Properties

Label 7098.2.a.bc.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} +1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{40} -1.00000 q^{42} -4.00000 q^{43} +2.00000 q^{45} +6.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +4.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{59} +2.00000 q^{60} +12.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} +2.00000 q^{68} +6.00000 q^{69} -2.00000 q^{70} +1.00000 q^{72} +14.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{80} +1.00000 q^{81} -14.0000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -4.00000 q^{89} +2.00000 q^{90} +6.00000 q^{92} -8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 6.00000 0.884652
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 2.00000 0.258199
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) −1.00000 −0.109109
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 2.00000 0.198030
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 4.00000 0.388514
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −2.00000 −0.183340
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) 12.0000 1.08643
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) −1.00000 −0.0890871
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −2.00000 −0.172774
\(135\) 2.00000 0.172133
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 6.00000 0.510754
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −2.00000 −0.169031
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 2.00000 0.158114
\(161\) −6.00000 −0.472866
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) −4.00000 −0.299813
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 2.00000 0.149071
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 6.00000 0.442326
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 8.00000 0.580381
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 12.0000 0.768221
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) −12.0000 −0.758947
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) −4.00000 −0.245256
\(267\) −4.00000 −0.244796
\(268\) −2.00000 −0.122169
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 2.00000 0.121716
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) −8.00000 −0.476393
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 14.0000 0.819288
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) 24.0000 1.37424
\(306\) 2.00000 0.114332
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 8.00000 0.451466
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 4.00000 0.224309
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) −6.00000 −0.334367
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.0000 0.775388
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −14.0000 −0.768350
\(333\) 2.00000 0.109599
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 12.0000 0.646058
\(346\) 6.00000 0.322562
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) −2.00000 −0.105851
\(358\) 20.0000 1.05703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −12.0000 −0.630706
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 28.0000 1.46559
\(366\) 12.0000 0.627250
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 8.00000 0.410391
\(381\) 12.0000 0.614779
\(382\) 2.00000 0.102329
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 10.0000 0.501255
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 2.00000 0.0990148
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −14.0000 −0.689730
\(413\) −6.00000 −0.295241
\(414\) 6.00000 0.294884
\(415\) −28.0000 −1.37447
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −28.0000 −1.36302
\(423\) −8.00000 −0.388973
\(424\) 4.00000 0.194257
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 24.0000 1.14808
\(438\) 14.0000 0.668946
\(439\) −30.0000 −1.43182 −0.715911 0.698192i \(-0.753988\pi\)
−0.715911 + 0.698192i \(0.753988\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) −8.00000 −0.379236
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) −1.00000 −0.0472456
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 12.0000 0.559503
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) −16.0000 −0.738025
\(471\) 8.00000 0.368621
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 4.00000 0.183147
\(478\) −16.0000 −0.731823
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) −6.00000 −0.273009
\(484\) −11.0000 −0.500000
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 12.0000 0.543214
\(489\) 14.0000 0.633102
\(490\) 2.00000 0.0903508
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −14.0000 −0.627355
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −12.0000 −0.536656
\(501\) 12.0000 0.536120
\(502\) 28.0000 1.24970
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 4.00000 0.177123
\(511\) −14.0000 −0.619324
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −18.0000 −0.793946
\(515\) −28.0000 −1.23383
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 8.00000 0.347498
\(531\) 6.00000 0.260378
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) −24.0000 −1.03761
\(536\) −2.00000 −0.0863868
\(537\) 20.0000 0.863064
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) −12.0000 −0.514969
\(544\) 2.00000 0.0857493
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000 0.0854358
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 4.00000 0.169791
\(556\) 20.0000 0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −8.00000 −0.336861
\(565\) 28.0000 1.17797
\(566\) 16.0000 0.672530
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 8.00000 0.335083
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 2.00000 0.0835512
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 14.0000 0.580818
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −6.00000 −0.245770
\(597\) 10.0000 0.409273
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 4.00000 0.163028
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) −2.00000 −0.0812444
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −14.0000 −0.563163
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −12.0000 −0.481156
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 4.00000 0.159490
\(630\) −2.00000 −0.0796819
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) −28.0000 −1.11290
\(634\) −2.00000 −0.0794301
\(635\) 24.0000 0.952411
\(636\) 4.00000 0.158610
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) −12.0000 −0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −6.00000 −0.236433
\(645\) −8.00000 −0.315000
\(646\) 8.00000 0.314756
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) −6.00000 −0.234619
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 8.00000 0.311872
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) −8.00000 −0.310227
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 16.0000 0.618596
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 22.0000 0.847408
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 14.0000 0.537667
\(679\) 2.00000 0.0767530
\(680\) 4.00000 0.153393
\(681\) −22.0000 −0.843042
\(682\) 0 0
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 4.00000 0.152944
\(685\) 4.00000 0.152832
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 12.0000 0.456832
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 40.0000 1.51729
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) −24.0000 −0.903252
\(707\) 2.00000 0.0752177
\(708\) 6.00000 0.225494
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.00000 −0.149906
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) −16.0000 −0.597531
\(718\) −24.0000 −0.895672
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 2.00000 0.0745356
\(721\) 14.0000 0.521387
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) −8.00000 −0.295891
\(732\) 12.0000 0.443533
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −22.0000 −0.812035
\(735\) 2.00000 0.0737711
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 34.0000 1.24483
\(747\) −14.0000 −0.512233
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −12.0000 −0.438178
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −8.00000 −0.291730
\(753\) 28.0000 1.02038
\(754\) 0 0
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −26.0000 −0.944363
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 12.0000 0.434714
\(763\) 6.00000 0.217215
\(764\) 2.00000 0.0723575
\(765\) 4.00000 0.144620
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 4.00000 0.143963
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −2.00000 −0.0717496
\(778\) −20.0000 −0.717035
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 12.0000 0.429119
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 16.0000 0.571064
\(786\) 12.0000 0.428026
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 18.0000 0.641223
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 8.00000 0.283731
\(796\) 10.0000 0.354441
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −4.00000 −0.141598
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) −4.00000 −0.141333
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) −2.00000 −0.0703598
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 2.00000 0.0702728
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 28.0000 0.980797
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 2.00000 0.0697580
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 6.00000 0.208514
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) −28.0000 −0.971894
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 20.0000 0.692543
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) −30.0000 −1.03387
\(843\) −30.0000 −1.03325
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 11.0000 0.377964
\(848\) 4.00000 0.137361
\(849\) 16.0000 0.549119
\(850\) −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 \(0.646836\pi\)
\(854\) −12.0000 −0.410632
\(855\) 8.00000 0.273594
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0000 0.408012
\(866\) 26.0000 0.883516
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 24.0000 0.811812
\(875\) 12.0000 0.405674
\(876\) 14.0000 0.473016
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −30.0000 −1.01245
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 1.00000 0.0336718
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 4.00000 0.134383
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 2.00000 0.0671156
\(889\) −12.0000 −0.402467
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) −32.0000 −1.07084
\(894\) −6.00000 −0.200670
\(895\) 40.0000 1.33705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 14.0000 0.465633
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −22.0000 −0.730096
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 8.00000 0.264616
\(915\) 24.0000 0.793416
\(916\) −14.0000 −0.462573
\(917\) −12.0000 −0.396275
\(918\) 2.00000 0.0660098
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 12.0000 0.395628
\(921\) −28.0000 −0.922631
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −16.0000 −0.525793
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) −12.0000 −0.392862
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 2.00000 0.0653023
\(939\) −6.00000 −0.195803
\(940\) −16.0000 −0.521862
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 8.00000 0.260654
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) −2.00000 −0.0648544
\(952\) −2.00000 −0.0648204
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 4.00000 0.129505
\(955\) 4.00000 0.129437
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −2.00000 −0.0645834
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 8.00000 0.257529
\(966\) −6.00000 −0.193047
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −11.0000 −0.353553
\(969\) 8.00000 0.256997
\(970\) −4.00000 −0.128432
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 1.00000 0.0320750
\(973\) −20.0000 −0.641171
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 14.0000 0.447671
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) −6.00000 −0.191565
\(982\) 8.00000 0.255290
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) −14.0000 −0.443607
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 14.0000 0.443162
\(999\) 2.00000 0.0632772
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 7098.2.a.bc.1.1 1
13.5 odd 4 546.2.c.b.337.1 2
13.8 odd 4 546.2.c.b.337.2 yes 2
13.12 even 2 7098.2.a.k.1.1 1
39.5 even 4 1638.2.c.b.883.2 2
39.8 even 4 1638.2.c.b.883.1 2
52.31 even 4 4368.2.h.f.337.1 2
52.47 even 4 4368.2.h.f.337.2 2
91.34 even 4 3822.2.c.c.883.2 2
91.83 even 4 3822.2.c.c.883.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.b.337.1 2 13.5 odd 4
546.2.c.b.337.2 yes 2 13.8 odd 4
1638.2.c.b.883.1 2 39.8 even 4
1638.2.c.b.883.2 2 39.5 even 4
3822.2.c.c.883.1 2 91.83 even 4
3822.2.c.c.883.2 2 91.34 even 4
4368.2.h.f.337.1 2 52.31 even 4
4368.2.h.f.337.2 2 52.47 even 4
7098.2.a.k.1.1 1 13.12 even 2
7098.2.a.bc.1.1 1 1.1 even 1 trivial