Properties

Label 135.4.a.c.1.1
Level $135$
Weight $4$
Character 135.1
Self dual yes
Analytic conductor $7.965$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 135.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} -6.00000 q^{7} -15.0000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} -6.00000 q^{7} -15.0000 q^{8} +5.00000 q^{10} -47.0000 q^{11} -5.00000 q^{13} -6.00000 q^{14} +41.0000 q^{16} -131.000 q^{17} -56.0000 q^{19} -35.0000 q^{20} -47.0000 q^{22} +3.00000 q^{23} +25.0000 q^{25} -5.00000 q^{26} +42.0000 q^{28} -157.000 q^{29} +225.000 q^{31} +161.000 q^{32} -131.000 q^{34} -30.0000 q^{35} -70.0000 q^{37} -56.0000 q^{38} -75.0000 q^{40} +140.000 q^{41} +397.000 q^{43} +329.000 q^{44} +3.00000 q^{46} -347.000 q^{47} -307.000 q^{49} +25.0000 q^{50} +35.0000 q^{52} +4.00000 q^{53} -235.000 q^{55} +90.0000 q^{56} -157.000 q^{58} +748.000 q^{59} -338.000 q^{61} +225.000 q^{62} -167.000 q^{64} -25.0000 q^{65} +492.000 q^{67} +917.000 q^{68} -30.0000 q^{70} +32.0000 q^{71} +970.000 q^{73} -70.0000 q^{74} +392.000 q^{76} +282.000 q^{77} -1257.00 q^{79} +205.000 q^{80} +140.000 q^{82} -102.000 q^{83} -655.000 q^{85} +397.000 q^{86} +705.000 q^{88} -1488.00 q^{89} +30.0000 q^{91} -21.0000 q^{92} -347.000 q^{94} -280.000 q^{95} +974.000 q^{97} -307.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.353553 0.176777 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) −15.0000 −0.662913
\(9\) 0 0
\(10\) 5.00000 0.158114
\(11\) −47.0000 −1.28828 −0.644138 0.764909i \(-0.722784\pi\)
−0.644138 + 0.764909i \(0.722784\pi\)
\(12\) 0 0
\(13\) −5.00000 −0.106673 −0.0533366 0.998577i \(-0.516986\pi\)
−0.0533366 + 0.998577i \(0.516986\pi\)
\(14\) −6.00000 −0.114541
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −131.000 −1.86895 −0.934475 0.356027i \(-0.884131\pi\)
−0.934475 + 0.356027i \(0.884131\pi\)
\(18\) 0 0
\(19\) −56.0000 −0.676173 −0.338086 0.941115i \(-0.609780\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) −35.0000 −0.391312
\(21\) 0 0
\(22\) −47.0000 −0.455474
\(23\) 3.00000 0.0271975 0.0135988 0.999908i \(-0.495671\pi\)
0.0135988 + 0.999908i \(0.495671\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −5.00000 −0.0377146
\(27\) 0 0
\(28\) 42.0000 0.283473
\(29\) −157.000 −1.00532 −0.502658 0.864485i \(-0.667645\pi\)
−0.502658 + 0.864485i \(0.667645\pi\)
\(30\) 0 0
\(31\) 225.000 1.30359 0.651793 0.758397i \(-0.274017\pi\)
0.651793 + 0.758397i \(0.274017\pi\)
\(32\) 161.000 0.889408
\(33\) 0 0
\(34\) −131.000 −0.660774
\(35\) −30.0000 −0.144884
\(36\) 0 0
\(37\) −70.0000 −0.311025 −0.155513 0.987834i \(-0.549703\pi\)
−0.155513 + 0.987834i \(0.549703\pi\)
\(38\) −56.0000 −0.239063
\(39\) 0 0
\(40\) −75.0000 −0.296464
\(41\) 140.000 0.533276 0.266638 0.963797i \(-0.414087\pi\)
0.266638 + 0.963797i \(0.414087\pi\)
\(42\) 0 0
\(43\) 397.000 1.40795 0.703976 0.710224i \(-0.251406\pi\)
0.703976 + 0.710224i \(0.251406\pi\)
\(44\) 329.000 1.12724
\(45\) 0 0
\(46\) 3.00000 0.00961578
\(47\) −347.000 −1.07692 −0.538459 0.842652i \(-0.680993\pi\)
−0.538459 + 0.842652i \(0.680993\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 25.0000 0.0707107
\(51\) 0 0
\(52\) 35.0000 0.0933390
\(53\) 4.00000 0.0103668 0.00518342 0.999987i \(-0.498350\pi\)
0.00518342 + 0.999987i \(0.498350\pi\)
\(54\) 0 0
\(55\) −235.000 −0.576134
\(56\) 90.0000 0.214763
\(57\) 0 0
\(58\) −157.000 −0.355433
\(59\) 748.000 1.65053 0.825265 0.564745i \(-0.191026\pi\)
0.825265 + 0.564745i \(0.191026\pi\)
\(60\) 0 0
\(61\) −338.000 −0.709450 −0.354725 0.934971i \(-0.615426\pi\)
−0.354725 + 0.934971i \(0.615426\pi\)
\(62\) 225.000 0.460888
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) −25.0000 −0.0477057
\(66\) 0 0
\(67\) 492.000 0.897125 0.448562 0.893751i \(-0.351936\pi\)
0.448562 + 0.893751i \(0.351936\pi\)
\(68\) 917.000 1.63533
\(69\) 0 0
\(70\) −30.0000 −0.0512241
\(71\) 32.0000 0.0534888 0.0267444 0.999642i \(-0.491486\pi\)
0.0267444 + 0.999642i \(0.491486\pi\)
\(72\) 0 0
\(73\) 970.000 1.55520 0.777602 0.628757i \(-0.216436\pi\)
0.777602 + 0.628757i \(0.216436\pi\)
\(74\) −70.0000 −0.109964
\(75\) 0 0
\(76\) 392.000 0.591651
\(77\) 282.000 0.417362
\(78\) 0 0
\(79\) −1257.00 −1.79017 −0.895086 0.445894i \(-0.852886\pi\)
−0.895086 + 0.445894i \(0.852886\pi\)
\(80\) 205.000 0.286496
\(81\) 0 0
\(82\) 140.000 0.188542
\(83\) −102.000 −0.134891 −0.0674455 0.997723i \(-0.521485\pi\)
−0.0674455 + 0.997723i \(0.521485\pi\)
\(84\) 0 0
\(85\) −655.000 −0.835820
\(86\) 397.000 0.497786
\(87\) 0 0
\(88\) 705.000 0.854014
\(89\) −1488.00 −1.77222 −0.886111 0.463474i \(-0.846603\pi\)
−0.886111 + 0.463474i \(0.846603\pi\)
\(90\) 0 0
\(91\) 30.0000 0.0345588
\(92\) −21.0000 −0.0237978
\(93\) 0 0
\(94\) −347.000 −0.380748
\(95\) −280.000 −0.302394
\(96\) 0 0
\(97\) 974.000 1.01953 0.509767 0.860313i \(-0.329732\pi\)
0.509767 + 0.860313i \(0.329732\pi\)
\(98\) −307.000 −0.316446
\(99\) 0 0
\(100\) −175.000 −0.175000
\(101\) −1335.00 −1.31522 −0.657611 0.753357i \(-0.728433\pi\)
−0.657611 + 0.753357i \(0.728433\pi\)
\(102\) 0 0
\(103\) 686.000 0.656248 0.328124 0.944635i \(-0.393584\pi\)
0.328124 + 0.944635i \(0.393584\pi\)
\(104\) 75.0000 0.0707150
\(105\) 0 0
\(106\) 4.00000 0.00366523
\(107\) −1098.00 −0.992034 −0.496017 0.868313i \(-0.665205\pi\)
−0.496017 + 0.868313i \(0.665205\pi\)
\(108\) 0 0
\(109\) −700.000 −0.615118 −0.307559 0.951529i \(-0.599512\pi\)
−0.307559 + 0.951529i \(0.599512\pi\)
\(110\) −235.000 −0.203694
\(111\) 0 0
\(112\) −246.000 −0.207543
\(113\) −1055.00 −0.878284 −0.439142 0.898418i \(-0.644717\pi\)
−0.439142 + 0.898418i \(0.644717\pi\)
\(114\) 0 0
\(115\) 15.0000 0.0121631
\(116\) 1099.00 0.879652
\(117\) 0 0
\(118\) 748.000 0.583551
\(119\) 786.000 0.605483
\(120\) 0 0
\(121\) 878.000 0.659654
\(122\) −338.000 −0.250829
\(123\) 0 0
\(124\) −1575.00 −1.14064
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1646.00 −1.15007 −0.575035 0.818129i \(-0.695012\pi\)
−0.575035 + 0.818129i \(0.695012\pi\)
\(128\) −1455.00 −1.00473
\(129\) 0 0
\(130\) −25.0000 −0.0168665
\(131\) −1833.00 −1.22252 −0.611259 0.791430i \(-0.709337\pi\)
−0.611259 + 0.791430i \(0.709337\pi\)
\(132\) 0 0
\(133\) 336.000 0.219059
\(134\) 492.000 0.317182
\(135\) 0 0
\(136\) 1965.00 1.23895
\(137\) 1098.00 0.684733 0.342367 0.939566i \(-0.388771\pi\)
0.342367 + 0.939566i \(0.388771\pi\)
\(138\) 0 0
\(139\) −1042.00 −0.635837 −0.317918 0.948118i \(-0.602984\pi\)
−0.317918 + 0.948118i \(0.602984\pi\)
\(140\) 210.000 0.126773
\(141\) 0 0
\(142\) 32.0000 0.0189111
\(143\) 235.000 0.137424
\(144\) 0 0
\(145\) −785.000 −0.449591
\(146\) 970.000 0.549848
\(147\) 0 0
\(148\) 490.000 0.272147
\(149\) −2941.00 −1.61702 −0.808510 0.588482i \(-0.799726\pi\)
−0.808510 + 0.588482i \(0.799726\pi\)
\(150\) 0 0
\(151\) 511.000 0.275395 0.137697 0.990474i \(-0.456030\pi\)
0.137697 + 0.990474i \(0.456030\pi\)
\(152\) 840.000 0.448243
\(153\) 0 0
\(154\) 282.000 0.147560
\(155\) 1125.00 0.582982
\(156\) 0 0
\(157\) −571.000 −0.290260 −0.145130 0.989413i \(-0.546360\pi\)
−0.145130 + 0.989413i \(0.546360\pi\)
\(158\) −1257.00 −0.632921
\(159\) 0 0
\(160\) 805.000 0.397755
\(161\) −18.0000 −0.00881117
\(162\) 0 0
\(163\) 713.000 0.342616 0.171308 0.985217i \(-0.445201\pi\)
0.171308 + 0.985217i \(0.445201\pi\)
\(164\) −980.000 −0.466617
\(165\) 0 0
\(166\) −102.000 −0.0476912
\(167\) 1596.00 0.739534 0.369767 0.929125i \(-0.379437\pi\)
0.369767 + 0.929125i \(0.379437\pi\)
\(168\) 0 0
\(169\) −2172.00 −0.988621
\(170\) −655.000 −0.295507
\(171\) 0 0
\(172\) −2779.00 −1.23196
\(173\) 4134.00 1.81678 0.908388 0.418129i \(-0.137314\pi\)
0.908388 + 0.418129i \(0.137314\pi\)
\(174\) 0 0
\(175\) −150.000 −0.0647939
\(176\) −1927.00 −0.825302
\(177\) 0 0
\(178\) −1488.00 −0.626575
\(179\) 1828.00 0.763302 0.381651 0.924306i \(-0.375356\pi\)
0.381651 + 0.924306i \(0.375356\pi\)
\(180\) 0 0
\(181\) −520.000 −0.213543 −0.106772 0.994284i \(-0.534051\pi\)
−0.106772 + 0.994284i \(0.534051\pi\)
\(182\) 30.0000 0.0122184
\(183\) 0 0
\(184\) −45.0000 −0.0180296
\(185\) −350.000 −0.139095
\(186\) 0 0
\(187\) 6157.00 2.40772
\(188\) 2429.00 0.942303
\(189\) 0 0
\(190\) −280.000 −0.106912
\(191\) 4826.00 1.82826 0.914129 0.405424i \(-0.132876\pi\)
0.914129 + 0.405424i \(0.132876\pi\)
\(192\) 0 0
\(193\) 1670.00 0.622846 0.311423 0.950271i \(-0.399194\pi\)
0.311423 + 0.950271i \(0.399194\pi\)
\(194\) 974.000 0.360459
\(195\) 0 0
\(196\) 2149.00 0.783163
\(197\) −1380.00 −0.499091 −0.249546 0.968363i \(-0.580281\pi\)
−0.249546 + 0.968363i \(0.580281\pi\)
\(198\) 0 0
\(199\) 4357.00 1.55206 0.776029 0.630697i \(-0.217231\pi\)
0.776029 + 0.630697i \(0.217231\pi\)
\(200\) −375.000 −0.132583
\(201\) 0 0
\(202\) −1335.00 −0.465001
\(203\) 942.000 0.325692
\(204\) 0 0
\(205\) 700.000 0.238488
\(206\) 686.000 0.232019
\(207\) 0 0
\(208\) −205.000 −0.0683375
\(209\) 2632.00 0.871097
\(210\) 0 0
\(211\) −4162.00 −1.35793 −0.678967 0.734169i \(-0.737572\pi\)
−0.678967 + 0.734169i \(0.737572\pi\)
\(212\) −28.0000 −0.00907098
\(213\) 0 0
\(214\) −1098.00 −0.350737
\(215\) 1985.00 0.629655
\(216\) 0 0
\(217\) −1350.00 −0.422322
\(218\) −700.000 −0.217477
\(219\) 0 0
\(220\) 1645.00 0.504118
\(221\) 655.000 0.199367
\(222\) 0 0
\(223\) −5956.00 −1.78853 −0.894267 0.447533i \(-0.852303\pi\)
−0.894267 + 0.447533i \(0.852303\pi\)
\(224\) −966.000 −0.288141
\(225\) 0 0
\(226\) −1055.00 −0.310520
\(227\) 4940.00 1.44440 0.722201 0.691683i \(-0.243130\pi\)
0.722201 + 0.691683i \(0.243130\pi\)
\(228\) 0 0
\(229\) 4344.00 1.25354 0.626768 0.779206i \(-0.284378\pi\)
0.626768 + 0.779206i \(0.284378\pi\)
\(230\) 15.0000 0.00430031
\(231\) 0 0
\(232\) 2355.00 0.666437
\(233\) −5202.00 −1.46264 −0.731318 0.682036i \(-0.761095\pi\)
−0.731318 + 0.682036i \(0.761095\pi\)
\(234\) 0 0
\(235\) −1735.00 −0.481612
\(236\) −5236.00 −1.44421
\(237\) 0 0
\(238\) 786.000 0.214071
\(239\) 1546.00 0.418420 0.209210 0.977871i \(-0.432911\pi\)
0.209210 + 0.977871i \(0.432911\pi\)
\(240\) 0 0
\(241\) −3659.00 −0.977995 −0.488998 0.872285i \(-0.662637\pi\)
−0.488998 + 0.872285i \(0.662637\pi\)
\(242\) 878.000 0.233223
\(243\) 0 0
\(244\) 2366.00 0.620769
\(245\) −1535.00 −0.400276
\(246\) 0 0
\(247\) 280.000 0.0721294
\(248\) −3375.00 −0.864164
\(249\) 0 0
\(250\) 125.000 0.0316228
\(251\) 1221.00 0.307047 0.153524 0.988145i \(-0.450938\pi\)
0.153524 + 0.988145i \(0.450938\pi\)
\(252\) 0 0
\(253\) −141.000 −0.0350379
\(254\) −1646.00 −0.406611
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −6255.00 −1.51820 −0.759098 0.650977i \(-0.774360\pi\)
−0.759098 + 0.650977i \(0.774360\pi\)
\(258\) 0 0
\(259\) 420.000 0.100763
\(260\) 175.000 0.0417425
\(261\) 0 0
\(262\) −1833.00 −0.432226
\(263\) 836.000 0.196007 0.0980037 0.995186i \(-0.468754\pi\)
0.0980037 + 0.995186i \(0.468754\pi\)
\(264\) 0 0
\(265\) 20.0000 0.00463619
\(266\) 336.000 0.0774492
\(267\) 0 0
\(268\) −3444.00 −0.784984
\(269\) −2231.00 −0.505675 −0.252837 0.967509i \(-0.581364\pi\)
−0.252837 + 0.967509i \(0.581364\pi\)
\(270\) 0 0
\(271\) −4832.00 −1.08311 −0.541556 0.840665i \(-0.682164\pi\)
−0.541556 + 0.840665i \(0.682164\pi\)
\(272\) −5371.00 −1.19730
\(273\) 0 0
\(274\) 1098.00 0.242090
\(275\) −1175.00 −0.257655
\(276\) 0 0
\(277\) 6450.00 1.39907 0.699536 0.714597i \(-0.253390\pi\)
0.699536 + 0.714597i \(0.253390\pi\)
\(278\) −1042.00 −0.224802
\(279\) 0 0
\(280\) 450.000 0.0960452
\(281\) −1050.00 −0.222910 −0.111455 0.993769i \(-0.535551\pi\)
−0.111455 + 0.993769i \(0.535551\pi\)
\(282\) 0 0
\(283\) −1584.00 −0.332717 −0.166359 0.986065i \(-0.553201\pi\)
−0.166359 + 0.986065i \(0.553201\pi\)
\(284\) −224.000 −0.0468027
\(285\) 0 0
\(286\) 235.000 0.0485869
\(287\) −840.000 −0.172765
\(288\) 0 0
\(289\) 12248.0 2.49298
\(290\) −785.000 −0.158954
\(291\) 0 0
\(292\) −6790.00 −1.36080
\(293\) 6594.00 1.31476 0.657382 0.753558i \(-0.271664\pi\)
0.657382 + 0.753558i \(0.271664\pi\)
\(294\) 0 0
\(295\) 3740.00 0.738140
\(296\) 1050.00 0.206182
\(297\) 0 0
\(298\) −2941.00 −0.571703
\(299\) −15.0000 −0.00290125
\(300\) 0 0
\(301\) −2382.00 −0.456134
\(302\) 511.000 0.0973667
\(303\) 0 0
\(304\) −2296.00 −0.433173
\(305\) −1690.00 −0.317276
\(306\) 0 0
\(307\) −4343.00 −0.807388 −0.403694 0.914894i \(-0.632274\pi\)
−0.403694 + 0.914894i \(0.632274\pi\)
\(308\) −1974.00 −0.365192
\(309\) 0 0
\(310\) 1125.00 0.206115
\(311\) −2124.00 −0.387270 −0.193635 0.981074i \(-0.562028\pi\)
−0.193635 + 0.981074i \(0.562028\pi\)
\(312\) 0 0
\(313\) −7516.00 −1.35728 −0.678641 0.734470i \(-0.737431\pi\)
−0.678641 + 0.734470i \(0.737431\pi\)
\(314\) −571.000 −0.102622
\(315\) 0 0
\(316\) 8799.00 1.56640
\(317\) −6880.00 −1.21899 −0.609494 0.792791i \(-0.708627\pi\)
−0.609494 + 0.792791i \(0.708627\pi\)
\(318\) 0 0
\(319\) 7379.00 1.29512
\(320\) −835.000 −0.145868
\(321\) 0 0
\(322\) −18.0000 −0.00311522
\(323\) 7336.00 1.26373
\(324\) 0 0
\(325\) −125.000 −0.0213346
\(326\) 713.000 0.121133
\(327\) 0 0
\(328\) −2100.00 −0.353516
\(329\) 2082.00 0.348889
\(330\) 0 0
\(331\) −4986.00 −0.827962 −0.413981 0.910286i \(-0.635862\pi\)
−0.413981 + 0.910286i \(0.635862\pi\)
\(332\) 714.000 0.118030
\(333\) 0 0
\(334\) 1596.00 0.261465
\(335\) 2460.00 0.401206
\(336\) 0 0
\(337\) 904.000 0.146125 0.0730623 0.997327i \(-0.476723\pi\)
0.0730623 + 0.997327i \(0.476723\pi\)
\(338\) −2172.00 −0.349530
\(339\) 0 0
\(340\) 4585.00 0.731343
\(341\) −10575.0 −1.67938
\(342\) 0 0
\(343\) 3900.00 0.613936
\(344\) −5955.00 −0.933349
\(345\) 0 0
\(346\) 4134.00 0.642327
\(347\) 8860.00 1.37069 0.685345 0.728218i \(-0.259651\pi\)
0.685345 + 0.728218i \(0.259651\pi\)
\(348\) 0 0
\(349\) −4454.00 −0.683144 −0.341572 0.939856i \(-0.610959\pi\)
−0.341572 + 0.939856i \(0.610959\pi\)
\(350\) −150.000 −0.0229081
\(351\) 0 0
\(352\) −7567.00 −1.14580
\(353\) −8781.00 −1.32398 −0.661991 0.749512i \(-0.730288\pi\)
−0.661991 + 0.749512i \(0.730288\pi\)
\(354\) 0 0
\(355\) 160.000 0.0239209
\(356\) 10416.0 1.55069
\(357\) 0 0
\(358\) 1828.00 0.269868
\(359\) 2928.00 0.430457 0.215228 0.976564i \(-0.430950\pi\)
0.215228 + 0.976564i \(0.430950\pi\)
\(360\) 0 0
\(361\) −3723.00 −0.542790
\(362\) −520.000 −0.0754989
\(363\) 0 0
\(364\) −210.000 −0.0302390
\(365\) 4850.00 0.695508
\(366\) 0 0
\(367\) 9102.00 1.29461 0.647303 0.762233i \(-0.275897\pi\)
0.647303 + 0.762233i \(0.275897\pi\)
\(368\) 123.000 0.0174234
\(369\) 0 0
\(370\) −350.000 −0.0491774
\(371\) −24.0000 −0.00335854
\(372\) 0 0
\(373\) −8183.00 −1.13592 −0.567962 0.823055i \(-0.692268\pi\)
−0.567962 + 0.823055i \(0.692268\pi\)
\(374\) 6157.00 0.851259
\(375\) 0 0
\(376\) 5205.00 0.713903
\(377\) 785.000 0.107240
\(378\) 0 0
\(379\) 6136.00 0.831623 0.415812 0.909451i \(-0.363498\pi\)
0.415812 + 0.909451i \(0.363498\pi\)
\(380\) 1960.00 0.264594
\(381\) 0 0
\(382\) 4826.00 0.646386
\(383\) 5643.00 0.752856 0.376428 0.926446i \(-0.377152\pi\)
0.376428 + 0.926446i \(0.377152\pi\)
\(384\) 0 0
\(385\) 1410.00 0.186650
\(386\) 1670.00 0.220209
\(387\) 0 0
\(388\) −6818.00 −0.892092
\(389\) 8991.00 1.17188 0.585941 0.810354i \(-0.300725\pi\)
0.585941 + 0.810354i \(0.300725\pi\)
\(390\) 0 0
\(391\) −393.000 −0.0508309
\(392\) 4605.00 0.593336
\(393\) 0 0
\(394\) −1380.00 −0.176455
\(395\) −6285.00 −0.800589
\(396\) 0 0
\(397\) −12449.0 −1.57380 −0.786898 0.617082i \(-0.788314\pi\)
−0.786898 + 0.617082i \(0.788314\pi\)
\(398\) 4357.00 0.548735
\(399\) 0 0
\(400\) 1025.00 0.128125
\(401\) 8076.00 1.00573 0.502863 0.864366i \(-0.332280\pi\)
0.502863 + 0.864366i \(0.332280\pi\)
\(402\) 0 0
\(403\) −1125.00 −0.139058
\(404\) 9345.00 1.15082
\(405\) 0 0
\(406\) 942.000 0.115149
\(407\) 3290.00 0.400686
\(408\) 0 0
\(409\) −2833.00 −0.342501 −0.171250 0.985228i \(-0.554781\pi\)
−0.171250 + 0.985228i \(0.554781\pi\)
\(410\) 700.000 0.0843184
\(411\) 0 0
\(412\) −4802.00 −0.574217
\(413\) −4488.00 −0.534722
\(414\) 0 0
\(415\) −510.000 −0.0603251
\(416\) −805.000 −0.0948759
\(417\) 0 0
\(418\) 2632.00 0.307979
\(419\) −4777.00 −0.556973 −0.278487 0.960440i \(-0.589833\pi\)
−0.278487 + 0.960440i \(0.589833\pi\)
\(420\) 0 0
\(421\) −6464.00 −0.748304 −0.374152 0.927367i \(-0.622066\pi\)
−0.374152 + 0.927367i \(0.622066\pi\)
\(422\) −4162.00 −0.480102
\(423\) 0 0
\(424\) −60.0000 −0.00687231
\(425\) −3275.00 −0.373790
\(426\) 0 0
\(427\) 2028.00 0.229840
\(428\) 7686.00 0.868030
\(429\) 0 0
\(430\) 1985.00 0.222617
\(431\) 10680.0 1.19359 0.596795 0.802394i \(-0.296440\pi\)
0.596795 + 0.802394i \(0.296440\pi\)
\(432\) 0 0
\(433\) 11566.0 1.28366 0.641832 0.766845i \(-0.278175\pi\)
0.641832 + 0.766845i \(0.278175\pi\)
\(434\) −1350.00 −0.149314
\(435\) 0 0
\(436\) 4900.00 0.538228
\(437\) −168.000 −0.0183902
\(438\) 0 0
\(439\) −1448.00 −0.157424 −0.0787122 0.996897i \(-0.525081\pi\)
−0.0787122 + 0.996897i \(0.525081\pi\)
\(440\) 3525.00 0.381927
\(441\) 0 0
\(442\) 655.000 0.0704868
\(443\) −2376.00 −0.254824 −0.127412 0.991850i \(-0.540667\pi\)
−0.127412 + 0.991850i \(0.540667\pi\)
\(444\) 0 0
\(445\) −7440.00 −0.792561
\(446\) −5956.00 −0.632343
\(447\) 0 0
\(448\) 1002.00 0.105670
\(449\) −14894.0 −1.56546 −0.782730 0.622362i \(-0.786173\pi\)
−0.782730 + 0.622362i \(0.786173\pi\)
\(450\) 0 0
\(451\) −6580.00 −0.687007
\(452\) 7385.00 0.768498
\(453\) 0 0
\(454\) 4940.00 0.510673
\(455\) 150.000 0.0154552
\(456\) 0 0
\(457\) 16204.0 1.65862 0.829312 0.558786i \(-0.188733\pi\)
0.829312 + 0.558786i \(0.188733\pi\)
\(458\) 4344.00 0.443192
\(459\) 0 0
\(460\) −105.000 −0.0106427
\(461\) −5082.00 −0.513432 −0.256716 0.966487i \(-0.582641\pi\)
−0.256716 + 0.966487i \(0.582641\pi\)
\(462\) 0 0
\(463\) −10326.0 −1.03648 −0.518240 0.855235i \(-0.673412\pi\)
−0.518240 + 0.855235i \(0.673412\pi\)
\(464\) −6437.00 −0.644031
\(465\) 0 0
\(466\) −5202.00 −0.517120
\(467\) 4184.00 0.414588 0.207294 0.978279i \(-0.433534\pi\)
0.207294 + 0.978279i \(0.433534\pi\)
\(468\) 0 0
\(469\) −2952.00 −0.290641
\(470\) −1735.00 −0.170276
\(471\) 0 0
\(472\) −11220.0 −1.09416
\(473\) −18659.0 −1.81383
\(474\) 0 0
\(475\) −1400.00 −0.135235
\(476\) −5502.00 −0.529798
\(477\) 0 0
\(478\) 1546.00 0.147934
\(479\) −15576.0 −1.48577 −0.742887 0.669417i \(-0.766544\pi\)
−0.742887 + 0.669417i \(0.766544\pi\)
\(480\) 0 0
\(481\) 350.000 0.0331780
\(482\) −3659.00 −0.345774
\(483\) 0 0
\(484\) −6146.00 −0.577198
\(485\) 4870.00 0.455949
\(486\) 0 0
\(487\) 10220.0 0.950949 0.475475 0.879729i \(-0.342276\pi\)
0.475475 + 0.879729i \(0.342276\pi\)
\(488\) 5070.00 0.470304
\(489\) 0 0
\(490\) −1535.00 −0.141519
\(491\) −2692.00 −0.247430 −0.123715 0.992318i \(-0.539481\pi\)
−0.123715 + 0.992318i \(0.539481\pi\)
\(492\) 0 0
\(493\) 20567.0 1.87889
\(494\) 280.000 0.0255016
\(495\) 0 0
\(496\) 9225.00 0.835110
\(497\) −192.000 −0.0173287
\(498\) 0 0
\(499\) 5764.00 0.517098 0.258549 0.965998i \(-0.416756\pi\)
0.258549 + 0.965998i \(0.416756\pi\)
\(500\) −875.000 −0.0782624
\(501\) 0 0
\(502\) 1221.00 0.108558
\(503\) −2437.00 −0.216025 −0.108012 0.994150i \(-0.534449\pi\)
−0.108012 + 0.994150i \(0.534449\pi\)
\(504\) 0 0
\(505\) −6675.00 −0.588185
\(506\) −141.000 −0.0123878
\(507\) 0 0
\(508\) 11522.0 1.00631
\(509\) −5849.00 −0.509337 −0.254668 0.967028i \(-0.581966\pi\)
−0.254668 + 0.967028i \(0.581966\pi\)
\(510\) 0 0
\(511\) −5820.00 −0.503839
\(512\) 11521.0 0.994455
\(513\) 0 0
\(514\) −6255.00 −0.536763
\(515\) 3430.00 0.293483
\(516\) 0 0
\(517\) 16309.0 1.38737
\(518\) 420.000 0.0356250
\(519\) 0 0
\(520\) 375.000 0.0316247
\(521\) −17032.0 −1.43222 −0.716109 0.697989i \(-0.754079\pi\)
−0.716109 + 0.697989i \(0.754079\pi\)
\(522\) 0 0
\(523\) 4147.00 0.346722 0.173361 0.984858i \(-0.444537\pi\)
0.173361 + 0.984858i \(0.444537\pi\)
\(524\) 12831.0 1.06970
\(525\) 0 0
\(526\) 836.000 0.0692991
\(527\) −29475.0 −2.43634
\(528\) 0 0
\(529\) −12158.0 −0.999260
\(530\) 20.0000 0.00163914
\(531\) 0 0
\(532\) −2352.00 −0.191677
\(533\) −700.000 −0.0568862
\(534\) 0 0
\(535\) −5490.00 −0.443651
\(536\) −7380.00 −0.594715
\(537\) 0 0
\(538\) −2231.00 −0.178783
\(539\) 14429.0 1.15306
\(540\) 0 0
\(541\) −3942.00 −0.313271 −0.156636 0.987656i \(-0.550065\pi\)
−0.156636 + 0.987656i \(0.550065\pi\)
\(542\) −4832.00 −0.382938
\(543\) 0 0
\(544\) −21091.0 −1.66226
\(545\) −3500.00 −0.275089
\(546\) 0 0
\(547\) −13751.0 −1.07486 −0.537432 0.843307i \(-0.680605\pi\)
−0.537432 + 0.843307i \(0.680605\pi\)
\(548\) −7686.00 −0.599142
\(549\) 0 0
\(550\) −1175.00 −0.0910949
\(551\) 8792.00 0.679767
\(552\) 0 0
\(553\) 7542.00 0.579961
\(554\) 6450.00 0.494647
\(555\) 0 0
\(556\) 7294.00 0.556357
\(557\) 7944.00 0.604305 0.302153 0.953260i \(-0.402295\pi\)
0.302153 + 0.953260i \(0.402295\pi\)
\(558\) 0 0
\(559\) −1985.00 −0.150191
\(560\) −1230.00 −0.0928160
\(561\) 0 0
\(562\) −1050.00 −0.0788106
\(563\) −6702.00 −0.501697 −0.250849 0.968026i \(-0.580710\pi\)
−0.250849 + 0.968026i \(0.580710\pi\)
\(564\) 0 0
\(565\) −5275.00 −0.392780
\(566\) −1584.00 −0.117633
\(567\) 0 0
\(568\) −480.000 −0.0354584
\(569\) −2760.00 −0.203348 −0.101674 0.994818i \(-0.532420\pi\)
−0.101674 + 0.994818i \(0.532420\pi\)
\(570\) 0 0
\(571\) 8930.00 0.654481 0.327241 0.944941i \(-0.393881\pi\)
0.327241 + 0.944941i \(0.393881\pi\)
\(572\) −1645.00 −0.120246
\(573\) 0 0
\(574\) −840.000 −0.0610817
\(575\) 75.0000 0.00543951
\(576\) 0 0
\(577\) 6944.00 0.501010 0.250505 0.968115i \(-0.419403\pi\)
0.250505 + 0.968115i \(0.419403\pi\)
\(578\) 12248.0 0.881401
\(579\) 0 0
\(580\) 5495.00 0.393392
\(581\) 612.000 0.0437006
\(582\) 0 0
\(583\) −188.000 −0.0133553
\(584\) −14550.0 −1.03096
\(585\) 0 0
\(586\) 6594.00 0.464839
\(587\) −4206.00 −0.295741 −0.147871 0.989007i \(-0.547242\pi\)
−0.147871 + 0.989007i \(0.547242\pi\)
\(588\) 0 0
\(589\) −12600.0 −0.881450
\(590\) 3740.00 0.260972
\(591\) 0 0
\(592\) −2870.00 −0.199250
\(593\) −6571.00 −0.455040 −0.227520 0.973773i \(-0.573062\pi\)
−0.227520 + 0.973773i \(0.573062\pi\)
\(594\) 0 0
\(595\) 3930.00 0.270780
\(596\) 20587.0 1.41489
\(597\) 0 0
\(598\) −15.0000 −0.00102575
\(599\) −9490.00 −0.647330 −0.323665 0.946172i \(-0.604915\pi\)
−0.323665 + 0.946172i \(0.604915\pi\)
\(600\) 0 0
\(601\) 11861.0 0.805025 0.402513 0.915414i \(-0.368137\pi\)
0.402513 + 0.915414i \(0.368137\pi\)
\(602\) −2382.00 −0.161268
\(603\) 0 0
\(604\) −3577.00 −0.240970
\(605\) 4390.00 0.295006
\(606\) 0 0
\(607\) −518.000 −0.0346375 −0.0173188 0.999850i \(-0.505513\pi\)
−0.0173188 + 0.999850i \(0.505513\pi\)
\(608\) −9016.00 −0.601393
\(609\) 0 0
\(610\) −1690.00 −0.112174
\(611\) 1735.00 0.114878
\(612\) 0 0
\(613\) 15163.0 0.999067 0.499533 0.866295i \(-0.333505\pi\)
0.499533 + 0.866295i \(0.333505\pi\)
\(614\) −4343.00 −0.285455
\(615\) 0 0
\(616\) −4230.00 −0.276675
\(617\) 19011.0 1.24044 0.620222 0.784426i \(-0.287042\pi\)
0.620222 + 0.784426i \(0.287042\pi\)
\(618\) 0 0
\(619\) −7906.00 −0.513359 −0.256679 0.966497i \(-0.582628\pi\)
−0.256679 + 0.966497i \(0.582628\pi\)
\(620\) −7875.00 −0.510109
\(621\) 0 0
\(622\) −2124.00 −0.136921
\(623\) 8928.00 0.574146
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −7516.00 −0.479872
\(627\) 0 0
\(628\) 3997.00 0.253977
\(629\) 9170.00 0.581291
\(630\) 0 0
\(631\) −3416.00 −0.215513 −0.107757 0.994177i \(-0.534367\pi\)
−0.107757 + 0.994177i \(0.534367\pi\)
\(632\) 18855.0 1.18673
\(633\) 0 0
\(634\) −6880.00 −0.430977
\(635\) −8230.00 −0.514327
\(636\) 0 0
\(637\) 1535.00 0.0954771
\(638\) 7379.00 0.457896
\(639\) 0 0
\(640\) −7275.00 −0.449328
\(641\) 4830.00 0.297619 0.148809 0.988866i \(-0.452456\pi\)
0.148809 + 0.988866i \(0.452456\pi\)
\(642\) 0 0
\(643\) 12549.0 0.769649 0.384824 0.922990i \(-0.374262\pi\)
0.384824 + 0.922990i \(0.374262\pi\)
\(644\) 126.000 0.00770978
\(645\) 0 0
\(646\) 7336.00 0.446797
\(647\) 8164.00 0.496074 0.248037 0.968751i \(-0.420215\pi\)
0.248037 + 0.968751i \(0.420215\pi\)
\(648\) 0 0
\(649\) −35156.0 −2.12634
\(650\) −125.000 −0.00754293
\(651\) 0 0
\(652\) −4991.00 −0.299789
\(653\) 23768.0 1.42437 0.712185 0.701992i \(-0.247706\pi\)
0.712185 + 0.701992i \(0.247706\pi\)
\(654\) 0 0
\(655\) −9165.00 −0.546727
\(656\) 5740.00 0.341630
\(657\) 0 0
\(658\) 2082.00 0.123351
\(659\) 21472.0 1.26924 0.634621 0.772824i \(-0.281156\pi\)
0.634621 + 0.772824i \(0.281156\pi\)
\(660\) 0 0
\(661\) 12982.0 0.763905 0.381953 0.924182i \(-0.375252\pi\)
0.381953 + 0.924182i \(0.375252\pi\)
\(662\) −4986.00 −0.292729
\(663\) 0 0
\(664\) 1530.00 0.0894210
\(665\) 1680.00 0.0979663
\(666\) 0 0
\(667\) −471.000 −0.0273421
\(668\) −11172.0 −0.647092
\(669\) 0 0
\(670\) 2460.00 0.141848
\(671\) 15886.0 0.913968
\(672\) 0 0
\(673\) −6006.00 −0.344003 −0.172002 0.985097i \(-0.555023\pi\)
−0.172002 + 0.985097i \(0.555023\pi\)
\(674\) 904.000 0.0516629
\(675\) 0 0
\(676\) 15204.0 0.865043
\(677\) 1164.00 0.0660800 0.0330400 0.999454i \(-0.489481\pi\)
0.0330400 + 0.999454i \(0.489481\pi\)
\(678\) 0 0
\(679\) −5844.00 −0.330298
\(680\) 9825.00 0.554076
\(681\) 0 0
\(682\) −10575.0 −0.593750
\(683\) 26496.0 1.48439 0.742197 0.670182i \(-0.233784\pi\)
0.742197 + 0.670182i \(0.233784\pi\)
\(684\) 0 0
\(685\) 5490.00 0.306222
\(686\) 3900.00 0.217059
\(687\) 0 0
\(688\) 16277.0 0.901969
\(689\) −20.0000 −0.00110586
\(690\) 0 0
\(691\) 17110.0 0.941961 0.470981 0.882144i \(-0.343900\pi\)
0.470981 + 0.882144i \(0.343900\pi\)
\(692\) −28938.0 −1.58968
\(693\) 0 0
\(694\) 8860.00 0.484612
\(695\) −5210.00 −0.284355
\(696\) 0 0
\(697\) −18340.0 −0.996667
\(698\) −4454.00 −0.241528
\(699\) 0 0
\(700\) 1050.00 0.0566947
\(701\) 30251.0 1.62991 0.814953 0.579527i \(-0.196763\pi\)
0.814953 + 0.579527i \(0.196763\pi\)
\(702\) 0 0
\(703\) 3920.00 0.210307
\(704\) 7849.00 0.420199
\(705\) 0 0
\(706\) −8781.00 −0.468098
\(707\) 8010.00 0.426092
\(708\) 0 0
\(709\) 18820.0 0.996897 0.498448 0.866919i \(-0.333903\pi\)
0.498448 + 0.866919i \(0.333903\pi\)
\(710\) 160.000 0.00845731
\(711\) 0 0
\(712\) 22320.0 1.17483
\(713\) 675.000 0.0354543
\(714\) 0 0
\(715\) 1175.00 0.0614581
\(716\) −12796.0 −0.667890
\(717\) 0 0
\(718\) 2928.00 0.152189
\(719\) 31890.0 1.65410 0.827049 0.562130i \(-0.190018\pi\)
0.827049 + 0.562130i \(0.190018\pi\)
\(720\) 0 0
\(721\) −4116.00 −0.212605
\(722\) −3723.00 −0.191905
\(723\) 0 0
\(724\) 3640.00 0.186850
\(725\) −3925.00 −0.201063
\(726\) 0 0
\(727\) −11452.0 −0.584224 −0.292112 0.956384i \(-0.594358\pi\)
−0.292112 + 0.956384i \(0.594358\pi\)
\(728\) −450.000 −0.0229095
\(729\) 0 0
\(730\) 4850.00 0.245899
\(731\) −52007.0 −2.63139
\(732\) 0 0
\(733\) 7094.00 0.357466 0.178733 0.983898i \(-0.442800\pi\)
0.178733 + 0.983898i \(0.442800\pi\)
\(734\) 9102.00 0.457712
\(735\) 0 0
\(736\) 483.000 0.0241897
\(737\) −23124.0 −1.15574
\(738\) 0 0
\(739\) −3200.00 −0.159288 −0.0796440 0.996823i \(-0.525378\pi\)
−0.0796440 + 0.996823i \(0.525378\pi\)
\(740\) 2450.00 0.121708
\(741\) 0 0
\(742\) −24.0000 −0.00118742
\(743\) −20831.0 −1.02855 −0.514277 0.857624i \(-0.671940\pi\)
−0.514277 + 0.857624i \(0.671940\pi\)
\(744\) 0 0
\(745\) −14705.0 −0.723154
\(746\) −8183.00 −0.401610
\(747\) 0 0
\(748\) −43099.0 −2.10676
\(749\) 6588.00 0.321389
\(750\) 0 0
\(751\) −15605.0 −0.758235 −0.379118 0.925349i \(-0.623772\pi\)
−0.379118 + 0.925349i \(0.623772\pi\)
\(752\) −14227.0 −0.689901
\(753\) 0 0
\(754\) 785.000 0.0379151
\(755\) 2555.00 0.123160
\(756\) 0 0
\(757\) 21349.0 1.02502 0.512512 0.858680i \(-0.328715\pi\)
0.512512 + 0.858680i \(0.328715\pi\)
\(758\) 6136.00 0.294023
\(759\) 0 0
\(760\) 4200.00 0.200461
\(761\) 3702.00 0.176343 0.0881717 0.996105i \(-0.471898\pi\)
0.0881717 + 0.996105i \(0.471898\pi\)
\(762\) 0 0
\(763\) 4200.00 0.199279
\(764\) −33782.0 −1.59972
\(765\) 0 0
\(766\) 5643.00 0.266175
\(767\) −3740.00 −0.176067
\(768\) 0 0
\(769\) −1393.00 −0.0653223 −0.0326612 0.999466i \(-0.510398\pi\)
−0.0326612 + 0.999466i \(0.510398\pi\)
\(770\) 1410.00 0.0659907
\(771\) 0 0
\(772\) −11690.0 −0.544990
\(773\) −6906.00 −0.321334 −0.160667 0.987009i \(-0.551365\pi\)
−0.160667 + 0.987009i \(0.551365\pi\)
\(774\) 0 0
\(775\) 5625.00 0.260717
\(776\) −14610.0 −0.675861
\(777\) 0 0
\(778\) 8991.00 0.414323
\(779\) −7840.00 −0.360587
\(780\) 0 0
\(781\) −1504.00 −0.0689083
\(782\) −393.000 −0.0179714
\(783\) 0 0
\(784\) −12587.0 −0.573387
\(785\) −2855.00 −0.129808
\(786\) 0 0
\(787\) 30493.0 1.38114 0.690571 0.723265i \(-0.257359\pi\)
0.690571 + 0.723265i \(0.257359\pi\)
\(788\) 9660.00 0.436705
\(789\) 0 0
\(790\) −6285.00 −0.283051
\(791\) 6330.00 0.284537
\(792\) 0 0
\(793\) 1690.00 0.0756793
\(794\) −12449.0 −0.556421
\(795\) 0 0
\(796\) −30499.0 −1.35805
\(797\) 33488.0 1.48834 0.744169 0.667991i \(-0.232846\pi\)
0.744169 + 0.667991i \(0.232846\pi\)
\(798\) 0 0
\(799\) 45457.0 2.01271
\(800\) 4025.00 0.177882
\(801\) 0 0
\(802\) 8076.00 0.355578
\(803\) −45590.0 −2.00353
\(804\) 0 0
\(805\) −90.0000 −0.00394048
\(806\) −1125.00 −0.0491643
\(807\) 0 0
\(808\) 20025.0 0.871878
\(809\) −15304.0 −0.665093 −0.332546 0.943087i \(-0.607908\pi\)
−0.332546 + 0.943087i \(0.607908\pi\)
\(810\) 0 0
\(811\) −40122.0 −1.73721 −0.868603 0.495509i \(-0.834982\pi\)
−0.868603 + 0.495509i \(0.834982\pi\)
\(812\) −6594.00 −0.284980
\(813\) 0 0
\(814\) 3290.00 0.141664
\(815\) 3565.00 0.153223
\(816\) 0 0
\(817\) −22232.0 −0.952019
\(818\) −2833.00 −0.121092
\(819\) 0 0
\(820\) −4900.00 −0.208677
\(821\) 25098.0 1.06690 0.533451 0.845831i \(-0.320895\pi\)
0.533451 + 0.845831i \(0.320895\pi\)
\(822\) 0 0
\(823\) −43492.0 −1.84208 −0.921042 0.389462i \(-0.872661\pi\)
−0.921042 + 0.389462i \(0.872661\pi\)
\(824\) −10290.0 −0.435035
\(825\) 0 0
\(826\) −4488.00 −0.189053
\(827\) 11206.0 0.471186 0.235593 0.971852i \(-0.424297\pi\)
0.235593 + 0.971852i \(0.424297\pi\)
\(828\) 0 0
\(829\) −23964.0 −1.00399 −0.501993 0.864872i \(-0.667400\pi\)
−0.501993 + 0.864872i \(0.667400\pi\)
\(830\) −510.000 −0.0213281
\(831\) 0 0
\(832\) 835.000 0.0347938
\(833\) 40217.0 1.67279
\(834\) 0 0
\(835\) 7980.00 0.330730
\(836\) −18424.0 −0.762210
\(837\) 0 0
\(838\) −4777.00 −0.196920
\(839\) −34606.0 −1.42399 −0.711997 0.702182i \(-0.752209\pi\)
−0.711997 + 0.702182i \(0.752209\pi\)
\(840\) 0 0
\(841\) 260.000 0.0106605
\(842\) −6464.00 −0.264566
\(843\) 0 0
\(844\) 29134.0 1.18819
\(845\) −10860.0 −0.442125
\(846\) 0 0
\(847\) −5268.00 −0.213708
\(848\) 164.000 0.00664125
\(849\) 0 0
\(850\) −3275.00 −0.132155
\(851\) −210.000 −0.00845912
\(852\) 0 0
\(853\) −18477.0 −0.741665 −0.370833 0.928700i \(-0.620928\pi\)
−0.370833 + 0.928700i \(0.620928\pi\)
\(854\) 2028.00 0.0812608
\(855\) 0 0
\(856\) 16470.0 0.657632
\(857\) −41342.0 −1.64786 −0.823930 0.566692i \(-0.808223\pi\)
−0.823930 + 0.566692i \(0.808223\pi\)
\(858\) 0 0
\(859\) 21898.0 0.869791 0.434895 0.900481i \(-0.356785\pi\)
0.434895 + 0.900481i \(0.356785\pi\)
\(860\) −13895.0 −0.550948
\(861\) 0 0
\(862\) 10680.0 0.421998
\(863\) −18487.0 −0.729206 −0.364603 0.931163i \(-0.618795\pi\)
−0.364603 + 0.931163i \(0.618795\pi\)
\(864\) 0 0
\(865\) 20670.0 0.812487
\(866\) 11566.0 0.453844
\(867\) 0 0
\(868\) 9450.00 0.369532
\(869\) 59079.0 2.30623
\(870\) 0 0
\(871\) −2460.00 −0.0956991
\(872\) 10500.0 0.407769
\(873\) 0 0
\(874\) −168.000 −0.00650193
\(875\) −750.000 −0.0289767
\(876\) 0 0
\(877\) 7593.00 0.292357 0.146179 0.989258i \(-0.453303\pi\)
0.146179 + 0.989258i \(0.453303\pi\)
\(878\) −1448.00 −0.0556579
\(879\) 0 0
\(880\) −9635.00 −0.369086
\(881\) −3038.00 −0.116178 −0.0580890 0.998311i \(-0.518501\pi\)
−0.0580890 + 0.998311i \(0.518501\pi\)
\(882\) 0 0
\(883\) −16732.0 −0.637686 −0.318843 0.947808i \(-0.603294\pi\)
−0.318843 + 0.947808i \(0.603294\pi\)
\(884\) −4585.00 −0.174446
\(885\) 0 0
\(886\) −2376.00 −0.0900940
\(887\) −8031.00 −0.304007 −0.152004 0.988380i \(-0.548573\pi\)
−0.152004 + 0.988380i \(0.548573\pi\)
\(888\) 0 0
\(889\) 9876.00 0.372588
\(890\) −7440.00 −0.280213
\(891\) 0 0
\(892\) 41692.0 1.56497
\(893\) 19432.0 0.728183
\(894\) 0 0
\(895\) 9140.00 0.341359
\(896\) 8730.00 0.325501
\(897\) 0 0
\(898\) −14894.0 −0.553474
\(899\) −35325.0 −1.31052
\(900\) 0 0
\(901\) −524.000 −0.0193751
\(902\) −6580.00 −0.242894
\(903\) 0 0
\(904\) 15825.0 0.582225
\(905\) −2600.00 −0.0954994
\(906\) 0 0
\(907\) −38487.0 −1.40897 −0.704487 0.709717i \(-0.748823\pi\)
−0.704487 + 0.709717i \(0.748823\pi\)
\(908\) −34580.0 −1.26385
\(909\) 0 0
\(910\) 150.000 0.00546423
\(911\) −5120.00 −0.186205 −0.0931027 0.995657i \(-0.529679\pi\)
−0.0931027 + 0.995657i \(0.529679\pi\)
\(912\) 0 0
\(913\) 4794.00 0.173777
\(914\) 16204.0 0.586412
\(915\) 0 0
\(916\) −30408.0 −1.09684
\(917\) 10998.0 0.396059
\(918\) 0 0
\(919\) 28075.0 1.00774 0.503868 0.863781i \(-0.331910\pi\)
0.503868 + 0.863781i \(0.331910\pi\)
\(920\) −225.000 −0.00806308
\(921\) 0 0
\(922\) −5082.00 −0.181526
\(923\) −160.000 −0.00570581
\(924\) 0 0
\(925\) −1750.00 −0.0622050
\(926\) −10326.0 −0.366451
\(927\) 0 0
\(928\) −25277.0 −0.894136
\(929\) −12856.0 −0.454028 −0.227014 0.973892i \(-0.572896\pi\)
−0.227014 + 0.973892i \(0.572896\pi\)
\(930\) 0 0
\(931\) 17192.0 0.605204
\(932\) 36414.0 1.27981
\(933\) 0 0
\(934\) 4184.00 0.146579
\(935\) 30785.0 1.07677
\(936\) 0 0
\(937\) −1374.00 −0.0479046 −0.0239523 0.999713i \(-0.507625\pi\)
−0.0239523 + 0.999713i \(0.507625\pi\)
\(938\) −2952.00 −0.102757
\(939\) 0 0
\(940\) 12145.0 0.421411
\(941\) 8543.00 0.295955 0.147978 0.988991i \(-0.452724\pi\)
0.147978 + 0.988991i \(0.452724\pi\)
\(942\) 0 0
\(943\) 420.000 0.0145038
\(944\) 30668.0 1.05737
\(945\) 0 0
\(946\) −18659.0 −0.641286
\(947\) 13506.0 0.463449 0.231724 0.972781i \(-0.425563\pi\)
0.231724 + 0.972781i \(0.425563\pi\)
\(948\) 0 0
\(949\) −4850.00 −0.165898
\(950\) −1400.00 −0.0478126
\(951\) 0 0
\(952\) −11790.0 −0.401382
\(953\) 21775.0 0.740148 0.370074 0.929002i \(-0.379332\pi\)
0.370074 + 0.929002i \(0.379332\pi\)
\(954\) 0 0
\(955\) 24130.0 0.817621
\(956\) −10822.0 −0.366118
\(957\) 0 0
\(958\) −15576.0 −0.525300
\(959\) −6588.00 −0.221833
\(960\) 0 0
\(961\) 20834.0 0.699339
\(962\) 350.000 0.0117302
\(963\) 0 0
\(964\) 25613.0 0.855746
\(965\) 8350.00 0.278545
\(966\) 0 0
\(967\) 3854.00 0.128166 0.0640829 0.997945i \(-0.479588\pi\)
0.0640829 + 0.997945i \(0.479588\pi\)
\(968\) −13170.0 −0.437293
\(969\) 0 0
\(970\) 4870.00 0.161202
\(971\) 12933.0 0.427435 0.213718 0.976895i \(-0.431443\pi\)
0.213718 + 0.976895i \(0.431443\pi\)
\(972\) 0 0
\(973\) 6252.00 0.205992
\(974\) 10220.0 0.336211
\(975\) 0 0
\(976\) −13858.0 −0.454492
\(977\) 17521.0 0.573743 0.286871 0.957969i \(-0.407385\pi\)
0.286871 + 0.957969i \(0.407385\pi\)
\(978\) 0 0
\(979\) 69936.0 2.28311
\(980\) 10745.0 0.350241
\(981\) 0 0
\(982\) −2692.00 −0.0874798
\(983\) −12573.0 −0.407952 −0.203976 0.978976i \(-0.565386\pi\)
−0.203976 + 0.978976i \(0.565386\pi\)
\(984\) 0 0
\(985\) −6900.00 −0.223200
\(986\) 20567.0 0.664287
\(987\) 0 0
\(988\) −1960.00 −0.0631133
\(989\) 1191.00 0.0382928
\(990\) 0 0
\(991\) 8945.00 0.286728 0.143364 0.989670i \(-0.454208\pi\)
0.143364 + 0.989670i \(0.454208\pi\)
\(992\) 36225.0 1.15942
\(993\) 0 0
\(994\) −192.000 −0.00612663
\(995\) 21785.0 0.694101
\(996\) 0 0
\(997\) −58179.0 −1.84809 −0.924046 0.382282i \(-0.875138\pi\)
−0.924046 + 0.382282i \(0.875138\pi\)
\(998\) 5764.00 0.182822
\(999\) 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 135.4.a.c.1.1 yes 1
3.2 odd 2 135.4.a.b.1.1 1
4.3 odd 2 2160.4.a.p.1.1 1
5.2 odd 4 675.4.b.e.649.2 2
5.3 odd 4 675.4.b.e.649.1 2
5.4 even 2 675.4.a.c.1.1 1
9.2 odd 6 405.4.e.h.271.1 2
9.4 even 3 405.4.e.f.136.1 2
9.5 odd 6 405.4.e.h.136.1 2
9.7 even 3 405.4.e.f.271.1 2
12.11 even 2 2160.4.a.f.1.1 1
15.2 even 4 675.4.b.f.649.1 2
15.8 even 4 675.4.b.f.649.2 2
15.14 odd 2 675.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.b.1.1 1 3.2 odd 2
135.4.a.c.1.1 yes 1 1.1 even 1 trivial
405.4.e.f.136.1 2 9.4 even 3
405.4.e.f.271.1 2 9.7 even 3
405.4.e.h.136.1 2 9.5 odd 6
405.4.e.h.271.1 2 9.2 odd 6
675.4.a.c.1.1 1 5.4 even 2
675.4.a.h.1.1 1 15.14 odd 2
675.4.b.e.649.1 2 5.3 odd 4
675.4.b.e.649.2 2 5.2 odd 4
675.4.b.f.649.1 2 15.2 even 4
675.4.b.f.649.2 2 15.8 even 4
2160.4.a.f.1.1 1 12.11 even 2
2160.4.a.p.1.1 1 4.3 odd 2