Properties

Label 126.4.a.e.1.1
Level $126$
Weight $4$
Character 126.1
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +22.0000 q^{5} -7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +22.0000 q^{5} -7.00000 q^{7} -8.00000 q^{8} -44.0000 q^{10} +26.0000 q^{11} -54.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +74.0000 q^{17} +116.000 q^{19} +88.0000 q^{20} -52.0000 q^{22} -58.0000 q^{23} +359.000 q^{25} +108.000 q^{26} -28.0000 q^{28} +208.000 q^{29} -252.000 q^{31} -32.0000 q^{32} -148.000 q^{34} -154.000 q^{35} +50.0000 q^{37} -232.000 q^{38} -176.000 q^{40} +126.000 q^{41} +164.000 q^{43} +104.000 q^{44} +116.000 q^{46} -444.000 q^{47} +49.0000 q^{49} -718.000 q^{50} -216.000 q^{52} +12.0000 q^{53} +572.000 q^{55} +56.0000 q^{56} -416.000 q^{58} +124.000 q^{59} -162.000 q^{61} +504.000 q^{62} +64.0000 q^{64} -1188.00 q^{65} -860.000 q^{67} +296.000 q^{68} +308.000 q^{70} -238.000 q^{71} -146.000 q^{73} -100.000 q^{74} +464.000 q^{76} -182.000 q^{77} -984.000 q^{79} +352.000 q^{80} -252.000 q^{82} +656.000 q^{83} +1628.00 q^{85} -328.000 q^{86} -208.000 q^{88} -954.000 q^{89} +378.000 q^{91} -232.000 q^{92} +888.000 q^{94} +2552.00 q^{95} +526.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
\(5\) 22.0000 1.96774 0.983870 0.178885i \(-0.0572491\pi\)
0.983870 + 0.178885i \(0.0572491\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −44.0000 −1.39140
\(11\) 26.0000 0.712663 0.356332 0.934360i \(-0.384027\pi\)
0.356332 + 0.934360i \(0.384027\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 74.0000 1.05574 0.527872 0.849324i \(-0.322990\pi\)
0.527872 + 0.849324i \(0.322990\pi\)
\(18\) 0 0
\(19\) 116.000 1.40064 0.700322 0.713827i \(-0.253040\pi\)
0.700322 + 0.713827i \(0.253040\pi\)
\(20\) 88.0000 0.983870
\(21\) 0 0
\(22\) −52.0000 −0.503929
\(23\) −58.0000 −0.525819 −0.262909 0.964821i \(-0.584682\pi\)
−0.262909 + 0.964821i \(0.584682\pi\)
\(24\) 0 0
\(25\) 359.000 2.87200
\(26\) 108.000 0.814636
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) 208.000 1.33188 0.665942 0.746004i \(-0.268030\pi\)
0.665942 + 0.746004i \(0.268030\pi\)
\(30\) 0 0
\(31\) −252.000 −1.46002 −0.730009 0.683438i \(-0.760484\pi\)
−0.730009 + 0.683438i \(0.760484\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −148.000 −0.746523
\(35\) −154.000 −0.743736
\(36\) 0 0
\(37\) 50.0000 0.222161 0.111080 0.993811i \(-0.464569\pi\)
0.111080 + 0.993811i \(0.464569\pi\)
\(38\) −232.000 −0.990404
\(39\) 0 0
\(40\) −176.000 −0.695701
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 104.000 0.356332
\(45\) 0 0
\(46\) 116.000 0.371810
\(47\) −444.000 −1.37796 −0.688979 0.724781i \(-0.741941\pi\)
−0.688979 + 0.724781i \(0.741941\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −718.000 −2.03081
\(51\) 0 0
\(52\) −216.000 −0.576035
\(53\) 12.0000 0.0311005 0.0155503 0.999879i \(-0.495050\pi\)
0.0155503 + 0.999879i \(0.495050\pi\)
\(54\) 0 0
\(55\) 572.000 1.40234
\(56\) 56.0000 0.133631
\(57\) 0 0
\(58\) −416.000 −0.941784
\(59\) 124.000 0.273617 0.136809 0.990597i \(-0.456315\pi\)
0.136809 + 0.990597i \(0.456315\pi\)
\(60\) 0 0
\(61\) −162.000 −0.340032 −0.170016 0.985441i \(-0.554382\pi\)
−0.170016 + 0.985441i \(0.554382\pi\)
\(62\) 504.000 1.03239
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −1188.00 −2.26697
\(66\) 0 0
\(67\) −860.000 −1.56815 −0.784073 0.620669i \(-0.786861\pi\)
−0.784073 + 0.620669i \(0.786861\pi\)
\(68\) 296.000 0.527872
\(69\) 0 0
\(70\) 308.000 0.525901
\(71\) −238.000 −0.397823 −0.198911 0.980017i \(-0.563741\pi\)
−0.198911 + 0.980017i \(0.563741\pi\)
\(72\) 0 0
\(73\) −146.000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −100.000 −0.157091
\(75\) 0 0
\(76\) 464.000 0.700322
\(77\) −182.000 −0.269361
\(78\) 0 0
\(79\) −984.000 −1.40138 −0.700688 0.713468i \(-0.747123\pi\)
−0.700688 + 0.713468i \(0.747123\pi\)
\(80\) 352.000 0.491935
\(81\) 0 0
\(82\) −252.000 −0.339375
\(83\) 656.000 0.867534 0.433767 0.901025i \(-0.357184\pi\)
0.433767 + 0.901025i \(0.357184\pi\)
\(84\) 0 0
\(85\) 1628.00 2.07743
\(86\) −328.000 −0.411269
\(87\) 0 0
\(88\) −208.000 −0.251964
\(89\) −954.000 −1.13622 −0.568111 0.822952i \(-0.692326\pi\)
−0.568111 + 0.822952i \(0.692326\pi\)
\(90\) 0 0
\(91\) 378.000 0.435441
\(92\) −232.000 −0.262909
\(93\) 0 0
\(94\) 888.000 0.974364
\(95\) 2552.00 2.75610
\(96\) 0 0
\(97\) 526.000 0.550590 0.275295 0.961360i \(-0.411225\pi\)
0.275295 + 0.961360i \(0.411225\pi\)
\(98\) −98.0000 −0.101015
\(99\) 0 0
\(100\) 1436.00 1.43600
\(101\) 1306.00 1.28665 0.643326 0.765592i \(-0.277554\pi\)
0.643326 + 0.765592i \(0.277554\pi\)
\(102\) 0 0
\(103\) −508.000 −0.485968 −0.242984 0.970030i \(-0.578126\pi\)
−0.242984 + 0.970030i \(0.578126\pi\)
\(104\) 432.000 0.407318
\(105\) 0 0
\(106\) −24.0000 −0.0219914
\(107\) 498.000 0.449939 0.224970 0.974366i \(-0.427772\pi\)
0.224970 + 0.974366i \(0.427772\pi\)
\(108\) 0 0
\(109\) −614.000 −0.539546 −0.269773 0.962924i \(-0.586949\pi\)
−0.269773 + 0.962924i \(0.586949\pi\)
\(110\) −1144.00 −0.991601
\(111\) 0 0
\(112\) −112.000 −0.0944911
\(113\) −1232.00 −1.02564 −0.512818 0.858498i \(-0.671398\pi\)
−0.512818 + 0.858498i \(0.671398\pi\)
\(114\) 0 0
\(115\) −1276.00 −1.03467
\(116\) 832.000 0.665942
\(117\) 0 0
\(118\) −248.000 −0.193477
\(119\) −518.000 −0.399033
\(120\) 0 0
\(121\) −655.000 −0.492111
\(122\) 324.000 0.240439
\(123\) 0 0
\(124\) −1008.00 −0.730009
\(125\) 5148.00 3.68361
\(126\) 0 0
\(127\) −2808.00 −1.96197 −0.980983 0.194093i \(-0.937824\pi\)
−0.980983 + 0.194093i \(0.937824\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 2376.00 1.60299
\(131\) 520.000 0.346814 0.173407 0.984850i \(-0.444522\pi\)
0.173407 + 0.984850i \(0.444522\pi\)
\(132\) 0 0
\(133\) −812.000 −0.529393
\(134\) 1720.00 1.10885
\(135\) 0 0
\(136\) −592.000 −0.373262
\(137\) −2516.00 −1.56902 −0.784512 0.620113i \(-0.787087\pi\)
−0.784512 + 0.620113i \(0.787087\pi\)
\(138\) 0 0
\(139\) −2672.00 −1.63048 −0.815238 0.579126i \(-0.803394\pi\)
−0.815238 + 0.579126i \(0.803394\pi\)
\(140\) −616.000 −0.371868
\(141\) 0 0
\(142\) 476.000 0.281303
\(143\) −1404.00 −0.821038
\(144\) 0 0
\(145\) 4576.00 2.62080
\(146\) 292.000 0.165521
\(147\) 0 0
\(148\) 200.000 0.111080
\(149\) −1164.00 −0.639991 −0.319995 0.947419i \(-0.603681\pi\)
−0.319995 + 0.947419i \(0.603681\pi\)
\(150\) 0 0
\(151\) 1672.00 0.901096 0.450548 0.892752i \(-0.351229\pi\)
0.450548 + 0.892752i \(0.351229\pi\)
\(152\) −928.000 −0.495202
\(153\) 0 0
\(154\) 364.000 0.190467
\(155\) −5544.00 −2.87293
\(156\) 0 0
\(157\) 446.000 0.226718 0.113359 0.993554i \(-0.463839\pi\)
0.113359 + 0.993554i \(0.463839\pi\)
\(158\) 1968.00 0.990922
\(159\) 0 0
\(160\) −704.000 −0.347851
\(161\) 406.000 0.198741
\(162\) 0 0
\(163\) 428.000 0.205666 0.102833 0.994699i \(-0.467209\pi\)
0.102833 + 0.994699i \(0.467209\pi\)
\(164\) 504.000 0.239974
\(165\) 0 0
\(166\) −1312.00 −0.613439
\(167\) 4.00000 0.00185347 0.000926734 1.00000i \(-0.499705\pi\)
0.000926734 1.00000i \(0.499705\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) −3256.00 −1.46896
\(171\) 0 0
\(172\) 656.000 0.290811
\(173\) −590.000 −0.259288 −0.129644 0.991561i \(-0.541383\pi\)
−0.129644 + 0.991561i \(0.541383\pi\)
\(174\) 0 0
\(175\) −2513.00 −1.08551
\(176\) 416.000 0.178166
\(177\) 0 0
\(178\) 1908.00 0.803431
\(179\) 3534.00 1.47566 0.737831 0.674985i \(-0.235850\pi\)
0.737831 + 0.674985i \(0.235850\pi\)
\(180\) 0 0
\(181\) 1098.00 0.450904 0.225452 0.974254i \(-0.427614\pi\)
0.225452 + 0.974254i \(0.427614\pi\)
\(182\) −756.000 −0.307904
\(183\) 0 0
\(184\) 464.000 0.185905
\(185\) 1100.00 0.437155
\(186\) 0 0
\(187\) 1924.00 0.752389
\(188\) −1776.00 −0.688979
\(189\) 0 0
\(190\) −5104.00 −1.94886
\(191\) −4854.00 −1.83886 −0.919432 0.393248i \(-0.871351\pi\)
−0.919432 + 0.393248i \(0.871351\pi\)
\(192\) 0 0
\(193\) −1498.00 −0.558696 −0.279348 0.960190i \(-0.590118\pi\)
−0.279348 + 0.960190i \(0.590118\pi\)
\(194\) −1052.00 −0.389326
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) 620.000 0.224229 0.112115 0.993695i \(-0.464238\pi\)
0.112115 + 0.993695i \(0.464238\pi\)
\(198\) 0 0
\(199\) 32.0000 0.0113991 0.00569955 0.999984i \(-0.498186\pi\)
0.00569955 + 0.999984i \(0.498186\pi\)
\(200\) −2872.00 −1.01541
\(201\) 0 0
\(202\) −2612.00 −0.909800
\(203\) −1456.00 −0.503405
\(204\) 0 0
\(205\) 2772.00 0.944414
\(206\) 1016.00 0.343631
\(207\) 0 0
\(208\) −864.000 −0.288017
\(209\) 3016.00 0.998187
\(210\) 0 0
\(211\) 4268.00 1.39252 0.696259 0.717791i \(-0.254847\pi\)
0.696259 + 0.717791i \(0.254847\pi\)
\(212\) 48.0000 0.0155503
\(213\) 0 0
\(214\) −996.000 −0.318155
\(215\) 3608.00 1.14448
\(216\) 0 0
\(217\) 1764.00 0.551835
\(218\) 1228.00 0.381517
\(219\) 0 0
\(220\) 2288.00 0.701168
\(221\) −3996.00 −1.21629
\(222\) 0 0
\(223\) 3464.00 1.04021 0.520104 0.854103i \(-0.325893\pi\)
0.520104 + 0.854103i \(0.325893\pi\)
\(224\) 224.000 0.0668153
\(225\) 0 0
\(226\) 2464.00 0.725234
\(227\) −3252.00 −0.950849 −0.475425 0.879756i \(-0.657706\pi\)
−0.475425 + 0.879756i \(0.657706\pi\)
\(228\) 0 0
\(229\) 418.000 0.120621 0.0603105 0.998180i \(-0.480791\pi\)
0.0603105 + 0.998180i \(0.480791\pi\)
\(230\) 2552.00 0.731626
\(231\) 0 0
\(232\) −1664.00 −0.470892
\(233\) 2084.00 0.585954 0.292977 0.956119i \(-0.405354\pi\)
0.292977 + 0.956119i \(0.405354\pi\)
\(234\) 0 0
\(235\) −9768.00 −2.71146
\(236\) 496.000 0.136809
\(237\) 0 0
\(238\) 1036.00 0.282159
\(239\) 1662.00 0.449815 0.224908 0.974380i \(-0.427792\pi\)
0.224908 + 0.974380i \(0.427792\pi\)
\(240\) 0 0
\(241\) 6182.00 1.65236 0.826178 0.563410i \(-0.190511\pi\)
0.826178 + 0.563410i \(0.190511\pi\)
\(242\) 1310.00 0.347975
\(243\) 0 0
\(244\) −648.000 −0.170016
\(245\) 1078.00 0.281106
\(246\) 0 0
\(247\) −6264.00 −1.61364
\(248\) 2016.00 0.516194
\(249\) 0 0
\(250\) −10296.0 −2.60470
\(251\) −996.000 −0.250466 −0.125233 0.992127i \(-0.539968\pi\)
−0.125233 + 0.992127i \(0.539968\pi\)
\(252\) 0 0
\(253\) −1508.00 −0.374732
\(254\) 5616.00 1.38732
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −5994.00 −1.45485 −0.727423 0.686189i \(-0.759282\pi\)
−0.727423 + 0.686189i \(0.759282\pi\)
\(258\) 0 0
\(259\) −350.000 −0.0839689
\(260\) −4752.00 −1.13349
\(261\) 0 0
\(262\) −1040.00 −0.245234
\(263\) 6414.00 1.50382 0.751909 0.659267i \(-0.229133\pi\)
0.751909 + 0.659267i \(0.229133\pi\)
\(264\) 0 0
\(265\) 264.000 0.0611977
\(266\) 1624.00 0.374338
\(267\) 0 0
\(268\) −3440.00 −0.784073
\(269\) 2686.00 0.608804 0.304402 0.952544i \(-0.401543\pi\)
0.304402 + 0.952544i \(0.401543\pi\)
\(270\) 0 0
\(271\) 5100.00 1.14318 0.571592 0.820538i \(-0.306326\pi\)
0.571592 + 0.820538i \(0.306326\pi\)
\(272\) 1184.00 0.263936
\(273\) 0 0
\(274\) 5032.00 1.10947
\(275\) 9334.00 2.04677
\(276\) 0 0
\(277\) −4426.00 −0.960045 −0.480023 0.877256i \(-0.659372\pi\)
−0.480023 + 0.877256i \(0.659372\pi\)
\(278\) 5344.00 1.15292
\(279\) 0 0
\(280\) 1232.00 0.262950
\(281\) 7508.00 1.59391 0.796957 0.604036i \(-0.206442\pi\)
0.796957 + 0.604036i \(0.206442\pi\)
\(282\) 0 0
\(283\) 3412.00 0.716687 0.358343 0.933590i \(-0.383342\pi\)
0.358343 + 0.933590i \(0.383342\pi\)
\(284\) −952.000 −0.198911
\(285\) 0 0
\(286\) 2808.00 0.580561
\(287\) −882.000 −0.181404
\(288\) 0 0
\(289\) 563.000 0.114594
\(290\) −9152.00 −1.85319
\(291\) 0 0
\(292\) −584.000 −0.117041
\(293\) −4734.00 −0.943902 −0.471951 0.881625i \(-0.656450\pi\)
−0.471951 + 0.881625i \(0.656450\pi\)
\(294\) 0 0
\(295\) 2728.00 0.538408
\(296\) −400.000 −0.0785457
\(297\) 0 0
\(298\) 2328.00 0.452542
\(299\) 3132.00 0.605780
\(300\) 0 0
\(301\) −1148.00 −0.219833
\(302\) −3344.00 −0.637171
\(303\) 0 0
\(304\) 1856.00 0.350161
\(305\) −3564.00 −0.669095
\(306\) 0 0
\(307\) 5836.00 1.08494 0.542472 0.840074i \(-0.317488\pi\)
0.542472 + 0.840074i \(0.317488\pi\)
\(308\) −728.000 −0.134681
\(309\) 0 0
\(310\) 11088.0 2.03147
\(311\) 5620.00 1.02470 0.512349 0.858777i \(-0.328775\pi\)
0.512349 + 0.858777i \(0.328775\pi\)
\(312\) 0 0
\(313\) 6082.00 1.09832 0.549161 0.835716i \(-0.314947\pi\)
0.549161 + 0.835716i \(0.314947\pi\)
\(314\) −892.000 −0.160314
\(315\) 0 0
\(316\) −3936.00 −0.700688
\(317\) −7308.00 −1.29482 −0.647410 0.762142i \(-0.724148\pi\)
−0.647410 + 0.762142i \(0.724148\pi\)
\(318\) 0 0
\(319\) 5408.00 0.949185
\(320\) 1408.00 0.245967
\(321\) 0 0
\(322\) −812.000 −0.140531
\(323\) 8584.00 1.47872
\(324\) 0 0
\(325\) −19386.0 −3.30874
\(326\) −856.000 −0.145428
\(327\) 0 0
\(328\) −1008.00 −0.169687
\(329\) 3108.00 0.520819
\(330\) 0 0
\(331\) −8020.00 −1.33178 −0.665890 0.746050i \(-0.731948\pi\)
−0.665890 + 0.746050i \(0.731948\pi\)
\(332\) 2624.00 0.433767
\(333\) 0 0
\(334\) −8.00000 −0.00131060
\(335\) −18920.0 −3.08570
\(336\) 0 0
\(337\) 4590.00 0.741938 0.370969 0.928645i \(-0.379026\pi\)
0.370969 + 0.928645i \(0.379026\pi\)
\(338\) −1438.00 −0.231411
\(339\) 0 0
\(340\) 6512.00 1.03871
\(341\) −6552.00 −1.04050
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −1312.00 −0.205635
\(345\) 0 0
\(346\) 1180.00 0.183344
\(347\) −6546.00 −1.01270 −0.506351 0.862327i \(-0.669006\pi\)
−0.506351 + 0.862327i \(0.669006\pi\)
\(348\) 0 0
\(349\) −7994.00 −1.22610 −0.613050 0.790044i \(-0.710058\pi\)
−0.613050 + 0.790044i \(0.710058\pi\)
\(350\) 5026.00 0.767574
\(351\) 0 0
\(352\) −832.000 −0.125982
\(353\) 4650.00 0.701118 0.350559 0.936541i \(-0.385992\pi\)
0.350559 + 0.936541i \(0.385992\pi\)
\(354\) 0 0
\(355\) −5236.00 −0.782811
\(356\) −3816.00 −0.568111
\(357\) 0 0
\(358\) −7068.00 −1.04345
\(359\) 346.000 0.0508668 0.0254334 0.999677i \(-0.491903\pi\)
0.0254334 + 0.999677i \(0.491903\pi\)
\(360\) 0 0
\(361\) 6597.00 0.961802
\(362\) −2196.00 −0.318838
\(363\) 0 0
\(364\) 1512.00 0.217721
\(365\) −3212.00 −0.460613
\(366\) 0 0
\(367\) −6784.00 −0.964910 −0.482455 0.875921i \(-0.660255\pi\)
−0.482455 + 0.875921i \(0.660255\pi\)
\(368\) −928.000 −0.131455
\(369\) 0 0
\(370\) −2200.00 −0.309115
\(371\) −84.0000 −0.0117549
\(372\) 0 0
\(373\) −6098.00 −0.846495 −0.423247 0.906014i \(-0.639110\pi\)
−0.423247 + 0.906014i \(0.639110\pi\)
\(374\) −3848.00 −0.532020
\(375\) 0 0
\(376\) 3552.00 0.487182
\(377\) −11232.0 −1.53442
\(378\) 0 0
\(379\) −2660.00 −0.360515 −0.180257 0.983619i \(-0.557693\pi\)
−0.180257 + 0.983619i \(0.557693\pi\)
\(380\) 10208.0 1.37805
\(381\) 0 0
\(382\) 9708.00 1.30027
\(383\) −760.000 −0.101395 −0.0506974 0.998714i \(-0.516144\pi\)
−0.0506974 + 0.998714i \(0.516144\pi\)
\(384\) 0 0
\(385\) −4004.00 −0.530033
\(386\) 2996.00 0.395058
\(387\) 0 0
\(388\) 2104.00 0.275295
\(389\) −104.000 −0.0135553 −0.00677765 0.999977i \(-0.502157\pi\)
−0.00677765 + 0.999977i \(0.502157\pi\)
\(390\) 0 0
\(391\) −4292.00 −0.555130
\(392\) −392.000 −0.0505076
\(393\) 0 0
\(394\) −1240.00 −0.158554
\(395\) −21648.0 −2.75754
\(396\) 0 0
\(397\) 4398.00 0.555993 0.277997 0.960582i \(-0.410330\pi\)
0.277997 + 0.960582i \(0.410330\pi\)
\(398\) −64.0000 −0.00806038
\(399\) 0 0
\(400\) 5744.00 0.718000
\(401\) 13236.0 1.64831 0.824157 0.566361i \(-0.191649\pi\)
0.824157 + 0.566361i \(0.191649\pi\)
\(402\) 0 0
\(403\) 13608.0 1.68204
\(404\) 5224.00 0.643326
\(405\) 0 0
\(406\) 2912.00 0.355961
\(407\) 1300.00 0.158326
\(408\) 0 0
\(409\) −9490.00 −1.14731 −0.573656 0.819097i \(-0.694475\pi\)
−0.573656 + 0.819097i \(0.694475\pi\)
\(410\) −5544.00 −0.667802
\(411\) 0 0
\(412\) −2032.00 −0.242984
\(413\) −868.000 −0.103418
\(414\) 0 0
\(415\) 14432.0 1.70708
\(416\) 1728.00 0.203659
\(417\) 0 0
\(418\) −6032.00 −0.705825
\(419\) 4236.00 0.493895 0.246948 0.969029i \(-0.420572\pi\)
0.246948 + 0.969029i \(0.420572\pi\)
\(420\) 0 0
\(421\) 918.000 0.106272 0.0531361 0.998587i \(-0.483078\pi\)
0.0531361 + 0.998587i \(0.483078\pi\)
\(422\) −8536.00 −0.984659
\(423\) 0 0
\(424\) −96.0000 −0.0109957
\(425\) 26566.0 3.03209
\(426\) 0 0
\(427\) 1134.00 0.128520
\(428\) 1992.00 0.224970
\(429\) 0 0
\(430\) −7216.00 −0.809271
\(431\) −11814.0 −1.32033 −0.660163 0.751123i \(-0.729513\pi\)
−0.660163 + 0.751123i \(0.729513\pi\)
\(432\) 0 0
\(433\) −8374.00 −0.929397 −0.464698 0.885469i \(-0.653837\pi\)
−0.464698 + 0.885469i \(0.653837\pi\)
\(434\) −3528.00 −0.390206
\(435\) 0 0
\(436\) −2456.00 −0.269773
\(437\) −6728.00 −0.736485
\(438\) 0 0
\(439\) 3840.00 0.417479 0.208739 0.977971i \(-0.433064\pi\)
0.208739 + 0.977971i \(0.433064\pi\)
\(440\) −4576.00 −0.495801
\(441\) 0 0
\(442\) 7992.00 0.860047
\(443\) 10166.0 1.09030 0.545148 0.838340i \(-0.316473\pi\)
0.545148 + 0.838340i \(0.316473\pi\)
\(444\) 0 0
\(445\) −20988.0 −2.23579
\(446\) −6928.00 −0.735539
\(447\) 0 0
\(448\) −448.000 −0.0472456
\(449\) 8200.00 0.861875 0.430938 0.902382i \(-0.358183\pi\)
0.430938 + 0.902382i \(0.358183\pi\)
\(450\) 0 0
\(451\) 3276.00 0.342042
\(452\) −4928.00 −0.512818
\(453\) 0 0
\(454\) 6504.00 0.672352
\(455\) 8316.00 0.856835
\(456\) 0 0
\(457\) −6074.00 −0.621728 −0.310864 0.950454i \(-0.600618\pi\)
−0.310864 + 0.950454i \(0.600618\pi\)
\(458\) −836.000 −0.0852920
\(459\) 0 0
\(460\) −5104.00 −0.517337
\(461\) −2006.00 −0.202665 −0.101333 0.994853i \(-0.532311\pi\)
−0.101333 + 0.994853i \(0.532311\pi\)
\(462\) 0 0
\(463\) −3728.00 −0.374201 −0.187100 0.982341i \(-0.559909\pi\)
−0.187100 + 0.982341i \(0.559909\pi\)
\(464\) 3328.00 0.332971
\(465\) 0 0
\(466\) −4168.00 −0.414332
\(467\) −6380.00 −0.632187 −0.316093 0.948728i \(-0.602371\pi\)
−0.316093 + 0.948728i \(0.602371\pi\)
\(468\) 0 0
\(469\) 6020.00 0.592703
\(470\) 19536.0 1.91729
\(471\) 0 0
\(472\) −992.000 −0.0967383
\(473\) 4264.00 0.414501
\(474\) 0 0
\(475\) 41644.0 4.02265
\(476\) −2072.00 −0.199517
\(477\) 0 0
\(478\) −3324.00 −0.318067
\(479\) 17180.0 1.63878 0.819389 0.573239i \(-0.194313\pi\)
0.819389 + 0.573239i \(0.194313\pi\)
\(480\) 0 0
\(481\) −2700.00 −0.255945
\(482\) −12364.0 −1.16839
\(483\) 0 0
\(484\) −2620.00 −0.246056
\(485\) 11572.0 1.08342
\(486\) 0 0
\(487\) −2728.00 −0.253835 −0.126917 0.991913i \(-0.540508\pi\)
−0.126917 + 0.991913i \(0.540508\pi\)
\(488\) 1296.00 0.120220
\(489\) 0 0
\(490\) −2156.00 −0.198772
\(491\) 2574.00 0.236585 0.118292 0.992979i \(-0.462258\pi\)
0.118292 + 0.992979i \(0.462258\pi\)
\(492\) 0 0
\(493\) 15392.0 1.40613
\(494\) 12528.0 1.14101
\(495\) 0 0
\(496\) −4032.00 −0.365004
\(497\) 1666.00 0.150363
\(498\) 0 0
\(499\) −7484.00 −0.671403 −0.335701 0.941969i \(-0.608973\pi\)
−0.335701 + 0.941969i \(0.608973\pi\)
\(500\) 20592.0 1.84180
\(501\) 0 0
\(502\) 1992.00 0.177106
\(503\) 7920.00 0.702058 0.351029 0.936365i \(-0.385832\pi\)
0.351029 + 0.936365i \(0.385832\pi\)
\(504\) 0 0
\(505\) 28732.0 2.53180
\(506\) 3016.00 0.264975
\(507\) 0 0
\(508\) −11232.0 −0.980983
\(509\) 7254.00 0.631685 0.315843 0.948812i \(-0.397713\pi\)
0.315843 + 0.948812i \(0.397713\pi\)
\(510\) 0 0
\(511\) 1022.00 0.0884748
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 11988.0 1.02873
\(515\) −11176.0 −0.956259
\(516\) 0 0
\(517\) −11544.0 −0.982020
\(518\) 700.000 0.0593750
\(519\) 0 0
\(520\) 9504.00 0.801496
\(521\) −17862.0 −1.50201 −0.751006 0.660295i \(-0.770431\pi\)
−0.751006 + 0.660295i \(0.770431\pi\)
\(522\) 0 0
\(523\) −592.000 −0.0494959 −0.0247479 0.999694i \(-0.507878\pi\)
−0.0247479 + 0.999694i \(0.507878\pi\)
\(524\) 2080.00 0.173407
\(525\) 0 0
\(526\) −12828.0 −1.06336
\(527\) −18648.0 −1.54140
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) −528.000 −0.0432733
\(531\) 0 0
\(532\) −3248.00 −0.264697
\(533\) −6804.00 −0.552934
\(534\) 0 0
\(535\) 10956.0 0.885363
\(536\) 6880.00 0.554423
\(537\) 0 0
\(538\) −5372.00 −0.430490
\(539\) 1274.00 0.101809
\(540\) 0 0
\(541\) −6402.00 −0.508768 −0.254384 0.967103i \(-0.581873\pi\)
−0.254384 + 0.967103i \(0.581873\pi\)
\(542\) −10200.0 −0.808353
\(543\) 0 0
\(544\) −2368.00 −0.186631
\(545\) −13508.0 −1.06169
\(546\) 0 0
\(547\) −8988.00 −0.702558 −0.351279 0.936271i \(-0.614253\pi\)
−0.351279 + 0.936271i \(0.614253\pi\)
\(548\) −10064.0 −0.784512
\(549\) 0 0
\(550\) −18668.0 −1.44728
\(551\) 24128.0 1.86549
\(552\) 0 0
\(553\) 6888.00 0.529670
\(554\) 8852.00 0.678855
\(555\) 0 0
\(556\) −10688.0 −0.815238
\(557\) 3244.00 0.246773 0.123387 0.992359i \(-0.460624\pi\)
0.123387 + 0.992359i \(0.460624\pi\)
\(558\) 0 0
\(559\) −8856.00 −0.670070
\(560\) −2464.00 −0.185934
\(561\) 0 0
\(562\) −15016.0 −1.12707
\(563\) 9812.00 0.734505 0.367253 0.930121i \(-0.380298\pi\)
0.367253 + 0.930121i \(0.380298\pi\)
\(564\) 0 0
\(565\) −27104.0 −2.01818
\(566\) −6824.00 −0.506774
\(567\) 0 0
\(568\) 1904.00 0.140652
\(569\) −12156.0 −0.895617 −0.447808 0.894130i \(-0.647795\pi\)
−0.447808 + 0.894130i \(0.647795\pi\)
\(570\) 0 0
\(571\) 6876.00 0.503943 0.251972 0.967735i \(-0.418921\pi\)
0.251972 + 0.967735i \(0.418921\pi\)
\(572\) −5616.00 −0.410519
\(573\) 0 0
\(574\) 1764.00 0.128272
\(575\) −20822.0 −1.51015
\(576\) 0 0
\(577\) 20002.0 1.44314 0.721572 0.692339i \(-0.243420\pi\)
0.721572 + 0.692339i \(0.243420\pi\)
\(578\) −1126.00 −0.0810301
\(579\) 0 0
\(580\) 18304.0 1.31040
\(581\) −4592.00 −0.327897
\(582\) 0 0
\(583\) 312.000 0.0221642
\(584\) 1168.00 0.0827606
\(585\) 0 0
\(586\) 9468.00 0.667439
\(587\) 18404.0 1.29406 0.647031 0.762464i \(-0.276010\pi\)
0.647031 + 0.762464i \(0.276010\pi\)
\(588\) 0 0
\(589\) −29232.0 −2.04496
\(590\) −5456.00 −0.380712
\(591\) 0 0
\(592\) 800.000 0.0555402
\(593\) −9846.00 −0.681833 −0.340916 0.940094i \(-0.610737\pi\)
−0.340916 + 0.940094i \(0.610737\pi\)
\(594\) 0 0
\(595\) −11396.0 −0.785194
\(596\) −4656.00 −0.319995
\(597\) 0 0
\(598\) −6264.00 −0.428351
\(599\) 9234.00 0.629868 0.314934 0.949114i \(-0.398018\pi\)
0.314934 + 0.949114i \(0.398018\pi\)
\(600\) 0 0
\(601\) −1510.00 −0.102486 −0.0512431 0.998686i \(-0.516318\pi\)
−0.0512431 + 0.998686i \(0.516318\pi\)
\(602\) 2296.00 0.155445
\(603\) 0 0
\(604\) 6688.00 0.450548
\(605\) −14410.0 −0.968347
\(606\) 0 0
\(607\) 17544.0 1.17313 0.586564 0.809903i \(-0.300480\pi\)
0.586564 + 0.809903i \(0.300480\pi\)
\(608\) −3712.00 −0.247601
\(609\) 0 0
\(610\) 7128.00 0.473122
\(611\) 23976.0 1.58750
\(612\) 0 0
\(613\) 9246.00 0.609205 0.304602 0.952480i \(-0.401476\pi\)
0.304602 + 0.952480i \(0.401476\pi\)
\(614\) −11672.0 −0.767172
\(615\) 0 0
\(616\) 1456.00 0.0952336
\(617\) −29212.0 −1.90605 −0.953023 0.302897i \(-0.902046\pi\)
−0.953023 + 0.302897i \(0.902046\pi\)
\(618\) 0 0
\(619\) 7096.00 0.460763 0.230382 0.973100i \(-0.426003\pi\)
0.230382 + 0.973100i \(0.426003\pi\)
\(620\) −22176.0 −1.43647
\(621\) 0 0
\(622\) −11240.0 −0.724571
\(623\) 6678.00 0.429452
\(624\) 0 0
\(625\) 68381.0 4.37638
\(626\) −12164.0 −0.776631
\(627\) 0 0
\(628\) 1784.00 0.113359
\(629\) 3700.00 0.234545
\(630\) 0 0
\(631\) 488.000 0.0307876 0.0153938 0.999882i \(-0.495100\pi\)
0.0153938 + 0.999882i \(0.495100\pi\)
\(632\) 7872.00 0.495461
\(633\) 0 0
\(634\) 14616.0 0.915577
\(635\) −61776.0 −3.86064
\(636\) 0 0
\(637\) −2646.00 −0.164581
\(638\) −10816.0 −0.671175
\(639\) 0 0
\(640\) −2816.00 −0.173925
\(641\) 8756.00 0.539534 0.269767 0.962926i \(-0.413053\pi\)
0.269767 + 0.962926i \(0.413053\pi\)
\(642\) 0 0
\(643\) −3364.00 −0.206319 −0.103160 0.994665i \(-0.532895\pi\)
−0.103160 + 0.994665i \(0.532895\pi\)
\(644\) 1624.00 0.0993704
\(645\) 0 0
\(646\) −17168.0 −1.04561
\(647\) 21804.0 1.32489 0.662445 0.749111i \(-0.269519\pi\)
0.662445 + 0.749111i \(0.269519\pi\)
\(648\) 0 0
\(649\) 3224.00 0.194997
\(650\) 38772.0 2.33964
\(651\) 0 0
\(652\) 1712.00 0.102833
\(653\) 13488.0 0.808310 0.404155 0.914691i \(-0.367566\pi\)
0.404155 + 0.914691i \(0.367566\pi\)
\(654\) 0 0
\(655\) 11440.0 0.682439
\(656\) 2016.00 0.119987
\(657\) 0 0
\(658\) −6216.00 −0.368275
\(659\) 28946.0 1.71104 0.855521 0.517769i \(-0.173237\pi\)
0.855521 + 0.517769i \(0.173237\pi\)
\(660\) 0 0
\(661\) −20642.0 −1.21465 −0.607323 0.794455i \(-0.707757\pi\)
−0.607323 + 0.794455i \(0.707757\pi\)
\(662\) 16040.0 0.941710
\(663\) 0 0
\(664\) −5248.00 −0.306720
\(665\) −17864.0 −1.04171
\(666\) 0 0
\(667\) −12064.0 −0.700330
\(668\) 16.0000 0.000926734 0
\(669\) 0 0
\(670\) 37840.0 2.18192
\(671\) −4212.00 −0.242329
\(672\) 0 0
\(673\) −17602.0 −1.00818 −0.504092 0.863650i \(-0.668173\pi\)
−0.504092 + 0.863650i \(0.668173\pi\)
\(674\) −9180.00 −0.524630
\(675\) 0 0
\(676\) 2876.00 0.163632
\(677\) 4266.00 0.242180 0.121090 0.992642i \(-0.461361\pi\)
0.121090 + 0.992642i \(0.461361\pi\)
\(678\) 0 0
\(679\) −3682.00 −0.208103
\(680\) −13024.0 −0.734482
\(681\) 0 0
\(682\) 13104.0 0.735745
\(683\) −26874.0 −1.50557 −0.752786 0.658266i \(-0.771290\pi\)
−0.752786 + 0.658266i \(0.771290\pi\)
\(684\) 0 0
\(685\) −55352.0 −3.08743
\(686\) 686.000 0.0381802
\(687\) 0 0
\(688\) 2624.00 0.145406
\(689\) −648.000 −0.0358299
\(690\) 0 0
\(691\) −17128.0 −0.942952 −0.471476 0.881879i \(-0.656279\pi\)
−0.471476 + 0.881879i \(0.656279\pi\)
\(692\) −2360.00 −0.129644
\(693\) 0 0
\(694\) 13092.0 0.716089
\(695\) −58784.0 −3.20835
\(696\) 0 0
\(697\) 9324.00 0.506703
\(698\) 15988.0 0.866984
\(699\) 0 0
\(700\) −10052.0 −0.542757
\(701\) 11968.0 0.644829 0.322414 0.946599i \(-0.395506\pi\)
0.322414 + 0.946599i \(0.395506\pi\)
\(702\) 0 0
\(703\) 5800.00 0.311168
\(704\) 1664.00 0.0890829
\(705\) 0 0
\(706\) −9300.00 −0.495765
\(707\) −9142.00 −0.486309
\(708\) 0 0
\(709\) −5278.00 −0.279576 −0.139788 0.990181i \(-0.544642\pi\)
−0.139788 + 0.990181i \(0.544642\pi\)
\(710\) 10472.0 0.553531
\(711\) 0 0
\(712\) 7632.00 0.401715
\(713\) 14616.0 0.767705
\(714\) 0 0
\(715\) −30888.0 −1.61559
\(716\) 14136.0 0.737831
\(717\) 0 0
\(718\) −692.000 −0.0359683
\(719\) 6720.00 0.348559 0.174279 0.984696i \(-0.444240\pi\)
0.174279 + 0.984696i \(0.444240\pi\)
\(720\) 0 0
\(721\) 3556.00 0.183679
\(722\) −13194.0 −0.680097
\(723\) 0 0
\(724\) 4392.00 0.225452
\(725\) 74672.0 3.82517
\(726\) 0 0
\(727\) 16804.0 0.857257 0.428629 0.903481i \(-0.358997\pi\)
0.428629 + 0.903481i \(0.358997\pi\)
\(728\) −3024.00 −0.153952
\(729\) 0 0
\(730\) 6424.00 0.325703
\(731\) 12136.0 0.614044
\(732\) 0 0
\(733\) 27522.0 1.38683 0.693416 0.720537i \(-0.256105\pi\)
0.693416 + 0.720537i \(0.256105\pi\)
\(734\) 13568.0 0.682294
\(735\) 0 0
\(736\) 1856.00 0.0929525
\(737\) −22360.0 −1.11756
\(738\) 0 0
\(739\) 21132.0 1.05190 0.525949 0.850516i \(-0.323710\pi\)
0.525949 + 0.850516i \(0.323710\pi\)
\(740\) 4400.00 0.218577
\(741\) 0 0
\(742\) 168.000 0.00831196
\(743\) −30.0000 −0.00148128 −0.000740641 1.00000i \(-0.500236\pi\)
−0.000740641 1.00000i \(0.500236\pi\)
\(744\) 0 0
\(745\) −25608.0 −1.25933
\(746\) 12196.0 0.598562
\(747\) 0 0
\(748\) 7696.00 0.376195
\(749\) −3486.00 −0.170061
\(750\) 0 0
\(751\) −15480.0 −0.752161 −0.376081 0.926587i \(-0.622728\pi\)
−0.376081 + 0.926587i \(0.622728\pi\)
\(752\) −7104.00 −0.344490
\(753\) 0 0
\(754\) 22464.0 1.08500
\(755\) 36784.0 1.77312
\(756\) 0 0
\(757\) 28770.0 1.38133 0.690663 0.723177i \(-0.257319\pi\)
0.690663 + 0.723177i \(0.257319\pi\)
\(758\) 5320.00 0.254922
\(759\) 0 0
\(760\) −20416.0 −0.974429
\(761\) −12418.0 −0.591527 −0.295764 0.955261i \(-0.595574\pi\)
−0.295764 + 0.955261i \(0.595574\pi\)
\(762\) 0 0
\(763\) 4298.00 0.203929
\(764\) −19416.0 −0.919432
\(765\) 0 0
\(766\) 1520.00 0.0716969
\(767\) −6696.00 −0.315226
\(768\) 0 0
\(769\) 12346.0 0.578944 0.289472 0.957186i \(-0.406520\pi\)
0.289472 + 0.957186i \(0.406520\pi\)
\(770\) 8008.00 0.374790
\(771\) 0 0
\(772\) −5992.00 −0.279348
\(773\) 38098.0 1.77269 0.886345 0.463025i \(-0.153236\pi\)
0.886345 + 0.463025i \(0.153236\pi\)
\(774\) 0 0
\(775\) −90468.0 −4.19317
\(776\) −4208.00 −0.194663
\(777\) 0 0
\(778\) 208.000 0.00958504
\(779\) 14616.0 0.672237
\(780\) 0 0
\(781\) −6188.00 −0.283514
\(782\) 8584.00 0.392536
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) 9812.00 0.446121
\(786\) 0 0
\(787\) 13824.0 0.626140 0.313070 0.949730i \(-0.398643\pi\)
0.313070 + 0.949730i \(0.398643\pi\)
\(788\) 2480.00 0.112115
\(789\) 0 0
\(790\) 43296.0 1.94988
\(791\) 8624.00 0.387654
\(792\) 0 0
\(793\) 8748.00 0.391741
\(794\) −8796.00 −0.393147
\(795\) 0 0
\(796\) 128.000 0.00569955
\(797\) 22170.0 0.985322 0.492661 0.870221i \(-0.336024\pi\)
0.492661 + 0.870221i \(0.336024\pi\)
\(798\) 0 0
\(799\) −32856.0 −1.45477
\(800\) −11488.0 −0.507703
\(801\) 0 0
\(802\) −26472.0 −1.16553
\(803\) −3796.00 −0.166822
\(804\) 0 0
\(805\) 8932.00 0.391070
\(806\) −27216.0 −1.18938
\(807\) 0 0
\(808\) −10448.0 −0.454900
\(809\) 16288.0 0.707856 0.353928 0.935273i \(-0.384846\pi\)
0.353928 + 0.935273i \(0.384846\pi\)
\(810\) 0 0
\(811\) −8720.00 −0.377559 −0.188780 0.982019i \(-0.560453\pi\)
−0.188780 + 0.982019i \(0.560453\pi\)
\(812\) −5824.00 −0.251702
\(813\) 0 0
\(814\) −2600.00 −0.111953
\(815\) 9416.00 0.404697
\(816\) 0 0
\(817\) 19024.0 0.814646
\(818\) 18980.0 0.811272
\(819\) 0 0
\(820\) 11088.0 0.472207
\(821\) −31372.0 −1.33361 −0.666803 0.745234i \(-0.732338\pi\)
−0.666803 + 0.745234i \(0.732338\pi\)
\(822\) 0 0
\(823\) −17648.0 −0.747473 −0.373737 0.927535i \(-0.621924\pi\)
−0.373737 + 0.927535i \(0.621924\pi\)
\(824\) 4064.00 0.171816
\(825\) 0 0
\(826\) 1736.00 0.0731273
\(827\) −2382.00 −0.100158 −0.0500788 0.998745i \(-0.515947\pi\)
−0.0500788 + 0.998745i \(0.515947\pi\)
\(828\) 0 0
\(829\) 26650.0 1.11652 0.558259 0.829667i \(-0.311470\pi\)
0.558259 + 0.829667i \(0.311470\pi\)
\(830\) −28864.0 −1.20709
\(831\) 0 0
\(832\) −3456.00 −0.144009
\(833\) 3626.00 0.150820
\(834\) 0 0
\(835\) 88.0000 0.00364714
\(836\) 12064.0 0.499093
\(837\) 0 0
\(838\) −8472.00 −0.349237
\(839\) 24092.0 0.991357 0.495678 0.868506i \(-0.334920\pi\)
0.495678 + 0.868506i \(0.334920\pi\)
\(840\) 0 0
\(841\) 18875.0 0.773914
\(842\) −1836.00 −0.0751458
\(843\) 0 0
\(844\) 17072.0 0.696259
\(845\) 15818.0 0.643971
\(846\) 0 0
\(847\) 4585.00 0.186001
\(848\) 192.000 0.00777513
\(849\) 0 0
\(850\) −53132.0 −2.14401
\(851\) −2900.00 −0.116816
\(852\) 0 0
\(853\) −8194.00 −0.328906 −0.164453 0.986385i \(-0.552586\pi\)
−0.164453 + 0.986385i \(0.552586\pi\)
\(854\) −2268.00 −0.0908775
\(855\) 0 0
\(856\) −3984.00 −0.159077
\(857\) −16962.0 −0.676092 −0.338046 0.941130i \(-0.609766\pi\)
−0.338046 + 0.941130i \(0.609766\pi\)
\(858\) 0 0
\(859\) −48556.0 −1.92865 −0.964324 0.264723i \(-0.914719\pi\)
−0.964324 + 0.264723i \(0.914719\pi\)
\(860\) 14432.0 0.572241
\(861\) 0 0
\(862\) 23628.0 0.933611
\(863\) −34274.0 −1.35191 −0.675956 0.736942i \(-0.736269\pi\)
−0.675956 + 0.736942i \(0.736269\pi\)
\(864\) 0 0
\(865\) −12980.0 −0.510212
\(866\) 16748.0 0.657183
\(867\) 0 0
\(868\) 7056.00 0.275917
\(869\) −25584.0 −0.998709
\(870\) 0 0
\(871\) 46440.0 1.80661
\(872\) 4912.00 0.190758
\(873\) 0 0
\(874\) 13456.0 0.520773
\(875\) −36036.0 −1.39227
\(876\) 0 0
\(877\) 7126.00 0.274376 0.137188 0.990545i \(-0.456194\pi\)
0.137188 + 0.990545i \(0.456194\pi\)
\(878\) −7680.00 −0.295202
\(879\) 0 0
\(880\) 9152.00 0.350584
\(881\) −9222.00 −0.352664 −0.176332 0.984331i \(-0.556423\pi\)
−0.176332 + 0.984331i \(0.556423\pi\)
\(882\) 0 0
\(883\) 37652.0 1.43498 0.717492 0.696567i \(-0.245290\pi\)
0.717492 + 0.696567i \(0.245290\pi\)
\(884\) −15984.0 −0.608145
\(885\) 0 0
\(886\) −20332.0 −0.770956
\(887\) −21996.0 −0.832642 −0.416321 0.909218i \(-0.636681\pi\)
−0.416321 + 0.909218i \(0.636681\pi\)
\(888\) 0 0
\(889\) 19656.0 0.741554
\(890\) 41976.0 1.58094
\(891\) 0 0
\(892\) 13856.0 0.520104
\(893\) −51504.0 −1.93003
\(894\) 0 0
\(895\) 77748.0 2.90372
\(896\) 896.000 0.0334077
\(897\) 0 0
\(898\) −16400.0 −0.609438
\(899\) −52416.0 −1.94457
\(900\) 0 0
\(901\) 888.000 0.0328342
\(902\) −6552.00 −0.241860
\(903\) 0 0
\(904\) 9856.00 0.362617
\(905\) 24156.0 0.887263
\(906\) 0 0
\(907\) 14844.0 0.543426 0.271713 0.962378i \(-0.412410\pi\)
0.271713 + 0.962378i \(0.412410\pi\)
\(908\) −13008.0 −0.475425
\(909\) 0 0
\(910\) −16632.0 −0.605874
\(911\) −19446.0 −0.707217 −0.353609 0.935394i \(-0.615045\pi\)
−0.353609 + 0.935394i \(0.615045\pi\)
\(912\) 0 0
\(913\) 17056.0 0.618260
\(914\) 12148.0 0.439628
\(915\) 0 0
\(916\) 1672.00 0.0603105
\(917\) −3640.00 −0.131083
\(918\) 0 0
\(919\) −39200.0 −1.40706 −0.703530 0.710665i \(-0.748394\pi\)
−0.703530 + 0.710665i \(0.748394\pi\)
\(920\) 10208.0 0.365813
\(921\) 0 0
\(922\) 4012.00 0.143306
\(923\) 12852.0 0.458319
\(924\) 0 0
\(925\) 17950.0 0.638046
\(926\) 7456.00 0.264600
\(927\) 0 0
\(928\) −6656.00 −0.235446
\(929\) −15954.0 −0.563438 −0.281719 0.959497i \(-0.590905\pi\)
−0.281719 + 0.959497i \(0.590905\pi\)
\(930\) 0 0
\(931\) 5684.00 0.200092
\(932\) 8336.00 0.292977
\(933\) 0 0
\(934\) 12760.0 0.447024
\(935\) 42328.0 1.48051
\(936\) 0 0
\(937\) 2546.00 0.0887665 0.0443832 0.999015i \(-0.485868\pi\)
0.0443832 + 0.999015i \(0.485868\pi\)
\(938\) −12040.0 −0.419104
\(939\) 0 0
\(940\) −39072.0 −1.35573
\(941\) 430.000 0.0148965 0.00744825 0.999972i \(-0.497629\pi\)
0.00744825 + 0.999972i \(0.497629\pi\)
\(942\) 0 0
\(943\) −7308.00 −0.252366
\(944\) 1984.00 0.0684043
\(945\) 0 0
\(946\) −8528.00 −0.293096
\(947\) 38266.0 1.31307 0.656535 0.754295i \(-0.272021\pi\)
0.656535 + 0.754295i \(0.272021\pi\)
\(948\) 0 0
\(949\) 7884.00 0.269679
\(950\) −83288.0 −2.84444
\(951\) 0 0
\(952\) 4144.00 0.141080
\(953\) −28216.0 −0.959083 −0.479541 0.877519i \(-0.659197\pi\)
−0.479541 + 0.877519i \(0.659197\pi\)
\(954\) 0 0
\(955\) −106788. −3.61841
\(956\) 6648.00 0.224908
\(957\) 0 0
\(958\) −34360.0 −1.15879
\(959\) 17612.0 0.593036
\(960\) 0 0
\(961\) 33713.0 1.13165
\(962\) 5400.00 0.180980
\(963\) 0 0
\(964\) 24728.0 0.826178
\(965\) −32956.0 −1.09937
\(966\) 0 0
\(967\) −27712.0 −0.921570 −0.460785 0.887512i \(-0.652432\pi\)
−0.460785 + 0.887512i \(0.652432\pi\)
\(968\) 5240.00 0.173988
\(969\) 0 0
\(970\) −23144.0 −0.766092
\(971\) 32976.0 1.08986 0.544928 0.838483i \(-0.316557\pi\)
0.544928 + 0.838483i \(0.316557\pi\)
\(972\) 0 0
\(973\) 18704.0 0.616262
\(974\) 5456.00 0.179488
\(975\) 0 0
\(976\) −2592.00 −0.0850081
\(977\) −8940.00 −0.292749 −0.146375 0.989229i \(-0.546760\pi\)
−0.146375 + 0.989229i \(0.546760\pi\)
\(978\) 0 0
\(979\) −24804.0 −0.809744
\(980\) 4312.00 0.140553
\(981\) 0 0
\(982\) −5148.00 −0.167291
\(983\) −3288.00 −0.106685 −0.0533423 0.998576i \(-0.516987\pi\)
−0.0533423 + 0.998576i \(0.516987\pi\)
\(984\) 0 0
\(985\) 13640.0 0.441225
\(986\) −30784.0 −0.994282
\(987\) 0 0
\(988\) −25056.0 −0.806819
\(989\) −9512.00 −0.305828
\(990\) 0 0
\(991\) 33944.0 1.08806 0.544030 0.839066i \(-0.316898\pi\)
0.544030 + 0.839066i \(0.316898\pi\)
\(992\) 8064.00 0.258097
\(993\) 0 0
\(994\) −3332.00 −0.106323
\(995\) 704.000 0.0224305
\(996\) 0 0
\(997\) −54562.0 −1.73320 −0.866598 0.499007i \(-0.833698\pi\)
−0.866598 + 0.499007i \(0.833698\pi\)
\(998\) 14968.0 0.474753
\(999\) 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.e.1.1 1
3.2 odd 2 126.4.a.f.1.1 yes 1
4.3 odd 2 1008.4.a.w.1.1 1
7.2 even 3 882.4.g.n.361.1 2
7.3 odd 6 882.4.g.x.667.1 2
7.4 even 3 882.4.g.n.667.1 2
7.5 odd 6 882.4.g.x.361.1 2
7.6 odd 2 882.4.a.a.1.1 1
12.11 even 2 1008.4.a.a.1.1 1
21.2 odd 6 882.4.g.m.361.1 2
21.5 even 6 882.4.g.a.361.1 2
21.11 odd 6 882.4.g.m.667.1 2
21.17 even 6 882.4.g.a.667.1 2
21.20 even 2 882.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.4.a.e.1.1 1 1.1 even 1 trivial
126.4.a.f.1.1 yes 1 3.2 odd 2
882.4.a.a.1.1 1 7.6 odd 2
882.4.a.s.1.1 1 21.20 even 2
882.4.g.a.361.1 2 21.5 even 6
882.4.g.a.667.1 2 21.17 even 6
882.4.g.m.361.1 2 21.2 odd 6
882.4.g.m.667.1 2 21.11 odd 6
882.4.g.n.361.1 2 7.2 even 3
882.4.g.n.667.1 2 7.4 even 3
882.4.g.x.361.1 2 7.5 odd 6
882.4.g.x.667.1 2 7.3 odd 6
1008.4.a.a.1.1 1 12.11 even 2
1008.4.a.w.1.1 1 4.3 odd 2