Properties

Label 1170.2.a.d.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 130)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1170.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -4.00000 q^{28} -2.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -4.00000 q^{35} -2.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} +4.00000 q^{56} +2.00000 q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -12.0000 q^{67} -2.00000 q^{68} +4.00000 q^{70} -10.0000 q^{71} +10.0000 q^{73} +2.00000 q^{74} +6.00000 q^{76} -8.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} +10.0000 q^{82} -2.00000 q^{85} +10.0000 q^{86} -2.00000 q^{88} +14.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -12.0000 q^{94} +6.00000 q^{95} +14.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 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
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −18.0000 −1.77359 −0.886796 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −24.0000 −2.08106
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 18.0000 1.25412
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 40.0000 2.36113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 40.0000 2.30556
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 24.0000 1.36975 0.684876 0.728659i \(-0.259856\pi\)
0.684876 + 0.728659i \(0.259856\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) −10.0000 −0.530745
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 30.0000 1.56599 0.782994 0.622030i \(-0.213692\pi\)
0.782994 + 0.622030i \(0.213692\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) −18.0000 −0.886796
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) −24.0000 −1.15204
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −10.0000 −0.462745 −0.231372 0.972865i \(-0.574322\pi\)
−0.231372 + 0.972865i \(0.574322\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) −26.0000 −1.18921
\(479\) −2.00000 −0.0913823 −0.0456912 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 40.0000 1.79425
\(498\) 0 0
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −40.0000 −1.76950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) −24.0000 −1.04053
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −40.0000 −1.66957
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −40.0000 −1.63028
\(603\) 0 0
\(604\) 6.00000 0.244137
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −56.0000 −2.24359
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −14.0000 −0.555573
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 4.00000 0.158362
\(639\) 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
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 48.0000 1.87123
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 14.0000 0.544125
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) −20.0000 −0.773823
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −38.0000 −1.44559 −0.722794 0.691063i \(-0.757142\pi\)
−0.722794 + 0.691063i \(0.757142\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 56.0000 2.10610
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 10.0000 0.375293
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 36.0000 1.34821
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −6.00000 −0.223918
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 72.0000 2.68142
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) −10.0000 −0.370117
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −30.0000 −1.10732
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −2.00000 −0.0728357
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −6.00000 −0.217930
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 10.0000 0.361079
\(768\) 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
\(772\) 14.0000 0.503871
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) −60.0000 −2.14972
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −38.0000 −1.34857
\(795\) 0 0
\(796\) 0 0
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) 14.0000 0.492518
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −60.0000 −2.09913
\(818\) −34.0000 −1.18878
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) −40.0000 −1.39178
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −20.0000 −0.692129
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −10.0000 −0.340997
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 38.0000 1.29129
\(867\) 0 0
\(868\) 24.0000 0.814613
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 36.0000 1.21772
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 38.0000 1.27880 0.639401 0.768874i \(-0.279182\pi\)
0.639401 + 0.768874i \(0.279182\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −14.0000 −0.470339
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 56.0000 1.87818
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 4.00000 0.133930
\(893\) 72.0000 2.40939
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 20.0000 0.665927
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −14.0000 −0.464862 −0.232431 0.972613i \(-0.574668\pi\)
−0.232431 + 0.972613i \(0.574668\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 6.00000 0.197814
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) −48.0000 −1.56726
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 60.0000 1.95387
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 0 0
\(958\) 2.00000 0.0646171
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 32.0000 1.02587
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 28.0000 0.894884
\(980\) 9.00000 0.287494
\(981\) 0 0
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −40.0000 −1.26872
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −38.0000 −1.20287
\(999\) 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.d.1.1 1
3.2 odd 2 130.2.a.c.1.1 1
4.3 odd 2 9360.2.a.by.1.1 1
5.2 odd 4 5850.2.e.u.5149.1 2
5.3 odd 4 5850.2.e.u.5149.2 2
5.4 even 2 5850.2.a.cb.1.1 1
12.11 even 2 1040.2.a.b.1.1 1
15.2 even 4 650.2.b.g.599.2 2
15.8 even 4 650.2.b.g.599.1 2
15.14 odd 2 650.2.a.c.1.1 1
21.20 even 2 6370.2.a.l.1.1 1
24.5 odd 2 4160.2.a.c.1.1 1
24.11 even 2 4160.2.a.t.1.1 1
39.2 even 12 1690.2.l.a.1161.2 4
39.5 even 4 1690.2.d.e.1351.1 2
39.8 even 4 1690.2.d.e.1351.2 2
39.11 even 12 1690.2.l.a.1161.1 4
39.17 odd 6 1690.2.e.g.991.1 2
39.20 even 12 1690.2.l.a.361.2 4
39.23 odd 6 1690.2.e.g.191.1 2
39.29 odd 6 1690.2.e.a.191.1 2
39.32 even 12 1690.2.l.a.361.1 4
39.35 odd 6 1690.2.e.a.991.1 2
39.38 odd 2 1690.2.a.e.1.1 1
60.59 even 2 5200.2.a.bd.1.1 1
195.194 odd 2 8450.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.2.a.c.1.1 1 3.2 odd 2
650.2.a.c.1.1 1 15.14 odd 2
650.2.b.g.599.1 2 15.8 even 4
650.2.b.g.599.2 2 15.2 even 4
1040.2.a.b.1.1 1 12.11 even 2
1170.2.a.d.1.1 1 1.1 even 1 trivial
1690.2.a.e.1.1 1 39.38 odd 2
1690.2.d.e.1351.1 2 39.5 even 4
1690.2.d.e.1351.2 2 39.8 even 4
1690.2.e.a.191.1 2 39.29 odd 6
1690.2.e.a.991.1 2 39.35 odd 6
1690.2.e.g.191.1 2 39.23 odd 6
1690.2.e.g.991.1 2 39.17 odd 6
1690.2.l.a.361.1 4 39.32 even 12
1690.2.l.a.361.2 4 39.20 even 12
1690.2.l.a.1161.1 4 39.11 even 12
1690.2.l.a.1161.2 4 39.2 even 12
4160.2.a.c.1.1 1 24.5 odd 2
4160.2.a.t.1.1 1 24.11 even 2
5200.2.a.bd.1.1 1 60.59 even 2
5850.2.a.cb.1.1 1 5.4 even 2
5850.2.e.u.5149.1 2 5.2 odd 4
5850.2.e.u.5149.2 2 5.3 odd 4
6370.2.a.l.1.1 1 21.20 even 2
8450.2.a.n.1.1 1 195.194 odd 2
9360.2.a.by.1.1 1 4.3 odd 2