Properties

Label 1170.2.a.j.1.1
Level $1170$
Weight $2$
Character 1170.1
Self dual yes
Analytic conductor $9.342$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.34249703649\)
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 \(\chi\) \(=\) 1170.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -2.00000 q^{19} -1.00000 q^{20} -4.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} -8.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} +2.00000 q^{35} +6.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{46} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +4.00000 q^{55} -2.00000 q^{56} -8.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -8.00000 q^{67} -4.00000 q^{68} +2.00000 q^{70} +6.00000 q^{74} -2.00000 q^{76} +8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -10.0000 q^{82} +12.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -4.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -2.00000 q^{92} +2.00000 q^{95} -8.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −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 \(0.661206\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 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
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −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 \(0.561934\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) 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 \(0.472772\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −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 \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −2.00000 −0.147442
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) −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 \(0.615940\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −20.0000 −1.40720
\(203\) 16.0000 1.12298
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −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
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) −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 \(0.630377\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) −28.0000 −1.83434 −0.917170 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 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 \(0.419729\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −14.0000 −0.864923
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 22.0000 1.27443
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −4.00000 −0.230174
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −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 \(0.482113\pi\)
\(318\) 0 0
\(319\) 32.0000 1.79166
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) −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
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 12.0000 0.610784
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −20.0000 −0.995037
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 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
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −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 \(0.438856\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 16.0000 0.752577
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 28.0000 1.30835
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) −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 \(0.589555\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −38.0000 −1.71492 −0.857458 0.514554i \(-0.827958\pi\)
−0.857458 + 0.514554i \(0.827958\pi\)
\(492\) 0 0
\(493\) 32.0000 1.44121
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −10.0000 −0.446322
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) 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
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 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 \(0.432049\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 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 \(0.579657\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 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 \(0.777217\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) −20.0000 −0.793676
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 32.0000 1.26689
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 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 \(0.384880\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 38.0000 1.48705 0.743527 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 0 0
\(655\) 14.0000 0.547025
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −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 \(0.361060\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) −28.0000 −1.05982
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 40.0000 1.50435
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −22.0000 −0.806018
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −14.0000 −0.508503
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) 12.0000 0.431889
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) 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 \(0.232994\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) −20.0000 −0.703598
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 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 \(0.222732\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 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 \(0.411893\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 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 \(0.467882\pi\)
\(888\) 0 0
\(889\) −40.0000 −1.34156
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) 0 0
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −32.0000 −1.06726
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 16.0000 0.532152
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 28.0000 0.924641
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 2.00000 0.0657241
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −28.0000 −0.917170
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 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 \(0.479298\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) 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 \(0.734500\pi\)
\(978\) 0 0
\(979\) −40.0000 −1.27841
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −38.0000 −1.21263
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −14.0000 −0.443162
\(999\) 0 0
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 1170.2.a.j.1.1 1
3.2 odd 2 390.2.a.b.1.1 1
4.3 odd 2 9360.2.a.v.1.1 1
5.2 odd 4 5850.2.e.h.5149.2 2
5.3 odd 4 5850.2.e.h.5149.1 2
5.4 even 2 5850.2.a.s.1.1 1
12.11 even 2 3120.2.a.y.1.1 1
15.2 even 4 1950.2.e.m.1249.1 2
15.8 even 4 1950.2.e.m.1249.2 2
15.14 odd 2 1950.2.a.ba.1.1 1
39.5 even 4 5070.2.b.f.1351.2 2
39.8 even 4 5070.2.b.f.1351.1 2
39.38 odd 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 3.2 odd 2
1170.2.a.j.1.1 1 1.1 even 1 trivial
1950.2.a.ba.1.1 1 15.14 odd 2
1950.2.e.m.1249.1 2 15.2 even 4
1950.2.e.m.1249.2 2 15.8 even 4
3120.2.a.y.1.1 1 12.11 even 2
5070.2.a.n.1.1 1 39.38 odd 2
5070.2.b.f.1351.1 2 39.8 even 4
5070.2.b.f.1351.2 2 39.5 even 4
5850.2.a.s.1.1 1 5.4 even 2
5850.2.e.h.5149.1 2 5.3 odd 4
5850.2.e.h.5149.2 2 5.2 odd 4
9360.2.a.v.1.1 1 4.3 odd 2