Properties

Label 1638.2.a.u.1.2
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(1\)
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 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.561553 q^{10} -2.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.68466 q^{17} +7.68466 q^{19} +0.561553 q^{20} +2.56155 q^{22} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} +5.68466 q^{29} -10.2462 q^{31} -1.00000 q^{32} +5.68466 q^{34} -0.561553 q^{35} -3.43845 q^{37} -7.68466 q^{38} -0.561553 q^{40} -7.12311 q^{41} -10.5616 q^{43} -2.56155 q^{44} -1.43845 q^{46} +1.00000 q^{49} +4.68466 q^{50} +1.00000 q^{52} +4.24621 q^{53} -1.43845 q^{55} +1.00000 q^{56} -5.68466 q^{58} +14.2462 q^{59} -5.68466 q^{61} +10.2462 q^{62} +1.00000 q^{64} +0.561553 q^{65} +1.12311 q^{67} -5.68466 q^{68} +0.561553 q^{70} -8.00000 q^{71} +0.561553 q^{73} +3.43845 q^{74} +7.68466 q^{76} +2.56155 q^{77} -2.87689 q^{79} +0.561553 q^{80} +7.12311 q^{82} -17.1231 q^{83} -3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +1.43845 q^{92} +4.31534 q^{95} -18.4924 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.561553 −0.177579
\(11\) −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
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.68466 −1.37873 −0.689366 0.724413i \(-0.742111\pi\)
−0.689366 + 0.724413i \(0.742111\pi\)
\(18\) 0 0
\(19\) 7.68466 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(20\) 0.561553 0.125567
\(21\) 0 0
\(22\) 2.56155 0.546125
\(23\) 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
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.68466 0.974911
\(35\) −0.561553 −0.0949197
\(36\) 0 0
\(37\) −3.43845 −0.565277 −0.282639 0.959226i \(-0.591210\pi\)
−0.282639 + 0.959226i \(0.591210\pi\)
\(38\) −7.68466 −1.24662
\(39\) 0 0
\(40\) −0.561553 −0.0887893
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) −10.5616 −1.61062 −0.805311 0.592853i \(-0.798002\pi\)
−0.805311 + 0.592853i \(0.798002\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0 0
\(46\) −1.43845 −0.212087
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.68466 0.662511
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 4.24621 0.583262 0.291631 0.956531i \(-0.405802\pi\)
0.291631 + 0.956531i \(0.405802\pi\)
\(54\) 0 0
\(55\) −1.43845 −0.193960
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −5.68466 −0.746432
\(59\) 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 \(0.618563\pi\)
\(62\) 10.2462 1.30127
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.561553 0.0696521
\(66\) 0 0
\(67\) 1.12311 0.137209 0.0686046 0.997644i \(-0.478145\pi\)
0.0686046 + 0.997644i \(0.478145\pi\)
\(68\) −5.68466 −0.689366
\(69\) 0 0
\(70\) 0.561553 0.0671184
\(71\) −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 \(0.489538\pi\)
\(74\) 3.43845 0.399711
\(75\) 0 0
\(76\) 7.68466 0.881491
\(77\) 2.56155 0.291916
\(78\) 0 0
\(79\) −2.87689 −0.323676 −0.161838 0.986817i \(-0.551742\pi\)
−0.161838 + 0.986817i \(0.551742\pi\)
\(80\) 0.561553 0.0627835
\(81\) 0 0
\(82\) 7.12311 0.786615
\(83\) −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
\(86\) 10.5616 1.13888
\(87\) 0 0
\(88\) 2.56155 0.273062
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 1.43845 0.149968
\(93\) 0 0
\(94\) 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 \(0.888071\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −4.68466 −0.468466
\(101\) −3.12311 −0.310761 −0.155380 0.987855i \(-0.549660\pi\)
−0.155380 + 0.987855i \(0.549660\pi\)
\(102\) 0 0
\(103\) 11.6847 1.15132 0.575662 0.817688i \(-0.304744\pi\)
0.575662 + 0.817688i \(0.304744\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −4.24621 −0.412428
\(107\) 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 \(0.306406\pi\)
\(110\) 1.43845 0.137151
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −12.2462 −1.15203 −0.576013 0.817440i \(-0.695392\pi\)
−0.576013 + 0.817440i \(0.695392\pi\)
\(114\) 0 0
\(115\) 0.807764 0.0753244
\(116\) 5.68466 0.527807
\(117\) 0 0
\(118\) −14.2462 −1.31147
\(119\) 5.68466 0.521112
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 5.68466 0.514665
\(123\) 0 0
\(124\) −10.2462 −0.920137
\(125\) −5.43845 −0.486430
\(126\) 0 0
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −0.561553 −0.0492514
\(131\) 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
\(134\) −1.12311 −0.0970215
\(135\) 0 0
\(136\) 5.68466 0.487455
\(137\) −0.561553 −0.0479767 −0.0239883 0.999712i \(-0.507636\pi\)
−0.0239883 + 0.999712i \(0.507636\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −0.561553 −0.0474599
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −2.56155 −0.214208
\(144\) 0 0
\(145\) 3.19224 0.265101
\(146\) −0.561553 −0.0464744
\(147\) 0 0
\(148\) −3.43845 −0.282639
\(149\) −23.6155 −1.93466 −0.967330 0.253522i \(-0.918411\pi\)
−0.967330 + 0.253522i \(0.918411\pi\)
\(150\) 0 0
\(151\) −8.80776 −0.716766 −0.358383 0.933575i \(-0.616672\pi\)
−0.358383 + 0.933575i \(0.616672\pi\)
\(152\) −7.68466 −0.623308
\(153\) 0 0
\(154\) −2.56155 −0.206416
\(155\) −5.75379 −0.462155
\(156\) 0 0
\(157\) −11.4384 −0.912887 −0.456444 0.889752i \(-0.650877\pi\)
−0.456444 + 0.889752i \(0.650877\pi\)
\(158\) 2.87689 0.228873
\(159\) 0 0
\(160\) −0.561553 −0.0443946
\(161\) −1.43845 −0.113366
\(162\) 0 0
\(163\) −25.1231 −1.96779 −0.983897 0.178738i \(-0.942799\pi\)
−0.983897 + 0.178738i \(0.942799\pi\)
\(164\) −7.12311 −0.556221
\(165\) 0 0
\(166\) 17.1231 1.32901
\(167\) 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
\(170\) 3.19224 0.244833
\(171\) 0 0
\(172\) −10.5616 −0.805311
\(173\) −3.75379 −0.285395 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(174\) 0 0
\(175\) 4.68466 0.354127
\(176\) −2.56155 −0.193084
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 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 \(0.293658\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) −1.43845 −0.106044
\(185\) −1.93087 −0.141960
\(186\) 0 0
\(187\) 14.5616 1.06485
\(188\) 0 0
\(189\) 0 0
\(190\) −4.31534 −0.313068
\(191\) 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 \(0.561902\pi\)
\(194\) 18.4924 1.32768
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −19.1231 −1.36246 −0.681232 0.732067i \(-0.738556\pi\)
−0.681232 + 0.732067i \(0.738556\pi\)
\(198\) 0 0
\(199\) −21.9309 −1.55464 −0.777319 0.629107i \(-0.783421\pi\)
−0.777319 + 0.629107i \(0.783421\pi\)
\(200\) 4.68466 0.331255
\(201\) 0 0
\(202\) 3.12311 0.219741
\(203\) −5.68466 −0.398985
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) −11.6847 −0.814109
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −19.6847 −1.36162
\(210\) 0 0
\(211\) 4.80776 0.330980 0.165490 0.986211i \(-0.447079\pi\)
0.165490 + 0.986211i \(0.447079\pi\)
\(212\) 4.24621 0.291631
\(213\) 0 0
\(214\) −17.1231 −1.17051
\(215\) −5.93087 −0.404482
\(216\) 0 0
\(217\) 10.2462 0.695558
\(218\) −11.9309 −0.808060
\(219\) 0 0
\(220\) −1.43845 −0.0969801
\(221\) −5.68466 −0.382392
\(222\) 0 0
\(223\) 17.6155 1.17962 0.589812 0.807541i \(-0.299202\pi\)
0.589812 + 0.807541i \(0.299202\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 12.2462 0.814606
\(227\) −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 \(0.373043\pi\)
\(230\) −0.807764 −0.0532624
\(231\) 0 0
\(232\) −5.68466 −0.373216
\(233\) −2.63068 −0.172342 −0.0861709 0.996280i \(-0.527463\pi\)
−0.0861709 + 0.996280i \(0.527463\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.2462 0.927349
\(237\) 0 0
\(238\) −5.68466 −0.368482
\(239\) −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 \(0.648900\pi\)
\(242\) 4.43845 0.285314
\(243\) 0 0
\(244\) −5.68466 −0.363923
\(245\) 0.561553 0.0358763
\(246\) 0 0
\(247\) 7.68466 0.488963
\(248\) 10.2462 0.650635
\(249\) 0 0
\(250\) 5.43845 0.343958
\(251\) −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
\(254\) 13.1231 0.823417
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.2462 −0.763898 −0.381949 0.924183i \(-0.624747\pi\)
−0.381949 + 0.924183i \(0.624747\pi\)
\(258\) 0 0
\(259\) 3.43845 0.213655
\(260\) 0.561553 0.0348260
\(261\) 0 0
\(262\) −5.43845 −0.335988
\(263\) 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
\(266\) 7.68466 0.471177
\(267\) 0 0
\(268\) 1.12311 0.0686046
\(269\) −3.75379 −0.228873 −0.114436 0.993431i \(-0.536506\pi\)
−0.114436 + 0.993431i \(0.536506\pi\)
\(270\) 0 0
\(271\) 13.1231 0.797172 0.398586 0.917131i \(-0.369501\pi\)
0.398586 + 0.917131i \(0.369501\pi\)
\(272\) −5.68466 −0.344683
\(273\) 0 0
\(274\) 0.561553 0.0339246
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 10.4924 0.630429 0.315214 0.949021i \(-0.397924\pi\)
0.315214 + 0.949021i \(0.397924\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 0.561553 0.0335592
\(281\) 0.246211 0.0146877 0.00734387 0.999973i \(-0.497662\pi\)
0.00734387 + 0.999973i \(0.497662\pi\)
\(282\) 0 0
\(283\) 1.75379 0.104252 0.0521260 0.998641i \(-0.483400\pi\)
0.0521260 + 0.998641i \(0.483400\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 2.56155 0.151468
\(287\) 7.12311 0.420464
\(288\) 0 0
\(289\) 15.3153 0.900902
\(290\) −3.19224 −0.187455
\(291\) 0 0
\(292\) 0.561553 0.0328624
\(293\) 24.7386 1.44525 0.722623 0.691242i \(-0.242936\pi\)
0.722623 + 0.691242i \(0.242936\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 3.43845 0.199856
\(297\) 0 0
\(298\) 23.6155 1.36801
\(299\) 1.43845 0.0831875
\(300\) 0 0
\(301\) 10.5616 0.608758
\(302\) 8.80776 0.506830
\(303\) 0 0
\(304\) 7.68466 0.440745
\(305\) −3.19224 −0.182787
\(306\) 0 0
\(307\) 9.75379 0.556678 0.278339 0.960483i \(-0.410216\pi\)
0.278339 + 0.960483i \(0.410216\pi\)
\(308\) 2.56155 0.145958
\(309\) 0 0
\(310\) 5.75379 0.326793
\(311\) 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 \(0.214984\pi\)
\(314\) 11.4384 0.645509
\(315\) 0 0
\(316\) −2.87689 −0.161838
\(317\) −11.1231 −0.624736 −0.312368 0.949961i \(-0.601122\pi\)
−0.312368 + 0.949961i \(0.601122\pi\)
\(318\) 0 0
\(319\) −14.5616 −0.815290
\(320\) 0.561553 0.0313918
\(321\) 0 0
\(322\) 1.43845 0.0801615
\(323\) −43.6847 −2.43068
\(324\) 0 0
\(325\) −4.68466 −0.259858
\(326\) 25.1231 1.39144
\(327\) 0 0
\(328\) 7.12311 0.393308
\(329\) 0 0
\(330\) 0 0
\(331\) −11.3693 −0.624914 −0.312457 0.949932i \(-0.601152\pi\)
−0.312457 + 0.949932i \(0.601152\pi\)
\(332\) −17.1231 −0.939753
\(333\) 0 0
\(334\) −5.93087 −0.324523
\(335\) 0.630683 0.0344579
\(336\) 0 0
\(337\) 8.56155 0.466377 0.233189 0.972431i \(-0.425084\pi\)
0.233189 + 0.972431i \(0.425084\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −3.19224 −0.173123
\(341\) 26.2462 1.42131
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 10.5616 0.569441
\(345\) 0 0
\(346\) 3.75379 0.201805
\(347\) 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 \(0.275215\pi\)
\(350\) −4.68466 −0.250406
\(351\) 0 0
\(352\) 2.56155 0.136531
\(353\) 2.49242 0.132658 0.0663291 0.997798i \(-0.478871\pi\)
0.0663291 + 0.997798i \(0.478871\pi\)
\(354\) 0 0
\(355\) −4.49242 −0.238433
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −16.4924 −0.871652
\(359\) −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
\(362\) −16.2462 −0.853882
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 0.315342 0.0165057
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 1.43845 0.0749842
\(369\) 0 0
\(370\) 1.93087 0.100381
\(371\) −4.24621 −0.220452
\(372\) 0 0
\(373\) 23.6155 1.22277 0.611383 0.791335i \(-0.290614\pi\)
0.611383 + 0.791335i \(0.290614\pi\)
\(374\) −14.5616 −0.752960
\(375\) 0 0
\(376\) 0 0
\(377\) 5.68466 0.292775
\(378\) 0 0
\(379\) 17.7538 0.911951 0.455975 0.889992i \(-0.349290\pi\)
0.455975 + 0.889992i \(0.349290\pi\)
\(380\) 4.31534 0.221372
\(381\) 0 0
\(382\) −13.9309 −0.712765
\(383\) 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
\(386\) 5.36932 0.273291
\(387\) 0 0
\(388\) −18.4924 −0.938811
\(389\) −20.7386 −1.05149 −0.525745 0.850642i \(-0.676213\pi\)
−0.525745 + 0.850642i \(0.676213\pi\)
\(390\) 0 0
\(391\) −8.17708 −0.413533
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 19.1231 0.963408
\(395\) −1.61553 −0.0812860
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 21.9309 1.09930
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) 8.24621 0.411796 0.205898 0.978573i \(-0.433988\pi\)
0.205898 + 0.978573i \(0.433988\pi\)
\(402\) 0 0
\(403\) −10.2462 −0.510400
\(404\) −3.12311 −0.155380
\(405\) 0 0
\(406\) 5.68466 0.282125
\(407\) 8.80776 0.436585
\(408\) 0 0
\(409\) 23.9309 1.18331 0.591653 0.806193i \(-0.298476\pi\)
0.591653 + 0.806193i \(0.298476\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) 11.6847 0.575662
\(413\) −14.2462 −0.701010
\(414\) 0 0
\(415\) −9.61553 −0.472008
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 19.6847 0.962808
\(419\) 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 \(0.664235\pi\)
\(422\) −4.80776 −0.234038
\(423\) 0 0
\(424\) −4.24621 −0.206214
\(425\) 26.6307 1.29178
\(426\) 0 0
\(427\) 5.68466 0.275100
\(428\) 17.1231 0.827677
\(429\) 0 0
\(430\) 5.93087 0.286012
\(431\) 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 \(0.550932\pi\)
\(434\) −10.2462 −0.491834
\(435\) 0 0
\(436\) 11.9309 0.571385
\(437\) 11.0540 0.528783
\(438\) 0 0
\(439\) 21.9309 1.04670 0.523352 0.852117i \(-0.324681\pi\)
0.523352 + 0.852117i \(0.324681\pi\)
\(440\) 1.43845 0.0685753
\(441\) 0 0
\(442\) 5.68466 0.270392
\(443\) −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
\(446\) −17.6155 −0.834119
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 1.68466 0.0795039 0.0397520 0.999210i \(-0.487343\pi\)
0.0397520 + 0.999210i \(0.487343\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) −12.2462 −0.576013
\(453\) 0 0
\(454\) 1.12311 0.0527100
\(455\) −0.561553 −0.0263260
\(456\) 0 0
\(457\) 2.63068 0.123058 0.0615291 0.998105i \(-0.480402\pi\)
0.0615291 + 0.998105i \(0.480402\pi\)
\(458\) −11.7538 −0.549218
\(459\) 0 0
\(460\) 0.807764 0.0376622
\(461\) −9.05398 −0.421686 −0.210843 0.977520i \(-0.567621\pi\)
−0.210843 + 0.977520i \(0.567621\pi\)
\(462\) 0 0
\(463\) 3.68466 0.171241 0.0856203 0.996328i \(-0.472713\pi\)
0.0856203 + 0.996328i \(0.472713\pi\)
\(464\) 5.68466 0.263904
\(465\) 0 0
\(466\) 2.63068 0.121864
\(467\) 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
\(471\) 0 0
\(472\) −14.2462 −0.655735
\(473\) 27.0540 1.24394
\(474\) 0 0
\(475\) −36.0000 −1.65179
\(476\) 5.68466 0.260556
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 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
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −4.43845 −0.201748
\(485\) −10.3845 −0.471535
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 5.68466 0.257332
\(489\) 0 0
\(490\) −0.561553 −0.0253684
\(491\) 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
\(494\) −7.68466 −0.345749
\(495\) 0 0
\(496\) −10.2462 −0.460068
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −5.43845 −0.243215
\(501\) 0 0
\(502\) 12.8078 0.571638
\(503\) −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
\(506\) 3.68466 0.163803
\(507\) 0 0
\(508\) −13.1231 −0.582244
\(509\) −25.0540 −1.11050 −0.555249 0.831684i \(-0.687377\pi\)
−0.555249 + 0.831684i \(0.687377\pi\)
\(510\) 0 0
\(511\) −0.561553 −0.0248416
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.2462 0.540157
\(515\) 6.56155 0.289137
\(516\) 0 0
\(517\) 0 0
\(518\) −3.43845 −0.151077
\(519\) 0 0
\(520\) −0.561553 −0.0246257
\(521\) 9.68466 0.424293 0.212146 0.977238i \(-0.431955\pi\)
0.212146 + 0.977238i \(0.431955\pi\)
\(522\) 0 0
\(523\) 38.2462 1.67239 0.836195 0.548432i \(-0.184775\pi\)
0.836195 + 0.548432i \(0.184775\pi\)
\(524\) 5.43845 0.237580
\(525\) 0 0
\(526\) −13.7538 −0.599694
\(527\) 58.2462 2.53724
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) −2.38447 −0.103575
\(531\) 0 0
\(532\) −7.68466 −0.333172
\(533\) −7.12311 −0.308536
\(534\) 0 0
\(535\) 9.61553 0.415716
\(536\) −1.12311 −0.0485108
\(537\) 0 0
\(538\) 3.75379 0.161837
\(539\) −2.56155 −0.110334
\(540\) 0 0
\(541\) −4.06913 −0.174946 −0.0874728 0.996167i \(-0.527879\pi\)
−0.0874728 + 0.996167i \(0.527879\pi\)
\(542\) −13.1231 −0.563686
\(543\) 0 0
\(544\) 5.68466 0.243728
\(545\) 6.69981 0.286988
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −0.561553 −0.0239883
\(549\) 0 0
\(550\) −12.0000 −0.511682
\(551\) 43.6847 1.86103
\(552\) 0 0
\(553\) 2.87689 0.122338
\(554\) −10.4924 −0.445780
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −21.3693 −0.905447 −0.452724 0.891651i \(-0.649548\pi\)
−0.452724 + 0.891651i \(0.649548\pi\)
\(558\) 0 0
\(559\) −10.5616 −0.446706
\(560\) −0.561553 −0.0237299
\(561\) 0 0
\(562\) −0.246211 −0.0103858
\(563\) −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
\(566\) −1.75379 −0.0737172
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 41.2311 1.72850 0.864248 0.503066i \(-0.167795\pi\)
0.864248 + 0.503066i \(0.167795\pi\)
\(570\) 0 0
\(571\) 1.75379 0.0733938 0.0366969 0.999326i \(-0.488316\pi\)
0.0366969 + 0.999326i \(0.488316\pi\)
\(572\) −2.56155 −0.107104
\(573\) 0 0
\(574\) −7.12311 −0.297313
\(575\) −6.73863 −0.281020
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −15.3153 −0.637034
\(579\) 0 0
\(580\) 3.19224 0.132550
\(581\) 17.1231 0.710386
\(582\) 0 0
\(583\) −10.8769 −0.450475
\(584\) −0.561553 −0.0232372
\(585\) 0 0
\(586\) −24.7386 −1.02194
\(587\) 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
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −3.43845 −0.141319
\(593\) 3.75379 0.154150 0.0770748 0.997025i \(-0.475442\pi\)
0.0770748 + 0.997025i \(0.475442\pi\)
\(594\) 0 0
\(595\) 3.19224 0.130869
\(596\) −23.6155 −0.967330
\(597\) 0 0
\(598\) −1.43845 −0.0588225
\(599\) −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 \(0.627533\pi\)
\(602\) −10.5616 −0.430457
\(603\) 0 0
\(604\) −8.80776 −0.358383
\(605\) −2.49242 −0.101331
\(606\) 0 0
\(607\) −22.5616 −0.915745 −0.457873 0.889018i \(-0.651388\pi\)
−0.457873 + 0.889018i \(0.651388\pi\)
\(608\) −7.68466 −0.311654
\(609\) 0 0
\(610\) 3.19224 0.129250
\(611\) 0 0
\(612\) 0 0
\(613\) −25.5464 −1.03181 −0.515905 0.856646i \(-0.672544\pi\)
−0.515905 + 0.856646i \(0.672544\pi\)
\(614\) −9.75379 −0.393631
\(615\) 0 0
\(616\) −2.56155 −0.103208
\(617\) −28.4233 −1.14428 −0.572139 0.820156i \(-0.693886\pi\)
−0.572139 + 0.820156i \(0.693886\pi\)
\(618\) 0 0
\(619\) 31.6847 1.27351 0.636757 0.771065i \(-0.280275\pi\)
0.636757 + 0.771065i \(0.280275\pi\)
\(620\) −5.75379 −0.231078
\(621\) 0 0
\(622\) −21.1231 −0.846959
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −27.6155 −1.10374
\(627\) 0 0
\(628\) −11.4384 −0.456444
\(629\) 19.5464 0.779366
\(630\) 0 0
\(631\) 5.93087 0.236104 0.118052 0.993007i \(-0.462335\pi\)
0.118052 + 0.993007i \(0.462335\pi\)
\(632\) 2.87689 0.114437
\(633\) 0 0
\(634\) 11.1231 0.441755
\(635\) −7.36932 −0.292442
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 14.5616 0.576497
\(639\) 0 0
\(640\) −0.561553 −0.0221973
\(641\) 3.75379 0.148266 0.0741329 0.997248i \(-0.476381\pi\)
0.0741329 + 0.997248i \(0.476381\pi\)
\(642\) 0 0
\(643\) 36.8078 1.45156 0.725778 0.687929i \(-0.241480\pi\)
0.725778 + 0.687929i \(0.241480\pi\)
\(644\) −1.43845 −0.0566828
\(645\) 0 0
\(646\) 43.6847 1.71875
\(647\) −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
\(650\) 4.68466 0.183747
\(651\) 0 0
\(652\) −25.1231 −0.983897
\(653\) −11.9309 −0.466891 −0.233446 0.972370i \(-0.575000\pi\)
−0.233446 + 0.972370i \(0.575000\pi\)
\(654\) 0 0
\(655\) 3.05398 0.119329
\(656\) −7.12311 −0.278111
\(657\) 0 0
\(658\) 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 \(0.397676\pi\)
\(662\) 11.3693 0.441881
\(663\) 0 0
\(664\) 17.1231 0.664505
\(665\) −4.31534 −0.167342
\(666\) 0 0
\(667\) 8.17708 0.316618
\(668\) 5.93087 0.229472
\(669\) 0 0
\(670\) −0.630683 −0.0243654
\(671\) 14.5616 0.562143
\(672\) 0 0
\(673\) −13.1922 −0.508523 −0.254262 0.967135i \(-0.581832\pi\)
−0.254262 + 0.967135i \(0.581832\pi\)
\(674\) −8.56155 −0.329779
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 30.4924 1.17192 0.585959 0.810340i \(-0.300718\pi\)
0.585959 + 0.810340i \(0.300718\pi\)
\(678\) 0 0
\(679\) 18.4924 0.709674
\(680\) 3.19224 0.122417
\(681\) 0 0
\(682\) −26.2462 −1.00502
\(683\) −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
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −10.5616 −0.402655
\(689\) 4.24621 0.161768
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) −3.75379 −0.142698
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 6.73863 0.255611
\(696\) 0 0
\(697\) 40.4924 1.53376
\(698\) −24.2462 −0.917733
\(699\) 0 0
\(700\) 4.68466 0.177063
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −26.4233 −0.996573
\(704\) −2.56155 −0.0965422
\(705\) 0 0
\(706\) −2.49242 −0.0938036
\(707\) 3.12311 0.117456
\(708\) 0 0
\(709\) 0.246211 0.00924666 0.00462333 0.999989i \(-0.498528\pi\)
0.00462333 + 0.999989i \(0.498528\pi\)
\(710\) 4.49242 0.168598
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −14.7386 −0.551966
\(714\) 0 0
\(715\) −1.43845 −0.0537949
\(716\) 16.4924 0.616351
\(717\) 0 0
\(718\) 22.7386 0.848598
\(719\) −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
\(722\) −40.0540 −1.49065
\(723\) 0 0
\(724\) 16.2462 0.603786
\(725\) −26.6307 −0.989039
\(726\) 0 0
\(727\) 21.9309 0.813371 0.406685 0.913568i \(-0.366684\pi\)
0.406685 + 0.913568i \(0.366684\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −0.315342 −0.0116713
\(731\) 60.0388 2.22062
\(732\) 0 0
\(733\) −0.384472 −0.0142008 −0.00710040 0.999975i \(-0.502260\pi\)
−0.00710040 + 0.999975i \(0.502260\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −1.43845 −0.0530219
\(737\) −2.87689 −0.105972
\(738\) 0 0
\(739\) 42.1080 1.54897 0.774483 0.632595i \(-0.218010\pi\)
0.774483 + 0.632595i \(0.218010\pi\)
\(740\) −1.93087 −0.0709802
\(741\) 0 0
\(742\) 4.24621 0.155883
\(743\) 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
\(746\) −23.6155 −0.864626
\(747\) 0 0
\(748\) 14.5616 0.532423
\(749\) −17.1231 −0.625665
\(750\) 0 0
\(751\) 10.2462 0.373890 0.186945 0.982370i \(-0.440141\pi\)
0.186945 + 0.982370i \(0.440141\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −5.68466 −0.207023
\(755\) −4.94602 −0.180004
\(756\) 0 0
\(757\) 1.50758 0.0547938 0.0273969 0.999625i \(-0.491278\pi\)
0.0273969 + 0.999625i \(0.491278\pi\)
\(758\) −17.7538 −0.644847
\(759\) 0 0
\(760\) −4.31534 −0.156534
\(761\) 3.12311 0.113212 0.0566062 0.998397i \(-0.481972\pi\)
0.0566062 + 0.998397i \(0.481972\pi\)
\(762\) 0 0
\(763\) −11.9309 −0.431926
\(764\) 13.9309 0.504001
\(765\) 0 0
\(766\) −34.4233 −1.24376
\(767\) 14.2462 0.514401
\(768\) 0 0
\(769\) −20.5616 −0.741469 −0.370734 0.928739i \(-0.620894\pi\)
−0.370734 + 0.928739i \(0.620894\pi\)
\(770\) −1.43845 −0.0518380
\(771\) 0 0
\(772\) −5.36932 −0.193246
\(773\) −24.0691 −0.865706 −0.432853 0.901464i \(-0.642493\pi\)
−0.432853 + 0.901464i \(0.642493\pi\)
\(774\) 0 0
\(775\) 48.0000 1.72421
\(776\) 18.4924 0.663839
\(777\) 0 0
\(778\) 20.7386 0.743516
\(779\) −54.7386 −1.96122
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 8.17708 0.292412
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −6.42329 −0.229257
\(786\) 0 0
\(787\) −33.3002 −1.18702 −0.593512 0.804825i \(-0.702259\pi\)
−0.593512 + 0.804825i \(0.702259\pi\)
\(788\) −19.1231 −0.681232
\(789\) 0 0
\(790\) 1.61553 0.0574779
\(791\) 12.2462 0.435425
\(792\) 0 0
\(793\) −5.68466 −0.201868
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −21.9309 −0.777319
\(797\) −6.63068 −0.234871 −0.117435 0.993081i \(-0.537467\pi\)
−0.117435 + 0.993081i \(0.537467\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.68466 0.165628
\(801\) 0 0
\(802\) −8.24621 −0.291184
\(803\) −1.43845 −0.0507617
\(804\) 0 0
\(805\) −0.807764 −0.0284699
\(806\) 10.2462 0.360907
\(807\) 0 0
\(808\) 3.12311 0.109870
\(809\) −30.4924 −1.07206 −0.536028 0.844200i \(-0.680076\pi\)
−0.536028 + 0.844200i \(0.680076\pi\)
\(810\) 0 0
\(811\) 37.4384 1.31464 0.657321 0.753611i \(-0.271690\pi\)
0.657321 + 0.753611i \(0.271690\pi\)
\(812\) −5.68466 −0.199492
\(813\) 0 0
\(814\) −8.80776 −0.308712
\(815\) −14.1080 −0.494180
\(816\) 0 0
\(817\) −81.1619 −2.83950
\(818\) −23.9309 −0.836723
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −55.6155 −1.94100 −0.970498 0.241111i \(-0.922488\pi\)
−0.970498 + 0.241111i \(0.922488\pi\)
\(822\) 0 0
\(823\) −28.4924 −0.993183 −0.496592 0.867984i \(-0.665415\pi\)
−0.496592 + 0.867984i \(0.665415\pi\)
\(824\) −11.6847 −0.407054
\(825\) 0 0
\(826\) 14.2462 0.495689
\(827\) 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 \(0.348929\pi\)
\(830\) 9.61553 0.333760
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −5.68466 −0.196962
\(834\) 0 0
\(835\) 3.33050 0.115257
\(836\) −19.6847 −0.680808
\(837\) 0 0
\(838\) −24.3153 −0.839960
\(839\) −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
\(842\) 20.2462 0.697731
\(843\) 0 0
\(844\) 4.80776 0.165490
\(845\) 0.561553 0.0193180
\(846\) 0 0
\(847\) 4.43845 0.152507
\(848\) 4.24621 0.145815
\(849\) 0 0
\(850\) −26.6307 −0.913425
\(851\) −4.94602 −0.169548
\(852\) 0 0
\(853\) −2.63068 −0.0900729 −0.0450364 0.998985i \(-0.514340\pi\)
−0.0450364 + 0.998985i \(0.514340\pi\)
\(854\) −5.68466 −0.194525
\(855\) 0 0
\(856\) −17.1231 −0.585256
\(857\) 1.50758 0.0514979 0.0257489 0.999668i \(-0.491803\pi\)
0.0257489 + 0.999668i \(0.491803\pi\)
\(858\) 0 0
\(859\) 22.2462 0.759031 0.379515 0.925185i \(-0.376091\pi\)
0.379515 + 0.925185i \(0.376091\pi\)
\(860\) −5.93087 −0.202241
\(861\) 0 0
\(862\) −8.63068 −0.293962
\(863\) −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
\(866\) 6.63068 0.225320
\(867\) 0 0
\(868\) 10.2462 0.347779
\(869\) 7.36932 0.249987
\(870\) 0 0
\(871\) 1.12311 0.0380550
\(872\) −11.9309 −0.404030
\(873\) 0 0
\(874\) −11.0540 −0.373906
\(875\) 5.43845 0.183853
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) −21.9309 −0.740131
\(879\) 0 0
\(880\) −1.43845 −0.0484900
\(881\) −26.1771 −0.881928 −0.440964 0.897525i \(-0.645363\pi\)
−0.440964 + 0.897525i \(0.645363\pi\)
\(882\) 0 0
\(883\) −2.56155 −0.0862031 −0.0431016 0.999071i \(-0.513724\pi\)
−0.0431016 + 0.999071i \(0.513724\pi\)
\(884\) −5.68466 −0.191196
\(885\) 0 0
\(886\) 19.3693 0.650725
\(887\) 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
\(890\) 5.61553 0.188233
\(891\) 0 0
\(892\) 17.6155 0.589812
\(893\) 0 0
\(894\) 0 0
\(895\) 9.26137 0.309573
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −1.68466 −0.0562178
\(899\) −58.2462 −1.94262
\(900\) 0 0
\(901\) −24.1383 −0.804162
\(902\) −18.2462 −0.607532
\(903\) 0 0
\(904\) 12.2462 0.407303
\(905\) 9.12311 0.303262
\(906\) 0 0
\(907\) −40.4924 −1.34453 −0.672264 0.740311i \(-0.734678\pi\)
−0.672264 + 0.740311i \(0.734678\pi\)
\(908\) −1.12311 −0.0372716
\(909\) 0 0
\(910\) 0.561553 0.0186153
\(911\) −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
\(914\) −2.63068 −0.0870153
\(915\) 0 0
\(916\) 11.7538 0.388356
\(917\) −5.43845 −0.179593
\(918\) 0 0
\(919\) 10.8769 0.358796 0.179398 0.983777i \(-0.442585\pi\)
0.179398 + 0.983777i \(0.442585\pi\)
\(920\) −0.807764 −0.0266312
\(921\) 0 0
\(922\) 9.05398 0.298177
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 16.1080 0.529626
\(926\) −3.68466 −0.121085
\(927\) 0 0
\(928\) −5.68466 −0.186608
\(929\) 15.6155 0.512329 0.256164 0.966633i \(-0.417541\pi\)
0.256164 + 0.966633i \(0.417541\pi\)
\(930\) 0 0
\(931\) 7.68466 0.251855
\(932\) −2.63068 −0.0861709
\(933\) 0 0
\(934\) −15.6847 −0.513218
\(935\) 8.17708 0.267419
\(936\) 0 0
\(937\) 18.9848 0.620208 0.310104 0.950703i \(-0.399636\pi\)
0.310104 + 0.950703i \(0.399636\pi\)
\(938\) 1.12311 0.0366707
\(939\) 0 0
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) −10.2462 −0.333663
\(944\) 14.2462 0.463675
\(945\) 0 0
\(946\) −27.0540 −0.879601
\(947\) −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
\(950\) 36.0000 1.16799
\(951\) 0 0
\(952\) −5.68466 −0.184241
\(953\) 13.3693 0.433075 0.216537 0.976274i \(-0.430524\pi\)
0.216537 + 0.976274i \(0.430524\pi\)
\(954\) 0 0
\(955\) 7.82292 0.253144
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −21.3002 −0.688178
\(959\) 0.561553 0.0181335
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 3.43845 0.110860
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) −3.01515 −0.0970613
\(966\) 0 0
\(967\) −17.4384 −0.560783 −0.280391 0.959886i \(-0.590464\pi\)
−0.280391 + 0.959886i \(0.590464\pi\)
\(968\) 4.43845 0.142657
\(969\) 0 0
\(970\) 10.3845 0.333425
\(971\) 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
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −5.68466 −0.181961
\(977\) 10.3153 0.330017 0.165009 0.986292i \(-0.447235\pi\)
0.165009 + 0.986292i \(0.447235\pi\)
\(978\) 0 0
\(979\) 25.6155 0.818676
\(980\) 0.561553 0.0179381
\(981\) 0 0
\(982\) −17.1231 −0.546420
\(983\) −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
\(986\) 32.3153 1.02913
\(987\) 0 0
\(988\) 7.68466 0.244482
\(989\) −15.1922 −0.483085
\(990\) 0 0
\(991\) 33.6155 1.06783 0.533916 0.845537i \(-0.320720\pi\)
0.533916 + 0.845537i \(0.320720\pi\)
\(992\) 10.2462 0.325318
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −12.3153 −0.390423
\(996\) 0 0
\(997\) −8.73863 −0.276755 −0.138378 0.990380i \(-0.544189\pi\)
−0.138378 + 0.990380i \(0.544189\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 1638.2.a.u.1.2 2
3.2 odd 2 546.2.a.j.1.1 2
12.11 even 2 4368.2.a.be.1.1 2
21.20 even 2 3822.2.a.bo.1.2 2
39.38 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 3.2 odd 2
1638.2.a.u.1.2 2 1.1 even 1 trivial
3822.2.a.bo.1.2 2 21.20 even 2
4368.2.a.be.1.1 2 12.11 even 2
7098.2.a.bl.1.2 2 39.38 odd 2