Properties

Label 1694.2.a.g.1.1
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5266581024\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1694.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{10} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{18} +6.00000 q^{19} -4.00000 q^{20} +4.00000 q^{23} +11.0000 q^{25} -2.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -4.00000 q^{40} -4.00000 q^{41} +8.00000 q^{43} +12.0000 q^{45} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} -2.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} -4.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} -4.00000 q^{80} +9.00000 q^{81} -4.00000 q^{82} +6.00000 q^{83} -16.0000 q^{85} +8.00000 q^{86} -6.00000 q^{89} +12.0000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +2.00000 q^{94} -24.0000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) −4.00000 −1.26491
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −3.00000 −0.707107
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −4.00000 −0.676123
\(36\) −3.00000 −0.500000
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 12.0000 1.78885
\(46\) 4.00000 0.589768
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −2.00000 −0.254000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.00000 −0.353553
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 9.00000 1.00000
\(82\) −4.00000 −0.441726
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 12.0000 1.26491
\(91\) −2.00000 −0.209657
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 18.0000 1.77359 0.886796 0.462160i \(-0.152926\pi\)
0.886796 + 0.462160i \(0.152926\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) −12.0000 −1.10469
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 0 0
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −24.0000 −2.14663
\(126\) −3.00000 −0.267261
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) −8.00000 −0.664364
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 6.00000 0.486664
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) 4.00000 0.315244
\(162\) 9.00000 0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −16.0000 −1.22714
\(171\) −18.0000 −1.37649
\(172\) 8.00000 0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 12.0000 0.894427
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −40.0000 −2.94086
\(186\) 0 0
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 11.0000 0.777817
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 16.0000 1.11749
\(206\) 18.0000 1.25412
\(207\) −12.0000 −0.834058
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) −32.0000 −2.18238
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 1.00000 0.0668153
\(225\) −33.0000 −2.20000
\(226\) 14.0000 0.931266
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −30.0000 −1.96537 −0.982683 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 6.00000 0.392232
\(235\) −8.00000 −0.521862
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) −4.00000 −0.255551
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 8.00000 0.496139
\(261\) −6.00000 −0.371391
\(262\) −6.00000 −0.370681
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) −14.0000 −0.839664
\(279\) 6.00000 0.359211
\(280\) −4.00000 −0.239046
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) −3.00000 −0.176777
\(289\) −1.00000 −0.0588235
\(290\) −8.00000 −0.469776
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 48.0000 2.79467
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 24.0000 1.38104
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) −56.0000 −3.20655
\(306\) −12.0000 −0.685994
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) −8.00000 −0.451466
\(315\) 12.0000 0.676123
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.00000 −0.223607
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 24.0000 1.33540
\(324\) 9.00000 0.500000
\(325\) −22.0000 −1.22034
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 6.00000 0.329293
\(333\) −30.0000 −1.64399
\(334\) −4.00000 −0.218870
\(335\) 48.0000 2.62252
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −16.0000 −0.867722
\(341\) 0 0
\(342\) −18.0000 −0.973329
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 11.0000 0.587975
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 32.0000 1.69838
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 12.0000 0.632456
\(361\) 17.0000 0.894737
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 4.00000 0.208514
\(369\) 12.0000 0.624695
\(370\) −40.0000 −2.07950
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −24.0000 −1.23117
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) −10.0000 −0.510976 −0.255488 0.966812i \(-0.582236\pi\)
−0.255488 + 0.966812i \(0.582236\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −24.0000 −1.21999
\(388\) −14.0000 −0.710742
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −6.00000 −0.298511
\(405\) −36.0000 −1.78885
\(406\) 2.00000 0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 16.0000 0.790184
\(411\) 0 0
\(412\) 18.0000 0.886796
\(413\) −12.0000 −0.590481
\(414\) −12.0000 −0.589768
\(415\) −24.0000 −1.17811
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −32.0000 −1.56330 −0.781651 0.623716i \(-0.785622\pi\)
−0.781651 + 0.623716i \(0.785622\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 8.00000 0.389434
\(423\) −6.00000 −0.291730
\(424\) 6.00000 0.291386
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) −32.0000 −1.54318
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −8.00000 −0.380521
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −33.0000 −1.55563
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 2.00000 0.0938647
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) −16.0000 −0.746004
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 6.00000 0.277350
\(469\) −12.0000 −0.554109
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 66.0000 3.02829
\(476\) 4.00000 0.183340
\(477\) −18.0000 −0.824163
\(478\) −16.0000 −0.731823
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −12.0000 −0.546585
\(483\) 0 0
\(484\) 0 0
\(485\) 56.0000 2.54283
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 14.0000 0.633750
\(489\) 0 0
\(490\) −4.00000 −0.180702
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −3.00000 −0.133631
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −72.0000 −3.17270
\(516\) 0 0
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −6.00000 −0.262613
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −24.0000 −1.04249
\(531\) 36.0000 1.56227
\(532\) 6.00000 0.260133
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −64.0000 −2.76696
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) −56.0000 −2.39878
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 6.00000 0.256307
\(549\) −42.0000 −1.79252
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 30.0000 1.27458
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 6.00000 0.254000
\(559\) −16.0000 −0.676728
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 34.0000 1.43293 0.716465 0.697623i \(-0.245759\pi\)
0.716465 + 0.697623i \(0.245759\pi\)
\(564\) 0 0
\(565\) −56.0000 −2.35594
\(566\) −6.00000 −0.252199
\(567\) 9.00000 0.377964
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 44.0000 1.83493
\(576\) −3.00000 −0.125000
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −8.00000 −0.332182
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) −24.0000 −0.992278
\(586\) −18.0000 −0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 48.0000 1.97613
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 8.00000 0.326056
\(603\) 36.0000 1.46603
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −56.0000 −2.26737
\(611\) −4.00000 −0.161823
\(612\) −12.0000 −0.485071
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 14.0000 0.561349
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 40.0000 1.59490
\(630\) 12.0000 0.478091
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 32.0000 1.26988
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) −4.00000 −0.158114
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) −22.0000 −0.862911
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) −4.00000 −0.156174
\(657\) 12.0000 0.468165
\(658\) 2.00000 0.0779681
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −24.0000 −0.930680
\(666\) −30.0000 −1.16248
\(667\) 8.00000 0.309761
\(668\) −4.00000 −0.154765
\(669\) 0 0
\(670\) 48.0000 1.85440
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −16.0000 −0.613572
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −18.0000 −0.688247
\(685\) −24.0000 −0.916993
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) 56.0000 2.12420
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 11.0000 0.415761
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 60.0000 2.26294
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 32.0000 1.20094
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 12.0000 0.447214
\(721\) 18.0000 0.670355
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) 22.0000 0.817059
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) −2.00000 −0.0741249
\(729\) −27.0000 −1.00000
\(730\) 16.0000 0.592187
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −40.0000 −1.47043
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 10.0000 0.366126
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) −96.0000 −3.49380
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) −4.00000 −0.144715
\(765\) 48.0000 1.73544
\(766\) −10.0000 −0.361315
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −48.0000 −1.72644 −0.863220 0.504828i \(-0.831556\pi\)
−0.863220 + 0.504828i \(0.831556\pi\)
\(774\) −24.0000 −0.862662
\(775\) −22.0000 −0.790263
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 24.0000 0.851728
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −16.0000 −0.566749 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 11.0000 0.388909
\(801\) 18.0000 0.635999
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −36.0000 −1.26491
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −16.0000 −0.559427
\(819\) 6.00000 0.209657
\(820\) 16.0000 0.558744
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) −12.0000 −0.417029
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) −24.0000 −0.833052
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) −32.0000 −1.10542
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 36.0000 1.23844
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 44.0000 1.50919
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 14.0000 0.479070
\(855\) 72.0000 2.46235
\(856\) 16.0000 0.546869
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −32.0000 −1.09119
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 14.0000 0.474100
\(873\) 42.0000 1.42148
\(874\) 24.0000 0.811812
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −28.0000 −0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −3.00000 −0.101015
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 24.0000 0.804482
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −4.00000 −0.133407
\(900\) −33.0000 −1.10000
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −80.0000 −2.65929
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 2.00000 0.0663723
\(909\) 18.0000 0.597022
\(910\) 8.00000 0.265197
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −16.0000 −0.527504
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 110.000 3.61678
\(926\) −32.0000 −1.05159
\(927\) −54.0000 −1.77359
\(928\) 2.00000 0.0656532
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −30.0000 −0.982683
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −12.0000 −0.392023 −0.196011 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 66.0000 2.14132
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −18.0000 −0.582772
\(955\) 16.0000 0.517748
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −20.0000 −0.644826
\(963\) −48.0000 −1.54678
\(964\) −12.0000 −0.386494
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 56.0000 1.79805
\(971\) 56.0000 1.79713 0.898563 0.438845i \(-0.144612\pi\)
0.898563 + 0.438845i \(0.144612\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.00000 −0.127775
\(981\) −42.0000 −1.34096
\(982\) 36.0000 1.14881
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 56.0000 1.77532
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 44.0000 1.39280
\(999\) 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 1694.2.a.g.1.1 1
11.10 odd 2 154.2.a.a.1.1 1
33.32 even 2 1386.2.a.l.1.1 1
44.43 even 2 1232.2.a.e.1.1 1
55.32 even 4 3850.2.c.j.1849.1 2
55.43 even 4 3850.2.c.j.1849.2 2
55.54 odd 2 3850.2.a.u.1.1 1
77.10 even 6 1078.2.e.i.177.1 2
77.32 odd 6 1078.2.e.j.177.1 2
77.54 even 6 1078.2.e.i.67.1 2
77.65 odd 6 1078.2.e.j.67.1 2
77.76 even 2 1078.2.a.d.1.1 1
88.21 odd 2 4928.2.a.v.1.1 1
88.43 even 2 4928.2.a.w.1.1 1
231.230 odd 2 9702.2.a.ba.1.1 1
308.307 odd 2 8624.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.a.1.1 1 11.10 odd 2
1078.2.a.d.1.1 1 77.76 even 2
1078.2.e.i.67.1 2 77.54 even 6
1078.2.e.i.177.1 2 77.10 even 6
1078.2.e.j.67.1 2 77.65 odd 6
1078.2.e.j.177.1 2 77.32 odd 6
1232.2.a.e.1.1 1 44.43 even 2
1386.2.a.l.1.1 1 33.32 even 2
1694.2.a.g.1.1 1 1.1 even 1 trivial
3850.2.a.u.1.1 1 55.54 odd 2
3850.2.c.j.1849.1 2 55.32 even 4
3850.2.c.j.1849.2 2 55.43 even 4
4928.2.a.v.1.1 1 88.21 odd 2
4928.2.a.w.1.1 1 88.43 even 2
8624.2.a.r.1.1 1 308.307 odd 2
9702.2.a.ba.1.1 1 231.230 odd 2