Properties

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

Related objects

Downloads

Learn more

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: \(1\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -1.00000 q^{18} +6.00000 q^{19} +6.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} -10.0000 q^{37} -6.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -8.00000 q^{43} -6.00000 q^{44} +6.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} -6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -4.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} +1.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +8.00000 q^{86} +2.00000 q^{87} +6.00000 q^{88} -10.0000 q^{89} -6.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −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
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 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
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −6.00000 −0.973329
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −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
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −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
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 2.00000 0.214423
\(88\) 6.00000 0.639602
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 7.00000 0.707107
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −1.00000 −0.0924500
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −7.00000 −0.577350
\(148\) −10.0000 −0.821995
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −16.0000 −1.27289
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 10.0000 0.751646
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 6.00000 0.426401
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) −6.00000 −0.417029
\(208\) −1.00000 −0.0693375
\(209\) −36.0000 −2.49017
\(210\) 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
\(213\) −8.00000 −0.548151
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) 10.0000 0.671156
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 8.00000 0.532152
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 6.00000 0.397360
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −28.0000 −1.81117 −0.905585 0.424165i \(-0.860568\pi\)
−0.905585 + 0.424165i \(0.860568\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −6.00000 −0.381771
\(248\) −4.00000 −0.254000
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −12.0000 −0.741362
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −4.00000 −0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −8.00000 −0.479808
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) −6.00000 −0.348155
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −2.00000 −0.114897
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 1.00000 0.0566139
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) −16.0000 −0.884802
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) −4.00000 −0.219529
\(333\) −10.0000 −0.547997
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 2.00000 0.107211
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 6.00000 0.319801
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) −6.00000 −0.313197 −0.156599 0.987662i \(-0.550053\pi\)
−0.156599 + 0.987662i \(0.550053\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 22.0000 1.13006 0.565032 0.825069i \(-0.308864\pi\)
0.565032 + 0.825069i \(0.308864\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 8.00000 0.409316
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 7.00000 0.353553
\(393\) 12.0000 0.605320
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 4.00000 0.199502
\(403\) −4.00000 −0.199254
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 60.0000 2.97409
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 8.00000 0.391762
\(418\) 36.0000 1.76082
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −4.00000 −0.194717
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −36.0000 −1.72211
\(438\) 6.00000 0.286691
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) −8.00000 −0.376288
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −20.0000 −0.934539
\(459\) 0 0
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) 40.0000 1.85098 0.925490 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −10.0000 −0.460287
\(473\) 48.0000 2.20704
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 28.0000 1.28069
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 6.00000 0.271607
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) −8.00000 −0.357057
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −36.0000 −1.60040
\(507\) 1.00000 0.0444116
\(508\) 18.0000 0.798621
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) −24.0000 −1.05859
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 48.0000 2.11104
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −12.0000 −0.517838
\(538\) −14.0000 −0.603583
\(539\) 42.0000 1.80907
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 16.0000 0.687259
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −4.00000 −0.169334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 6.00000 0.250873
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −36.0000 −1.49097
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −7.00000 −0.288675
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) −10.0000 −0.410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) −6.00000 −0.245358
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −10.0000 −0.402259
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) −36.0000 −1.43770
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −16.0000 −0.636446
\(633\) 4.00000 0.158986
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 7.00000 0.277350
\(638\) 12.0000 0.475085
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 16.0000 0.631470
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.0000 −0.550397 −0.275198 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −60.0000 −2.35521
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) −12.0000 −0.464642
\(668\) −8.00000 −0.309529
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 8.00000 0.307238
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 24.0000 0.919007
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −8.00000 −0.304997
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) −8.00000 −0.302804
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 1.00000 0.0377426
\(703\) −60.0000 −2.26294
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 0 0
\(708\) 10.0000 0.375823
\(709\) −44.0000 −1.65245 −0.826227 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 10.0000 0.374766
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −28.0000 −1.04568
\(718\) 12.0000 0.447836
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −22.0000 −0.818189
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −6.00000 −0.221766
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 6.00000 0.221464
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 24.0000 0.884051
\(738\) 6.00000 0.220863
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) −52.0000 −1.90769 −0.953847 0.300291i \(-0.902916\pi\)
−0.953847 + 0.300291i \(0.902916\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 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
\(753\) 8.00000 0.291536
\(754\) 2.00000 0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −22.0000 −0.799076
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) −10.0000 −0.361079
\(768\) 1.00000 0.0360844
\(769\) −54.0000 −1.94729 −0.973645 0.228069i \(-0.926759\pi\)
−0.973645 + 0.228069i \(0.926759\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) −10.0000 −0.359908
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 22.0000 0.783718
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 6.00000 0.213066
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) −14.0000 −0.494357
\(803\) 36.0000 1.27041
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 14.0000 0.492823
\(808\) 2.00000 0.0703598
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) −60.0000 −2.10300
\(815\) 0 0
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 0 0
\(821\) 4.00000 0.139601 0.0698005 0.997561i \(-0.477764\pi\)
0.0698005 + 0.997561i \(0.477764\pi\)
\(822\) 2.00000 0.0697580
\(823\) 54.0000 1.88232 0.941161 0.337959i \(-0.109737\pi\)
0.941161 + 0.337959i \(0.109737\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −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
\(831\) −26.0000 −0.901930
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −36.0000 −1.24509
\(837\) 4.00000 0.138260
\(838\) −28.0000 −0.967244
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 8.00000 0.275698
\(843\) −10.0000 −0.344418
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) −8.00000 −0.274075
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 52.0000 1.77629 0.888143 0.459567i \(-0.151995\pi\)
0.888143 + 0.459567i \(0.151995\pi\)
\(858\) −6.00000 −0.204837
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 24.0000 0.815553
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −96.0000 −3.25658
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 16.0000 0.541828
\(873\) −2.00000 −0.0676897
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −16.0000 −0.539974
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 7.00000 0.235702
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −32.0000 −1.07506
\(887\) 26.0000 0.872995 0.436497 0.899706i \(-0.356219\pi\)
0.436497 + 0.899706i \(0.356219\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −4.00000 −0.133930
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 34.0000 1.13459
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 0 0
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) 8.00000 0.266076
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 20.0000 0.663723
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 6.00000 0.198680
\(913\) 24.0000 0.794284
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) −36.0000 −1.18560
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) 10.0000 0.328443
\(928\) −2.00000 −0.0656532
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 8.00000 0.262049
\(933\) 0 0
\(934\) −40.0000 −1.30884
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 36.0000 1.17232
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 16.0000 0.519656
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −28.0000 −0.905585
\(957\) −12.0000 −0.387905
\(958\) −20.0000 −0.646171
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −10.0000 −0.322413
\(963\) −16.0000 −0.515593
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 12.0000 0.383718
\(979\) 60.0000 1.91761
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 12.0000 0.382935
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −4.00000 −0.127000
\(993\) −26.0000 −0.825085
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 6.00000 0.189927
\(999\) −10.0000 −0.316386
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.i.1.1 1
3.2 odd 2 5850.2.a.bs.1.1 1
5.2 odd 4 390.2.e.a.79.1 2
5.3 odd 4 390.2.e.a.79.2 yes 2
5.4 even 2 1950.2.a.r.1.1 1
15.2 even 4 1170.2.e.d.469.2 2
15.8 even 4 1170.2.e.d.469.1 2
15.14 odd 2 5850.2.a.o.1.1 1
20.3 even 4 3120.2.l.a.1249.1 2
20.7 even 4 3120.2.l.a.1249.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.a.79.1 2 5.2 odd 4
390.2.e.a.79.2 yes 2 5.3 odd 4
1170.2.e.d.469.1 2 15.8 even 4
1170.2.e.d.469.2 2 15.2 even 4
1950.2.a.i.1.1 1 1.1 even 1 trivial
1950.2.a.r.1.1 1 5.4 even 2
3120.2.l.a.1249.1 2 20.3 even 4
3120.2.l.a.1249.2 2 20.7 even 4
5850.2.a.o.1.1 1 15.14 odd 2
5850.2.a.bs.1.1 1 3.2 odd 2