Properties

Label 63.4.a.b.1.1
Level $63$
Weight $4$
Character 63.1
Self dual yes
Analytic conductor $3.717$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.71712033036\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 63.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -7.00000 q^{4} -16.0000 q^{5} -7.00000 q^{7} -15.0000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -7.00000 q^{4} -16.0000 q^{5} -7.00000 q^{7} -15.0000 q^{8} -16.0000 q^{10} +8.00000 q^{11} +28.0000 q^{13} -7.00000 q^{14} +41.0000 q^{16} -54.0000 q^{17} -110.000 q^{19} +112.000 q^{20} +8.00000 q^{22} -48.0000 q^{23} +131.000 q^{25} +28.0000 q^{26} +49.0000 q^{28} +110.000 q^{29} +12.0000 q^{31} +161.000 q^{32} -54.0000 q^{34} +112.000 q^{35} -246.000 q^{37} -110.000 q^{38} +240.000 q^{40} -182.000 q^{41} +128.000 q^{43} -56.0000 q^{44} -48.0000 q^{46} -324.000 q^{47} +49.0000 q^{49} +131.000 q^{50} -196.000 q^{52} +162.000 q^{53} -128.000 q^{55} +105.000 q^{56} +110.000 q^{58} -810.000 q^{59} -488.000 q^{61} +12.0000 q^{62} -167.000 q^{64} -448.000 q^{65} +244.000 q^{67} +378.000 q^{68} +112.000 q^{70} +768.000 q^{71} -702.000 q^{73} -246.000 q^{74} +770.000 q^{76} -56.0000 q^{77} +440.000 q^{79} -656.000 q^{80} -182.000 q^{82} +1302.00 q^{83} +864.000 q^{85} +128.000 q^{86} -120.000 q^{88} -730.000 q^{89} -196.000 q^{91} +336.000 q^{92} -324.000 q^{94} +1760.00 q^{95} +294.000 q^{97} +49.0000 q^{98} +O(q^{100})\)

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 0.353553 0.176777 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) −16.0000 −1.43108 −0.715542 0.698570i \(-0.753820\pi\)
−0.715542 + 0.698570i \(0.753820\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −15.0000 −0.662913
\(9\) 0 0
\(10\) −16.0000 −0.505964
\(11\) 8.00000 0.219281 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(12\) 0 0
\(13\) 28.0000 0.597369 0.298685 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) −7.00000 −0.133631
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 0 0
\(19\) −110.000 −1.32820 −0.664098 0.747645i \(-0.731184\pi\)
−0.664098 + 0.747645i \(0.731184\pi\)
\(20\) 112.000 1.25220
\(21\) 0 0
\(22\) 8.00000 0.0775275
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) 0 0
\(25\) 131.000 1.04800
\(26\) 28.0000 0.211202
\(27\) 0 0
\(28\) 49.0000 0.330719
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) 0 0
\(31\) 12.0000 0.0695246 0.0347623 0.999396i \(-0.488933\pi\)
0.0347623 + 0.999396i \(0.488933\pi\)
\(32\) 161.000 0.889408
\(33\) 0 0
\(34\) −54.0000 −0.272380
\(35\) 112.000 0.540899
\(36\) 0 0
\(37\) −246.000 −1.09303 −0.546516 0.837449i \(-0.684046\pi\)
−0.546516 + 0.837449i \(0.684046\pi\)
\(38\) −110.000 −0.469588
\(39\) 0 0
\(40\) 240.000 0.948683
\(41\) −182.000 −0.693259 −0.346630 0.938002i \(-0.612674\pi\)
−0.346630 + 0.938002i \(0.612674\pi\)
\(42\) 0 0
\(43\) 128.000 0.453949 0.226975 0.973901i \(-0.427117\pi\)
0.226975 + 0.973901i \(0.427117\pi\)
\(44\) −56.0000 −0.191871
\(45\) 0 0
\(46\) −48.0000 −0.153852
\(47\) −324.000 −1.00554 −0.502769 0.864421i \(-0.667685\pi\)
−0.502769 + 0.864421i \(0.667685\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 131.000 0.370524
\(51\) 0 0
\(52\) −196.000 −0.522698
\(53\) 162.000 0.419857 0.209928 0.977717i \(-0.432677\pi\)
0.209928 + 0.977717i \(0.432677\pi\)
\(54\) 0 0
\(55\) −128.000 −0.313809
\(56\) 105.000 0.250557
\(57\) 0 0
\(58\) 110.000 0.249029
\(59\) −810.000 −1.78734 −0.893670 0.448725i \(-0.851878\pi\)
−0.893670 + 0.448725i \(0.851878\pi\)
\(60\) 0 0
\(61\) −488.000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 12.0000 0.0245807
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) −448.000 −0.854886
\(66\) 0 0
\(67\) 244.000 0.444916 0.222458 0.974942i \(-0.428592\pi\)
0.222458 + 0.974942i \(0.428592\pi\)
\(68\) 378.000 0.674106
\(69\) 0 0
\(70\) 112.000 0.191237
\(71\) 768.000 1.28373 0.641865 0.766818i \(-0.278161\pi\)
0.641865 + 0.766818i \(0.278161\pi\)
\(72\) 0 0
\(73\) −702.000 −1.12552 −0.562759 0.826621i \(-0.690260\pi\)
−0.562759 + 0.826621i \(0.690260\pi\)
\(74\) −246.000 −0.386445
\(75\) 0 0
\(76\) 770.000 1.16217
\(77\) −56.0000 −0.0828804
\(78\) 0 0
\(79\) 440.000 0.626631 0.313316 0.949649i \(-0.398560\pi\)
0.313316 + 0.949649i \(0.398560\pi\)
\(80\) −656.000 −0.916788
\(81\) 0 0
\(82\) −182.000 −0.245104
\(83\) 1302.00 1.72184 0.860922 0.508737i \(-0.169887\pi\)
0.860922 + 0.508737i \(0.169887\pi\)
\(84\) 0 0
\(85\) 864.000 1.10252
\(86\) 128.000 0.160495
\(87\) 0 0
\(88\) −120.000 −0.145364
\(89\) −730.000 −0.869436 −0.434718 0.900567i \(-0.643152\pi\)
−0.434718 + 0.900567i \(0.643152\pi\)
\(90\) 0 0
\(91\) −196.000 −0.225784
\(92\) 336.000 0.380765
\(93\) 0 0
\(94\) −324.000 −0.355511
\(95\) 1760.00 1.90076
\(96\) 0 0
\(97\) 294.000 0.307744 0.153872 0.988091i \(-0.450826\pi\)
0.153872 + 0.988091i \(0.450826\pi\)
\(98\) 49.0000 0.0505076
\(99\) 0 0
\(100\) −917.000 −0.917000
\(101\) 688.000 0.677808 0.338904 0.940821i \(-0.389944\pi\)
0.338904 + 0.940821i \(0.389944\pi\)
\(102\) 0 0
\(103\) 1388.00 1.32780 0.663901 0.747820i \(-0.268899\pi\)
0.663901 + 0.747820i \(0.268899\pi\)
\(104\) −420.000 −0.396004
\(105\) 0 0
\(106\) 162.000 0.148442
\(107\) −244.000 −0.220452 −0.110226 0.993907i \(-0.535157\pi\)
−0.110226 + 0.993907i \(0.535157\pi\)
\(108\) 0 0
\(109\) 90.0000 0.0790866 0.0395433 0.999218i \(-0.487410\pi\)
0.0395433 + 0.999218i \(0.487410\pi\)
\(110\) −128.000 −0.110948
\(111\) 0 0
\(112\) −287.000 −0.242133
\(113\) −1318.00 −1.09723 −0.548615 0.836075i \(-0.684845\pi\)
−0.548615 + 0.836075i \(0.684845\pi\)
\(114\) 0 0
\(115\) 768.000 0.622751
\(116\) −770.000 −0.616316
\(117\) 0 0
\(118\) −810.000 −0.631920
\(119\) 378.000 0.291187
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) −488.000 −0.362143
\(123\) 0 0
\(124\) −84.0000 −0.0608341
\(125\) −96.0000 −0.0686920
\(126\) 0 0
\(127\) −1776.00 −1.24090 −0.620451 0.784245i \(-0.713050\pi\)
−0.620451 + 0.784245i \(0.713050\pi\)
\(128\) −1455.00 −1.00473
\(129\) 0 0
\(130\) −448.000 −0.302248
\(131\) 1118.00 0.745650 0.372825 0.927902i \(-0.378389\pi\)
0.372825 + 0.927902i \(0.378389\pi\)
\(132\) 0 0
\(133\) 770.000 0.502011
\(134\) 244.000 0.157301
\(135\) 0 0
\(136\) 810.000 0.510713
\(137\) −2274.00 −1.41811 −0.709054 0.705154i \(-0.750878\pi\)
−0.709054 + 0.705154i \(0.750878\pi\)
\(138\) 0 0
\(139\) −210.000 −0.128144 −0.0640718 0.997945i \(-0.520409\pi\)
−0.0640718 + 0.997945i \(0.520409\pi\)
\(140\) −784.000 −0.473286
\(141\) 0 0
\(142\) 768.000 0.453867
\(143\) 224.000 0.130992
\(144\) 0 0
\(145\) −1760.00 −1.00800
\(146\) −702.000 −0.397931
\(147\) 0 0
\(148\) 1722.00 0.956402
\(149\) 2010.00 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(150\) 0 0
\(151\) 1112.00 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(152\) 1650.00 0.880478
\(153\) 0 0
\(154\) −56.0000 −0.0293027
\(155\) −192.000 −0.0994956
\(156\) 0 0
\(157\) 124.000 0.0630336 0.0315168 0.999503i \(-0.489966\pi\)
0.0315168 + 0.999503i \(0.489966\pi\)
\(158\) 440.000 0.221548
\(159\) 0 0
\(160\) −2576.00 −1.27282
\(161\) 336.000 0.164475
\(162\) 0 0
\(163\) 2008.00 0.964900 0.482450 0.875924i \(-0.339747\pi\)
0.482450 + 0.875924i \(0.339747\pi\)
\(164\) 1274.00 0.606602
\(165\) 0 0
\(166\) 1302.00 0.608764
\(167\) −2884.00 −1.33635 −0.668176 0.744004i \(-0.732924\pi\)
−0.668176 + 0.744004i \(0.732924\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 864.000 0.389799
\(171\) 0 0
\(172\) −896.000 −0.397206
\(173\) −2228.00 −0.979143 −0.489571 0.871963i \(-0.662847\pi\)
−0.489571 + 0.871963i \(0.662847\pi\)
\(174\) 0 0
\(175\) −917.000 −0.396107
\(176\) 328.000 0.140477
\(177\) 0 0
\(178\) −730.000 −0.307392
\(179\) 820.000 0.342400 0.171200 0.985236i \(-0.445236\pi\)
0.171200 + 0.985236i \(0.445236\pi\)
\(180\) 0 0
\(181\) 3892.00 1.59829 0.799144 0.601140i \(-0.205287\pi\)
0.799144 + 0.601140i \(0.205287\pi\)
\(182\) −196.000 −0.0798268
\(183\) 0 0
\(184\) 720.000 0.288473
\(185\) 3936.00 1.56422
\(186\) 0 0
\(187\) −432.000 −0.168936
\(188\) 2268.00 0.879845
\(189\) 0 0
\(190\) 1760.00 0.672020
\(191\) 5048.00 1.91236 0.956179 0.292782i \(-0.0945810\pi\)
0.956179 + 0.292782i \(0.0945810\pi\)
\(192\) 0 0
\(193\) −2962.00 −1.10471 −0.552356 0.833608i \(-0.686271\pi\)
−0.552356 + 0.833608i \(0.686271\pi\)
\(194\) 294.000 0.108804
\(195\) 0 0
\(196\) −343.000 −0.125000
\(197\) −3334.00 −1.20577 −0.602887 0.797826i \(-0.705983\pi\)
−0.602887 + 0.797826i \(0.705983\pi\)
\(198\) 0 0
\(199\) 1860.00 0.662572 0.331286 0.943530i \(-0.392517\pi\)
0.331286 + 0.943530i \(0.392517\pi\)
\(200\) −1965.00 −0.694732
\(201\) 0 0
\(202\) 688.000 0.239641
\(203\) −770.000 −0.266224
\(204\) 0 0
\(205\) 2912.00 0.992112
\(206\) 1388.00 0.469449
\(207\) 0 0
\(208\) 1148.00 0.382690
\(209\) −880.000 −0.291248
\(210\) 0 0
\(211\) −4268.00 −1.39252 −0.696259 0.717791i \(-0.745153\pi\)
−0.696259 + 0.717791i \(0.745153\pi\)
\(212\) −1134.00 −0.367375
\(213\) 0 0
\(214\) −244.000 −0.0779416
\(215\) −2048.00 −0.649639
\(216\) 0 0
\(217\) −84.0000 −0.0262778
\(218\) 90.0000 0.0279613
\(219\) 0 0
\(220\) 896.000 0.274583
\(221\) −1512.00 −0.460218
\(222\) 0 0
\(223\) −5432.00 −1.63118 −0.815591 0.578629i \(-0.803588\pi\)
−0.815591 + 0.578629i \(0.803588\pi\)
\(224\) −1127.00 −0.336165
\(225\) 0 0
\(226\) −1318.00 −0.387929
\(227\) 2046.00 0.598228 0.299114 0.954217i \(-0.403309\pi\)
0.299114 + 0.954217i \(0.403309\pi\)
\(228\) 0 0
\(229\) −2980.00 −0.859930 −0.429965 0.902846i \(-0.641474\pi\)
−0.429965 + 0.902846i \(0.641474\pi\)
\(230\) 768.000 0.220176
\(231\) 0 0
\(232\) −1650.00 −0.466930
\(233\) −4458.00 −1.25345 −0.626724 0.779241i \(-0.715605\pi\)
−0.626724 + 0.779241i \(0.715605\pi\)
\(234\) 0 0
\(235\) 5184.00 1.43901
\(236\) 5670.00 1.56392
\(237\) 0 0
\(238\) 378.000 0.102950
\(239\) −4440.00 −1.20167 −0.600836 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) 0 0
\(241\) 3302.00 0.882575 0.441287 0.897366i \(-0.354522\pi\)
0.441287 + 0.897366i \(0.354522\pi\)
\(242\) −1267.00 −0.336553
\(243\) 0 0
\(244\) 3416.00 0.896258
\(245\) −784.000 −0.204441
\(246\) 0 0
\(247\) −3080.00 −0.793424
\(248\) −180.000 −0.0460888
\(249\) 0 0
\(250\) −96.0000 −0.0242863
\(251\) −1582.00 −0.397829 −0.198914 0.980017i \(-0.563742\pi\)
−0.198914 + 0.980017i \(0.563742\pi\)
\(252\) 0 0
\(253\) −384.000 −0.0954224
\(254\) −1776.00 −0.438725
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −2354.00 −0.571356 −0.285678 0.958326i \(-0.592219\pi\)
−0.285678 + 0.958326i \(0.592219\pi\)
\(258\) 0 0
\(259\) 1722.00 0.413127
\(260\) 3136.00 0.748025
\(261\) 0 0
\(262\) 1118.00 0.263627
\(263\) 3872.00 0.907824 0.453912 0.891046i \(-0.350028\pi\)
0.453912 + 0.891046i \(0.350028\pi\)
\(264\) 0 0
\(265\) −2592.00 −0.600850
\(266\) 770.000 0.177488
\(267\) 0 0
\(268\) −1708.00 −0.389301
\(269\) −180.000 −0.0407985 −0.0203992 0.999792i \(-0.506494\pi\)
−0.0203992 + 0.999792i \(0.506494\pi\)
\(270\) 0 0
\(271\) 2032.00 0.455480 0.227740 0.973722i \(-0.426866\pi\)
0.227740 + 0.973722i \(0.426866\pi\)
\(272\) −2214.00 −0.493542
\(273\) 0 0
\(274\) −2274.00 −0.501377
\(275\) 1048.00 0.229806
\(276\) 0 0
\(277\) −5426.00 −1.17696 −0.588478 0.808513i \(-0.700273\pi\)
−0.588478 + 0.808513i \(0.700273\pi\)
\(278\) −210.000 −0.0453056
\(279\) 0 0
\(280\) −1680.00 −0.358569
\(281\) −842.000 −0.178753 −0.0893764 0.995998i \(-0.528487\pi\)
−0.0893764 + 0.995998i \(0.528487\pi\)
\(282\) 0 0
\(283\) −3782.00 −0.794405 −0.397202 0.917731i \(-0.630019\pi\)
−0.397202 + 0.917731i \(0.630019\pi\)
\(284\) −5376.00 −1.12326
\(285\) 0 0
\(286\) 224.000 0.0463126
\(287\) 1274.00 0.262027
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) −1760.00 −0.356382
\(291\) 0 0
\(292\) 4914.00 0.984829
\(293\) 4312.00 0.859760 0.429880 0.902886i \(-0.358556\pi\)
0.429880 + 0.902886i \(0.358556\pi\)
\(294\) 0 0
\(295\) 12960.0 2.55783
\(296\) 3690.00 0.724584
\(297\) 0 0
\(298\) 2010.00 0.390725
\(299\) −1344.00 −0.259952
\(300\) 0 0
\(301\) −896.000 −0.171577
\(302\) 1112.00 0.211882
\(303\) 0 0
\(304\) −4510.00 −0.850876
\(305\) 7808.00 1.46585
\(306\) 0 0
\(307\) 2674.00 0.497112 0.248556 0.968618i \(-0.420044\pi\)
0.248556 + 0.968618i \(0.420044\pi\)
\(308\) 392.000 0.0725204
\(309\) 0 0
\(310\) −192.000 −0.0351770
\(311\) 3768.00 0.687021 0.343511 0.939149i \(-0.388384\pi\)
0.343511 + 0.939149i \(0.388384\pi\)
\(312\) 0 0
\(313\) 2438.00 0.440268 0.220134 0.975470i \(-0.429351\pi\)
0.220134 + 0.975470i \(0.429351\pi\)
\(314\) 124.000 0.0222857
\(315\) 0 0
\(316\) −3080.00 −0.548302
\(317\) 3186.00 0.564491 0.282245 0.959342i \(-0.408921\pi\)
0.282245 + 0.959342i \(0.408921\pi\)
\(318\) 0 0
\(319\) 880.000 0.154453
\(320\) 2672.00 0.466779
\(321\) 0 0
\(322\) 336.000 0.0581508
\(323\) 5940.00 1.02325
\(324\) 0 0
\(325\) 3668.00 0.626043
\(326\) 2008.00 0.341144
\(327\) 0 0
\(328\) 2730.00 0.459570
\(329\) 2268.00 0.380057
\(330\) 0 0
\(331\) 8672.00 1.44005 0.720025 0.693949i \(-0.244131\pi\)
0.720025 + 0.693949i \(0.244131\pi\)
\(332\) −9114.00 −1.50661
\(333\) 0 0
\(334\) −2884.00 −0.472471
\(335\) −3904.00 −0.636711
\(336\) 0 0
\(337\) 814.000 0.131577 0.0657884 0.997834i \(-0.479044\pi\)
0.0657884 + 0.997834i \(0.479044\pi\)
\(338\) −1413.00 −0.227388
\(339\) 0 0
\(340\) −6048.00 −0.964703
\(341\) 96.0000 0.0152454
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −1920.00 −0.300929
\(345\) 0 0
\(346\) −2228.00 −0.346179
\(347\) −9344.00 −1.44557 −0.722784 0.691074i \(-0.757138\pi\)
−0.722784 + 0.691074i \(0.757138\pi\)
\(348\) 0 0
\(349\) −5180.00 −0.794496 −0.397248 0.917711i \(-0.630035\pi\)
−0.397248 + 0.917711i \(0.630035\pi\)
\(350\) −917.000 −0.140045
\(351\) 0 0
\(352\) 1288.00 0.195030
\(353\) −12178.0 −1.83617 −0.918087 0.396379i \(-0.870267\pi\)
−0.918087 + 0.396379i \(0.870267\pi\)
\(354\) 0 0
\(355\) −12288.0 −1.83712
\(356\) 5110.00 0.760757
\(357\) 0 0
\(358\) 820.000 0.121057
\(359\) −440.000 −0.0646861 −0.0323431 0.999477i \(-0.510297\pi\)
−0.0323431 + 0.999477i \(0.510297\pi\)
\(360\) 0 0
\(361\) 5241.00 0.764106
\(362\) 3892.00 0.565080
\(363\) 0 0
\(364\) 1372.00 0.197561
\(365\) 11232.0 1.61071
\(366\) 0 0
\(367\) −9816.00 −1.39616 −0.698080 0.716019i \(-0.745962\pi\)
−0.698080 + 0.716019i \(0.745962\pi\)
\(368\) −1968.00 −0.278775
\(369\) 0 0
\(370\) 3936.00 0.553035
\(371\) −1134.00 −0.158691
\(372\) 0 0
\(373\) −442.000 −0.0613563 −0.0306781 0.999529i \(-0.509767\pi\)
−0.0306781 + 0.999529i \(0.509767\pi\)
\(374\) −432.000 −0.0597278
\(375\) 0 0
\(376\) 4860.00 0.666583
\(377\) 3080.00 0.420764
\(378\) 0 0
\(379\) −3960.00 −0.536706 −0.268353 0.963321i \(-0.586479\pi\)
−0.268353 + 0.963321i \(0.586479\pi\)
\(380\) −12320.0 −1.66316
\(381\) 0 0
\(382\) 5048.00 0.676121
\(383\) −6708.00 −0.894942 −0.447471 0.894298i \(-0.647675\pi\)
−0.447471 + 0.894298i \(0.647675\pi\)
\(384\) 0 0
\(385\) 896.000 0.118609
\(386\) −2962.00 −0.390575
\(387\) 0 0
\(388\) −2058.00 −0.269276
\(389\) 13350.0 1.74003 0.870015 0.493025i \(-0.164109\pi\)
0.870015 + 0.493025i \(0.164109\pi\)
\(390\) 0 0
\(391\) 2592.00 0.335251
\(392\) −735.000 −0.0947018
\(393\) 0 0
\(394\) −3334.00 −0.426306
\(395\) −7040.00 −0.896762
\(396\) 0 0
\(397\) −1356.00 −0.171425 −0.0857125 0.996320i \(-0.527317\pi\)
−0.0857125 + 0.996320i \(0.527317\pi\)
\(398\) 1860.00 0.234255
\(399\) 0 0
\(400\) 5371.00 0.671375
\(401\) −6222.00 −0.774843 −0.387421 0.921903i \(-0.626634\pi\)
−0.387421 + 0.921903i \(0.626634\pi\)
\(402\) 0 0
\(403\) 336.000 0.0415319
\(404\) −4816.00 −0.593082
\(405\) 0 0
\(406\) −770.000 −0.0941243
\(407\) −1968.00 −0.239681
\(408\) 0 0
\(409\) 5150.00 0.622619 0.311309 0.950309i \(-0.399232\pi\)
0.311309 + 0.950309i \(0.399232\pi\)
\(410\) 2912.00 0.350764
\(411\) 0 0
\(412\) −9716.00 −1.16183
\(413\) 5670.00 0.675551
\(414\) 0 0
\(415\) −20832.0 −2.46410
\(416\) 4508.00 0.531305
\(417\) 0 0
\(418\) −880.000 −0.102972
\(419\) −2310.00 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(420\) 0 0
\(421\) 1262.00 0.146095 0.0730476 0.997328i \(-0.476727\pi\)
0.0730476 + 0.997328i \(0.476727\pi\)
\(422\) −4268.00 −0.492329
\(423\) 0 0
\(424\) −2430.00 −0.278328
\(425\) −7074.00 −0.807387
\(426\) 0 0
\(427\) 3416.00 0.387147
\(428\) 1708.00 0.192896
\(429\) 0 0
\(430\) −2048.00 −0.229682
\(431\) 4488.00 0.501576 0.250788 0.968042i \(-0.419310\pi\)
0.250788 + 0.968042i \(0.419310\pi\)
\(432\) 0 0
\(433\) 17038.0 1.89098 0.945490 0.325652i \(-0.105584\pi\)
0.945490 + 0.325652i \(0.105584\pi\)
\(434\) −84.0000 −0.00929062
\(435\) 0 0
\(436\) −630.000 −0.0692008
\(437\) 5280.00 0.577979
\(438\) 0 0
\(439\) 16200.0 1.76124 0.880619 0.473824i \(-0.157127\pi\)
0.880619 + 0.473824i \(0.157127\pi\)
\(440\) 1920.00 0.208028
\(441\) 0 0
\(442\) −1512.00 −0.162712
\(443\) 8772.00 0.940791 0.470395 0.882456i \(-0.344111\pi\)
0.470395 + 0.882456i \(0.344111\pi\)
\(444\) 0 0
\(445\) 11680.0 1.24424
\(446\) −5432.00 −0.576710
\(447\) 0 0
\(448\) 1169.00 0.123281
\(449\) −2130.00 −0.223877 −0.111939 0.993715i \(-0.535706\pi\)
−0.111939 + 0.993715i \(0.535706\pi\)
\(450\) 0 0
\(451\) −1456.00 −0.152019
\(452\) 9226.00 0.960076
\(453\) 0 0
\(454\) 2046.00 0.211506
\(455\) 3136.00 0.323116
\(456\) 0 0
\(457\) 10534.0 1.07825 0.539124 0.842226i \(-0.318755\pi\)
0.539124 + 0.842226i \(0.318755\pi\)
\(458\) −2980.00 −0.304031
\(459\) 0 0
\(460\) −5376.00 −0.544907
\(461\) 9268.00 0.936342 0.468171 0.883638i \(-0.344913\pi\)
0.468171 + 0.883638i \(0.344913\pi\)
\(462\) 0 0
\(463\) −9392.00 −0.942728 −0.471364 0.881939i \(-0.656238\pi\)
−0.471364 + 0.881939i \(0.656238\pi\)
\(464\) 4510.00 0.451232
\(465\) 0 0
\(466\) −4458.00 −0.443161
\(467\) 10806.0 1.07075 0.535377 0.844613i \(-0.320170\pi\)
0.535377 + 0.844613i \(0.320170\pi\)
\(468\) 0 0
\(469\) −1708.00 −0.168162
\(470\) 5184.00 0.508766
\(471\) 0 0
\(472\) 12150.0 1.18485
\(473\) 1024.00 0.0995424
\(474\) 0 0
\(475\) −14410.0 −1.39195
\(476\) −2646.00 −0.254788
\(477\) 0 0
\(478\) −4440.00 −0.424855
\(479\) −4940.00 −0.471220 −0.235610 0.971848i \(-0.575709\pi\)
−0.235610 + 0.971848i \(0.575709\pi\)
\(480\) 0 0
\(481\) −6888.00 −0.652943
\(482\) 3302.00 0.312037
\(483\) 0 0
\(484\) 8869.00 0.832926
\(485\) −4704.00 −0.440407
\(486\) 0 0
\(487\) −5216.00 −0.485338 −0.242669 0.970109i \(-0.578023\pi\)
−0.242669 + 0.970109i \(0.578023\pi\)
\(488\) 7320.00 0.679018
\(489\) 0 0
\(490\) −784.000 −0.0722806
\(491\) −4412.00 −0.405521 −0.202760 0.979228i \(-0.564991\pi\)
−0.202760 + 0.979228i \(0.564991\pi\)
\(492\) 0 0
\(493\) −5940.00 −0.542645
\(494\) −3080.00 −0.280518
\(495\) 0 0
\(496\) 492.000 0.0445392
\(497\) −5376.00 −0.485204
\(498\) 0 0
\(499\) 19060.0 1.70991 0.854953 0.518706i \(-0.173586\pi\)
0.854953 + 0.518706i \(0.173586\pi\)
\(500\) 672.000 0.0601055
\(501\) 0 0
\(502\) −1582.00 −0.140654
\(503\) −12768.0 −1.13180 −0.565902 0.824473i \(-0.691472\pi\)
−0.565902 + 0.824473i \(0.691472\pi\)
\(504\) 0 0
\(505\) −11008.0 −0.969999
\(506\) −384.000 −0.0337369
\(507\) 0 0
\(508\) 12432.0 1.08579
\(509\) 5500.00 0.478945 0.239473 0.970903i \(-0.423025\pi\)
0.239473 + 0.970903i \(0.423025\pi\)
\(510\) 0 0
\(511\) 4914.00 0.425406
\(512\) 11521.0 0.994455
\(513\) 0 0
\(514\) −2354.00 −0.202005
\(515\) −22208.0 −1.90020
\(516\) 0 0
\(517\) −2592.00 −0.220495
\(518\) 1722.00 0.146062
\(519\) 0 0
\(520\) 6720.00 0.566714
\(521\) 7338.00 0.617051 0.308526 0.951216i \(-0.400164\pi\)
0.308526 + 0.951216i \(0.400164\pi\)
\(522\) 0 0
\(523\) −17582.0 −1.46999 −0.734997 0.678070i \(-0.762817\pi\)
−0.734997 + 0.678070i \(0.762817\pi\)
\(524\) −7826.00 −0.652444
\(525\) 0 0
\(526\) 3872.00 0.320964
\(527\) −648.000 −0.0535623
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) −2592.00 −0.212433
\(531\) 0 0
\(532\) −5390.00 −0.439260
\(533\) −5096.00 −0.414132
\(534\) 0 0
\(535\) 3904.00 0.315485
\(536\) −3660.00 −0.294940
\(537\) 0 0
\(538\) −180.000 −0.0144244
\(539\) 392.000 0.0313259
\(540\) 0 0
\(541\) −1618.00 −0.128583 −0.0642914 0.997931i \(-0.520479\pi\)
−0.0642914 + 0.997931i \(0.520479\pi\)
\(542\) 2032.00 0.161037
\(543\) 0 0
\(544\) −8694.00 −0.685206
\(545\) −1440.00 −0.113179
\(546\) 0 0
\(547\) 16144.0 1.26192 0.630958 0.775817i \(-0.282662\pi\)
0.630958 + 0.775817i \(0.282662\pi\)
\(548\) 15918.0 1.24085
\(549\) 0 0
\(550\) 1048.00 0.0812489
\(551\) −12100.0 −0.935531
\(552\) 0 0
\(553\) −3080.00 −0.236844
\(554\) −5426.00 −0.416117
\(555\) 0 0
\(556\) 1470.00 0.112126
\(557\) −4654.00 −0.354033 −0.177016 0.984208i \(-0.556645\pi\)
−0.177016 + 0.984208i \(0.556645\pi\)
\(558\) 0 0
\(559\) 3584.00 0.271175
\(560\) 4592.00 0.346513
\(561\) 0 0
\(562\) −842.000 −0.0631986
\(563\) −10078.0 −0.754418 −0.377209 0.926128i \(-0.623116\pi\)
−0.377209 + 0.926128i \(0.623116\pi\)
\(564\) 0 0
\(565\) 21088.0 1.57023
\(566\) −3782.00 −0.280865
\(567\) 0 0
\(568\) −11520.0 −0.851001
\(569\) 5930.00 0.436904 0.218452 0.975848i \(-0.429899\pi\)
0.218452 + 0.975848i \(0.429899\pi\)
\(570\) 0 0
\(571\) −19048.0 −1.39603 −0.698016 0.716082i \(-0.745933\pi\)
−0.698016 + 0.716082i \(0.745933\pi\)
\(572\) −1568.00 −0.114618
\(573\) 0 0
\(574\) 1274.00 0.0926406
\(575\) −6288.00 −0.456048
\(576\) 0 0
\(577\) −14366.0 −1.03651 −0.518253 0.855227i \(-0.673418\pi\)
−0.518253 + 0.855227i \(0.673418\pi\)
\(578\) −1997.00 −0.143710
\(579\) 0 0
\(580\) 12320.0 0.882000
\(581\) −9114.00 −0.650796
\(582\) 0 0
\(583\) 1296.00 0.0920666
\(584\) 10530.0 0.746121
\(585\) 0 0
\(586\) 4312.00 0.303971
\(587\) 3626.00 0.254959 0.127480 0.991841i \(-0.459311\pi\)
0.127480 + 0.991841i \(0.459311\pi\)
\(588\) 0 0
\(589\) −1320.00 −0.0923424
\(590\) 12960.0 0.904330
\(591\) 0 0
\(592\) −10086.0 −0.700223
\(593\) 1062.00 0.0735432 0.0367716 0.999324i \(-0.488293\pi\)
0.0367716 + 0.999324i \(0.488293\pi\)
\(594\) 0 0
\(595\) −6048.00 −0.416712
\(596\) −14070.0 −0.966996
\(597\) 0 0
\(598\) −1344.00 −0.0919068
\(599\) 10200.0 0.695761 0.347880 0.937539i \(-0.386902\pi\)
0.347880 + 0.937539i \(0.386902\pi\)
\(600\) 0 0
\(601\) −25158.0 −1.70751 −0.853757 0.520671i \(-0.825682\pi\)
−0.853757 + 0.520671i \(0.825682\pi\)
\(602\) −896.000 −0.0606615
\(603\) 0 0
\(604\) −7784.00 −0.524382
\(605\) 20272.0 1.36227
\(606\) 0 0
\(607\) 25664.0 1.71609 0.858047 0.513570i \(-0.171677\pi\)
0.858047 + 0.513570i \(0.171677\pi\)
\(608\) −17710.0 −1.18131
\(609\) 0 0
\(610\) 7808.00 0.518257
\(611\) −9072.00 −0.600677
\(612\) 0 0
\(613\) 19018.0 1.25307 0.626533 0.779395i \(-0.284473\pi\)
0.626533 + 0.779395i \(0.284473\pi\)
\(614\) 2674.00 0.175755
\(615\) 0 0
\(616\) 840.000 0.0549425
\(617\) −17334.0 −1.13102 −0.565511 0.824741i \(-0.691321\pi\)
−0.565511 + 0.824741i \(0.691321\pi\)
\(618\) 0 0
\(619\) 18730.0 1.21619 0.608096 0.793864i \(-0.291934\pi\)
0.608096 + 0.793864i \(0.291934\pi\)
\(620\) 1344.00 0.0870586
\(621\) 0 0
\(622\) 3768.00 0.242899
\(623\) 5110.00 0.328616
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) 2438.00 0.155658
\(627\) 0 0
\(628\) −868.000 −0.0551544
\(629\) 13284.0 0.842079
\(630\) 0 0
\(631\) −6928.00 −0.437083 −0.218541 0.975828i \(-0.570130\pi\)
−0.218541 + 0.975828i \(0.570130\pi\)
\(632\) −6600.00 −0.415402
\(633\) 0 0
\(634\) 3186.00 0.199578
\(635\) 28416.0 1.77583
\(636\) 0 0
\(637\) 1372.00 0.0853385
\(638\) 880.000 0.0546074
\(639\) 0 0
\(640\) 23280.0 1.43785
\(641\) −16302.0 −1.00451 −0.502255 0.864720i \(-0.667496\pi\)
−0.502255 + 0.864720i \(0.667496\pi\)
\(642\) 0 0
\(643\) 4718.00 0.289362 0.144681 0.989478i \(-0.453784\pi\)
0.144681 + 0.989478i \(0.453784\pi\)
\(644\) −2352.00 −0.143916
\(645\) 0 0
\(646\) 5940.00 0.361774
\(647\) 21436.0 1.30253 0.651264 0.758851i \(-0.274239\pi\)
0.651264 + 0.758851i \(0.274239\pi\)
\(648\) 0 0
\(649\) −6480.00 −0.391930
\(650\) 3668.00 0.221340
\(651\) 0 0
\(652\) −14056.0 −0.844287
\(653\) −4458.00 −0.267159 −0.133580 0.991038i \(-0.542647\pi\)
−0.133580 + 0.991038i \(0.542647\pi\)
\(654\) 0 0
\(655\) −17888.0 −1.06709
\(656\) −7462.00 −0.444119
\(657\) 0 0
\(658\) 2268.00 0.134371
\(659\) 26640.0 1.57473 0.787365 0.616487i \(-0.211445\pi\)
0.787365 + 0.616487i \(0.211445\pi\)
\(660\) 0 0
\(661\) 7432.00 0.437324 0.218662 0.975801i \(-0.429831\pi\)
0.218662 + 0.975801i \(0.429831\pi\)
\(662\) 8672.00 0.509134
\(663\) 0 0
\(664\) −19530.0 −1.14143
\(665\) −12320.0 −0.718420
\(666\) 0 0
\(667\) −5280.00 −0.306510
\(668\) 20188.0 1.16931
\(669\) 0 0
\(670\) −3904.00 −0.225111
\(671\) −3904.00 −0.224608
\(672\) 0 0
\(673\) 58.0000 0.00332204 0.00166102 0.999999i \(-0.499471\pi\)
0.00166102 + 0.999999i \(0.499471\pi\)
\(674\) 814.000 0.0465194
\(675\) 0 0
\(676\) 9891.00 0.562756
\(677\) 21516.0 1.22146 0.610729 0.791840i \(-0.290876\pi\)
0.610729 + 0.791840i \(0.290876\pi\)
\(678\) 0 0
\(679\) −2058.00 −0.116316
\(680\) −12960.0 −0.730873
\(681\) 0 0
\(682\) 96.0000 0.00539007
\(683\) −18108.0 −1.01447 −0.507235 0.861808i \(-0.669332\pi\)
−0.507235 + 0.861808i \(0.669332\pi\)
\(684\) 0 0
\(685\) 36384.0 2.02943
\(686\) −343.000 −0.0190901
\(687\) 0 0
\(688\) 5248.00 0.290811
\(689\) 4536.00 0.250810
\(690\) 0 0
\(691\) −10078.0 −0.554827 −0.277413 0.960751i \(-0.589477\pi\)
−0.277413 + 0.960751i \(0.589477\pi\)
\(692\) 15596.0 0.856750
\(693\) 0 0
\(694\) −9344.00 −0.511086
\(695\) 3360.00 0.183384
\(696\) 0 0
\(697\) 9828.00 0.534092
\(698\) −5180.00 −0.280897
\(699\) 0 0
\(700\) 6419.00 0.346593
\(701\) −18762.0 −1.01089 −0.505443 0.862860i \(-0.668671\pi\)
−0.505443 + 0.862860i \(0.668671\pi\)
\(702\) 0 0
\(703\) 27060.0 1.45176
\(704\) −1336.00 −0.0715233
\(705\) 0 0
\(706\) −12178.0 −0.649186
\(707\) −4816.00 −0.256187
\(708\) 0 0
\(709\) 6810.00 0.360726 0.180363 0.983600i \(-0.442273\pi\)
0.180363 + 0.983600i \(0.442273\pi\)
\(710\) −12288.0 −0.649522
\(711\) 0 0
\(712\) 10950.0 0.576360
\(713\) −576.000 −0.0302544
\(714\) 0 0
\(715\) −3584.00 −0.187460
\(716\) −5740.00 −0.299600
\(717\) 0 0
\(718\) −440.000 −0.0228700
\(719\) −4860.00 −0.252083 −0.126041 0.992025i \(-0.540227\pi\)
−0.126041 + 0.992025i \(0.540227\pi\)
\(720\) 0 0
\(721\) −9716.00 −0.501862
\(722\) 5241.00 0.270152
\(723\) 0 0
\(724\) −27244.0 −1.39850
\(725\) 14410.0 0.738171
\(726\) 0 0
\(727\) −13636.0 −0.695641 −0.347821 0.937561i \(-0.613078\pi\)
−0.347821 + 0.937561i \(0.613078\pi\)
\(728\) 2940.00 0.149675
\(729\) 0 0
\(730\) 11232.0 0.569473
\(731\) −6912.00 −0.349726
\(732\) 0 0
\(733\) 2088.00 0.105214 0.0526071 0.998615i \(-0.483247\pi\)
0.0526071 + 0.998615i \(0.483247\pi\)
\(734\) −9816.00 −0.493617
\(735\) 0 0
\(736\) −7728.00 −0.387035
\(737\) 1952.00 0.0975615
\(738\) 0 0
\(739\) −5160.00 −0.256852 −0.128426 0.991719i \(-0.540992\pi\)
−0.128426 + 0.991719i \(0.540992\pi\)
\(740\) −27552.0 −1.36869
\(741\) 0 0
\(742\) −1134.00 −0.0561057
\(743\) 28152.0 1.39004 0.695018 0.718992i \(-0.255396\pi\)
0.695018 + 0.718992i \(0.255396\pi\)
\(744\) 0 0
\(745\) −32160.0 −1.58155
\(746\) −442.000 −0.0216927
\(747\) 0 0
\(748\) 3024.00 0.147819
\(749\) 1708.00 0.0833230
\(750\) 0 0
\(751\) −16808.0 −0.816688 −0.408344 0.912828i \(-0.633894\pi\)
−0.408344 + 0.912828i \(0.633894\pi\)
\(752\) −13284.0 −0.644172
\(753\) 0 0
\(754\) 3080.00 0.148763
\(755\) −17792.0 −0.857639
\(756\) 0 0
\(757\) 21674.0 1.04063 0.520314 0.853975i \(-0.325815\pi\)
0.520314 + 0.853975i \(0.325815\pi\)
\(758\) −3960.00 −0.189754
\(759\) 0 0
\(760\) −26400.0 −1.26004
\(761\) −7422.00 −0.353544 −0.176772 0.984252i \(-0.556566\pi\)
−0.176772 + 0.984252i \(0.556566\pi\)
\(762\) 0 0
\(763\) −630.000 −0.0298919
\(764\) −35336.0 −1.67331
\(765\) 0 0
\(766\) −6708.00 −0.316410
\(767\) −22680.0 −1.06770
\(768\) 0 0
\(769\) 13790.0 0.646658 0.323329 0.946287i \(-0.395198\pi\)
0.323329 + 0.946287i \(0.395198\pi\)
\(770\) 896.000 0.0419345
\(771\) 0 0
\(772\) 20734.0 0.966623
\(773\) 6232.00 0.289973 0.144987 0.989434i \(-0.453686\pi\)
0.144987 + 0.989434i \(0.453686\pi\)
\(774\) 0 0
\(775\) 1572.00 0.0728618
\(776\) −4410.00 −0.204007
\(777\) 0 0
\(778\) 13350.0 0.615194
\(779\) 20020.0 0.920784
\(780\) 0 0
\(781\) 6144.00 0.281498
\(782\) 2592.00 0.118529
\(783\) 0 0
\(784\) 2009.00 0.0915179
\(785\) −1984.00 −0.0902064
\(786\) 0 0
\(787\) −1766.00 −0.0799887 −0.0399943 0.999200i \(-0.512734\pi\)
−0.0399943 + 0.999200i \(0.512734\pi\)
\(788\) 23338.0 1.05505
\(789\) 0 0
\(790\) −7040.00 −0.317053
\(791\) 9226.00 0.414714
\(792\) 0 0
\(793\) −13664.0 −0.611883
\(794\) −1356.00 −0.0606079
\(795\) 0 0
\(796\) −13020.0 −0.579751
\(797\) −1204.00 −0.0535105 −0.0267552 0.999642i \(-0.508517\pi\)
−0.0267552 + 0.999642i \(0.508517\pi\)
\(798\) 0 0
\(799\) 17496.0 0.774673
\(800\) 21091.0 0.932099
\(801\) 0 0
\(802\) −6222.00 −0.273948
\(803\) −5616.00 −0.246805
\(804\) 0 0
\(805\) −5376.00 −0.235378
\(806\) 336.000 0.0146837
\(807\) 0 0
\(808\) −10320.0 −0.449327
\(809\) 7050.00 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(810\) 0 0
\(811\) 23282.0 1.00807 0.504033 0.863684i \(-0.331849\pi\)
0.504033 + 0.863684i \(0.331849\pi\)
\(812\) 5390.00 0.232946
\(813\) 0 0
\(814\) −1968.00 −0.0847400
\(815\) −32128.0 −1.38085
\(816\) 0 0
\(817\) −14080.0 −0.602934
\(818\) 5150.00 0.220129
\(819\) 0 0
\(820\) −20384.0 −0.868098
\(821\) −10142.0 −0.431131 −0.215565 0.976489i \(-0.569159\pi\)
−0.215565 + 0.976489i \(0.569159\pi\)
\(822\) 0 0
\(823\) −9192.00 −0.389323 −0.194662 0.980870i \(-0.562361\pi\)
−0.194662 + 0.980870i \(0.562361\pi\)
\(824\) −20820.0 −0.880217
\(825\) 0 0
\(826\) 5670.00 0.238843
\(827\) 46716.0 1.96430 0.982149 0.188104i \(-0.0602344\pi\)
0.982149 + 0.188104i \(0.0602344\pi\)
\(828\) 0 0
\(829\) 11240.0 0.470906 0.235453 0.971886i \(-0.424343\pi\)
0.235453 + 0.971886i \(0.424343\pi\)
\(830\) −20832.0 −0.871192
\(831\) 0 0
\(832\) −4676.00 −0.194845
\(833\) −2646.00 −0.110058
\(834\) 0 0
\(835\) 46144.0 1.91243
\(836\) 6160.00 0.254842
\(837\) 0 0
\(838\) −2310.00 −0.0952239
\(839\) −700.000 −0.0288042 −0.0144021 0.999896i \(-0.504584\pi\)
−0.0144021 + 0.999896i \(0.504584\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) 1262.00 0.0516525
\(843\) 0 0
\(844\) 29876.0 1.21845
\(845\) 22608.0 0.920401
\(846\) 0 0
\(847\) 8869.00 0.359790
\(848\) 6642.00 0.268971
\(849\) 0 0
\(850\) −7074.00 −0.285454
\(851\) 11808.0 0.475644
\(852\) 0 0
\(853\) −37492.0 −1.50493 −0.752463 0.658635i \(-0.771134\pi\)
−0.752463 + 0.658635i \(0.771134\pi\)
\(854\) 3416.00 0.136877
\(855\) 0 0
\(856\) 3660.00 0.146140
\(857\) −28894.0 −1.15169 −0.575846 0.817558i \(-0.695327\pi\)
−0.575846 + 0.817558i \(0.695327\pi\)
\(858\) 0 0
\(859\) −2770.00 −0.110025 −0.0550123 0.998486i \(-0.517520\pi\)
−0.0550123 + 0.998486i \(0.517520\pi\)
\(860\) 14336.0 0.568434
\(861\) 0 0
\(862\) 4488.00 0.177334
\(863\) −17688.0 −0.697690 −0.348845 0.937180i \(-0.613426\pi\)
−0.348845 + 0.937180i \(0.613426\pi\)
\(864\) 0 0
\(865\) 35648.0 1.40124
\(866\) 17038.0 0.668562
\(867\) 0 0
\(868\) 588.000 0.0229931
\(869\) 3520.00 0.137408
\(870\) 0 0
\(871\) 6832.00 0.265779
\(872\) −1350.00 −0.0524275
\(873\) 0 0
\(874\) 5280.00 0.204346
\(875\) 672.000 0.0259631
\(876\) 0 0
\(877\) −33566.0 −1.29241 −0.646205 0.763164i \(-0.723645\pi\)
−0.646205 + 0.763164i \(0.723645\pi\)
\(878\) 16200.0 0.622692
\(879\) 0 0
\(880\) −5248.00 −0.201034
\(881\) 16758.0 0.640853 0.320426 0.947273i \(-0.396174\pi\)
0.320426 + 0.947273i \(0.396174\pi\)
\(882\) 0 0
\(883\) 11468.0 0.437066 0.218533 0.975830i \(-0.429873\pi\)
0.218533 + 0.975830i \(0.429873\pi\)
\(884\) 10584.0 0.402691
\(885\) 0 0
\(886\) 8772.00 0.332620
\(887\) 50356.0 1.90619 0.953094 0.302674i \(-0.0978793\pi\)
0.953094 + 0.302674i \(0.0978793\pi\)
\(888\) 0 0
\(889\) 12432.0 0.469017
\(890\) 11680.0 0.439904
\(891\) 0 0
\(892\) 38024.0 1.42728
\(893\) 35640.0 1.33555
\(894\) 0 0
\(895\) −13120.0 −0.490004
\(896\) 10185.0 0.379751
\(897\) 0 0
\(898\) −2130.00 −0.0791526
\(899\) 1320.00 0.0489705
\(900\) 0 0
\(901\) −8748.00 −0.323461
\(902\) −1456.00 −0.0537467
\(903\) 0 0
\(904\) 19770.0 0.727368
\(905\) −62272.0 −2.28728
\(906\) 0 0
\(907\) −8716.00 −0.319085 −0.159542 0.987191i \(-0.551002\pi\)
−0.159542 + 0.987191i \(0.551002\pi\)
\(908\) −14322.0 −0.523450
\(909\) 0 0
\(910\) 3136.00 0.114239
\(911\) −7632.00 −0.277563 −0.138781 0.990323i \(-0.544318\pi\)
−0.138781 + 0.990323i \(0.544318\pi\)
\(912\) 0 0
\(913\) 10416.0 0.377568
\(914\) 10534.0 0.381219
\(915\) 0 0
\(916\) 20860.0 0.752439
\(917\) −7826.00 −0.281829
\(918\) 0 0
\(919\) −23080.0 −0.828443 −0.414221 0.910176i \(-0.635946\pi\)
−0.414221 + 0.910176i \(0.635946\pi\)
\(920\) −11520.0 −0.412830
\(921\) 0 0
\(922\) 9268.00 0.331047
\(923\) 21504.0 0.766861
\(924\) 0 0
\(925\) −32226.0 −1.14550
\(926\) −9392.00 −0.333305
\(927\) 0 0
\(928\) 17710.0 0.626465
\(929\) −45110.0 −1.59312 −0.796561 0.604558i \(-0.793350\pi\)
−0.796561 + 0.604558i \(0.793350\pi\)
\(930\) 0 0
\(931\) −5390.00 −0.189742
\(932\) 31206.0 1.09677
\(933\) 0 0
\(934\) 10806.0 0.378569
\(935\) 6912.00 0.241761
\(936\) 0 0
\(937\) 16674.0 0.581340 0.290670 0.956823i \(-0.406122\pi\)
0.290670 + 0.956823i \(0.406122\pi\)
\(938\) −1708.00 −0.0594543
\(939\) 0 0
\(940\) −36288.0 −1.25913
\(941\) −43832.0 −1.51847 −0.759236 0.650815i \(-0.774427\pi\)
−0.759236 + 0.650815i \(0.774427\pi\)
\(942\) 0 0
\(943\) 8736.00 0.301679
\(944\) −33210.0 −1.14501
\(945\) 0 0
\(946\) 1024.00 0.0351936
\(947\) 736.000 0.0252553 0.0126277 0.999920i \(-0.495980\pi\)
0.0126277 + 0.999920i \(0.495980\pi\)
\(948\) 0 0
\(949\) −19656.0 −0.672351
\(950\) −14410.0 −0.492129
\(951\) 0 0
\(952\) −5670.00 −0.193031
\(953\) −38138.0 −1.29634 −0.648169 0.761496i \(-0.724465\pi\)
−0.648169 + 0.761496i \(0.724465\pi\)
\(954\) 0 0
\(955\) −80768.0 −2.73674
\(956\) 31080.0 1.05146
\(957\) 0 0
\(958\) −4940.00 −0.166601
\(959\) 15918.0 0.535995
\(960\) 0 0
\(961\) −29647.0 −0.995166
\(962\) −6888.00 −0.230850
\(963\) 0 0
\(964\) −23114.0 −0.772253
\(965\) 47392.0 1.58094
\(966\) 0 0
\(967\) 26224.0 0.872086 0.436043 0.899926i \(-0.356380\pi\)
0.436043 + 0.899926i \(0.356380\pi\)
\(968\) 19005.0 0.631037
\(969\) 0 0
\(970\) −4704.00 −0.155708
\(971\) −18762.0 −0.620084 −0.310042 0.950723i \(-0.600343\pi\)
−0.310042 + 0.950723i \(0.600343\pi\)
\(972\) 0 0
\(973\) 1470.00 0.0484337
\(974\) −5216.00 −0.171593
\(975\) 0 0
\(976\) −20008.0 −0.656189
\(977\) −38394.0 −1.25725 −0.628625 0.777709i \(-0.716382\pi\)
−0.628625 + 0.777709i \(0.716382\pi\)
\(978\) 0 0
\(979\) −5840.00 −0.190651
\(980\) 5488.00 0.178885
\(981\) 0 0
\(982\) −4412.00 −0.143373
\(983\) −5388.00 −0.174822 −0.0874112 0.996172i \(-0.527859\pi\)
−0.0874112 + 0.996172i \(0.527859\pi\)
\(984\) 0 0
\(985\) 53344.0 1.72556
\(986\) −5940.00 −0.191854
\(987\) 0 0
\(988\) 21560.0 0.694246
\(989\) −6144.00 −0.197541
\(990\) 0 0
\(991\) 25472.0 0.816493 0.408247 0.912872i \(-0.366140\pi\)
0.408247 + 0.912872i \(0.366140\pi\)
\(992\) 1932.00 0.0618357
\(993\) 0 0
\(994\) −5376.00 −0.171546
\(995\) −29760.0 −0.948196
\(996\) 0 0
\(997\) −17096.0 −0.543065 −0.271532 0.962429i \(-0.587530\pi\)
−0.271532 + 0.962429i \(0.587530\pi\)
\(998\) 19060.0 0.604543
\(999\) 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 63.4.a.b.1.1 1
3.2 odd 2 7.4.a.a.1.1 1
4.3 odd 2 1008.4.a.c.1.1 1
5.4 even 2 1575.4.a.e.1.1 1
7.2 even 3 441.4.e.h.361.1 2
7.3 odd 6 441.4.e.e.226.1 2
7.4 even 3 441.4.e.h.226.1 2
7.5 odd 6 441.4.e.e.361.1 2
7.6 odd 2 441.4.a.i.1.1 1
12.11 even 2 112.4.a.f.1.1 1
15.2 even 4 175.4.b.b.99.1 2
15.8 even 4 175.4.b.b.99.2 2
15.14 odd 2 175.4.a.b.1.1 1
21.2 odd 6 49.4.c.c.18.1 2
21.5 even 6 49.4.c.b.18.1 2
21.11 odd 6 49.4.c.c.30.1 2
21.17 even 6 49.4.c.b.30.1 2
21.20 even 2 49.4.a.b.1.1 1
24.5 odd 2 448.4.a.i.1.1 1
24.11 even 2 448.4.a.e.1.1 1
33.32 even 2 847.4.a.b.1.1 1
39.38 odd 2 1183.4.a.b.1.1 1
51.50 odd 2 2023.4.a.a.1.1 1
84.83 odd 2 784.4.a.g.1.1 1
105.104 even 2 1225.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 3.2 odd 2
49.4.a.b.1.1 1 21.20 even 2
49.4.c.b.18.1 2 21.5 even 6
49.4.c.b.30.1 2 21.17 even 6
49.4.c.c.18.1 2 21.2 odd 6
49.4.c.c.30.1 2 21.11 odd 6
63.4.a.b.1.1 1 1.1 even 1 trivial
112.4.a.f.1.1 1 12.11 even 2
175.4.a.b.1.1 1 15.14 odd 2
175.4.b.b.99.1 2 15.2 even 4
175.4.b.b.99.2 2 15.8 even 4
441.4.a.i.1.1 1 7.6 odd 2
441.4.e.e.226.1 2 7.3 odd 6
441.4.e.e.361.1 2 7.5 odd 6
441.4.e.h.226.1 2 7.4 even 3
441.4.e.h.361.1 2 7.2 even 3
448.4.a.e.1.1 1 24.11 even 2
448.4.a.i.1.1 1 24.5 odd 2
784.4.a.g.1.1 1 84.83 odd 2
847.4.a.b.1.1 1 33.32 even 2
1008.4.a.c.1.1 1 4.3 odd 2
1183.4.a.b.1.1 1 39.38 odd 2
1225.4.a.j.1.1 1 105.104 even 2
1575.4.a.e.1.1 1 5.4 even 2
2023.4.a.a.1.1 1 51.50 odd 2