Properties

Label 336.4.a.k.1.1
Level $336$
Weight $4$
Character 336.1
Self dual yes
Analytic conductor $19.825$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 336.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +62.0000 q^{13} +18.0000 q^{15} +114.000 q^{17} +76.0000 q^{19} -21.0000 q^{21} +24.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +54.0000 q^{29} +112.000 q^{31} -108.000 q^{33} -42.0000 q^{35} -178.000 q^{37} +186.000 q^{39} +378.000 q^{41} +172.000 q^{43} +54.0000 q^{45} +192.000 q^{47} +49.0000 q^{49} +342.000 q^{51} -402.000 q^{53} -216.000 q^{55} +228.000 q^{57} -396.000 q^{59} +254.000 q^{61} -63.0000 q^{63} +372.000 q^{65} +1012.00 q^{67} +72.0000 q^{69} -840.000 q^{71} +890.000 q^{73} -267.000 q^{75} +252.000 q^{77} -80.0000 q^{79} +81.0000 q^{81} +108.000 q^{83} +684.000 q^{85} +162.000 q^{87} -1638.00 q^{89} -434.000 q^{91} +336.000 q^{93} +456.000 q^{95} +1010.00 q^{97} -324.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 62.0000 1.32275 0.661373 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) 0 0
\(15\) 18.0000 0.309839
\(16\) 0 0
\(17\) 114.000 1.62642 0.813208 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 24.0000 0.217580 0.108790 0.994065i \(-0.465302\pi\)
0.108790 + 0.994065i \(0.465302\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) 112.000 0.648897 0.324448 0.945903i \(-0.394821\pi\)
0.324448 + 0.945903i \(0.394821\pi\)
\(32\) 0 0
\(33\) −108.000 −0.569709
\(34\) 0 0
\(35\) −42.0000 −0.202837
\(36\) 0 0
\(37\) −178.000 −0.790892 −0.395446 0.918489i \(-0.629410\pi\)
−0.395446 + 0.918489i \(0.629410\pi\)
\(38\) 0 0
\(39\) 186.000 0.763688
\(40\) 0 0
\(41\) 378.000 1.43985 0.719923 0.694054i \(-0.244177\pi\)
0.719923 + 0.694054i \(0.244177\pi\)
\(42\) 0 0
\(43\) 172.000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 54.0000 0.178885
\(46\) 0 0
\(47\) 192.000 0.595874 0.297937 0.954586i \(-0.403701\pi\)
0.297937 + 0.954586i \(0.403701\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 342.000 0.939011
\(52\) 0 0
\(53\) −402.000 −1.04187 −0.520933 0.853597i \(-0.674416\pi\)
−0.520933 + 0.853597i \(0.674416\pi\)
\(54\) 0 0
\(55\) −216.000 −0.529553
\(56\) 0 0
\(57\) 228.000 0.529813
\(58\) 0 0
\(59\) −396.000 −0.873810 −0.436905 0.899508i \(-0.643925\pi\)
−0.436905 + 0.899508i \(0.643925\pi\)
\(60\) 0 0
\(61\) 254.000 0.533137 0.266569 0.963816i \(-0.414110\pi\)
0.266569 + 0.963816i \(0.414110\pi\)
\(62\) 0 0
\(63\) −63.0000 −0.125988
\(64\) 0 0
\(65\) 372.000 0.709860
\(66\) 0 0
\(67\) 1012.00 1.84531 0.922653 0.385632i \(-0.126016\pi\)
0.922653 + 0.385632i \(0.126016\pi\)
\(68\) 0 0
\(69\) 72.0000 0.125620
\(70\) 0 0
\(71\) −840.000 −1.40408 −0.702040 0.712138i \(-0.747727\pi\)
−0.702040 + 0.712138i \(0.747727\pi\)
\(72\) 0 0
\(73\) 890.000 1.42694 0.713470 0.700686i \(-0.247122\pi\)
0.713470 + 0.700686i \(0.247122\pi\)
\(74\) 0 0
\(75\) −267.000 −0.411073
\(76\) 0 0
\(77\) 252.000 0.372962
\(78\) 0 0
\(79\) −80.0000 −0.113933 −0.0569665 0.998376i \(-0.518143\pi\)
−0.0569665 + 0.998376i \(0.518143\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 108.000 0.142826 0.0714129 0.997447i \(-0.477249\pi\)
0.0714129 + 0.997447i \(0.477249\pi\)
\(84\) 0 0
\(85\) 684.000 0.872826
\(86\) 0 0
\(87\) 162.000 0.199635
\(88\) 0 0
\(89\) −1638.00 −1.95087 −0.975436 0.220282i \(-0.929302\pi\)
−0.975436 + 0.220282i \(0.929302\pi\)
\(90\) 0 0
\(91\) −434.000 −0.499951
\(92\) 0 0
\(93\) 336.000 0.374641
\(94\) 0 0
\(95\) 456.000 0.492470
\(96\) 0 0
\(97\) 1010.00 1.05722 0.528608 0.848866i \(-0.322714\pi\)
0.528608 + 0.848866i \(0.322714\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 6.00000 0.00591111 0.00295556 0.999996i \(-0.499059\pi\)
0.00295556 + 0.999996i \(0.499059\pi\)
\(102\) 0 0
\(103\) 472.000 0.451530 0.225765 0.974182i \(-0.427512\pi\)
0.225765 + 0.974182i \(0.427512\pi\)
\(104\) 0 0
\(105\) −126.000 −0.117108
\(106\) 0 0
\(107\) 972.000 0.878194 0.439097 0.898440i \(-0.355298\pi\)
0.439097 + 0.898440i \(0.355298\pi\)
\(108\) 0 0
\(109\) −1786.00 −1.56943 −0.784715 0.619857i \(-0.787190\pi\)
−0.784715 + 0.619857i \(0.787190\pi\)
\(110\) 0 0
\(111\) −534.000 −0.456622
\(112\) 0 0
\(113\) −2286.00 −1.90309 −0.951543 0.307515i \(-0.900503\pi\)
−0.951543 + 0.307515i \(0.900503\pi\)
\(114\) 0 0
\(115\) 144.000 0.116766
\(116\) 0 0
\(117\) 558.000 0.440916
\(118\) 0 0
\(119\) −798.000 −0.614727
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 1134.00 0.831295
\(124\) 0 0
\(125\) −1284.00 −0.918756
\(126\) 0 0
\(127\) −1328.00 −0.927881 −0.463941 0.885866i \(-0.653565\pi\)
−0.463941 + 0.885866i \(0.653565\pi\)
\(128\) 0 0
\(129\) 516.000 0.352180
\(130\) 0 0
\(131\) 1212.00 0.808343 0.404171 0.914683i \(-0.367560\pi\)
0.404171 + 0.914683i \(0.367560\pi\)
\(132\) 0 0
\(133\) −532.000 −0.346844
\(134\) 0 0
\(135\) 162.000 0.103280
\(136\) 0 0
\(137\) −1254.00 −0.782018 −0.391009 0.920387i \(-0.627874\pi\)
−0.391009 + 0.920387i \(0.627874\pi\)
\(138\) 0 0
\(139\) 340.000 0.207471 0.103735 0.994605i \(-0.466921\pi\)
0.103735 + 0.994605i \(0.466921\pi\)
\(140\) 0 0
\(141\) 576.000 0.344028
\(142\) 0 0
\(143\) −2232.00 −1.30524
\(144\) 0 0
\(145\) 324.000 0.185564
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 1038.00 0.570713 0.285357 0.958421i \(-0.407888\pi\)
0.285357 + 0.958421i \(0.407888\pi\)
\(150\) 0 0
\(151\) −2936.00 −1.58231 −0.791153 0.611618i \(-0.790519\pi\)
−0.791153 + 0.611618i \(0.790519\pi\)
\(152\) 0 0
\(153\) 1026.00 0.542138
\(154\) 0 0
\(155\) 672.000 0.348234
\(156\) 0 0
\(157\) −1330.00 −0.676086 −0.338043 0.941131i \(-0.609765\pi\)
−0.338043 + 0.941131i \(0.609765\pi\)
\(158\) 0 0
\(159\) −1206.00 −0.601522
\(160\) 0 0
\(161\) −168.000 −0.0822376
\(162\) 0 0
\(163\) 3364.00 1.61650 0.808248 0.588842i \(-0.200416\pi\)
0.808248 + 0.588842i \(0.200416\pi\)
\(164\) 0 0
\(165\) −648.000 −0.305738
\(166\) 0 0
\(167\) −3048.00 −1.41234 −0.706172 0.708041i \(-0.749579\pi\)
−0.706172 + 0.708041i \(0.749579\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) 0 0
\(171\) 684.000 0.305888
\(172\) 0 0
\(173\) −2706.00 −1.18921 −0.594605 0.804018i \(-0.702692\pi\)
−0.594605 + 0.804018i \(0.702692\pi\)
\(174\) 0 0
\(175\) 623.000 0.269111
\(176\) 0 0
\(177\) −1188.00 −0.504495
\(178\) 0 0
\(179\) −4716.00 −1.96922 −0.984610 0.174766i \(-0.944083\pi\)
−0.984610 + 0.174766i \(0.944083\pi\)
\(180\) 0 0
\(181\) 1910.00 0.784360 0.392180 0.919888i \(-0.371721\pi\)
0.392180 + 0.919888i \(0.371721\pi\)
\(182\) 0 0
\(183\) 762.000 0.307807
\(184\) 0 0
\(185\) −1068.00 −0.424437
\(186\) 0 0
\(187\) −4104.00 −1.60489
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −4080.00 −1.54565 −0.772823 0.634621i \(-0.781156\pi\)
−0.772823 + 0.634621i \(0.781156\pi\)
\(192\) 0 0
\(193\) −2686.00 −1.00177 −0.500887 0.865512i \(-0.666993\pi\)
−0.500887 + 0.865512i \(0.666993\pi\)
\(194\) 0 0
\(195\) 1116.00 0.409838
\(196\) 0 0
\(197\) 510.000 0.184447 0.0922233 0.995738i \(-0.470603\pi\)
0.0922233 + 0.995738i \(0.470603\pi\)
\(198\) 0 0
\(199\) −1352.00 −0.481612 −0.240806 0.970573i \(-0.577412\pi\)
−0.240806 + 0.970573i \(0.577412\pi\)
\(200\) 0 0
\(201\) 3036.00 1.06539
\(202\) 0 0
\(203\) −378.000 −0.130692
\(204\) 0 0
\(205\) 2268.00 0.772702
\(206\) 0 0
\(207\) 216.000 0.0725268
\(208\) 0 0
\(209\) −2736.00 −0.905517
\(210\) 0 0
\(211\) 3364.00 1.09757 0.548785 0.835963i \(-0.315091\pi\)
0.548785 + 0.835963i \(0.315091\pi\)
\(212\) 0 0
\(213\) −2520.00 −0.810646
\(214\) 0 0
\(215\) 1032.00 0.327357
\(216\) 0 0
\(217\) −784.000 −0.245260
\(218\) 0 0
\(219\) 2670.00 0.823844
\(220\) 0 0
\(221\) 7068.00 2.15134
\(222\) 0 0
\(223\) 4768.00 1.43179 0.715894 0.698209i \(-0.246019\pi\)
0.715894 + 0.698209i \(0.246019\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) −420.000 −0.122803 −0.0614017 0.998113i \(-0.519557\pi\)
−0.0614017 + 0.998113i \(0.519557\pi\)
\(228\) 0 0
\(229\) −1882.00 −0.543083 −0.271542 0.962427i \(-0.587533\pi\)
−0.271542 + 0.962427i \(0.587533\pi\)
\(230\) 0 0
\(231\) 756.000 0.215330
\(232\) 0 0
\(233\) 5082.00 1.42890 0.714448 0.699688i \(-0.246678\pi\)
0.714448 + 0.699688i \(0.246678\pi\)
\(234\) 0 0
\(235\) 1152.00 0.319780
\(236\) 0 0
\(237\) −240.000 −0.0657792
\(238\) 0 0
\(239\) 5424.00 1.46799 0.733995 0.679155i \(-0.237654\pi\)
0.733995 + 0.679155i \(0.237654\pi\)
\(240\) 0 0
\(241\) −2590.00 −0.692268 −0.346134 0.938185i \(-0.612506\pi\)
−0.346134 + 0.938185i \(0.612506\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 294.000 0.0766652
\(246\) 0 0
\(247\) 4712.00 1.21384
\(248\) 0 0
\(249\) 324.000 0.0824605
\(250\) 0 0
\(251\) 4932.00 1.24026 0.620130 0.784499i \(-0.287080\pi\)
0.620130 + 0.784499i \(0.287080\pi\)
\(252\) 0 0
\(253\) −864.000 −0.214700
\(254\) 0 0
\(255\) 2052.00 0.503926
\(256\) 0 0
\(257\) −3438.00 −0.834461 −0.417231 0.908801i \(-0.636999\pi\)
−0.417231 + 0.908801i \(0.636999\pi\)
\(258\) 0 0
\(259\) 1246.00 0.298929
\(260\) 0 0
\(261\) 486.000 0.115259
\(262\) 0 0
\(263\) −6120.00 −1.43489 −0.717444 0.696617i \(-0.754688\pi\)
−0.717444 + 0.696617i \(0.754688\pi\)
\(264\) 0 0
\(265\) −2412.00 −0.559124
\(266\) 0 0
\(267\) −4914.00 −1.12634
\(268\) 0 0
\(269\) −18.0000 −0.00407985 −0.00203992 0.999998i \(-0.500649\pi\)
−0.00203992 + 0.999998i \(0.500649\pi\)
\(270\) 0 0
\(271\) −6896.00 −1.54576 −0.772882 0.634549i \(-0.781186\pi\)
−0.772882 + 0.634549i \(0.781186\pi\)
\(272\) 0 0
\(273\) −1302.00 −0.288647
\(274\) 0 0
\(275\) 3204.00 0.702576
\(276\) 0 0
\(277\) 6254.00 1.35656 0.678279 0.734805i \(-0.262726\pi\)
0.678279 + 0.734805i \(0.262726\pi\)
\(278\) 0 0
\(279\) 1008.00 0.216299
\(280\) 0 0
\(281\) 1194.00 0.253481 0.126740 0.991936i \(-0.459549\pi\)
0.126740 + 0.991936i \(0.459549\pi\)
\(282\) 0 0
\(283\) 7156.00 1.50311 0.751555 0.659671i \(-0.229304\pi\)
0.751555 + 0.659671i \(0.229304\pi\)
\(284\) 0 0
\(285\) 1368.00 0.284327
\(286\) 0 0
\(287\) −2646.00 −0.544211
\(288\) 0 0
\(289\) 8083.00 1.64523
\(290\) 0 0
\(291\) 3030.00 0.610384
\(292\) 0 0
\(293\) −3738.00 −0.745312 −0.372656 0.927970i \(-0.621553\pi\)
−0.372656 + 0.927970i \(0.621553\pi\)
\(294\) 0 0
\(295\) −2376.00 −0.468936
\(296\) 0 0
\(297\) −972.000 −0.189903
\(298\) 0 0
\(299\) 1488.00 0.287804
\(300\) 0 0
\(301\) −1204.00 −0.230556
\(302\) 0 0
\(303\) 18.0000 0.00341278
\(304\) 0 0
\(305\) 1524.00 0.286111
\(306\) 0 0
\(307\) 844.000 0.156904 0.0784522 0.996918i \(-0.475002\pi\)
0.0784522 + 0.996918i \(0.475002\pi\)
\(308\) 0 0
\(309\) 1416.00 0.260691
\(310\) 0 0
\(311\) −6312.00 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(312\) 0 0
\(313\) 8282.00 1.49561 0.747806 0.663918i \(-0.231108\pi\)
0.747806 + 0.663918i \(0.231108\pi\)
\(314\) 0 0
\(315\) −378.000 −0.0676123
\(316\) 0 0
\(317\) 9318.00 1.65095 0.825475 0.564439i \(-0.190907\pi\)
0.825475 + 0.564439i \(0.190907\pi\)
\(318\) 0 0
\(319\) −1944.00 −0.341201
\(320\) 0 0
\(321\) 2916.00 0.507026
\(322\) 0 0
\(323\) 8664.00 1.49250
\(324\) 0 0
\(325\) −5518.00 −0.941796
\(326\) 0 0
\(327\) −5358.00 −0.906110
\(328\) 0 0
\(329\) −1344.00 −0.225219
\(330\) 0 0
\(331\) −1652.00 −0.274327 −0.137163 0.990548i \(-0.543799\pi\)
−0.137163 + 0.990548i \(0.543799\pi\)
\(332\) 0 0
\(333\) −1602.00 −0.263631
\(334\) 0 0
\(335\) 6072.00 0.990295
\(336\) 0 0
\(337\) −1294.00 −0.209165 −0.104583 0.994516i \(-0.533351\pi\)
−0.104583 + 0.994516i \(0.533351\pi\)
\(338\) 0 0
\(339\) −6858.00 −1.09875
\(340\) 0 0
\(341\) −4032.00 −0.640308
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 432.000 0.0674148
\(346\) 0 0
\(347\) −3636.00 −0.562509 −0.281255 0.959633i \(-0.590751\pi\)
−0.281255 + 0.959633i \(0.590751\pi\)
\(348\) 0 0
\(349\) 10478.0 1.60709 0.803545 0.595244i \(-0.202945\pi\)
0.803545 + 0.595244i \(0.202945\pi\)
\(350\) 0 0
\(351\) 1674.00 0.254563
\(352\) 0 0
\(353\) −7566.00 −1.14079 −0.570393 0.821372i \(-0.693209\pi\)
−0.570393 + 0.821372i \(0.693209\pi\)
\(354\) 0 0
\(355\) −5040.00 −0.753508
\(356\) 0 0
\(357\) −2394.00 −0.354913
\(358\) 0 0
\(359\) 8040.00 1.18199 0.590996 0.806675i \(-0.298735\pi\)
0.590996 + 0.806675i \(0.298735\pi\)
\(360\) 0 0
\(361\) −1083.00 −0.157895
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 5340.00 0.765776
\(366\) 0 0
\(367\) −7568.00 −1.07642 −0.538210 0.842811i \(-0.680899\pi\)
−0.538210 + 0.842811i \(0.680899\pi\)
\(368\) 0 0
\(369\) 3402.00 0.479949
\(370\) 0 0
\(371\) 2814.00 0.393789
\(372\) 0 0
\(373\) −13522.0 −1.87706 −0.938529 0.345200i \(-0.887811\pi\)
−0.938529 + 0.345200i \(0.887811\pi\)
\(374\) 0 0
\(375\) −3852.00 −0.530444
\(376\) 0 0
\(377\) 3348.00 0.457376
\(378\) 0 0
\(379\) −2468.00 −0.334492 −0.167246 0.985915i \(-0.553487\pi\)
−0.167246 + 0.985915i \(0.553487\pi\)
\(380\) 0 0
\(381\) −3984.00 −0.535713
\(382\) 0 0
\(383\) 12336.0 1.64580 0.822898 0.568189i \(-0.192356\pi\)
0.822898 + 0.568189i \(0.192356\pi\)
\(384\) 0 0
\(385\) 1512.00 0.200152
\(386\) 0 0
\(387\) 1548.00 0.203331
\(388\) 0 0
\(389\) −3762.00 −0.490337 −0.245168 0.969481i \(-0.578843\pi\)
−0.245168 + 0.969481i \(0.578843\pi\)
\(390\) 0 0
\(391\) 2736.00 0.353876
\(392\) 0 0
\(393\) 3636.00 0.466697
\(394\) 0 0
\(395\) −480.000 −0.0611428
\(396\) 0 0
\(397\) −8770.00 −1.10870 −0.554350 0.832284i \(-0.687033\pi\)
−0.554350 + 0.832284i \(0.687033\pi\)
\(398\) 0 0
\(399\) −1596.00 −0.200250
\(400\) 0 0
\(401\) 6642.00 0.827146 0.413573 0.910471i \(-0.364281\pi\)
0.413573 + 0.910471i \(0.364281\pi\)
\(402\) 0 0
\(403\) 6944.00 0.858326
\(404\) 0 0
\(405\) 486.000 0.0596285
\(406\) 0 0
\(407\) 6408.00 0.780424
\(408\) 0 0
\(409\) −1510.00 −0.182554 −0.0912771 0.995826i \(-0.529095\pi\)
−0.0912771 + 0.995826i \(0.529095\pi\)
\(410\) 0 0
\(411\) −3762.00 −0.451498
\(412\) 0 0
\(413\) 2772.00 0.330269
\(414\) 0 0
\(415\) 648.000 0.0766484
\(416\) 0 0
\(417\) 1020.00 0.119783
\(418\) 0 0
\(419\) 1260.00 0.146909 0.0734547 0.997299i \(-0.476598\pi\)
0.0734547 + 0.997299i \(0.476598\pi\)
\(420\) 0 0
\(421\) 3998.00 0.462828 0.231414 0.972855i \(-0.425665\pi\)
0.231414 + 0.972855i \(0.425665\pi\)
\(422\) 0 0
\(423\) 1728.00 0.198625
\(424\) 0 0
\(425\) −10146.0 −1.15801
\(426\) 0 0
\(427\) −1778.00 −0.201507
\(428\) 0 0
\(429\) −6696.00 −0.753580
\(430\) 0 0
\(431\) 2736.00 0.305774 0.152887 0.988244i \(-0.451143\pi\)
0.152887 + 0.988244i \(0.451143\pi\)
\(432\) 0 0
\(433\) 2690.00 0.298552 0.149276 0.988796i \(-0.452306\pi\)
0.149276 + 0.988796i \(0.452306\pi\)
\(434\) 0 0
\(435\) 972.000 0.107135
\(436\) 0 0
\(437\) 1824.00 0.199665
\(438\) 0 0
\(439\) 1240.00 0.134811 0.0674054 0.997726i \(-0.478528\pi\)
0.0674054 + 0.997726i \(0.478528\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 3900.00 0.418272 0.209136 0.977887i \(-0.432935\pi\)
0.209136 + 0.977887i \(0.432935\pi\)
\(444\) 0 0
\(445\) −9828.00 −1.04695
\(446\) 0 0
\(447\) 3114.00 0.329501
\(448\) 0 0
\(449\) −10878.0 −1.14335 −0.571675 0.820480i \(-0.693706\pi\)
−0.571675 + 0.820480i \(0.693706\pi\)
\(450\) 0 0
\(451\) −13608.0 −1.42079
\(452\) 0 0
\(453\) −8808.00 −0.913545
\(454\) 0 0
\(455\) −2604.00 −0.268302
\(456\) 0 0
\(457\) 2330.00 0.238496 0.119248 0.992864i \(-0.461952\pi\)
0.119248 + 0.992864i \(0.461952\pi\)
\(458\) 0 0
\(459\) 3078.00 0.313004
\(460\) 0 0
\(461\) 15150.0 1.53060 0.765299 0.643675i \(-0.222591\pi\)
0.765299 + 0.643675i \(0.222591\pi\)
\(462\) 0 0
\(463\) 2992.00 0.300324 0.150162 0.988661i \(-0.452020\pi\)
0.150162 + 0.988661i \(0.452020\pi\)
\(464\) 0 0
\(465\) 2016.00 0.201053
\(466\) 0 0
\(467\) −8724.00 −0.864451 −0.432225 0.901766i \(-0.642272\pi\)
−0.432225 + 0.901766i \(0.642272\pi\)
\(468\) 0 0
\(469\) −7084.00 −0.697460
\(470\) 0 0
\(471\) −3990.00 −0.390339
\(472\) 0 0
\(473\) −6192.00 −0.601921
\(474\) 0 0
\(475\) −6764.00 −0.653376
\(476\) 0 0
\(477\) −3618.00 −0.347289
\(478\) 0 0
\(479\) −9744.00 −0.929467 −0.464734 0.885451i \(-0.653850\pi\)
−0.464734 + 0.885451i \(0.653850\pi\)
\(480\) 0 0
\(481\) −11036.0 −1.04615
\(482\) 0 0
\(483\) −504.000 −0.0474799
\(484\) 0 0
\(485\) 6060.00 0.567362
\(486\) 0 0
\(487\) −4136.00 −0.384846 −0.192423 0.981312i \(-0.561635\pi\)
−0.192423 + 0.981312i \(0.561635\pi\)
\(488\) 0 0
\(489\) 10092.0 0.933284
\(490\) 0 0
\(491\) −16212.0 −1.49010 −0.745048 0.667011i \(-0.767574\pi\)
−0.745048 + 0.667011i \(0.767574\pi\)
\(492\) 0 0
\(493\) 6156.00 0.562378
\(494\) 0 0
\(495\) −1944.00 −0.176518
\(496\) 0 0
\(497\) 5880.00 0.530692
\(498\) 0 0
\(499\) −2396.00 −0.214949 −0.107475 0.994208i \(-0.534276\pi\)
−0.107475 + 0.994208i \(0.534276\pi\)
\(500\) 0 0
\(501\) −9144.00 −0.815417
\(502\) 0 0
\(503\) −13128.0 −1.16371 −0.581857 0.813291i \(-0.697674\pi\)
−0.581857 + 0.813291i \(0.697674\pi\)
\(504\) 0 0
\(505\) 36.0000 0.00317224
\(506\) 0 0
\(507\) 4941.00 0.432816
\(508\) 0 0
\(509\) 12798.0 1.11446 0.557231 0.830357i \(-0.311864\pi\)
0.557231 + 0.830357i \(0.311864\pi\)
\(510\) 0 0
\(511\) −6230.00 −0.539333
\(512\) 0 0
\(513\) 2052.00 0.176604
\(514\) 0 0
\(515\) 2832.00 0.242316
\(516\) 0 0
\(517\) −6912.00 −0.587987
\(518\) 0 0
\(519\) −8118.00 −0.686591
\(520\) 0 0
\(521\) 7386.00 0.621087 0.310544 0.950559i \(-0.399489\pi\)
0.310544 + 0.950559i \(0.399489\pi\)
\(522\) 0 0
\(523\) −5180.00 −0.433089 −0.216545 0.976273i \(-0.569479\pi\)
−0.216545 + 0.976273i \(0.569479\pi\)
\(524\) 0 0
\(525\) 1869.00 0.155371
\(526\) 0 0
\(527\) 12768.0 1.05538
\(528\) 0 0
\(529\) −11591.0 −0.952659
\(530\) 0 0
\(531\) −3564.00 −0.291270
\(532\) 0 0
\(533\) 23436.0 1.90455
\(534\) 0 0
\(535\) 5832.00 0.471288
\(536\) 0 0
\(537\) −14148.0 −1.13693
\(538\) 0 0
\(539\) −1764.00 −0.140966
\(540\) 0 0
\(541\) 4070.00 0.323444 0.161722 0.986836i \(-0.448295\pi\)
0.161722 + 0.986836i \(0.448295\pi\)
\(542\) 0 0
\(543\) 5730.00 0.452851
\(544\) 0 0
\(545\) −10716.0 −0.842244
\(546\) 0 0
\(547\) −14780.0 −1.15530 −0.577648 0.816286i \(-0.696029\pi\)
−0.577648 + 0.816286i \(0.696029\pi\)
\(548\) 0 0
\(549\) 2286.00 0.177712
\(550\) 0 0
\(551\) 4104.00 0.317307
\(552\) 0 0
\(553\) 560.000 0.0430626
\(554\) 0 0
\(555\) −3204.00 −0.245049
\(556\) 0 0
\(557\) −6858.00 −0.521693 −0.260846 0.965380i \(-0.584002\pi\)
−0.260846 + 0.965380i \(0.584002\pi\)
\(558\) 0 0
\(559\) 10664.0 0.806868
\(560\) 0 0
\(561\) −12312.0 −0.926583
\(562\) 0 0
\(563\) −6660.00 −0.498553 −0.249277 0.968432i \(-0.580193\pi\)
−0.249277 + 0.968432i \(0.580193\pi\)
\(564\) 0 0
\(565\) −13716.0 −1.02130
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) −150.000 −0.0110515 −0.00552577 0.999985i \(-0.501759\pi\)
−0.00552577 + 0.999985i \(0.501759\pi\)
\(570\) 0 0
\(571\) 8188.00 0.600100 0.300050 0.953923i \(-0.402997\pi\)
0.300050 + 0.953923i \(0.402997\pi\)
\(572\) 0 0
\(573\) −12240.0 −0.892379
\(574\) 0 0
\(575\) −2136.00 −0.154917
\(576\) 0 0
\(577\) −5854.00 −0.422366 −0.211183 0.977447i \(-0.567732\pi\)
−0.211183 + 0.977447i \(0.567732\pi\)
\(578\) 0 0
\(579\) −8058.00 −0.578375
\(580\) 0 0
\(581\) −756.000 −0.0539831
\(582\) 0 0
\(583\) 14472.0 1.02808
\(584\) 0 0
\(585\) 3348.00 0.236620
\(586\) 0 0
\(587\) −17580.0 −1.23612 −0.618062 0.786130i \(-0.712082\pi\)
−0.618062 + 0.786130i \(0.712082\pi\)
\(588\) 0 0
\(589\) 8512.00 0.595468
\(590\) 0 0
\(591\) 1530.00 0.106490
\(592\) 0 0
\(593\) 17154.0 1.18791 0.593955 0.804498i \(-0.297566\pi\)
0.593955 + 0.804498i \(0.297566\pi\)
\(594\) 0 0
\(595\) −4788.00 −0.329897
\(596\) 0 0
\(597\) −4056.00 −0.278059
\(598\) 0 0
\(599\) 18120.0 1.23600 0.617999 0.786179i \(-0.287943\pi\)
0.617999 + 0.786179i \(0.287943\pi\)
\(600\) 0 0
\(601\) 17546.0 1.19088 0.595438 0.803401i \(-0.296979\pi\)
0.595438 + 0.803401i \(0.296979\pi\)
\(602\) 0 0
\(603\) 9108.00 0.615102
\(604\) 0 0
\(605\) −210.000 −0.0141119
\(606\) 0 0
\(607\) 14560.0 0.973595 0.486798 0.873515i \(-0.338165\pi\)
0.486798 + 0.873515i \(0.338165\pi\)
\(608\) 0 0
\(609\) −1134.00 −0.0754548
\(610\) 0 0
\(611\) 11904.0 0.788190
\(612\) 0 0
\(613\) −4498.00 −0.296366 −0.148183 0.988960i \(-0.547343\pi\)
−0.148183 + 0.988960i \(0.547343\pi\)
\(614\) 0 0
\(615\) 6804.00 0.446120
\(616\) 0 0
\(617\) −5478.00 −0.357433 −0.178716 0.983901i \(-0.557194\pi\)
−0.178716 + 0.983901i \(0.557194\pi\)
\(618\) 0 0
\(619\) −6044.00 −0.392454 −0.196227 0.980559i \(-0.562869\pi\)
−0.196227 + 0.980559i \(0.562869\pi\)
\(620\) 0 0
\(621\) 648.000 0.0418733
\(622\) 0 0
\(623\) 11466.0 0.737360
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) −8208.00 −0.522801
\(628\) 0 0
\(629\) −20292.0 −1.28632
\(630\) 0 0
\(631\) 15352.0 0.968547 0.484274 0.874917i \(-0.339084\pi\)
0.484274 + 0.874917i \(0.339084\pi\)
\(632\) 0 0
\(633\) 10092.0 0.633682
\(634\) 0 0
\(635\) −7968.00 −0.497953
\(636\) 0 0
\(637\) 3038.00 0.188964
\(638\) 0 0
\(639\) −7560.00 −0.468027
\(640\) 0 0
\(641\) −22398.0 −1.38014 −0.690068 0.723744i \(-0.742420\pi\)
−0.690068 + 0.723744i \(0.742420\pi\)
\(642\) 0 0
\(643\) −3764.00 −0.230852 −0.115426 0.993316i \(-0.536823\pi\)
−0.115426 + 0.993316i \(0.536823\pi\)
\(644\) 0 0
\(645\) 3096.00 0.189000
\(646\) 0 0
\(647\) 17688.0 1.07479 0.537393 0.843332i \(-0.319409\pi\)
0.537393 + 0.843332i \(0.319409\pi\)
\(648\) 0 0
\(649\) 14256.0 0.862245
\(650\) 0 0
\(651\) −2352.00 −0.141601
\(652\) 0 0
\(653\) 19878.0 1.19125 0.595625 0.803263i \(-0.296904\pi\)
0.595625 + 0.803263i \(0.296904\pi\)
\(654\) 0 0
\(655\) 7272.00 0.433802
\(656\) 0 0
\(657\) 8010.00 0.475647
\(658\) 0 0
\(659\) 20004.0 1.18247 0.591233 0.806501i \(-0.298641\pi\)
0.591233 + 0.806501i \(0.298641\pi\)
\(660\) 0 0
\(661\) −1306.00 −0.0768495 −0.0384247 0.999261i \(-0.512234\pi\)
−0.0384247 + 0.999261i \(0.512234\pi\)
\(662\) 0 0
\(663\) 21204.0 1.24207
\(664\) 0 0
\(665\) −3192.00 −0.186136
\(666\) 0 0
\(667\) 1296.00 0.0752344
\(668\) 0 0
\(669\) 14304.0 0.826644
\(670\) 0 0
\(671\) −9144.00 −0.526081
\(672\) 0 0
\(673\) −13054.0 −0.747689 −0.373845 0.927491i \(-0.621961\pi\)
−0.373845 + 0.927491i \(0.621961\pi\)
\(674\) 0 0
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) 5046.00 0.286460 0.143230 0.989689i \(-0.454251\pi\)
0.143230 + 0.989689i \(0.454251\pi\)
\(678\) 0 0
\(679\) −7070.00 −0.399590
\(680\) 0 0
\(681\) −1260.00 −0.0709006
\(682\) 0 0
\(683\) −12468.0 −0.698499 −0.349249 0.937030i \(-0.613563\pi\)
−0.349249 + 0.937030i \(0.613563\pi\)
\(684\) 0 0
\(685\) −7524.00 −0.419675
\(686\) 0 0
\(687\) −5646.00 −0.313549
\(688\) 0 0
\(689\) −24924.0 −1.37813
\(690\) 0 0
\(691\) 23212.0 1.27790 0.638948 0.769250i \(-0.279370\pi\)
0.638948 + 0.769250i \(0.279370\pi\)
\(692\) 0 0
\(693\) 2268.00 0.124321
\(694\) 0 0
\(695\) 2040.00 0.111340
\(696\) 0 0
\(697\) 43092.0 2.34179
\(698\) 0 0
\(699\) 15246.0 0.824974
\(700\) 0 0
\(701\) 35958.0 1.93740 0.968698 0.248241i \(-0.0798526\pi\)
0.968698 + 0.248241i \(0.0798526\pi\)
\(702\) 0 0
\(703\) −13528.0 −0.725773
\(704\) 0 0
\(705\) 3456.00 0.184625
\(706\) 0 0
\(707\) −42.0000 −0.00223419
\(708\) 0 0
\(709\) 6446.00 0.341445 0.170723 0.985319i \(-0.445390\pi\)
0.170723 + 0.985319i \(0.445390\pi\)
\(710\) 0 0
\(711\) −720.000 −0.0379777
\(712\) 0 0
\(713\) 2688.00 0.141187
\(714\) 0 0
\(715\) −13392.0 −0.700465
\(716\) 0 0
\(717\) 16272.0 0.847544
\(718\) 0 0
\(719\) 4704.00 0.243991 0.121996 0.992531i \(-0.461071\pi\)
0.121996 + 0.992531i \(0.461071\pi\)
\(720\) 0 0
\(721\) −3304.00 −0.170662
\(722\) 0 0
\(723\) −7770.00 −0.399681
\(724\) 0 0
\(725\) −4806.00 −0.246194
\(726\) 0 0
\(727\) 10600.0 0.540760 0.270380 0.962754i \(-0.412851\pi\)
0.270380 + 0.962754i \(0.412851\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19608.0 0.992104
\(732\) 0 0
\(733\) 12542.0 0.631991 0.315995 0.948761i \(-0.397662\pi\)
0.315995 + 0.948761i \(0.397662\pi\)
\(734\) 0 0
\(735\) 882.000 0.0442627
\(736\) 0 0
\(737\) −36432.0 −1.82088
\(738\) 0 0
\(739\) −23324.0 −1.16101 −0.580506 0.814256i \(-0.697145\pi\)
−0.580506 + 0.814256i \(0.697145\pi\)
\(740\) 0 0
\(741\) 14136.0 0.700808
\(742\) 0 0
\(743\) 6312.00 0.311662 0.155831 0.987784i \(-0.450194\pi\)
0.155831 + 0.987784i \(0.450194\pi\)
\(744\) 0 0
\(745\) 6228.00 0.306277
\(746\) 0 0
\(747\) 972.000 0.0476086
\(748\) 0 0
\(749\) −6804.00 −0.331926
\(750\) 0 0
\(751\) −35840.0 −1.74144 −0.870719 0.491781i \(-0.836346\pi\)
−0.870719 + 0.491781i \(0.836346\pi\)
\(752\) 0 0
\(753\) 14796.0 0.716064
\(754\) 0 0
\(755\) −17616.0 −0.849155
\(756\) 0 0
\(757\) −34594.0 −1.66095 −0.830476 0.557055i \(-0.811931\pi\)
−0.830476 + 0.557055i \(0.811931\pi\)
\(758\) 0 0
\(759\) −2592.00 −0.123957
\(760\) 0 0
\(761\) 23946.0 1.14066 0.570330 0.821416i \(-0.306815\pi\)
0.570330 + 0.821416i \(0.306815\pi\)
\(762\) 0 0
\(763\) 12502.0 0.593188
\(764\) 0 0
\(765\) 6156.00 0.290942
\(766\) 0 0
\(767\) −24552.0 −1.15583
\(768\) 0 0
\(769\) 18770.0 0.880187 0.440093 0.897952i \(-0.354945\pi\)
0.440093 + 0.897952i \(0.354945\pi\)
\(770\) 0 0
\(771\) −10314.0 −0.481776
\(772\) 0 0
\(773\) 30342.0 1.41181 0.705903 0.708309i \(-0.250541\pi\)
0.705903 + 0.708309i \(0.250541\pi\)
\(774\) 0 0
\(775\) −9968.00 −0.462014
\(776\) 0 0
\(777\) 3738.00 0.172587
\(778\) 0 0
\(779\) 28728.0 1.32129
\(780\) 0 0
\(781\) 30240.0 1.38550
\(782\) 0 0
\(783\) 1458.00 0.0665449
\(784\) 0 0
\(785\) −7980.00 −0.362826
\(786\) 0 0
\(787\) 26188.0 1.18615 0.593076 0.805147i \(-0.297913\pi\)
0.593076 + 0.805147i \(0.297913\pi\)
\(788\) 0 0
\(789\) −18360.0 −0.828433
\(790\) 0 0
\(791\) 16002.0 0.719299
\(792\) 0 0
\(793\) 15748.0 0.705205
\(794\) 0 0
\(795\) −7236.00 −0.322811
\(796\) 0 0
\(797\) −34818.0 −1.54745 −0.773724 0.633522i \(-0.781609\pi\)
−0.773724 + 0.633522i \(0.781609\pi\)
\(798\) 0 0
\(799\) 21888.0 0.969139
\(800\) 0 0
\(801\) −14742.0 −0.650291
\(802\) 0 0
\(803\) −32040.0 −1.40805
\(804\) 0 0
\(805\) −1008.00 −0.0441333
\(806\) 0 0
\(807\) −54.0000 −0.00235550
\(808\) 0 0
\(809\) −21702.0 −0.943142 −0.471571 0.881828i \(-0.656313\pi\)
−0.471571 + 0.881828i \(0.656313\pi\)
\(810\) 0 0
\(811\) 20356.0 0.881376 0.440688 0.897660i \(-0.354735\pi\)
0.440688 + 0.897660i \(0.354735\pi\)
\(812\) 0 0
\(813\) −20688.0 −0.892448
\(814\) 0 0
\(815\) 20184.0 0.867503
\(816\) 0 0
\(817\) 13072.0 0.559769
\(818\) 0 0
\(819\) −3906.00 −0.166650
\(820\) 0 0
\(821\) −19890.0 −0.845513 −0.422756 0.906243i \(-0.638937\pi\)
−0.422756 + 0.906243i \(0.638937\pi\)
\(822\) 0 0
\(823\) −4232.00 −0.179245 −0.0896223 0.995976i \(-0.528566\pi\)
−0.0896223 + 0.995976i \(0.528566\pi\)
\(824\) 0 0
\(825\) 9612.00 0.405633
\(826\) 0 0
\(827\) −9636.00 −0.405171 −0.202586 0.979265i \(-0.564934\pi\)
−0.202586 + 0.979265i \(0.564934\pi\)
\(828\) 0 0
\(829\) 35294.0 1.47866 0.739331 0.673342i \(-0.235142\pi\)
0.739331 + 0.673342i \(0.235142\pi\)
\(830\) 0 0
\(831\) 18762.0 0.783209
\(832\) 0 0
\(833\) 5586.00 0.232345
\(834\) 0 0
\(835\) −18288.0 −0.757943
\(836\) 0 0
\(837\) 3024.00 0.124880
\(838\) 0 0
\(839\) 3768.00 0.155049 0.0775243 0.996990i \(-0.475298\pi\)
0.0775243 + 0.996990i \(0.475298\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) 3582.00 0.146347
\(844\) 0 0
\(845\) 9882.00 0.402309
\(846\) 0 0
\(847\) 245.000 0.00993896
\(848\) 0 0
\(849\) 21468.0 0.867821
\(850\) 0 0
\(851\) −4272.00 −0.172083
\(852\) 0 0
\(853\) −39466.0 −1.58416 −0.792081 0.610416i \(-0.791002\pi\)
−0.792081 + 0.610416i \(0.791002\pi\)
\(854\) 0 0
\(855\) 4104.00 0.164157
\(856\) 0 0
\(857\) −34038.0 −1.35673 −0.678364 0.734726i \(-0.737311\pi\)
−0.678364 + 0.734726i \(0.737311\pi\)
\(858\) 0 0
\(859\) 3364.00 0.133618 0.0668092 0.997766i \(-0.478718\pi\)
0.0668092 + 0.997766i \(0.478718\pi\)
\(860\) 0 0
\(861\) −7938.00 −0.314200
\(862\) 0 0
\(863\) −13104.0 −0.516878 −0.258439 0.966028i \(-0.583208\pi\)
−0.258439 + 0.966028i \(0.583208\pi\)
\(864\) 0 0
\(865\) −16236.0 −0.638197
\(866\) 0 0
\(867\) 24249.0 0.949872
\(868\) 0 0
\(869\) 2880.00 0.112425
\(870\) 0 0
\(871\) 62744.0 2.44087
\(872\) 0 0
\(873\) 9090.00 0.352405
\(874\) 0 0
\(875\) 8988.00 0.347257
\(876\) 0 0
\(877\) −40858.0 −1.57318 −0.786589 0.617477i \(-0.788155\pi\)
−0.786589 + 0.617477i \(0.788155\pi\)
\(878\) 0 0
\(879\) −11214.0 −0.430306
\(880\) 0 0
\(881\) −37374.0 −1.42924 −0.714621 0.699512i \(-0.753401\pi\)
−0.714621 + 0.699512i \(0.753401\pi\)
\(882\) 0 0
\(883\) −9788.00 −0.373038 −0.186519 0.982451i \(-0.559721\pi\)
−0.186519 + 0.982451i \(0.559721\pi\)
\(884\) 0 0
\(885\) −7128.00 −0.270740
\(886\) 0 0
\(887\) −50424.0 −1.90876 −0.954381 0.298591i \(-0.903483\pi\)
−0.954381 + 0.298591i \(0.903483\pi\)
\(888\) 0 0
\(889\) 9296.00 0.350706
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 0 0
\(893\) 14592.0 0.546811
\(894\) 0 0
\(895\) −28296.0 −1.05679
\(896\) 0 0
\(897\) 4464.00 0.166163
\(898\) 0 0
\(899\) 6048.00 0.224374
\(900\) 0 0
\(901\) −45828.0 −1.69451
\(902\) 0 0
\(903\) −3612.00 −0.133112
\(904\) 0 0
\(905\) 11460.0 0.420932
\(906\) 0 0
\(907\) 12412.0 0.454392 0.227196 0.973849i \(-0.427044\pi\)
0.227196 + 0.973849i \(0.427044\pi\)
\(908\) 0 0
\(909\) 54.0000 0.00197037
\(910\) 0 0
\(911\) −6576.00 −0.239158 −0.119579 0.992825i \(-0.538154\pi\)
−0.119579 + 0.992825i \(0.538154\pi\)
\(912\) 0 0
\(913\) −3888.00 −0.140935
\(914\) 0 0
\(915\) 4572.00 0.165187
\(916\) 0 0
\(917\) −8484.00 −0.305525
\(918\) 0 0
\(919\) −8264.00 −0.296631 −0.148316 0.988940i \(-0.547385\pi\)
−0.148316 + 0.988940i \(0.547385\pi\)
\(920\) 0 0
\(921\) 2532.00 0.0905887
\(922\) 0 0
\(923\) −52080.0 −1.85724
\(924\) 0 0
\(925\) 15842.0 0.563115
\(926\) 0 0
\(927\) 4248.00 0.150510
\(928\) 0 0
\(929\) 39426.0 1.39238 0.696192 0.717855i \(-0.254876\pi\)
0.696192 + 0.717855i \(0.254876\pi\)
\(930\) 0 0
\(931\) 3724.00 0.131095
\(932\) 0 0
\(933\) −18936.0 −0.664455
\(934\) 0 0
\(935\) −24624.0 −0.861274
\(936\) 0 0
\(937\) −4678.00 −0.163099 −0.0815494 0.996669i \(-0.525987\pi\)
−0.0815494 + 0.996669i \(0.525987\pi\)
\(938\) 0 0
\(939\) 24846.0 0.863492
\(940\) 0 0
\(941\) −17346.0 −0.600918 −0.300459 0.953795i \(-0.597140\pi\)
−0.300459 + 0.953795i \(0.597140\pi\)
\(942\) 0 0
\(943\) 9072.00 0.313282
\(944\) 0 0
\(945\) −1134.00 −0.0390360
\(946\) 0 0
\(947\) −19452.0 −0.667482 −0.333741 0.942665i \(-0.608311\pi\)
−0.333741 + 0.942665i \(0.608311\pi\)
\(948\) 0 0
\(949\) 55180.0 1.88748
\(950\) 0 0
\(951\) 27954.0 0.953176
\(952\) 0 0
\(953\) 4458.00 0.151531 0.0757654 0.997126i \(-0.475860\pi\)
0.0757654 + 0.997126i \(0.475860\pi\)
\(954\) 0 0
\(955\) −24480.0 −0.829481
\(956\) 0 0
\(957\) −5832.00 −0.196992
\(958\) 0 0
\(959\) 8778.00 0.295575
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) 8748.00 0.292731
\(964\) 0 0
\(965\) −16116.0 −0.537609
\(966\) 0 0
\(967\) −52520.0 −1.74657 −0.873283 0.487213i \(-0.838013\pi\)
−0.873283 + 0.487213i \(0.838013\pi\)
\(968\) 0 0
\(969\) 25992.0 0.861696
\(970\) 0 0
\(971\) 10404.0 0.343852 0.171926 0.985110i \(-0.445001\pi\)
0.171926 + 0.985110i \(0.445001\pi\)
\(972\) 0 0
\(973\) −2380.00 −0.0784165
\(974\) 0 0
\(975\) −16554.0 −0.543746
\(976\) 0 0
\(977\) −7566.00 −0.247756 −0.123878 0.992297i \(-0.539533\pi\)
−0.123878 + 0.992297i \(0.539533\pi\)
\(978\) 0 0
\(979\) 58968.0 1.92505
\(980\) 0 0
\(981\) −16074.0 −0.523143
\(982\) 0 0
\(983\) −44376.0 −1.43985 −0.719926 0.694051i \(-0.755824\pi\)
−0.719926 + 0.694051i \(0.755824\pi\)
\(984\) 0 0
\(985\) 3060.00 0.0989845
\(986\) 0 0
\(987\) −4032.00 −0.130030
\(988\) 0 0
\(989\) 4128.00 0.132723
\(990\) 0 0
\(991\) 27328.0 0.875986 0.437993 0.898978i \(-0.355689\pi\)
0.437993 + 0.898978i \(0.355689\pi\)
\(992\) 0 0
\(993\) −4956.00 −0.158383
\(994\) 0 0
\(995\) −8112.00 −0.258460
\(996\) 0 0
\(997\) 2774.00 0.0881178 0.0440589 0.999029i \(-0.485971\pi\)
0.0440589 + 0.999029i \(0.485971\pi\)
\(998\) 0 0
\(999\) −4806.00 −0.152207
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 336.4.a.k.1.1 1
3.2 odd 2 1008.4.a.h.1.1 1
4.3 odd 2 84.4.a.a.1.1 1
7.6 odd 2 2352.4.a.d.1.1 1
8.3 odd 2 1344.4.a.q.1.1 1
8.5 even 2 1344.4.a.d.1.1 1
12.11 even 2 252.4.a.b.1.1 1
20.3 even 4 2100.4.k.j.1849.1 2
20.7 even 4 2100.4.k.j.1849.2 2
20.19 odd 2 2100.4.a.l.1.1 1
28.3 even 6 588.4.i.c.373.1 2
28.11 odd 6 588.4.i.f.373.1 2
28.19 even 6 588.4.i.c.361.1 2
28.23 odd 6 588.4.i.f.361.1 2
28.27 even 2 588.4.a.d.1.1 1
84.11 even 6 1764.4.k.l.1549.1 2
84.23 even 6 1764.4.k.l.361.1 2
84.47 odd 6 1764.4.k.f.361.1 2
84.59 odd 6 1764.4.k.f.1549.1 2
84.83 odd 2 1764.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.a.1.1 1 4.3 odd 2
252.4.a.b.1.1 1 12.11 even 2
336.4.a.k.1.1 1 1.1 even 1 trivial
588.4.a.d.1.1 1 28.27 even 2
588.4.i.c.361.1 2 28.19 even 6
588.4.i.c.373.1 2 28.3 even 6
588.4.i.f.361.1 2 28.23 odd 6
588.4.i.f.373.1 2 28.11 odd 6
1008.4.a.h.1.1 1 3.2 odd 2
1344.4.a.d.1.1 1 8.5 even 2
1344.4.a.q.1.1 1 8.3 odd 2
1764.4.a.j.1.1 1 84.83 odd 2
1764.4.k.f.361.1 2 84.47 odd 6
1764.4.k.f.1549.1 2 84.59 odd 6
1764.4.k.l.361.1 2 84.23 even 6
1764.4.k.l.1549.1 2 84.11 even 6
2100.4.a.l.1.1 1 20.19 odd 2
2100.4.k.j.1849.1 2 20.3 even 4
2100.4.k.j.1849.2 2 20.7 even 4
2352.4.a.d.1.1 1 7.6 odd 2