Properties

Label 8004.2.a.g.1.11
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 1.11
Root \(-2.64977\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} -1.24093 q^{11} -1.48851 q^{13} -2.64977 q^{15} -4.24112 q^{17} +2.85565 q^{19} +1.93393 q^{21} +1.00000 q^{23} +2.02127 q^{25} -1.00000 q^{27} -1.00000 q^{29} +8.48867 q^{31} +1.24093 q^{33} -5.12446 q^{35} +0.788034 q^{37} +1.48851 q^{39} -12.2376 q^{41} +11.0116 q^{43} +2.64977 q^{45} -4.51365 q^{47} -3.25992 q^{49} +4.24112 q^{51} +10.2067 q^{53} -3.28817 q^{55} -2.85565 q^{57} -1.07637 q^{59} -1.98032 q^{61} -1.93393 q^{63} -3.94421 q^{65} +0.801106 q^{67} -1.00000 q^{69} +1.24243 q^{71} +1.47031 q^{73} -2.02127 q^{75} +2.39986 q^{77} -12.7792 q^{79} +1.00000 q^{81} +3.32257 q^{83} -11.2380 q^{85} +1.00000 q^{87} +2.02980 q^{89} +2.87868 q^{91} -8.48867 q^{93} +7.56682 q^{95} +9.39099 q^{97} -1.24093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) 2.64977 1.18501 0.592506 0.805566i \(-0.298139\pi\)
0.592506 + 0.805566i \(0.298139\pi\)
\(6\) 0 0
\(7\) −1.93393 −0.730956 −0.365478 0.930820i \(-0.619094\pi\)
−0.365478 + 0.930820i \(0.619094\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.24093 −0.374154 −0.187077 0.982345i \(-0.559901\pi\)
−0.187077 + 0.982345i \(0.559901\pi\)
\(12\) 0 0
\(13\) −1.48851 −0.412839 −0.206420 0.978464i \(-0.566181\pi\)
−0.206420 + 0.978464i \(0.566181\pi\)
\(14\) 0 0
\(15\) −2.64977 −0.684167
\(16\) 0 0
\(17\) −4.24112 −1.02862 −0.514311 0.857604i \(-0.671952\pi\)
−0.514311 + 0.857604i \(0.671952\pi\)
\(18\) 0 0
\(19\) 2.85565 0.655132 0.327566 0.944828i \(-0.393772\pi\)
0.327566 + 0.944828i \(0.393772\pi\)
\(20\) 0 0
\(21\) 1.93393 0.422018
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 2.02127 0.404254
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 8.48867 1.52461 0.762305 0.647217i \(-0.224067\pi\)
0.762305 + 0.647217i \(0.224067\pi\)
\(32\) 0 0
\(33\) 1.24093 0.216018
\(34\) 0 0
\(35\) −5.12446 −0.866192
\(36\) 0 0
\(37\) 0.788034 0.129552 0.0647760 0.997900i \(-0.479367\pi\)
0.0647760 + 0.997900i \(0.479367\pi\)
\(38\) 0 0
\(39\) 1.48851 0.238353
\(40\) 0 0
\(41\) −12.2376 −1.91119 −0.955597 0.294676i \(-0.904788\pi\)
−0.955597 + 0.294676i \(0.904788\pi\)
\(42\) 0 0
\(43\) 11.0116 1.67925 0.839625 0.543166i \(-0.182775\pi\)
0.839625 + 0.543166i \(0.182775\pi\)
\(44\) 0 0
\(45\) 2.64977 0.395004
\(46\) 0 0
\(47\) −4.51365 −0.658383 −0.329192 0.944263i \(-0.606776\pi\)
−0.329192 + 0.944263i \(0.606776\pi\)
\(48\) 0 0
\(49\) −3.25992 −0.465703
\(50\) 0 0
\(51\) 4.24112 0.593876
\(52\) 0 0
\(53\) 10.2067 1.40199 0.700997 0.713164i \(-0.252739\pi\)
0.700997 + 0.713164i \(0.252739\pi\)
\(54\) 0 0
\(55\) −3.28817 −0.443377
\(56\) 0 0
\(57\) −2.85565 −0.378241
\(58\) 0 0
\(59\) −1.07637 −0.140131 −0.0700656 0.997542i \(-0.522321\pi\)
−0.0700656 + 0.997542i \(0.522321\pi\)
\(60\) 0 0
\(61\) −1.98032 −0.253554 −0.126777 0.991931i \(-0.540463\pi\)
−0.126777 + 0.991931i \(0.540463\pi\)
\(62\) 0 0
\(63\) −1.93393 −0.243652
\(64\) 0 0
\(65\) −3.94421 −0.489219
\(66\) 0 0
\(67\) 0.801106 0.0978707 0.0489353 0.998802i \(-0.484417\pi\)
0.0489353 + 0.998802i \(0.484417\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 1.24243 0.147449 0.0737246 0.997279i \(-0.476511\pi\)
0.0737246 + 0.997279i \(0.476511\pi\)
\(72\) 0 0
\(73\) 1.47031 0.172087 0.0860436 0.996291i \(-0.472578\pi\)
0.0860436 + 0.996291i \(0.472578\pi\)
\(74\) 0 0
\(75\) −2.02127 −0.233396
\(76\) 0 0
\(77\) 2.39986 0.273490
\(78\) 0 0
\(79\) −12.7792 −1.43777 −0.718885 0.695129i \(-0.755347\pi\)
−0.718885 + 0.695129i \(0.755347\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.32257 0.364699 0.182350 0.983234i \(-0.441630\pi\)
0.182350 + 0.983234i \(0.441630\pi\)
\(84\) 0 0
\(85\) −11.2380 −1.21893
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 2.02980 0.215158 0.107579 0.994197i \(-0.465690\pi\)
0.107579 + 0.994197i \(0.465690\pi\)
\(90\) 0 0
\(91\) 2.87868 0.301767
\(92\) 0 0
\(93\) −8.48867 −0.880234
\(94\) 0 0
\(95\) 7.56682 0.776339
\(96\) 0 0
\(97\) 9.39099 0.953511 0.476755 0.879036i \(-0.341813\pi\)
0.476755 + 0.879036i \(0.341813\pi\)
\(98\) 0 0
\(99\) −1.24093 −0.124718
\(100\) 0 0
\(101\) 4.57021 0.454753 0.227376 0.973807i \(-0.426985\pi\)
0.227376 + 0.973807i \(0.426985\pi\)
\(102\) 0 0
\(103\) −17.8586 −1.75966 −0.879828 0.475292i \(-0.842342\pi\)
−0.879828 + 0.475292i \(0.842342\pi\)
\(104\) 0 0
\(105\) 5.12446 0.500096
\(106\) 0 0
\(107\) 0.391113 0.0378104 0.0189052 0.999821i \(-0.493982\pi\)
0.0189052 + 0.999821i \(0.493982\pi\)
\(108\) 0 0
\(109\) 4.81471 0.461166 0.230583 0.973053i \(-0.425937\pi\)
0.230583 + 0.973053i \(0.425937\pi\)
\(110\) 0 0
\(111\) −0.788034 −0.0747968
\(112\) 0 0
\(113\) −15.6910 −1.47609 −0.738044 0.674752i \(-0.764251\pi\)
−0.738044 + 0.674752i \(0.764251\pi\)
\(114\) 0 0
\(115\) 2.64977 0.247092
\(116\) 0 0
\(117\) −1.48851 −0.137613
\(118\) 0 0
\(119\) 8.20202 0.751878
\(120\) 0 0
\(121\) −9.46010 −0.860009
\(122\) 0 0
\(123\) 12.2376 1.10343
\(124\) 0 0
\(125\) −7.89294 −0.705966
\(126\) 0 0
\(127\) −17.7894 −1.57856 −0.789279 0.614035i \(-0.789546\pi\)
−0.789279 + 0.614035i \(0.789546\pi\)
\(128\) 0 0
\(129\) −11.0116 −0.969516
\(130\) 0 0
\(131\) −5.98647 −0.523040 −0.261520 0.965198i \(-0.584224\pi\)
−0.261520 + 0.965198i \(0.584224\pi\)
\(132\) 0 0
\(133\) −5.52263 −0.478873
\(134\) 0 0
\(135\) −2.64977 −0.228056
\(136\) 0 0
\(137\) −15.5240 −1.32630 −0.663151 0.748486i \(-0.730781\pi\)
−0.663151 + 0.748486i \(0.730781\pi\)
\(138\) 0 0
\(139\) −15.7224 −1.33355 −0.666777 0.745257i \(-0.732327\pi\)
−0.666777 + 0.745257i \(0.732327\pi\)
\(140\) 0 0
\(141\) 4.51365 0.380118
\(142\) 0 0
\(143\) 1.84714 0.154465
\(144\) 0 0
\(145\) −2.64977 −0.220051
\(146\) 0 0
\(147\) 3.25992 0.268874
\(148\) 0 0
\(149\) −8.87792 −0.727307 −0.363654 0.931534i \(-0.618471\pi\)
−0.363654 + 0.931534i \(0.618471\pi\)
\(150\) 0 0
\(151\) 4.12277 0.335506 0.167753 0.985829i \(-0.446349\pi\)
0.167753 + 0.985829i \(0.446349\pi\)
\(152\) 0 0
\(153\) −4.24112 −0.342874
\(154\) 0 0
\(155\) 22.4930 1.80668
\(156\) 0 0
\(157\) −12.9881 −1.03656 −0.518281 0.855211i \(-0.673428\pi\)
−0.518281 + 0.855211i \(0.673428\pi\)
\(158\) 0 0
\(159\) −10.2067 −0.809441
\(160\) 0 0
\(161\) −1.93393 −0.152415
\(162\) 0 0
\(163\) 6.56405 0.514136 0.257068 0.966393i \(-0.417244\pi\)
0.257068 + 0.966393i \(0.417244\pi\)
\(164\) 0 0
\(165\) 3.28817 0.255984
\(166\) 0 0
\(167\) 19.7215 1.52610 0.763050 0.646340i \(-0.223701\pi\)
0.763050 + 0.646340i \(0.223701\pi\)
\(168\) 0 0
\(169\) −10.7843 −0.829564
\(170\) 0 0
\(171\) 2.85565 0.218377
\(172\) 0 0
\(173\) 18.6036 1.41440 0.707202 0.707012i \(-0.249957\pi\)
0.707202 + 0.707012i \(0.249957\pi\)
\(174\) 0 0
\(175\) −3.90899 −0.295492
\(176\) 0 0
\(177\) 1.07637 0.0809047
\(178\) 0 0
\(179\) 12.7500 0.952976 0.476488 0.879181i \(-0.341909\pi\)
0.476488 + 0.879181i \(0.341909\pi\)
\(180\) 0 0
\(181\) −20.1955 −1.50112 −0.750561 0.660801i \(-0.770217\pi\)
−0.750561 + 0.660801i \(0.770217\pi\)
\(182\) 0 0
\(183\) 1.98032 0.146389
\(184\) 0 0
\(185\) 2.08811 0.153521
\(186\) 0 0
\(187\) 5.26292 0.384863
\(188\) 0 0
\(189\) 1.93393 0.140673
\(190\) 0 0
\(191\) −2.95262 −0.213644 −0.106822 0.994278i \(-0.534068\pi\)
−0.106822 + 0.994278i \(0.534068\pi\)
\(192\) 0 0
\(193\) 10.9562 0.788648 0.394324 0.918972i \(-0.370979\pi\)
0.394324 + 0.918972i \(0.370979\pi\)
\(194\) 0 0
\(195\) 3.94421 0.282451
\(196\) 0 0
\(197\) −27.2203 −1.93936 −0.969682 0.244372i \(-0.921418\pi\)
−0.969682 + 0.244372i \(0.921418\pi\)
\(198\) 0 0
\(199\) −1.73732 −0.123156 −0.0615778 0.998102i \(-0.519613\pi\)
−0.0615778 + 0.998102i \(0.519613\pi\)
\(200\) 0 0
\(201\) −0.801106 −0.0565057
\(202\) 0 0
\(203\) 1.93393 0.135735
\(204\) 0 0
\(205\) −32.4268 −2.26479
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.54366 −0.245120
\(210\) 0 0
\(211\) 15.0518 1.03621 0.518105 0.855317i \(-0.326638\pi\)
0.518105 + 0.855317i \(0.326638\pi\)
\(212\) 0 0
\(213\) −1.24243 −0.0851298
\(214\) 0 0
\(215\) 29.1781 1.98993
\(216\) 0 0
\(217\) −16.4165 −1.11442
\(218\) 0 0
\(219\) −1.47031 −0.0993546
\(220\) 0 0
\(221\) 6.31296 0.424656
\(222\) 0 0
\(223\) −9.96082 −0.667026 −0.333513 0.942746i \(-0.608234\pi\)
−0.333513 + 0.942746i \(0.608234\pi\)
\(224\) 0 0
\(225\) 2.02127 0.134751
\(226\) 0 0
\(227\) −6.11865 −0.406109 −0.203054 0.979167i \(-0.565087\pi\)
−0.203054 + 0.979167i \(0.565087\pi\)
\(228\) 0 0
\(229\) −14.5804 −0.963500 −0.481750 0.876309i \(-0.659999\pi\)
−0.481750 + 0.876309i \(0.659999\pi\)
\(230\) 0 0
\(231\) −2.39986 −0.157899
\(232\) 0 0
\(233\) −15.2715 −1.00047 −0.500234 0.865890i \(-0.666753\pi\)
−0.500234 + 0.865890i \(0.666753\pi\)
\(234\) 0 0
\(235\) −11.9601 −0.780192
\(236\) 0 0
\(237\) 12.7792 0.830097
\(238\) 0 0
\(239\) −5.61982 −0.363516 −0.181758 0.983343i \(-0.558179\pi\)
−0.181758 + 0.983343i \(0.558179\pi\)
\(240\) 0 0
\(241\) 15.6192 1.00612 0.503059 0.864252i \(-0.332208\pi\)
0.503059 + 0.864252i \(0.332208\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −8.63804 −0.551864
\(246\) 0 0
\(247\) −4.25068 −0.270464
\(248\) 0 0
\(249\) −3.32257 −0.210559
\(250\) 0 0
\(251\) −10.7037 −0.675613 −0.337807 0.941216i \(-0.609685\pi\)
−0.337807 + 0.941216i \(0.609685\pi\)
\(252\) 0 0
\(253\) −1.24093 −0.0780164
\(254\) 0 0
\(255\) 11.2380 0.703750
\(256\) 0 0
\(257\) −3.36904 −0.210155 −0.105077 0.994464i \(-0.533509\pi\)
−0.105077 + 0.994464i \(0.533509\pi\)
\(258\) 0 0
\(259\) −1.52400 −0.0946967
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −16.2821 −1.00399 −0.501997 0.864869i \(-0.667401\pi\)
−0.501997 + 0.864869i \(0.667401\pi\)
\(264\) 0 0
\(265\) 27.0453 1.66138
\(266\) 0 0
\(267\) −2.02980 −0.124222
\(268\) 0 0
\(269\) −1.07002 −0.0652406 −0.0326203 0.999468i \(-0.510385\pi\)
−0.0326203 + 0.999468i \(0.510385\pi\)
\(270\) 0 0
\(271\) 27.4839 1.66953 0.834765 0.550606i \(-0.185603\pi\)
0.834765 + 0.550606i \(0.185603\pi\)
\(272\) 0 0
\(273\) −2.87868 −0.174225
\(274\) 0 0
\(275\) −2.50825 −0.151253
\(276\) 0 0
\(277\) 15.6321 0.939245 0.469622 0.882867i \(-0.344390\pi\)
0.469622 + 0.882867i \(0.344390\pi\)
\(278\) 0 0
\(279\) 8.48867 0.508204
\(280\) 0 0
\(281\) −2.26311 −0.135006 −0.0675028 0.997719i \(-0.521503\pi\)
−0.0675028 + 0.997719i \(0.521503\pi\)
\(282\) 0 0
\(283\) 31.0378 1.84501 0.922503 0.385991i \(-0.126140\pi\)
0.922503 + 0.385991i \(0.126140\pi\)
\(284\) 0 0
\(285\) −7.56682 −0.448220
\(286\) 0 0
\(287\) 23.6667 1.39700
\(288\) 0 0
\(289\) 0.987100 0.0580647
\(290\) 0 0
\(291\) −9.39099 −0.550510
\(292\) 0 0
\(293\) −7.01724 −0.409952 −0.204976 0.978767i \(-0.565712\pi\)
−0.204976 + 0.978767i \(0.565712\pi\)
\(294\) 0 0
\(295\) −2.85212 −0.166057
\(296\) 0 0
\(297\) 1.24093 0.0720059
\(298\) 0 0
\(299\) −1.48851 −0.0860829
\(300\) 0 0
\(301\) −21.2956 −1.22746
\(302\) 0 0
\(303\) −4.57021 −0.262552
\(304\) 0 0
\(305\) −5.24739 −0.300464
\(306\) 0 0
\(307\) 7.12070 0.406400 0.203200 0.979137i \(-0.434866\pi\)
0.203200 + 0.979137i \(0.434866\pi\)
\(308\) 0 0
\(309\) 17.8586 1.01594
\(310\) 0 0
\(311\) 3.14032 0.178071 0.0890357 0.996028i \(-0.471621\pi\)
0.0890357 + 0.996028i \(0.471621\pi\)
\(312\) 0 0
\(313\) 21.0426 1.18940 0.594700 0.803947i \(-0.297271\pi\)
0.594700 + 0.803947i \(0.297271\pi\)
\(314\) 0 0
\(315\) −5.12446 −0.288731
\(316\) 0 0
\(317\) 7.50308 0.421415 0.210708 0.977549i \(-0.432423\pi\)
0.210708 + 0.977549i \(0.432423\pi\)
\(318\) 0 0
\(319\) 1.24093 0.0694786
\(320\) 0 0
\(321\) −0.391113 −0.0218298
\(322\) 0 0
\(323\) −12.1112 −0.673884
\(324\) 0 0
\(325\) −3.00869 −0.166892
\(326\) 0 0
\(327\) −4.81471 −0.266254
\(328\) 0 0
\(329\) 8.72907 0.481249
\(330\) 0 0
\(331\) 10.2367 0.562661 0.281331 0.959611i \(-0.409224\pi\)
0.281331 + 0.959611i \(0.409224\pi\)
\(332\) 0 0
\(333\) 0.788034 0.0431840
\(334\) 0 0
\(335\) 2.12274 0.115978
\(336\) 0 0
\(337\) −23.9208 −1.30305 −0.651525 0.758628i \(-0.725870\pi\)
−0.651525 + 0.758628i \(0.725870\pi\)
\(338\) 0 0
\(339\) 15.6910 0.852220
\(340\) 0 0
\(341\) −10.5338 −0.570439
\(342\) 0 0
\(343\) 19.8420 1.07136
\(344\) 0 0
\(345\) −2.64977 −0.142659
\(346\) 0 0
\(347\) 31.6145 1.69716 0.848578 0.529071i \(-0.177459\pi\)
0.848578 + 0.529071i \(0.177459\pi\)
\(348\) 0 0
\(349\) −10.5683 −0.565710 −0.282855 0.959163i \(-0.591282\pi\)
−0.282855 + 0.959163i \(0.591282\pi\)
\(350\) 0 0
\(351\) 1.48851 0.0794509
\(352\) 0 0
\(353\) −18.2584 −0.971794 −0.485897 0.874016i \(-0.661507\pi\)
−0.485897 + 0.874016i \(0.661507\pi\)
\(354\) 0 0
\(355\) 3.29215 0.174729
\(356\) 0 0
\(357\) −8.20202 −0.434097
\(358\) 0 0
\(359\) −24.9980 −1.31935 −0.659673 0.751553i \(-0.729305\pi\)
−0.659673 + 0.751553i \(0.729305\pi\)
\(360\) 0 0
\(361\) −10.8452 −0.570802
\(362\) 0 0
\(363\) 9.46010 0.496526
\(364\) 0 0
\(365\) 3.89599 0.203925
\(366\) 0 0
\(367\) −11.2238 −0.585879 −0.292939 0.956131i \(-0.594633\pi\)
−0.292939 + 0.956131i \(0.594633\pi\)
\(368\) 0 0
\(369\) −12.2376 −0.637065
\(370\) 0 0
\(371\) −19.7390 −1.02480
\(372\) 0 0
\(373\) 17.8707 0.925307 0.462654 0.886539i \(-0.346897\pi\)
0.462654 + 0.886539i \(0.346897\pi\)
\(374\) 0 0
\(375\) 7.89294 0.407590
\(376\) 0 0
\(377\) 1.48851 0.0766623
\(378\) 0 0
\(379\) −11.7746 −0.604820 −0.302410 0.953178i \(-0.597791\pi\)
−0.302410 + 0.953178i \(0.597791\pi\)
\(380\) 0 0
\(381\) 17.7894 0.911381
\(382\) 0 0
\(383\) −11.8807 −0.607076 −0.303538 0.952819i \(-0.598168\pi\)
−0.303538 + 0.952819i \(0.598168\pi\)
\(384\) 0 0
\(385\) 6.35908 0.324089
\(386\) 0 0
\(387\) 11.0116 0.559750
\(388\) 0 0
\(389\) −34.3701 −1.74264 −0.871318 0.490719i \(-0.836734\pi\)
−0.871318 + 0.490719i \(0.836734\pi\)
\(390\) 0 0
\(391\) −4.24112 −0.214483
\(392\) 0 0
\(393\) 5.98647 0.301978
\(394\) 0 0
\(395\) −33.8619 −1.70378
\(396\) 0 0
\(397\) −26.4584 −1.32791 −0.663955 0.747772i \(-0.731123\pi\)
−0.663955 + 0.747772i \(0.731123\pi\)
\(398\) 0 0
\(399\) 5.52263 0.276477
\(400\) 0 0
\(401\) 27.3683 1.36671 0.683354 0.730087i \(-0.260520\pi\)
0.683354 + 0.730087i \(0.260520\pi\)
\(402\) 0 0
\(403\) −12.6355 −0.629419
\(404\) 0 0
\(405\) 2.64977 0.131668
\(406\) 0 0
\(407\) −0.977892 −0.0484723
\(408\) 0 0
\(409\) 2.46181 0.121729 0.0608644 0.998146i \(-0.480614\pi\)
0.0608644 + 0.998146i \(0.480614\pi\)
\(410\) 0 0
\(411\) 15.5240 0.765741
\(412\) 0 0
\(413\) 2.08162 0.102430
\(414\) 0 0
\(415\) 8.80403 0.432173
\(416\) 0 0
\(417\) 15.7224 0.769928
\(418\) 0 0
\(419\) 6.50269 0.317677 0.158839 0.987305i \(-0.449225\pi\)
0.158839 + 0.987305i \(0.449225\pi\)
\(420\) 0 0
\(421\) −18.3109 −0.892419 −0.446210 0.894928i \(-0.647226\pi\)
−0.446210 + 0.894928i \(0.647226\pi\)
\(422\) 0 0
\(423\) −4.51365 −0.219461
\(424\) 0 0
\(425\) −8.57245 −0.415825
\(426\) 0 0
\(427\) 3.82979 0.185337
\(428\) 0 0
\(429\) −1.84714 −0.0891806
\(430\) 0 0
\(431\) −16.9720 −0.817513 −0.408757 0.912643i \(-0.634038\pi\)
−0.408757 + 0.912643i \(0.634038\pi\)
\(432\) 0 0
\(433\) −24.3836 −1.17180 −0.585900 0.810383i \(-0.699259\pi\)
−0.585900 + 0.810383i \(0.699259\pi\)
\(434\) 0 0
\(435\) 2.64977 0.127047
\(436\) 0 0
\(437\) 2.85565 0.136604
\(438\) 0 0
\(439\) −4.70699 −0.224653 −0.112326 0.993671i \(-0.535830\pi\)
−0.112326 + 0.993671i \(0.535830\pi\)
\(440\) 0 0
\(441\) −3.25992 −0.155234
\(442\) 0 0
\(443\) −8.69144 −0.412943 −0.206471 0.978453i \(-0.566198\pi\)
−0.206471 + 0.978453i \(0.566198\pi\)
\(444\) 0 0
\(445\) 5.37850 0.254965
\(446\) 0 0
\(447\) 8.87792 0.419911
\(448\) 0 0
\(449\) −34.5410 −1.63009 −0.815045 0.579398i \(-0.803288\pi\)
−0.815045 + 0.579398i \(0.803288\pi\)
\(450\) 0 0
\(451\) 15.1860 0.715080
\(452\) 0 0
\(453\) −4.12277 −0.193705
\(454\) 0 0
\(455\) 7.62782 0.357598
\(456\) 0 0
\(457\) −26.5728 −1.24302 −0.621511 0.783405i \(-0.713481\pi\)
−0.621511 + 0.783405i \(0.713481\pi\)
\(458\) 0 0
\(459\) 4.24112 0.197959
\(460\) 0 0
\(461\) 13.7477 0.640295 0.320147 0.947368i \(-0.396268\pi\)
0.320147 + 0.947368i \(0.396268\pi\)
\(462\) 0 0
\(463\) 25.9111 1.20419 0.602095 0.798424i \(-0.294333\pi\)
0.602095 + 0.798424i \(0.294333\pi\)
\(464\) 0 0
\(465\) −22.4930 −1.04309
\(466\) 0 0
\(467\) 21.1330 0.977920 0.488960 0.872306i \(-0.337376\pi\)
0.488960 + 0.872306i \(0.337376\pi\)
\(468\) 0 0
\(469\) −1.54928 −0.0715391
\(470\) 0 0
\(471\) 12.9881 0.598459
\(472\) 0 0
\(473\) −13.6646 −0.628298
\(474\) 0 0
\(475\) 5.77205 0.264840
\(476\) 0 0
\(477\) 10.2067 0.467331
\(478\) 0 0
\(479\) −20.4547 −0.934597 −0.467299 0.884100i \(-0.654773\pi\)
−0.467299 + 0.884100i \(0.654773\pi\)
\(480\) 0 0
\(481\) −1.17300 −0.0534841
\(482\) 0 0
\(483\) 1.93393 0.0879968
\(484\) 0 0
\(485\) 24.8840 1.12992
\(486\) 0 0
\(487\) −8.57065 −0.388373 −0.194187 0.980965i \(-0.562207\pi\)
−0.194187 + 0.980965i \(0.562207\pi\)
\(488\) 0 0
\(489\) −6.56405 −0.296837
\(490\) 0 0
\(491\) −40.2177 −1.81500 −0.907500 0.420051i \(-0.862012\pi\)
−0.907500 + 0.420051i \(0.862012\pi\)
\(492\) 0 0
\(493\) 4.24112 0.191010
\(494\) 0 0
\(495\) −3.28817 −0.147792
\(496\) 0 0
\(497\) −2.40277 −0.107779
\(498\) 0 0
\(499\) −26.5415 −1.18816 −0.594080 0.804406i \(-0.702484\pi\)
−0.594080 + 0.804406i \(0.702484\pi\)
\(500\) 0 0
\(501\) −19.7215 −0.881094
\(502\) 0 0
\(503\) 1.45303 0.0647872 0.0323936 0.999475i \(-0.489687\pi\)
0.0323936 + 0.999475i \(0.489687\pi\)
\(504\) 0 0
\(505\) 12.1100 0.538887
\(506\) 0 0
\(507\) 10.7843 0.478949
\(508\) 0 0
\(509\) −36.2225 −1.60553 −0.802767 0.596293i \(-0.796640\pi\)
−0.802767 + 0.596293i \(0.796640\pi\)
\(510\) 0 0
\(511\) −2.84348 −0.125788
\(512\) 0 0
\(513\) −2.85565 −0.126080
\(514\) 0 0
\(515\) −47.3210 −2.08521
\(516\) 0 0
\(517\) 5.60111 0.246337
\(518\) 0 0
\(519\) −18.6036 −0.816606
\(520\) 0 0
\(521\) 31.9431 1.39945 0.699727 0.714410i \(-0.253305\pi\)
0.699727 + 0.714410i \(0.253305\pi\)
\(522\) 0 0
\(523\) −9.46920 −0.414059 −0.207030 0.978335i \(-0.566380\pi\)
−0.207030 + 0.978335i \(0.566380\pi\)
\(524\) 0 0
\(525\) 3.90899 0.170602
\(526\) 0 0
\(527\) −36.0015 −1.56825
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.07637 −0.0467104
\(532\) 0 0
\(533\) 18.2158 0.789016
\(534\) 0 0
\(535\) 1.03636 0.0448057
\(536\) 0 0
\(537\) −12.7500 −0.550201
\(538\) 0 0
\(539\) 4.04533 0.174245
\(540\) 0 0
\(541\) −21.0013 −0.902917 −0.451458 0.892292i \(-0.649096\pi\)
−0.451458 + 0.892292i \(0.649096\pi\)
\(542\) 0 0
\(543\) 20.1955 0.866673
\(544\) 0 0
\(545\) 12.7579 0.546487
\(546\) 0 0
\(547\) 10.7155 0.458161 0.229080 0.973408i \(-0.426428\pi\)
0.229080 + 0.973408i \(0.426428\pi\)
\(548\) 0 0
\(549\) −1.98032 −0.0845180
\(550\) 0 0
\(551\) −2.85565 −0.121655
\(552\) 0 0
\(553\) 24.7140 1.05095
\(554\) 0 0
\(555\) −2.08811 −0.0886352
\(556\) 0 0
\(557\) −3.64952 −0.154635 −0.0773175 0.997007i \(-0.524636\pi\)
−0.0773175 + 0.997007i \(0.524636\pi\)
\(558\) 0 0
\(559\) −16.3909 −0.693260
\(560\) 0 0
\(561\) −5.26292 −0.222201
\(562\) 0 0
\(563\) 31.4639 1.32605 0.663023 0.748599i \(-0.269273\pi\)
0.663023 + 0.748599i \(0.269273\pi\)
\(564\) 0 0
\(565\) −41.5776 −1.74918
\(566\) 0 0
\(567\) −1.93393 −0.0812173
\(568\) 0 0
\(569\) 7.53105 0.315718 0.157859 0.987462i \(-0.449541\pi\)
0.157859 + 0.987462i \(0.449541\pi\)
\(570\) 0 0
\(571\) −15.5206 −0.649519 −0.324759 0.945797i \(-0.605283\pi\)
−0.324759 + 0.945797i \(0.605283\pi\)
\(572\) 0 0
\(573\) 2.95262 0.123348
\(574\) 0 0
\(575\) 2.02127 0.0842928
\(576\) 0 0
\(577\) −28.8794 −1.20226 −0.601132 0.799150i \(-0.705283\pi\)
−0.601132 + 0.799150i \(0.705283\pi\)
\(578\) 0 0
\(579\) −10.9562 −0.455326
\(580\) 0 0
\(581\) −6.42560 −0.266579
\(582\) 0 0
\(583\) −12.6657 −0.524561
\(584\) 0 0
\(585\) −3.94421 −0.163073
\(586\) 0 0
\(587\) 8.03989 0.331842 0.165921 0.986139i \(-0.446940\pi\)
0.165921 + 0.986139i \(0.446940\pi\)
\(588\) 0 0
\(589\) 24.2407 0.998821
\(590\) 0 0
\(591\) 27.2203 1.11969
\(592\) 0 0
\(593\) −38.0061 −1.56073 −0.780363 0.625327i \(-0.784966\pi\)
−0.780363 + 0.625327i \(0.784966\pi\)
\(594\) 0 0
\(595\) 21.7334 0.890985
\(596\) 0 0
\(597\) 1.73732 0.0711039
\(598\) 0 0
\(599\) 23.9171 0.977225 0.488613 0.872501i \(-0.337503\pi\)
0.488613 + 0.872501i \(0.337503\pi\)
\(600\) 0 0
\(601\) 29.4240 1.20023 0.600116 0.799913i \(-0.295121\pi\)
0.600116 + 0.799913i \(0.295121\pi\)
\(602\) 0 0
\(603\) 0.801106 0.0326236
\(604\) 0 0
\(605\) −25.0671 −1.01912
\(606\) 0 0
\(607\) 42.3530 1.71905 0.859527 0.511090i \(-0.170758\pi\)
0.859527 + 0.511090i \(0.170758\pi\)
\(608\) 0 0
\(609\) −1.93393 −0.0783667
\(610\) 0 0
\(611\) 6.71862 0.271806
\(612\) 0 0
\(613\) 31.1855 1.25957 0.629784 0.776770i \(-0.283143\pi\)
0.629784 + 0.776770i \(0.283143\pi\)
\(614\) 0 0
\(615\) 32.4268 1.30758
\(616\) 0 0
\(617\) −24.3940 −0.982065 −0.491032 0.871141i \(-0.663380\pi\)
−0.491032 + 0.871141i \(0.663380\pi\)
\(618\) 0 0
\(619\) −0.273497 −0.0109928 −0.00549639 0.999985i \(-0.501750\pi\)
−0.00549639 + 0.999985i \(0.501750\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −3.92549 −0.157271
\(624\) 0 0
\(625\) −31.0208 −1.24083
\(626\) 0 0
\(627\) 3.54366 0.141520
\(628\) 0 0
\(629\) −3.34214 −0.133260
\(630\) 0 0
\(631\) −5.91729 −0.235564 −0.117782 0.993039i \(-0.537578\pi\)
−0.117782 + 0.993039i \(0.537578\pi\)
\(632\) 0 0
\(633\) −15.0518 −0.598256
\(634\) 0 0
\(635\) −47.1379 −1.87061
\(636\) 0 0
\(637\) 4.85244 0.192261
\(638\) 0 0
\(639\) 1.24243 0.0491497
\(640\) 0 0
\(641\) −29.9256 −1.18199 −0.590994 0.806676i \(-0.701264\pi\)
−0.590994 + 0.806676i \(0.701264\pi\)
\(642\) 0 0
\(643\) 20.9811 0.827413 0.413706 0.910410i \(-0.364234\pi\)
0.413706 + 0.910410i \(0.364234\pi\)
\(644\) 0 0
\(645\) −29.1781 −1.14889
\(646\) 0 0
\(647\) −15.5729 −0.612236 −0.306118 0.951994i \(-0.599030\pi\)
−0.306118 + 0.951994i \(0.599030\pi\)
\(648\) 0 0
\(649\) 1.33569 0.0524306
\(650\) 0 0
\(651\) 16.4165 0.643413
\(652\) 0 0
\(653\) 33.4932 1.31069 0.655345 0.755330i \(-0.272523\pi\)
0.655345 + 0.755330i \(0.272523\pi\)
\(654\) 0 0
\(655\) −15.8628 −0.619809
\(656\) 0 0
\(657\) 1.47031 0.0573624
\(658\) 0 0
\(659\) 36.6739 1.42861 0.714305 0.699834i \(-0.246743\pi\)
0.714305 + 0.699834i \(0.246743\pi\)
\(660\) 0 0
\(661\) −19.4626 −0.757007 −0.378504 0.925600i \(-0.623561\pi\)
−0.378504 + 0.925600i \(0.623561\pi\)
\(662\) 0 0
\(663\) −6.31296 −0.245175
\(664\) 0 0
\(665\) −14.6337 −0.567470
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 9.96082 0.385108
\(670\) 0 0
\(671\) 2.45743 0.0948681
\(672\) 0 0
\(673\) 7.20582 0.277764 0.138882 0.990309i \(-0.455649\pi\)
0.138882 + 0.990309i \(0.455649\pi\)
\(674\) 0 0
\(675\) −2.02127 −0.0777987
\(676\) 0 0
\(677\) −19.7475 −0.758959 −0.379479 0.925200i \(-0.623897\pi\)
−0.379479 + 0.925200i \(0.623897\pi\)
\(678\) 0 0
\(679\) −18.1615 −0.696974
\(680\) 0 0
\(681\) 6.11865 0.234467
\(682\) 0 0
\(683\) −17.0182 −0.651183 −0.325592 0.945510i \(-0.605563\pi\)
−0.325592 + 0.945510i \(0.605563\pi\)
\(684\) 0 0
\(685\) −41.1349 −1.57168
\(686\) 0 0
\(687\) 14.5804 0.556277
\(688\) 0 0
\(689\) −15.1927 −0.578798
\(690\) 0 0
\(691\) −6.06240 −0.230625 −0.115312 0.993329i \(-0.536787\pi\)
−0.115312 + 0.993329i \(0.536787\pi\)
\(692\) 0 0
\(693\) 2.39986 0.0911633
\(694\) 0 0
\(695\) −41.6606 −1.58028
\(696\) 0 0
\(697\) 51.9012 1.96590
\(698\) 0 0
\(699\) 15.2715 0.577621
\(700\) 0 0
\(701\) 10.3633 0.391415 0.195708 0.980662i \(-0.437300\pi\)
0.195708 + 0.980662i \(0.437300\pi\)
\(702\) 0 0
\(703\) 2.25035 0.0848736
\(704\) 0 0
\(705\) 11.9601 0.450444
\(706\) 0 0
\(707\) −8.83845 −0.332404
\(708\) 0 0
\(709\) 21.2000 0.796184 0.398092 0.917345i \(-0.369672\pi\)
0.398092 + 0.917345i \(0.369672\pi\)
\(710\) 0 0
\(711\) −12.7792 −0.479257
\(712\) 0 0
\(713\) 8.48867 0.317903
\(714\) 0 0
\(715\) 4.89448 0.183043
\(716\) 0 0
\(717\) 5.61982 0.209876
\(718\) 0 0
\(719\) 24.3265 0.907225 0.453612 0.891199i \(-0.350135\pi\)
0.453612 + 0.891199i \(0.350135\pi\)
\(720\) 0 0
\(721\) 34.5372 1.28623
\(722\) 0 0
\(723\) −15.6192 −0.580883
\(724\) 0 0
\(725\) −2.02127 −0.0750681
\(726\) 0 0
\(727\) 47.5490 1.76350 0.881748 0.471720i \(-0.156367\pi\)
0.881748 + 0.471720i \(0.156367\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −46.7014 −1.72732
\(732\) 0 0
\(733\) 43.8561 1.61986 0.809931 0.586526i \(-0.199505\pi\)
0.809931 + 0.586526i \(0.199505\pi\)
\(734\) 0 0
\(735\) 8.63804 0.318619
\(736\) 0 0
\(737\) −0.994114 −0.0366187
\(738\) 0 0
\(739\) −34.7321 −1.27764 −0.638821 0.769356i \(-0.720577\pi\)
−0.638821 + 0.769356i \(0.720577\pi\)
\(740\) 0 0
\(741\) 4.25068 0.156152
\(742\) 0 0
\(743\) 22.0528 0.809040 0.404520 0.914529i \(-0.367439\pi\)
0.404520 + 0.914529i \(0.367439\pi\)
\(744\) 0 0
\(745\) −23.5244 −0.861868
\(746\) 0 0
\(747\) 3.32257 0.121566
\(748\) 0 0
\(749\) −0.756385 −0.0276377
\(750\) 0 0
\(751\) −39.6205 −1.44577 −0.722887 0.690966i \(-0.757185\pi\)
−0.722887 + 0.690966i \(0.757185\pi\)
\(752\) 0 0
\(753\) 10.7037 0.390065
\(754\) 0 0
\(755\) 10.9244 0.397579
\(756\) 0 0
\(757\) −16.6428 −0.604893 −0.302447 0.953166i \(-0.597803\pi\)
−0.302447 + 0.953166i \(0.597803\pi\)
\(758\) 0 0
\(759\) 1.24093 0.0450428
\(760\) 0 0
\(761\) 10.9063 0.395354 0.197677 0.980267i \(-0.436660\pi\)
0.197677 + 0.980267i \(0.436660\pi\)
\(762\) 0 0
\(763\) −9.31130 −0.337092
\(764\) 0 0
\(765\) −11.2380 −0.406310
\(766\) 0 0
\(767\) 1.60219 0.0578516
\(768\) 0 0
\(769\) −43.4487 −1.56680 −0.783400 0.621518i \(-0.786516\pi\)
−0.783400 + 0.621518i \(0.786516\pi\)
\(770\) 0 0
\(771\) 3.36904 0.121333
\(772\) 0 0
\(773\) −36.5429 −1.31436 −0.657178 0.753736i \(-0.728250\pi\)
−0.657178 + 0.753736i \(0.728250\pi\)
\(774\) 0 0
\(775\) 17.1579 0.616330
\(776\) 0 0
\(777\) 1.52400 0.0546732
\(778\) 0 0
\(779\) −34.9464 −1.25208
\(780\) 0 0
\(781\) −1.54176 −0.0551687
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −34.4154 −1.22834
\(786\) 0 0
\(787\) −33.1757 −1.18259 −0.591293 0.806457i \(-0.701382\pi\)
−0.591293 + 0.806457i \(0.701382\pi\)
\(788\) 0 0
\(789\) 16.2821 0.579657
\(790\) 0 0
\(791\) 30.3453 1.07896
\(792\) 0 0
\(793\) 2.94773 0.104677
\(794\) 0 0
\(795\) −27.0453 −0.959198
\(796\) 0 0
\(797\) 28.4906 1.00919 0.504595 0.863356i \(-0.331642\pi\)
0.504595 + 0.863356i \(0.331642\pi\)
\(798\) 0 0
\(799\) 19.1429 0.677228
\(800\) 0 0
\(801\) 2.02980 0.0717195
\(802\) 0 0
\(803\) −1.82455 −0.0643871
\(804\) 0 0
\(805\) −5.12446 −0.180613
\(806\) 0 0
\(807\) 1.07002 0.0376667
\(808\) 0 0
\(809\) 28.0357 0.985684 0.492842 0.870119i \(-0.335958\pi\)
0.492842 + 0.870119i \(0.335958\pi\)
\(810\) 0 0
\(811\) 4.54499 0.159596 0.0797981 0.996811i \(-0.474572\pi\)
0.0797981 + 0.996811i \(0.474572\pi\)
\(812\) 0 0
\(813\) −27.4839 −0.963904
\(814\) 0 0
\(815\) 17.3932 0.609258
\(816\) 0 0
\(817\) 31.4453 1.10013
\(818\) 0 0
\(819\) 2.87868 0.100589
\(820\) 0 0
\(821\) −16.8343 −0.587522 −0.293761 0.955879i \(-0.594907\pi\)
−0.293761 + 0.955879i \(0.594907\pi\)
\(822\) 0 0
\(823\) −11.0980 −0.386850 −0.193425 0.981115i \(-0.561960\pi\)
−0.193425 + 0.981115i \(0.561960\pi\)
\(824\) 0 0
\(825\) 2.50825 0.0873261
\(826\) 0 0
\(827\) 22.3267 0.776375 0.388188 0.921580i \(-0.373101\pi\)
0.388188 + 0.921580i \(0.373101\pi\)
\(828\) 0 0
\(829\) −20.6403 −0.716867 −0.358433 0.933555i \(-0.616689\pi\)
−0.358433 + 0.933555i \(0.616689\pi\)
\(830\) 0 0
\(831\) −15.6321 −0.542273
\(832\) 0 0
\(833\) 13.8257 0.479033
\(834\) 0 0
\(835\) 52.2575 1.80845
\(836\) 0 0
\(837\) −8.48867 −0.293411
\(838\) 0 0
\(839\) −39.9378 −1.37881 −0.689403 0.724378i \(-0.742127\pi\)
−0.689403 + 0.724378i \(0.742127\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 2.26311 0.0779455
\(844\) 0 0
\(845\) −28.5760 −0.983043
\(846\) 0 0
\(847\) 18.2951 0.628629
\(848\) 0 0
\(849\) −31.0378 −1.06521
\(850\) 0 0
\(851\) 0.788034 0.0270134
\(852\) 0 0
\(853\) 11.6903 0.400270 0.200135 0.979768i \(-0.435862\pi\)
0.200135 + 0.979768i \(0.435862\pi\)
\(854\) 0 0
\(855\) 7.56682 0.258780
\(856\) 0 0
\(857\) −53.8039 −1.83791 −0.918953 0.394367i \(-0.870964\pi\)
−0.918953 + 0.394367i \(0.870964\pi\)
\(858\) 0 0
\(859\) −9.42914 −0.321718 −0.160859 0.986977i \(-0.551426\pi\)
−0.160859 + 0.986977i \(0.551426\pi\)
\(860\) 0 0
\(861\) −23.6667 −0.806558
\(862\) 0 0
\(863\) 6.06532 0.206466 0.103233 0.994657i \(-0.467081\pi\)
0.103233 + 0.994657i \(0.467081\pi\)
\(864\) 0 0
\(865\) 49.2951 1.67609
\(866\) 0 0
\(867\) −0.987100 −0.0335237
\(868\) 0 0
\(869\) 15.8580 0.537947
\(870\) 0 0
\(871\) −1.19246 −0.0404048
\(872\) 0 0
\(873\) 9.39099 0.317837
\(874\) 0 0
\(875\) 15.2644 0.516030
\(876\) 0 0
\(877\) 55.5669 1.87636 0.938180 0.346148i \(-0.112510\pi\)
0.938180 + 0.346148i \(0.112510\pi\)
\(878\) 0 0
\(879\) 7.01724 0.236686
\(880\) 0 0
\(881\) −52.5638 −1.77092 −0.885460 0.464715i \(-0.846157\pi\)
−0.885460 + 0.464715i \(0.846157\pi\)
\(882\) 0 0
\(883\) 1.12724 0.0379346 0.0189673 0.999820i \(-0.493962\pi\)
0.0189673 + 0.999820i \(0.493962\pi\)
\(884\) 0 0
\(885\) 2.85212 0.0958731
\(886\) 0 0
\(887\) −20.6619 −0.693759 −0.346879 0.937910i \(-0.612759\pi\)
−0.346879 + 0.937910i \(0.612759\pi\)
\(888\) 0 0
\(889\) 34.4035 1.15386
\(890\) 0 0
\(891\) −1.24093 −0.0415726
\(892\) 0 0
\(893\) −12.8894 −0.431328
\(894\) 0 0
\(895\) 33.7844 1.12929
\(896\) 0 0
\(897\) 1.48851 0.0497000
\(898\) 0 0
\(899\) −8.48867 −0.283113
\(900\) 0 0
\(901\) −43.2877 −1.44212
\(902\) 0 0
\(903\) 21.2956 0.708673
\(904\) 0 0
\(905\) −53.5135 −1.77885
\(906\) 0 0
\(907\) 24.7191 0.820784 0.410392 0.911909i \(-0.365392\pi\)
0.410392 + 0.911909i \(0.365392\pi\)
\(908\) 0 0
\(909\) 4.57021 0.151584
\(910\) 0 0
\(911\) −39.4313 −1.30642 −0.653208 0.757178i \(-0.726577\pi\)
−0.653208 + 0.757178i \(0.726577\pi\)
\(912\) 0 0
\(913\) −4.12307 −0.136454
\(914\) 0 0
\(915\) 5.24739 0.173473
\(916\) 0 0
\(917\) 11.5774 0.382319
\(918\) 0 0
\(919\) 43.6803 1.44088 0.720440 0.693518i \(-0.243940\pi\)
0.720440 + 0.693518i \(0.243940\pi\)
\(920\) 0 0
\(921\) −7.12070 −0.234635
\(922\) 0 0
\(923\) −1.84937 −0.0608728
\(924\) 0 0
\(925\) 1.59283 0.0523719
\(926\) 0 0
\(927\) −17.8586 −0.586552
\(928\) 0 0
\(929\) 0.457221 0.0150009 0.00750047 0.999972i \(-0.497613\pi\)
0.00750047 + 0.999972i \(0.497613\pi\)
\(930\) 0 0
\(931\) −9.30921 −0.305097
\(932\) 0 0
\(933\) −3.14032 −0.102810
\(934\) 0 0
\(935\) 13.9455 0.456067
\(936\) 0 0
\(937\) 45.8896 1.49915 0.749574 0.661920i \(-0.230258\pi\)
0.749574 + 0.661920i \(0.230258\pi\)
\(938\) 0 0
\(939\) −21.0426 −0.686701
\(940\) 0 0
\(941\) 21.0845 0.687336 0.343668 0.939091i \(-0.388331\pi\)
0.343668 + 0.939091i \(0.388331\pi\)
\(942\) 0 0
\(943\) −12.2376 −0.398512
\(944\) 0 0
\(945\) 5.12446 0.166699
\(946\) 0 0
\(947\) −30.3315 −0.985640 −0.492820 0.870131i \(-0.664034\pi\)
−0.492820 + 0.870131i \(0.664034\pi\)
\(948\) 0 0
\(949\) −2.18858 −0.0710443
\(950\) 0 0
\(951\) −7.50308 −0.243304
\(952\) 0 0
\(953\) −28.0954 −0.910100 −0.455050 0.890466i \(-0.650379\pi\)
−0.455050 + 0.890466i \(0.650379\pi\)
\(954\) 0 0
\(955\) −7.82376 −0.253171
\(956\) 0 0
\(957\) −1.24093 −0.0401135
\(958\) 0 0
\(959\) 30.0222 0.969468
\(960\) 0 0
\(961\) 41.0576 1.32444
\(962\) 0 0
\(963\) 0.391113 0.0126035
\(964\) 0 0
\(965\) 29.0315 0.934557
\(966\) 0 0
\(967\) 27.4178 0.881697 0.440848 0.897582i \(-0.354678\pi\)
0.440848 + 0.897582i \(0.354678\pi\)
\(968\) 0 0
\(969\) 12.1112 0.389067
\(970\) 0 0
\(971\) −8.24362 −0.264550 −0.132275 0.991213i \(-0.542228\pi\)
−0.132275 + 0.991213i \(0.542228\pi\)
\(972\) 0 0
\(973\) 30.4059 0.974769
\(974\) 0 0
\(975\) 3.00869 0.0963551
\(976\) 0 0
\(977\) 28.5065 0.912004 0.456002 0.889979i \(-0.349281\pi\)
0.456002 + 0.889979i \(0.349281\pi\)
\(978\) 0 0
\(979\) −2.51884 −0.0805023
\(980\) 0 0
\(981\) 4.81471 0.153722
\(982\) 0 0
\(983\) −51.7100 −1.64929 −0.824646 0.565649i \(-0.808626\pi\)
−0.824646 + 0.565649i \(0.808626\pi\)
\(984\) 0 0
\(985\) −72.1274 −2.29817
\(986\) 0 0
\(987\) −8.72907 −0.277849
\(988\) 0 0
\(989\) 11.0116 0.350148
\(990\) 0 0
\(991\) −39.0995 −1.24204 −0.621019 0.783796i \(-0.713281\pi\)
−0.621019 + 0.783796i \(0.713281\pi\)
\(992\) 0 0
\(993\) −10.2367 −0.324853
\(994\) 0 0
\(995\) −4.60351 −0.145941
\(996\) 0 0
\(997\) 60.2082 1.90681 0.953407 0.301688i \(-0.0975500\pi\)
0.953407 + 0.301688i \(0.0975500\pi\)
\(998\) 0 0
\(999\) −0.788034 −0.0249323
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.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.11 12 1.1 even 1 trivial