Properties

Label 8004.2.a.g.1.3
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 1.3
Root \(3.06377\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +3.57105 q^{11} +1.11209 q^{13} +3.06377 q^{15} -5.35191 q^{17} -1.67014 q^{19} -3.59303 q^{21} +1.00000 q^{23} +4.38666 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.98746 q^{31} -3.57105 q^{33} -11.0082 q^{35} -4.76622 q^{37} -1.11209 q^{39} +3.76801 q^{41} -11.7805 q^{43} -3.06377 q^{45} -6.41650 q^{47} +5.90985 q^{49} +5.35191 q^{51} -0.261608 q^{53} -10.9408 q^{55} +1.67014 q^{57} -7.33982 q^{59} -3.81452 q^{61} +3.59303 q^{63} -3.40720 q^{65} -0.212601 q^{67} -1.00000 q^{69} -15.4656 q^{71} +15.2479 q^{73} -4.38666 q^{75} +12.8309 q^{77} +14.1479 q^{79} +1.00000 q^{81} +4.48671 q^{83} +16.3970 q^{85} +1.00000 q^{87} +7.32038 q^{89} +3.99579 q^{91} -5.98746 q^{93} +5.11693 q^{95} +5.66075 q^{97} +3.57105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) −3.06377 −1.37016 −0.685079 0.728469i \(-0.740232\pi\)
−0.685079 + 0.728469i \(0.740232\pi\)
\(6\) 0 0
\(7\) 3.59303 1.35804 0.679018 0.734121i \(-0.262406\pi\)
0.679018 + 0.734121i \(0.262406\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.57105 1.07671 0.538355 0.842718i \(-0.319046\pi\)
0.538355 + 0.842718i \(0.319046\pi\)
\(12\) 0 0
\(13\) 1.11209 0.308440 0.154220 0.988037i \(-0.450714\pi\)
0.154220 + 0.988037i \(0.450714\pi\)
\(14\) 0 0
\(15\) 3.06377 0.791061
\(16\) 0 0
\(17\) −5.35191 −1.29803 −0.649014 0.760776i \(-0.724818\pi\)
−0.649014 + 0.760776i \(0.724818\pi\)
\(18\) 0 0
\(19\) −1.67014 −0.383157 −0.191579 0.981477i \(-0.561361\pi\)
−0.191579 + 0.981477i \(0.561361\pi\)
\(20\) 0 0
\(21\) −3.59303 −0.784063
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.38666 0.877332
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 5.98746 1.07538 0.537690 0.843143i \(-0.319297\pi\)
0.537690 + 0.843143i \(0.319297\pi\)
\(32\) 0 0
\(33\) −3.57105 −0.621639
\(34\) 0 0
\(35\) −11.0082 −1.86072
\(36\) 0 0
\(37\) −4.76622 −0.783562 −0.391781 0.920059i \(-0.628141\pi\)
−0.391781 + 0.920059i \(0.628141\pi\)
\(38\) 0 0
\(39\) −1.11209 −0.178078
\(40\) 0 0
\(41\) 3.76801 0.588464 0.294232 0.955734i \(-0.404936\pi\)
0.294232 + 0.955734i \(0.404936\pi\)
\(42\) 0 0
\(43\) −11.7805 −1.79651 −0.898256 0.439473i \(-0.855165\pi\)
−0.898256 + 0.439473i \(0.855165\pi\)
\(44\) 0 0
\(45\) −3.06377 −0.456719
\(46\) 0 0
\(47\) −6.41650 −0.935943 −0.467971 0.883744i \(-0.655015\pi\)
−0.467971 + 0.883744i \(0.655015\pi\)
\(48\) 0 0
\(49\) 5.90985 0.844264
\(50\) 0 0
\(51\) 5.35191 0.749417
\(52\) 0 0
\(53\) −0.261608 −0.0359346 −0.0179673 0.999839i \(-0.505719\pi\)
−0.0179673 + 0.999839i \(0.505719\pi\)
\(54\) 0 0
\(55\) −10.9408 −1.47526
\(56\) 0 0
\(57\) 1.67014 0.221216
\(58\) 0 0
\(59\) −7.33982 −0.955563 −0.477781 0.878479i \(-0.658559\pi\)
−0.477781 + 0.878479i \(0.658559\pi\)
\(60\) 0 0
\(61\) −3.81452 −0.488399 −0.244199 0.969725i \(-0.578525\pi\)
−0.244199 + 0.969725i \(0.578525\pi\)
\(62\) 0 0
\(63\) 3.59303 0.452679
\(64\) 0 0
\(65\) −3.40720 −0.422611
\(66\) 0 0
\(67\) −0.212601 −0.0259734 −0.0129867 0.999916i \(-0.504134\pi\)
−0.0129867 + 0.999916i \(0.504134\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −15.4656 −1.83542 −0.917712 0.397245i \(-0.869966\pi\)
−0.917712 + 0.397245i \(0.869966\pi\)
\(72\) 0 0
\(73\) 15.2479 1.78464 0.892318 0.451408i \(-0.149078\pi\)
0.892318 + 0.451408i \(0.149078\pi\)
\(74\) 0 0
\(75\) −4.38666 −0.506528
\(76\) 0 0
\(77\) 12.8309 1.46221
\(78\) 0 0
\(79\) 14.1479 1.59176 0.795879 0.605455i \(-0.207009\pi\)
0.795879 + 0.605455i \(0.207009\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.48671 0.492481 0.246240 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(84\) 0 0
\(85\) 16.3970 1.77850
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 7.32038 0.775958 0.387979 0.921668i \(-0.373173\pi\)
0.387979 + 0.921668i \(0.373173\pi\)
\(90\) 0 0
\(91\) 3.99579 0.418872
\(92\) 0 0
\(93\) −5.98746 −0.620871
\(94\) 0 0
\(95\) 5.11693 0.524986
\(96\) 0 0
\(97\) 5.66075 0.574762 0.287381 0.957816i \(-0.407215\pi\)
0.287381 + 0.957816i \(0.407215\pi\)
\(98\) 0 0
\(99\) 3.57105 0.358904
\(100\) 0 0
\(101\) −0.0231709 −0.00230559 −0.00115279 0.999999i \(-0.500367\pi\)
−0.00115279 + 0.999999i \(0.500367\pi\)
\(102\) 0 0
\(103\) −5.51046 −0.542962 −0.271481 0.962444i \(-0.587513\pi\)
−0.271481 + 0.962444i \(0.587513\pi\)
\(104\) 0 0
\(105\) 11.0082 1.07429
\(106\) 0 0
\(107\) 3.17665 0.307099 0.153549 0.988141i \(-0.450930\pi\)
0.153549 + 0.988141i \(0.450930\pi\)
\(108\) 0 0
\(109\) −11.5216 −1.10357 −0.551783 0.833987i \(-0.686053\pi\)
−0.551783 + 0.833987i \(0.686053\pi\)
\(110\) 0 0
\(111\) 4.76622 0.452390
\(112\) 0 0
\(113\) −0.0994853 −0.00935879 −0.00467940 0.999989i \(-0.501490\pi\)
−0.00467940 + 0.999989i \(0.501490\pi\)
\(114\) 0 0
\(115\) −3.06377 −0.285698
\(116\) 0 0
\(117\) 1.11209 0.102813
\(118\) 0 0
\(119\) −19.2296 −1.76277
\(120\) 0 0
\(121\) 1.75237 0.159306
\(122\) 0 0
\(123\) −3.76801 −0.339750
\(124\) 0 0
\(125\) 1.87914 0.168075
\(126\) 0 0
\(127\) 4.69936 0.417001 0.208500 0.978022i \(-0.433142\pi\)
0.208500 + 0.978022i \(0.433142\pi\)
\(128\) 0 0
\(129\) 11.7805 1.03722
\(130\) 0 0
\(131\) −15.2208 −1.32985 −0.664924 0.746911i \(-0.731536\pi\)
−0.664924 + 0.746911i \(0.731536\pi\)
\(132\) 0 0
\(133\) −6.00087 −0.520342
\(134\) 0 0
\(135\) 3.06377 0.263687
\(136\) 0 0
\(137\) −15.8868 −1.35730 −0.678651 0.734461i \(-0.737435\pi\)
−0.678651 + 0.734461i \(0.737435\pi\)
\(138\) 0 0
\(139\) −4.61451 −0.391397 −0.195699 0.980664i \(-0.562697\pi\)
−0.195699 + 0.980664i \(0.562697\pi\)
\(140\) 0 0
\(141\) 6.41650 0.540367
\(142\) 0 0
\(143\) 3.97134 0.332100
\(144\) 0 0
\(145\) 3.06377 0.254432
\(146\) 0 0
\(147\) −5.90985 −0.487436
\(148\) 0 0
\(149\) 6.38885 0.523395 0.261697 0.965150i \(-0.415718\pi\)
0.261697 + 0.965150i \(0.415718\pi\)
\(150\) 0 0
\(151\) −7.13056 −0.580276 −0.290138 0.956985i \(-0.593701\pi\)
−0.290138 + 0.956985i \(0.593701\pi\)
\(152\) 0 0
\(153\) −5.35191 −0.432676
\(154\) 0 0
\(155\) −18.3442 −1.47344
\(156\) 0 0
\(157\) 19.2868 1.53926 0.769628 0.638492i \(-0.220442\pi\)
0.769628 + 0.638492i \(0.220442\pi\)
\(158\) 0 0
\(159\) 0.261608 0.0207468
\(160\) 0 0
\(161\) 3.59303 0.283170
\(162\) 0 0
\(163\) 14.3483 1.12384 0.561922 0.827190i \(-0.310062\pi\)
0.561922 + 0.827190i \(0.310062\pi\)
\(164\) 0 0
\(165\) 10.9408 0.851744
\(166\) 0 0
\(167\) −5.56450 −0.430594 −0.215297 0.976549i \(-0.569072\pi\)
−0.215297 + 0.976549i \(0.569072\pi\)
\(168\) 0 0
\(169\) −11.7632 −0.904865
\(170\) 0 0
\(171\) −1.67014 −0.127719
\(172\) 0 0
\(173\) 2.66312 0.202473 0.101236 0.994862i \(-0.467720\pi\)
0.101236 + 0.994862i \(0.467720\pi\)
\(174\) 0 0
\(175\) 15.7614 1.19145
\(176\) 0 0
\(177\) 7.33982 0.551694
\(178\) 0 0
\(179\) −5.69011 −0.425299 −0.212649 0.977129i \(-0.568209\pi\)
−0.212649 + 0.977129i \(0.568209\pi\)
\(180\) 0 0
\(181\) −7.03851 −0.523168 −0.261584 0.965181i \(-0.584245\pi\)
−0.261584 + 0.965181i \(0.584245\pi\)
\(182\) 0 0
\(183\) 3.81452 0.281977
\(184\) 0 0
\(185\) 14.6026 1.07360
\(186\) 0 0
\(187\) −19.1119 −1.39760
\(188\) 0 0
\(189\) −3.59303 −0.261354
\(190\) 0 0
\(191\) −21.7486 −1.57367 −0.786836 0.617162i \(-0.788282\pi\)
−0.786836 + 0.617162i \(0.788282\pi\)
\(192\) 0 0
\(193\) 4.55878 0.328148 0.164074 0.986448i \(-0.447536\pi\)
0.164074 + 0.986448i \(0.447536\pi\)
\(194\) 0 0
\(195\) 3.40720 0.243994
\(196\) 0 0
\(197\) 2.74978 0.195914 0.0979569 0.995191i \(-0.468769\pi\)
0.0979569 + 0.995191i \(0.468769\pi\)
\(198\) 0 0
\(199\) 24.3927 1.72915 0.864575 0.502503i \(-0.167587\pi\)
0.864575 + 0.502503i \(0.167587\pi\)
\(200\) 0 0
\(201\) 0.212601 0.0149957
\(202\) 0 0
\(203\) −3.59303 −0.252181
\(204\) 0 0
\(205\) −11.5443 −0.806289
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −5.96416 −0.412550
\(210\) 0 0
\(211\) −9.28159 −0.638971 −0.319485 0.947591i \(-0.603510\pi\)
−0.319485 + 0.947591i \(0.603510\pi\)
\(212\) 0 0
\(213\) 15.4656 1.05968
\(214\) 0 0
\(215\) 36.0927 2.46150
\(216\) 0 0
\(217\) 21.5131 1.46041
\(218\) 0 0
\(219\) −15.2479 −1.03036
\(220\) 0 0
\(221\) −5.95183 −0.400363
\(222\) 0 0
\(223\) 19.8162 1.32699 0.663497 0.748179i \(-0.269072\pi\)
0.663497 + 0.748179i \(0.269072\pi\)
\(224\) 0 0
\(225\) 4.38666 0.292444
\(226\) 0 0
\(227\) 7.50724 0.498273 0.249136 0.968468i \(-0.419853\pi\)
0.249136 + 0.968468i \(0.419853\pi\)
\(228\) 0 0
\(229\) −8.32203 −0.549935 −0.274968 0.961453i \(-0.588667\pi\)
−0.274968 + 0.961453i \(0.588667\pi\)
\(230\) 0 0
\(231\) −12.8309 −0.844209
\(232\) 0 0
\(233\) 13.5592 0.888293 0.444146 0.895954i \(-0.353507\pi\)
0.444146 + 0.895954i \(0.353507\pi\)
\(234\) 0 0
\(235\) 19.6586 1.28239
\(236\) 0 0
\(237\) −14.1479 −0.919002
\(238\) 0 0
\(239\) −5.02517 −0.325051 −0.162526 0.986704i \(-0.551964\pi\)
−0.162526 + 0.986704i \(0.551964\pi\)
\(240\) 0 0
\(241\) −2.24679 −0.144729 −0.0723644 0.997378i \(-0.523054\pi\)
−0.0723644 + 0.997378i \(0.523054\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −18.1064 −1.15677
\(246\) 0 0
\(247\) −1.85736 −0.118181
\(248\) 0 0
\(249\) −4.48671 −0.284334
\(250\) 0 0
\(251\) 3.86119 0.243716 0.121858 0.992548i \(-0.461115\pi\)
0.121858 + 0.992548i \(0.461115\pi\)
\(252\) 0 0
\(253\) 3.57105 0.224510
\(254\) 0 0
\(255\) −16.3970 −1.02682
\(256\) 0 0
\(257\) −20.1469 −1.25673 −0.628365 0.777919i \(-0.716275\pi\)
−0.628365 + 0.777919i \(0.716275\pi\)
\(258\) 0 0
\(259\) −17.1252 −1.06411
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −32.1602 −1.98308 −0.991542 0.129788i \(-0.958570\pi\)
−0.991542 + 0.129788i \(0.958570\pi\)
\(264\) 0 0
\(265\) 0.801505 0.0492360
\(266\) 0 0
\(267\) −7.32038 −0.448000
\(268\) 0 0
\(269\) −10.6391 −0.648678 −0.324339 0.945941i \(-0.605142\pi\)
−0.324339 + 0.945941i \(0.605142\pi\)
\(270\) 0 0
\(271\) −5.40339 −0.328233 −0.164116 0.986441i \(-0.552477\pi\)
−0.164116 + 0.986441i \(0.552477\pi\)
\(272\) 0 0
\(273\) −3.99579 −0.241836
\(274\) 0 0
\(275\) 15.6650 0.944633
\(276\) 0 0
\(277\) −17.0246 −1.02291 −0.511454 0.859311i \(-0.670893\pi\)
−0.511454 + 0.859311i \(0.670893\pi\)
\(278\) 0 0
\(279\) 5.98746 0.358460
\(280\) 0 0
\(281\) 2.47389 0.147580 0.0737898 0.997274i \(-0.476491\pi\)
0.0737898 + 0.997274i \(0.476491\pi\)
\(282\) 0 0
\(283\) 18.6982 1.11150 0.555748 0.831351i \(-0.312432\pi\)
0.555748 + 0.831351i \(0.312432\pi\)
\(284\) 0 0
\(285\) −5.11693 −0.303101
\(286\) 0 0
\(287\) 13.5386 0.799156
\(288\) 0 0
\(289\) 11.6429 0.684878
\(290\) 0 0
\(291\) −5.66075 −0.331839
\(292\) 0 0
\(293\) −19.6458 −1.14772 −0.573861 0.818953i \(-0.694555\pi\)
−0.573861 + 0.818953i \(0.694555\pi\)
\(294\) 0 0
\(295\) 22.4875 1.30927
\(296\) 0 0
\(297\) −3.57105 −0.207213
\(298\) 0 0
\(299\) 1.11209 0.0643141
\(300\) 0 0
\(301\) −42.3277 −2.43973
\(302\) 0 0
\(303\) 0.0231709 0.00133113
\(304\) 0 0
\(305\) 11.6868 0.669183
\(306\) 0 0
\(307\) −12.5654 −0.717144 −0.358572 0.933502i \(-0.616736\pi\)
−0.358572 + 0.933502i \(0.616736\pi\)
\(308\) 0 0
\(309\) 5.51046 0.313479
\(310\) 0 0
\(311\) −11.0036 −0.623954 −0.311977 0.950090i \(-0.600991\pi\)
−0.311977 + 0.950090i \(0.600991\pi\)
\(312\) 0 0
\(313\) 15.4938 0.875762 0.437881 0.899033i \(-0.355729\pi\)
0.437881 + 0.899033i \(0.355729\pi\)
\(314\) 0 0
\(315\) −11.0082 −0.620241
\(316\) 0 0
\(317\) 4.06492 0.228309 0.114154 0.993463i \(-0.463584\pi\)
0.114154 + 0.993463i \(0.463584\pi\)
\(318\) 0 0
\(319\) −3.57105 −0.199940
\(320\) 0 0
\(321\) −3.17665 −0.177304
\(322\) 0 0
\(323\) 8.93846 0.497349
\(324\) 0 0
\(325\) 4.87838 0.270604
\(326\) 0 0
\(327\) 11.5216 0.637145
\(328\) 0 0
\(329\) −23.0547 −1.27104
\(330\) 0 0
\(331\) −14.8554 −0.816524 −0.408262 0.912865i \(-0.633865\pi\)
−0.408262 + 0.912865i \(0.633865\pi\)
\(332\) 0 0
\(333\) −4.76622 −0.261187
\(334\) 0 0
\(335\) 0.651360 0.0355876
\(336\) 0 0
\(337\) −15.2412 −0.830242 −0.415121 0.909766i \(-0.636261\pi\)
−0.415121 + 0.909766i \(0.636261\pi\)
\(338\) 0 0
\(339\) 0.0994853 0.00540330
\(340\) 0 0
\(341\) 21.3815 1.15787
\(342\) 0 0
\(343\) −3.91694 −0.211495
\(344\) 0 0
\(345\) 3.06377 0.164948
\(346\) 0 0
\(347\) −3.98566 −0.213962 −0.106981 0.994261i \(-0.534118\pi\)
−0.106981 + 0.994261i \(0.534118\pi\)
\(348\) 0 0
\(349\) −21.3660 −1.14370 −0.571848 0.820360i \(-0.693773\pi\)
−0.571848 + 0.820360i \(0.693773\pi\)
\(350\) 0 0
\(351\) −1.11209 −0.0593592
\(352\) 0 0
\(353\) 1.96364 0.104514 0.0522569 0.998634i \(-0.483359\pi\)
0.0522569 + 0.998634i \(0.483359\pi\)
\(354\) 0 0
\(355\) 47.3829 2.51482
\(356\) 0 0
\(357\) 19.2296 1.01774
\(358\) 0 0
\(359\) −16.1743 −0.853646 −0.426823 0.904335i \(-0.640367\pi\)
−0.426823 + 0.904335i \(0.640367\pi\)
\(360\) 0 0
\(361\) −16.2106 −0.853191
\(362\) 0 0
\(363\) −1.75237 −0.0919756
\(364\) 0 0
\(365\) −46.7161 −2.44523
\(366\) 0 0
\(367\) 8.17933 0.426957 0.213479 0.976948i \(-0.431521\pi\)
0.213479 + 0.976948i \(0.431521\pi\)
\(368\) 0 0
\(369\) 3.76801 0.196155
\(370\) 0 0
\(371\) −0.939964 −0.0488005
\(372\) 0 0
\(373\) −18.4633 −0.955994 −0.477997 0.878361i \(-0.658637\pi\)
−0.477997 + 0.878361i \(0.658637\pi\)
\(374\) 0 0
\(375\) −1.87914 −0.0970381
\(376\) 0 0
\(377\) −1.11209 −0.0572758
\(378\) 0 0
\(379\) −9.89915 −0.508485 −0.254243 0.967140i \(-0.581826\pi\)
−0.254243 + 0.967140i \(0.581826\pi\)
\(380\) 0 0
\(381\) −4.69936 −0.240755
\(382\) 0 0
\(383\) −8.58947 −0.438901 −0.219451 0.975624i \(-0.570426\pi\)
−0.219451 + 0.975624i \(0.570426\pi\)
\(384\) 0 0
\(385\) −39.3108 −2.00346
\(386\) 0 0
\(387\) −11.7805 −0.598837
\(388\) 0 0
\(389\) −10.3169 −0.523088 −0.261544 0.965191i \(-0.584232\pi\)
−0.261544 + 0.965191i \(0.584232\pi\)
\(390\) 0 0
\(391\) −5.35191 −0.270658
\(392\) 0 0
\(393\) 15.2208 0.767788
\(394\) 0 0
\(395\) −43.3457 −2.18096
\(396\) 0 0
\(397\) −5.68400 −0.285272 −0.142636 0.989775i \(-0.545558\pi\)
−0.142636 + 0.989775i \(0.545558\pi\)
\(398\) 0 0
\(399\) 6.00087 0.300419
\(400\) 0 0
\(401\) 6.02992 0.301120 0.150560 0.988601i \(-0.451892\pi\)
0.150560 + 0.988601i \(0.451892\pi\)
\(402\) 0 0
\(403\) 6.65863 0.331690
\(404\) 0 0
\(405\) −3.06377 −0.152240
\(406\) 0 0
\(407\) −17.0204 −0.843669
\(408\) 0 0
\(409\) 8.16164 0.403567 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(410\) 0 0
\(411\) 15.8868 0.783638
\(412\) 0 0
\(413\) −26.3722 −1.29769
\(414\) 0 0
\(415\) −13.7462 −0.674776
\(416\) 0 0
\(417\) 4.61451 0.225973
\(418\) 0 0
\(419\) −11.6677 −0.570007 −0.285003 0.958527i \(-0.591995\pi\)
−0.285003 + 0.958527i \(0.591995\pi\)
\(420\) 0 0
\(421\) −1.91906 −0.0935293 −0.0467647 0.998906i \(-0.514891\pi\)
−0.0467647 + 0.998906i \(0.514891\pi\)
\(422\) 0 0
\(423\) −6.41650 −0.311981
\(424\) 0 0
\(425\) −23.4770 −1.13880
\(426\) 0 0
\(427\) −13.7057 −0.663264
\(428\) 0 0
\(429\) −3.97134 −0.191738
\(430\) 0 0
\(431\) 16.5396 0.796684 0.398342 0.917237i \(-0.369586\pi\)
0.398342 + 0.917237i \(0.369586\pi\)
\(432\) 0 0
\(433\) 25.1753 1.20985 0.604923 0.796284i \(-0.293204\pi\)
0.604923 + 0.796284i \(0.293204\pi\)
\(434\) 0 0
\(435\) −3.06377 −0.146896
\(436\) 0 0
\(437\) −1.67014 −0.0798938
\(438\) 0 0
\(439\) 2.53610 0.121042 0.0605208 0.998167i \(-0.480724\pi\)
0.0605208 + 0.998167i \(0.480724\pi\)
\(440\) 0 0
\(441\) 5.90985 0.281421
\(442\) 0 0
\(443\) −20.1892 −0.959219 −0.479610 0.877482i \(-0.659222\pi\)
−0.479610 + 0.877482i \(0.659222\pi\)
\(444\) 0 0
\(445\) −22.4279 −1.06319
\(446\) 0 0
\(447\) −6.38885 −0.302182
\(448\) 0 0
\(449\) −11.7302 −0.553584 −0.276792 0.960930i \(-0.589271\pi\)
−0.276792 + 0.960930i \(0.589271\pi\)
\(450\) 0 0
\(451\) 13.4557 0.633606
\(452\) 0 0
\(453\) 7.13056 0.335023
\(454\) 0 0
\(455\) −12.2422 −0.573921
\(456\) 0 0
\(457\) −9.00150 −0.421072 −0.210536 0.977586i \(-0.567521\pi\)
−0.210536 + 0.977586i \(0.567521\pi\)
\(458\) 0 0
\(459\) 5.35191 0.249806
\(460\) 0 0
\(461\) −19.8985 −0.926764 −0.463382 0.886159i \(-0.653364\pi\)
−0.463382 + 0.886159i \(0.653364\pi\)
\(462\) 0 0
\(463\) −3.02956 −0.140796 −0.0703978 0.997519i \(-0.522427\pi\)
−0.0703978 + 0.997519i \(0.522427\pi\)
\(464\) 0 0
\(465\) 18.3442 0.850691
\(466\) 0 0
\(467\) −5.74598 −0.265892 −0.132946 0.991123i \(-0.542444\pi\)
−0.132946 + 0.991123i \(0.542444\pi\)
\(468\) 0 0
\(469\) −0.763882 −0.0352728
\(470\) 0 0
\(471\) −19.2868 −0.888690
\(472\) 0 0
\(473\) −42.0688 −1.93432
\(474\) 0 0
\(475\) −7.32635 −0.336156
\(476\) 0 0
\(477\) −0.261608 −0.0119782
\(478\) 0 0
\(479\) 14.9178 0.681614 0.340807 0.940133i \(-0.389300\pi\)
0.340807 + 0.940133i \(0.389300\pi\)
\(480\) 0 0
\(481\) −5.30049 −0.241681
\(482\) 0 0
\(483\) −3.59303 −0.163488
\(484\) 0 0
\(485\) −17.3432 −0.787514
\(486\) 0 0
\(487\) −29.1042 −1.31884 −0.659418 0.751776i \(-0.729197\pi\)
−0.659418 + 0.751776i \(0.729197\pi\)
\(488\) 0 0
\(489\) −14.3483 −0.648851
\(490\) 0 0
\(491\) −38.5168 −1.73824 −0.869120 0.494602i \(-0.835314\pi\)
−0.869120 + 0.494602i \(0.835314\pi\)
\(492\) 0 0
\(493\) 5.35191 0.241038
\(494\) 0 0
\(495\) −10.9408 −0.491755
\(496\) 0 0
\(497\) −55.5682 −2.49257
\(498\) 0 0
\(499\) −7.15238 −0.320185 −0.160092 0.987102i \(-0.551179\pi\)
−0.160092 + 0.987102i \(0.551179\pi\)
\(500\) 0 0
\(501\) 5.56450 0.248604
\(502\) 0 0
\(503\) −37.5510 −1.67432 −0.837158 0.546961i \(-0.815785\pi\)
−0.837158 + 0.546961i \(0.815785\pi\)
\(504\) 0 0
\(505\) 0.0709901 0.00315902
\(506\) 0 0
\(507\) 11.7632 0.522424
\(508\) 0 0
\(509\) −2.74755 −0.121783 −0.0608914 0.998144i \(-0.519394\pi\)
−0.0608914 + 0.998144i \(0.519394\pi\)
\(510\) 0 0
\(511\) 54.7862 2.42360
\(512\) 0 0
\(513\) 1.67014 0.0737386
\(514\) 0 0
\(515\) 16.8828 0.743943
\(516\) 0 0
\(517\) −22.9136 −1.00774
\(518\) 0 0
\(519\) −2.66312 −0.116898
\(520\) 0 0
\(521\) −25.5716 −1.12031 −0.560156 0.828387i \(-0.689259\pi\)
−0.560156 + 0.828387i \(0.689259\pi\)
\(522\) 0 0
\(523\) 17.5183 0.766024 0.383012 0.923743i \(-0.374887\pi\)
0.383012 + 0.923743i \(0.374887\pi\)
\(524\) 0 0
\(525\) −15.7614 −0.687883
\(526\) 0 0
\(527\) −32.0444 −1.39587
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.33982 −0.318521
\(532\) 0 0
\(533\) 4.19039 0.181506
\(534\) 0 0
\(535\) −9.73252 −0.420774
\(536\) 0 0
\(537\) 5.69011 0.245546
\(538\) 0 0
\(539\) 21.1043 0.909029
\(540\) 0 0
\(541\) −30.9368 −1.33008 −0.665039 0.746809i \(-0.731585\pi\)
−0.665039 + 0.746809i \(0.731585\pi\)
\(542\) 0 0
\(543\) 7.03851 0.302051
\(544\) 0 0
\(545\) 35.2994 1.51206
\(546\) 0 0
\(547\) 27.4338 1.17299 0.586493 0.809955i \(-0.300508\pi\)
0.586493 + 0.809955i \(0.300508\pi\)
\(548\) 0 0
\(549\) −3.81452 −0.162800
\(550\) 0 0
\(551\) 1.67014 0.0711505
\(552\) 0 0
\(553\) 50.8337 2.16167
\(554\) 0 0
\(555\) −14.6026 −0.619845
\(556\) 0 0
\(557\) 18.3290 0.776627 0.388313 0.921527i \(-0.373058\pi\)
0.388313 + 0.921527i \(0.373058\pi\)
\(558\) 0 0
\(559\) −13.1010 −0.554115
\(560\) 0 0
\(561\) 19.1119 0.806906
\(562\) 0 0
\(563\) 6.98199 0.294256 0.147128 0.989117i \(-0.452997\pi\)
0.147128 + 0.989117i \(0.452997\pi\)
\(564\) 0 0
\(565\) 0.304800 0.0128230
\(566\) 0 0
\(567\) 3.59303 0.150893
\(568\) 0 0
\(569\) 39.0041 1.63514 0.817569 0.575831i \(-0.195321\pi\)
0.817569 + 0.575831i \(0.195321\pi\)
\(570\) 0 0
\(571\) 14.1398 0.591730 0.295865 0.955230i \(-0.404392\pi\)
0.295865 + 0.955230i \(0.404392\pi\)
\(572\) 0 0
\(573\) 21.7486 0.908560
\(574\) 0 0
\(575\) 4.38666 0.182936
\(576\) 0 0
\(577\) −41.1894 −1.71474 −0.857369 0.514702i \(-0.827903\pi\)
−0.857369 + 0.514702i \(0.827903\pi\)
\(578\) 0 0
\(579\) −4.55878 −0.189456
\(580\) 0 0
\(581\) 16.1209 0.668807
\(582\) 0 0
\(583\) −0.934213 −0.0386912
\(584\) 0 0
\(585\) −3.40720 −0.140870
\(586\) 0 0
\(587\) 27.7763 1.14645 0.573225 0.819398i \(-0.305692\pi\)
0.573225 + 0.819398i \(0.305692\pi\)
\(588\) 0 0
\(589\) −9.99992 −0.412040
\(590\) 0 0
\(591\) −2.74978 −0.113111
\(592\) 0 0
\(593\) 5.43304 0.223108 0.111554 0.993758i \(-0.464417\pi\)
0.111554 + 0.993758i \(0.464417\pi\)
\(594\) 0 0
\(595\) 58.9149 2.41527
\(596\) 0 0
\(597\) −24.3927 −0.998326
\(598\) 0 0
\(599\) −1.30603 −0.0533631 −0.0266816 0.999644i \(-0.508494\pi\)
−0.0266816 + 0.999644i \(0.508494\pi\)
\(600\) 0 0
\(601\) −34.8724 −1.42248 −0.711238 0.702951i \(-0.751865\pi\)
−0.711238 + 0.702951i \(0.751865\pi\)
\(602\) 0 0
\(603\) −0.212601 −0.00865779
\(604\) 0 0
\(605\) −5.36885 −0.218275
\(606\) 0 0
\(607\) 42.4139 1.72153 0.860764 0.509004i \(-0.169986\pi\)
0.860764 + 0.509004i \(0.169986\pi\)
\(608\) 0 0
\(609\) 3.59303 0.145597
\(610\) 0 0
\(611\) −7.13576 −0.288682
\(612\) 0 0
\(613\) −8.94979 −0.361479 −0.180739 0.983531i \(-0.557849\pi\)
−0.180739 + 0.983531i \(0.557849\pi\)
\(614\) 0 0
\(615\) 11.5443 0.465511
\(616\) 0 0
\(617\) 25.1866 1.01397 0.506986 0.861954i \(-0.330759\pi\)
0.506986 + 0.861954i \(0.330759\pi\)
\(618\) 0 0
\(619\) −19.2846 −0.775112 −0.387556 0.921846i \(-0.626681\pi\)
−0.387556 + 0.921846i \(0.626681\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 26.3023 1.05378
\(624\) 0 0
\(625\) −27.6905 −1.10762
\(626\) 0 0
\(627\) 5.96416 0.238186
\(628\) 0 0
\(629\) 25.5084 1.01709
\(630\) 0 0
\(631\) 42.6072 1.69616 0.848082 0.529865i \(-0.177757\pi\)
0.848082 + 0.529865i \(0.177757\pi\)
\(632\) 0 0
\(633\) 9.28159 0.368910
\(634\) 0 0
\(635\) −14.3977 −0.571356
\(636\) 0 0
\(637\) 6.57231 0.260405
\(638\) 0 0
\(639\) −15.4656 −0.611808
\(640\) 0 0
\(641\) −32.7436 −1.29329 −0.646647 0.762789i \(-0.723829\pi\)
−0.646647 + 0.762789i \(0.723829\pi\)
\(642\) 0 0
\(643\) −20.2307 −0.797821 −0.398911 0.916990i \(-0.630612\pi\)
−0.398911 + 0.916990i \(0.630612\pi\)
\(644\) 0 0
\(645\) −36.0927 −1.42115
\(646\) 0 0
\(647\) 19.7182 0.775201 0.387601 0.921827i \(-0.373304\pi\)
0.387601 + 0.921827i \(0.373304\pi\)
\(648\) 0 0
\(649\) −26.2108 −1.02886
\(650\) 0 0
\(651\) −21.5131 −0.843166
\(652\) 0 0
\(653\) 37.4049 1.46377 0.731884 0.681430i \(-0.238641\pi\)
0.731884 + 0.681430i \(0.238641\pi\)
\(654\) 0 0
\(655\) 46.6330 1.82210
\(656\) 0 0
\(657\) 15.2479 0.594878
\(658\) 0 0
\(659\) −34.1613 −1.33074 −0.665368 0.746516i \(-0.731725\pi\)
−0.665368 + 0.746516i \(0.731725\pi\)
\(660\) 0 0
\(661\) 30.1100 1.17114 0.585572 0.810620i \(-0.300870\pi\)
0.585572 + 0.810620i \(0.300870\pi\)
\(662\) 0 0
\(663\) 5.95183 0.231150
\(664\) 0 0
\(665\) 18.3853 0.712950
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −19.8162 −0.766140
\(670\) 0 0
\(671\) −13.6218 −0.525864
\(672\) 0 0
\(673\) −14.3930 −0.554811 −0.277405 0.960753i \(-0.589475\pi\)
−0.277405 + 0.960753i \(0.589475\pi\)
\(674\) 0 0
\(675\) −4.38666 −0.168843
\(676\) 0 0
\(677\) 9.90351 0.380623 0.190311 0.981724i \(-0.439050\pi\)
0.190311 + 0.981724i \(0.439050\pi\)
\(678\) 0 0
\(679\) 20.3392 0.780548
\(680\) 0 0
\(681\) −7.50724 −0.287678
\(682\) 0 0
\(683\) 24.6562 0.943442 0.471721 0.881748i \(-0.343633\pi\)
0.471721 + 0.881748i \(0.343633\pi\)
\(684\) 0 0
\(685\) 48.6735 1.85972
\(686\) 0 0
\(687\) 8.32203 0.317505
\(688\) 0 0
\(689\) −0.290933 −0.0110836
\(690\) 0 0
\(691\) −13.2353 −0.503496 −0.251748 0.967793i \(-0.581005\pi\)
−0.251748 + 0.967793i \(0.581005\pi\)
\(692\) 0 0
\(693\) 12.8309 0.487404
\(694\) 0 0
\(695\) 14.1378 0.536276
\(696\) 0 0
\(697\) −20.1661 −0.763844
\(698\) 0 0
\(699\) −13.5592 −0.512856
\(700\) 0 0
\(701\) 0.304707 0.0115086 0.00575432 0.999983i \(-0.498168\pi\)
0.00575432 + 0.999983i \(0.498168\pi\)
\(702\) 0 0
\(703\) 7.96027 0.300227
\(704\) 0 0
\(705\) −19.6586 −0.740388
\(706\) 0 0
\(707\) −0.0832536 −0.00313107
\(708\) 0 0
\(709\) 37.7708 1.41851 0.709256 0.704951i \(-0.249031\pi\)
0.709256 + 0.704951i \(0.249031\pi\)
\(710\) 0 0
\(711\) 14.1479 0.530586
\(712\) 0 0
\(713\) 5.98746 0.224232
\(714\) 0 0
\(715\) −12.1673 −0.455030
\(716\) 0 0
\(717\) 5.02517 0.187668
\(718\) 0 0
\(719\) 10.6927 0.398769 0.199385 0.979921i \(-0.436106\pi\)
0.199385 + 0.979921i \(0.436106\pi\)
\(720\) 0 0
\(721\) −19.7992 −0.737362
\(722\) 0 0
\(723\) 2.24679 0.0835592
\(724\) 0 0
\(725\) −4.38666 −0.162916
\(726\) 0 0
\(727\) −22.2928 −0.826795 −0.413397 0.910551i \(-0.635658\pi\)
−0.413397 + 0.910551i \(0.635658\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 63.0482 2.33192
\(732\) 0 0
\(733\) −38.1613 −1.40952 −0.704759 0.709447i \(-0.748945\pi\)
−0.704759 + 0.709447i \(0.748945\pi\)
\(734\) 0 0
\(735\) 18.1064 0.667864
\(736\) 0 0
\(737\) −0.759209 −0.0279658
\(738\) 0 0
\(739\) −45.0827 −1.65839 −0.829197 0.558957i \(-0.811202\pi\)
−0.829197 + 0.558957i \(0.811202\pi\)
\(740\) 0 0
\(741\) 1.85736 0.0682318
\(742\) 0 0
\(743\) 32.4523 1.19056 0.595280 0.803518i \(-0.297041\pi\)
0.595280 + 0.803518i \(0.297041\pi\)
\(744\) 0 0
\(745\) −19.5739 −0.717133
\(746\) 0 0
\(747\) 4.48671 0.164160
\(748\) 0 0
\(749\) 11.4138 0.417051
\(750\) 0 0
\(751\) −51.5238 −1.88013 −0.940065 0.340994i \(-0.889236\pi\)
−0.940065 + 0.340994i \(0.889236\pi\)
\(752\) 0 0
\(753\) −3.86119 −0.140709
\(754\) 0 0
\(755\) 21.8464 0.795070
\(756\) 0 0
\(757\) 24.0031 0.872409 0.436205 0.899848i \(-0.356322\pi\)
0.436205 + 0.899848i \(0.356322\pi\)
\(758\) 0 0
\(759\) −3.57105 −0.129621
\(760\) 0 0
\(761\) −11.1063 −0.402605 −0.201302 0.979529i \(-0.564517\pi\)
−0.201302 + 0.979529i \(0.564517\pi\)
\(762\) 0 0
\(763\) −41.3973 −1.49868
\(764\) 0 0
\(765\) 16.3970 0.592835
\(766\) 0 0
\(767\) −8.16257 −0.294733
\(768\) 0 0
\(769\) 30.5474 1.10157 0.550784 0.834648i \(-0.314329\pi\)
0.550784 + 0.834648i \(0.314329\pi\)
\(770\) 0 0
\(771\) 20.1469 0.725573
\(772\) 0 0
\(773\) −35.2240 −1.26692 −0.633460 0.773775i \(-0.718366\pi\)
−0.633460 + 0.773775i \(0.718366\pi\)
\(774\) 0 0
\(775\) 26.2650 0.943465
\(776\) 0 0
\(777\) 17.1252 0.614362
\(778\) 0 0
\(779\) −6.29312 −0.225474
\(780\) 0 0
\(781\) −55.2282 −1.97622
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −59.0903 −2.10902
\(786\) 0 0
\(787\) −8.36305 −0.298111 −0.149055 0.988829i \(-0.547623\pi\)
−0.149055 + 0.988829i \(0.547623\pi\)
\(788\) 0 0
\(789\) 32.1602 1.14493
\(790\) 0 0
\(791\) −0.357454 −0.0127096
\(792\) 0 0
\(793\) −4.24210 −0.150642
\(794\) 0 0
\(795\) −0.801505 −0.0284264
\(796\) 0 0
\(797\) −39.6446 −1.40429 −0.702143 0.712036i \(-0.747773\pi\)
−0.702143 + 0.712036i \(0.747773\pi\)
\(798\) 0 0
\(799\) 34.3405 1.21488
\(800\) 0 0
\(801\) 7.32038 0.258653
\(802\) 0 0
\(803\) 54.4511 1.92154
\(804\) 0 0
\(805\) −11.0082 −0.387988
\(806\) 0 0
\(807\) 10.6391 0.374514
\(808\) 0 0
\(809\) 0.856343 0.0301074 0.0150537 0.999887i \(-0.495208\pi\)
0.0150537 + 0.999887i \(0.495208\pi\)
\(810\) 0 0
\(811\) −34.9984 −1.22896 −0.614480 0.788932i \(-0.710634\pi\)
−0.614480 + 0.788932i \(0.710634\pi\)
\(812\) 0 0
\(813\) 5.40339 0.189505
\(814\) 0 0
\(815\) −43.9597 −1.53984
\(816\) 0 0
\(817\) 19.6752 0.688346
\(818\) 0 0
\(819\) 3.99579 0.139624
\(820\) 0 0
\(821\) −6.38652 −0.222891 −0.111445 0.993771i \(-0.535548\pi\)
−0.111445 + 0.993771i \(0.535548\pi\)
\(822\) 0 0
\(823\) −5.71924 −0.199360 −0.0996802 0.995020i \(-0.531782\pi\)
−0.0996802 + 0.995020i \(0.531782\pi\)
\(824\) 0 0
\(825\) −15.6650 −0.545384
\(826\) 0 0
\(827\) 8.17831 0.284388 0.142194 0.989839i \(-0.454584\pi\)
0.142194 + 0.989839i \(0.454584\pi\)
\(828\) 0 0
\(829\) 33.1778 1.15231 0.576156 0.817340i \(-0.304552\pi\)
0.576156 + 0.817340i \(0.304552\pi\)
\(830\) 0 0
\(831\) 17.0246 0.590576
\(832\) 0 0
\(833\) −31.6290 −1.09588
\(834\) 0 0
\(835\) 17.0483 0.589982
\(836\) 0 0
\(837\) −5.98746 −0.206957
\(838\) 0 0
\(839\) 22.5616 0.778914 0.389457 0.921045i \(-0.372663\pi\)
0.389457 + 0.921045i \(0.372663\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −2.47389 −0.0852052
\(844\) 0 0
\(845\) 36.0398 1.23981
\(846\) 0 0
\(847\) 6.29632 0.216344
\(848\) 0 0
\(849\) −18.6982 −0.641722
\(850\) 0 0
\(851\) −4.76622 −0.163384
\(852\) 0 0
\(853\) 15.1924 0.520177 0.260088 0.965585i \(-0.416248\pi\)
0.260088 + 0.965585i \(0.416248\pi\)
\(854\) 0 0
\(855\) 5.11693 0.174995
\(856\) 0 0
\(857\) 6.28683 0.214754 0.107377 0.994218i \(-0.465755\pi\)
0.107377 + 0.994218i \(0.465755\pi\)
\(858\) 0 0
\(859\) 9.15460 0.312351 0.156175 0.987729i \(-0.450083\pi\)
0.156175 + 0.987729i \(0.450083\pi\)
\(860\) 0 0
\(861\) −13.5386 −0.461393
\(862\) 0 0
\(863\) −35.4016 −1.20508 −0.602542 0.798087i \(-0.705845\pi\)
−0.602542 + 0.798087i \(0.705845\pi\)
\(864\) 0 0
\(865\) −8.15916 −0.277420
\(866\) 0 0
\(867\) −11.6429 −0.395415
\(868\) 0 0
\(869\) 50.5227 1.71386
\(870\) 0 0
\(871\) −0.236433 −0.00801122
\(872\) 0 0
\(873\) 5.66075 0.191587
\(874\) 0 0
\(875\) 6.75179 0.228252
\(876\) 0 0
\(877\) −45.5745 −1.53894 −0.769471 0.638682i \(-0.779480\pi\)
−0.769471 + 0.638682i \(0.779480\pi\)
\(878\) 0 0
\(879\) 19.6458 0.662638
\(880\) 0 0
\(881\) −3.80769 −0.128284 −0.0641422 0.997941i \(-0.520431\pi\)
−0.0641422 + 0.997941i \(0.520431\pi\)
\(882\) 0 0
\(883\) −50.1228 −1.68677 −0.843384 0.537312i \(-0.819440\pi\)
−0.843384 + 0.537312i \(0.819440\pi\)
\(884\) 0 0
\(885\) −22.4875 −0.755908
\(886\) 0 0
\(887\) 29.2728 0.982884 0.491442 0.870910i \(-0.336470\pi\)
0.491442 + 0.870910i \(0.336470\pi\)
\(888\) 0 0
\(889\) 16.8849 0.566302
\(890\) 0 0
\(891\) 3.57105 0.119635
\(892\) 0 0
\(893\) 10.7165 0.358613
\(894\) 0 0
\(895\) 17.4332 0.582727
\(896\) 0 0
\(897\) −1.11209 −0.0371318
\(898\) 0 0
\(899\) −5.98746 −0.199693
\(900\) 0 0
\(901\) 1.40010 0.0466441
\(902\) 0 0
\(903\) 42.3277 1.40858
\(904\) 0 0
\(905\) 21.5643 0.716823
\(906\) 0 0
\(907\) 0.0482385 0.00160173 0.000800866 1.00000i \(-0.499745\pi\)
0.000800866 1.00000i \(0.499745\pi\)
\(908\) 0 0
\(909\) −0.0231709 −0.000768529 0
\(910\) 0 0
\(911\) −59.2456 −1.96289 −0.981447 0.191734i \(-0.938589\pi\)
−0.981447 + 0.191734i \(0.938589\pi\)
\(912\) 0 0
\(913\) 16.0223 0.530259
\(914\) 0 0
\(915\) −11.6868 −0.386353
\(916\) 0 0
\(917\) −54.6888 −1.80598
\(918\) 0 0
\(919\) 32.5343 1.07321 0.536604 0.843834i \(-0.319707\pi\)
0.536604 + 0.843834i \(0.319707\pi\)
\(920\) 0 0
\(921\) 12.5654 0.414043
\(922\) 0 0
\(923\) −17.1992 −0.566118
\(924\) 0 0
\(925\) −20.9078 −0.687443
\(926\) 0 0
\(927\) −5.51046 −0.180987
\(928\) 0 0
\(929\) 37.1472 1.21876 0.609380 0.792878i \(-0.291418\pi\)
0.609380 + 0.792878i \(0.291418\pi\)
\(930\) 0 0
\(931\) −9.87030 −0.323486
\(932\) 0 0
\(933\) 11.0036 0.360240
\(934\) 0 0
\(935\) 58.5544 1.91493
\(936\) 0 0
\(937\) −13.3829 −0.437199 −0.218599 0.975815i \(-0.570149\pi\)
−0.218599 + 0.975815i \(0.570149\pi\)
\(938\) 0 0
\(939\) −15.4938 −0.505622
\(940\) 0 0
\(941\) 3.67507 0.119804 0.0599019 0.998204i \(-0.480921\pi\)
0.0599019 + 0.998204i \(0.480921\pi\)
\(942\) 0 0
\(943\) 3.76801 0.122703
\(944\) 0 0
\(945\) 11.0082 0.358097
\(946\) 0 0
\(947\) 15.2302 0.494916 0.247458 0.968899i \(-0.420405\pi\)
0.247458 + 0.968899i \(0.420405\pi\)
\(948\) 0 0
\(949\) 16.9571 0.550452
\(950\) 0 0
\(951\) −4.06492 −0.131814
\(952\) 0 0
\(953\) 23.3029 0.754855 0.377428 0.926039i \(-0.376809\pi\)
0.377428 + 0.926039i \(0.376809\pi\)
\(954\) 0 0
\(955\) 66.6326 2.15618
\(956\) 0 0
\(957\) 3.57105 0.115436
\(958\) 0 0
\(959\) −57.0817 −1.84327
\(960\) 0 0
\(961\) 4.84972 0.156442
\(962\) 0 0
\(963\) 3.17665 0.102366
\(964\) 0 0
\(965\) −13.9670 −0.449614
\(966\) 0 0
\(967\) −38.5543 −1.23982 −0.619911 0.784672i \(-0.712831\pi\)
−0.619911 + 0.784672i \(0.712831\pi\)
\(968\) 0 0
\(969\) −8.93846 −0.287145
\(970\) 0 0
\(971\) 46.9412 1.50642 0.753208 0.657782i \(-0.228505\pi\)
0.753208 + 0.657782i \(0.228505\pi\)
\(972\) 0 0
\(973\) −16.5800 −0.531532
\(974\) 0 0
\(975\) −4.87838 −0.156233
\(976\) 0 0
\(977\) −16.4403 −0.525973 −0.262986 0.964800i \(-0.584707\pi\)
−0.262986 + 0.964800i \(0.584707\pi\)
\(978\) 0 0
\(979\) 26.1414 0.835483
\(980\) 0 0
\(981\) −11.5216 −0.367856
\(982\) 0 0
\(983\) 14.4253 0.460095 0.230048 0.973179i \(-0.426112\pi\)
0.230048 + 0.973179i \(0.426112\pi\)
\(984\) 0 0
\(985\) −8.42468 −0.268433
\(986\) 0 0
\(987\) 23.0547 0.733838
\(988\) 0 0
\(989\) −11.7805 −0.374599
\(990\) 0 0
\(991\) 30.6353 0.973162 0.486581 0.873635i \(-0.338244\pi\)
0.486581 + 0.873635i \(0.338244\pi\)
\(992\) 0 0
\(993\) 14.8554 0.471420
\(994\) 0 0
\(995\) −74.7334 −2.36921
\(996\) 0 0
\(997\) 40.4323 1.28051 0.640253 0.768164i \(-0.278830\pi\)
0.640253 + 0.768164i \(0.278830\pi\)
\(998\) 0 0
\(999\) 4.76622 0.150797
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.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.3 12 1.1 even 1 trivial