Properties

Label 4650.2.a.bl.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
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 \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +6.00000 q^{51} +4.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +4.00000 q^{58} +6.00000 q^{59} +10.0000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +10.0000 q^{67} +6.00000 q^{68} -14.0000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} +8.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +8.00000 q^{86} +4.00000 q^{87} -4.00000 q^{88} -8.00000 q^{89} -8.00000 q^{91} +1.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −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 \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −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 \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 4.00000 0.554700
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 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
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 4.00000 0.428845
\(88\) −4.00000 −0.426401
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 6.00000 0.594089
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 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 \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −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
\(117\) 4.00000 0.369800
\(118\) 6.00000 0.552345
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) −4.00000 −0.348155
\(133\) 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 \(0.669953\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −14.0000 −1.17485
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −8.00000 −0.636446
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 6.00000 0.450988
\(178\) −8.00000 −0.599625
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 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
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) −24.0000 −1.75505
\(188\) 12.0000 0.875190
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 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 \(0.522698\pi\)
\(198\) −4.00000 −0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 14.0000 0.985037
\(203\) −8.00000 −0.561490
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −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
\(213\) −14.0000 −0.959264
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 18.0000 1.21911
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −4.00000 −0.268462
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −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
\(231\) 8.00000 0.526361
\(232\) 4.00000 0.262613
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −8.00000 −0.519656
\(238\) −12.0000 −0.777844
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 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
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) −2.00000 −0.125988
\(253\) 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 \(0.480131\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 14.0000 0.864923
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 10.0000 0.610847
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 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
\(273\) −8.00000 −0.484182
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −12.0000 −0.719712
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 12.0000 0.714590
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −4.00000 −0.234082
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) −4.00000 −0.232104
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 8.00000 0.455842
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 4.00000 0.226455
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 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 \(0.760551\pi\)
\(318\) 2.00000 0.112154
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 18.0000 0.995402
\(328\) −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
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 3.00000 0.163178
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −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 \(0.534242\pi\)
\(348\) 4.00000 0.214423
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) −4.00000 −0.213201
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) −12.0000 −0.635107
\(358\) −4.00000 −0.211407
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000 0.946059
\(363\) 5.00000 0.262432
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 1.00000 0.0518476
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 16.0000 0.824042
\(378\) −2.00000 −0.102869
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) −10.0000 −0.511645
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 14.0000 0.706207
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 10.0000 0.498755
\(403\) 4.00000 0.199254
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 16.0000 0.793091
\(408\) 6.00000 0.297044
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) −14.0000 −0.689730
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 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
\(423\) 12.0000 0.583460
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −14.0000 −0.678302
\(427\) −20.0000 −0.967868
\(428\) 16.0000 0.773389
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 24.0000 1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 14.0000 0.662177
\(448\) −2.00000 −0.0944911
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) −26.0000 −1.21490
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 8.00000 0.372194
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 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 \(0.312606\pi\)
\(468\) 4.00000 0.184900
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 6.00000 0.276172
\(473\) −32.0000 −1.47136
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 2.00000 0.0915737
\(478\) 20.0000 0.914779
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 10.0000 0.452679
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −6.00000 −0.270501
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 28.0000 1.25597
\(498\) 4.00000 0.179244
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −28.0000 −1.24970
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 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
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −48.0000 −2.11104
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 4.00000 0.175075
\(523\) 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
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) −8.00000 −0.346194
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) −4.00000 −0.172613
\(538\) −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
\(543\) 18.0000 0.772454
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −46.0000 −1.96682 −0.983409 0.181402i \(-0.941936\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 18.0000 0.768922
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 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 \(0.375462\pi\)
\(558\) 1.00000 0.0423334
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 6.00000 0.253095
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) −2.00000 −0.0839921
\(568\) −14.0000 −0.587427
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 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
\(573\) −10.0000 −0.417756
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 2.00000 0.0829027
\(583\) −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 \(0.420343\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −4.00000 −0.164399
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 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
\(603\) 10.0000 0.407231
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 6.00000 0.242536
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) −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 \(0.675322\pi\)
\(618\) −14.0000 −0.563163
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 16.0000 0.641026
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 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
\(633\) 16.0000 0.635943
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) −12.0000 −0.475457
\(638\) −16.0000 −0.633446
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 16.0000 0.631470
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) −6.00000 −0.234978
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −4.00000 −0.156055
\(658\) −24.0000 −0.935617
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 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
\(663\) 24.0000 0.932083
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) −2.00000 −0.0771517
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −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 \(0.273358\pi\)
\(678\) 6.00000 0.230429
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) −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
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −26.0000 −0.991962
\(688\) 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
\(693\) 8.00000 0.303895
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) −36.0000 −1.36360
\(698\) 2.00000 0.0757011
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 4.00000 0.150970
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) −28.0000 −1.05305
\(708\) 6.00000 0.225494
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 20.0000 0.746914
\(718\) 30.0000 1.11959
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) −19.0000 −0.707107
\(723\) 10.0000 0.371904
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 10.0000 0.369611
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) −6.00000 −0.220863
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) 4.00000 0.146352
\(748\) −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
\(753\) −28.0000 −1.02038
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 12.0000 0.434714
\(763\) −36.0000 −1.30329
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 2.00000 0.0719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 8.00000 0.286998
\(778\) 36.0000 1.29066
\(779\) 0 0
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) −4.00000 −0.142134
\(793\) 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
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) −4.00000 −0.141245
\(803\) 16.0000 0.564628
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −12.0000 −0.422420
\(808\) 14.0000 0.492518
\(809\) 20.0000 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) 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
\(813\) −24.0000 −0.841717
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 18.0000 0.627822
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −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 \(0.488942\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 4.00000 0.138675
\(833\) −18.0000 −0.623663
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) −30.0000 −1.03633
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −30.0000 −1.03387
\(843\) 6.00000 0.206651
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −10.0000 −0.343604
\(848\) 2.00000 0.0686803
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 0 0
\(852\) −14.0000 −0.479632
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −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 \(0.754643\pi\)
\(858\) −16.0000 −0.546231
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −10.0000 −0.340601
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) 19.0000 0.645274
\(868\) −2.00000 −0.0678844
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 18.0000 0.609557
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 24.0000 0.809961
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −56.0000 −1.88669 −0.943344 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) −3.00000 −0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 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 \(0.632005\pi\)
\(888\) −4.00000 −0.134231
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) −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
\(903\) −16.0000 −0.532447
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) −58.0000 −1.92586 −0.962929 0.269754i \(-0.913058\pi\)
−0.962929 + 0.269754i \(0.913058\pi\)
\(908\) −20.0000 −0.663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 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
\(918\) 6.00000 0.198030
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 24.0000 0.790398
\(923\) −56.0000 −1.84326
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) −14.0000 −0.459820
\(928\) 4.00000 0.131306
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 6.00000 0.196431
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −20.0000 −0.653023
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 6.00000 0.195491
\(943\) 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 \(0.349661\pi\)
\(948\) −8.00000 −0.259828
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −26.0000 −0.843108
\(952\) −12.0000 −0.388922
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) −16.0000 −0.517207
\(958\) −14.0000 −0.452319
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −16.0000 −0.515861
\(963\) 16.0000 0.515593
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 1.00000 0.0320750
\(973\) 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 \(0.340676\pi\)
\(978\) −6.00000 −0.191859
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) −24.0000 −0.763928
\(988\) 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
\(993\) −28.0000 −0.888553
\(994\) 28.0000 0.888106
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −4.00000 −0.126618
\(999\) −4.00000 −0.126554
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 4650.2.a.bl.1.1 1
5.2 odd 4 4650.2.d.p.3349.2 2
5.3 odd 4 4650.2.d.p.3349.1 2
5.4 even 2 930.2.a.d.1.1 1
15.14 odd 2 2790.2.a.t.1.1 1
20.19 odd 2 7440.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.d.1.1 1 5.4 even 2
2790.2.a.t.1.1 1 15.14 odd 2
4650.2.a.bl.1.1 1 1.1 even 1 trivial
4650.2.d.p.3349.1 2 5.3 odd 4
4650.2.d.p.3349.2 2 5.2 odd 4
7440.2.a.y.1.1 1 20.19 odd 2