even 4 2880.2.f.e.1729.2 2
120.83 odd 4 2880.2.f.c.1729.1 2
120.107 odd 4 2880.2.f.c.1729.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.c.a.19.1 2 5.3 odd 4
30.2.c.a.19.2 yes 2 5.2 odd 4
90.2.c.a.19.1 2 15.2 even 4
90.2.c.a.19.2 2 15.8 even 4
150.2.a.a.1.1 1 5.4 even 2
150.2.a.c.1.1 1 1.1 even 1 trivial
240.2.f.a.49.1 2 20.3 even 4
240.2.f.a.49.2 2 20.7 even 4
450.2.a.b.1.1 1 3.2 odd 2
450.2.a.f.1.1 1 15.14 odd 2
720.2.f.f.289.1 2 60.47 odd 4
720.2.f.f.289.2 2 60.23 odd 4
810.2.i.b.109.1 4 45.2 even 12
810.2.i.b.109.2 4 45.38 even 12
810.2.i.b.379.1 4 45.23 even 12
810.2.i.b.379.2 4 45.32 even 12
810.2.i.e.109.1 4 45.43 odd 12
810.2.i.e.109.2 4 45.7 odd 12
810.2.i.e.379.1 4 45.22 odd 12
810.2.i.e.379.2 4 45.13 odd 12
960.2.f.h.769.1 2 40.13 odd 4
960.2.f.h.769.2 2 40.37 odd 4
960.2.f.i.769.1 2 40.27 even 4
960.2.f.i.769.2 2 40.3 even 4
1200.2.a.g.1.1 1 4.3 odd 2
1200.2.a.m.1.1 1 20.19 odd 2
1470.2.g.g.589.1 2 35.13 even 4
1470.2.g.g.589.2 2 35.27 even 4
1470.2.n.a.79.1 4 35.17 even 12
1470.2.n.a.79.2 4 35.3 even 12
1470.2.n.a.949.1 4 35.33 even 12
1470.2.n.a.949.2 4 35.12 even 12
1470.2.n.h.79.1 4 35.32 odd 12
1470.2.n.h.79.2 4 35.18 odd 12
1470.2.n.h.949.1 4 35.23 odd 12
1470.2.n.h.949.2 4 35.2 odd 12
2880.2.f.c.1729.1 2 120.83 odd 4
2880.2.f.c.1729.2 2 120.107 odd 4
2880.2.f.e.1729.1 2 120.53 even 4
2880.2.f.e.1729.2 2 120.77 even 4
3600.2.a.o.1.1 1 60.59 even 2
3600.2.a.bg.1.1 1 12.11 even 2
3840.2.d.g.2689.1 2 80.37 odd 4
3840.2.d.g.2689.2 2 80.13 odd 4
3840.2.d.j.2689.1 2 80.43 even 4
3840.2.d.j.2689.2 2 80.67 even 4
3840.2.d.x.2689.1 2 80.27 even 4
3840.2.d.x.2689.2 2 80.3 even 4
3840.2.d.y.2689.1 2 80.53 odd 4
3840.2.d.y.2689.2 2 80.77 odd 4
4800.2.a.l.1.1 1 8.5 even 2
4800.2.a.m.1.1 1 40.19 odd 2
4800.2.a.cg.1.1 1 40.29 even 2
4800.2.a.cj.1.1 1 8.3 odd 2
7350.2.a.bg.1.1 1 35.34 odd 2
7350.2.a.cc.1.1 1 7.6 odd 2