Properties

Label 2790.2.a.t.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -4.00000 q^{26} +2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +4.00000 q^{37} -1.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +12.0000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} +2.00000 q^{53} -4.00000 q^{55} +2.00000 q^{56} -4.00000 q^{58} -6.00000 q^{59} +10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +6.00000 q^{68} -2.00000 q^{70} +14.0000 q^{71} +4.00000 q^{73} +4.00000 q^{74} +8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} +8.00000 q^{89} -8.00000 q^{91} +12.0000 q^{94} -2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 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
\(52\) −4.00000 −0.554700
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 14.0000 1.17485
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −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
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 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
\(205\) −6.00000 −0.419058
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −14.0000 −0.864923
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −14.0000 −0.743043
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.0000 0.511645
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 28.0000 1.25597
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 48.0000 2.11104
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 46.0000 1.96682 0.983409 0.181402i \(-0.0580636\pi\)
0.983409 + 0.181402i \(0.0580636\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 14.0000 0.587427
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.00000 −0.159872
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −26.0000 −1.03259
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −16.0000 −0.633446
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) 14.0000 0.547025
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 10.0000 0.386334
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −14.0000 −0.525411
\(711\) 0 0
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −30.0000 −1.11264 −0.556319 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) −4.00000 −0.148047
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 18.0000 0.659027
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 16.0000 0.582686
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 36.0000 1.30329
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) 0 0
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 0 0
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) 16.0000 0.564628
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) −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
\(826\) −12.0000 −0.417533
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −30.0000 −1.03387
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 10.0000 0.340601
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −4.00000 −0.135926
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 18.0000 0.609557
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 56.0000 1.88669 0.943344 0.331816i \(-0.107661\pi\)
0.943344 + 0.331816i \(0.107661\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 58.0000 1.92586 0.962929 0.269754i \(-0.0869425\pi\)
0.962929 + 0.269754i \(0.0869425\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −4.00000 −0.132308
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) −28.0000 −0.924641
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −24.0000 −0.790398
\(923\) −56.0000 −1.84326
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 14.0000 0.452319
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) −24.0000 −0.769405
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 32.0000 1.02272
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 28.0000 0.888106
\(995\) 0 0
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 2790.2.a.t.1.1 1
3.2 odd 2 930.2.a.d.1.1 1
12.11 even 2 7440.2.a.y.1.1 1
15.2 even 4 4650.2.d.p.3349.1 2
15.8 even 4 4650.2.d.p.3349.2 2
15.14 odd 2 4650.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.d.1.1 1 3.2 odd 2
2790.2.a.t.1.1 1 1.1 even 1 trivial
4650.2.a.bl.1.1 1 15.14 odd 2
4650.2.d.p.3349.1 2 15.2 even 4
4650.2.d.p.3349.2 2 15.8 even 4
7440.2.a.y.1.1 1 12.11 even 2