Properties

Label 1950.2.a.ba.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{21} +4.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} +2.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -2.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -2.00000 q^{57} +8.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} -2.00000 q^{69} +1.00000 q^{72} -6.00000 q^{74} -2.00000 q^{76} +8.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +8.00000 q^{87} +4.00000 q^{88} -10.0000 q^{89} +2.00000 q^{91} -2.00000 q^{92} +4.00000 q^{93} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 4.00000 0.852803
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 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
\(33\) 4.00000 0.696311
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −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 \(0.214777\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) −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
\(57\) −2.00000 −0.264906
\(58\) 8.00000 1.05045
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −4.00000 −0.485071
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 8.00000 0.911685
\(78\) 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
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 8.00000 0.857690
\(88\) 4.00000 0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −2.00000 −0.208514
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 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
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) −4.00000 −0.396059
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 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
\(112\) 2.00000 0.188982
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 8.00000 0.742781
\(117\) 1.00000 0.0924500
\(118\) −12.0000 −1.10469
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −2.00000 −0.170251
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) −6.00000 −0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −2.00000 −0.162221
\(153\) −4.00000 −0.323381
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 2.00000 0.154303
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −12.0000 −0.901975
\(178\) −10.0000 −0.749532
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 2.00000 0.148250
\(183\) −2.00000 −0.147844
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −16.0000 −1.17004
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 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
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 4.00000 0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 20.0000 1.40720
\(203\) 16.0000 1.12298
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −2.00000 −0.139010
\(208\) 1.00000 0.0693375
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) −6.00000 −0.402694
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 16.0000 1.06430
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −2.00000 −0.132453
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 8.00000 0.525226
\(233\) −28.0000 −1.83434 −0.917170 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) −2.00000 −0.127257
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 2.00000 0.125988
\(253\) −8.00000 −0.502956
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) −4.00000 −0.249029
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 14.0000 0.864923
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) 8.00000 0.488678
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −4.00000 −0.242536
\(273\) 2.00000 0.121046
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −16.0000 −0.959616
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) −22.0000 −1.27443
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −4.00000 −0.230174
\(303\) 20.0000 1.14897
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 8.00000 0.455842
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 1.00000 0.0566139
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) 32.0000 1.79166
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) −4.00000 −0.222911
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 4.00000 0.221201
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 12.0000 0.658586
\(333\) −6.00000 −0.328798
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 1.00000 0.0543928
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 8.00000 0.428845
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.00000 0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −8.00000 −0.423405
\(358\) 2.00000 0.105703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −22.0000 −1.15629
\(363\) 5.00000 0.262432
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −2.00000 −0.104257
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 4.00000 0.207390
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 2.00000 0.102869
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) −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
\(386\) −12.0000 −0.610784
\(387\) −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
\(393\) 14.0000 0.706207
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 8.00000 0.399004
\(403\) 4.00000 0.199254
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) −24.0000 −1.18964
\(408\) −4.00000 −0.198030
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −4.00000 −0.197066
\(413\) −24.0000 −1.18096
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −16.0000 −0.783523
\(418\) −8.00000 −0.391293
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −4.00000 −0.193347
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −4.00000 −0.190261
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) −22.0000 −1.04056
\(448\) 2.00000 0.0944911
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 16.0000 0.752577
\(453\) −4.00000 −0.187936
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) 28.0000 1.30835
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 8.00000 0.372194
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) −28.0000 −1.29707
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 1.00000 0.0462250
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −6.00000 −0.274721
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −22.0000 −1.00207
\(483\) −4.00000 −0.182006
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 38.0000 1.71492 0.857458 0.514554i \(-0.172042\pi\)
0.857458 + 0.514554i \(0.172042\pi\)
\(492\) 10.0000 0.450835
\(493\) −32.0000 −1.44121
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 10.0000 0.446322
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 1.00000 0.0444116
\(508\) −20.0000 −0.887357
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) 8.00000 0.352865
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 8.00000 0.350150
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) −16.0000 −0.696971
\(528\) 4.00000 0.174078
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) 10.0000 0.433148
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 2.00000 0.0863064
\(538\) −4.00000 −0.172452
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) −28.0000 −1.20270
\(543\) −22.0000 −0.944110
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) −2.00000 −0.0851257
\(553\) −16.0000 −0.680389
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 4.00000 0.169334
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) −10.0000 −0.421825
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 4.00000 0.167248
\(573\) −8.00000 −0.334205
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 8.00000 0.331611
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 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
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) −6.00000 −0.246598
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 24.0000 0.982255
\(598\) −2.00000 −0.0817861
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) −8.00000 −0.326056
\(603\) 8.00000 0.325785
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 20.0000 0.812444
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −4.00000 −0.160904
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) −16.0000 −0.641542
\(623\) −20.0000 −0.801283
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) −8.00000 −0.319489
\(628\) 10.0000 0.399043
\(629\) 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
\(637\) −3.00000 −0.118864
\(638\) 32.0000 1.26689
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) −12.0000 −0.469956
\(653\) 38.0000 1.48705 0.743527 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) −10.0000 −0.388661
\(663\) −4.00000 −0.155347
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −16.0000 −0.619522
\(668\) 12.0000 0.464294
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 2.00000 0.0771517
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 16.0000 0.614476
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 16.0000 0.612672
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 28.0000 1.06827
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) −14.0000 −0.532200
\(693\) 8.00000 0.303895
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 8.00000 0.303239
\(697\) −40.0000 −1.51511
\(698\) −28.0000 −1.05982
\(699\) −28.0000 −1.05906
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 1.00000 0.0377426
\(703\) 12.0000 0.452589
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 40.0000 1.50435
\(708\) −12.0000 −0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −10.0000 −0.374766
\(713\) −8.00000 −0.299602
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) −8.00000 −0.298765
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −15.0000 −0.558242
\(723\) −22.0000 −0.818189
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −2.00000 −0.0739221
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 32.0000 1.17874
\(738\) 10.0000 0.368105
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) −12.0000 −0.440534
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 12.0000 0.439057
\(748\) −16.0000 −0.585018
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 10.0000 0.364420
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −14.0000 −0.508503
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −46.0000 −1.66750 −0.833749 0.552143i \(-0.813810\pi\)
−0.833749 + 0.552143i \(0.813810\pi\)
\(762\) −20.0000 −0.724524
\(763\) 8.00000 0.289619
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) −12.0000 −0.433295
\(768\) 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
\(771\) 8.00000 0.288113
\(772\) −12.0000 −0.431889
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) −12.0000 −0.430498
\(778\) 20.0000 0.717035
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 8.00000 0.285897
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −10.0000 −0.356235
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 4.00000 0.142134
\(793\) −2.00000 −0.0710221
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −4.00000 −0.140807
\(808\) 20.0000 0.703598
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 16.0000 0.561490
\(813\) −28.0000 −0.982003
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 8.00000 0.279885
\(818\) −18.0000 −0.629355
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 2.00000 0.0697580
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 1.00000 0.0346688
\(833\) 12.0000 0.415775
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 4.00000 0.138260
\(838\) 14.0000 0.483622
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −16.0000 −0.551396
\(843\) −10.0000 −0.344418
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −6.00000 −0.206041
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 4.00000 0.136558
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 16.0000 0.544962
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) −1.00000 −0.0339618
\(868\) 8.00000 0.271538
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 4.00000 0.135457
\(873\) 8.00000 0.270759
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 8.00000 0.269987
\(879\) 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
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 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
\(891\) 4.00000 0.134005
\(892\) 22.0000 0.736614
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) −2.00000 −0.0667781
\(898\) −18.0000 −0.600668
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 40.0000 1.33185
\(903\) −8.00000 −0.266223
\(904\) 16.0000 0.532152
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −12.0000 −0.398234
\(909\) 20.0000 0.663358
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 48.0000 1.58857
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 28.0000 0.924641
\(918\) −4.00000 −0.132020
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) −2.00000 −0.0657241
\(927\) −4.00000 −0.131377
\(928\) 8.00000 0.262613
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −28.0000 −0.917170
\(933\) −16.0000 −0.523816
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 16.0000 0.522419
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 10.0000 0.325818
\(943\) −20.0000 −0.651290
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) −8.00000 −0.259281
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 32.0000 1.03441
\(958\) 24.0000 0.775405
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −6.00000 −0.193448
\(963\) −4.00000 −0.128898
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) 5.00000 0.160706
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 1.00000 0.0320750
\(973\) −32.0000 −1.02587
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −12.0000 −0.383718
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 38.0000 1.21263
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −14.0000 −0.443162
\(999\) −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.ba.1.1 1
3.2 odd 2 5850.2.a.s.1.1 1
5.2 odd 4 1950.2.e.m.1249.2 2
5.3 odd 4 1950.2.e.m.1249.1 2
5.4 even 2 390.2.a.b.1.1 1
15.2 even 4 5850.2.e.h.5149.1 2
15.8 even 4 5850.2.e.h.5149.2 2
15.14 odd 2 1170.2.a.j.1.1 1
20.19 odd 2 3120.2.a.y.1.1 1
60.59 even 2 9360.2.a.v.1.1 1
65.34 odd 4 5070.2.b.f.1351.1 2
65.44 odd 4 5070.2.b.f.1351.2 2
65.64 even 2 5070.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 5.4 even 2
1170.2.a.j.1.1 1 15.14 odd 2
1950.2.a.ba.1.1 1 1.1 even 1 trivial
1950.2.e.m.1249.1 2 5.3 odd 4
1950.2.e.m.1249.2 2 5.2 odd 4
3120.2.a.y.1.1 1 20.19 odd 2
5070.2.a.n.1.1 1 65.64 even 2
5070.2.b.f.1351.1 2 65.34 odd 4
5070.2.b.f.1351.2 2 65.44 odd 4
5850.2.a.s.1.1 1 3.2 odd 2
5850.2.e.h.5149.1 2 15.2 even 4
5850.2.e.h.5149.2 2 15.8 even 4
9360.2.a.v.1.1 1 60.59 even 2