Properties

Label 126.4.a.b.1.1
Level $126$
Weight $4$
Character 126.1
Self dual yes
Analytic conductor $7.434$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.43424066072\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 126.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +12.0000 q^{10} -30.0000 q^{11} +2.00000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -66.0000 q^{17} -52.0000 q^{19} -24.0000 q^{20} +60.0000 q^{22} -114.000 q^{23} -89.0000 q^{25} -4.00000 q^{26} +28.0000 q^{28} -72.0000 q^{29} -196.000 q^{31} -32.0000 q^{32} +132.000 q^{34} -42.0000 q^{35} -286.000 q^{37} +104.000 q^{38} +48.0000 q^{40} +378.000 q^{41} +164.000 q^{43} -120.000 q^{44} +228.000 q^{46} +228.000 q^{47} +49.0000 q^{49} +178.000 q^{50} +8.00000 q^{52} +348.000 q^{53} +180.000 q^{55} -56.0000 q^{56} +144.000 q^{58} +348.000 q^{59} -106.000 q^{61} +392.000 q^{62} +64.0000 q^{64} -12.0000 q^{65} +596.000 q^{67} -264.000 q^{68} +84.0000 q^{70} -630.000 q^{71} -1042.00 q^{73} +572.000 q^{74} -208.000 q^{76} -210.000 q^{77} -88.0000 q^{79} -96.0000 q^{80} -756.000 q^{82} +1440.00 q^{83} +396.000 q^{85} -328.000 q^{86} +240.000 q^{88} -1374.00 q^{89} +14.0000 q^{91} -456.000 q^{92} -456.000 q^{94} +312.000 q^{95} -34.0000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 12.0000 0.379473
\(11\) −30.0000 −0.822304 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 0 0
\(13\) 2.00000 0.0426692 0.0213346 0.999772i \(-0.493208\pi\)
0.0213346 + 0.999772i \(0.493208\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) −24.0000 −0.268328
\(21\) 0 0
\(22\) 60.0000 0.581456
\(23\) −114.000 −1.03351 −0.516753 0.856134i \(-0.672859\pi\)
−0.516753 + 0.856134i \(0.672859\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) −4.00000 −0.0301717
\(27\) 0 0
\(28\) 28.0000 0.188982
\(29\) −72.0000 −0.461037 −0.230518 0.973068i \(-0.574042\pi\)
−0.230518 + 0.973068i \(0.574042\pi\)
\(30\) 0 0
\(31\) −196.000 −1.13557 −0.567785 0.823177i \(-0.692199\pi\)
−0.567785 + 0.823177i \(0.692199\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 132.000 0.665818
\(35\) −42.0000 −0.202837
\(36\) 0 0
\(37\) −286.000 −1.27076 −0.635380 0.772200i \(-0.719156\pi\)
−0.635380 + 0.772200i \(0.719156\pi\)
\(38\) 104.000 0.443974
\(39\) 0 0
\(40\) 48.0000 0.189737
\(41\) 378.000 1.43985 0.719923 0.694054i \(-0.244177\pi\)
0.719923 + 0.694054i \(0.244177\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) −120.000 −0.411152
\(45\) 0 0
\(46\) 228.000 0.730799
\(47\) 228.000 0.707600 0.353800 0.935321i \(-0.384889\pi\)
0.353800 + 0.935321i \(0.384889\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 178.000 0.503460
\(51\) 0 0
\(52\) 8.00000 0.0213346
\(53\) 348.000 0.901915 0.450957 0.892546i \(-0.351083\pi\)
0.450957 + 0.892546i \(0.351083\pi\)
\(54\) 0 0
\(55\) 180.000 0.441294
\(56\) −56.0000 −0.133631
\(57\) 0 0
\(58\) 144.000 0.326002
\(59\) 348.000 0.767894 0.383947 0.923355i \(-0.374565\pi\)
0.383947 + 0.923355i \(0.374565\pi\)
\(60\) 0 0
\(61\) −106.000 −0.222490 −0.111245 0.993793i \(-0.535484\pi\)
−0.111245 + 0.993793i \(0.535484\pi\)
\(62\) 392.000 0.802969
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −12.0000 −0.0228987
\(66\) 0 0
\(67\) 596.000 1.08676 0.543381 0.839487i \(-0.317144\pi\)
0.543381 + 0.839487i \(0.317144\pi\)
\(68\) −264.000 −0.470804
\(69\) 0 0
\(70\) 84.0000 0.143427
\(71\) −630.000 −1.05306 −0.526530 0.850157i \(-0.676507\pi\)
−0.526530 + 0.850157i \(0.676507\pi\)
\(72\) 0 0
\(73\) −1042.00 −1.67064 −0.835321 0.549762i \(-0.814718\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(74\) 572.000 0.898563
\(75\) 0 0
\(76\) −208.000 −0.313937
\(77\) −210.000 −0.310802
\(78\) 0 0
\(79\) −88.0000 −0.125326 −0.0626631 0.998035i \(-0.519959\pi\)
−0.0626631 + 0.998035i \(0.519959\pi\)
\(80\) −96.0000 −0.134164
\(81\) 0 0
\(82\) −756.000 −1.01812
\(83\) 1440.00 1.90434 0.952172 0.305563i \(-0.0988446\pi\)
0.952172 + 0.305563i \(0.0988446\pi\)
\(84\) 0 0
\(85\) 396.000 0.505320
\(86\) −328.000 −0.411269
\(87\) 0 0
\(88\) 240.000 0.290728
\(89\) −1374.00 −1.63645 −0.818223 0.574901i \(-0.805041\pi\)
−0.818223 + 0.574901i \(0.805041\pi\)
\(90\) 0 0
\(91\) 14.0000 0.0161275
\(92\) −456.000 −0.516753
\(93\) 0 0
\(94\) −456.000 −0.500349
\(95\) 312.000 0.336953
\(96\) 0 0
\(97\) −34.0000 −0.0355895 −0.0177947 0.999842i \(-0.505665\pi\)
−0.0177947 + 0.999842i \(0.505665\pi\)
\(98\) −98.0000 −0.101015
\(99\) 0 0
\(100\) −356.000 −0.356000
\(101\) 438.000 0.431511 0.215756 0.976447i \(-0.430779\pi\)
0.215756 + 0.976447i \(0.430779\pi\)
\(102\) 0 0
\(103\) 1676.00 1.60331 0.801656 0.597785i \(-0.203952\pi\)
0.801656 + 0.597785i \(0.203952\pi\)
\(104\) −16.0000 −0.0150859
\(105\) 0 0
\(106\) −696.000 −0.637750
\(107\) −2022.00 −1.82686 −0.913430 0.406995i \(-0.866577\pi\)
−0.913430 + 0.406995i \(0.866577\pi\)
\(108\) 0 0
\(109\) −502.000 −0.441127 −0.220564 0.975373i \(-0.570790\pi\)
−0.220564 + 0.975373i \(0.570790\pi\)
\(110\) −360.000 −0.312042
\(111\) 0 0
\(112\) 112.000 0.0944911
\(113\) −2016.00 −1.67831 −0.839156 0.543890i \(-0.816951\pi\)
−0.839156 + 0.543890i \(0.816951\pi\)
\(114\) 0 0
\(115\) 684.000 0.554638
\(116\) −288.000 −0.230518
\(117\) 0 0
\(118\) −696.000 −0.542983
\(119\) −462.000 −0.355895
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 212.000 0.157324
\(123\) 0 0
\(124\) −784.000 −0.567785
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 1784.00 1.24649 0.623246 0.782026i \(-0.285814\pi\)
0.623246 + 0.782026i \(0.285814\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 24.0000 0.0161918
\(131\) −1608.00 −1.07246 −0.536228 0.844073i \(-0.680151\pi\)
−0.536228 + 0.844073i \(0.680151\pi\)
\(132\) 0 0
\(133\) −364.000 −0.237314
\(134\) −1192.00 −0.768456
\(135\) 0 0
\(136\) 528.000 0.332909
\(137\) 2580.00 1.60894 0.804468 0.593996i \(-0.202450\pi\)
0.804468 + 0.593996i \(0.202450\pi\)
\(138\) 0 0
\(139\) 2144.00 1.30829 0.654143 0.756371i \(-0.273030\pi\)
0.654143 + 0.756371i \(0.273030\pi\)
\(140\) −168.000 −0.101419
\(141\) 0 0
\(142\) 1260.00 0.744626
\(143\) −60.0000 −0.0350871
\(144\) 0 0
\(145\) 432.000 0.247418
\(146\) 2084.00 1.18132
\(147\) 0 0
\(148\) −1144.00 −0.635380
\(149\) −1500.00 −0.824730 −0.412365 0.911019i \(-0.635297\pi\)
−0.412365 + 0.911019i \(0.635297\pi\)
\(150\) 0 0
\(151\) −1240.00 −0.668277 −0.334138 0.942524i \(-0.608445\pi\)
−0.334138 + 0.942524i \(0.608445\pi\)
\(152\) 416.000 0.221987
\(153\) 0 0
\(154\) 420.000 0.219770
\(155\) 1176.00 0.609410
\(156\) 0 0
\(157\) 614.000 0.312118 0.156059 0.987748i \(-0.450121\pi\)
0.156059 + 0.987748i \(0.450121\pi\)
\(158\) 176.000 0.0886190
\(159\) 0 0
\(160\) 192.000 0.0948683
\(161\) −798.000 −0.390629
\(162\) 0 0
\(163\) 92.0000 0.0442086 0.0221043 0.999756i \(-0.492963\pi\)
0.0221043 + 0.999756i \(0.492963\pi\)
\(164\) 1512.00 0.719923
\(165\) 0 0
\(166\) −2880.00 −1.34657
\(167\) 3924.00 1.81825 0.909126 0.416520i \(-0.136750\pi\)
0.909126 + 0.416520i \(0.136750\pi\)
\(168\) 0 0
\(169\) −2193.00 −0.998179
\(170\) −792.000 −0.357315
\(171\) 0 0
\(172\) 656.000 0.290811
\(173\) 1902.00 0.835875 0.417938 0.908476i \(-0.362753\pi\)
0.417938 + 0.908476i \(0.362753\pi\)
\(174\) 0 0
\(175\) −623.000 −0.269111
\(176\) −480.000 −0.205576
\(177\) 0 0
\(178\) 2748.00 1.15714
\(179\) 6.00000 0.00250537 0.00125268 0.999999i \(-0.499601\pi\)
0.00125268 + 0.999999i \(0.499601\pi\)
\(180\) 0 0
\(181\) −2878.00 −1.18188 −0.590939 0.806716i \(-0.701243\pi\)
−0.590939 + 0.806716i \(0.701243\pi\)
\(182\) −28.0000 −0.0114038
\(183\) 0 0
\(184\) 912.000 0.365400
\(185\) 1716.00 0.681961
\(186\) 0 0
\(187\) 1980.00 0.774288
\(188\) 912.000 0.353800
\(189\) 0 0
\(190\) −624.000 −0.238262
\(191\) 354.000 0.134108 0.0670538 0.997749i \(-0.478640\pi\)
0.0670538 + 0.997749i \(0.478640\pi\)
\(192\) 0 0
\(193\) −4858.00 −1.81185 −0.905924 0.423441i \(-0.860822\pi\)
−0.905924 + 0.423441i \(0.860822\pi\)
\(194\) 68.0000 0.0251655
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) 396.000 0.143217 0.0716087 0.997433i \(-0.477187\pi\)
0.0716087 + 0.997433i \(0.477187\pi\)
\(198\) 0 0
\(199\) 1712.00 0.609852 0.304926 0.952376i \(-0.401368\pi\)
0.304926 + 0.952376i \(0.401368\pi\)
\(200\) 712.000 0.251730
\(201\) 0 0
\(202\) −876.000 −0.305124
\(203\) −504.000 −0.174255
\(204\) 0 0
\(205\) −2268.00 −0.772702
\(206\) −3352.00 −1.13371
\(207\) 0 0
\(208\) 32.0000 0.0106673
\(209\) 1560.00 0.516304
\(210\) 0 0
\(211\) −772.000 −0.251880 −0.125940 0.992038i \(-0.540195\pi\)
−0.125940 + 0.992038i \(0.540195\pi\)
\(212\) 1392.00 0.450957
\(213\) 0 0
\(214\) 4044.00 1.29179
\(215\) −984.000 −0.312131
\(216\) 0 0
\(217\) −1372.00 −0.429205
\(218\) 1004.00 0.311924
\(219\) 0 0
\(220\) 720.000 0.220647
\(221\) −132.000 −0.0401777
\(222\) 0 0
\(223\) 776.000 0.233026 0.116513 0.993189i \(-0.462828\pi\)
0.116513 + 0.993189i \(0.462828\pi\)
\(224\) −224.000 −0.0668153
\(225\) 0 0
\(226\) 4032.00 1.18675
\(227\) 1788.00 0.522792 0.261396 0.965232i \(-0.415817\pi\)
0.261396 + 0.965232i \(0.415817\pi\)
\(228\) 0 0
\(229\) 5402.00 1.55884 0.779420 0.626502i \(-0.215514\pi\)
0.779420 + 0.626502i \(0.215514\pi\)
\(230\) −1368.00 −0.392188
\(231\) 0 0
\(232\) 576.000 0.163001
\(233\) −3012.00 −0.846878 −0.423439 0.905925i \(-0.639177\pi\)
−0.423439 + 0.905925i \(0.639177\pi\)
\(234\) 0 0
\(235\) −1368.00 −0.379738
\(236\) 1392.00 0.383947
\(237\) 0 0
\(238\) 924.000 0.251656
\(239\) −3546.00 −0.959714 −0.479857 0.877347i \(-0.659311\pi\)
−0.479857 + 0.877347i \(0.659311\pi\)
\(240\) 0 0
\(241\) −3562.00 −0.952069 −0.476034 0.879427i \(-0.657926\pi\)
−0.476034 + 0.879427i \(0.657926\pi\)
\(242\) 862.000 0.228973
\(243\) 0 0
\(244\) −424.000 −0.111245
\(245\) −294.000 −0.0766652
\(246\) 0 0
\(247\) −104.000 −0.0267909
\(248\) 1568.00 0.401484
\(249\) 0 0
\(250\) −2568.00 −0.649658
\(251\) −3348.00 −0.841928 −0.420964 0.907077i \(-0.638308\pi\)
−0.420964 + 0.907077i \(0.638308\pi\)
\(252\) 0 0
\(253\) 3420.00 0.849856
\(254\) −3568.00 −0.881402
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −366.000 −0.0888344 −0.0444172 0.999013i \(-0.514143\pi\)
−0.0444172 + 0.999013i \(0.514143\pi\)
\(258\) 0 0
\(259\) −2002.00 −0.480302
\(260\) −48.0000 −0.0114494
\(261\) 0 0
\(262\) 3216.00 0.758340
\(263\) −4170.00 −0.977693 −0.488846 0.872370i \(-0.662582\pi\)
−0.488846 + 0.872370i \(0.662582\pi\)
\(264\) 0 0
\(265\) −2088.00 −0.484018
\(266\) 728.000 0.167807
\(267\) 0 0
\(268\) 2384.00 0.543381
\(269\) −6078.00 −1.37763 −0.688814 0.724938i \(-0.741869\pi\)
−0.688814 + 0.724938i \(0.741869\pi\)
\(270\) 0 0
\(271\) 2468.00 0.553212 0.276606 0.960983i \(-0.410790\pi\)
0.276606 + 0.960983i \(0.410790\pi\)
\(272\) −1056.00 −0.235402
\(273\) 0 0
\(274\) −5160.00 −1.13769
\(275\) 2670.00 0.585480
\(276\) 0 0
\(277\) −394.000 −0.0854627 −0.0427313 0.999087i \(-0.513606\pi\)
−0.0427313 + 0.999087i \(0.513606\pi\)
\(278\) −4288.00 −0.925098
\(279\) 0 0
\(280\) 336.000 0.0717137
\(281\) 396.000 0.0840690 0.0420345 0.999116i \(-0.486616\pi\)
0.0420345 + 0.999116i \(0.486616\pi\)
\(282\) 0 0
\(283\) −1348.00 −0.283146 −0.141573 0.989928i \(-0.545216\pi\)
−0.141573 + 0.989928i \(0.545216\pi\)
\(284\) −2520.00 −0.526530
\(285\) 0 0
\(286\) 120.000 0.0248103
\(287\) 2646.00 0.544211
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) −864.000 −0.174951
\(291\) 0 0
\(292\) −4168.00 −0.835321
\(293\) −7506.00 −1.49660 −0.748302 0.663358i \(-0.769131\pi\)
−0.748302 + 0.663358i \(0.769131\pi\)
\(294\) 0 0
\(295\) −2088.00 −0.412095
\(296\) 2288.00 0.449281
\(297\) 0 0
\(298\) 3000.00 0.583172
\(299\) −228.000 −0.0440989
\(300\) 0 0
\(301\) 1148.00 0.219833
\(302\) 2480.00 0.472543
\(303\) 0 0
\(304\) −832.000 −0.156969
\(305\) 636.000 0.119401
\(306\) 0 0
\(307\) 1748.00 0.324963 0.162481 0.986712i \(-0.448050\pi\)
0.162481 + 0.986712i \(0.448050\pi\)
\(308\) −840.000 −0.155401
\(309\) 0 0
\(310\) −2352.00 −0.430918
\(311\) 1140.00 0.207857 0.103928 0.994585i \(-0.466859\pi\)
0.103928 + 0.994585i \(0.466859\pi\)
\(312\) 0 0
\(313\) 146.000 0.0263655 0.0131828 0.999913i \(-0.495804\pi\)
0.0131828 + 0.999913i \(0.495804\pi\)
\(314\) −1228.00 −0.220701
\(315\) 0 0
\(316\) −352.000 −0.0626631
\(317\) 8148.00 1.44365 0.721825 0.692075i \(-0.243303\pi\)
0.721825 + 0.692075i \(0.243303\pi\)
\(318\) 0 0
\(319\) 2160.00 0.379112
\(320\) −384.000 −0.0670820
\(321\) 0 0
\(322\) 1596.00 0.276216
\(323\) 3432.00 0.591212
\(324\) 0 0
\(325\) −178.000 −0.0303805
\(326\) −184.000 −0.0312602
\(327\) 0 0
\(328\) −3024.00 −0.509062
\(329\) 1596.00 0.267448
\(330\) 0 0
\(331\) −9700.00 −1.61076 −0.805378 0.592762i \(-0.798038\pi\)
−0.805378 + 0.592762i \(0.798038\pi\)
\(332\) 5760.00 0.952172
\(333\) 0 0
\(334\) −7848.00 −1.28570
\(335\) −3576.00 −0.583217
\(336\) 0 0
\(337\) 8174.00 1.32126 0.660632 0.750710i \(-0.270288\pi\)
0.660632 + 0.750710i \(0.270288\pi\)
\(338\) 4386.00 0.705819
\(339\) 0 0
\(340\) 1584.00 0.252660
\(341\) 5880.00 0.933783
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −1312.00 −0.205635
\(345\) 0 0
\(346\) −3804.00 −0.591053
\(347\) 4038.00 0.624701 0.312350 0.949967i \(-0.398884\pi\)
0.312350 + 0.949967i \(0.398884\pi\)
\(348\) 0 0
\(349\) 10766.0 1.65126 0.825631 0.564210i \(-0.190819\pi\)
0.825631 + 0.564210i \(0.190819\pi\)
\(350\) 1246.00 0.190290
\(351\) 0 0
\(352\) 960.000 0.145364
\(353\) −3666.00 −0.552752 −0.276376 0.961050i \(-0.589134\pi\)
−0.276376 + 0.961050i \(0.589134\pi\)
\(354\) 0 0
\(355\) 3780.00 0.565131
\(356\) −5496.00 −0.818223
\(357\) 0 0
\(358\) −12.0000 −0.00177156
\(359\) 5106.00 0.750653 0.375326 0.926893i \(-0.377531\pi\)
0.375326 + 0.926893i \(0.377531\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 5756.00 0.835714
\(363\) 0 0
\(364\) 56.0000 0.00806373
\(365\) 6252.00 0.896561
\(366\) 0 0
\(367\) −5776.00 −0.821539 −0.410769 0.911739i \(-0.634740\pi\)
−0.410769 + 0.911739i \(0.634740\pi\)
\(368\) −1824.00 −0.258377
\(369\) 0 0
\(370\) −3432.00 −0.482219
\(371\) 2436.00 0.340892
\(372\) 0 0
\(373\) 8462.00 1.17465 0.587327 0.809350i \(-0.300180\pi\)
0.587327 + 0.809350i \(0.300180\pi\)
\(374\) −3960.00 −0.547505
\(375\) 0 0
\(376\) −1824.00 −0.250175
\(377\) −144.000 −0.0196721
\(378\) 0 0
\(379\) 6860.00 0.929748 0.464874 0.885377i \(-0.346100\pi\)
0.464874 + 0.885377i \(0.346100\pi\)
\(380\) 1248.00 0.168476
\(381\) 0 0
\(382\) −708.000 −0.0948284
\(383\) 696.000 0.0928562 0.0464281 0.998922i \(-0.485216\pi\)
0.0464281 + 0.998922i \(0.485216\pi\)
\(384\) 0 0
\(385\) 1260.00 0.166794
\(386\) 9716.00 1.28117
\(387\) 0 0
\(388\) −136.000 −0.0177947
\(389\) −11136.0 −1.45146 −0.725730 0.687980i \(-0.758498\pi\)
−0.725730 + 0.687980i \(0.758498\pi\)
\(390\) 0 0
\(391\) 7524.00 0.973159
\(392\) −392.000 −0.0505076
\(393\) 0 0
\(394\) −792.000 −0.101270
\(395\) 528.000 0.0672571
\(396\) 0 0
\(397\) 10838.0 1.37014 0.685068 0.728480i \(-0.259773\pi\)
0.685068 + 0.728480i \(0.259773\pi\)
\(398\) −3424.00 −0.431230
\(399\) 0 0
\(400\) −1424.00 −0.178000
\(401\) 8364.00 1.04159 0.520796 0.853681i \(-0.325635\pi\)
0.520796 + 0.853681i \(0.325635\pi\)
\(402\) 0 0
\(403\) −392.000 −0.0484539
\(404\) 1752.00 0.215756
\(405\) 0 0
\(406\) 1008.00 0.123217
\(407\) 8580.00 1.04495
\(408\) 0 0
\(409\) −1762.00 −0.213020 −0.106510 0.994312i \(-0.533968\pi\)
−0.106510 + 0.994312i \(0.533968\pi\)
\(410\) 4536.00 0.546383
\(411\) 0 0
\(412\) 6704.00 0.801656
\(413\) 2436.00 0.290237
\(414\) 0 0
\(415\) −8640.00 −1.02198
\(416\) −64.0000 −0.00754293
\(417\) 0 0
\(418\) −3120.00 −0.365082
\(419\) −14580.0 −1.69995 −0.849976 0.526822i \(-0.823383\pi\)
−0.849976 + 0.526822i \(0.823383\pi\)
\(420\) 0 0
\(421\) 8534.00 0.987938 0.493969 0.869480i \(-0.335546\pi\)
0.493969 + 0.869480i \(0.335546\pi\)
\(422\) 1544.00 0.178106
\(423\) 0 0
\(424\) −2784.00 −0.318875
\(425\) 5874.00 0.670426
\(426\) 0 0
\(427\) −742.000 −0.0840934
\(428\) −8088.00 −0.913430
\(429\) 0 0
\(430\) 1968.00 0.220710
\(431\) −5934.00 −0.663180 −0.331590 0.943424i \(-0.607585\pi\)
−0.331590 + 0.943424i \(0.607585\pi\)
\(432\) 0 0
\(433\) −14758.0 −1.63793 −0.818966 0.573843i \(-0.805452\pi\)
−0.818966 + 0.573843i \(0.805452\pi\)
\(434\) 2744.00 0.303494
\(435\) 0 0
\(436\) −2008.00 −0.220564
\(437\) 5928.00 0.648912
\(438\) 0 0
\(439\) −11392.0 −1.23852 −0.619260 0.785186i \(-0.712567\pi\)
−0.619260 + 0.785186i \(0.712567\pi\)
\(440\) −1440.00 −0.156021
\(441\) 0 0
\(442\) 264.000 0.0284100
\(443\) −7026.00 −0.753533 −0.376767 0.926308i \(-0.622964\pi\)
−0.376767 + 0.926308i \(0.622964\pi\)
\(444\) 0 0
\(445\) 8244.00 0.878209
\(446\) −1552.00 −0.164774
\(447\) 0 0
\(448\) 448.000 0.0472456
\(449\) 3384.00 0.355681 0.177841 0.984059i \(-0.443089\pi\)
0.177841 + 0.984059i \(0.443089\pi\)
\(450\) 0 0
\(451\) −11340.0 −1.18399
\(452\) −8064.00 −0.839156
\(453\) 0 0
\(454\) −3576.00 −0.369670
\(455\) −84.0000 −0.00865490
\(456\) 0 0
\(457\) −4282.00 −0.438301 −0.219150 0.975691i \(-0.570329\pi\)
−0.219150 + 0.975691i \(0.570329\pi\)
\(458\) −10804.0 −1.10227
\(459\) 0 0
\(460\) 2736.00 0.277319
\(461\) −16650.0 −1.68214 −0.841071 0.540924i \(-0.818075\pi\)
−0.841071 + 0.540924i \(0.818075\pi\)
\(462\) 0 0
\(463\) −9664.00 −0.970031 −0.485015 0.874506i \(-0.661186\pi\)
−0.485015 + 0.874506i \(0.661186\pi\)
\(464\) −1152.00 −0.115259
\(465\) 0 0
\(466\) 6024.00 0.598834
\(467\) 12324.0 1.22117 0.610585 0.791950i \(-0.290934\pi\)
0.610585 + 0.791950i \(0.290934\pi\)
\(468\) 0 0
\(469\) 4172.00 0.410757
\(470\) 2736.00 0.268515
\(471\) 0 0
\(472\) −2784.00 −0.271491
\(473\) −4920.00 −0.478270
\(474\) 0 0
\(475\) 4628.00 0.447047
\(476\) −1848.00 −0.177947
\(477\) 0 0
\(478\) 7092.00 0.678620
\(479\) −18660.0 −1.77995 −0.889976 0.456007i \(-0.849279\pi\)
−0.889976 + 0.456007i \(0.849279\pi\)
\(480\) 0 0
\(481\) −572.000 −0.0542224
\(482\) 7124.00 0.673214
\(483\) 0 0
\(484\) −1724.00 −0.161908
\(485\) 204.000 0.0190993
\(486\) 0 0
\(487\) −3400.00 −0.316363 −0.158181 0.987410i \(-0.550563\pi\)
−0.158181 + 0.987410i \(0.550563\pi\)
\(488\) 848.000 0.0786622
\(489\) 0 0
\(490\) 588.000 0.0542105
\(491\) −2970.00 −0.272982 −0.136491 0.990641i \(-0.543582\pi\)
−0.136491 + 0.990641i \(0.543582\pi\)
\(492\) 0 0
\(493\) 4752.00 0.434116
\(494\) 208.000 0.0189441
\(495\) 0 0
\(496\) −3136.00 −0.283892
\(497\) −4410.00 −0.398019
\(498\) 0 0
\(499\) −988.000 −0.0886352 −0.0443176 0.999017i \(-0.514111\pi\)
−0.0443176 + 0.999017i \(0.514111\pi\)
\(500\) 5136.00 0.459378
\(501\) 0 0
\(502\) 6696.00 0.595333
\(503\) −5184.00 −0.459529 −0.229765 0.973246i \(-0.573796\pi\)
−0.229765 + 0.973246i \(0.573796\pi\)
\(504\) 0 0
\(505\) −2628.00 −0.231573
\(506\) −6840.00 −0.600939
\(507\) 0 0
\(508\) 7136.00 0.623246
\(509\) −16854.0 −1.46766 −0.733831 0.679332i \(-0.762270\pi\)
−0.733831 + 0.679332i \(0.762270\pi\)
\(510\) 0 0
\(511\) −7294.00 −0.631443
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 732.000 0.0628154
\(515\) −10056.0 −0.860428
\(516\) 0 0
\(517\) −6840.00 −0.581862
\(518\) 4004.00 0.339625
\(519\) 0 0
\(520\) 96.0000 0.00809592
\(521\) 4398.00 0.369827 0.184914 0.982755i \(-0.440800\pi\)
0.184914 + 0.982755i \(0.440800\pi\)
\(522\) 0 0
\(523\) −10672.0 −0.892264 −0.446132 0.894967i \(-0.647199\pi\)
−0.446132 + 0.894967i \(0.647199\pi\)
\(524\) −6432.00 −0.536228
\(525\) 0 0
\(526\) 8340.00 0.691333
\(527\) 12936.0 1.06926
\(528\) 0 0
\(529\) 829.000 0.0681351
\(530\) 4176.00 0.342253
\(531\) 0 0
\(532\) −1456.00 −0.118657
\(533\) 756.000 0.0614371
\(534\) 0 0
\(535\) 12132.0 0.980396
\(536\) −4768.00 −0.384228
\(537\) 0 0
\(538\) 12156.0 0.974131
\(539\) −1470.00 −0.117472
\(540\) 0 0
\(541\) 20702.0 1.64519 0.822596 0.568627i \(-0.192525\pi\)
0.822596 + 0.568627i \(0.192525\pi\)
\(542\) −4936.00 −0.391180
\(543\) 0 0
\(544\) 2112.00 0.166455
\(545\) 3012.00 0.236734
\(546\) 0 0
\(547\) −22876.0 −1.78813 −0.894065 0.447937i \(-0.852159\pi\)
−0.894065 + 0.447937i \(0.852159\pi\)
\(548\) 10320.0 0.804468
\(549\) 0 0
\(550\) −5340.00 −0.413997
\(551\) 3744.00 0.289473
\(552\) 0 0
\(553\) −616.000 −0.0473689
\(554\) 788.000 0.0604312
\(555\) 0 0
\(556\) 8576.00 0.654143
\(557\) 12876.0 0.979486 0.489743 0.871867i \(-0.337091\pi\)
0.489743 + 0.871867i \(0.337091\pi\)
\(558\) 0 0
\(559\) 328.000 0.0248174
\(560\) −672.000 −0.0507093
\(561\) 0 0
\(562\) −792.000 −0.0594457
\(563\) 6900.00 0.516519 0.258260 0.966076i \(-0.416851\pi\)
0.258260 + 0.966076i \(0.416851\pi\)
\(564\) 0 0
\(565\) 12096.0 0.900677
\(566\) 2696.00 0.200214
\(567\) 0 0
\(568\) 5040.00 0.372313
\(569\) −14676.0 −1.08128 −0.540641 0.841253i \(-0.681818\pi\)
−0.540641 + 0.841253i \(0.681818\pi\)
\(570\) 0 0
\(571\) 380.000 0.0278503 0.0139251 0.999903i \(-0.495567\pi\)
0.0139251 + 0.999903i \(0.495567\pi\)
\(572\) −240.000 −0.0175435
\(573\) 0 0
\(574\) −5292.00 −0.384815
\(575\) 10146.0 0.735856
\(576\) 0 0
\(577\) −11806.0 −0.851803 −0.425901 0.904770i \(-0.640043\pi\)
−0.425901 + 0.904770i \(0.640043\pi\)
\(578\) 1114.00 0.0801666
\(579\) 0 0
\(580\) 1728.00 0.123709
\(581\) 10080.0 0.719774
\(582\) 0 0
\(583\) −10440.0 −0.741648
\(584\) 8336.00 0.590661
\(585\) 0 0
\(586\) 15012.0 1.05826
\(587\) 19188.0 1.34919 0.674594 0.738189i \(-0.264319\pi\)
0.674594 + 0.738189i \(0.264319\pi\)
\(588\) 0 0
\(589\) 10192.0 0.712995
\(590\) 4176.00 0.291395
\(591\) 0 0
\(592\) −4576.00 −0.317690
\(593\) −690.000 −0.0477823 −0.0238912 0.999715i \(-0.507606\pi\)
−0.0238912 + 0.999715i \(0.507606\pi\)
\(594\) 0 0
\(595\) 2772.00 0.190993
\(596\) −6000.00 −0.412365
\(597\) 0 0
\(598\) 456.000 0.0311827
\(599\) 20490.0 1.39766 0.698830 0.715287i \(-0.253704\pi\)
0.698830 + 0.715287i \(0.253704\pi\)
\(600\) 0 0
\(601\) −11590.0 −0.786632 −0.393316 0.919403i \(-0.628672\pi\)
−0.393316 + 0.919403i \(0.628672\pi\)
\(602\) −2296.00 −0.155445
\(603\) 0 0
\(604\) −4960.00 −0.334138
\(605\) 2586.00 0.173778
\(606\) 0 0
\(607\) −6424.00 −0.429559 −0.214779 0.976663i \(-0.568903\pi\)
−0.214779 + 0.976663i \(0.568903\pi\)
\(608\) 1664.00 0.110994
\(609\) 0 0
\(610\) −1272.00 −0.0844291
\(611\) 456.000 0.0301928
\(612\) 0 0
\(613\) −9682.00 −0.637932 −0.318966 0.947766i \(-0.603336\pi\)
−0.318966 + 0.947766i \(0.603336\pi\)
\(614\) −3496.00 −0.229784
\(615\) 0 0
\(616\) 1680.00 0.109885
\(617\) −5076.00 −0.331203 −0.165601 0.986193i \(-0.552956\pi\)
−0.165601 + 0.986193i \(0.552956\pi\)
\(618\) 0 0
\(619\) 22664.0 1.47164 0.735818 0.677179i \(-0.236798\pi\)
0.735818 + 0.677179i \(0.236798\pi\)
\(620\) 4704.00 0.304705
\(621\) 0 0
\(622\) −2280.00 −0.146977
\(623\) −9618.00 −0.618519
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) −292.000 −0.0186432
\(627\) 0 0
\(628\) 2456.00 0.156059
\(629\) 18876.0 1.19656
\(630\) 0 0
\(631\) −8584.00 −0.541559 −0.270779 0.962641i \(-0.587281\pi\)
−0.270779 + 0.962641i \(0.587281\pi\)
\(632\) 704.000 0.0443095
\(633\) 0 0
\(634\) −16296.0 −1.02082
\(635\) −10704.0 −0.668937
\(636\) 0 0
\(637\) 98.0000 0.00609561
\(638\) −4320.00 −0.268073
\(639\) 0 0
\(640\) 768.000 0.0474342
\(641\) −372.000 −0.0229222 −0.0114611 0.999934i \(-0.503648\pi\)
−0.0114611 + 0.999934i \(0.503648\pi\)
\(642\) 0 0
\(643\) 3188.00 0.195525 0.0977624 0.995210i \(-0.468831\pi\)
0.0977624 + 0.995210i \(0.468831\pi\)
\(644\) −3192.00 −0.195314
\(645\) 0 0
\(646\) −6864.00 −0.418050
\(647\) 12732.0 0.773642 0.386821 0.922155i \(-0.373573\pi\)
0.386821 + 0.922155i \(0.373573\pi\)
\(648\) 0 0
\(649\) −10440.0 −0.631442
\(650\) 356.000 0.0214823
\(651\) 0 0
\(652\) 368.000 0.0221043
\(653\) 3576.00 0.214303 0.107151 0.994243i \(-0.465827\pi\)
0.107151 + 0.994243i \(0.465827\pi\)
\(654\) 0 0
\(655\) 9648.00 0.575540
\(656\) 6048.00 0.359961
\(657\) 0 0
\(658\) −3192.00 −0.189114
\(659\) −11430.0 −0.675644 −0.337822 0.941210i \(-0.609690\pi\)
−0.337822 + 0.941210i \(0.609690\pi\)
\(660\) 0 0
\(661\) 22646.0 1.33257 0.666284 0.745698i \(-0.267884\pi\)
0.666284 + 0.745698i \(0.267884\pi\)
\(662\) 19400.0 1.13898
\(663\) 0 0
\(664\) −11520.0 −0.673287
\(665\) 2184.00 0.127356
\(666\) 0 0
\(667\) 8208.00 0.476484
\(668\) 15696.0 0.909126
\(669\) 0 0
\(670\) 7152.00 0.412397
\(671\) 3180.00 0.182955
\(672\) 0 0
\(673\) −13570.0 −0.777244 −0.388622 0.921397i \(-0.627049\pi\)
−0.388622 + 0.921397i \(0.627049\pi\)
\(674\) −16348.0 −0.934275
\(675\) 0 0
\(676\) −8772.00 −0.499090
\(677\) 2838.00 0.161113 0.0805563 0.996750i \(-0.474330\pi\)
0.0805563 + 0.996750i \(0.474330\pi\)
\(678\) 0 0
\(679\) −238.000 −0.0134515
\(680\) −3168.00 −0.178658
\(681\) 0 0
\(682\) −11760.0 −0.660284
\(683\) 6558.00 0.367401 0.183701 0.982982i \(-0.441192\pi\)
0.183701 + 0.982982i \(0.441192\pi\)
\(684\) 0 0
\(685\) −15480.0 −0.863446
\(686\) −686.000 −0.0381802
\(687\) 0 0
\(688\) 2624.00 0.145406
\(689\) 696.000 0.0384840
\(690\) 0 0
\(691\) −21832.0 −1.20192 −0.600961 0.799278i \(-0.705215\pi\)
−0.600961 + 0.799278i \(0.705215\pi\)
\(692\) 7608.00 0.417938
\(693\) 0 0
\(694\) −8076.00 −0.441730
\(695\) −12864.0 −0.702100
\(696\) 0 0
\(697\) −24948.0 −1.35577
\(698\) −21532.0 −1.16762
\(699\) 0 0
\(700\) −2492.00 −0.134555
\(701\) −16200.0 −0.872847 −0.436423 0.899741i \(-0.643755\pi\)
−0.436423 + 0.899741i \(0.643755\pi\)
\(702\) 0 0
\(703\) 14872.0 0.797878
\(704\) −1920.00 −0.102788
\(705\) 0 0
\(706\) 7332.00 0.390855
\(707\) 3066.00 0.163096
\(708\) 0 0
\(709\) 36722.0 1.94517 0.972584 0.232553i \(-0.0747080\pi\)
0.972584 + 0.232553i \(0.0747080\pi\)
\(710\) −7560.00 −0.399608
\(711\) 0 0
\(712\) 10992.0 0.578571
\(713\) 22344.0 1.17362
\(714\) 0 0
\(715\) 360.000 0.0188297
\(716\) 24.0000 0.00125268
\(717\) 0 0
\(718\) −10212.0 −0.530792
\(719\) −13776.0 −0.714545 −0.357273 0.934000i \(-0.616293\pi\)
−0.357273 + 0.934000i \(0.616293\pi\)
\(720\) 0 0
\(721\) 11732.0 0.605995
\(722\) 8310.00 0.428347
\(723\) 0 0
\(724\) −11512.0 −0.590939
\(725\) 6408.00 0.328258
\(726\) 0 0
\(727\) 34220.0 1.74574 0.872868 0.487957i \(-0.162258\pi\)
0.872868 + 0.487957i \(0.162258\pi\)
\(728\) −112.000 −0.00570192
\(729\) 0 0
\(730\) −12504.0 −0.633964
\(731\) −10824.0 −0.547661
\(732\) 0 0
\(733\) −13750.0 −0.692862 −0.346431 0.938075i \(-0.612607\pi\)
−0.346431 + 0.938075i \(0.612607\pi\)
\(734\) 11552.0 0.580916
\(735\) 0 0
\(736\) 3648.00 0.182700
\(737\) −17880.0 −0.893648
\(738\) 0 0
\(739\) 39836.0 1.98294 0.991469 0.130344i \(-0.0416081\pi\)
0.991469 + 0.130344i \(0.0416081\pi\)
\(740\) 6864.00 0.340981
\(741\) 0 0
\(742\) −4872.00 −0.241047
\(743\) −34470.0 −1.70199 −0.850997 0.525170i \(-0.824002\pi\)
−0.850997 + 0.525170i \(0.824002\pi\)
\(744\) 0 0
\(745\) 9000.00 0.442597
\(746\) −16924.0 −0.830605
\(747\) 0 0
\(748\) 7920.00 0.387144
\(749\) −14154.0 −0.690489
\(750\) 0 0
\(751\) 5240.00 0.254608 0.127304 0.991864i \(-0.459368\pi\)
0.127304 + 0.991864i \(0.459368\pi\)
\(752\) 3648.00 0.176900
\(753\) 0 0
\(754\) 288.000 0.0139103
\(755\) 7440.00 0.358635
\(756\) 0 0
\(757\) 18578.0 0.891980 0.445990 0.895038i \(-0.352852\pi\)
0.445990 + 0.895038i \(0.352852\pi\)
\(758\) −13720.0 −0.657431
\(759\) 0 0
\(760\) −2496.00 −0.119131
\(761\) −30534.0 −1.45448 −0.727238 0.686385i \(-0.759196\pi\)
−0.727238 + 0.686385i \(0.759196\pi\)
\(762\) 0 0
\(763\) −3514.00 −0.166730
\(764\) 1416.00 0.0670538
\(765\) 0 0
\(766\) −1392.00 −0.0656593
\(767\) 696.000 0.0327655
\(768\) 0 0
\(769\) −39958.0 −1.87376 −0.936881 0.349650i \(-0.886301\pi\)
−0.936881 + 0.349650i \(0.886301\pi\)
\(770\) −2520.00 −0.117941
\(771\) 0 0
\(772\) −19432.0 −0.905924
\(773\) 3966.00 0.184537 0.0922685 0.995734i \(-0.470588\pi\)
0.0922685 + 0.995734i \(0.470588\pi\)
\(774\) 0 0
\(775\) 17444.0 0.808525
\(776\) 272.000 0.0125828
\(777\) 0 0
\(778\) 22272.0 1.02634
\(779\) −19656.0 −0.904043
\(780\) 0 0
\(781\) 18900.0 0.865935
\(782\) −15048.0 −0.688127
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) −3684.00 −0.167500
\(786\) 0 0
\(787\) −3760.00 −0.170304 −0.0851522 0.996368i \(-0.527138\pi\)
−0.0851522 + 0.996368i \(0.527138\pi\)
\(788\) 1584.00 0.0716087
\(789\) 0 0
\(790\) −1056.00 −0.0475580
\(791\) −14112.0 −0.634343
\(792\) 0 0
\(793\) −212.000 −0.00949349
\(794\) −21676.0 −0.968832
\(795\) 0 0
\(796\) 6848.00 0.304926
\(797\) 24102.0 1.07119 0.535594 0.844476i \(-0.320088\pi\)
0.535594 + 0.844476i \(0.320088\pi\)
\(798\) 0 0
\(799\) −15048.0 −0.666283
\(800\) 2848.00 0.125865
\(801\) 0 0
\(802\) −16728.0 −0.736516
\(803\) 31260.0 1.37378
\(804\) 0 0
\(805\) 4788.00 0.209633
\(806\) 784.000 0.0342621
\(807\) 0 0
\(808\) −3504.00 −0.152562
\(809\) −11712.0 −0.508989 −0.254494 0.967074i \(-0.581909\pi\)
−0.254494 + 0.967074i \(0.581909\pi\)
\(810\) 0 0
\(811\) 37424.0 1.62039 0.810194 0.586162i \(-0.199362\pi\)
0.810194 + 0.586162i \(0.199362\pi\)
\(812\) −2016.00 −0.0871277
\(813\) 0 0
\(814\) −17160.0 −0.738892
\(815\) −552.000 −0.0237248
\(816\) 0 0
\(817\) −8528.00 −0.365186
\(818\) 3524.00 0.150628
\(819\) 0 0
\(820\) −9072.00 −0.386351
\(821\) −13452.0 −0.571837 −0.285918 0.958254i \(-0.592299\pi\)
−0.285918 + 0.958254i \(0.592299\pi\)
\(822\) 0 0
\(823\) 20432.0 0.865389 0.432694 0.901541i \(-0.357563\pi\)
0.432694 + 0.901541i \(0.357563\pi\)
\(824\) −13408.0 −0.566857
\(825\) 0 0
\(826\) −4872.00 −0.205228
\(827\) −24390.0 −1.02554 −0.512771 0.858525i \(-0.671381\pi\)
−0.512771 + 0.858525i \(0.671381\pi\)
\(828\) 0 0
\(829\) −28510.0 −1.19444 −0.597221 0.802076i \(-0.703729\pi\)
−0.597221 + 0.802076i \(0.703729\pi\)
\(830\) 17280.0 0.722648
\(831\) 0 0
\(832\) 128.000 0.00533366
\(833\) −3234.00 −0.134516
\(834\) 0 0
\(835\) −23544.0 −0.975777
\(836\) 6240.00 0.258152
\(837\) 0 0
\(838\) 29160.0 1.20205
\(839\) 36972.0 1.52135 0.760677 0.649131i \(-0.224867\pi\)
0.760677 + 0.649131i \(0.224867\pi\)
\(840\) 0 0
\(841\) −19205.0 −0.787445
\(842\) −17068.0 −0.698577
\(843\) 0 0
\(844\) −3088.00 −0.125940
\(845\) 13158.0 0.535679
\(846\) 0 0
\(847\) −3017.00 −0.122391
\(848\) 5568.00 0.225479
\(849\) 0 0
\(850\) −11748.0 −0.474062
\(851\) 32604.0 1.31334
\(852\) 0 0
\(853\) −14074.0 −0.564929 −0.282465 0.959278i \(-0.591152\pi\)
−0.282465 + 0.959278i \(0.591152\pi\)
\(854\) 1484.00 0.0594630
\(855\) 0 0
\(856\) 16176.0 0.645893
\(857\) 8826.00 0.351797 0.175899 0.984408i \(-0.443717\pi\)
0.175899 + 0.984408i \(0.443717\pi\)
\(858\) 0 0
\(859\) −20500.0 −0.814262 −0.407131 0.913370i \(-0.633471\pi\)
−0.407131 + 0.913370i \(0.633471\pi\)
\(860\) −3936.00 −0.156066
\(861\) 0 0
\(862\) 11868.0 0.468939
\(863\) −7674.00 −0.302695 −0.151348 0.988481i \(-0.548361\pi\)
−0.151348 + 0.988481i \(0.548361\pi\)
\(864\) 0 0
\(865\) −11412.0 −0.448578
\(866\) 29516.0 1.15819
\(867\) 0 0
\(868\) −5488.00 −0.214602
\(869\) 2640.00 0.103056
\(870\) 0 0
\(871\) 1192.00 0.0463713
\(872\) 4016.00 0.155962
\(873\) 0 0
\(874\) −11856.0 −0.458850
\(875\) 8988.00 0.347257
\(876\) 0 0
\(877\) −8890.00 −0.342296 −0.171148 0.985245i \(-0.554748\pi\)
−0.171148 + 0.985245i \(0.554748\pi\)
\(878\) 22784.0 0.875766
\(879\) 0 0
\(880\) 2880.00 0.110324
\(881\) −738.000 −0.0282223 −0.0141112 0.999900i \(-0.504492\pi\)
−0.0141112 + 0.999900i \(0.504492\pi\)
\(882\) 0 0
\(883\) 20.0000 0.000762235 0 0.000381118 1.00000i \(-0.499879\pi\)
0.000381118 1.00000i \(0.499879\pi\)
\(884\) −528.000 −0.0200889
\(885\) 0 0
\(886\) 14052.0 0.532829
\(887\) −39804.0 −1.50675 −0.753375 0.657591i \(-0.771576\pi\)
−0.753375 + 0.657591i \(0.771576\pi\)
\(888\) 0 0
\(889\) 12488.0 0.471129
\(890\) −16488.0 −0.620988
\(891\) 0 0
\(892\) 3104.00 0.116513
\(893\) −11856.0 −0.444284
\(894\) 0 0
\(895\) −36.0000 −0.00134452
\(896\) −896.000 −0.0334077
\(897\) 0 0
\(898\) −6768.00 −0.251505
\(899\) 14112.0 0.523539
\(900\) 0 0
\(901\) −22968.0 −0.849251
\(902\) 22680.0 0.837208
\(903\) 0 0
\(904\) 16128.0 0.593373
\(905\) 17268.0 0.634263
\(906\) 0 0
\(907\) 29180.0 1.06825 0.534127 0.845404i \(-0.320640\pi\)
0.534127 + 0.845404i \(0.320640\pi\)
\(908\) 7152.00 0.261396
\(909\) 0 0
\(910\) 168.000 0.00611994
\(911\) 48258.0 1.75506 0.877530 0.479523i \(-0.159190\pi\)
0.877530 + 0.479523i \(0.159190\pi\)
\(912\) 0 0
\(913\) −43200.0 −1.56595
\(914\) 8564.00 0.309926
\(915\) 0 0
\(916\) 21608.0 0.779420
\(917\) −11256.0 −0.405350
\(918\) 0 0
\(919\) 25760.0 0.924640 0.462320 0.886713i \(-0.347017\pi\)
0.462320 + 0.886713i \(0.347017\pi\)
\(920\) −5472.00 −0.196094
\(921\) 0 0
\(922\) 33300.0 1.18945
\(923\) −1260.00 −0.0449333
\(924\) 0 0
\(925\) 25454.0 0.904781
\(926\) 19328.0 0.685915
\(927\) 0 0
\(928\) 2304.00 0.0815005
\(929\) 44778.0 1.58140 0.790699 0.612205i \(-0.209717\pi\)
0.790699 + 0.612205i \(0.209717\pi\)
\(930\) 0 0
\(931\) −2548.00 −0.0896964
\(932\) −12048.0 −0.423439
\(933\) 0 0
\(934\) −24648.0 −0.863498
\(935\) −11880.0 −0.415527
\(936\) 0 0
\(937\) −44494.0 −1.55129 −0.775643 0.631171i \(-0.782574\pi\)
−0.775643 + 0.631171i \(0.782574\pi\)
\(938\) −8344.00 −0.290449
\(939\) 0 0
\(940\) −5472.00 −0.189869
\(941\) 7458.00 0.258368 0.129184 0.991621i \(-0.458764\pi\)
0.129184 + 0.991621i \(0.458764\pi\)
\(942\) 0 0
\(943\) −43092.0 −1.48809
\(944\) 5568.00 0.191973
\(945\) 0 0
\(946\) 9840.00 0.338188
\(947\) −17790.0 −0.610451 −0.305226 0.952280i \(-0.598732\pi\)
−0.305226 + 0.952280i \(0.598732\pi\)
\(948\) 0 0
\(949\) −2084.00 −0.0712850
\(950\) −9256.00 −0.316110
\(951\) 0 0
\(952\) 3696.00 0.125828
\(953\) 5832.00 0.198234 0.0991170 0.995076i \(-0.468398\pi\)
0.0991170 + 0.995076i \(0.468398\pi\)
\(954\) 0 0
\(955\) −2124.00 −0.0719697
\(956\) −14184.0 −0.479857
\(957\) 0 0
\(958\) 37320.0 1.25862
\(959\) 18060.0 0.608121
\(960\) 0 0
\(961\) 8625.00 0.289517
\(962\) 1144.00 0.0383410
\(963\) 0 0
\(964\) −14248.0 −0.476034
\(965\) 29148.0 0.972339
\(966\) 0 0
\(967\) −13264.0 −0.441098 −0.220549 0.975376i \(-0.570785\pi\)
−0.220549 + 0.975376i \(0.570785\pi\)
\(968\) 3448.00 0.114486
\(969\) 0 0
\(970\) −408.000 −0.0135052
\(971\) −3984.00 −0.131671 −0.0658356 0.997830i \(-0.520971\pi\)
−0.0658356 + 0.997830i \(0.520971\pi\)
\(972\) 0 0
\(973\) 15008.0 0.494485
\(974\) 6800.00 0.223702
\(975\) 0 0
\(976\) −1696.00 −0.0556226
\(977\) 38940.0 1.27513 0.637564 0.770397i \(-0.279942\pi\)
0.637564 + 0.770397i \(0.279942\pi\)
\(978\) 0 0
\(979\) 41220.0 1.34566
\(980\) −1176.00 −0.0383326
\(981\) 0 0
\(982\) 5940.00 0.193028
\(983\) 8808.00 0.285790 0.142895 0.989738i \(-0.454359\pi\)
0.142895 + 0.989738i \(0.454359\pi\)
\(984\) 0 0
\(985\) −2376.00 −0.0768585
\(986\) −9504.00 −0.306967
\(987\) 0 0
\(988\) −416.000 −0.0133955
\(989\) −18696.0 −0.601110
\(990\) 0 0
\(991\) 18488.0 0.592624 0.296312 0.955091i \(-0.404243\pi\)
0.296312 + 0.955091i \(0.404243\pi\)
\(992\) 6272.00 0.200742
\(993\) 0 0
\(994\) 8820.00 0.281442
\(995\) −10272.0 −0.327281
\(996\) 0 0
\(997\) −5002.00 −0.158892 −0.0794458 0.996839i \(-0.525315\pi\)
−0.0794458 + 0.996839i \(0.525315\pi\)
\(998\) 1976.00 0.0626745
\(999\) 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 126.4.a.b.1.1 1
3.2 odd 2 126.4.a.g.1.1 yes 1
4.3 odd 2 1008.4.a.g.1.1 1
7.2 even 3 882.4.g.t.361.1 2
7.3 odd 6 882.4.g.q.667.1 2
7.4 even 3 882.4.g.t.667.1 2
7.5 odd 6 882.4.g.q.361.1 2
7.6 odd 2 882.4.a.e.1.1 1
12.11 even 2 1008.4.a.n.1.1 1
21.2 odd 6 882.4.g.e.361.1 2
21.5 even 6 882.4.g.h.361.1 2
21.11 odd 6 882.4.g.e.667.1 2
21.17 even 6 882.4.g.h.667.1 2
21.20 even 2 882.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.4.a.b.1.1 1 1.1 even 1 trivial
126.4.a.g.1.1 yes 1 3.2 odd 2
882.4.a.e.1.1 1 7.6 odd 2
882.4.a.m.1.1 1 21.20 even 2
882.4.g.e.361.1 2 21.2 odd 6
882.4.g.e.667.1 2 21.11 odd 6
882.4.g.h.361.1 2 21.5 even 6
882.4.g.h.667.1 2 21.17 even 6
882.4.g.q.361.1 2 7.5 odd 6
882.4.g.q.667.1 2 7.3 odd 6
882.4.g.t.361.1 2 7.2 even 3
882.4.g.t.667.1 2 7.4 even 3
1008.4.a.g.1.1 1 4.3 odd 2
1008.4.a.n.1.1 1 12.11 even 2