Properties

Label 3042.2.a.n.1.1
Level $3042$
Weight $2$
Character 3042.1
Self dual yes
Analytic conductor $24.290$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3042.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{28} +10.0000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +8.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{53} -2.00000 q^{56} +10.0000 q^{58} -4.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +2.00000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -4.00000 q^{73} +8.00000 q^{74} +6.00000 q^{76} +2.00000 q^{80} +10.0000 q^{82} -4.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} +12.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 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 \(0.258380\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 20.0000 1.66091
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) −20.0000 −1.60644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 16.0000 1.17634
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) −4.00000 −0.260378
\(237\) 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
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) −10.0000 −0.635001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −12.0000 −0.735767
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −20.0000 −1.19952
\(279\) 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
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 20.0000 1.17444
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −20.0000 −1.13592
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 16.0000 0.831800
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 0 0
\(388\) 12.0000 0.609208
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 0 0
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 20.0000 0.987730
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −28.0000 −1.24970
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) 0 0
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) −10.0000 −0.429537
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) 0 0
\(551\) 60.0000 2.55609
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −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
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 0 0
\(565\) −28.0000 −1.17797
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 20.0000 0.830455
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 0 0
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) −20.0000 −0.803219
\(621\) 0 0
\(622\) −28.0000 −1.12270
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −6.00000 −0.236617 −0.118308 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) −16.0000 −0.605609
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) −36.0000 −1.35201 −0.676004 0.736898i \(-0.736290\pi\)
−0.676004 + 0.736898i \(0.736290\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −4.00000 −0.149279
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −10.0000 −0.371391
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 16.0000 0.588172
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) −6.00000 −0.219676
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −34.0000 −1.23494
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 0 0
\(768\) 0 0
\(769\) −24.0000 −0.865462 −0.432731 0.901523i \(-0.642450\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.0000 0.575853
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) 0 0
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) 2.00000 0.0703598
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) −10.0000 −0.351147 −0.175574 0.984466i \(-0.556178\pi\)
−0.175574 + 0.984466i \(0.556178\pi\)
\(812\) −20.0000 −0.701862
\(813\) 0 0
\(814\) 0 0
\(815\) −28.0000 −0.980797
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) −40.0000 −1.38178
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000 0.755929
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 56.0000 1.91740 0.958702 0.284413i \(-0.0917988\pi\)
0.958702 + 0.284413i \(0.0917988\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −20.0000 −0.681203
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 26.0000 0.883516
\(867\) 0 0
\(868\) 20.0000 0.678844
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 72.0000 2.40939
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) −100.000 −3.33519
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −44.0000 −1.46261
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 6.00000 0.197172
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) 32.0000 1.03012
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 40.0000 1.28234
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −44.0000 −1.40196
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −10.0000 −0.317500
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −14.0000 −0.443162
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3042.2.a.n.1.1 1
3.2 odd 2 1014.2.a.b.1.1 1
12.11 even 2 8112.2.a.g.1.1 1
13.5 odd 4 234.2.b.a.181.1 2
13.8 odd 4 234.2.b.a.181.2 2
13.12 even 2 3042.2.a.c.1.1 1
39.2 even 12 1014.2.i.c.823.1 4
39.5 even 4 78.2.b.a.25.2 yes 2
39.8 even 4 78.2.b.a.25.1 2
39.11 even 12 1014.2.i.c.823.2 4
39.17 odd 6 1014.2.e.b.991.1 2
39.20 even 12 1014.2.i.c.361.1 4
39.23 odd 6 1014.2.e.b.529.1 2
39.29 odd 6 1014.2.e.e.529.1 2
39.32 even 12 1014.2.i.c.361.2 4
39.35 odd 6 1014.2.e.e.991.1 2
39.38 odd 2 1014.2.a.g.1.1 1
52.31 even 4 1872.2.c.b.1585.1 2
52.47 even 4 1872.2.c.b.1585.2 2
156.47 odd 4 624.2.c.a.337.1 2
156.83 odd 4 624.2.c.a.337.2 2
156.155 even 2 8112.2.a.j.1.1 1
195.8 odd 4 1950.2.f.d.649.2 2
195.44 even 4 1950.2.b.c.1351.1 2
195.47 odd 4 1950.2.f.g.649.1 2
195.83 odd 4 1950.2.f.g.649.2 2
195.122 odd 4 1950.2.f.d.649.1 2
195.164 even 4 1950.2.b.c.1351.2 2
273.83 odd 4 3822.2.c.d.883.2 2
273.125 odd 4 3822.2.c.d.883.1 2
312.5 even 4 2496.2.c.f.961.1 2
312.83 odd 4 2496.2.c.m.961.1 2
312.125 even 4 2496.2.c.f.961.2 2
312.203 odd 4 2496.2.c.m.961.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 2 39.8 even 4
78.2.b.a.25.2 yes 2 39.5 even 4
234.2.b.a.181.1 2 13.5 odd 4
234.2.b.a.181.2 2 13.8 odd 4
624.2.c.a.337.1 2 156.47 odd 4
624.2.c.a.337.2 2 156.83 odd 4
1014.2.a.b.1.1 1 3.2 odd 2
1014.2.a.g.1.1 1 39.38 odd 2
1014.2.e.b.529.1 2 39.23 odd 6
1014.2.e.b.991.1 2 39.17 odd 6
1014.2.e.e.529.1 2 39.29 odd 6
1014.2.e.e.991.1 2 39.35 odd 6
1014.2.i.c.361.1 4 39.20 even 12
1014.2.i.c.361.2 4 39.32 even 12
1014.2.i.c.823.1 4 39.2 even 12
1014.2.i.c.823.2 4 39.11 even 12
1872.2.c.b.1585.1 2 52.31 even 4
1872.2.c.b.1585.2 2 52.47 even 4
1950.2.b.c.1351.1 2 195.44 even 4
1950.2.b.c.1351.2 2 195.164 even 4
1950.2.f.d.649.1 2 195.122 odd 4
1950.2.f.d.649.2 2 195.8 odd 4
1950.2.f.g.649.1 2 195.47 odd 4
1950.2.f.g.649.2 2 195.83 odd 4
2496.2.c.f.961.1 2 312.5 even 4
2496.2.c.f.961.2 2 312.125 even 4
2496.2.c.m.961.1 2 312.83 odd 4
2496.2.c.m.961.2 2 312.203 odd 4
3042.2.a.c.1.1 1 13.12 even 2
3042.2.a.n.1.1 1 1.1 even 1 trivial
3822.2.c.d.883.1 2 273.125 odd 4
3822.2.c.d.883.2 2 273.83 odd 4
8112.2.a.g.1.1 1 12.11 even 2
8112.2.a.j.1.1 1 156.155 even 2