Properties

Label 8034.2.a.i.1.1
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{22} +1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +10.0000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{38} +1.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +6.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -5.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +2.00000 q^{66} -10.0000 q^{67} -2.00000 q^{68} +6.00000 q^{71} +1.00000 q^{72} -5.00000 q^{75} +8.00000 q^{76} +1.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{86} -2.00000 q^{87} +2.00000 q^{88} +8.00000 q^{89} +10.0000 q^{93} +6.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.00000 1.29777
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) −5.00000 −0.707107
\(51\) −2.00000 −0.280056
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −5.00000 −0.577350
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 2.00000 0.213201
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) 2.00000 0.201008
\(100\) −5.00000 −0.500000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 1.00000 0.0924500
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 2.00000 0.180334
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 6.00000 0.503509
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −5.00000 −0.408248
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 12.0000 0.901975
\(178\) 8.00000 0.599625
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) −4.00000 −0.292509
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 2.00000 0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −5.00000 −0.353553
\(201\) −10.0000 −0.705346
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −1.00000 −0.0696733
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −6.00000 −0.412082
\(213\) 6.00000 0.411113
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −20.0000 −1.35457
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −2.00000 −0.133038
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 8.00000 0.529813
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 8.00000 0.509028
\(248\) 10.0000 0.635001
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −10.0000 −0.610847
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) −10.0000 −0.603023
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 20.0000 1.19952
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 6.00000 0.357295
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) 6.00000 0.344691
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 1.00000 0.0566139
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) 24.0000 1.32924
\(327\) −20.0000 −1.10600
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 1.00000 0.0543928
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 2.00000 0.106600
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 10.0000 0.518476
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 12.0000 0.613973
\(383\) −10.0000 −0.510976 −0.255488 0.966812i \(-0.582236\pi\)
−0.255488 + 0.966812i \(0.582236\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) 12.0000 0.605320
\(394\) −16.0000 −0.806068
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −10.0000 −0.498755
\(403\) 10.0000 0.498135
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 20.0000 0.979404
\(418\) 16.0000 0.782586
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −16.0000 −0.778868
\(423\) 6.00000 0.291730
\(424\) −6.00000 −0.291386
\(425\) 10.0000 0.485071
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −2.00000 −0.0951303
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) −5.00000 −0.235702
\(451\) 4.00000 0.188353
\(452\) −2.00000 −0.0940721
\(453\) 10.0000 0.469841
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 22.0000 1.02799
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −40.0000 −1.83533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 16.0000 0.731823
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 12.0000 0.546585
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 2.00000 0.0905357
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 2.00000 0.0901670
\(493\) 4.00000 0.180151
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 16.0000 0.714115
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) −4.00000 −0.177471
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) −20.0000 −0.871214
\(528\) 2.00000 0.0870388
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −10.0000 −0.431934
\(537\) −20.0000 −0.863064
\(538\) −30.0000 −1.29339
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 2.00000 0.0859074
\(543\) 10.0000 0.429141
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) 2.00000 0.0853579
\(550\) −10.0000 −0.426401
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 10.0000 0.423334
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −12.0000 −0.506189
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 2.00000 0.0836242
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) −13.0000 −0.540729
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −7.00000 −0.288675
\(589\) 80.0000 3.29634
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) −5.00000 −0.204124
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) −2.00000 −0.0808452
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) 0 0
\(617\) 16.0000 0.644136 0.322068 0.946717i \(-0.395622\pi\)
0.322068 + 0.946717i \(0.395622\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 25.0000 1.00000
\(626\) 10.0000 0.399680
\(627\) 16.0000 0.638978
\(628\) −18.0000 −0.718278
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −16.0000 −0.635943
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −7.00000 −0.277350
\(638\) −4.00000 −0.158362
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 12.0000 0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 1.00000 0.0392837
\(649\) 24.0000 0.942082
\(650\) −5.00000 −0.196116
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) −2.00000 −0.0777322
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −14.0000 −0.539260
\(675\) −5.00000 −0.192450
\(676\) 1.00000 0.0384615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 20.0000 0.765840
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) −4.00000 −0.151511
\(698\) 16.0000 0.605609
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 1.00000 0.0377426
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 16.0000 0.597531
\(718\) −20.0000 −0.746393
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) 12.0000 0.446285
\(724\) 10.0000 0.371647
\(725\) 10.0000 0.371391
\(726\) −7.00000 −0.259794
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) −20.0000 −0.736709
\(738\) 2.00000 0.0736210
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 6.00000 0.218797
\(753\) 16.0000 0.583072
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) −4.00000 −0.144905
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −10.0000 −0.361315
\(767\) 12.0000 0.433295
\(768\) 1.00000 0.0360844
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −4.00000 −0.143963
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) −4.00000 −0.143777
\(775\) −50.0000 −1.79605
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −16.0000 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) −16.0000 −0.569976
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 2.00000 0.0710221
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) −5.00000 −0.176777
\(801\) 8.00000 0.282666
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) −30.0000 −1.05605
\(808\) 6.00000 0.211079
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) −10.0000 −0.348790
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) 50.0000 1.73867 0.869335 0.494223i \(-0.164547\pi\)
0.869335 + 0.494223i \(0.164547\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 1.00000 0.0346688
\(833\) 14.0000 0.485071
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 10.0000 0.345651
\(838\) 20.0000 0.690889
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000 0.0689246
\(843\) −12.0000 −0.413302
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 10.0000 0.342997
\(851\) 0 0
\(852\) 6.00000 0.205557
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 2.00000 0.0682789
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 4.00000 0.136241
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) −20.0000 −0.677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) −7.00000 −0.235702
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 4.00000 0.133930
\(893\) 48.0000 1.60626
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 4.00000 0.133482
\(899\) −20.0000 −0.667037
\(900\) −5.00000 −0.166667
\(901\) 12.0000 0.399778
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 2.00000 0.0663723
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) 10.0000 0.329332
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) −1.00000 −0.0328443
\(928\) −2.00000 −0.0656532
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −56.0000 −1.83533
\(932\) −26.0000 −0.851658
\(933\) −24.0000 −0.785725
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) −18.0000 −0.586472
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −40.0000 −1.29777
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −4.00000 −0.129302
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 12.0000 0.386494
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) −7.00000 −0.224989
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −18.0000 −0.576757
\(975\) −5.00000 −0.160128
\(976\) 2.00000 0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 24.0000 0.767435
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 4.00000 0.127645
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 10.0000 0.317500
\(993\) −2.00000 −0.0634681
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −38.0000 −1.20287
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8034.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.i.1.1 1 1.1 even 1 trivial