Properties

Label 8004.2.a.g.1.4
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.52462\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +0.526219 q^{11} -1.93131 q^{13} +2.52462 q^{15} +0.739721 q^{17} -1.99953 q^{19} +2.76852 q^{21} +1.00000 q^{23} +1.37371 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.82478 q^{31} -0.526219 q^{33} +6.98946 q^{35} +0.747593 q^{37} +1.93131 q^{39} +3.52897 q^{41} +2.13846 q^{43} -2.52462 q^{45} +7.82942 q^{47} +0.664694 q^{49} -0.739721 q^{51} +7.39345 q^{53} -1.32850 q^{55} +1.99953 q^{57} +12.9991 q^{59} +10.0143 q^{61} -2.76852 q^{63} +4.87583 q^{65} -3.29209 q^{67} -1.00000 q^{69} -14.3864 q^{71} -4.34510 q^{73} -1.37371 q^{75} -1.45685 q^{77} -0.963327 q^{79} +1.00000 q^{81} -13.7019 q^{83} -1.86752 q^{85} +1.00000 q^{87} -1.29398 q^{89} +5.34687 q^{91} -3.82478 q^{93} +5.04806 q^{95} +3.48364 q^{97} +0.526219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) −2.52462 −1.12904 −0.564522 0.825418i \(-0.690940\pi\)
−0.564522 + 0.825418i \(0.690940\pi\)
\(6\) 0 0
\(7\) −2.76852 −1.04640 −0.523201 0.852209i \(-0.675262\pi\)
−0.523201 + 0.852209i \(0.675262\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.526219 0.158661 0.0793305 0.996848i \(-0.474722\pi\)
0.0793305 + 0.996848i \(0.474722\pi\)
\(12\) 0 0
\(13\) −1.93131 −0.535649 −0.267825 0.963468i \(-0.586305\pi\)
−0.267825 + 0.963468i \(0.586305\pi\)
\(14\) 0 0
\(15\) 2.52462 0.651854
\(16\) 0 0
\(17\) 0.739721 0.179409 0.0897044 0.995968i \(-0.471408\pi\)
0.0897044 + 0.995968i \(0.471408\pi\)
\(18\) 0 0
\(19\) −1.99953 −0.458724 −0.229362 0.973341i \(-0.573664\pi\)
−0.229362 + 0.973341i \(0.573664\pi\)
\(20\) 0 0
\(21\) 2.76852 0.604140
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.37371 0.274742
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 3.82478 0.686950 0.343475 0.939162i \(-0.388396\pi\)
0.343475 + 0.939162i \(0.388396\pi\)
\(32\) 0 0
\(33\) −0.526219 −0.0916030
\(34\) 0 0
\(35\) 6.98946 1.18143
\(36\) 0 0
\(37\) 0.747593 0.122904 0.0614518 0.998110i \(-0.480427\pi\)
0.0614518 + 0.998110i \(0.480427\pi\)
\(38\) 0 0
\(39\) 1.93131 0.309257
\(40\) 0 0
\(41\) 3.52897 0.551133 0.275566 0.961282i \(-0.411135\pi\)
0.275566 + 0.961282i \(0.411135\pi\)
\(42\) 0 0
\(43\) 2.13846 0.326113 0.163056 0.986617i \(-0.447865\pi\)
0.163056 + 0.986617i \(0.447865\pi\)
\(44\) 0 0
\(45\) −2.52462 −0.376348
\(46\) 0 0
\(47\) 7.82942 1.14204 0.571019 0.820937i \(-0.306548\pi\)
0.571019 + 0.820937i \(0.306548\pi\)
\(48\) 0 0
\(49\) 0.664694 0.0949562
\(50\) 0 0
\(51\) −0.739721 −0.103582
\(52\) 0 0
\(53\) 7.39345 1.01557 0.507785 0.861484i \(-0.330465\pi\)
0.507785 + 0.861484i \(0.330465\pi\)
\(54\) 0 0
\(55\) −1.32850 −0.179135
\(56\) 0 0
\(57\) 1.99953 0.264844
\(58\) 0 0
\(59\) 12.9991 1.69234 0.846172 0.532909i \(-0.178901\pi\)
0.846172 + 0.532909i \(0.178901\pi\)
\(60\) 0 0
\(61\) 10.0143 1.28220 0.641098 0.767459i \(-0.278479\pi\)
0.641098 + 0.767459i \(0.278479\pi\)
\(62\) 0 0
\(63\) −2.76852 −0.348801
\(64\) 0 0
\(65\) 4.87583 0.604772
\(66\) 0 0
\(67\) −3.29209 −0.402192 −0.201096 0.979572i \(-0.564450\pi\)
−0.201096 + 0.979572i \(0.564450\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −14.3864 −1.70735 −0.853677 0.520803i \(-0.825633\pi\)
−0.853677 + 0.520803i \(0.825633\pi\)
\(72\) 0 0
\(73\) −4.34510 −0.508556 −0.254278 0.967131i \(-0.581838\pi\)
−0.254278 + 0.967131i \(0.581838\pi\)
\(74\) 0 0
\(75\) −1.37371 −0.158622
\(76\) 0 0
\(77\) −1.45685 −0.166023
\(78\) 0 0
\(79\) −0.963327 −0.108383 −0.0541914 0.998531i \(-0.517258\pi\)
−0.0541914 + 0.998531i \(0.517258\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.7019 −1.50398 −0.751990 0.659175i \(-0.770906\pi\)
−0.751990 + 0.659175i \(0.770906\pi\)
\(84\) 0 0
\(85\) −1.86752 −0.202560
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −1.29398 −0.137161 −0.0685806 0.997646i \(-0.521847\pi\)
−0.0685806 + 0.997646i \(0.521847\pi\)
\(90\) 0 0
\(91\) 5.34687 0.560504
\(92\) 0 0
\(93\) −3.82478 −0.396611
\(94\) 0 0
\(95\) 5.04806 0.517920
\(96\) 0 0
\(97\) 3.48364 0.353710 0.176855 0.984237i \(-0.443408\pi\)
0.176855 + 0.984237i \(0.443408\pi\)
\(98\) 0 0
\(99\) 0.526219 0.0528870
\(100\) 0 0
\(101\) −10.3219 −1.02707 −0.513535 0.858068i \(-0.671664\pi\)
−0.513535 + 0.858068i \(0.671664\pi\)
\(102\) 0 0
\(103\) 15.2741 1.50500 0.752499 0.658594i \(-0.228848\pi\)
0.752499 + 0.658594i \(0.228848\pi\)
\(104\) 0 0
\(105\) −6.98946 −0.682101
\(106\) 0 0
\(107\) −13.8318 −1.33717 −0.668583 0.743637i \(-0.733099\pi\)
−0.668583 + 0.743637i \(0.733099\pi\)
\(108\) 0 0
\(109\) −0.610729 −0.0584972 −0.0292486 0.999572i \(-0.509311\pi\)
−0.0292486 + 0.999572i \(0.509311\pi\)
\(110\) 0 0
\(111\) −0.747593 −0.0709584
\(112\) 0 0
\(113\) 10.9256 1.02780 0.513898 0.857851i \(-0.328201\pi\)
0.513898 + 0.857851i \(0.328201\pi\)
\(114\) 0 0
\(115\) −2.52462 −0.235422
\(116\) 0 0
\(117\) −1.93131 −0.178550
\(118\) 0 0
\(119\) −2.04793 −0.187734
\(120\) 0 0
\(121\) −10.7231 −0.974827
\(122\) 0 0
\(123\) −3.52897 −0.318197
\(124\) 0 0
\(125\) 9.15501 0.818849
\(126\) 0 0
\(127\) 20.6821 1.83524 0.917621 0.397457i \(-0.130107\pi\)
0.917621 + 0.397457i \(0.130107\pi\)
\(128\) 0 0
\(129\) −2.13846 −0.188281
\(130\) 0 0
\(131\) −1.82772 −0.159688 −0.0798441 0.996807i \(-0.525442\pi\)
−0.0798441 + 0.996807i \(0.525442\pi\)
\(132\) 0 0
\(133\) 5.53574 0.480009
\(134\) 0 0
\(135\) 2.52462 0.217285
\(136\) 0 0
\(137\) 20.1737 1.72356 0.861779 0.507285i \(-0.169351\pi\)
0.861779 + 0.507285i \(0.169351\pi\)
\(138\) 0 0
\(139\) −6.98799 −0.592714 −0.296357 0.955077i \(-0.595772\pi\)
−0.296357 + 0.955077i \(0.595772\pi\)
\(140\) 0 0
\(141\) −7.82942 −0.659356
\(142\) 0 0
\(143\) −1.01629 −0.0849867
\(144\) 0 0
\(145\) 2.52462 0.209658
\(146\) 0 0
\(147\) −0.664694 −0.0548230
\(148\) 0 0
\(149\) −7.38340 −0.604872 −0.302436 0.953170i \(-0.597800\pi\)
−0.302436 + 0.953170i \(0.597800\pi\)
\(150\) 0 0
\(151\) 15.4647 1.25850 0.629248 0.777204i \(-0.283363\pi\)
0.629248 + 0.777204i \(0.283363\pi\)
\(152\) 0 0
\(153\) 0.739721 0.0598029
\(154\) 0 0
\(155\) −9.65611 −0.775598
\(156\) 0 0
\(157\) −0.360889 −0.0288020 −0.0144010 0.999896i \(-0.504584\pi\)
−0.0144010 + 0.999896i \(0.504584\pi\)
\(158\) 0 0
\(159\) −7.39345 −0.586339
\(160\) 0 0
\(161\) −2.76852 −0.218190
\(162\) 0 0
\(163\) 10.3594 0.811409 0.405704 0.914004i \(-0.367026\pi\)
0.405704 + 0.914004i \(0.367026\pi\)
\(164\) 0 0
\(165\) 1.32850 0.103424
\(166\) 0 0
\(167\) 5.80202 0.448973 0.224487 0.974477i \(-0.427929\pi\)
0.224487 + 0.974477i \(0.427929\pi\)
\(168\) 0 0
\(169\) −9.27004 −0.713080
\(170\) 0 0
\(171\) −1.99953 −0.152908
\(172\) 0 0
\(173\) −11.2699 −0.856835 −0.428418 0.903581i \(-0.640929\pi\)
−0.428418 + 0.903581i \(0.640929\pi\)
\(174\) 0 0
\(175\) −3.80314 −0.287491
\(176\) 0 0
\(177\) −12.9991 −0.977076
\(178\) 0 0
\(179\) 24.2983 1.81614 0.908071 0.418816i \(-0.137555\pi\)
0.908071 + 0.418816i \(0.137555\pi\)
\(180\) 0 0
\(181\) −0.434698 −0.0323108 −0.0161554 0.999869i \(-0.505143\pi\)
−0.0161554 + 0.999869i \(0.505143\pi\)
\(182\) 0 0
\(183\) −10.0143 −0.740277
\(184\) 0 0
\(185\) −1.88739 −0.138764
\(186\) 0 0
\(187\) 0.389255 0.0284652
\(188\) 0 0
\(189\) 2.76852 0.201380
\(190\) 0 0
\(191\) −26.2461 −1.89910 −0.949549 0.313619i \(-0.898459\pi\)
−0.949549 + 0.313619i \(0.898459\pi\)
\(192\) 0 0
\(193\) −8.48991 −0.611117 −0.305559 0.952173i \(-0.598843\pi\)
−0.305559 + 0.952173i \(0.598843\pi\)
\(194\) 0 0
\(195\) −4.87583 −0.349165
\(196\) 0 0
\(197\) −13.1347 −0.935809 −0.467904 0.883779i \(-0.654991\pi\)
−0.467904 + 0.883779i \(0.654991\pi\)
\(198\) 0 0
\(199\) 7.99311 0.566617 0.283308 0.959029i \(-0.408568\pi\)
0.283308 + 0.959029i \(0.408568\pi\)
\(200\) 0 0
\(201\) 3.29209 0.232206
\(202\) 0 0
\(203\) 2.76852 0.194312
\(204\) 0 0
\(205\) −8.90931 −0.622253
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −1.05219 −0.0727816
\(210\) 0 0
\(211\) 0.868521 0.0597914 0.0298957 0.999553i \(-0.490482\pi\)
0.0298957 + 0.999553i \(0.490482\pi\)
\(212\) 0 0
\(213\) 14.3864 0.985741
\(214\) 0 0
\(215\) −5.39881 −0.368196
\(216\) 0 0
\(217\) −10.5890 −0.718826
\(218\) 0 0
\(219\) 4.34510 0.293615
\(220\) 0 0
\(221\) −1.42863 −0.0961002
\(222\) 0 0
\(223\) −2.31567 −0.155068 −0.0775342 0.996990i \(-0.524705\pi\)
−0.0775342 + 0.996990i \(0.524705\pi\)
\(224\) 0 0
\(225\) 1.37371 0.0915807
\(226\) 0 0
\(227\) 3.21958 0.213691 0.106845 0.994276i \(-0.465925\pi\)
0.106845 + 0.994276i \(0.465925\pi\)
\(228\) 0 0
\(229\) −9.23186 −0.610059 −0.305029 0.952343i \(-0.598666\pi\)
−0.305029 + 0.952343i \(0.598666\pi\)
\(230\) 0 0
\(231\) 1.45685 0.0958535
\(232\) 0 0
\(233\) −22.9381 −1.50273 −0.751363 0.659889i \(-0.770603\pi\)
−0.751363 + 0.659889i \(0.770603\pi\)
\(234\) 0 0
\(235\) −19.7663 −1.28941
\(236\) 0 0
\(237\) 0.963327 0.0625748
\(238\) 0 0
\(239\) −15.9646 −1.03266 −0.516331 0.856389i \(-0.672702\pi\)
−0.516331 + 0.856389i \(0.672702\pi\)
\(240\) 0 0
\(241\) −25.1441 −1.61967 −0.809837 0.586655i \(-0.800445\pi\)
−0.809837 + 0.586655i \(0.800445\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.67810 −0.107210
\(246\) 0 0
\(247\) 3.86172 0.245715
\(248\) 0 0
\(249\) 13.7019 0.868323
\(250\) 0 0
\(251\) −21.4340 −1.35290 −0.676452 0.736487i \(-0.736483\pi\)
−0.676452 + 0.736487i \(0.736483\pi\)
\(252\) 0 0
\(253\) 0.526219 0.0330831
\(254\) 0 0
\(255\) 1.86752 0.116948
\(256\) 0 0
\(257\) −18.8959 −1.17869 −0.589346 0.807881i \(-0.700615\pi\)
−0.589346 + 0.807881i \(0.700615\pi\)
\(258\) 0 0
\(259\) −2.06973 −0.128606
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 26.6795 1.64513 0.822563 0.568673i \(-0.192543\pi\)
0.822563 + 0.568673i \(0.192543\pi\)
\(264\) 0 0
\(265\) −18.6657 −1.14662
\(266\) 0 0
\(267\) 1.29398 0.0791900
\(268\) 0 0
\(269\) 13.7876 0.840647 0.420323 0.907374i \(-0.361917\pi\)
0.420323 + 0.907374i \(0.361917\pi\)
\(270\) 0 0
\(271\) 12.4101 0.753860 0.376930 0.926242i \(-0.376980\pi\)
0.376930 + 0.926242i \(0.376980\pi\)
\(272\) 0 0
\(273\) −5.34687 −0.323607
\(274\) 0 0
\(275\) 0.722873 0.0435909
\(276\) 0 0
\(277\) −5.28861 −0.317762 −0.158881 0.987298i \(-0.550789\pi\)
−0.158881 + 0.987298i \(0.550789\pi\)
\(278\) 0 0
\(279\) 3.82478 0.228983
\(280\) 0 0
\(281\) 19.8312 1.18303 0.591516 0.806293i \(-0.298530\pi\)
0.591516 + 0.806293i \(0.298530\pi\)
\(282\) 0 0
\(283\) 23.5538 1.40013 0.700064 0.714080i \(-0.253155\pi\)
0.700064 + 0.714080i \(0.253155\pi\)
\(284\) 0 0
\(285\) −5.04806 −0.299021
\(286\) 0 0
\(287\) −9.77002 −0.576706
\(288\) 0 0
\(289\) −16.4528 −0.967813
\(290\) 0 0
\(291\) −3.48364 −0.204215
\(292\) 0 0
\(293\) 7.00209 0.409066 0.204533 0.978860i \(-0.434432\pi\)
0.204533 + 0.978860i \(0.434432\pi\)
\(294\) 0 0
\(295\) −32.8179 −1.91073
\(296\) 0 0
\(297\) −0.526219 −0.0305343
\(298\) 0 0
\(299\) −1.93131 −0.111691
\(300\) 0 0
\(301\) −5.92037 −0.341245
\(302\) 0 0
\(303\) 10.3219 0.592980
\(304\) 0 0
\(305\) −25.2822 −1.44766
\(306\) 0 0
\(307\) −22.4598 −1.28185 −0.640925 0.767604i \(-0.721449\pi\)
−0.640925 + 0.767604i \(0.721449\pi\)
\(308\) 0 0
\(309\) −15.2741 −0.868911
\(310\) 0 0
\(311\) −27.7082 −1.57119 −0.785595 0.618741i \(-0.787643\pi\)
−0.785595 + 0.618741i \(0.787643\pi\)
\(312\) 0 0
\(313\) −0.503230 −0.0284442 −0.0142221 0.999899i \(-0.504527\pi\)
−0.0142221 + 0.999899i \(0.504527\pi\)
\(314\) 0 0
\(315\) 6.98946 0.393811
\(316\) 0 0
\(317\) −14.6482 −0.822724 −0.411362 0.911472i \(-0.634947\pi\)
−0.411362 + 0.911472i \(0.634947\pi\)
\(318\) 0 0
\(319\) −0.526219 −0.0294626
\(320\) 0 0
\(321\) 13.8318 0.772014
\(322\) 0 0
\(323\) −1.47909 −0.0822990
\(324\) 0 0
\(325\) −2.65306 −0.147165
\(326\) 0 0
\(327\) 0.610729 0.0337734
\(328\) 0 0
\(329\) −21.6759 −1.19503
\(330\) 0 0
\(331\) 22.3670 1.22940 0.614702 0.788760i \(-0.289276\pi\)
0.614702 + 0.788760i \(0.289276\pi\)
\(332\) 0 0
\(333\) 0.747593 0.0409679
\(334\) 0 0
\(335\) 8.31127 0.454093
\(336\) 0 0
\(337\) 29.7405 1.62007 0.810034 0.586383i \(-0.199449\pi\)
0.810034 + 0.586383i \(0.199449\pi\)
\(338\) 0 0
\(339\) −10.9256 −0.593398
\(340\) 0 0
\(341\) 2.01267 0.108992
\(342\) 0 0
\(343\) 17.5394 0.947039
\(344\) 0 0
\(345\) 2.52462 0.135921
\(346\) 0 0
\(347\) −33.6176 −1.80469 −0.902343 0.431019i \(-0.858154\pi\)
−0.902343 + 0.431019i \(0.858154\pi\)
\(348\) 0 0
\(349\) 21.0513 1.12685 0.563424 0.826168i \(-0.309484\pi\)
0.563424 + 0.826168i \(0.309484\pi\)
\(350\) 0 0
\(351\) 1.93131 0.103086
\(352\) 0 0
\(353\) −9.93922 −0.529011 −0.264506 0.964384i \(-0.585209\pi\)
−0.264506 + 0.964384i \(0.585209\pi\)
\(354\) 0 0
\(355\) 36.3202 1.92768
\(356\) 0 0
\(357\) 2.04793 0.108388
\(358\) 0 0
\(359\) 5.50696 0.290646 0.145323 0.989384i \(-0.453578\pi\)
0.145323 + 0.989384i \(0.453578\pi\)
\(360\) 0 0
\(361\) −15.0019 −0.789573
\(362\) 0 0
\(363\) 10.7231 0.562816
\(364\) 0 0
\(365\) 10.9697 0.574182
\(366\) 0 0
\(367\) −13.1984 −0.688953 −0.344477 0.938795i \(-0.611944\pi\)
−0.344477 + 0.938795i \(0.611944\pi\)
\(368\) 0 0
\(369\) 3.52897 0.183711
\(370\) 0 0
\(371\) −20.4689 −1.06269
\(372\) 0 0
\(373\) 3.43321 0.177765 0.0888825 0.996042i \(-0.471670\pi\)
0.0888825 + 0.996042i \(0.471670\pi\)
\(374\) 0 0
\(375\) −9.15501 −0.472763
\(376\) 0 0
\(377\) 1.93131 0.0994676
\(378\) 0 0
\(379\) −32.5768 −1.67336 −0.836680 0.547692i \(-0.815507\pi\)
−0.836680 + 0.547692i \(0.815507\pi\)
\(380\) 0 0
\(381\) −20.6821 −1.05958
\(382\) 0 0
\(383\) −22.4133 −1.14527 −0.572633 0.819812i \(-0.694078\pi\)
−0.572633 + 0.819812i \(0.694078\pi\)
\(384\) 0 0
\(385\) 3.67799 0.187448
\(386\) 0 0
\(387\) 2.13846 0.108704
\(388\) 0 0
\(389\) −27.6519 −1.40201 −0.701003 0.713159i \(-0.747264\pi\)
−0.701003 + 0.713159i \(0.747264\pi\)
\(390\) 0 0
\(391\) 0.739721 0.0374093
\(392\) 0 0
\(393\) 1.82772 0.0921961
\(394\) 0 0
\(395\) 2.43204 0.122369
\(396\) 0 0
\(397\) −3.62435 −0.181901 −0.0909505 0.995855i \(-0.528991\pi\)
−0.0909505 + 0.995855i \(0.528991\pi\)
\(398\) 0 0
\(399\) −5.53574 −0.277133
\(400\) 0 0
\(401\) −14.4203 −0.720115 −0.360057 0.932930i \(-0.617243\pi\)
−0.360057 + 0.932930i \(0.617243\pi\)
\(402\) 0 0
\(403\) −7.38684 −0.367965
\(404\) 0 0
\(405\) −2.52462 −0.125449
\(406\) 0 0
\(407\) 0.393398 0.0195000
\(408\) 0 0
\(409\) −17.4810 −0.864381 −0.432190 0.901782i \(-0.642259\pi\)
−0.432190 + 0.901782i \(0.642259\pi\)
\(410\) 0 0
\(411\) −20.1737 −0.995096
\(412\) 0 0
\(413\) −35.9884 −1.77087
\(414\) 0 0
\(415\) 34.5921 1.69806
\(416\) 0 0
\(417\) 6.98799 0.342203
\(418\) 0 0
\(419\) −26.6543 −1.30215 −0.651074 0.759014i \(-0.725681\pi\)
−0.651074 + 0.759014i \(0.725681\pi\)
\(420\) 0 0
\(421\) −28.6872 −1.39813 −0.699065 0.715058i \(-0.746400\pi\)
−0.699065 + 0.715058i \(0.746400\pi\)
\(422\) 0 0
\(423\) 7.82942 0.380679
\(424\) 0 0
\(425\) 1.01616 0.0492911
\(426\) 0 0
\(427\) −27.7247 −1.34169
\(428\) 0 0
\(429\) 1.01629 0.0490671
\(430\) 0 0
\(431\) 24.2099 1.16615 0.583076 0.812418i \(-0.301849\pi\)
0.583076 + 0.812418i \(0.301849\pi\)
\(432\) 0 0
\(433\) 32.0026 1.53795 0.768974 0.639279i \(-0.220767\pi\)
0.768974 + 0.639279i \(0.220767\pi\)
\(434\) 0 0
\(435\) −2.52462 −0.121046
\(436\) 0 0
\(437\) −1.99953 −0.0956505
\(438\) 0 0
\(439\) −29.8370 −1.42404 −0.712020 0.702159i \(-0.752220\pi\)
−0.712020 + 0.702159i \(0.752220\pi\)
\(440\) 0 0
\(441\) 0.664694 0.0316521
\(442\) 0 0
\(443\) −29.4781 −1.40054 −0.700272 0.713876i \(-0.746938\pi\)
−0.700272 + 0.713876i \(0.746938\pi\)
\(444\) 0 0
\(445\) 3.26680 0.154861
\(446\) 0 0
\(447\) 7.38340 0.349223
\(448\) 0 0
\(449\) −13.5353 −0.638772 −0.319386 0.947625i \(-0.603477\pi\)
−0.319386 + 0.947625i \(0.603477\pi\)
\(450\) 0 0
\(451\) 1.85701 0.0874433
\(452\) 0 0
\(453\) −15.4647 −0.726593
\(454\) 0 0
\(455\) −13.4988 −0.632835
\(456\) 0 0
\(457\) 33.2067 1.55334 0.776672 0.629906i \(-0.216906\pi\)
0.776672 + 0.629906i \(0.216906\pi\)
\(458\) 0 0
\(459\) −0.739721 −0.0345272
\(460\) 0 0
\(461\) 10.0870 0.469797 0.234899 0.972020i \(-0.424524\pi\)
0.234899 + 0.972020i \(0.424524\pi\)
\(462\) 0 0
\(463\) 41.6137 1.93395 0.966977 0.254864i \(-0.0820306\pi\)
0.966977 + 0.254864i \(0.0820306\pi\)
\(464\) 0 0
\(465\) 9.65611 0.447791
\(466\) 0 0
\(467\) 33.9829 1.57254 0.786270 0.617883i \(-0.212009\pi\)
0.786270 + 0.617883i \(0.212009\pi\)
\(468\) 0 0
\(469\) 9.11420 0.420855
\(470\) 0 0
\(471\) 0.360889 0.0166289
\(472\) 0 0
\(473\) 1.12530 0.0517414
\(474\) 0 0
\(475\) −2.74678 −0.126031
\(476\) 0 0
\(477\) 7.39345 0.338523
\(478\) 0 0
\(479\) 2.40384 0.109834 0.0549170 0.998491i \(-0.482511\pi\)
0.0549170 + 0.998491i \(0.482511\pi\)
\(480\) 0 0
\(481\) −1.44384 −0.0658332
\(482\) 0 0
\(483\) 2.76852 0.125972
\(484\) 0 0
\(485\) −8.79487 −0.399355
\(486\) 0 0
\(487\) −22.0887 −1.00093 −0.500467 0.865756i \(-0.666838\pi\)
−0.500467 + 0.865756i \(0.666838\pi\)
\(488\) 0 0
\(489\) −10.3594 −0.468467
\(490\) 0 0
\(491\) −31.9350 −1.44121 −0.720603 0.693348i \(-0.756135\pi\)
−0.720603 + 0.693348i \(0.756135\pi\)
\(492\) 0 0
\(493\) −0.739721 −0.0333154
\(494\) 0 0
\(495\) −1.32850 −0.0597118
\(496\) 0 0
\(497\) 39.8291 1.78658
\(498\) 0 0
\(499\) −20.2643 −0.907155 −0.453578 0.891217i \(-0.649853\pi\)
−0.453578 + 0.891217i \(0.649853\pi\)
\(500\) 0 0
\(501\) −5.80202 −0.259215
\(502\) 0 0
\(503\) 8.20416 0.365805 0.182903 0.983131i \(-0.441451\pi\)
0.182903 + 0.983131i \(0.441451\pi\)
\(504\) 0 0
\(505\) 26.0590 1.15961
\(506\) 0 0
\(507\) 9.27004 0.411697
\(508\) 0 0
\(509\) −19.2432 −0.852938 −0.426469 0.904502i \(-0.640243\pi\)
−0.426469 + 0.904502i \(0.640243\pi\)
\(510\) 0 0
\(511\) 12.0295 0.532154
\(512\) 0 0
\(513\) 1.99953 0.0882814
\(514\) 0 0
\(515\) −38.5612 −1.69921
\(516\) 0 0
\(517\) 4.11999 0.181197
\(518\) 0 0
\(519\) 11.2699 0.494694
\(520\) 0 0
\(521\) −42.9060 −1.87975 −0.939873 0.341524i \(-0.889057\pi\)
−0.939873 + 0.341524i \(0.889057\pi\)
\(522\) 0 0
\(523\) −9.21145 −0.402789 −0.201394 0.979510i \(-0.564547\pi\)
−0.201394 + 0.979510i \(0.564547\pi\)
\(524\) 0 0
\(525\) 3.80314 0.165983
\(526\) 0 0
\(527\) 2.82927 0.123245
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.9991 0.564115
\(532\) 0 0
\(533\) −6.81554 −0.295214
\(534\) 0 0
\(535\) 34.9200 1.50972
\(536\) 0 0
\(537\) −24.2983 −1.04855
\(538\) 0 0
\(539\) 0.349775 0.0150659
\(540\) 0 0
\(541\) 19.0764 0.820161 0.410080 0.912049i \(-0.365501\pi\)
0.410080 + 0.912049i \(0.365501\pi\)
\(542\) 0 0
\(543\) 0.434698 0.0186547
\(544\) 0 0
\(545\) 1.54186 0.0660460
\(546\) 0 0
\(547\) 34.9038 1.49238 0.746189 0.665734i \(-0.231882\pi\)
0.746189 + 0.665734i \(0.231882\pi\)
\(548\) 0 0
\(549\) 10.0143 0.427399
\(550\) 0 0
\(551\) 1.99953 0.0851828
\(552\) 0 0
\(553\) 2.66699 0.113412
\(554\) 0 0
\(555\) 1.88739 0.0801152
\(556\) 0 0
\(557\) −18.3979 −0.779545 −0.389772 0.920911i \(-0.627446\pi\)
−0.389772 + 0.920911i \(0.627446\pi\)
\(558\) 0 0
\(559\) −4.13004 −0.174682
\(560\) 0 0
\(561\) −0.389255 −0.0164344
\(562\) 0 0
\(563\) 21.6776 0.913601 0.456801 0.889569i \(-0.348995\pi\)
0.456801 + 0.889569i \(0.348995\pi\)
\(564\) 0 0
\(565\) −27.5830 −1.16043
\(566\) 0 0
\(567\) −2.76852 −0.116267
\(568\) 0 0
\(569\) 35.3588 1.48232 0.741160 0.671329i \(-0.234276\pi\)
0.741160 + 0.671329i \(0.234276\pi\)
\(570\) 0 0
\(571\) −37.9152 −1.58670 −0.793350 0.608765i \(-0.791665\pi\)
−0.793350 + 0.608765i \(0.791665\pi\)
\(572\) 0 0
\(573\) 26.2461 1.09644
\(574\) 0 0
\(575\) 1.37371 0.0572877
\(576\) 0 0
\(577\) −25.2547 −1.05137 −0.525684 0.850680i \(-0.676191\pi\)
−0.525684 + 0.850680i \(0.676191\pi\)
\(578\) 0 0
\(579\) 8.48991 0.352829
\(580\) 0 0
\(581\) 37.9340 1.57377
\(582\) 0 0
\(583\) 3.89058 0.161131
\(584\) 0 0
\(585\) 4.87583 0.201591
\(586\) 0 0
\(587\) −25.7576 −1.06313 −0.531566 0.847017i \(-0.678396\pi\)
−0.531566 + 0.847017i \(0.678396\pi\)
\(588\) 0 0
\(589\) −7.64776 −0.315120
\(590\) 0 0
\(591\) 13.1347 0.540289
\(592\) 0 0
\(593\) 24.6936 1.01404 0.507022 0.861933i \(-0.330746\pi\)
0.507022 + 0.861933i \(0.330746\pi\)
\(594\) 0 0
\(595\) 5.17025 0.211960
\(596\) 0 0
\(597\) −7.99311 −0.327136
\(598\) 0 0
\(599\) 31.6646 1.29378 0.646889 0.762584i \(-0.276070\pi\)
0.646889 + 0.762584i \(0.276070\pi\)
\(600\) 0 0
\(601\) 18.6588 0.761109 0.380554 0.924759i \(-0.375733\pi\)
0.380554 + 0.924759i \(0.375733\pi\)
\(602\) 0 0
\(603\) −3.29209 −0.134064
\(604\) 0 0
\(605\) 27.0717 1.10062
\(606\) 0 0
\(607\) 8.67966 0.352296 0.176148 0.984364i \(-0.443636\pi\)
0.176148 + 0.984364i \(0.443636\pi\)
\(608\) 0 0
\(609\) −2.76852 −0.112186
\(610\) 0 0
\(611\) −15.1210 −0.611732
\(612\) 0 0
\(613\) −46.5278 −1.87924 −0.939620 0.342220i \(-0.888821\pi\)
−0.939620 + 0.342220i \(0.888821\pi\)
\(614\) 0 0
\(615\) 8.90931 0.359258
\(616\) 0 0
\(617\) −43.1480 −1.73707 −0.868537 0.495625i \(-0.834939\pi\)
−0.868537 + 0.495625i \(0.834939\pi\)
\(618\) 0 0
\(619\) 17.5317 0.704658 0.352329 0.935876i \(-0.385390\pi\)
0.352329 + 0.935876i \(0.385390\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 3.58239 0.143526
\(624\) 0 0
\(625\) −29.9815 −1.19926
\(626\) 0 0
\(627\) 1.05219 0.0420205
\(628\) 0 0
\(629\) 0.553010 0.0220500
\(630\) 0 0
\(631\) −22.9146 −0.912216 −0.456108 0.889925i \(-0.650757\pi\)
−0.456108 + 0.889925i \(0.650757\pi\)
\(632\) 0 0
\(633\) −0.868521 −0.0345206
\(634\) 0 0
\(635\) −52.2145 −2.07207
\(636\) 0 0
\(637\) −1.28373 −0.0508633
\(638\) 0 0
\(639\) −14.3864 −0.569118
\(640\) 0 0
\(641\) −7.61469 −0.300762 −0.150381 0.988628i \(-0.548050\pi\)
−0.150381 + 0.988628i \(0.548050\pi\)
\(642\) 0 0
\(643\) −5.73510 −0.226170 −0.113085 0.993585i \(-0.536073\pi\)
−0.113085 + 0.993585i \(0.536073\pi\)
\(644\) 0 0
\(645\) 5.39881 0.212578
\(646\) 0 0
\(647\) −40.0251 −1.57355 −0.786775 0.617239i \(-0.788251\pi\)
−0.786775 + 0.617239i \(0.788251\pi\)
\(648\) 0 0
\(649\) 6.84040 0.268509
\(650\) 0 0
\(651\) 10.5890 0.415014
\(652\) 0 0
\(653\) −46.5044 −1.81986 −0.909929 0.414764i \(-0.863864\pi\)
−0.909929 + 0.414764i \(0.863864\pi\)
\(654\) 0 0
\(655\) 4.61429 0.180295
\(656\) 0 0
\(657\) −4.34510 −0.169519
\(658\) 0 0
\(659\) 29.8314 1.16207 0.581034 0.813879i \(-0.302648\pi\)
0.581034 + 0.813879i \(0.302648\pi\)
\(660\) 0 0
\(661\) −19.3456 −0.752455 −0.376227 0.926527i \(-0.622779\pi\)
−0.376227 + 0.926527i \(0.622779\pi\)
\(662\) 0 0
\(663\) 1.42863 0.0554835
\(664\) 0 0
\(665\) −13.9756 −0.541952
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 2.31567 0.0895288
\(670\) 0 0
\(671\) 5.26970 0.203435
\(672\) 0 0
\(673\) 5.04602 0.194510 0.0972550 0.995260i \(-0.468994\pi\)
0.0972550 + 0.995260i \(0.468994\pi\)
\(674\) 0 0
\(675\) −1.37371 −0.0528741
\(676\) 0 0
\(677\) −13.3579 −0.513387 −0.256693 0.966493i \(-0.582633\pi\)
−0.256693 + 0.966493i \(0.582633\pi\)
\(678\) 0 0
\(679\) −9.64452 −0.370123
\(680\) 0 0
\(681\) −3.21958 −0.123374
\(682\) 0 0
\(683\) 14.7302 0.563635 0.281817 0.959468i \(-0.409063\pi\)
0.281817 + 0.959468i \(0.409063\pi\)
\(684\) 0 0
\(685\) −50.9310 −1.94597
\(686\) 0 0
\(687\) 9.23186 0.352218
\(688\) 0 0
\(689\) −14.2791 −0.543989
\(690\) 0 0
\(691\) 33.0276 1.25643 0.628214 0.778040i \(-0.283786\pi\)
0.628214 + 0.778040i \(0.283786\pi\)
\(692\) 0 0
\(693\) −1.45685 −0.0553411
\(694\) 0 0
\(695\) 17.6420 0.669200
\(696\) 0 0
\(697\) 2.61045 0.0988780
\(698\) 0 0
\(699\) 22.9381 0.867600
\(700\) 0 0
\(701\) −38.1941 −1.44257 −0.721285 0.692638i \(-0.756448\pi\)
−0.721285 + 0.692638i \(0.756448\pi\)
\(702\) 0 0
\(703\) −1.49484 −0.0563788
\(704\) 0 0
\(705\) 19.7663 0.744443
\(706\) 0 0
\(707\) 28.5765 1.07473
\(708\) 0 0
\(709\) 30.4902 1.14508 0.572542 0.819875i \(-0.305957\pi\)
0.572542 + 0.819875i \(0.305957\pi\)
\(710\) 0 0
\(711\) −0.963327 −0.0361276
\(712\) 0 0
\(713\) 3.82478 0.143239
\(714\) 0 0
\(715\) 2.56576 0.0959538
\(716\) 0 0
\(717\) 15.9646 0.596207
\(718\) 0 0
\(719\) 19.3654 0.722209 0.361104 0.932525i \(-0.382400\pi\)
0.361104 + 0.932525i \(0.382400\pi\)
\(720\) 0 0
\(721\) −42.2865 −1.57483
\(722\) 0 0
\(723\) 25.1441 0.935119
\(724\) 0 0
\(725\) −1.37371 −0.0510183
\(726\) 0 0
\(727\) 41.2825 1.53108 0.765541 0.643387i \(-0.222472\pi\)
0.765541 + 0.643387i \(0.222472\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.58187 0.0585074
\(732\) 0 0
\(733\) −25.6687 −0.948097 −0.474048 0.880499i \(-0.657208\pi\)
−0.474048 + 0.880499i \(0.657208\pi\)
\(734\) 0 0
\(735\) 1.67810 0.0618976
\(736\) 0 0
\(737\) −1.73236 −0.0638123
\(738\) 0 0
\(739\) 30.9111 1.13708 0.568541 0.822655i \(-0.307508\pi\)
0.568541 + 0.822655i \(0.307508\pi\)
\(740\) 0 0
\(741\) −3.86172 −0.141864
\(742\) 0 0
\(743\) 16.4627 0.603959 0.301979 0.953314i \(-0.402353\pi\)
0.301979 + 0.953314i \(0.402353\pi\)
\(744\) 0 0
\(745\) 18.6403 0.682928
\(746\) 0 0
\(747\) −13.7019 −0.501327
\(748\) 0 0
\(749\) 38.2935 1.39921
\(750\) 0 0
\(751\) 50.3553 1.83749 0.918746 0.394850i \(-0.129203\pi\)
0.918746 + 0.394850i \(0.129203\pi\)
\(752\) 0 0
\(753\) 21.4340 0.781099
\(754\) 0 0
\(755\) −39.0424 −1.42090
\(756\) 0 0
\(757\) −37.5477 −1.36469 −0.682347 0.731029i \(-0.739041\pi\)
−0.682347 + 0.731029i \(0.739041\pi\)
\(758\) 0 0
\(759\) −0.526219 −0.0191005
\(760\) 0 0
\(761\) 16.6558 0.603771 0.301886 0.953344i \(-0.402384\pi\)
0.301886 + 0.953344i \(0.402384\pi\)
\(762\) 0 0
\(763\) 1.69081 0.0612116
\(764\) 0 0
\(765\) −1.86752 −0.0675202
\(766\) 0 0
\(767\) −25.1054 −0.906504
\(768\) 0 0
\(769\) 11.1202 0.401005 0.200502 0.979693i \(-0.435743\pi\)
0.200502 + 0.979693i \(0.435743\pi\)
\(770\) 0 0
\(771\) 18.8959 0.680519
\(772\) 0 0
\(773\) 28.0186 1.00776 0.503879 0.863774i \(-0.331906\pi\)
0.503879 + 0.863774i \(0.331906\pi\)
\(774\) 0 0
\(775\) 5.25414 0.188734
\(776\) 0 0
\(777\) 2.06973 0.0742510
\(778\) 0 0
\(779\) −7.05628 −0.252818
\(780\) 0 0
\(781\) −7.57041 −0.270891
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 0.911107 0.0325188
\(786\) 0 0
\(787\) 44.0485 1.57016 0.785080 0.619394i \(-0.212622\pi\)
0.785080 + 0.619394i \(0.212622\pi\)
\(788\) 0 0
\(789\) −26.6795 −0.949814
\(790\) 0 0
\(791\) −30.2478 −1.07549
\(792\) 0 0
\(793\) −19.3407 −0.686808
\(794\) 0 0
\(795\) 18.6657 0.662003
\(796\) 0 0
\(797\) 6.11994 0.216779 0.108390 0.994108i \(-0.465431\pi\)
0.108390 + 0.994108i \(0.465431\pi\)
\(798\) 0 0
\(799\) 5.79159 0.204892
\(800\) 0 0
\(801\) −1.29398 −0.0457204
\(802\) 0 0
\(803\) −2.28648 −0.0806880
\(804\) 0 0
\(805\) 6.98946 0.246346
\(806\) 0 0
\(807\) −13.7876 −0.485348
\(808\) 0 0
\(809\) −11.5618 −0.406490 −0.203245 0.979128i \(-0.565149\pi\)
−0.203245 + 0.979128i \(0.565149\pi\)
\(810\) 0 0
\(811\) −11.0513 −0.388065 −0.194032 0.980995i \(-0.562157\pi\)
−0.194032 + 0.980995i \(0.562157\pi\)
\(812\) 0 0
\(813\) −12.4101 −0.435241
\(814\) 0 0
\(815\) −26.1535 −0.916117
\(816\) 0 0
\(817\) −4.27592 −0.149596
\(818\) 0 0
\(819\) 5.34687 0.186835
\(820\) 0 0
\(821\) 12.6542 0.441636 0.220818 0.975315i \(-0.429127\pi\)
0.220818 + 0.975315i \(0.429127\pi\)
\(822\) 0 0
\(823\) 49.0282 1.70902 0.854509 0.519437i \(-0.173858\pi\)
0.854509 + 0.519437i \(0.173858\pi\)
\(824\) 0 0
\(825\) −0.722873 −0.0251672
\(826\) 0 0
\(827\) −13.9071 −0.483599 −0.241799 0.970326i \(-0.577738\pi\)
−0.241799 + 0.970326i \(0.577738\pi\)
\(828\) 0 0
\(829\) 22.9863 0.798347 0.399174 0.916875i \(-0.369297\pi\)
0.399174 + 0.916875i \(0.369297\pi\)
\(830\) 0 0
\(831\) 5.28861 0.183460
\(832\) 0 0
\(833\) 0.491688 0.0170360
\(834\) 0 0
\(835\) −14.6479 −0.506911
\(836\) 0 0
\(837\) −3.82478 −0.132204
\(838\) 0 0
\(839\) 43.7697 1.51110 0.755549 0.655092i \(-0.227370\pi\)
0.755549 + 0.655092i \(0.227370\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −19.8312 −0.683024
\(844\) 0 0
\(845\) 23.4033 0.805099
\(846\) 0 0
\(847\) 29.6871 1.02006
\(848\) 0 0
\(849\) −23.5538 −0.808364
\(850\) 0 0
\(851\) 0.747593 0.0256272
\(852\) 0 0
\(853\) 29.6457 1.01505 0.507524 0.861637i \(-0.330561\pi\)
0.507524 + 0.861637i \(0.330561\pi\)
\(854\) 0 0
\(855\) 5.04806 0.172640
\(856\) 0 0
\(857\) 25.3516 0.865995 0.432997 0.901395i \(-0.357456\pi\)
0.432997 + 0.901395i \(0.357456\pi\)
\(858\) 0 0
\(859\) −28.6168 −0.976394 −0.488197 0.872734i \(-0.662345\pi\)
−0.488197 + 0.872734i \(0.662345\pi\)
\(860\) 0 0
\(861\) 9.77002 0.332961
\(862\) 0 0
\(863\) 42.6297 1.45113 0.725565 0.688153i \(-0.241578\pi\)
0.725565 + 0.688153i \(0.241578\pi\)
\(864\) 0 0
\(865\) 28.4522 0.967405
\(866\) 0 0
\(867\) 16.4528 0.558767
\(868\) 0 0
\(869\) −0.506921 −0.0171961
\(870\) 0 0
\(871\) 6.35804 0.215434
\(872\) 0 0
\(873\) 3.48364 0.117903
\(874\) 0 0
\(875\) −25.3458 −0.856845
\(876\) 0 0
\(877\) −6.42553 −0.216975 −0.108487 0.994098i \(-0.534601\pi\)
−0.108487 + 0.994098i \(0.534601\pi\)
\(878\) 0 0
\(879\) −7.00209 −0.236174
\(880\) 0 0
\(881\) −14.8442 −0.500113 −0.250056 0.968231i \(-0.580449\pi\)
−0.250056 + 0.968231i \(0.580449\pi\)
\(882\) 0 0
\(883\) −43.4782 −1.46316 −0.731579 0.681757i \(-0.761216\pi\)
−0.731579 + 0.681757i \(0.761216\pi\)
\(884\) 0 0
\(885\) 32.8179 1.10316
\(886\) 0 0
\(887\) 4.64900 0.156098 0.0780490 0.996950i \(-0.475131\pi\)
0.0780490 + 0.996950i \(0.475131\pi\)
\(888\) 0 0
\(889\) −57.2588 −1.92040
\(890\) 0 0
\(891\) 0.526219 0.0176290
\(892\) 0 0
\(893\) −15.6552 −0.523880
\(894\) 0 0
\(895\) −61.3441 −2.05051
\(896\) 0 0
\(897\) 1.93131 0.0644846
\(898\) 0 0
\(899\) −3.82478 −0.127563
\(900\) 0 0
\(901\) 5.46909 0.182202
\(902\) 0 0
\(903\) 5.92037 0.197018
\(904\) 0 0
\(905\) 1.09745 0.0364804
\(906\) 0 0
\(907\) −15.5997 −0.517981 −0.258990 0.965880i \(-0.583390\pi\)
−0.258990 + 0.965880i \(0.583390\pi\)
\(908\) 0 0
\(909\) −10.3219 −0.342357
\(910\) 0 0
\(911\) 42.0282 1.39245 0.696227 0.717821i \(-0.254861\pi\)
0.696227 + 0.717821i \(0.254861\pi\)
\(912\) 0 0
\(913\) −7.21021 −0.238623
\(914\) 0 0
\(915\) 25.2822 0.835805
\(916\) 0 0
\(917\) 5.06006 0.167098
\(918\) 0 0
\(919\) 20.0000 0.659738 0.329869 0.944027i \(-0.392995\pi\)
0.329869 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 22.4598 0.740076
\(922\) 0 0
\(923\) 27.7847 0.914543
\(924\) 0 0
\(925\) 1.02698 0.0337668
\(926\) 0 0
\(927\) 15.2741 0.501666
\(928\) 0 0
\(929\) −22.5550 −0.740007 −0.370004 0.929030i \(-0.620644\pi\)
−0.370004 + 0.929030i \(0.620644\pi\)
\(930\) 0 0
\(931\) −1.32907 −0.0435587
\(932\) 0 0
\(933\) 27.7082 0.907127
\(934\) 0 0
\(935\) −0.982722 −0.0321385
\(936\) 0 0
\(937\) −33.3680 −1.09008 −0.545042 0.838409i \(-0.683486\pi\)
−0.545042 + 0.838409i \(0.683486\pi\)
\(938\) 0 0
\(939\) 0.503230 0.0164223
\(940\) 0 0
\(941\) 30.2255 0.985324 0.492662 0.870221i \(-0.336024\pi\)
0.492662 + 0.870221i \(0.336024\pi\)
\(942\) 0 0
\(943\) 3.52897 0.114919
\(944\) 0 0
\(945\) −6.98946 −0.227367
\(946\) 0 0
\(947\) −19.5974 −0.636830 −0.318415 0.947951i \(-0.603151\pi\)
−0.318415 + 0.947951i \(0.603151\pi\)
\(948\) 0 0
\(949\) 8.39175 0.272408
\(950\) 0 0
\(951\) 14.6482 0.475000
\(952\) 0 0
\(953\) −1.14762 −0.0371749 −0.0185875 0.999827i \(-0.505917\pi\)
−0.0185875 + 0.999827i \(0.505917\pi\)
\(954\) 0 0
\(955\) 66.2613 2.14417
\(956\) 0 0
\(957\) 0.526219 0.0170103
\(958\) 0 0
\(959\) −55.8513 −1.80353
\(960\) 0 0
\(961\) −16.3711 −0.528099
\(962\) 0 0
\(963\) −13.8318 −0.445722
\(964\) 0 0
\(965\) 21.4338 0.689979
\(966\) 0 0
\(967\) 6.81919 0.219290 0.109645 0.993971i \(-0.465029\pi\)
0.109645 + 0.993971i \(0.465029\pi\)
\(968\) 0 0
\(969\) 1.47909 0.0475154
\(970\) 0 0
\(971\) −20.5144 −0.658337 −0.329169 0.944271i \(-0.606768\pi\)
−0.329169 + 0.944271i \(0.606768\pi\)
\(972\) 0 0
\(973\) 19.3464 0.620216
\(974\) 0 0
\(975\) 2.65306 0.0849660
\(976\) 0 0
\(977\) −25.1984 −0.806168 −0.403084 0.915163i \(-0.632062\pi\)
−0.403084 + 0.915163i \(0.632062\pi\)
\(978\) 0 0
\(979\) −0.680915 −0.0217621
\(980\) 0 0
\(981\) −0.610729 −0.0194991
\(982\) 0 0
\(983\) −17.2132 −0.549017 −0.274509 0.961585i \(-0.588515\pi\)
−0.274509 + 0.961585i \(0.588515\pi\)
\(984\) 0 0
\(985\) 33.1601 1.05657
\(986\) 0 0
\(987\) 21.6759 0.689951
\(988\) 0 0
\(989\) 2.13846 0.0679992
\(990\) 0 0
\(991\) 7.10587 0.225725 0.112863 0.993611i \(-0.463998\pi\)
0.112863 + 0.993611i \(0.463998\pi\)
\(992\) 0 0
\(993\) −22.3670 −0.709796
\(994\) 0 0
\(995\) −20.1796 −0.639735
\(996\) 0 0
\(997\) 20.5726 0.651539 0.325770 0.945449i \(-0.394377\pi\)
0.325770 + 0.945449i \(0.394377\pi\)
\(998\) 0 0
\(999\) −0.747593 −0.0236528
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 8004.2.a.g.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.4 12 1.1 even 1 trivial