Properties

Label 441.4.a.j.1.1
Level 441
Weight 4
Character 441.1
Self dual yes
Analytic conductor 26.020
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
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 \(\chi\) \(=\) 441.1

$q$-expansion

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

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 \(-0.797826\pi\)
−0.804984 + 0.593296i \(0.797826\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −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 \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 42.0000 0.599206 0.299603 0.954064i \(-0.403146\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) −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
\(28\) 0 0
\(29\) −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 \(-0.346594\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(32\) −45.0000 −0.248592
\(33\) 0 0
\(34\) 126.000 0.635554
\(35\) 0 0
\(36\) 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 \(-0.707087\pi\)
−0.605649 + 0.795732i \(0.707087\pi\)
\(42\) 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 \(-0.378528\pi\)
0.372421 + 0.928064i \(0.378528\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 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
\(56\) 0 0
\(57\) 0 0
\(58\) −306.000 −0.692755
\(59\) −132.000 −0.291270 −0.145635 0.989338i \(-0.546523\pi\)
−0.145635 + 0.989338i \(0.546523\pi\)
\(60\) 0 0
\(61\) −398.000 −0.835388 −0.417694 0.908588i \(-0.637162\pi\)
−0.417694 + 0.908588i \(0.637162\pi\)
\(62\) 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
\(70\) 0 0
\(71\) 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 \(-0.368166\pi\)
0.402429 + 0.915451i \(0.368166\pi\)
\(74\) 1194.00 1.87567
\(75\) 0 0
\(76\) 124.000 0.187155
\(77\) 0 0
\(78\) 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 \(-0.543068\pi\)
−0.134891 + 0.990860i \(0.543068\pi\)
\(84\) 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 \(-0.432390\pi\)
0.210809 + 0.977527i \(0.432390\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 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 \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 199.000 0.199000
\(101\) 414.000 0.407867 0.203933 0.978985i \(-0.434627\pi\)
0.203933 + 0.978985i \(0.434627\pi\)
\(102\) 0 0
\(103\) −56.0000 −0.0535713 −0.0267857 0.999641i \(-0.508527\pi\)
−0.0267857 + 0.999641i \(0.508527\pi\)
\(104\) −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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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 \(-0.299817\pi\)
0.588250 + 0.808679i \(0.299817\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 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 \(-0.494950\pi\)
0.0158654 + 0.999874i \(0.494950\pi\)
\(140\) 0 0
\(141\) 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
\(154\) 0 0
\(155\) −2880.00 −1.49243
\(156\) 0 0
\(157\) −1694.00 −0.861120 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(158\) −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 \(-0.412163\pi\)
0.272460 + 0.962167i \(0.412163\pi\)
\(168\) 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 \(-0.438772\pi\)
0.191170 + 0.981557i \(0.438772\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −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 \(-0.493071\pi\)
0.0217650 + 0.999763i \(0.493071\pi\)
\(182\) 0 0
\(183\) 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
\(196\) 0 0
\(197\) 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 \(-0.862128\pi\)
−0.907653 + 0.419722i \(0.862128\pi\)
\(200\) −4179.00 −1.47750
\(201\) 0 0
\(202\) 1242.00 0.432608
\(203\) 0 0
\(204\) 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
\(217\) 0 0
\(218\) 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 \(-0.408513\pi\)
0.283475 + 0.958980i \(0.408513\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1206.00 −0.354964
\(227\) −4716.00 −1.37891 −0.689454 0.724330i \(-0.742149\pi\)
−0.689454 + 0.724330i \(0.742149\pi\)
\(228\) 0 0
\(229\) 1690.00 0.487678 0.243839 0.969816i \(-0.421593\pi\)
0.243839 + 0.969816i \(0.421593\pi\)
\(230\) 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
\(238\) 0 0
\(239\) −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 \(-0.340330\pi\)
0.480846 + 0.876805i \(0.340330\pi\)
\(242\) −105.000 −0.0278911
\(243\) 0 0
\(244\) −398.000 −0.104424
\(245\) 0 0
\(246\) 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 \(-0.625713\pi\)
−0.384752 + 0.923020i \(0.625713\pi\)
\(252\) 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 \(-0.810469\pi\)
−0.827908 + 0.560864i \(0.810469\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −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
\(266\) 0 0
\(267\) 0 0
\(268\) 92.0000 0.0209694
\(269\) 8214.00 1.86177 0.930886 0.365311i \(-0.119037\pi\)
0.930886 + 0.365311i \(0.119037\pi\)
\(270\) 0 0
\(271\) 5344.00 1.19788 0.598939 0.800795i \(-0.295589\pi\)
0.598939 + 0.800795i \(0.295589\pi\)
\(272\) −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
\(280\) 0 0
\(281\) −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 \(-0.611233\pi\)
−0.342380 + 0.939562i \(0.611233\pi\)
\(284\) 720.000 0.150437
\(285\) 0 0
\(286\) 3672.00 0.759195
\(287\) 0 0
\(288\) 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 \(-0.329559\pi\)
0.510233 + 0.860036i \(0.329559\pi\)
\(294\) 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
\(301\) 0 0
\(302\) −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 \(-0.513378\pi\)
−0.0420147 + 0.999117i \(0.513378\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8640.00 −1.58296
\(311\) 5016.00 0.914570 0.457285 0.889320i \(-0.348822\pi\)
0.457285 + 0.889320i \(0.348822\pi\)
\(312\) 0 0
\(313\) −5402.00 −0.975524 −0.487762 0.872977i \(-0.662187\pi\)
−0.487762 + 0.872977i \(0.662187\pi\)
\(314\) −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
\(329\) 0 0
\(330\) 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
\(343\) 0 0
\(344\) 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 \(-0.0986888\pi\)
0.952321 + 0.305097i \(0.0986888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1620.00 −0.245302
\(353\) −7830.00 −1.18059 −0.590296 0.807187i \(-0.700989\pi\)
−0.590296 + 0.807187i \(0.700989\pi\)
\(354\) 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
\(364\) 0 0
\(365\) −9036.00 −1.29580
\(366\) 0 0
\(367\) 3760.00 0.534797 0.267398 0.963586i \(-0.413836\pi\)
0.267398 + 0.963586i \(0.413836\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −21492.0 −3.01977
\(371\) 0 0
\(372\) 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 \(-0.453975\pi\)
0.144087 + 0.989565i \(0.453975\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) 0 0
\(394\) 3942.00 0.504048
\(395\) 18432.0 2.34788
\(396\) 0 0
\(397\) 6514.00 0.823497 0.411748 0.911298i \(-0.364918\pi\)
0.411748 + 0.911298i \(0.364918\pi\)
\(398\) −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
\(406\) 0 0
\(407\) 14328.0 1.74499
\(408\) 0 0
\(409\) 5398.00 0.652601 0.326301 0.945266i \(-0.394198\pi\)
0.326301 + 0.945266i \(0.394198\pi\)
\(410\) 17172.0 2.06845
\(411\) 0 0
\(412\) −56.0000 −0.00669641
\(413\) 0 0
\(414\) 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 \(-0.223613\pi\)
0.763229 + 0.646128i \(0.223613\pi\)
\(420\) 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
\(427\) 0 0
\(428\) −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 \(-0.577095\pi\)
−0.239841 + 0.970812i \(0.577095\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1478.00 0.162347
\(437\) 0 0
\(438\) 0 0
\(439\) 9016.00 0.980205 0.490103 0.871665i \(-0.336959\pi\)
0.490103 + 0.871665i \(0.336959\pi\)
\(440\) 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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) 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 \(-0.587442\pi\)
−0.271264 + 0.962505i \(0.587442\pi\)
\(462\) 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 \(-0.427655\pi\)
0.225328 + 0.974283i \(0.427655\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −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
\(476\) 0 0
\(477\) 0 0
\(478\) −5688.00 −0.544274
\(479\) −8064.00 −0.769214 −0.384607 0.923080i \(-0.625663\pi\)
−0.384607 + 0.923080i \(0.625663\pi\)
\(480\) 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
\(490\) 0 0
\(491\) 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
\(497\) 0 0
\(498\) 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 \(-0.593787\pi\)
−0.290397 + 0.956906i \(0.593787\pi\)
\(504\) 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 \(-0.461237\pi\)
0.121478 + 0.992594i \(0.461237\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 0 0
\(520\) 12852.0 1.08384
\(521\) −14862.0 −1.24974 −0.624871 0.780728i \(-0.714849\pi\)
−0.624871 + 0.780728i \(0.714849\pi\)
\(522\) 0 0
\(523\) −17660.0 −1.47652 −0.738258 0.674518i \(-0.764351\pi\)
−0.738258 + 0.674518i \(0.764351\pi\)
\(524\) 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
\(532\) 0 0
\(533\) −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
\(539\) 0 0
\(540\) 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
\(553\) 0 0
\(554\) −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
\(560\) 0 0
\(561\) 0 0
\(562\) −19854.0 −1.49020
\(563\) −17772.0 −1.33037 −0.665187 0.746677i \(-0.731648\pi\)
−0.665187 + 0.746677i \(0.731648\pi\)
\(564\) 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
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.00000 −0.000144300 0 −7.21500e−5 1.00000i \(-0.500023\pi\)
−7.21500e−5 1.00000i \(0.500023\pi\)
\(578\) −9447.00 −0.679833
\(579\) 0 0
\(580\) 1836.00 0.131441
\(581\) 0 0
\(582\) 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 \(-0.122479\pi\)
0.926881 + 0.375354i \(0.122479\pi\)
\(588\) 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 \(-0.474646\pi\)
0.0795679 + 0.996829i \(0.474646\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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 \(-0.813528\pi\)
−0.833260 + 0.552881i \(0.813528\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −232.000 −0.0156290
\(605\) 630.000 0.0423358
\(606\) 0 0
\(607\) −16832.0 −1.12552 −0.562759 0.826621i \(-0.690260\pi\)
−0.562759 + 0.826621i \(0.690260\pi\)
\(608\) −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
\(616\) 0 0
\(617\) 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 \(-0.332623\pi\)
0.501930 + 0.864908i \(0.332623\pi\)
\(620\) −2880.00 −0.186554
\(621\) 0 0
\(622\) 15048.0 0.970048
\(623\) 0 0
\(624\) 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
\(637\) 0 0
\(638\) −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 \(-0.377734\pi\)
0.374735 + 0.927132i \(0.377734\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 15624.0 0.951576
\(647\) 13560.0 0.823955 0.411977 0.911194i \(-0.364838\pi\)
0.411977 + 0.911194i \(0.364838\pi\)
\(648\) 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
\(658\) 0 0
\(659\) −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 \(-0.671748\pi\)
−0.513762 + 0.857933i \(0.671748\pi\)
\(662\) −24132.0 −1.41679
\(663\) 0 0
\(664\) 4284.00 0.250379
\(665\) 0 0
\(666\) 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 \(-0.758321\pi\)
−0.725347 + 0.688383i \(0.758321\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) 0 0
\(688\) 19028.0 1.05441
\(689\) 16932.0 0.936223
\(690\) 0 0
\(691\) −27380.0 −1.50736 −0.753679 0.657243i \(-0.771723\pi\)
−0.753679 + 0.657243i \(0.771723\pi\)
\(692\) 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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) 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 \(-0.627831\pi\)
−0.390884 + 0.920440i \(0.627831\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 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 \(-0.182726\pi\)
0.839708 + 0.543038i \(0.182726\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −27108.0 −1.37440
\(731\) −11256.0 −0.569519
\(732\) 0 0
\(733\) 6946.00 0.350009 0.175004 0.984568i \(-0.444006\pi\)
0.175004 + 0.984568i \(0.444006\pi\)
\(734\) 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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) 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 \(-0.411387\pi\)
0.274804 + 0.961500i \(0.411387\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 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 \(-0.563850\pi\)
−0.199249 + 0.979949i \(0.563850\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4034.00 0.188066
\(773\) −32322.0 −1.50393 −0.751967 0.659200i \(-0.770895\pi\)
−0.751967 + 0.659200i \(0.770895\pi\)
\(774\) 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
\(784\) 0 0
\(785\) 30492.0 1.38638
\(786\) 0 0
\(787\) −26228.0 −1.18796 −0.593982 0.804479i \(-0.702445\pi\)
−0.593982 + 0.804479i \(0.702445\pi\)
\(788\) 1314.00 0.0594027
\(789\) 0 0
\(790\) 55296.0 2.49031
\(791\) 0 0
\(792\) 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 \(-0.913207\pi\)
−0.963056 + 0.269302i \(0.913207\pi\)
\(798\) 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 \(-0.700114\pi\)
−0.588075 + 0.808807i \(0.700114\pi\)
\(812\) 0 0
\(813\) 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
\(826\) 0 0
\(827\) −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 \(-0.490170\pi\)
0.0308770 + 0.999523i \(0.490170\pi\)
\(830\) 11016.0 0.460688
\(831\) 0 0
\(832\) 14722.0 0.613454
\(833\) 0 0
\(834\) 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 \(-0.257690\pi\)
0.689818 + 0.723983i \(0.257690\pi\)
\(840\) 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
\(847\) 0 0
\(848\) −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 \(-0.507603\pi\)
−0.0238832 + 0.999715i \(0.507603\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 252.000 0.0100621
\(857\) 34578.0 1.37825 0.689126 0.724642i \(-0.257995\pi\)
0.689126 + 0.724642i \(0.257995\pi\)
\(858\) 0 0
\(859\) 44404.0 1.76373 0.881865 0.471501i \(-0.156288\pi\)
0.881865 + 0.471501i \(0.156288\pi\)
\(860\) 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
\(868\) 0 0
\(869\) −36864.0 −1.43904
\(870\) 0 0
\(871\) 3128.00 0.121686
\(872\) −31038.0 −1.20537
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 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 \(-0.733752\pi\)
−0.670108 + 0.742264i \(0.733752\pi\)
\(882\) 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 \(-0.664714\pi\)
−0.494679 + 0.869076i \(0.664714\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −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
\(896\) 0 0
\(897\) 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
\(910\) 0 0
\(911\) −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
\(917\) 0 0
\(918\) 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 \(-0.359243\pi\)
0.427929 + 0.903812i \(0.359243\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −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 \(-0.422128\pi\)
0.242208 + 0.970224i \(0.422128\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4320.00 −0.149897
\(941\) 46758.0 1.61984 0.809919 0.586542i \(-0.199511\pi\)
0.809919 + 0.586542i \(0.199511\pi\)
\(942\) 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
\(952\) 0 0
\(953\) 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
\(959\) 0 0
\(960\) 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 \(-0.913780\pi\)
−0.963539 + 0.267568i \(0.913780\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) 0 0
\(982\) 21420.0 0.696069
\(983\) 15768.0 0.511619 0.255809 0.966727i \(-0.417658\pi\)
0.255809 + 0.966727i \(0.417658\pi\)
\(984\) 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
\(994\) 0 0
\(995\) 91728.0 2.92259
\(996\) 0 0
\(997\) 29338.0 0.931940 0.465970 0.884801i \(-0.345706\pi\)
0.465970 + 0.884801i \(0.345706\pi\)
\(998\) −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 441.4.a.j.1.1 1
3.2 odd 2 147.4.a.c.1.1 1
7.2 even 3 441.4.e.d.361.1 2
7.3 odd 6 441.4.e.b.226.1 2
7.4 even 3 441.4.e.d.226.1 2
7.5 odd 6 441.4.e.b.361.1 2
7.6 odd 2 63.4.a.c.1.1 1
12.11 even 2 2352.4.a.r.1.1 1
21.2 odd 6 147.4.e.g.67.1 2
21.5 even 6 147.4.e.i.67.1 2
21.11 odd 6 147.4.e.g.79.1 2
21.17 even 6 147.4.e.i.79.1 2
21.20 even 2 21.4.a.a.1.1 1
28.27 even 2 1008.4.a.v.1.1 1
35.34 odd 2 1575.4.a.b.1.1 1
84.83 odd 2 336.4.a.f.1.1 1
105.62 odd 4 525.4.d.c.274.1 2
105.83 odd 4 525.4.d.c.274.2 2
105.104 even 2 525.4.a.g.1.1 1
168.83 odd 2 1344.4.a.n.1.1 1
168.125 even 2 1344.4.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 21.20 even 2
63.4.a.c.1.1 1 7.6 odd 2
147.4.a.c.1.1 1 3.2 odd 2
147.4.e.g.67.1 2 21.2 odd 6
147.4.e.g.79.1 2 21.11 odd 6
147.4.e.i.67.1 2 21.5 even 6
147.4.e.i.79.1 2 21.17 even 6
336.4.a.f.1.1 1 84.83 odd 2
441.4.a.j.1.1 1 1.1 even 1 trivial
441.4.e.b.226.1 2 7.3 odd 6
441.4.e.b.361.1 2 7.5 odd 6
441.4.e.d.226.1 2 7.4 even 3
441.4.e.d.361.1 2 7.2 even 3
525.4.a.g.1.1 1 105.104 even 2
525.4.d.c.274.1 2 105.62 odd 4
525.4.d.c.274.2 2 105.83 odd 4
1008.4.a.v.1.1 1 28.27 even 2
1344.4.a.n.1.1 1 168.83 odd 2
1344.4.a.ba.1.1 1 168.125 even 2
1575.4.a.b.1.1 1 35.34 odd 2
2352.4.a.r.1.1 1 12.11 even 2