Properties

Label 4368.2.a.be.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.56155 q^{11} +1.00000 q^{13} +0.561553 q^{15} +5.68466 q^{17} -7.68466 q^{19} -1.00000 q^{21} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{27} -5.68466 q^{29} +10.2462 q^{31} +2.56155 q^{33} -0.561553 q^{35} -3.43845 q^{37} -1.00000 q^{39} +7.12311 q^{41} +10.5616 q^{43} -0.561553 q^{45} +1.00000 q^{49} -5.68466 q^{51} -4.24621 q^{53} +1.43845 q^{55} +7.68466 q^{57} +14.2462 q^{59} -5.68466 q^{61} +1.00000 q^{63} -0.561553 q^{65} -1.12311 q^{67} -1.43845 q^{69} -8.00000 q^{71} +0.561553 q^{73} +4.68466 q^{75} -2.56155 q^{77} +2.87689 q^{79} +1.00000 q^{81} -17.1231 q^{83} -3.19224 q^{85} +5.68466 q^{87} +10.0000 q^{89} +1.00000 q^{91} -10.2462 q^{93} +4.31534 q^{95} -18.4924 q^{97} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 3q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 3q^{5} + 2q^{7} + 2q^{9} - q^{11} + 2q^{13} - 3q^{15} - q^{17} - 3q^{19} - 2q^{21} + 7q^{23} + 3q^{25} - 2q^{27} + q^{29} + 4q^{31} + q^{33} + 3q^{35} - 11q^{37} - 2q^{39} + 6q^{41} + 17q^{43} + 3q^{45} + 2q^{49} + q^{51} + 8q^{53} + 7q^{55} + 3q^{57} + 12q^{59} + q^{61} + 2q^{63} + 3q^{65} + 6q^{67} - 7q^{69} - 16q^{71} - 3q^{73} - 3q^{75} - q^{77} + 14q^{79} + 2q^{81} - 26q^{83} - 27q^{85} - q^{87} + 20q^{89} + 2q^{91} - 4q^{93} + 21q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0.561553 0.144992
\(16\) 0 0
\(17\) 5.68466 1.37873 0.689366 0.724413i \(-0.257889\pi\)
0.689366 + 0.724413i \(0.257889\pi\)
\(18\) 0 0
\(19\) −7.68466 −1.76298 −0.881491 0.472201i \(-0.843460\pi\)
−0.881491 + 0.472201i \(0.843460\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.43845 0.299937 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 0 0
\(31\) 10.2462 1.84027 0.920137 0.391597i \(-0.128077\pi\)
0.920137 + 0.391597i \(0.128077\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) −0.561553 −0.0949197
\(36\) 0 0
\(37\) −3.43845 −0.565277 −0.282639 0.959226i \(-0.591210\pi\)
−0.282639 + 0.959226i \(0.591210\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 0 0
\(43\) 10.5616 1.61062 0.805311 0.592853i \(-0.201998\pi\)
0.805311 + 0.592853i \(0.201998\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.68466 −0.796011
\(52\) 0 0
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) 0 0
\(55\) 1.43845 0.193960
\(56\) 0 0
\(57\) 7.68466 1.01786
\(58\) 0 0
\(59\) 14.2462 1.85470 0.927349 0.374197i \(-0.122082\pi\)
0.927349 + 0.374197i \(0.122082\pi\)
\(60\) 0 0
\(61\) −5.68466 −0.727846 −0.363923 0.931429i \(-0.618563\pi\)
−0.363923 + 0.931429i \(0.618563\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −0.561553 −0.0696521
\(66\) 0 0
\(67\) −1.12311 −0.137209 −0.0686046 0.997644i \(-0.521855\pi\)
−0.0686046 + 0.997644i \(0.521855\pi\)
\(68\) 0 0
\(69\) −1.43845 −0.173169
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 0.561553 0.0657248 0.0328624 0.999460i \(-0.489538\pi\)
0.0328624 + 0.999460i \(0.489538\pi\)
\(74\) 0 0
\(75\) 4.68466 0.540938
\(76\) 0 0
\(77\) −2.56155 −0.291916
\(78\) 0 0
\(79\) 2.87689 0.323676 0.161838 0.986817i \(-0.448258\pi\)
0.161838 + 0.986817i \(0.448258\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −17.1231 −1.87951 −0.939753 0.341856i \(-0.888945\pi\)
−0.939753 + 0.341856i \(0.888945\pi\)
\(84\) 0 0
\(85\) −3.19224 −0.346247
\(86\) 0 0
\(87\) 5.68466 0.609459
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −10.2462 −1.06248
\(94\) 0 0
\(95\) 4.31534 0.442745
\(96\) 0 0
\(97\) −18.4924 −1.87762 −0.938811 0.344434i \(-0.888071\pi\)
−0.938811 + 0.344434i \(0.888071\pi\)
\(98\) 0 0
\(99\) −2.56155 −0.257446
\(100\) 0 0
\(101\) 3.12311 0.310761 0.155380 0.987855i \(-0.450340\pi\)
0.155380 + 0.987855i \(0.450340\pi\)
\(102\) 0 0
\(103\) −11.6847 −1.15132 −0.575662 0.817688i \(-0.695256\pi\)
−0.575662 + 0.817688i \(0.695256\pi\)
\(104\) 0 0
\(105\) 0.561553 0.0548019
\(106\) 0 0
\(107\) 17.1231 1.65535 0.827677 0.561205i \(-0.189662\pi\)
0.827677 + 0.561205i \(0.189662\pi\)
\(108\) 0 0
\(109\) 11.9309 1.14277 0.571385 0.820682i \(-0.306406\pi\)
0.571385 + 0.820682i \(0.306406\pi\)
\(110\) 0 0
\(111\) 3.43845 0.326363
\(112\) 0 0
\(113\) 12.2462 1.15203 0.576013 0.817440i \(-0.304608\pi\)
0.576013 + 0.817440i \(0.304608\pi\)
\(114\) 0 0
\(115\) −0.807764 −0.0753244
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 5.68466 0.521112
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) −7.12311 −0.642269
\(124\) 0 0
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) 13.1231 1.16449 0.582244 0.813014i \(-0.302175\pi\)
0.582244 + 0.813014i \(0.302175\pi\)
\(128\) 0 0
\(129\) −10.5616 −0.929893
\(130\) 0 0
\(131\) 5.43845 0.475159 0.237580 0.971368i \(-0.423646\pi\)
0.237580 + 0.971368i \(0.423646\pi\)
\(132\) 0 0
\(133\) −7.68466 −0.666344
\(134\) 0 0
\(135\) 0.561553 0.0483308
\(136\) 0 0
\(137\) 0.561553 0.0479767 0.0239883 0.999712i \(-0.492364\pi\)
0.0239883 + 0.999712i \(0.492364\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.56155 −0.214208
\(144\) 0 0
\(145\) 3.19224 0.265101
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 23.6155 1.93466 0.967330 0.253522i \(-0.0815889\pi\)
0.967330 + 0.253522i \(0.0815889\pi\)
\(150\) 0 0
\(151\) 8.80776 0.716766 0.358383 0.933575i \(-0.383328\pi\)
0.358383 + 0.933575i \(0.383328\pi\)
\(152\) 0 0
\(153\) 5.68466 0.459577
\(154\) 0 0
\(155\) −5.75379 −0.462155
\(156\) 0 0
\(157\) −11.4384 −0.912887 −0.456444 0.889752i \(-0.650877\pi\)
−0.456444 + 0.889752i \(0.650877\pi\)
\(158\) 0 0
\(159\) 4.24621 0.336746
\(160\) 0 0
\(161\) 1.43845 0.113366
\(162\) 0 0
\(163\) 25.1231 1.96779 0.983897 0.178738i \(-0.0572014\pi\)
0.983897 + 0.178738i \(0.0572014\pi\)
\(164\) 0 0
\(165\) −1.43845 −0.111983
\(166\) 0 0
\(167\) 5.93087 0.458944 0.229472 0.973315i \(-0.426300\pi\)
0.229472 + 0.973315i \(0.426300\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.68466 −0.587661
\(172\) 0 0
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) 0 0
\(175\) −4.68466 −0.354127
\(176\) 0 0
\(177\) −14.2462 −1.07081
\(178\) 0 0
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) 0 0
\(181\) 16.2462 1.20757 0.603786 0.797147i \(-0.293658\pi\)
0.603786 + 0.797147i \(0.293658\pi\)
\(182\) 0 0
\(183\) 5.68466 0.420222
\(184\) 0 0
\(185\) 1.93087 0.141960
\(186\) 0 0
\(187\) −14.5616 −1.06485
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 13.9309 1.00800 0.504001 0.863703i \(-0.331861\pi\)
0.504001 + 0.863703i \(0.331861\pi\)
\(192\) 0 0
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) 0 0
\(195\) 0.561553 0.0402136
\(196\) 0 0
\(197\) 19.1231 1.36246 0.681232 0.732067i \(-0.261444\pi\)
0.681232 + 0.732067i \(0.261444\pi\)
\(198\) 0 0
\(199\) 21.9309 1.55464 0.777319 0.629107i \(-0.216579\pi\)
0.777319 + 0.629107i \(0.216579\pi\)
\(200\) 0 0
\(201\) 1.12311 0.0792178
\(202\) 0 0
\(203\) −5.68466 −0.398985
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 1.43845 0.0999790
\(208\) 0 0
\(209\) 19.6847 1.36162
\(210\) 0 0
\(211\) −4.80776 −0.330980 −0.165490 0.986211i \(-0.552921\pi\)
−0.165490 + 0.986211i \(0.552921\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −5.93087 −0.404482
\(216\) 0 0
\(217\) 10.2462 0.695558
\(218\) 0 0
\(219\) −0.561553 −0.0379462
\(220\) 0 0
\(221\) 5.68466 0.382392
\(222\) 0 0
\(223\) −17.6155 −1.17962 −0.589812 0.807541i \(-0.700798\pi\)
−0.589812 + 0.807541i \(0.700798\pi\)
\(224\) 0 0
\(225\) −4.68466 −0.312311
\(226\) 0 0
\(227\) −1.12311 −0.0745431 −0.0372716 0.999305i \(-0.511867\pi\)
−0.0372716 + 0.999305i \(0.511867\pi\)
\(228\) 0 0
\(229\) 11.7538 0.776712 0.388356 0.921509i \(-0.373043\pi\)
0.388356 + 0.921509i \(0.373043\pi\)
\(230\) 0 0
\(231\) 2.56155 0.168538
\(232\) 0 0
\(233\) 2.63068 0.172342 0.0861709 0.996280i \(-0.472537\pi\)
0.0861709 + 0.996280i \(0.472537\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.87689 −0.186874
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.561553 −0.0358763
\(246\) 0 0
\(247\) −7.68466 −0.488963
\(248\) 0 0
\(249\) 17.1231 1.08513
\(250\) 0 0
\(251\) −12.8078 −0.808419 −0.404209 0.914666i \(-0.632453\pi\)
−0.404209 + 0.914666i \(0.632453\pi\)
\(252\) 0 0
\(253\) −3.68466 −0.231652
\(254\) 0 0
\(255\) 3.19224 0.199906
\(256\) 0 0
\(257\) 12.2462 0.763898 0.381949 0.924183i \(-0.375253\pi\)
0.381949 + 0.924183i \(0.375253\pi\)
\(258\) 0 0
\(259\) −3.43845 −0.213655
\(260\) 0 0
\(261\) −5.68466 −0.351872
\(262\) 0 0
\(263\) 13.7538 0.848095 0.424047 0.905640i \(-0.360609\pi\)
0.424047 + 0.905640i \(0.360609\pi\)
\(264\) 0 0
\(265\) 2.38447 0.146477
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 3.75379 0.228873 0.114436 0.993431i \(-0.463494\pi\)
0.114436 + 0.993431i \(0.463494\pi\)
\(270\) 0 0
\(271\) −13.1231 −0.797172 −0.398586 0.917131i \(-0.630499\pi\)
−0.398586 + 0.917131i \(0.630499\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 10.4924 0.630429 0.315214 0.949021i \(-0.397924\pi\)
0.315214 + 0.949021i \(0.397924\pi\)
\(278\) 0 0
\(279\) 10.2462 0.613425
\(280\) 0 0
\(281\) −0.246211 −0.0146877 −0.00734387 0.999973i \(-0.502338\pi\)
−0.00734387 + 0.999973i \(0.502338\pi\)
\(282\) 0 0
\(283\) −1.75379 −0.104252 −0.0521260 0.998641i \(-0.516600\pi\)
−0.0521260 + 0.998641i \(0.516600\pi\)
\(284\) 0 0
\(285\) −4.31534 −0.255619
\(286\) 0 0
\(287\) 7.12311 0.420464
\(288\) 0 0
\(289\) 15.3153 0.900902
\(290\) 0 0
\(291\) 18.4924 1.08405
\(292\) 0 0
\(293\) −24.7386 −1.44525 −0.722623 0.691242i \(-0.757064\pi\)
−0.722623 + 0.691242i \(0.757064\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 2.56155 0.148636
\(298\) 0 0
\(299\) 1.43845 0.0831875
\(300\) 0 0
\(301\) 10.5616 0.608758
\(302\) 0 0
\(303\) −3.12311 −0.179418
\(304\) 0 0
\(305\) 3.19224 0.182787
\(306\) 0 0
\(307\) −9.75379 −0.556678 −0.278339 0.960483i \(-0.589784\pi\)
−0.278339 + 0.960483i \(0.589784\pi\)
\(308\) 0 0
\(309\) 11.6847 0.664717
\(310\) 0 0
\(311\) 21.1231 1.19778 0.598891 0.800831i \(-0.295608\pi\)
0.598891 + 0.800831i \(0.295608\pi\)
\(312\) 0 0
\(313\) 27.6155 1.56092 0.780461 0.625205i \(-0.214984\pi\)
0.780461 + 0.625205i \(0.214984\pi\)
\(314\) 0 0
\(315\) −0.561553 −0.0316399
\(316\) 0 0
\(317\) 11.1231 0.624736 0.312368 0.949961i \(-0.398878\pi\)
0.312368 + 0.949961i \(0.398878\pi\)
\(318\) 0 0
\(319\) 14.5616 0.815290
\(320\) 0 0
\(321\) −17.1231 −0.955719
\(322\) 0 0
\(323\) −43.6847 −2.43068
\(324\) 0 0
\(325\) −4.68466 −0.259858
\(326\) 0 0
\(327\) −11.9309 −0.659779
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 11.3693 0.624914 0.312457 0.949932i \(-0.398848\pi\)
0.312457 + 0.949932i \(0.398848\pi\)
\(332\) 0 0
\(333\) −3.43845 −0.188426
\(334\) 0 0
\(335\) 0.630683 0.0344579
\(336\) 0 0
\(337\) 8.56155 0.466377 0.233189 0.972431i \(-0.425084\pi\)
0.233189 + 0.972431i \(0.425084\pi\)
\(338\) 0 0
\(339\) −12.2462 −0.665123
\(340\) 0 0
\(341\) −26.2462 −1.42131
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.807764 0.0434886
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 24.2462 1.29787 0.648935 0.760844i \(-0.275215\pi\)
0.648935 + 0.760844i \(0.275215\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −2.49242 −0.132658 −0.0663291 0.997798i \(-0.521129\pi\)
−0.0663291 + 0.997798i \(0.521129\pi\)
\(354\) 0 0
\(355\) 4.49242 0.238433
\(356\) 0 0
\(357\) −5.68466 −0.300864
\(358\) 0 0
\(359\) −22.7386 −1.20010 −0.600050 0.799963i \(-0.704852\pi\)
−0.600050 + 0.799963i \(0.704852\pi\)
\(360\) 0 0
\(361\) 40.0540 2.10810
\(362\) 0 0
\(363\) 4.43845 0.232958
\(364\) 0 0
\(365\) −0.315342 −0.0165057
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 7.12311 0.370814
\(370\) 0 0
\(371\) −4.24621 −0.220452
\(372\) 0 0
\(373\) 23.6155 1.22277 0.611383 0.791335i \(-0.290614\pi\)
0.611383 + 0.791335i \(0.290614\pi\)
\(374\) 0 0
\(375\) −5.43845 −0.280840
\(376\) 0 0
\(377\) −5.68466 −0.292775
\(378\) 0 0
\(379\) −17.7538 −0.911951 −0.455975 0.889992i \(-0.650710\pi\)
−0.455975 + 0.889992i \(0.650710\pi\)
\(380\) 0 0
\(381\) −13.1231 −0.672317
\(382\) 0 0
\(383\) 34.4233 1.75895 0.879474 0.475947i \(-0.157895\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(384\) 0 0
\(385\) 1.43845 0.0733101
\(386\) 0 0
\(387\) 10.5616 0.536874
\(388\) 0 0
\(389\) 20.7386 1.05149 0.525745 0.850642i \(-0.323787\pi\)
0.525745 + 0.850642i \(0.323787\pi\)
\(390\) 0 0
\(391\) 8.17708 0.413533
\(392\) 0 0
\(393\) −5.43845 −0.274333
\(394\) 0 0
\(395\) −1.61553 −0.0812860
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 7.68466 0.384714
\(400\) 0 0
\(401\) −8.24621 −0.411796 −0.205898 0.978573i \(-0.566012\pi\)
−0.205898 + 0.978573i \(0.566012\pi\)
\(402\) 0 0
\(403\) 10.2462 0.510400
\(404\) 0 0
\(405\) −0.561553 −0.0279038
\(406\) 0 0
\(407\) 8.80776 0.436585
\(408\) 0 0
\(409\) 23.9309 1.18331 0.591653 0.806193i \(-0.298476\pi\)
0.591653 + 0.806193i \(0.298476\pi\)
\(410\) 0 0
\(411\) −0.561553 −0.0276994
\(412\) 0 0
\(413\) 14.2462 0.701010
\(414\) 0 0
\(415\) 9.61553 0.472008
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 24.3153 1.18788 0.593941 0.804509i \(-0.297571\pi\)
0.593941 + 0.804509i \(0.297571\pi\)
\(420\) 0 0
\(421\) −20.2462 −0.986740 −0.493370 0.869820i \(-0.664235\pi\)
−0.493370 + 0.869820i \(0.664235\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −26.6307 −1.29178
\(426\) 0 0
\(427\) −5.68466 −0.275100
\(428\) 0 0
\(429\) 2.56155 0.123673
\(430\) 0 0
\(431\) 8.63068 0.415725 0.207863 0.978158i \(-0.433349\pi\)
0.207863 + 0.978158i \(0.433349\pi\)
\(432\) 0 0
\(433\) −6.63068 −0.318650 −0.159325 0.987226i \(-0.550932\pi\)
−0.159325 + 0.987226i \(0.550932\pi\)
\(434\) 0 0
\(435\) −3.19224 −0.153056
\(436\) 0 0
\(437\) −11.0540 −0.528783
\(438\) 0 0
\(439\) −21.9309 −1.04670 −0.523352 0.852117i \(-0.675319\pi\)
−0.523352 + 0.852117i \(0.675319\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −19.3693 −0.920264 −0.460132 0.887851i \(-0.652198\pi\)
−0.460132 + 0.887851i \(0.652198\pi\)
\(444\) 0 0
\(445\) −5.61553 −0.266202
\(446\) 0 0
\(447\) −23.6155 −1.11698
\(448\) 0 0
\(449\) −1.68466 −0.0795039 −0.0397520 0.999210i \(-0.512657\pi\)
−0.0397520 + 0.999210i \(0.512657\pi\)
\(450\) 0 0
\(451\) −18.2462 −0.859181
\(452\) 0 0
\(453\) −8.80776 −0.413825
\(454\) 0 0
\(455\) −0.561553 −0.0263260
\(456\) 0 0
\(457\) 2.63068 0.123058 0.0615291 0.998105i \(-0.480402\pi\)
0.0615291 + 0.998105i \(0.480402\pi\)
\(458\) 0 0
\(459\) −5.68466 −0.265337
\(460\) 0 0
\(461\) 9.05398 0.421686 0.210843 0.977520i \(-0.432379\pi\)
0.210843 + 0.977520i \(0.432379\pi\)
\(462\) 0 0
\(463\) −3.68466 −0.171241 −0.0856203 0.996328i \(-0.527287\pi\)
−0.0856203 + 0.996328i \(0.527287\pi\)
\(464\) 0 0
\(465\) 5.75379 0.266826
\(466\) 0 0
\(467\) 15.6847 0.725799 0.362900 0.931828i \(-0.381787\pi\)
0.362900 + 0.931828i \(0.381787\pi\)
\(468\) 0 0
\(469\) −1.12311 −0.0518602
\(470\) 0 0
\(471\) 11.4384 0.527056
\(472\) 0 0
\(473\) −27.0540 −1.24394
\(474\) 0 0
\(475\) 36.0000 1.65179
\(476\) 0 0
\(477\) −4.24621 −0.194421
\(478\) 0 0
\(479\) 21.3002 0.973230 0.486615 0.873616i \(-0.338231\pi\)
0.486615 + 0.873616i \(0.338231\pi\)
\(480\) 0 0
\(481\) −3.43845 −0.156780
\(482\) 0 0
\(483\) −1.43845 −0.0654516
\(484\) 0 0
\(485\) 10.3845 0.471535
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −25.1231 −1.13611
\(490\) 0 0
\(491\) 17.1231 0.772755 0.386377 0.922341i \(-0.373726\pi\)
0.386377 + 0.922341i \(0.373726\pi\)
\(492\) 0 0
\(493\) −32.3153 −1.45541
\(494\) 0 0
\(495\) 1.43845 0.0646534
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −5.93087 −0.264972
\(502\) 0 0
\(503\) −5.12311 −0.228428 −0.114214 0.993456i \(-0.536435\pi\)
−0.114214 + 0.993456i \(0.536435\pi\)
\(504\) 0 0
\(505\) −1.75379 −0.0780426
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 25.0540 1.11050 0.555249 0.831684i \(-0.312623\pi\)
0.555249 + 0.831684i \(0.312623\pi\)
\(510\) 0 0
\(511\) 0.561553 0.0248416
\(512\) 0 0
\(513\) 7.68466 0.339286
\(514\) 0 0
\(515\) 6.56155 0.289137
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −3.75379 −0.164773
\(520\) 0 0
\(521\) −9.68466 −0.424293 −0.212146 0.977238i \(-0.568045\pi\)
−0.212146 + 0.977238i \(0.568045\pi\)
\(522\) 0 0
\(523\) −38.2462 −1.67239 −0.836195 0.548432i \(-0.815225\pi\)
−0.836195 + 0.548432i \(0.815225\pi\)
\(524\) 0 0
\(525\) 4.68466 0.204455
\(526\) 0 0
\(527\) 58.2462 2.53724
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) 7.12311 0.308536
\(534\) 0 0
\(535\) −9.61553 −0.415716
\(536\) 0 0
\(537\) −16.4924 −0.711701
\(538\) 0 0
\(539\) −2.56155 −0.110334
\(540\) 0 0
\(541\) −4.06913 −0.174946 −0.0874728 0.996167i \(-0.527879\pi\)
−0.0874728 + 0.996167i \(0.527879\pi\)
\(542\) 0 0
\(543\) −16.2462 −0.697192
\(544\) 0 0
\(545\) −6.69981 −0.286988
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −5.68466 −0.242615
\(550\) 0 0
\(551\) 43.6847 1.86103
\(552\) 0 0
\(553\) 2.87689 0.122338
\(554\) 0 0
\(555\) −1.93087 −0.0819609
\(556\) 0 0
\(557\) 21.3693 0.905447 0.452724 0.891651i \(-0.350452\pi\)
0.452724 + 0.891651i \(0.350452\pi\)
\(558\) 0 0
\(559\) 10.5616 0.446706
\(560\) 0 0
\(561\) 14.5616 0.614789
\(562\) 0 0
\(563\) −31.0540 −1.30877 −0.654385 0.756162i \(-0.727072\pi\)
−0.654385 + 0.756162i \(0.727072\pi\)
\(564\) 0 0
\(565\) −6.87689 −0.289313
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −41.2311 −1.72850 −0.864248 0.503066i \(-0.832205\pi\)
−0.864248 + 0.503066i \(0.832205\pi\)
\(570\) 0 0
\(571\) −1.75379 −0.0733938 −0.0366969 0.999326i \(-0.511684\pi\)
−0.0366969 + 0.999326i \(0.511684\pi\)
\(572\) 0 0
\(573\) −13.9309 −0.581970
\(574\) 0 0
\(575\) −6.73863 −0.281020
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) 5.36932 0.223141
\(580\) 0 0
\(581\) −17.1231 −0.710386
\(582\) 0 0
\(583\) 10.8769 0.450475
\(584\) 0 0
\(585\) −0.561553 −0.0232174
\(586\) 0 0
\(587\) 11.3693 0.469262 0.234631 0.972085i \(-0.424612\pi\)
0.234631 + 0.972085i \(0.424612\pi\)
\(588\) 0 0
\(589\) −78.7386 −3.24437
\(590\) 0 0
\(591\) −19.1231 −0.786619
\(592\) 0 0
\(593\) −3.75379 −0.154150 −0.0770748 0.997025i \(-0.524558\pi\)
−0.0770748 + 0.997025i \(0.524558\pi\)
\(594\) 0 0
\(595\) −3.19224 −0.130869
\(596\) 0 0
\(597\) −21.9309 −0.897571
\(598\) 0 0
\(599\) −17.4384 −0.712516 −0.356258 0.934388i \(-0.615948\pi\)
−0.356258 + 0.934388i \(0.615948\pi\)
\(600\) 0 0
\(601\) −19.1231 −0.780048 −0.390024 0.920805i \(-0.627533\pi\)
−0.390024 + 0.920805i \(0.627533\pi\)
\(602\) 0 0
\(603\) −1.12311 −0.0457364
\(604\) 0 0
\(605\) 2.49242 0.101331
\(606\) 0 0
\(607\) 22.5616 0.915745 0.457873 0.889018i \(-0.348612\pi\)
0.457873 + 0.889018i \(0.348612\pi\)
\(608\) 0 0
\(609\) 5.68466 0.230354
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −25.5464 −1.03181 −0.515905 0.856646i \(-0.672544\pi\)
−0.515905 + 0.856646i \(0.672544\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 28.4233 1.14428 0.572139 0.820156i \(-0.306114\pi\)
0.572139 + 0.820156i \(0.306114\pi\)
\(618\) 0 0
\(619\) −31.6847 −1.27351 −0.636757 0.771065i \(-0.719725\pi\)
−0.636757 + 0.771065i \(0.719725\pi\)
\(620\) 0 0
\(621\) −1.43845 −0.0577229
\(622\) 0 0
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) −19.6847 −0.786130
\(628\) 0 0
\(629\) −19.5464 −0.779366
\(630\) 0 0
\(631\) −5.93087 −0.236104 −0.118052 0.993007i \(-0.537665\pi\)
−0.118052 + 0.993007i \(0.537665\pi\)
\(632\) 0 0
\(633\) 4.80776 0.191091
\(634\) 0 0
\(635\) −7.36932 −0.292442
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −3.75379 −0.148266 −0.0741329 0.997248i \(-0.523619\pi\)
−0.0741329 + 0.997248i \(0.523619\pi\)
\(642\) 0 0
\(643\) −36.8078 −1.45156 −0.725778 0.687929i \(-0.758520\pi\)
−0.725778 + 0.687929i \(0.758520\pi\)
\(644\) 0 0
\(645\) 5.93087 0.233528
\(646\) 0 0
\(647\) −2.24621 −0.0883077 −0.0441538 0.999025i \(-0.514059\pi\)
−0.0441538 + 0.999025i \(0.514059\pi\)
\(648\) 0 0
\(649\) −36.4924 −1.43245
\(650\) 0 0
\(651\) −10.2462 −0.401581
\(652\) 0 0
\(653\) 11.9309 0.466891 0.233446 0.972370i \(-0.425000\pi\)
0.233446 + 0.972370i \(0.425000\pi\)
\(654\) 0 0
\(655\) −3.05398 −0.119329
\(656\) 0 0
\(657\) 0.561553 0.0219083
\(658\) 0 0
\(659\) 3.36932 0.131250 0.0656250 0.997844i \(-0.479096\pi\)
0.0656250 + 0.997844i \(0.479096\pi\)
\(660\) 0 0
\(661\) 16.2462 0.631904 0.315952 0.948775i \(-0.397676\pi\)
0.315952 + 0.948775i \(0.397676\pi\)
\(662\) 0 0
\(663\) −5.68466 −0.220774
\(664\) 0 0
\(665\) 4.31534 0.167342
\(666\) 0 0
\(667\) −8.17708 −0.316618
\(668\) 0 0
\(669\) 17.6155 0.681056
\(670\) 0 0
\(671\) 14.5616 0.562143
\(672\) 0 0
\(673\) −13.1922 −0.508523 −0.254262 0.967135i \(-0.581832\pi\)
−0.254262 + 0.967135i \(0.581832\pi\)
\(674\) 0 0
\(675\) 4.68466 0.180313
\(676\) 0 0
\(677\) −30.4924 −1.17192 −0.585959 0.810340i \(-0.699282\pi\)
−0.585959 + 0.810340i \(0.699282\pi\)
\(678\) 0 0
\(679\) −18.4924 −0.709674
\(680\) 0 0
\(681\) 1.12311 0.0430375
\(682\) 0 0
\(683\) −47.6847 −1.82460 −0.912301 0.409519i \(-0.865696\pi\)
−0.912301 + 0.409519i \(0.865696\pi\)
\(684\) 0 0
\(685\) −0.315342 −0.0120486
\(686\) 0 0
\(687\) −11.7538 −0.448435
\(688\) 0 0
\(689\) −4.24621 −0.161768
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) −2.56155 −0.0973053
\(694\) 0 0
\(695\) 6.73863 0.255611
\(696\) 0 0
\(697\) 40.4924 1.53376
\(698\) 0 0
\(699\) −2.63068 −0.0995016
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 26.4233 0.996573
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.12311 0.117456
\(708\) 0 0
\(709\) 0.246211 0.00924666 0.00462333 0.999989i \(-0.498528\pi\)
0.00462333 + 0.999989i \(0.498528\pi\)
\(710\) 0 0
\(711\) 2.87689 0.107892
\(712\) 0 0
\(713\) 14.7386 0.551966
\(714\) 0 0
\(715\) 1.43845 0.0537949
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) −34.8769 −1.30069 −0.650344 0.759640i \(-0.725375\pi\)
−0.650344 + 0.759640i \(0.725375\pi\)
\(720\) 0 0
\(721\) −11.6847 −0.435159
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) 26.6307 0.989039
\(726\) 0 0
\(727\) −21.9309 −0.813371 −0.406685 0.913568i \(-0.633316\pi\)
−0.406685 + 0.913568i \(0.633316\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 60.0388 2.22062
\(732\) 0 0
\(733\) −0.384472 −0.0142008 −0.00710040 0.999975i \(-0.502260\pi\)
−0.00710040 + 0.999975i \(0.502260\pi\)
\(734\) 0 0
\(735\) 0.561553 0.0207132
\(736\) 0 0
\(737\) 2.87689 0.105972
\(738\) 0 0
\(739\) −42.1080 −1.54897 −0.774483 0.632595i \(-0.781990\pi\)
−0.774483 + 0.632595i \(0.781990\pi\)
\(740\) 0 0
\(741\) 7.68466 0.282303
\(742\) 0 0
\(743\) 46.1080 1.69154 0.845768 0.533550i \(-0.179143\pi\)
0.845768 + 0.533550i \(0.179143\pi\)
\(744\) 0 0
\(745\) −13.2614 −0.485859
\(746\) 0 0
\(747\) −17.1231 −0.626502
\(748\) 0 0
\(749\) 17.1231 0.625665
\(750\) 0 0
\(751\) −10.2462 −0.373890 −0.186945 0.982370i \(-0.559859\pi\)
−0.186945 + 0.982370i \(0.559859\pi\)
\(752\) 0 0
\(753\) 12.8078 0.466741
\(754\) 0 0
\(755\) −4.94602 −0.180004
\(756\) 0 0
\(757\) 1.50758 0.0547938 0.0273969 0.999625i \(-0.491278\pi\)
0.0273969 + 0.999625i \(0.491278\pi\)
\(758\) 0 0
\(759\) 3.68466 0.133745
\(760\) 0 0
\(761\) −3.12311 −0.113212 −0.0566062 0.998397i \(-0.518028\pi\)
−0.0566062 + 0.998397i \(0.518028\pi\)
\(762\) 0 0
\(763\) 11.9309 0.431926
\(764\) 0 0
\(765\) −3.19224 −0.115416
\(766\) 0 0
\(767\) 14.2462 0.514401
\(768\) 0 0
\(769\) −20.5616 −0.741469 −0.370734 0.928739i \(-0.620894\pi\)
−0.370734 + 0.928739i \(0.620894\pi\)
\(770\) 0 0
\(771\) −12.2462 −0.441037
\(772\) 0 0
\(773\) 24.0691 0.865706 0.432853 0.901464i \(-0.357507\pi\)
0.432853 + 0.901464i \(0.357507\pi\)
\(774\) 0 0
\(775\) −48.0000 −1.72421
\(776\) 0 0
\(777\) 3.43845 0.123354
\(778\) 0 0
\(779\) −54.7386 −1.96122
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 0 0
\(783\) 5.68466 0.203153
\(784\) 0 0
\(785\) 6.42329 0.229257
\(786\) 0 0
\(787\) 33.3002 1.18702 0.593512 0.804825i \(-0.297741\pi\)
0.593512 + 0.804825i \(0.297741\pi\)
\(788\) 0 0
\(789\) −13.7538 −0.489648
\(790\) 0 0
\(791\) 12.2462 0.435425
\(792\) 0 0
\(793\) −5.68466 −0.201868
\(794\) 0 0
\(795\) −2.38447 −0.0845685
\(796\) 0 0
\(797\) 6.63068 0.234871 0.117435 0.993081i \(-0.462533\pi\)
0.117435 + 0.993081i \(0.462533\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) −1.43845 −0.0507617
\(804\) 0 0
\(805\) −0.807764 −0.0284699
\(806\) 0 0
\(807\) −3.75379 −0.132140
\(808\) 0 0
\(809\) 30.4924 1.07206 0.536028 0.844200i \(-0.319924\pi\)
0.536028 + 0.844200i \(0.319924\pi\)
\(810\) 0 0
\(811\) −37.4384 −1.31464 −0.657321 0.753611i \(-0.728310\pi\)
−0.657321 + 0.753611i \(0.728310\pi\)
\(812\) 0 0
\(813\) 13.1231 0.460247
\(814\) 0 0
\(815\) −14.1080 −0.494180
\(816\) 0 0
\(817\) −81.1619 −2.83950
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 55.6155 1.94100 0.970498 0.241111i \(-0.0775116\pi\)
0.970498 + 0.241111i \(0.0775116\pi\)
\(822\) 0 0
\(823\) 28.4924 0.993183 0.496592 0.867984i \(-0.334585\pi\)
0.496592 + 0.867984i \(0.334585\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 1.93087 0.0671429 0.0335715 0.999436i \(-0.489312\pi\)
0.0335715 + 0.999436i \(0.489312\pi\)
\(828\) 0 0
\(829\) 26.3153 0.913970 0.456985 0.889475i \(-0.348929\pi\)
0.456985 + 0.889475i \(0.348929\pi\)
\(830\) 0 0
\(831\) −10.4924 −0.363978
\(832\) 0 0
\(833\) 5.68466 0.196962
\(834\) 0 0
\(835\) −3.33050 −0.115257
\(836\) 0 0
\(837\) −10.2462 −0.354161
\(838\) 0 0
\(839\) −38.7386 −1.33741 −0.668703 0.743530i \(-0.733150\pi\)
−0.668703 + 0.743530i \(0.733150\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) 0.246211 0.00847997
\(844\) 0 0
\(845\) −0.561553 −0.0193180
\(846\) 0 0
\(847\) −4.43845 −0.152507
\(848\) 0 0
\(849\) 1.75379 0.0601899
\(850\) 0 0
\(851\) −4.94602 −0.169548
\(852\) 0 0
\(853\) −2.63068 −0.0900729 −0.0450364 0.998985i \(-0.514340\pi\)
−0.0450364 + 0.998985i \(0.514340\pi\)
\(854\) 0 0
\(855\) 4.31534 0.147582
\(856\) 0 0
\(857\) −1.50758 −0.0514979 −0.0257489 0.999668i \(-0.508197\pi\)
−0.0257489 + 0.999668i \(0.508197\pi\)
\(858\) 0 0
\(859\) −22.2462 −0.759031 −0.379515 0.925185i \(-0.623909\pi\)
−0.379515 + 0.925185i \(0.623909\pi\)
\(860\) 0 0
\(861\) −7.12311 −0.242755
\(862\) 0 0
\(863\) −7.36932 −0.250854 −0.125427 0.992103i \(-0.540030\pi\)
−0.125427 + 0.992103i \(0.540030\pi\)
\(864\) 0 0
\(865\) −2.10795 −0.0716725
\(866\) 0 0
\(867\) −15.3153 −0.520136
\(868\) 0 0
\(869\) −7.36932 −0.249987
\(870\) 0 0
\(871\) −1.12311 −0.0380550
\(872\) 0 0
\(873\) −18.4924 −0.625874
\(874\) 0 0
\(875\) 5.43845 0.183853
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) 0 0
\(879\) 24.7386 0.834413
\(880\) 0 0
\(881\) 26.1771 0.881928 0.440964 0.897525i \(-0.354637\pi\)
0.440964 + 0.897525i \(0.354637\pi\)
\(882\) 0 0
\(883\) 2.56155 0.0862031 0.0431016 0.999071i \(-0.486276\pi\)
0.0431016 + 0.999071i \(0.486276\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 19.5076 0.655000 0.327500 0.944851i \(-0.393794\pi\)
0.327500 + 0.944851i \(0.393794\pi\)
\(888\) 0 0
\(889\) 13.1231 0.440135
\(890\) 0 0
\(891\) −2.56155 −0.0858152
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −9.26137 −0.309573
\(896\) 0 0
\(897\) −1.43845 −0.0480284
\(898\) 0 0
\(899\) −58.2462 −1.94262
\(900\) 0 0
\(901\) −24.1383 −0.804162
\(902\) 0 0
\(903\) −10.5616 −0.351466
\(904\) 0 0
\(905\) −9.12311 −0.303262
\(906\) 0 0
\(907\) 40.4924 1.34453 0.672264 0.740311i \(-0.265322\pi\)
0.672264 + 0.740311i \(0.265322\pi\)
\(908\) 0 0
\(909\) 3.12311 0.103587
\(910\) 0 0
\(911\) −50.4233 −1.67060 −0.835299 0.549796i \(-0.814706\pi\)
−0.835299 + 0.549796i \(0.814706\pi\)
\(912\) 0 0
\(913\) 43.8617 1.45161
\(914\) 0 0
\(915\) −3.19224 −0.105532
\(916\) 0 0
\(917\) 5.43845 0.179593
\(918\) 0 0
\(919\) −10.8769 −0.358796 −0.179398 0.983777i \(-0.557415\pi\)
−0.179398 + 0.983777i \(0.557415\pi\)
\(920\) 0 0
\(921\) 9.75379 0.321398
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 16.1080 0.529626
\(926\) 0 0
\(927\) −11.6847 −0.383775
\(928\) 0 0
\(929\) −15.6155 −0.512329 −0.256164 0.966633i \(-0.582459\pi\)
−0.256164 + 0.966633i \(0.582459\pi\)
\(930\) 0 0
\(931\) −7.68466 −0.251855
\(932\) 0 0
\(933\) −21.1231 −0.691539
\(934\) 0 0
\(935\) 8.17708 0.267419
\(936\) 0 0
\(937\) 18.9848 0.620208 0.310104 0.950703i \(-0.399636\pi\)
0.310104 + 0.950703i \(0.399636\pi\)
\(938\) 0 0
\(939\) −27.6155 −0.901199
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) 10.2462 0.333663
\(944\) 0 0
\(945\) 0.561553 0.0182673
\(946\) 0 0
\(947\) −45.7926 −1.48806 −0.744030 0.668146i \(-0.767088\pi\)
−0.744030 + 0.668146i \(0.767088\pi\)
\(948\) 0 0
\(949\) 0.561553 0.0182288
\(950\) 0 0
\(951\) −11.1231 −0.360691
\(952\) 0 0
\(953\) −13.3693 −0.433075 −0.216537 0.976274i \(-0.569476\pi\)
−0.216537 + 0.976274i \(0.569476\pi\)
\(954\) 0 0
\(955\) −7.82292 −0.253144
\(956\) 0 0
\(957\) −14.5616 −0.470708
\(958\) 0 0
\(959\) 0.561553 0.0181335
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) 17.1231 0.551784
\(964\) 0 0
\(965\) 3.01515 0.0970613
\(966\) 0 0
\(967\) 17.4384 0.560783 0.280391 0.959886i \(-0.409536\pi\)
0.280391 + 0.959886i \(0.409536\pi\)
\(968\) 0 0
\(969\) 43.6847 1.40335
\(970\) 0 0
\(971\) 14.2462 0.457183 0.228591 0.973522i \(-0.426588\pi\)
0.228591 + 0.973522i \(0.426588\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 4.68466 0.150029
\(976\) 0 0
\(977\) −10.3153 −0.330017 −0.165009 0.986292i \(-0.552765\pi\)
−0.165009 + 0.986292i \(0.552765\pi\)
\(978\) 0 0
\(979\) −25.6155 −0.818676
\(980\) 0 0
\(981\) 11.9309 0.380923
\(982\) 0 0
\(983\) −32.1771 −1.02629 −0.513145 0.858302i \(-0.671520\pi\)
−0.513145 + 0.858302i \(0.671520\pi\)
\(984\) 0 0
\(985\) −10.7386 −0.342161
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 15.1922 0.483085
\(990\) 0 0
\(991\) −33.6155 −1.06783 −0.533916 0.845537i \(-0.679280\pi\)
−0.533916 + 0.845537i \(0.679280\pi\)
\(992\) 0 0
\(993\) −11.3693 −0.360794
\(994\) 0 0
\(995\) −12.3153 −0.390423
\(996\) 0 0
\(997\) −8.73863 −0.276755 −0.138378 0.990380i \(-0.544189\pi\)
−0.138378 + 0.990380i \(0.544189\pi\)
\(998\) 0 0
\(999\) 3.43845 0.108788
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 4368.2.a.be.1.1 2
4.3 odd 2 546.2.a.j.1.1 2
12.11 even 2 1638.2.a.u.1.2 2
28.27 even 2 3822.2.a.bo.1.2 2
52.51 odd 2 7098.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.1 2 4.3 odd 2
1638.2.a.u.1.2 2 12.11 even 2
3822.2.a.bo.1.2 2 28.27 even 2
4368.2.a.be.1.1 2 1.1 even 1 trivial
7098.2.a.bl.1.2 2 52.51 odd 2