Properties

Label 63.4.a.c.1.1
Level $63$
Weight $4$
Character 63.1
Self dual yes
Analytic conductor $3.717$
Analytic rank $0$
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: \(0\)
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 \(\chi\) \(=\) 63.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{2} +1.00000 q^{4} +18.0000 q^{5} +7.00000 q^{7} -21.0000 q^{8} +O(q^{10})\) \(q+3.00000 q^{2} +1.00000 q^{4} +18.0000 q^{5} +7.00000 q^{7} -21.0000 q^{8} +54.0000 q^{10} +36.0000 q^{11} -34.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} -42.0000 q^{17} -124.000 q^{19} +18.0000 q^{20} +108.000 q^{22} +199.000 q^{25} -102.000 q^{26} +7.00000 q^{28} -102.000 q^{29} -160.000 q^{31} -45.0000 q^{32} -126.000 q^{34} +126.000 q^{35} +398.000 q^{37} -372.000 q^{38} -378.000 q^{40} +318.000 q^{41} -268.000 q^{43} +36.0000 q^{44} -240.000 q^{47} +49.0000 q^{49} +597.000 q^{50} -34.0000 q^{52} +498.000 q^{53} +648.000 q^{55} -147.000 q^{56} -306.000 q^{58} +132.000 q^{59} +398.000 q^{61} -480.000 q^{62} +433.000 q^{64} -612.000 q^{65} +92.0000 q^{67} -42.0000 q^{68} +378.000 q^{70} +720.000 q^{71} -502.000 q^{73} +1194.00 q^{74} -124.000 q^{76} +252.000 q^{77} -1024.00 q^{79} -1278.00 q^{80} +954.000 q^{82} +204.000 q^{83} -756.000 q^{85} -804.000 q^{86} -756.000 q^{88} -354.000 q^{89} -238.000 q^{91} -720.000 q^{94} -2232.00 q^{95} -286.000 q^{97} +147.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −21.0000 −0.928078
\(9\) 0 0
\(10\) 54.0000 1.70763
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) −34.0000 −0.725377 −0.362689 0.931910i \(-0.618141\pi\)
−0.362689 + 0.931910i \(0.618141\pi\)
\(14\) 21.0000 0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −42.0000 −0.599206 −0.299603 0.954064i \(-0.596854\pi\)
−0.299603 + 0.954064i \(0.596854\pi\)
\(18\) 0 0
\(19\) −124.000 −1.49724 −0.748620 0.663000i \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) 18.0000 0.201246
\(21\) 0 0
\(22\) 108.000 1.04662
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) −102.000 −0.769379
\(27\) 0 0
\(28\) 7.00000 0.0472456
\(29\) −102.000 −0.653135 −0.326568 0.945174i \(-0.605892\pi\)
−0.326568 + 0.945174i \(0.605892\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) −45.0000 −0.248592
\(33\) 0 0
\(34\) −126.000 −0.635554
\(35\) 126.000 0.608511
\(36\) 0 0
\(37\) 398.000 1.76840 0.884200 0.467109i \(-0.154704\pi\)
0.884200 + 0.467109i \(0.154704\pi\)
\(38\) −372.000 −1.58806
\(39\) 0 0
\(40\) −378.000 −1.49418
\(41\) 318.000 1.21130 0.605649 0.795732i \(-0.292913\pi\)
0.605649 + 0.795732i \(0.292913\pi\)
\(42\) 0 0
\(43\) −268.000 −0.950456 −0.475228 0.879863i \(-0.657634\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(44\) 36.0000 0.123346
\(45\) 0 0
\(46\) 0 0
\(47\) −240.000 −0.744843 −0.372421 0.928064i \(-0.621472\pi\)
−0.372421 + 0.928064i \(0.621472\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 597.000 1.68857
\(51\) 0 0
\(52\) −34.0000 −0.0906721
\(53\) 498.000 1.29067 0.645335 0.763899i \(-0.276718\pi\)
0.645335 + 0.763899i \(0.276718\pi\)
\(54\) 0 0
\(55\) 648.000 1.58866
\(56\) −147.000 −0.350780
\(57\) 0 0
\(58\) −306.000 −0.692755
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) 0 0
\(61\) 398.000 0.835388 0.417694 0.908588i \(-0.362838\pi\)
0.417694 + 0.908588i \(0.362838\pi\)
\(62\) −480.000 −0.983227
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) −612.000 −1.16783
\(66\) 0 0
\(67\) 92.0000 0.167755 0.0838775 0.996476i \(-0.473270\pi\)
0.0838775 + 0.996476i \(0.473270\pi\)
\(68\) −42.0000 −0.0749007
\(69\) 0 0
\(70\) 378.000 0.645423
\(71\) 720.000 1.20350 0.601748 0.798686i \(-0.294471\pi\)
0.601748 + 0.798686i \(0.294471\pi\)
\(72\) 0 0
\(73\) −502.000 −0.804858 −0.402429 0.915451i \(-0.631834\pi\)
−0.402429 + 0.915451i \(0.631834\pi\)
\(74\) 1194.00 1.87567
\(75\) 0 0
\(76\) −124.000 −0.187155
\(77\) 252.000 0.372962
\(78\) 0 0
\(79\) −1024.00 −1.45834 −0.729171 0.684332i \(-0.760094\pi\)
−0.729171 + 0.684332i \(0.760094\pi\)
\(80\) −1278.00 −1.78606
\(81\) 0 0
\(82\) 954.000 1.28478
\(83\) 204.000 0.269782 0.134891 0.990860i \(-0.456932\pi\)
0.134891 + 0.990860i \(0.456932\pi\)
\(84\) 0 0
\(85\) −756.000 −0.964703
\(86\) −804.000 −1.00811
\(87\) 0 0
\(88\) −756.000 −0.915794
\(89\) −354.000 −0.421617 −0.210809 0.977527i \(-0.567610\pi\)
−0.210809 + 0.977527i \(0.567610\pi\)
\(90\) 0 0
\(91\) −238.000 −0.274167
\(92\) 0 0
\(93\) 0 0
\(94\) −720.000 −0.790025
\(95\) −2232.00 −2.41051
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 147.000 0.151523
\(99\) 0 0
\(100\) 199.000 0.199000
\(101\) −414.000 −0.407867 −0.203933 0.978985i \(-0.565373\pi\)
−0.203933 + 0.978985i \(0.565373\pi\)
\(102\) 0 0
\(103\) 56.0000 0.0535713 0.0267857 0.999641i \(-0.491473\pi\)
0.0267857 + 0.999641i \(0.491473\pi\)
\(104\) 714.000 0.673206
\(105\) 0 0
\(106\) 1494.00 1.36896
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 0 0
\(109\) 1478.00 1.29878 0.649389 0.760457i \(-0.275025\pi\)
0.649389 + 0.760457i \(0.275025\pi\)
\(110\) 1944.00 1.68503
\(111\) 0 0
\(112\) −497.000 −0.419304
\(113\) −402.000 −0.334664 −0.167332 0.985901i \(-0.553515\pi\)
−0.167332 + 0.985901i \(0.553515\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −102.000 −0.0816419
\(117\) 0 0
\(118\) 396.000 0.308939
\(119\) −294.000 −0.226478
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 1194.00 0.886063
\(123\) 0 0
\(124\) −160.000 −0.115874
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) 1280.00 0.894344 0.447172 0.894448i \(-0.352431\pi\)
0.447172 + 0.894448i \(0.352431\pi\)
\(128\) 1659.00 1.14560
\(129\) 0 0
\(130\) −1836.00 −1.23868
\(131\) −1764.00 −1.17650 −0.588250 0.808679i \(-0.700183\pi\)
−0.588250 + 0.808679i \(0.700183\pi\)
\(132\) 0 0
\(133\) −868.000 −0.565903
\(134\) 276.000 0.177931
\(135\) 0 0
\(136\) 882.000 0.556109
\(137\) 2358.00 1.47049 0.735246 0.677800i \(-0.237066\pi\)
0.735246 + 0.677800i \(0.237066\pi\)
\(138\) 0 0
\(139\) −52.0000 −0.0317308 −0.0158654 0.999874i \(-0.505050\pi\)
−0.0158654 + 0.999874i \(0.505050\pi\)
\(140\) 126.000 0.0760639
\(141\) 0 0
\(142\) 2160.00 1.27650
\(143\) −1224.00 −0.715776
\(144\) 0 0
\(145\) −1836.00 −1.05153
\(146\) −1506.00 −0.853681
\(147\) 0 0
\(148\) 398.000 0.221050
\(149\) 1746.00 0.959986 0.479993 0.877272i \(-0.340639\pi\)
0.479993 + 0.877272i \(0.340639\pi\)
\(150\) 0 0
\(151\) −232.000 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(152\) 2604.00 1.38955
\(153\) 0 0
\(154\) 756.000 0.395586
\(155\) −2880.00 −1.49243
\(156\) 0 0
\(157\) 1694.00 0.861120 0.430560 0.902562i \(-0.358316\pi\)
0.430560 + 0.902562i \(0.358316\pi\)
\(158\) −3072.00 −1.54681
\(159\) 0 0
\(160\) −810.000 −0.400226
\(161\) 0 0
\(162\) 0 0
\(163\) −2932.00 −1.40891 −0.704454 0.709750i \(-0.748808\pi\)
−0.704454 + 0.709750i \(0.748808\pi\)
\(164\) 318.000 0.151412
\(165\) 0 0
\(166\) 612.000 0.286147
\(167\) −1176.00 −0.544920 −0.272460 0.962167i \(-0.587837\pi\)
−0.272460 + 0.962167i \(0.587837\pi\)
\(168\) 0 0
\(169\) −1041.00 −0.473828
\(170\) −2268.00 −1.02322
\(171\) 0 0
\(172\) −268.000 −0.118807
\(173\) −870.000 −0.382340 −0.191170 0.981557i \(-0.561228\pi\)
−0.191170 + 0.981557i \(0.561228\pi\)
\(174\) 0 0
\(175\) 1393.00 0.601719
\(176\) −2556.00 −1.09469
\(177\) 0 0
\(178\) −1062.00 −0.447193
\(179\) 2316.00 0.967072 0.483536 0.875324i \(-0.339352\pi\)
0.483536 + 0.875324i \(0.339352\pi\)
\(180\) 0 0
\(181\) −106.000 −0.0435299 −0.0217650 0.999763i \(-0.506929\pi\)
−0.0217650 + 0.999763i \(0.506929\pi\)
\(182\) −714.000 −0.290798
\(183\) 0 0
\(184\) 0 0
\(185\) 7164.00 2.84707
\(186\) 0 0
\(187\) −1512.00 −0.591275
\(188\) −240.000 −0.0931053
\(189\) 0 0
\(190\) −6696.00 −2.55673
\(191\) 1128.00 0.427326 0.213663 0.976907i \(-0.431461\pi\)
0.213663 + 0.976907i \(0.431461\pi\)
\(192\) 0 0
\(193\) 4034.00 1.50453 0.752263 0.658862i \(-0.228962\pi\)
0.752263 + 0.658862i \(0.228962\pi\)
\(194\) −858.000 −0.317530
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) 1314.00 0.475221 0.237611 0.971360i \(-0.423636\pi\)
0.237611 + 0.971360i \(0.423636\pi\)
\(198\) 0 0
\(199\) 5096.00 1.81531 0.907653 0.419722i \(-0.137872\pi\)
0.907653 + 0.419722i \(0.137872\pi\)
\(200\) −4179.00 −1.47750
\(201\) 0 0
\(202\) −1242.00 −0.432608
\(203\) −714.000 −0.246862
\(204\) 0 0
\(205\) 5724.00 1.95015
\(206\) 168.000 0.0568209
\(207\) 0 0
\(208\) 2414.00 0.804715
\(209\) −4464.00 −1.47742
\(210\) 0 0
\(211\) −3076.00 −1.00360 −0.501802 0.864982i \(-0.667330\pi\)
−0.501802 + 0.864982i \(0.667330\pi\)
\(212\) 498.000 0.161334
\(213\) 0 0
\(214\) −36.0000 −0.0114996
\(215\) −4824.00 −1.53020
\(216\) 0 0
\(217\) −1120.00 −0.350371
\(218\) 4434.00 1.37756
\(219\) 0 0
\(220\) 648.000 0.198583
\(221\) 1428.00 0.434650
\(222\) 0 0
\(223\) −1888.00 −0.566950 −0.283475 0.958980i \(-0.591487\pi\)
−0.283475 + 0.958980i \(0.591487\pi\)
\(224\) −315.000 −0.0939590
\(225\) 0 0
\(226\) −1206.00 −0.354964
\(227\) 4716.00 1.37891 0.689454 0.724330i \(-0.257851\pi\)
0.689454 + 0.724330i \(0.257851\pi\)
\(228\) 0 0
\(229\) −1690.00 −0.487678 −0.243839 0.969816i \(-0.578407\pi\)
−0.243839 + 0.969816i \(0.578407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2142.00 0.606160
\(233\) −138.000 −0.0388012 −0.0194006 0.999812i \(-0.506176\pi\)
−0.0194006 + 0.999812i \(0.506176\pi\)
\(234\) 0 0
\(235\) −4320.00 −1.19917
\(236\) 132.000 0.0364088
\(237\) 0 0
\(238\) −882.000 −0.240217
\(239\) −1896.00 −0.513147 −0.256573 0.966525i \(-0.582594\pi\)
−0.256573 + 0.966525i \(0.582594\pi\)
\(240\) 0 0
\(241\) −3598.00 −0.961691 −0.480846 0.876805i \(-0.659670\pi\)
−0.480846 + 0.876805i \(0.659670\pi\)
\(242\) −105.000 −0.0278911
\(243\) 0 0
\(244\) 398.000 0.104424
\(245\) 882.000 0.229996
\(246\) 0 0
\(247\) 4216.00 1.08606
\(248\) 3360.00 0.860323
\(249\) 0 0
\(250\) 3996.00 1.01092
\(251\) 3060.00 0.769504 0.384752 0.923020i \(-0.374287\pi\)
0.384752 + 0.923020i \(0.374287\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 3840.00 0.948595
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) 6822.00 1.65582 0.827908 0.560864i \(-0.189531\pi\)
0.827908 + 0.560864i \(0.189531\pi\)
\(258\) 0 0
\(259\) 2786.00 0.668392
\(260\) −612.000 −0.145979
\(261\) 0 0
\(262\) −5292.00 −1.24787
\(263\) −2592.00 −0.607717 −0.303858 0.952717i \(-0.598275\pi\)
−0.303858 + 0.952717i \(0.598275\pi\)
\(264\) 0 0
\(265\) 8964.00 2.07794
\(266\) −2604.00 −0.600231
\(267\) 0 0
\(268\) 92.0000 0.0209694
\(269\) −8214.00 −1.86177 −0.930886 0.365311i \(-0.880963\pi\)
−0.930886 + 0.365311i \(0.880963\pi\)
\(270\) 0 0
\(271\) −5344.00 −1.19788 −0.598939 0.800795i \(-0.704411\pi\)
−0.598939 + 0.800795i \(0.704411\pi\)
\(272\) 2982.00 0.664744
\(273\) 0 0
\(274\) 7074.00 1.55969
\(275\) 7164.00 1.57093
\(276\) 0 0
\(277\) −6514.00 −1.41295 −0.706477 0.707736i \(-0.749717\pi\)
−0.706477 + 0.707736i \(0.749717\pi\)
\(278\) −156.000 −0.0336556
\(279\) 0 0
\(280\) −2646.00 −0.564746
\(281\) −6618.00 −1.40497 −0.702485 0.711698i \(-0.747926\pi\)
−0.702485 + 0.711698i \(0.747926\pi\)
\(282\) 0 0
\(283\) 3260.00 0.684759 0.342380 0.939562i \(-0.388767\pi\)
0.342380 + 0.939562i \(0.388767\pi\)
\(284\) 720.000 0.150437
\(285\) 0 0
\(286\) −3672.00 −0.759195
\(287\) 2226.00 0.457828
\(288\) 0 0
\(289\) −3149.00 −0.640953
\(290\) −5508.00 −1.11531
\(291\) 0 0
\(292\) −502.000 −0.100607
\(293\) −5118.00 −1.02047 −0.510233 0.860036i \(-0.670441\pi\)
−0.510233 + 0.860036i \(0.670441\pi\)
\(294\) 0 0
\(295\) 2376.00 0.468936
\(296\) −8358.00 −1.64121
\(297\) 0 0
\(298\) 5238.00 1.01822
\(299\) 0 0
\(300\) 0 0
\(301\) −1876.00 −0.359239
\(302\) −696.000 −0.132617
\(303\) 0 0
\(304\) 8804.00 1.66100
\(305\) 7164.00 1.34495
\(306\) 0 0
\(307\) 452.000 0.0840293 0.0420147 0.999117i \(-0.486622\pi\)
0.0420147 + 0.999117i \(0.486622\pi\)
\(308\) 252.000 0.0466202
\(309\) 0 0
\(310\) −8640.00 −1.58296
\(311\) −5016.00 −0.914570 −0.457285 0.889320i \(-0.651178\pi\)
−0.457285 + 0.889320i \(0.651178\pi\)
\(312\) 0 0
\(313\) 5402.00 0.975524 0.487762 0.872977i \(-0.337813\pi\)
0.487762 + 0.872977i \(0.337813\pi\)
\(314\) 5082.00 0.913356
\(315\) 0 0
\(316\) −1024.00 −0.182293
\(317\) −10086.0 −1.78702 −0.893511 0.449041i \(-0.851766\pi\)
−0.893511 + 0.449041i \(0.851766\pi\)
\(318\) 0 0
\(319\) −3672.00 −0.644491
\(320\) 7794.00 1.36156
\(321\) 0 0
\(322\) 0 0
\(323\) 5208.00 0.897154
\(324\) 0 0
\(325\) −6766.00 −1.15480
\(326\) −8796.00 −1.49437
\(327\) 0 0
\(328\) −6678.00 −1.12418
\(329\) −1680.00 −0.281524
\(330\) 0 0
\(331\) −8044.00 −1.33577 −0.667883 0.744267i \(-0.732799\pi\)
−0.667883 + 0.744267i \(0.732799\pi\)
\(332\) 204.000 0.0337228
\(333\) 0 0
\(334\) −3528.00 −0.577975
\(335\) 1656.00 0.270080
\(336\) 0 0
\(337\) 4178.00 0.675342 0.337671 0.941264i \(-0.390361\pi\)
0.337671 + 0.941264i \(0.390361\pi\)
\(338\) −3123.00 −0.502570
\(339\) 0 0
\(340\) −756.000 −0.120588
\(341\) −5760.00 −0.914726
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 5628.00 0.882097
\(345\) 0 0
\(346\) −2610.00 −0.405533
\(347\) −156.000 −0.0241341 −0.0120670 0.999927i \(-0.503841\pi\)
−0.0120670 + 0.999927i \(0.503841\pi\)
\(348\) 0 0
\(349\) −12418.0 −1.90464 −0.952321 0.305097i \(-0.901311\pi\)
−0.952321 + 0.305097i \(0.901311\pi\)
\(350\) 4179.00 0.638220
\(351\) 0 0
\(352\) −1620.00 −0.245302
\(353\) 7830.00 1.18059 0.590296 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(354\) 0 0
\(355\) 12960.0 1.93759
\(356\) −354.000 −0.0527021
\(357\) 0 0
\(358\) 6948.00 1.02574
\(359\) 9312.00 1.36899 0.684497 0.729016i \(-0.260022\pi\)
0.684497 + 0.729016i \(0.260022\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) −318.000 −0.0461705
\(363\) 0 0
\(364\) −238.000 −0.0342709
\(365\) −9036.00 −1.29580
\(366\) 0 0
\(367\) −3760.00 −0.534797 −0.267398 0.963586i \(-0.586164\pi\)
−0.267398 + 0.963586i \(0.586164\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 21492.0 3.01977
\(371\) 3486.00 0.487828
\(372\) 0 0
\(373\) 5870.00 0.814845 0.407422 0.913240i \(-0.366428\pi\)
0.407422 + 0.913240i \(0.366428\pi\)
\(374\) −4536.00 −0.627142
\(375\) 0 0
\(376\) 5040.00 0.691272
\(377\) 3468.00 0.473769
\(378\) 0 0
\(379\) −1852.00 −0.251005 −0.125502 0.992093i \(-0.540054\pi\)
−0.125502 + 0.992093i \(0.540054\pi\)
\(380\) −2232.00 −0.301314
\(381\) 0 0
\(382\) 3384.00 0.453247
\(383\) −2160.00 −0.288175 −0.144087 0.989565i \(-0.546025\pi\)
−0.144087 + 0.989565i \(0.546025\pi\)
\(384\) 0 0
\(385\) 4536.00 0.600457
\(386\) 12102.0 1.59579
\(387\) 0 0
\(388\) −286.000 −0.0374213
\(389\) 6786.00 0.884483 0.442241 0.896896i \(-0.354183\pi\)
0.442241 + 0.896896i \(0.354183\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1029.00 −0.132583
\(393\) 0 0
\(394\) 3942.00 0.504048
\(395\) −18432.0 −2.34788
\(396\) 0 0
\(397\) −6514.00 −0.823497 −0.411748 0.911298i \(-0.635082\pi\)
−0.411748 + 0.911298i \(0.635082\pi\)
\(398\) 15288.0 1.92542
\(399\) 0 0
\(400\) −14129.0 −1.76612
\(401\) −3330.00 −0.414694 −0.207347 0.978267i \(-0.566483\pi\)
−0.207347 + 0.978267i \(0.566483\pi\)
\(402\) 0 0
\(403\) 5440.00 0.672421
\(404\) −414.000 −0.0509833
\(405\) 0 0
\(406\) −2142.00 −0.261837
\(407\) 14328.0 1.74499
\(408\) 0 0
\(409\) −5398.00 −0.652601 −0.326301 0.945266i \(-0.605802\pi\)
−0.326301 + 0.945266i \(0.605802\pi\)
\(410\) 17172.0 2.06845
\(411\) 0 0
\(412\) 56.0000 0.00669641
\(413\) 924.000 0.110090
\(414\) 0 0
\(415\) 3672.00 0.434341
\(416\) 1530.00 0.180323
\(417\) 0 0
\(418\) −13392.0 −1.56704
\(419\) −13092.0 −1.52646 −0.763229 0.646128i \(-0.776387\pi\)
−0.763229 + 0.646128i \(0.776387\pi\)
\(420\) 0 0
\(421\) −322.000 −0.0372763 −0.0186381 0.999826i \(-0.505933\pi\)
−0.0186381 + 0.999826i \(0.505933\pi\)
\(422\) −9228.00 −1.06448
\(423\) 0 0
\(424\) −10458.0 −1.19784
\(425\) −8358.00 −0.953935
\(426\) 0 0
\(427\) 2786.00 0.315747
\(428\) −12.0000 −0.00135524
\(429\) 0 0
\(430\) −14472.0 −1.62303
\(431\) −2616.00 −0.292363 −0.146181 0.989258i \(-0.546698\pi\)
−0.146181 + 0.989258i \(0.546698\pi\)
\(432\) 0 0
\(433\) 4322.00 0.479681 0.239841 0.970812i \(-0.422905\pi\)
0.239841 + 0.970812i \(0.422905\pi\)
\(434\) −3360.00 −0.371625
\(435\) 0 0
\(436\) 1478.00 0.162347
\(437\) 0 0
\(438\) 0 0
\(439\) −9016.00 −0.980205 −0.490103 0.871665i \(-0.663041\pi\)
−0.490103 + 0.871665i \(0.663041\pi\)
\(440\) −13608.0 −1.47440
\(441\) 0 0
\(442\) 4284.00 0.461016
\(443\) 5268.00 0.564989 0.282495 0.959269i \(-0.408838\pi\)
0.282495 + 0.959269i \(0.408838\pi\)
\(444\) 0 0
\(445\) −6372.00 −0.678790
\(446\) −5664.00 −0.601341
\(447\) 0 0
\(448\) 3031.00 0.319646
\(449\) 5310.00 0.558117 0.279058 0.960274i \(-0.409978\pi\)
0.279058 + 0.960274i \(0.409978\pi\)
\(450\) 0 0
\(451\) 11448.0 1.19527
\(452\) −402.000 −0.0418329
\(453\) 0 0
\(454\) 14148.0 1.46255
\(455\) −4284.00 −0.441400
\(456\) 0 0
\(457\) 15770.0 1.61420 0.807100 0.590415i \(-0.201036\pi\)
0.807100 + 0.590415i \(0.201036\pi\)
\(458\) −5070.00 −0.517261
\(459\) 0 0
\(460\) 0 0
\(461\) 5370.00 0.542529 0.271264 0.962505i \(-0.412558\pi\)
0.271264 + 0.962505i \(0.412558\pi\)
\(462\) 0 0
\(463\) −3328.00 −0.334050 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(464\) 7242.00 0.724572
\(465\) 0 0
\(466\) −414.000 −0.0411549
\(467\) −4548.00 −0.450656 −0.225328 0.974283i \(-0.572345\pi\)
−0.225328 + 0.974283i \(0.572345\pi\)
\(468\) 0 0
\(469\) 644.000 0.0634055
\(470\) −12960.0 −1.27192
\(471\) 0 0
\(472\) −2772.00 −0.270321
\(473\) −9648.00 −0.937876
\(474\) 0 0
\(475\) −24676.0 −2.38361
\(476\) −294.000 −0.0283098
\(477\) 0 0
\(478\) −5688.00 −0.544274
\(479\) 8064.00 0.769214 0.384607 0.923080i \(-0.374337\pi\)
0.384607 + 0.923080i \(0.374337\pi\)
\(480\) 0 0
\(481\) −13532.0 −1.28276
\(482\) −10794.0 −1.02003
\(483\) 0 0
\(484\) −35.0000 −0.00328700
\(485\) −5148.00 −0.481977
\(486\) 0 0
\(487\) 16616.0 1.54608 0.773042 0.634355i \(-0.218734\pi\)
0.773042 + 0.634355i \(0.218734\pi\)
\(488\) −8358.00 −0.775305
\(489\) 0 0
\(490\) 2646.00 0.243947
\(491\) 7140.00 0.656260 0.328130 0.944633i \(-0.393582\pi\)
0.328130 + 0.944633i \(0.393582\pi\)
\(492\) 0 0
\(493\) 4284.00 0.391362
\(494\) 12648.0 1.15194
\(495\) 0 0
\(496\) 11360.0 1.02839
\(497\) 5040.00 0.454879
\(498\) 0 0
\(499\) −9124.00 −0.818530 −0.409265 0.912416i \(-0.634215\pi\)
−0.409265 + 0.912416i \(0.634215\pi\)
\(500\) 1332.00 0.119138
\(501\) 0 0
\(502\) 9180.00 0.816182
\(503\) 6552.00 0.580794 0.290397 0.956906i \(-0.406213\pi\)
0.290397 + 0.956906i \(0.406213\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) 0 0
\(508\) 1280.00 0.111793
\(509\) −2790.00 −0.242956 −0.121478 0.992594i \(-0.538763\pi\)
−0.121478 + 0.992594i \(0.538763\pi\)
\(510\) 0 0
\(511\) −3514.00 −0.304208
\(512\) −8733.00 −0.753804
\(513\) 0 0
\(514\) 20466.0 1.75626
\(515\) 1008.00 0.0862481
\(516\) 0 0
\(517\) −8640.00 −0.734984
\(518\) 8358.00 0.708937
\(519\) 0 0
\(520\) 12852.0 1.08384
\(521\) 14862.0 1.24974 0.624871 0.780728i \(-0.285151\pi\)
0.624871 + 0.780728i \(0.285151\pi\)
\(522\) 0 0
\(523\) 17660.0 1.47652 0.738258 0.674518i \(-0.235649\pi\)
0.738258 + 0.674518i \(0.235649\pi\)
\(524\) −1764.00 −0.147062
\(525\) 0 0
\(526\) −7776.00 −0.644581
\(527\) 6720.00 0.555461
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 26892.0 2.20399
\(531\) 0 0
\(532\) −868.000 −0.0707379
\(533\) −10812.0 −0.878649
\(534\) 0 0
\(535\) −216.000 −0.0174551
\(536\) −1932.00 −0.155690
\(537\) 0 0
\(538\) −24642.0 −1.97471
\(539\) 1764.00 0.140966
\(540\) 0 0
\(541\) −19834.0 −1.57621 −0.788106 0.615540i \(-0.788938\pi\)
−0.788106 + 0.615540i \(0.788938\pi\)
\(542\) −16032.0 −1.27054
\(543\) 0 0
\(544\) 1890.00 0.148958
\(545\) 26604.0 2.09099
\(546\) 0 0
\(547\) 20972.0 1.63930 0.819651 0.572863i \(-0.194167\pi\)
0.819651 + 0.572863i \(0.194167\pi\)
\(548\) 2358.00 0.183812
\(549\) 0 0
\(550\) 21492.0 1.66622
\(551\) 12648.0 0.977900
\(552\) 0 0
\(553\) −7168.00 −0.551201
\(554\) −19542.0 −1.49866
\(555\) 0 0
\(556\) −52.0000 −0.00396635
\(557\) −21174.0 −1.61072 −0.805360 0.592786i \(-0.798028\pi\)
−0.805360 + 0.592786i \(0.798028\pi\)
\(558\) 0 0
\(559\) 9112.00 0.689439
\(560\) −8946.00 −0.675067
\(561\) 0 0
\(562\) −19854.0 −1.49020
\(563\) 17772.0 1.33037 0.665187 0.746677i \(-0.268352\pi\)
0.665187 + 0.746677i \(0.268352\pi\)
\(564\) 0 0
\(565\) −7236.00 −0.538798
\(566\) 9780.00 0.726297
\(567\) 0 0
\(568\) −15120.0 −1.11694
\(569\) −8250.00 −0.607835 −0.303917 0.952698i \(-0.598295\pi\)
−0.303917 + 0.952698i \(0.598295\pi\)
\(570\) 0 0
\(571\) 20756.0 1.52121 0.760606 0.649214i \(-0.224902\pi\)
0.760606 + 0.649214i \(0.224902\pi\)
\(572\) −1224.00 −0.0894720
\(573\) 0 0
\(574\) 6678.00 0.485600
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.000144300 0 7.21500e−5 1.00000i \(-0.499977\pi\)
7.21500e−5 1.00000i \(0.499977\pi\)
\(578\) −9447.00 −0.679833
\(579\) 0 0
\(580\) −1836.00 −0.131441
\(581\) 1428.00 0.101968
\(582\) 0 0
\(583\) 17928.0 1.27359
\(584\) 10542.0 0.746971
\(585\) 0 0
\(586\) −15354.0 −1.08237
\(587\) −26364.0 −1.85376 −0.926881 0.375354i \(-0.877521\pi\)
−0.926881 + 0.375354i \(0.877521\pi\)
\(588\) 0 0
\(589\) 19840.0 1.38793
\(590\) 7128.00 0.497382
\(591\) 0 0
\(592\) −28258.0 −1.96182
\(593\) −2298.00 −0.159136 −0.0795679 0.996829i \(-0.525354\pi\)
−0.0795679 + 0.996829i \(0.525354\pi\)
\(594\) 0 0
\(595\) −5292.00 −0.364623
\(596\) 1746.00 0.119998
\(597\) 0 0
\(598\) 0 0
\(599\) −3072.00 −0.209547 −0.104773 0.994496i \(-0.533412\pi\)
−0.104773 + 0.994496i \(0.533412\pi\)
\(600\) 0 0
\(601\) 24554.0 1.66652 0.833260 0.552881i \(-0.186472\pi\)
0.833260 + 0.552881i \(0.186472\pi\)
\(602\) −5628.00 −0.381030
\(603\) 0 0
\(604\) −232.000 −0.0156290
\(605\) −630.000 −0.0423358
\(606\) 0 0
\(607\) 16832.0 1.12552 0.562759 0.826621i \(-0.309740\pi\)
0.562759 + 0.826621i \(0.309740\pi\)
\(608\) 5580.00 0.372202
\(609\) 0 0
\(610\) 21492.0 1.42653
\(611\) 8160.00 0.540292
\(612\) 0 0
\(613\) −2482.00 −0.163535 −0.0817676 0.996651i \(-0.526057\pi\)
−0.0817676 + 0.996651i \(0.526057\pi\)
\(614\) 1356.00 0.0891266
\(615\) 0 0
\(616\) −5292.00 −0.346138
\(617\) 15798.0 1.03080 0.515400 0.856950i \(-0.327643\pi\)
0.515400 + 0.856950i \(0.327643\pi\)
\(618\) 0 0
\(619\) −15460.0 −1.00386 −0.501930 0.864908i \(-0.667377\pi\)
−0.501930 + 0.864908i \(0.667377\pi\)
\(620\) −2880.00 −0.186554
\(621\) 0 0
\(622\) −15048.0 −0.970048
\(623\) −2478.00 −0.159356
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 16206.0 1.03470
\(627\) 0 0
\(628\) 1694.00 0.107640
\(629\) −16716.0 −1.05964
\(630\) 0 0
\(631\) −7720.00 −0.487050 −0.243525 0.969895i \(-0.578304\pi\)
−0.243525 + 0.969895i \(0.578304\pi\)
\(632\) 21504.0 1.35345
\(633\) 0 0
\(634\) −30258.0 −1.89542
\(635\) 23040.0 1.43987
\(636\) 0 0
\(637\) −1666.00 −0.103625
\(638\) −11016.0 −0.683586
\(639\) 0 0
\(640\) 29862.0 1.84437
\(641\) 17262.0 1.06366 0.531832 0.846850i \(-0.321504\pi\)
0.531832 + 0.846850i \(0.321504\pi\)
\(642\) 0 0
\(643\) −12220.0 −0.749471 −0.374735 0.927132i \(-0.622266\pi\)
−0.374735 + 0.927132i \(0.622266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 15624.0 0.951576
\(647\) −13560.0 −0.823955 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(648\) 0 0
\(649\) 4752.00 0.287415
\(650\) −20298.0 −1.22485
\(651\) 0 0
\(652\) −2932.00 −0.176113
\(653\) −23094.0 −1.38398 −0.691989 0.721908i \(-0.743265\pi\)
−0.691989 + 0.721908i \(0.743265\pi\)
\(654\) 0 0
\(655\) −31752.0 −1.89413
\(656\) −22578.0 −1.34378
\(657\) 0 0
\(658\) −5040.00 −0.298601
\(659\) −22548.0 −1.33285 −0.666423 0.745574i \(-0.732175\pi\)
−0.666423 + 0.745574i \(0.732175\pi\)
\(660\) 0 0
\(661\) 17462.0 1.02752 0.513762 0.857933i \(-0.328252\pi\)
0.513762 + 0.857933i \(0.328252\pi\)
\(662\) −24132.0 −1.41679
\(663\) 0 0
\(664\) −4284.00 −0.250379
\(665\) −15624.0 −0.911087
\(666\) 0 0
\(667\) 0 0
\(668\) −1176.00 −0.0681150
\(669\) 0 0
\(670\) 4968.00 0.286464
\(671\) 14328.0 0.824331
\(672\) 0 0
\(673\) −22462.0 −1.28655 −0.643274 0.765636i \(-0.722424\pi\)
−0.643274 + 0.765636i \(0.722424\pi\)
\(674\) 12534.0 0.716308
\(675\) 0 0
\(676\) −1041.00 −0.0592285
\(677\) 25554.0 1.45069 0.725347 0.688383i \(-0.241679\pi\)
0.725347 + 0.688383i \(0.241679\pi\)
\(678\) 0 0
\(679\) −2002.00 −0.113151
\(680\) 15876.0 0.895319
\(681\) 0 0
\(682\) −17280.0 −0.970213
\(683\) −9276.00 −0.519672 −0.259836 0.965653i \(-0.583669\pi\)
−0.259836 + 0.965653i \(0.583669\pi\)
\(684\) 0 0
\(685\) 42444.0 2.36745
\(686\) 1029.00 0.0572703
\(687\) 0 0
\(688\) 19028.0 1.05441
\(689\) −16932.0 −0.936223
\(690\) 0 0
\(691\) 27380.0 1.50736 0.753679 0.657243i \(-0.228277\pi\)
0.753679 + 0.657243i \(0.228277\pi\)
\(692\) −870.000 −0.0477925
\(693\) 0 0
\(694\) −468.000 −0.0255980
\(695\) −936.000 −0.0510856
\(696\) 0 0
\(697\) −13356.0 −0.725817
\(698\) −37254.0 −2.02018
\(699\) 0 0
\(700\) 1393.00 0.0752149
\(701\) −25830.0 −1.39171 −0.695853 0.718184i \(-0.744973\pi\)
−0.695853 + 0.718184i \(0.744973\pi\)
\(702\) 0 0
\(703\) −49352.0 −2.64772
\(704\) 15588.0 0.834510
\(705\) 0 0
\(706\) 23490.0 1.25221
\(707\) −2898.00 −0.154159
\(708\) 0 0
\(709\) −6226.00 −0.329792 −0.164896 0.986311i \(-0.552729\pi\)
−0.164896 + 0.986311i \(0.552729\pi\)
\(710\) 38880.0 2.05513
\(711\) 0 0
\(712\) 7434.00 0.391293
\(713\) 0 0
\(714\) 0 0
\(715\) −22032.0 −1.15238
\(716\) 2316.00 0.120884
\(717\) 0 0
\(718\) 27936.0 1.45204
\(719\) 15072.0 0.781767 0.390884 0.920440i \(-0.372169\pi\)
0.390884 + 0.920440i \(0.372169\pi\)
\(720\) 0 0
\(721\) 392.000 0.0202480
\(722\) 25551.0 1.31705
\(723\) 0 0
\(724\) −106.000 −0.00544124
\(725\) −20298.0 −1.03979
\(726\) 0 0
\(727\) −32920.0 −1.67942 −0.839708 0.543038i \(-0.817274\pi\)
−0.839708 + 0.543038i \(0.817274\pi\)
\(728\) 4998.00 0.254448
\(729\) 0 0
\(730\) −27108.0 −1.37440
\(731\) 11256.0 0.569519
\(732\) 0 0
\(733\) −6946.00 −0.350009 −0.175004 0.984568i \(-0.555994\pi\)
−0.175004 + 0.984568i \(0.555994\pi\)
\(734\) −11280.0 −0.567238
\(735\) 0 0
\(736\) 0 0
\(737\) 3312.00 0.165535
\(738\) 0 0
\(739\) −2356.00 −0.117276 −0.0586379 0.998279i \(-0.518676\pi\)
−0.0586379 + 0.998279i \(0.518676\pi\)
\(740\) 7164.00 0.355884
\(741\) 0 0
\(742\) 10458.0 0.517419
\(743\) 23520.0 1.16133 0.580663 0.814144i \(-0.302793\pi\)
0.580663 + 0.814144i \(0.302793\pi\)
\(744\) 0 0
\(745\) 31428.0 1.54555
\(746\) 17610.0 0.864273
\(747\) 0 0
\(748\) −1512.00 −0.0739094
\(749\) −84.0000 −0.00409785
\(750\) 0 0
\(751\) 3008.00 0.146156 0.0730782 0.997326i \(-0.476718\pi\)
0.0730782 + 0.997326i \(0.476718\pi\)
\(752\) 17040.0 0.826310
\(753\) 0 0
\(754\) 10404.0 0.502508
\(755\) −4176.00 −0.201298
\(756\) 0 0
\(757\) −20770.0 −0.997224 −0.498612 0.866825i \(-0.666157\pi\)
−0.498612 + 0.866825i \(0.666157\pi\)
\(758\) −5556.00 −0.266231
\(759\) 0 0
\(760\) 46872.0 2.23714
\(761\) −11538.0 −0.549609 −0.274804 0.961500i \(-0.588613\pi\)
−0.274804 + 0.961500i \(0.588613\pi\)
\(762\) 0 0
\(763\) 10346.0 0.490892
\(764\) 1128.00 0.0534157
\(765\) 0 0
\(766\) −6480.00 −0.305655
\(767\) −4488.00 −0.211281
\(768\) 0 0
\(769\) 8498.00 0.398499 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(770\) 13608.0 0.636881
\(771\) 0 0
\(772\) 4034.00 0.188066
\(773\) 32322.0 1.50393 0.751967 0.659200i \(-0.229105\pi\)
0.751967 + 0.659200i \(0.229105\pi\)
\(774\) 0 0
\(775\) −31840.0 −1.47578
\(776\) 6006.00 0.277839
\(777\) 0 0
\(778\) 20358.0 0.938136
\(779\) −39432.0 −1.81360
\(780\) 0 0
\(781\) 25920.0 1.18757
\(782\) 0 0
\(783\) 0 0
\(784\) −3479.00 −0.158482
\(785\) 30492.0 1.38638
\(786\) 0 0
\(787\) 26228.0 1.18796 0.593982 0.804479i \(-0.297555\pi\)
0.593982 + 0.804479i \(0.297555\pi\)
\(788\) 1314.00 0.0594027
\(789\) 0 0
\(790\) −55296.0 −2.49031
\(791\) −2814.00 −0.126491
\(792\) 0 0
\(793\) −13532.0 −0.605972
\(794\) −19542.0 −0.873450
\(795\) 0 0
\(796\) 5096.00 0.226913
\(797\) 43338.0 1.92611 0.963056 0.269302i \(-0.0867931\pi\)
0.963056 + 0.269302i \(0.0867931\pi\)
\(798\) 0 0
\(799\) 10080.0 0.446314
\(800\) −8955.00 −0.395759
\(801\) 0 0
\(802\) −9990.00 −0.439849
\(803\) −18072.0 −0.794206
\(804\) 0 0
\(805\) 0 0
\(806\) 16320.0 0.713210
\(807\) 0 0
\(808\) 8694.00 0.378532
\(809\) 28902.0 1.25604 0.628022 0.778195i \(-0.283865\pi\)
0.628022 + 0.778195i \(0.283865\pi\)
\(810\) 0 0
\(811\) 27164.0 1.17615 0.588075 0.808807i \(-0.299886\pi\)
0.588075 + 0.808807i \(0.299886\pi\)
\(812\) −714.000 −0.0308577
\(813\) 0 0
\(814\) 42984.0 1.85085
\(815\) −52776.0 −2.26830
\(816\) 0 0
\(817\) 33232.0 1.42306
\(818\) −16194.0 −0.692188
\(819\) 0 0
\(820\) 5724.00 0.243769
\(821\) 17202.0 0.731247 0.365624 0.930763i \(-0.380856\pi\)
0.365624 + 0.930763i \(0.380856\pi\)
\(822\) 0 0
\(823\) −5992.00 −0.253789 −0.126894 0.991916i \(-0.540501\pi\)
−0.126894 + 0.991916i \(0.540501\pi\)
\(824\) −1176.00 −0.0497183
\(825\) 0 0
\(826\) 2772.00 0.116768
\(827\) −25884.0 −1.08836 −0.544181 0.838968i \(-0.683159\pi\)
−0.544181 + 0.838968i \(0.683159\pi\)
\(828\) 0 0
\(829\) −1474.00 −0.0617541 −0.0308770 0.999523i \(-0.509830\pi\)
−0.0308770 + 0.999523i \(0.509830\pi\)
\(830\) 11016.0 0.460688
\(831\) 0 0
\(832\) −14722.0 −0.613454
\(833\) −2058.00 −0.0856008
\(834\) 0 0
\(835\) −21168.0 −0.877304
\(836\) −4464.00 −0.184678
\(837\) 0 0
\(838\) −39276.0 −1.61905
\(839\) −33528.0 −1.37964 −0.689818 0.723983i \(-0.742310\pi\)
−0.689818 + 0.723983i \(0.742310\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) −966.000 −0.0395375
\(843\) 0 0
\(844\) −3076.00 −0.125451
\(845\) −18738.0 −0.762848
\(846\) 0 0
\(847\) −245.000 −0.00993896
\(848\) −35358.0 −1.43184
\(849\) 0 0
\(850\) −25074.0 −1.01180
\(851\) 0 0
\(852\) 0 0
\(853\) 1190.00 0.0477665 0.0238832 0.999715i \(-0.492397\pi\)
0.0238832 + 0.999715i \(0.492397\pi\)
\(854\) 8358.00 0.334900
\(855\) 0 0
\(856\) 252.000 0.0100621
\(857\) −34578.0 −1.37825 −0.689126 0.724642i \(-0.742005\pi\)
−0.689126 + 0.724642i \(0.742005\pi\)
\(858\) 0 0
\(859\) −44404.0 −1.76373 −0.881865 0.471501i \(-0.843712\pi\)
−0.881865 + 0.471501i \(0.843712\pi\)
\(860\) −4824.00 −0.191276
\(861\) 0 0
\(862\) −7848.00 −0.310097
\(863\) 38328.0 1.51182 0.755910 0.654676i \(-0.227195\pi\)
0.755910 + 0.654676i \(0.227195\pi\)
\(864\) 0 0
\(865\) −15660.0 −0.615556
\(866\) 12966.0 0.508779
\(867\) 0 0
\(868\) −1120.00 −0.0437964
\(869\) −36864.0 −1.43904
\(870\) 0 0
\(871\) −3128.00 −0.121686
\(872\) −31038.0 −1.20537
\(873\) 0 0
\(874\) 0 0
\(875\) 9324.00 0.360239
\(876\) 0 0
\(877\) −38842.0 −1.49555 −0.747777 0.663950i \(-0.768879\pi\)
−0.747777 + 0.663950i \(0.768879\pi\)
\(878\) −27048.0 −1.03966
\(879\) 0 0
\(880\) −46008.0 −1.76242
\(881\) 35046.0 1.34022 0.670108 0.742264i \(-0.266248\pi\)
0.670108 + 0.742264i \(0.266248\pi\)
\(882\) 0 0
\(883\) 14204.0 0.541339 0.270670 0.962672i \(-0.412755\pi\)
0.270670 + 0.962672i \(0.412755\pi\)
\(884\) 1428.00 0.0543313
\(885\) 0 0
\(886\) 15804.0 0.599262
\(887\) 26136.0 0.989359 0.494679 0.869076i \(-0.335286\pi\)
0.494679 + 0.869076i \(0.335286\pi\)
\(888\) 0 0
\(889\) 8960.00 0.338030
\(890\) −19116.0 −0.719966
\(891\) 0 0
\(892\) −1888.00 −0.0708687
\(893\) 29760.0 1.11521
\(894\) 0 0
\(895\) 41688.0 1.55696
\(896\) 11613.0 0.432995
\(897\) 0 0
\(898\) 15930.0 0.591972
\(899\) 16320.0 0.605453
\(900\) 0 0
\(901\) −20916.0 −0.773377
\(902\) 34344.0 1.26777
\(903\) 0 0
\(904\) 8442.00 0.310594
\(905\) −1908.00 −0.0700818
\(906\) 0 0
\(907\) −9052.00 −0.331386 −0.165693 0.986177i \(-0.552986\pi\)
−0.165693 + 0.986177i \(0.552986\pi\)
\(908\) 4716.00 0.172363
\(909\) 0 0
\(910\) −12852.0 −0.468175
\(911\) −5016.00 −0.182423 −0.0912116 0.995832i \(-0.529074\pi\)
−0.0912116 + 0.995832i \(0.529074\pi\)
\(912\) 0 0
\(913\) 7344.00 0.266211
\(914\) 47310.0 1.71212
\(915\) 0 0
\(916\) −1690.00 −0.0609598
\(917\) −12348.0 −0.444675
\(918\) 0 0
\(919\) 44552.0 1.59917 0.799584 0.600555i \(-0.205054\pi\)
0.799584 + 0.600555i \(0.205054\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 16110.0 0.575439
\(923\) −24480.0 −0.872989
\(924\) 0 0
\(925\) 79202.0 2.81529
\(926\) −9984.00 −0.354314
\(927\) 0 0
\(928\) 4590.00 0.162364
\(929\) −24234.0 −0.855858 −0.427929 0.903812i \(-0.640757\pi\)
−0.427929 + 0.903812i \(0.640757\pi\)
\(930\) 0 0
\(931\) −6076.00 −0.213891
\(932\) −138.000 −0.00485015
\(933\) 0 0
\(934\) −13644.0 −0.477993
\(935\) −27216.0 −0.951934
\(936\) 0 0
\(937\) −13894.0 −0.484415 −0.242208 0.970224i \(-0.577872\pi\)
−0.242208 + 0.970224i \(0.577872\pi\)
\(938\) 1932.00 0.0672516
\(939\) 0 0
\(940\) −4320.00 −0.149897
\(941\) −46758.0 −1.61984 −0.809919 0.586542i \(-0.800489\pi\)
−0.809919 + 0.586542i \(0.800489\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −9372.00 −0.323128
\(945\) 0 0
\(946\) −28944.0 −0.994768
\(947\) −13812.0 −0.473949 −0.236974 0.971516i \(-0.576156\pi\)
−0.236974 + 0.971516i \(0.576156\pi\)
\(948\) 0 0
\(949\) 17068.0 0.583826
\(950\) −74028.0 −2.52820
\(951\) 0 0
\(952\) 6174.00 0.210190
\(953\) 58518.0 1.98907 0.994535 0.104402i \(-0.0332930\pi\)
0.994535 + 0.104402i \(0.0332930\pi\)
\(954\) 0 0
\(955\) 20304.0 0.687981
\(956\) −1896.00 −0.0641433
\(957\) 0 0
\(958\) 24192.0 0.815875
\(959\) 16506.0 0.555794
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) −40596.0 −1.36057
\(963\) 0 0
\(964\) −3598.00 −0.120211
\(965\) 72612.0 2.42224
\(966\) 0 0
\(967\) 19640.0 0.653133 0.326567 0.945174i \(-0.394108\pi\)
0.326567 + 0.945174i \(0.394108\pi\)
\(968\) 735.000 0.0244047
\(969\) 0 0
\(970\) −15444.0 −0.511213
\(971\) 58308.0 1.92708 0.963539 0.267568i \(-0.0862200\pi\)
0.963539 + 0.267568i \(0.0862200\pi\)
\(972\) 0 0
\(973\) −364.000 −0.0119931
\(974\) 49848.0 1.63987
\(975\) 0 0
\(976\) −28258.0 −0.926759
\(977\) 23550.0 0.771168 0.385584 0.922673i \(-0.374000\pi\)
0.385584 + 0.922673i \(0.374000\pi\)
\(978\) 0 0
\(979\) −12744.0 −0.416037
\(980\) 882.000 0.0287494
\(981\) 0 0
\(982\) 21420.0 0.696069
\(983\) −15768.0 −0.511619 −0.255809 0.966727i \(-0.582342\pi\)
−0.255809 + 0.966727i \(0.582342\pi\)
\(984\) 0 0
\(985\) 23652.0 0.765092
\(986\) 12852.0 0.415102
\(987\) 0 0
\(988\) 4216.00 0.135758
\(989\) 0 0
\(990\) 0 0
\(991\) 35264.0 1.13037 0.565186 0.824964i \(-0.308805\pi\)
0.565186 + 0.824964i \(0.308805\pi\)
\(992\) 7200.00 0.230444
\(993\) 0 0
\(994\) 15120.0 0.482472
\(995\) 91728.0 2.92259
\(996\) 0 0
\(997\) −29338.0 −0.931940 −0.465970 0.884801i \(-0.654294\pi\)
−0.465970 + 0.884801i \(0.654294\pi\)
\(998\) −27372.0 −0.868182
\(999\) 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.c.1.1 1
3.2 odd 2 21.4.a.a.1.1 1
4.3 odd 2 1008.4.a.v.1.1 1
5.4 even 2 1575.4.a.b.1.1 1
7.2 even 3 441.4.e.b.361.1 2
7.3 odd 6 441.4.e.d.226.1 2
7.4 even 3 441.4.e.b.226.1 2
7.5 odd 6 441.4.e.d.361.1 2
7.6 odd 2 441.4.a.j.1.1 1
12.11 even 2 336.4.a.f.1.1 1
15.2 even 4 525.4.d.c.274.1 2
15.8 even 4 525.4.d.c.274.2 2
15.14 odd 2 525.4.a.g.1.1 1
21.2 odd 6 147.4.e.i.67.1 2
21.5 even 6 147.4.e.g.67.1 2
21.11 odd 6 147.4.e.i.79.1 2
21.17 even 6 147.4.e.g.79.1 2
21.20 even 2 147.4.a.c.1.1 1
24.5 odd 2 1344.4.a.ba.1.1 1
24.11 even 2 1344.4.a.n.1.1 1
84.83 odd 2 2352.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 3.2 odd 2
63.4.a.c.1.1 1 1.1 even 1 trivial
147.4.a.c.1.1 1 21.20 even 2
147.4.e.g.67.1 2 21.5 even 6
147.4.e.g.79.1 2 21.17 even 6
147.4.e.i.67.1 2 21.2 odd 6
147.4.e.i.79.1 2 21.11 odd 6
336.4.a.f.1.1 1 12.11 even 2
441.4.a.j.1.1 1 7.6 odd 2
441.4.e.b.226.1 2 7.4 even 3
441.4.e.b.361.1 2 7.2 even 3
441.4.e.d.226.1 2 7.3 odd 6
441.4.e.d.361.1 2 7.5 odd 6
525.4.a.g.1.1 1 15.14 odd 2
525.4.d.c.274.1 2 15.2 even 4
525.4.d.c.274.2 2 15.8 even 4
1008.4.a.v.1.1 1 4.3 odd 2
1344.4.a.n.1.1 1 24.11 even 2
1344.4.a.ba.1.1 1 24.5 odd 2
1575.4.a.b.1.1 1 5.4 even 2
2352.4.a.r.1.1 1 84.83 odd 2