Properties

Label 1386.2.a.l.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
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 \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} +10.0000 q^{37} -6.00000 q^{38} +4.00000 q^{40} -4.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +4.00000 q^{73} +10.0000 q^{74} -6.00000 q^{76} -1.00000 q^{77} +4.00000 q^{80} -4.00000 q^{82} +6.00000 q^{83} +16.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -2.00000 q^{94} -24.0000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −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 \(0.338794\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 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
\(36\) 0 0
\(37\) 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 \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 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 \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 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 \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) −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
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −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 \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 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 \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 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 \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −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
\(162\) 0 0
\(163\) 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
\(171\) 0 0
\(172\) −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
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 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
\(187\) 4.00000 0.292509
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −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
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −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
\(220\) 4.00000 0.269680
\(221\) 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
\(225\) 0 0
\(226\) −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 \(0.940675\pi\)
\(234\) 0 0
\(235\) −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 \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 1.00000 0.0642824
\(243\) 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 \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) −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 \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) −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 \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −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
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 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
\(333\) 0 0
\(334\) −4.00000 −0.218870
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 16.0000 0.867722
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) −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 \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −11.0000 −0.587975
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 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 \(0.361250\pi\)
\(360\) 0 0
\(361\) 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
\(369\) 0 0
\(370\) 40.0000 2.07950
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 4.00000 0.206835
\(375\) 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 \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 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 \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) −16.0000 −0.790184
\(411\) 0 0
\(412\) 18.0000 0.886796
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 32.0000 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −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 \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 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 \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) −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 \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 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 \(0.5\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −66.0000 −3.02829
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) −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
\(484\) 1.00000 0.0454545
\(485\) −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 \(0.203460\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 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
\(517\) −2.00000 −0.0879599
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −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
\(531\) 0 0
\(532\) 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
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −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 \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 11.0000 0.469042
\(551\) −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 \(0.595853\pi\)
\(558\) 0 0
\(559\) −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
\(567\) 0 0
\(568\) 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 \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) −44.0000 −1.83493
\(576\) 0 0
\(577\) 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
\(583\) −6.00000 −0.248495
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 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 \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) −24.0000 −0.976546
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −56.0000 −2.26737
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) −10.0000 −0.403567
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 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
\(630\) 0 0
\(631\) −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
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 4.00000 0.158114
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 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 \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 22.0000 0.862911
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) 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
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −4.00000 −0.154765
\(669\) 0 0
\(670\) −48.0000 −1.85440
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) −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
\(682\) −2.00000 −0.0765840
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −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
\(704\) 1.00000 0.0376889
\(705\) 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
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) −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
\(729\) 0 0
\(730\) 16.0000 0.592187
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 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
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) −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
\(765\) 0 0
\(766\) 10.0000 0.361315
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −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
\(781\) 8.00000 0.286263
\(782\) −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 \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −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 \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 11.0000 0.388909
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 4.00000 0.141157
\(804\) 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 \(0.323177\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 10.0000 0.350500
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 16.0000 0.559427
\(819\) 0 0
\(820\) −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 \(0.613049\pi\)
\(828\) 0 0
\(829\) −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
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 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
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −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 \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 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 \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 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
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −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
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) −4.00000 −0.133185
\(903\) 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
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 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 \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −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
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −30.0000 −0.982683
\(933\) 0 0
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 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
\(946\) −8.00000 −0.260102
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 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 \(0.384030\pi\)
\(954\) 0 0
\(955\) 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
\(963\) 0 0
\(964\) 12.0000 0.386494
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −56.0000 −1.79805
\(971\) −56.0000 −1.79713 −0.898563 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) 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 \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 4.00000 0.127775
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 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 \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 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 1386.2.a.l.1.1 1
3.2 odd 2 154.2.a.a.1.1 1
7.6 odd 2 9702.2.a.ba.1.1 1
12.11 even 2 1232.2.a.e.1.1 1
15.2 even 4 3850.2.c.j.1849.1 2
15.8 even 4 3850.2.c.j.1849.2 2
15.14 odd 2 3850.2.a.u.1.1 1
21.2 odd 6 1078.2.e.j.67.1 2
21.5 even 6 1078.2.e.i.67.1 2
21.11 odd 6 1078.2.e.j.177.1 2
21.17 even 6 1078.2.e.i.177.1 2
21.20 even 2 1078.2.a.d.1.1 1
24.5 odd 2 4928.2.a.v.1.1 1
24.11 even 2 4928.2.a.w.1.1 1
33.32 even 2 1694.2.a.g.1.1 1
84.83 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 3.2 odd 2
1078.2.a.d.1.1 1 21.20 even 2
1078.2.e.i.67.1 2 21.5 even 6
1078.2.e.i.177.1 2 21.17 even 6
1078.2.e.j.67.1 2 21.2 odd 6
1078.2.e.j.177.1 2 21.11 odd 6
1232.2.a.e.1.1 1 12.11 even 2
1386.2.a.l.1.1 1 1.1 even 1 trivial
1694.2.a.g.1.1 1 33.32 even 2
3850.2.a.u.1.1 1 15.14 odd 2
3850.2.c.j.1849.1 2 15.2 even 4
3850.2.c.j.1849.2 2 15.8 even 4
4928.2.a.v.1.1 1 24.5 odd 2
4928.2.a.w.1.1 1 24.11 even 2
8624.2.a.r.1.1 1 84.83 odd 2
9702.2.a.ba.1.1 1 7.6 odd 2