Properties

Label 2898.2.a.ba.1.2
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
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 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.56155 q^{10} -5.12311 q^{11} -3.56155 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.12311 q^{17} -5.12311 q^{19} +1.56155 q^{20} -5.12311 q^{22} -1.00000 q^{23} -2.56155 q^{25} -3.56155 q^{26} -1.00000 q^{28} -7.56155 q^{29} +3.12311 q^{31} +1.00000 q^{32} +1.12311 q^{34} -1.56155 q^{35} -1.56155 q^{37} -5.12311 q^{38} +1.56155 q^{40} -3.56155 q^{41} +6.68466 q^{43} -5.12311 q^{44} -1.00000 q^{46} -2.43845 q^{47} +1.00000 q^{49} -2.56155 q^{50} -3.56155 q^{52} -14.2462 q^{53} -8.00000 q^{55} -1.00000 q^{56} -7.56155 q^{58} -4.87689 q^{59} +0.876894 q^{61} +3.12311 q^{62} +1.00000 q^{64} -5.56155 q^{65} -1.12311 q^{67} +1.12311 q^{68} -1.56155 q^{70} +9.36932 q^{71} +9.12311 q^{73} -1.56155 q^{74} -5.12311 q^{76} +5.12311 q^{77} +14.2462 q^{79} +1.56155 q^{80} -3.56155 q^{82} +9.12311 q^{83} +1.75379 q^{85} +6.68466 q^{86} -5.12311 q^{88} -14.0000 q^{89} +3.56155 q^{91} -1.00000 q^{92} -2.43845 q^{94} -8.00000 q^{95} -12.4384 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} - q^{10} - 2q^{11} - 3q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 2q^{19} - q^{20} - 2q^{22} - 2q^{23} - q^{25} - 3q^{26} - 2q^{28} - 11q^{29} - 2q^{31} + 2q^{32} - 6q^{34} + q^{35} + q^{37} - 2q^{38} - q^{40} - 3q^{41} + q^{43} - 2q^{44} - 2q^{46} - 9q^{47} + 2q^{49} - q^{50} - 3q^{52} - 12q^{53} - 16q^{55} - 2q^{56} - 11q^{58} - 18q^{59} + 10q^{61} - 2q^{62} + 2q^{64} - 7q^{65} + 6q^{67} - 6q^{68} + q^{70} - 6q^{71} + 10q^{73} + q^{74} - 2q^{76} + 2q^{77} + 12q^{79} - q^{80} - 3q^{82} + 10q^{83} + 20q^{85} + q^{86} - 2q^{88} - 28q^{89} + 3q^{91} - 2q^{92} - 9q^{94} - 16q^{95} - 29q^{97} + 2q^{98} + O(q^{100}) \)

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
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.56155 0.493806
\(11\) −5.12311 −1.54467 −0.772337 0.635213i \(-0.780912\pi\)
−0.772337 + 0.635213i \(0.780912\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.12311 0.272393 0.136197 0.990682i \(-0.456512\pi\)
0.136197 + 0.990682i \(0.456512\pi\)
\(18\) 0 0
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 1.56155 0.349174
\(21\) 0 0
\(22\) −5.12311 −1.09225
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) −3.56155 −0.698478
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −7.56155 −1.40415 −0.702073 0.712105i \(-0.747742\pi\)
−0.702073 + 0.712105i \(0.747742\pi\)
\(30\) 0 0
\(31\) 3.12311 0.560926 0.280463 0.959865i \(-0.409512\pi\)
0.280463 + 0.959865i \(0.409512\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.12311 0.192611
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) −1.56155 −0.256718 −0.128359 0.991728i \(-0.540971\pi\)
−0.128359 + 0.991728i \(0.540971\pi\)
\(38\) −5.12311 −0.831077
\(39\) 0 0
\(40\) 1.56155 0.246903
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) 6.68466 1.01940 0.509700 0.860352i \(-0.329756\pi\)
0.509700 + 0.860352i \(0.329756\pi\)
\(44\) −5.12311 −0.772337
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −2.43845 −0.355684 −0.177842 0.984059i \(-0.556912\pi\)
−0.177842 + 0.984059i \(0.556912\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.56155 −0.362258
\(51\) 0 0
\(52\) −3.56155 −0.493899
\(53\) −14.2462 −1.95687 −0.978434 0.206561i \(-0.933773\pi\)
−0.978434 + 0.206561i \(0.933773\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −7.56155 −0.992881
\(59\) −4.87689 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 3.12311 0.396635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.56155 −0.689826
\(66\) 0 0
\(67\) −1.12311 −0.137209 −0.0686046 0.997644i \(-0.521855\pi\)
−0.0686046 + 0.997644i \(0.521855\pi\)
\(68\) 1.12311 0.136197
\(69\) 0 0
\(70\) −1.56155 −0.186641
\(71\) 9.36932 1.11193 0.555967 0.831205i \(-0.312348\pi\)
0.555967 + 0.831205i \(0.312348\pi\)
\(72\) 0 0
\(73\) 9.12311 1.06778 0.533889 0.845554i \(-0.320730\pi\)
0.533889 + 0.845554i \(0.320730\pi\)
\(74\) −1.56155 −0.181527
\(75\) 0 0
\(76\) −5.12311 −0.587661
\(77\) 5.12311 0.583832
\(78\) 0 0
\(79\) 14.2462 1.60282 0.801412 0.598113i \(-0.204082\pi\)
0.801412 + 0.598113i \(0.204082\pi\)
\(80\) 1.56155 0.174587
\(81\) 0 0
\(82\) −3.56155 −0.393308
\(83\) 9.12311 1.00139 0.500695 0.865624i \(-0.333078\pi\)
0.500695 + 0.865624i \(0.333078\pi\)
\(84\) 0 0
\(85\) 1.75379 0.190225
\(86\) 6.68466 0.720825
\(87\) 0 0
\(88\) −5.12311 −0.546125
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 3.56155 0.373352
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −2.43845 −0.251507
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −12.4384 −1.26293 −0.631466 0.775403i \(-0.717547\pi\)
−0.631466 + 0.775403i \(0.717547\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −2.56155 −0.256155
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) 18.9309 1.86531 0.932657 0.360764i \(-0.117484\pi\)
0.932657 + 0.360764i \(0.117484\pi\)
\(104\) −3.56155 −0.349239
\(105\) 0 0
\(106\) −14.2462 −1.38371
\(107\) 5.12311 0.495269 0.247635 0.968853i \(-0.420347\pi\)
0.247635 + 0.968853i \(0.420347\pi\)
\(108\) 0 0
\(109\) 10.4384 0.999822 0.499911 0.866077i \(-0.333366\pi\)
0.499911 + 0.866077i \(0.333366\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 2.68466 0.252551 0.126276 0.991995i \(-0.459698\pi\)
0.126276 + 0.991995i \(0.459698\pi\)
\(114\) 0 0
\(115\) −1.56155 −0.145616
\(116\) −7.56155 −0.702073
\(117\) 0 0
\(118\) −4.87689 −0.448955
\(119\) −1.12311 −0.102955
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0.876894 0.0793903
\(123\) 0 0
\(124\) 3.12311 0.280463
\(125\) −11.8078 −1.05612
\(126\) 0 0
\(127\) −16.6847 −1.48052 −0.740262 0.672318i \(-0.765299\pi\)
−0.740262 + 0.672318i \(0.765299\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −5.56155 −0.487780
\(131\) −4.87689 −0.426096 −0.213048 0.977042i \(-0.568339\pi\)
−0.213048 + 0.977042i \(0.568339\pi\)
\(132\) 0 0
\(133\) 5.12311 0.444230
\(134\) −1.12311 −0.0970215
\(135\) 0 0
\(136\) 1.12311 0.0963055
\(137\) 0.438447 0.0374591 0.0187295 0.999825i \(-0.494038\pi\)
0.0187295 + 0.999825i \(0.494038\pi\)
\(138\) 0 0
\(139\) 3.80776 0.322970 0.161485 0.986875i \(-0.448372\pi\)
0.161485 + 0.986875i \(0.448372\pi\)
\(140\) −1.56155 −0.131975
\(141\) 0 0
\(142\) 9.36932 0.786256
\(143\) 18.2462 1.52582
\(144\) 0 0
\(145\) −11.8078 −0.980581
\(146\) 9.12311 0.755034
\(147\) 0 0
\(148\) −1.56155 −0.128359
\(149\) 6.24621 0.511710 0.255855 0.966715i \(-0.417643\pi\)
0.255855 + 0.966715i \(0.417643\pi\)
\(150\) 0 0
\(151\) 7.31534 0.595314 0.297657 0.954673i \(-0.403795\pi\)
0.297657 + 0.954673i \(0.403795\pi\)
\(152\) −5.12311 −0.415539
\(153\) 0 0
\(154\) 5.12311 0.412832
\(155\) 4.87689 0.391722
\(156\) 0 0
\(157\) 14.2462 1.13697 0.568486 0.822693i \(-0.307529\pi\)
0.568486 + 0.822693i \(0.307529\pi\)
\(158\) 14.2462 1.13337
\(159\) 0 0
\(160\) 1.56155 0.123452
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −3.56155 −0.278111
\(165\) 0 0
\(166\) 9.12311 0.708090
\(167\) 14.2462 1.10240 0.551202 0.834372i \(-0.314169\pi\)
0.551202 + 0.834372i \(0.314169\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 1.75379 0.134509
\(171\) 0 0
\(172\) 6.68466 0.509700
\(173\) −18.4924 −1.40595 −0.702976 0.711213i \(-0.748146\pi\)
−0.702976 + 0.711213i \(0.748146\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) −5.12311 −0.386169
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 3.80776 0.284606 0.142303 0.989823i \(-0.454549\pi\)
0.142303 + 0.989823i \(0.454549\pi\)
\(180\) 0 0
\(181\) −15.1231 −1.12409 −0.562046 0.827106i \(-0.689986\pi\)
−0.562046 + 0.827106i \(0.689986\pi\)
\(182\) 3.56155 0.264000
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −2.43845 −0.179278
\(186\) 0 0
\(187\) −5.75379 −0.420759
\(188\) −2.43845 −0.177842
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 2.68466 0.193246 0.0966230 0.995321i \(-0.469196\pi\)
0.0966230 + 0.995321i \(0.469196\pi\)
\(194\) −12.4384 −0.893028
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −24.9309 −1.77625 −0.888125 0.459601i \(-0.847992\pi\)
−0.888125 + 0.459601i \(0.847992\pi\)
\(198\) 0 0
\(199\) −21.1771 −1.50120 −0.750602 0.660755i \(-0.770236\pi\)
−0.750602 + 0.660755i \(0.770236\pi\)
\(200\) −2.56155 −0.181129
\(201\) 0 0
\(202\) −16.2462 −1.14308
\(203\) 7.56155 0.530717
\(204\) 0 0
\(205\) −5.56155 −0.388436
\(206\) 18.9309 1.31898
\(207\) 0 0
\(208\) −3.56155 −0.246949
\(209\) 26.2462 1.81549
\(210\) 0 0
\(211\) −16.4924 −1.13539 −0.567693 0.823241i \(-0.692164\pi\)
−0.567693 + 0.823241i \(0.692164\pi\)
\(212\) −14.2462 −0.978434
\(213\) 0 0
\(214\) 5.12311 0.350208
\(215\) 10.4384 0.711896
\(216\) 0 0
\(217\) −3.12311 −0.212010
\(218\) 10.4384 0.706981
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 3.12311 0.209139 0.104569 0.994518i \(-0.466654\pi\)
0.104569 + 0.994518i \(0.466654\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.68466 0.178581
\(227\) 6.19224 0.410993 0.205497 0.978658i \(-0.434119\pi\)
0.205497 + 0.978658i \(0.434119\pi\)
\(228\) 0 0
\(229\) 2.24621 0.148434 0.0742169 0.997242i \(-0.476354\pi\)
0.0742169 + 0.997242i \(0.476354\pi\)
\(230\) −1.56155 −0.102966
\(231\) 0 0
\(232\) −7.56155 −0.496440
\(233\) 18.4924 1.21148 0.605739 0.795663i \(-0.292877\pi\)
0.605739 + 0.795663i \(0.292877\pi\)
\(234\) 0 0
\(235\) −3.80776 −0.248391
\(236\) −4.87689 −0.317459
\(237\) 0 0
\(238\) −1.12311 −0.0728001
\(239\) −4.49242 −0.290591 −0.145295 0.989388i \(-0.546413\pi\)
−0.145295 + 0.989388i \(0.546413\pi\)
\(240\) 0 0
\(241\) −20.0540 −1.29179 −0.645895 0.763426i \(-0.723516\pi\)
−0.645895 + 0.763426i \(0.723516\pi\)
\(242\) 15.2462 0.980064
\(243\) 0 0
\(244\) 0.876894 0.0561374
\(245\) 1.56155 0.0997639
\(246\) 0 0
\(247\) 18.2462 1.16098
\(248\) 3.12311 0.198317
\(249\) 0 0
\(250\) −11.8078 −0.746789
\(251\) 0.438447 0.0276745 0.0138373 0.999904i \(-0.495595\pi\)
0.0138373 + 0.999904i \(0.495595\pi\)
\(252\) 0 0
\(253\) 5.12311 0.322087
\(254\) −16.6847 −1.04689
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 1.56155 0.0970302
\(260\) −5.56155 −0.344913
\(261\) 0 0
\(262\) −4.87689 −0.301296
\(263\) −22.9309 −1.41398 −0.706989 0.707225i \(-0.749947\pi\)
−0.706989 + 0.707225i \(0.749947\pi\)
\(264\) 0 0
\(265\) −22.2462 −1.36657
\(266\) 5.12311 0.314118
\(267\) 0 0
\(268\) −1.12311 −0.0686046
\(269\) −12.2462 −0.746665 −0.373332 0.927698i \(-0.621785\pi\)
−0.373332 + 0.927698i \(0.621785\pi\)
\(270\) 0 0
\(271\) 14.2462 0.865396 0.432698 0.901539i \(-0.357562\pi\)
0.432698 + 0.901539i \(0.357562\pi\)
\(272\) 1.12311 0.0680983
\(273\) 0 0
\(274\) 0.438447 0.0264876
\(275\) 13.1231 0.791353
\(276\) 0 0
\(277\) 13.1231 0.788491 0.394245 0.919005i \(-0.371006\pi\)
0.394245 + 0.919005i \(0.371006\pi\)
\(278\) 3.80776 0.228375
\(279\) 0 0
\(280\) −1.56155 −0.0933206
\(281\) −4.93087 −0.294151 −0.147076 0.989125i \(-0.546986\pi\)
−0.147076 + 0.989125i \(0.546986\pi\)
\(282\) 0 0
\(283\) −10.4924 −0.623710 −0.311855 0.950130i \(-0.600950\pi\)
−0.311855 + 0.950130i \(0.600950\pi\)
\(284\) 9.36932 0.555967
\(285\) 0 0
\(286\) 18.2462 1.07892
\(287\) 3.56155 0.210232
\(288\) 0 0
\(289\) −15.7386 −0.925802
\(290\) −11.8078 −0.693376
\(291\) 0 0
\(292\) 9.12311 0.533889
\(293\) −27.6155 −1.61332 −0.806658 0.591018i \(-0.798726\pi\)
−0.806658 + 0.591018i \(0.798726\pi\)
\(294\) 0 0
\(295\) −7.61553 −0.443393
\(296\) −1.56155 −0.0907634
\(297\) 0 0
\(298\) 6.24621 0.361833
\(299\) 3.56155 0.205970
\(300\) 0 0
\(301\) −6.68466 −0.385297
\(302\) 7.31534 0.420951
\(303\) 0 0
\(304\) −5.12311 −0.293830
\(305\) 1.36932 0.0784069
\(306\) 0 0
\(307\) 2.93087 0.167274 0.0836368 0.996496i \(-0.473346\pi\)
0.0836368 + 0.996496i \(0.473346\pi\)
\(308\) 5.12311 0.291916
\(309\) 0 0
\(310\) 4.87689 0.276989
\(311\) −18.7386 −1.06257 −0.531285 0.847193i \(-0.678291\pi\)
−0.531285 + 0.847193i \(0.678291\pi\)
\(312\) 0 0
\(313\) 4.24621 0.240010 0.120005 0.992773i \(-0.461709\pi\)
0.120005 + 0.992773i \(0.461709\pi\)
\(314\) 14.2462 0.803960
\(315\) 0 0
\(316\) 14.2462 0.801412
\(317\) 17.8078 1.00018 0.500092 0.865972i \(-0.333300\pi\)
0.500092 + 0.865972i \(0.333300\pi\)
\(318\) 0 0
\(319\) 38.7386 2.16895
\(320\) 1.56155 0.0872935
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) −5.75379 −0.320149
\(324\) 0 0
\(325\) 9.12311 0.506059
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) −3.56155 −0.196654
\(329\) 2.43845 0.134436
\(330\) 0 0
\(331\) −24.4924 −1.34623 −0.673113 0.739540i \(-0.735043\pi\)
−0.673113 + 0.739540i \(0.735043\pi\)
\(332\) 9.12311 0.500695
\(333\) 0 0
\(334\) 14.2462 0.779518
\(335\) −1.75379 −0.0958197
\(336\) 0 0
\(337\) 27.3693 1.49090 0.745451 0.666561i \(-0.232234\pi\)
0.745451 + 0.666561i \(0.232234\pi\)
\(338\) −0.315342 −0.0171523
\(339\) 0 0
\(340\) 1.75379 0.0951125
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.68466 0.360413
\(345\) 0 0
\(346\) −18.4924 −0.994159
\(347\) −14.9309 −0.801531 −0.400766 0.916181i \(-0.631256\pi\)
−0.400766 + 0.916181i \(0.631256\pi\)
\(348\) 0 0
\(349\) −16.2462 −0.869640 −0.434820 0.900517i \(-0.643188\pi\)
−0.434820 + 0.900517i \(0.643188\pi\)
\(350\) 2.56155 0.136921
\(351\) 0 0
\(352\) −5.12311 −0.273062
\(353\) 33.8078 1.79941 0.899703 0.436503i \(-0.143783\pi\)
0.899703 + 0.436503i \(0.143783\pi\)
\(354\) 0 0
\(355\) 14.6307 0.776516
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 3.80776 0.201247
\(359\) 6.43845 0.339808 0.169904 0.985461i \(-0.445654\pi\)
0.169904 + 0.985461i \(0.445654\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) −15.1231 −0.794853
\(363\) 0 0
\(364\) 3.56155 0.186676
\(365\) 14.2462 0.745681
\(366\) 0 0
\(367\) −18.4384 −0.962479 −0.481240 0.876589i \(-0.659813\pi\)
−0.481240 + 0.876589i \(0.659813\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −2.43845 −0.126769
\(371\) 14.2462 0.739626
\(372\) 0 0
\(373\) 15.1231 0.783045 0.391522 0.920169i \(-0.371949\pi\)
0.391522 + 0.920169i \(0.371949\pi\)
\(374\) −5.75379 −0.297521
\(375\) 0 0
\(376\) −2.43845 −0.125753
\(377\) 26.9309 1.38701
\(378\) 0 0
\(379\) 20.0540 1.03010 0.515052 0.857159i \(-0.327773\pi\)
0.515052 + 0.857159i \(0.327773\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 2.68466 0.136646
\(387\) 0 0
\(388\) −12.4384 −0.631466
\(389\) 29.3693 1.48908 0.744542 0.667576i \(-0.232668\pi\)
0.744542 + 0.667576i \(0.232668\pi\)
\(390\) 0 0
\(391\) −1.12311 −0.0567979
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −24.9309 −1.25600
\(395\) 22.2462 1.11933
\(396\) 0 0
\(397\) −7.75379 −0.389152 −0.194576 0.980887i \(-0.562333\pi\)
−0.194576 + 0.980887i \(0.562333\pi\)
\(398\) −21.1771 −1.06151
\(399\) 0 0
\(400\) −2.56155 −0.128078
\(401\) −16.2462 −0.811297 −0.405649 0.914029i \(-0.632954\pi\)
−0.405649 + 0.914029i \(0.632954\pi\)
\(402\) 0 0
\(403\) −11.1231 −0.554081
\(404\) −16.2462 −0.808279
\(405\) 0 0
\(406\) 7.56155 0.375274
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 33.6155 1.66218 0.831090 0.556137i \(-0.187717\pi\)
0.831090 + 0.556137i \(0.187717\pi\)
\(410\) −5.56155 −0.274666
\(411\) 0 0
\(412\) 18.9309 0.932657
\(413\) 4.87689 0.239976
\(414\) 0 0
\(415\) 14.2462 0.699319
\(416\) −3.56155 −0.174619
\(417\) 0 0
\(418\) 26.2462 1.28374
\(419\) −23.8617 −1.16572 −0.582861 0.812572i \(-0.698067\pi\)
−0.582861 + 0.812572i \(0.698067\pi\)
\(420\) 0 0
\(421\) 33.1771 1.61695 0.808476 0.588529i \(-0.200293\pi\)
0.808476 + 0.588529i \(0.200293\pi\)
\(422\) −16.4924 −0.802839
\(423\) 0 0
\(424\) −14.2462 −0.691857
\(425\) −2.87689 −0.139550
\(426\) 0 0
\(427\) −0.876894 −0.0424359
\(428\) 5.12311 0.247635
\(429\) 0 0
\(430\) 10.4384 0.503387
\(431\) 3.31534 0.159694 0.0798472 0.996807i \(-0.474557\pi\)
0.0798472 + 0.996807i \(0.474557\pi\)
\(432\) 0 0
\(433\) 14.6847 0.705700 0.352850 0.935680i \(-0.385213\pi\)
0.352850 + 0.935680i \(0.385213\pi\)
\(434\) −3.12311 −0.149914
\(435\) 0 0
\(436\) 10.4384 0.499911
\(437\) 5.12311 0.245071
\(438\) 0 0
\(439\) −23.6155 −1.12711 −0.563554 0.826079i \(-0.690566\pi\)
−0.563554 + 0.826079i \(0.690566\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −9.06913 −0.430887 −0.215444 0.976516i \(-0.569120\pi\)
−0.215444 + 0.976516i \(0.569120\pi\)
\(444\) 0 0
\(445\) −21.8617 −1.03635
\(446\) 3.12311 0.147883
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 0.246211 0.0116194 0.00580971 0.999983i \(-0.498151\pi\)
0.00580971 + 0.999983i \(0.498151\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) 2.68466 0.126276
\(453\) 0 0
\(454\) 6.19224 0.290616
\(455\) 5.56155 0.260730
\(456\) 0 0
\(457\) 7.36932 0.344722 0.172361 0.985034i \(-0.444860\pi\)
0.172361 + 0.985034i \(0.444860\pi\)
\(458\) 2.24621 0.104959
\(459\) 0 0
\(460\) −1.56155 −0.0728078
\(461\) −40.7386 −1.89739 −0.948694 0.316197i \(-0.897594\pi\)
−0.948694 + 0.316197i \(0.897594\pi\)
\(462\) 0 0
\(463\) −13.5616 −0.630259 −0.315129 0.949049i \(-0.602048\pi\)
−0.315129 + 0.949049i \(0.602048\pi\)
\(464\) −7.56155 −0.351036
\(465\) 0 0
\(466\) 18.4924 0.856645
\(467\) −2.68466 −0.124231 −0.0621156 0.998069i \(-0.519785\pi\)
−0.0621156 + 0.998069i \(0.519785\pi\)
\(468\) 0 0
\(469\) 1.12311 0.0518602
\(470\) −3.80776 −0.175639
\(471\) 0 0
\(472\) −4.87689 −0.224477
\(473\) −34.2462 −1.57464
\(474\) 0 0
\(475\) 13.1231 0.602129
\(476\) −1.12311 −0.0514775
\(477\) 0 0
\(478\) −4.49242 −0.205479
\(479\) −31.6155 −1.44455 −0.722275 0.691606i \(-0.756904\pi\)
−0.722275 + 0.691606i \(0.756904\pi\)
\(480\) 0 0
\(481\) 5.56155 0.253585
\(482\) −20.0540 −0.913434
\(483\) 0 0
\(484\) 15.2462 0.693010
\(485\) −19.4233 −0.881966
\(486\) 0 0
\(487\) −5.56155 −0.252018 −0.126009 0.992029i \(-0.540217\pi\)
−0.126009 + 0.992029i \(0.540217\pi\)
\(488\) 0.876894 0.0396951
\(489\) 0 0
\(490\) 1.56155 0.0705438
\(491\) 26.7386 1.20670 0.603349 0.797477i \(-0.293833\pi\)
0.603349 + 0.797477i \(0.293833\pi\)
\(492\) 0 0
\(493\) −8.49242 −0.382479
\(494\) 18.2462 0.820936
\(495\) 0 0
\(496\) 3.12311 0.140232
\(497\) −9.36932 −0.420271
\(498\) 0 0
\(499\) 41.3693 1.85194 0.925972 0.377591i \(-0.123247\pi\)
0.925972 + 0.377591i \(0.123247\pi\)
\(500\) −11.8078 −0.528059
\(501\) 0 0
\(502\) 0.438447 0.0195689
\(503\) 32.1080 1.43162 0.715811 0.698294i \(-0.246057\pi\)
0.715811 + 0.698294i \(0.246057\pi\)
\(504\) 0 0
\(505\) −25.3693 −1.12892
\(506\) 5.12311 0.227750
\(507\) 0 0
\(508\) −16.6847 −0.740262
\(509\) 3.36932 0.149342 0.0746712 0.997208i \(-0.476209\pi\)
0.0746712 + 0.997208i \(0.476209\pi\)
\(510\) 0 0
\(511\) −9.12311 −0.403582
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 29.5616 1.30264
\(516\) 0 0
\(517\) 12.4924 0.549416
\(518\) 1.56155 0.0686107
\(519\) 0 0
\(520\) −5.56155 −0.243890
\(521\) −0.246211 −0.0107867 −0.00539336 0.999985i \(-0.501717\pi\)
−0.00539336 + 0.999985i \(0.501717\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −4.87689 −0.213048
\(525\) 0 0
\(526\) −22.9309 −0.999833
\(527\) 3.50758 0.152792
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −22.2462 −0.966314
\(531\) 0 0
\(532\) 5.12311 0.222115
\(533\) 12.6847 0.549434
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −1.12311 −0.0485108
\(537\) 0 0
\(538\) −12.2462 −0.527972
\(539\) −5.12311 −0.220668
\(540\) 0 0
\(541\) −15.3693 −0.660779 −0.330389 0.943845i \(-0.607180\pi\)
−0.330389 + 0.943845i \(0.607180\pi\)
\(542\) 14.2462 0.611927
\(543\) 0 0
\(544\) 1.12311 0.0481528
\(545\) 16.3002 0.698223
\(546\) 0 0
\(547\) −3.61553 −0.154589 −0.0772944 0.997008i \(-0.524628\pi\)
−0.0772944 + 0.997008i \(0.524628\pi\)
\(548\) 0.438447 0.0187295
\(549\) 0 0
\(550\) 13.1231 0.559571
\(551\) 38.7386 1.65032
\(552\) 0 0
\(553\) −14.2462 −0.605811
\(554\) 13.1231 0.557547
\(555\) 0 0
\(556\) 3.80776 0.161485
\(557\) 28.9848 1.22813 0.614064 0.789257i \(-0.289534\pi\)
0.614064 + 0.789257i \(0.289534\pi\)
\(558\) 0 0
\(559\) −23.8078 −1.00696
\(560\) −1.56155 −0.0659877
\(561\) 0 0
\(562\) −4.93087 −0.207996
\(563\) −19.5616 −0.824421 −0.412211 0.911089i \(-0.635243\pi\)
−0.412211 + 0.911089i \(0.635243\pi\)
\(564\) 0 0
\(565\) 4.19224 0.176369
\(566\) −10.4924 −0.441029
\(567\) 0 0
\(568\) 9.36932 0.393128
\(569\) 44.5464 1.86748 0.933741 0.357949i \(-0.116524\pi\)
0.933741 + 0.357949i \(0.116524\pi\)
\(570\) 0 0
\(571\) 43.3693 1.81495 0.907475 0.420107i \(-0.138007\pi\)
0.907475 + 0.420107i \(0.138007\pi\)
\(572\) 18.2462 0.762912
\(573\) 0 0
\(574\) 3.56155 0.148656
\(575\) 2.56155 0.106824
\(576\) 0 0
\(577\) −19.7538 −0.822361 −0.411180 0.911554i \(-0.634883\pi\)
−0.411180 + 0.911554i \(0.634883\pi\)
\(578\) −15.7386 −0.654641
\(579\) 0 0
\(580\) −11.8078 −0.490291
\(581\) −9.12311 −0.378490
\(582\) 0 0
\(583\) 72.9848 3.02272
\(584\) 9.12311 0.377517
\(585\) 0 0
\(586\) −27.6155 −1.14079
\(587\) 24.9848 1.03123 0.515617 0.856819i \(-0.327563\pi\)
0.515617 + 0.856819i \(0.327563\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) −7.61553 −0.313526
\(591\) 0 0
\(592\) −1.56155 −0.0641794
\(593\) −30.3002 −1.24428 −0.622140 0.782906i \(-0.713736\pi\)
−0.622140 + 0.782906i \(0.713736\pi\)
\(594\) 0 0
\(595\) −1.75379 −0.0718983
\(596\) 6.24621 0.255855
\(597\) 0 0
\(598\) 3.56155 0.145643
\(599\) 10.7386 0.438769 0.219384 0.975639i \(-0.429595\pi\)
0.219384 + 0.975639i \(0.429595\pi\)
\(600\) 0 0
\(601\) −3.36932 −0.137437 −0.0687187 0.997636i \(-0.521891\pi\)
−0.0687187 + 0.997636i \(0.521891\pi\)
\(602\) −6.68466 −0.272446
\(603\) 0 0
\(604\) 7.31534 0.297657
\(605\) 23.8078 0.967923
\(606\) 0 0
\(607\) −47.6155 −1.93265 −0.966327 0.257316i \(-0.917162\pi\)
−0.966327 + 0.257316i \(0.917162\pi\)
\(608\) −5.12311 −0.207769
\(609\) 0 0
\(610\) 1.36932 0.0554420
\(611\) 8.68466 0.351344
\(612\) 0 0
\(613\) 3.80776 0.153794 0.0768971 0.997039i \(-0.475499\pi\)
0.0768971 + 0.997039i \(0.475499\pi\)
\(614\) 2.93087 0.118280
\(615\) 0 0
\(616\) 5.12311 0.206416
\(617\) −31.7538 −1.27836 −0.639180 0.769057i \(-0.720726\pi\)
−0.639180 + 0.769057i \(0.720726\pi\)
\(618\) 0 0
\(619\) 8.24621 0.331443 0.165722 0.986173i \(-0.447005\pi\)
0.165722 + 0.986173i \(0.447005\pi\)
\(620\) 4.87689 0.195861
\(621\) 0 0
\(622\) −18.7386 −0.751351
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 4.24621 0.169713
\(627\) 0 0
\(628\) 14.2462 0.568486
\(629\) −1.75379 −0.0699281
\(630\) 0 0
\(631\) 8.49242 0.338078 0.169039 0.985609i \(-0.445934\pi\)
0.169039 + 0.985609i \(0.445934\pi\)
\(632\) 14.2462 0.566684
\(633\) 0 0
\(634\) 17.8078 0.707237
\(635\) −26.0540 −1.03392
\(636\) 0 0
\(637\) −3.56155 −0.141114
\(638\) 38.7386 1.53368
\(639\) 0 0
\(640\) 1.56155 0.0617258
\(641\) 13.3153 0.525924 0.262962 0.964806i \(-0.415301\pi\)
0.262962 + 0.964806i \(0.415301\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) −5.75379 −0.226380
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) 0 0
\(649\) 24.9848 0.980741
\(650\) 9.12311 0.357838
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −36.5464 −1.43017 −0.715086 0.699037i \(-0.753612\pi\)
−0.715086 + 0.699037i \(0.753612\pi\)
\(654\) 0 0
\(655\) −7.61553 −0.297563
\(656\) −3.56155 −0.139055
\(657\) 0 0
\(658\) 2.43845 0.0950606
\(659\) −5.61553 −0.218750 −0.109375 0.994001i \(-0.534885\pi\)
−0.109375 + 0.994001i \(0.534885\pi\)
\(660\) 0 0
\(661\) 32.8769 1.27876 0.639381 0.768890i \(-0.279190\pi\)
0.639381 + 0.768890i \(0.279190\pi\)
\(662\) −24.4924 −0.951925
\(663\) 0 0
\(664\) 9.12311 0.354045
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 7.56155 0.292784
\(668\) 14.2462 0.551202
\(669\) 0 0
\(670\) −1.75379 −0.0677548
\(671\) −4.49242 −0.173428
\(672\) 0 0
\(673\) 22.6847 0.874429 0.437215 0.899357i \(-0.355965\pi\)
0.437215 + 0.899357i \(0.355965\pi\)
\(674\) 27.3693 1.05423
\(675\) 0 0
\(676\) −0.315342 −0.0121285
\(677\) −2.63068 −0.101105 −0.0505527 0.998721i \(-0.516098\pi\)
−0.0505527 + 0.998721i \(0.516098\pi\)
\(678\) 0 0
\(679\) 12.4384 0.477344
\(680\) 1.75379 0.0672547
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 7.50758 0.287269 0.143635 0.989631i \(-0.454121\pi\)
0.143635 + 0.989631i \(0.454121\pi\)
\(684\) 0 0
\(685\) 0.684658 0.0261595
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 6.68466 0.254850
\(689\) 50.7386 1.93299
\(690\) 0 0
\(691\) −50.0540 −1.90414 −0.952071 0.305876i \(-0.901051\pi\)
−0.952071 + 0.305876i \(0.901051\pi\)
\(692\) −18.4924 −0.702976
\(693\) 0 0
\(694\) −14.9309 −0.566768
\(695\) 5.94602 0.225546
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) −16.2462 −0.614928
\(699\) 0 0
\(700\) 2.56155 0.0968176
\(701\) −34.2462 −1.29346 −0.646731 0.762718i \(-0.723864\pi\)
−0.646731 + 0.762718i \(0.723864\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) −5.12311 −0.193084
\(705\) 0 0
\(706\) 33.8078 1.27237
\(707\) 16.2462 0.611002
\(708\) 0 0
\(709\) 10.6307 0.399244 0.199622 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(710\) 14.6307 0.549080
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) −3.12311 −0.116961
\(714\) 0 0
\(715\) 28.4924 1.06556
\(716\) 3.80776 0.142303
\(717\) 0 0
\(718\) 6.43845 0.240281
\(719\) −10.4384 −0.389288 −0.194644 0.980874i \(-0.562355\pi\)
−0.194644 + 0.980874i \(0.562355\pi\)
\(720\) 0 0
\(721\) −18.9309 −0.705022
\(722\) 7.24621 0.269676
\(723\) 0 0
\(724\) −15.1231 −0.562046
\(725\) 19.3693 0.719358
\(726\) 0 0
\(727\) −1.75379 −0.0650444 −0.0325222 0.999471i \(-0.510354\pi\)
−0.0325222 + 0.999471i \(0.510354\pi\)
\(728\) 3.56155 0.132000
\(729\) 0 0
\(730\) 14.2462 0.527276
\(731\) 7.50758 0.277678
\(732\) 0 0
\(733\) 25.7538 0.951238 0.475619 0.879651i \(-0.342224\pi\)
0.475619 + 0.879651i \(0.342224\pi\)
\(734\) −18.4384 −0.680576
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 5.75379 0.211944
\(738\) 0 0
\(739\) 23.6155 0.868711 0.434356 0.900741i \(-0.356976\pi\)
0.434356 + 0.900741i \(0.356976\pi\)
\(740\) −2.43845 −0.0896391
\(741\) 0 0
\(742\) 14.2462 0.522995
\(743\) 24.9848 0.916605 0.458303 0.888796i \(-0.348458\pi\)
0.458303 + 0.888796i \(0.348458\pi\)
\(744\) 0 0
\(745\) 9.75379 0.357351
\(746\) 15.1231 0.553696
\(747\) 0 0
\(748\) −5.75379 −0.210379
\(749\) −5.12311 −0.187194
\(750\) 0 0
\(751\) 24.4924 0.893741 0.446871 0.894599i \(-0.352538\pi\)
0.446871 + 0.894599i \(0.352538\pi\)
\(752\) −2.43845 −0.0889210
\(753\) 0 0
\(754\) 26.9309 0.980764
\(755\) 11.4233 0.415736
\(756\) 0 0
\(757\) −25.3693 −0.922064 −0.461032 0.887384i \(-0.652521\pi\)
−0.461032 + 0.887384i \(0.652521\pi\)
\(758\) 20.0540 0.728393
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 3.26137 0.118224 0.0591122 0.998251i \(-0.481173\pi\)
0.0591122 + 0.998251i \(0.481173\pi\)
\(762\) 0 0
\(763\) −10.4384 −0.377897
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) 6.24621 0.225685
\(767\) 17.3693 0.627170
\(768\) 0 0
\(769\) −47.1771 −1.70125 −0.850625 0.525774i \(-0.823776\pi\)
−0.850625 + 0.525774i \(0.823776\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 2.68466 0.0966230
\(773\) 32.3002 1.16176 0.580878 0.813990i \(-0.302709\pi\)
0.580878 + 0.813990i \(0.302709\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −12.4384 −0.446514
\(777\) 0 0
\(778\) 29.3693 1.05294
\(779\) 18.2462 0.653738
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) −1.12311 −0.0401622
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 22.2462 0.794001
\(786\) 0 0
\(787\) −10.8769 −0.387719 −0.193860 0.981029i \(-0.562101\pi\)
−0.193860 + 0.981029i \(0.562101\pi\)
\(788\) −24.9309 −0.888125
\(789\) 0 0
\(790\) 22.2462 0.791485
\(791\) −2.68466 −0.0954555
\(792\) 0 0
\(793\) −3.12311 −0.110905
\(794\) −7.75379 −0.275172
\(795\) 0 0
\(796\) −21.1771 −0.750602
\(797\) −27.4233 −0.971383 −0.485691 0.874130i \(-0.661432\pi\)
−0.485691 + 0.874130i \(0.661432\pi\)
\(798\) 0 0
\(799\) −2.73863 −0.0968859
\(800\) −2.56155 −0.0905646
\(801\) 0 0
\(802\) −16.2462 −0.573674
\(803\) −46.7386 −1.64937
\(804\) 0 0
\(805\) 1.56155 0.0550375
\(806\) −11.1231 −0.391795
\(807\) 0 0
\(808\) −16.2462 −0.571540
\(809\) −23.8617 −0.838934 −0.419467 0.907771i \(-0.637783\pi\)
−0.419467 + 0.907771i \(0.637783\pi\)
\(810\) 0 0
\(811\) −24.6847 −0.866796 −0.433398 0.901203i \(-0.642685\pi\)
−0.433398 + 0.901203i \(0.642685\pi\)
\(812\) 7.56155 0.265358
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −12.4924 −0.437590
\(816\) 0 0
\(817\) −34.2462 −1.19812
\(818\) 33.6155 1.17534
\(819\) 0 0
\(820\) −5.56155 −0.194218
\(821\) 55.4773 1.93617 0.968085 0.250622i \(-0.0806352\pi\)
0.968085 + 0.250622i \(0.0806352\pi\)
\(822\) 0 0
\(823\) 23.3153 0.812722 0.406361 0.913713i \(-0.366798\pi\)
0.406361 + 0.913713i \(0.366798\pi\)
\(824\) 18.9309 0.659488
\(825\) 0 0
\(826\) 4.87689 0.169689
\(827\) −48.7386 −1.69481 −0.847404 0.530948i \(-0.821836\pi\)
−0.847404 + 0.530948i \(0.821836\pi\)
\(828\) 0 0
\(829\) 24.7386 0.859208 0.429604 0.903017i \(-0.358653\pi\)
0.429604 + 0.903017i \(0.358653\pi\)
\(830\) 14.2462 0.494493
\(831\) 0 0
\(832\) −3.56155 −0.123475
\(833\) 1.12311 0.0389133
\(834\) 0 0
\(835\) 22.2462 0.769862
\(836\) 26.2462 0.907744
\(837\) 0 0
\(838\) −23.8617 −0.824290
\(839\) −7.61553 −0.262917 −0.131459 0.991322i \(-0.541966\pi\)
−0.131459 + 0.991322i \(0.541966\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) 33.1771 1.14336
\(843\) 0 0
\(844\) −16.4924 −0.567693
\(845\) −0.492423 −0.0169398
\(846\) 0 0
\(847\) −15.2462 −0.523866
\(848\) −14.2462 −0.489217
\(849\) 0 0
\(850\) −2.87689 −0.0986767
\(851\) 1.56155 0.0535293
\(852\) 0 0
\(853\) −0.438447 −0.0150121 −0.00750607 0.999972i \(-0.502389\pi\)
−0.00750607 + 0.999972i \(0.502389\pi\)
\(854\) −0.876894 −0.0300067
\(855\) 0 0
\(856\) 5.12311 0.175104
\(857\) −1.31534 −0.0449312 −0.0224656 0.999748i \(-0.507152\pi\)
−0.0224656 + 0.999748i \(0.507152\pi\)
\(858\) 0 0
\(859\) −33.1771 −1.13199 −0.565994 0.824410i \(-0.691507\pi\)
−0.565994 + 0.824410i \(0.691507\pi\)
\(860\) 10.4384 0.355948
\(861\) 0 0
\(862\) 3.31534 0.112921
\(863\) −10.7386 −0.365547 −0.182774 0.983155i \(-0.558508\pi\)
−0.182774 + 0.983155i \(0.558508\pi\)
\(864\) 0 0
\(865\) −28.8769 −0.981844
\(866\) 14.6847 0.499005
\(867\) 0 0
\(868\) −3.12311 −0.106005
\(869\) −72.9848 −2.47584
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 10.4384 0.353490
\(873\) 0 0
\(874\) 5.12311 0.173292
\(875\) 11.8078 0.399175
\(876\) 0 0
\(877\) −35.8617 −1.21096 −0.605482 0.795859i \(-0.707020\pi\)
−0.605482 + 0.795859i \(0.707020\pi\)
\(878\) −23.6155 −0.796985
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −8.87689 −0.298731 −0.149366 0.988782i \(-0.547723\pi\)
−0.149366 + 0.988782i \(0.547723\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −9.06913 −0.304683
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 16.6847 0.559585
\(890\) −21.8617 −0.732807
\(891\) 0 0
\(892\) 3.12311 0.104569
\(893\) 12.4924 0.418043
\(894\) 0 0
\(895\) 5.94602 0.198754
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 0.246211 0.00821618
\(899\) −23.6155 −0.787622
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 18.2462 0.607532
\(903\) 0 0
\(904\) 2.68466 0.0892904
\(905\) −23.6155 −0.785007
\(906\) 0 0
\(907\) 52.5464 1.74477 0.872387 0.488815i \(-0.162571\pi\)
0.872387 + 0.488815i \(0.162571\pi\)
\(908\) 6.19224 0.205497
\(909\) 0 0
\(910\) 5.56155 0.184364
\(911\) 1.94602 0.0644747 0.0322373 0.999480i \(-0.489737\pi\)
0.0322373 + 0.999480i \(0.489737\pi\)
\(912\) 0 0
\(913\) −46.7386 −1.54682
\(914\) 7.36932 0.243755
\(915\) 0 0
\(916\) 2.24621 0.0742169
\(917\) 4.87689 0.161049
\(918\) 0 0
\(919\) −50.7386 −1.67371 −0.836857 0.547422i \(-0.815609\pi\)
−0.836857 + 0.547422i \(0.815609\pi\)
\(920\) −1.56155 −0.0514829
\(921\) 0 0
\(922\) −40.7386 −1.34166
\(923\) −33.3693 −1.09836
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −13.5616 −0.445660
\(927\) 0 0
\(928\) −7.56155 −0.248220
\(929\) 36.4384 1.19551 0.597753 0.801680i \(-0.296060\pi\)
0.597753 + 0.801680i \(0.296060\pi\)
\(930\) 0 0
\(931\) −5.12311 −0.167903
\(932\) 18.4924 0.605739
\(933\) 0 0
\(934\) −2.68466 −0.0878447
\(935\) −8.98485 −0.293836
\(936\) 0 0
\(937\) −27.1771 −0.887837 −0.443918 0.896067i \(-0.646412\pi\)
−0.443918 + 0.896067i \(0.646412\pi\)
\(938\) 1.12311 0.0366707
\(939\) 0 0
\(940\) −3.80776 −0.124196
\(941\) −1.94602 −0.0634386 −0.0317193 0.999497i \(-0.510098\pi\)
−0.0317193 + 0.999497i \(0.510098\pi\)
\(942\) 0 0
\(943\) 3.56155 0.115980
\(944\) −4.87689 −0.158729
\(945\) 0 0
\(946\) −34.2462 −1.11344
\(947\) 54.9309 1.78501 0.892507 0.451034i \(-0.148945\pi\)
0.892507 + 0.451034i \(0.148945\pi\)
\(948\) 0 0
\(949\) −32.4924 −1.05475
\(950\) 13.1231 0.425770
\(951\) 0 0
\(952\) −1.12311 −0.0364001
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) 31.2311 1.01061
\(956\) −4.49242 −0.145295
\(957\) 0 0
\(958\) −31.6155 −1.02145
\(959\) −0.438447 −0.0141582
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 5.56155 0.179312
\(963\) 0 0
\(964\) −20.0540 −0.645895
\(965\) 4.19224 0.134953
\(966\) 0 0
\(967\) −9.75379 −0.313661 −0.156830 0.987626i \(-0.550128\pi\)
−0.156830 + 0.987626i \(0.550128\pi\)
\(968\) 15.2462 0.490032
\(969\) 0 0
\(970\) −19.4233 −0.623644
\(971\) 1.61553 0.0518448 0.0259224 0.999664i \(-0.491748\pi\)
0.0259224 + 0.999664i \(0.491748\pi\)
\(972\) 0 0
\(973\) −3.80776 −0.122071
\(974\) −5.56155 −0.178204
\(975\) 0 0
\(976\) 0.876894 0.0280687
\(977\) 26.1922 0.837964 0.418982 0.907995i \(-0.362387\pi\)
0.418982 + 0.907995i \(0.362387\pi\)
\(978\) 0 0
\(979\) 71.7235 2.29229
\(980\) 1.56155 0.0498820
\(981\) 0 0
\(982\) 26.7386 0.853264
\(983\) −33.8617 −1.08002 −0.540011 0.841658i \(-0.681580\pi\)
−0.540011 + 0.841658i \(0.681580\pi\)
\(984\) 0 0
\(985\) −38.9309 −1.24044
\(986\) −8.49242 −0.270454
\(987\) 0 0
\(988\) 18.2462 0.580489
\(989\) −6.68466 −0.212560
\(990\) 0 0
\(991\) −6.24621 −0.198417 −0.0992087 0.995067i \(-0.531631\pi\)
−0.0992087 + 0.995067i \(0.531631\pi\)
\(992\) 3.12311 0.0991587
\(993\) 0 0
\(994\) −9.36932 −0.297177
\(995\) −33.0691 −1.04836
\(996\) 0 0
\(997\) 51.4773 1.63030 0.815151 0.579249i \(-0.196654\pi\)
0.815151 + 0.579249i \(0.196654\pi\)
\(998\) 41.3693 1.30952
\(999\) 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 2898.2.a.ba.1.2 2
3.2 odd 2 966.2.a.l.1.1 2
12.11 even 2 7728.2.a.bm.1.1 2
21.20 even 2 6762.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.l.1.1 2 3.2 odd 2
2898.2.a.ba.1.2 2 1.1 even 1 trivial
6762.2.a.bw.1.2 2 21.20 even 2
7728.2.a.bm.1.1 2 12.11 even 2