Properties

Label 525.2.a.d.1.1
Level $525$
Weight $2$
Character 525.1
Self dual yes
Analytic conductor $4.192$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +5.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} -1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{48} +1.00000 q^{49} -6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -3.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} +1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} -3.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} +4.00000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -14.0000 q^{89} +2.00000 q^{91} -5.00000 q^{96} -18.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 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
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) −2.00000 −0.226455
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) −12.0000 −1.27920
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 1.00000 0.101015
\(99\) 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
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −3.00000 −0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000 0.348155
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −12.0000 −0.973329
\(153\) 6.00000 0.485071
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −16.0000 −1.27289
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) −14.0000 −1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 2.00000 0.148250
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −7.00000 −0.505181
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 4.00000 0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 14.0000 0.985037
\(203\) −2.00000 −0.140372
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −4.00000 −0.249029
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 4.00000 0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 14.0000 0.856786
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 2.00000 0.118056
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) −6.00000 −0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 8.00000 0.460348
\(303\) −14.0000 −0.804279
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −4.00000 −0.227921
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 6.00000 0.339683
\(313\) −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
\(316\) 16.0000 0.900070
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 18.0000 0.995402
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 20.0000 1.06600
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −6.00000 −0.317554
\(358\) −4.00000 −0.211407
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −26.0000 −1.36653
\(363\) −5.00000 −0.262432
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) −1.00000 −0.0514344
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) 18.0000 0.913812
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −4.00000 −0.201773
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −24.0000 −1.18964
\(408\) 18.0000 0.891133
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) −12.0000 −0.587643
\(418\) 16.0000 0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −6.00000 −0.283790
\(448\) 7.00000 0.330719
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −14.0000 −0.658505
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −10.0000 −0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −4.00000 −0.186097
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) 16.0000 0.735681
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) −20.0000 −0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) −26.0000 −1.14681
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) 4.00000 0.173259
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) 6.00000 0.258678
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 16.0000 0.687259
\(543\) 26.0000 1.11577
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −22.0000 −0.928014
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 8.00000 0.334205
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 18.0000 0.746124
\(583\) −24.0000 −0.993978
\(584\) −18.0000 −0.744845
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 20.0000 0.811107
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 8.00000 0.321807
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −14.0000 −0.560898
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 48.0000 1.90934
\(633\) −4.00000 −0.158986
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) −8.00000 −0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) −3.00000 −0.117851
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) −36.0000 −1.39707
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) −5.00000 −0.192879
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −14.0000 −0.537667
\(679\) −18.0000 −0.690777
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −10.0000 −0.380143
\(693\) 4.00000 0.151947
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) −2.00000 −0.0757011
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 14.0000 0.526524
\(708\) 12.0000 0.450988
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 42.0000 1.57402
\(713\) 0 0
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −24.0000 −0.896296
\(718\) 32.0000 1.19423
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) −2.00000 −0.0739221
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −6.00000 −0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 12.0000 0.439057
\(748\) −24.0000 −0.877527
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −18.0000 −0.651644
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 17.0000 0.613435
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 2.00000 0.0719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 54.0000 1.93849
\(777\) 6.00000 0.215249
\(778\) 6.00000 0.215110
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 22.0000 0.783718
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) −12.0000 −0.426401
\(793\) −4.00000 −0.142044
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) −30.0000 −1.05934
\(803\) 24.0000 0.846942
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) −42.0000 −1.47755
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 2.00000 0.0701862
\(813\) −16.0000 −0.561144
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −22.0000 −0.769212
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) −6.00000 −0.209274
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 38.0000 1.30957
\(843\) 22.0000 0.757720
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 6.00000 0.206041
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) −8.00000 −0.273115
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 54.0000 1.82867
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −24.0000 −0.809961
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000 0.0336718
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −18.0000 −0.604040
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 8.00000 0.266371
\(903\) −4.00000 −0.133112
\(904\) −42.0000 −1.39690
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −12.0000 −0.398234
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 4.00000 0.132453
\(913\) 48.0000 1.58857
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 4.00000 0.132092
\(918\) −6.00000 −0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −10.0000 −0.329332
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) −8.00000 −0.262754
\(928\) −10.0000 −0.328266
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −4.00000 −0.130605
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) −16.0000 −0.519656
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) −18.0000 −0.583383
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 8.00000 0.258603
\(958\) −16.0000 −0.516937
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −15.0000 −0.482118
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 4.00000 0.127906
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 20.0000 0.638226
\(983\) −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
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 4.00000 0.126618
\(999\) 6.00000 0.189832
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 525.2.a.d.1.1 1
3.2 odd 2 1575.2.a.c.1.1 1
4.3 odd 2 8400.2.a.bn.1.1 1
5.2 odd 4 525.2.d.a.274.2 2
5.3 odd 4 525.2.d.a.274.1 2
5.4 even 2 21.2.a.a.1.1 1
7.6 odd 2 3675.2.a.n.1.1 1
15.2 even 4 1575.2.d.a.1324.1 2
15.8 even 4 1575.2.d.a.1324.2 2
15.14 odd 2 63.2.a.a.1.1 1
20.19 odd 2 336.2.a.a.1.1 1
35.4 even 6 147.2.e.b.79.1 2
35.9 even 6 147.2.e.b.67.1 2
35.19 odd 6 147.2.e.c.67.1 2
35.24 odd 6 147.2.e.c.79.1 2
35.34 odd 2 147.2.a.a.1.1 1
40.19 odd 2 1344.2.a.s.1.1 1
40.29 even 2 1344.2.a.g.1.1 1
45.4 even 6 567.2.f.g.379.1 2
45.14 odd 6 567.2.f.b.379.1 2
45.29 odd 6 567.2.f.b.190.1 2
45.34 even 6 567.2.f.g.190.1 2
55.54 odd 2 2541.2.a.j.1.1 1
60.59 even 2 1008.2.a.l.1.1 1
65.64 even 2 3549.2.a.c.1.1 1
80.19 odd 4 5376.2.c.l.2689.1 2
80.29 even 4 5376.2.c.r.2689.2 2
80.59 odd 4 5376.2.c.l.2689.2 2
80.69 even 4 5376.2.c.r.2689.1 2
85.84 even 2 6069.2.a.b.1.1 1
95.94 odd 2 7581.2.a.d.1.1 1
105.44 odd 6 441.2.e.a.361.1 2
105.59 even 6 441.2.e.b.226.1 2
105.74 odd 6 441.2.e.a.226.1 2
105.89 even 6 441.2.e.b.361.1 2
105.104 even 2 441.2.a.f.1.1 1
120.29 odd 2 4032.2.a.h.1.1 1
120.59 even 2 4032.2.a.k.1.1 1
140.19 even 6 2352.2.q.e.1537.1 2
140.39 odd 6 2352.2.q.x.961.1 2
140.59 even 6 2352.2.q.e.961.1 2
140.79 odd 6 2352.2.q.x.1537.1 2
140.139 even 2 2352.2.a.v.1.1 1
165.164 even 2 7623.2.a.g.1.1 1
280.69 odd 2 9408.2.a.bv.1.1 1
280.139 even 2 9408.2.a.m.1.1 1
420.419 odd 2 7056.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 5.4 even 2
63.2.a.a.1.1 1 15.14 odd 2
147.2.a.a.1.1 1 35.34 odd 2
147.2.e.b.67.1 2 35.9 even 6
147.2.e.b.79.1 2 35.4 even 6
147.2.e.c.67.1 2 35.19 odd 6
147.2.e.c.79.1 2 35.24 odd 6
336.2.a.a.1.1 1 20.19 odd 2
441.2.a.f.1.1 1 105.104 even 2
441.2.e.a.226.1 2 105.74 odd 6
441.2.e.a.361.1 2 105.44 odd 6
441.2.e.b.226.1 2 105.59 even 6
441.2.e.b.361.1 2 105.89 even 6
525.2.a.d.1.1 1 1.1 even 1 trivial
525.2.d.a.274.1 2 5.3 odd 4
525.2.d.a.274.2 2 5.2 odd 4
567.2.f.b.190.1 2 45.29 odd 6
567.2.f.b.379.1 2 45.14 odd 6
567.2.f.g.190.1 2 45.34 even 6
567.2.f.g.379.1 2 45.4 even 6
1008.2.a.l.1.1 1 60.59 even 2
1344.2.a.g.1.1 1 40.29 even 2
1344.2.a.s.1.1 1 40.19 odd 2
1575.2.a.c.1.1 1 3.2 odd 2
1575.2.d.a.1324.1 2 15.2 even 4
1575.2.d.a.1324.2 2 15.8 even 4
2352.2.a.v.1.1 1 140.139 even 2
2352.2.q.e.961.1 2 140.59 even 6
2352.2.q.e.1537.1 2 140.19 even 6
2352.2.q.x.961.1 2 140.39 odd 6
2352.2.q.x.1537.1 2 140.79 odd 6
2541.2.a.j.1.1 1 55.54 odd 2
3549.2.a.c.1.1 1 65.64 even 2
3675.2.a.n.1.1 1 7.6 odd 2
4032.2.a.h.1.1 1 120.29 odd 2
4032.2.a.k.1.1 1 120.59 even 2
5376.2.c.l.2689.1 2 80.19 odd 4
5376.2.c.l.2689.2 2 80.59 odd 4
5376.2.c.r.2689.1 2 80.69 even 4
5376.2.c.r.2689.2 2 80.29 even 4
6069.2.a.b.1.1 1 85.84 even 2
7056.2.a.p.1.1 1 420.419 odd 2
7581.2.a.d.1.1 1 95.94 odd 2
7623.2.a.g.1.1 1 165.164 even 2
8400.2.a.bn.1.1 1 4.3 odd 2
9408.2.a.m.1.1 1 280.139 even 2
9408.2.a.bv.1.1 1 280.69 odd 2