Properties

Label 4368.2.a.bg.1.2
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.701562 q^{11} +1.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} +0.701562 q^{19} -1.00000 q^{21} -4.70156 q^{23} +2.29844 q^{25} +1.00000 q^{27} +2.70156 q^{29} +0.701562 q^{33} -2.70156 q^{35} +10.7016 q^{37} +1.00000 q^{39} +3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.70156 q^{51} -2.00000 q^{53} +1.89531 q^{55} +0.701562 q^{57} +14.8062 q^{59} +1.29844 q^{61} -1.00000 q^{63} +2.70156 q^{65} -5.40312 q^{67} -4.70156 q^{69} +8.00000 q^{71} -1.29844 q^{73} +2.29844 q^{75} -0.701562 q^{77} -9.40312 q^{79} +1.00000 q^{81} +13.4031 q^{83} -7.29844 q^{85} +2.70156 q^{87} -8.80625 q^{89} -1.00000 q^{91} +1.89531 q^{95} -8.80625 q^{97} +0.701562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - q^{5} - 2q^{7} + 2q^{9} - 5q^{11} + 2q^{13} - q^{15} + q^{17} - 5q^{19} - 2q^{21} - 3q^{23} + 11q^{25} + 2q^{27} - q^{29} - 5q^{33} + q^{35} + 15q^{37} + 2q^{39} - 6q^{41} + q^{43} - q^{45} + 16q^{47} + 2q^{49} + q^{51} - 4q^{53} + 23q^{55} - 5q^{57} + 4q^{59} + 9q^{61} - 2q^{63} - q^{65} + 2q^{67} - 3q^{69} + 16q^{71} - 9q^{73} + 11q^{75} + 5q^{77} - 6q^{79} + 2q^{81} + 14q^{83} - 21q^{85} - q^{87} + 8q^{89} - 2q^{91} + 23q^{95} + 8q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) 2.70156 1.20818 0.604088 0.796918i \(-0.293538\pi\)
0.604088 + 0.796918i \(0.293538\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.701562 0.211529 0.105764 0.994391i \(-0.466271\pi\)
0.105764 + 0.994391i \(0.466271\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.70156 0.697540
\(16\) 0 0
\(17\) −2.70156 −0.655225 −0.327613 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) 0 0
\(19\) 0.701562 0.160949 0.0804747 0.996757i \(-0.474356\pi\)
0.0804747 + 0.996757i \(0.474356\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.70156 −0.980343 −0.490172 0.871626i \(-0.663066\pi\)
−0.490172 + 0.871626i \(0.663066\pi\)
\(24\) 0 0
\(25\) 2.29844 0.459688
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.70156 0.501667 0.250834 0.968030i \(-0.419295\pi\)
0.250834 + 0.968030i \(0.419295\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0.701562 0.122126
\(34\) 0 0
\(35\) −2.70156 −0.456647
\(36\) 0 0
\(37\) 10.7016 1.75933 0.879663 0.475598i \(-0.157768\pi\)
0.879663 + 0.475598i \(0.157768\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.40312 0.531479 0.265739 0.964045i \(-0.414384\pi\)
0.265739 + 0.964045i \(0.414384\pi\)
\(42\) 0 0
\(43\) 10.1047 1.54095 0.770475 0.637470i \(-0.220019\pi\)
0.770475 + 0.637470i \(0.220019\pi\)
\(44\) 0 0
\(45\) 2.70156 0.402725
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.70156 −0.378294
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 1.89531 0.255564
\(56\) 0 0
\(57\) 0.701562 0.0929242
\(58\) 0 0
\(59\) 14.8062 1.92761 0.963805 0.266609i \(-0.0859033\pi\)
0.963805 + 0.266609i \(0.0859033\pi\)
\(60\) 0 0
\(61\) 1.29844 0.166248 0.0831240 0.996539i \(-0.473510\pi\)
0.0831240 + 0.996539i \(0.473510\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 2.70156 0.335088
\(66\) 0 0
\(67\) −5.40312 −0.660097 −0.330048 0.943964i \(-0.607065\pi\)
−0.330048 + 0.943964i \(0.607065\pi\)
\(68\) 0 0
\(69\) −4.70156 −0.566002
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −1.29844 −0.151971 −0.0759853 0.997109i \(-0.524210\pi\)
−0.0759853 + 0.997109i \(0.524210\pi\)
\(74\) 0 0
\(75\) 2.29844 0.265401
\(76\) 0 0
\(77\) −0.701562 −0.0799504
\(78\) 0 0
\(79\) −9.40312 −1.05793 −0.528967 0.848642i \(-0.677421\pi\)
−0.528967 + 0.848642i \(0.677421\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.4031 1.47118 0.735592 0.677425i \(-0.236904\pi\)
0.735592 + 0.677425i \(0.236904\pi\)
\(84\) 0 0
\(85\) −7.29844 −0.791627
\(86\) 0 0
\(87\) 2.70156 0.289638
\(88\) 0 0
\(89\) −8.80625 −0.933460 −0.466730 0.884400i \(-0.654568\pi\)
−0.466730 + 0.884400i \(0.654568\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.89531 0.194455
\(96\) 0 0
\(97\) −8.80625 −0.894139 −0.447070 0.894499i \(-0.647532\pi\)
−0.447070 + 0.894499i \(0.647532\pi\)
\(98\) 0 0
\(99\) 0.701562 0.0705096
\(100\) 0 0
\(101\) −3.40312 −0.338624 −0.169312 0.985563i \(-0.554154\pi\)
−0.169312 + 0.985563i \(0.554154\pi\)
\(102\) 0 0
\(103\) 3.29844 0.325005 0.162502 0.986708i \(-0.448043\pi\)
0.162502 + 0.986708i \(0.448043\pi\)
\(104\) 0 0
\(105\) −2.70156 −0.263645
\(106\) 0 0
\(107\) 5.40312 0.522340 0.261170 0.965293i \(-0.415892\pi\)
0.261170 + 0.965293i \(0.415892\pi\)
\(108\) 0 0
\(109\) 9.29844 0.890629 0.445314 0.895374i \(-0.353092\pi\)
0.445314 + 0.895374i \(0.353092\pi\)
\(110\) 0 0
\(111\) 10.7016 1.01575
\(112\) 0 0
\(113\) 4.80625 0.452134 0.226067 0.974112i \(-0.427413\pi\)
0.226067 + 0.974112i \(0.427413\pi\)
\(114\) 0 0
\(115\) −12.7016 −1.18443
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 2.70156 0.247652
\(120\) 0 0
\(121\) −10.5078 −0.955256
\(122\) 0 0
\(123\) 3.40312 0.306849
\(124\) 0 0
\(125\) −7.29844 −0.652792
\(126\) 0 0
\(127\) 6.59688 0.585378 0.292689 0.956208i \(-0.405450\pi\)
0.292689 + 0.956208i \(0.405450\pi\)
\(128\) 0 0
\(129\) 10.1047 0.889668
\(130\) 0 0
\(131\) 7.29844 0.637667 0.318834 0.947811i \(-0.396709\pi\)
0.318834 + 0.947811i \(0.396709\pi\)
\(132\) 0 0
\(133\) −0.701562 −0.0608332
\(134\) 0 0
\(135\) 2.70156 0.232513
\(136\) 0 0
\(137\) −18.7016 −1.59778 −0.798891 0.601476i \(-0.794580\pi\)
−0.798891 + 0.601476i \(0.794580\pi\)
\(138\) 0 0
\(139\) −6.80625 −0.577298 −0.288649 0.957435i \(-0.593206\pi\)
−0.288649 + 0.957435i \(0.593206\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 0.701562 0.0586676
\(144\) 0 0
\(145\) 7.29844 0.606102
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 15.4031 1.26187 0.630937 0.775834i \(-0.282671\pi\)
0.630937 + 0.775834i \(0.282671\pi\)
\(150\) 0 0
\(151\) 4.70156 0.382608 0.191304 0.981531i \(-0.438728\pi\)
0.191304 + 0.981531i \(0.438728\pi\)
\(152\) 0 0
\(153\) −2.70156 −0.218408
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.1047 1.60453 0.802264 0.596969i \(-0.203628\pi\)
0.802264 + 0.596969i \(0.203628\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 4.70156 0.370535
\(162\) 0 0
\(163\) −5.40312 −0.423205 −0.211603 0.977356i \(-0.567868\pi\)
−0.211603 + 0.977356i \(0.567868\pi\)
\(164\) 0 0
\(165\) 1.89531 0.147550
\(166\) 0 0
\(167\) −3.29844 −0.255241 −0.127620 0.991823i \(-0.540734\pi\)
−0.127620 + 0.991823i \(0.540734\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.701562 0.0536498
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −2.29844 −0.173746
\(176\) 0 0
\(177\) 14.8062 1.11291
\(178\) 0 0
\(179\) −14.8062 −1.10667 −0.553335 0.832958i \(-0.686645\pi\)
−0.553335 + 0.832958i \(0.686645\pi\)
\(180\) 0 0
\(181\) 8.80625 0.654563 0.327282 0.944927i \(-0.393867\pi\)
0.327282 + 0.944927i \(0.393867\pi\)
\(182\) 0 0
\(183\) 1.29844 0.0959833
\(184\) 0 0
\(185\) 28.9109 2.12557
\(186\) 0 0
\(187\) −1.89531 −0.138599
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.7016 −0.919053 −0.459526 0.888164i \(-0.651981\pi\)
−0.459526 + 0.888164i \(0.651981\pi\)
\(192\) 0 0
\(193\) 11.4031 0.820815 0.410407 0.911902i \(-0.365386\pi\)
0.410407 + 0.911902i \(0.365386\pi\)
\(194\) 0 0
\(195\) 2.70156 0.193463
\(196\) 0 0
\(197\) −3.40312 −0.242463 −0.121231 0.992624i \(-0.538684\pi\)
−0.121231 + 0.992624i \(0.538684\pi\)
\(198\) 0 0
\(199\) −22.1047 −1.56696 −0.783480 0.621417i \(-0.786557\pi\)
−0.783480 + 0.621417i \(0.786557\pi\)
\(200\) 0 0
\(201\) −5.40312 −0.381107
\(202\) 0 0
\(203\) −2.70156 −0.189612
\(204\) 0 0
\(205\) 9.19375 0.642119
\(206\) 0 0
\(207\) −4.70156 −0.326781
\(208\) 0 0
\(209\) 0.492189 0.0340455
\(210\) 0 0
\(211\) 24.7016 1.70053 0.850263 0.526358i \(-0.176443\pi\)
0.850263 + 0.526358i \(0.176443\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 27.2984 1.86174
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.29844 −0.0877403
\(220\) 0 0
\(221\) −2.70156 −0.181727
\(222\) 0 0
\(223\) −9.40312 −0.629680 −0.314840 0.949145i \(-0.601951\pi\)
−0.314840 + 0.949145i \(0.601951\pi\)
\(224\) 0 0
\(225\) 2.29844 0.153229
\(226\) 0 0
\(227\) −21.4031 −1.42058 −0.710288 0.703912i \(-0.751435\pi\)
−0.710288 + 0.703912i \(0.751435\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −0.701562 −0.0461594
\(232\) 0 0
\(233\) −18.2094 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(234\) 0 0
\(235\) 21.6125 1.40984
\(236\) 0 0
\(237\) −9.40312 −0.610799
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −24.8062 −1.59791 −0.798955 0.601390i \(-0.794614\pi\)
−0.798955 + 0.601390i \(0.794614\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.70156 0.172596
\(246\) 0 0
\(247\) 0.701562 0.0446393
\(248\) 0 0
\(249\) 13.4031 0.849388
\(250\) 0 0
\(251\) −3.50781 −0.221411 −0.110706 0.993853i \(-0.535311\pi\)
−0.110706 + 0.993853i \(0.535311\pi\)
\(252\) 0 0
\(253\) −3.29844 −0.207371
\(254\) 0 0
\(255\) −7.29844 −0.457046
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −10.7016 −0.664963
\(260\) 0 0
\(261\) 2.70156 0.167222
\(262\) 0 0
\(263\) 26.8062 1.65294 0.826472 0.562978i \(-0.190344\pi\)
0.826472 + 0.562978i \(0.190344\pi\)
\(264\) 0 0
\(265\) −5.40312 −0.331911
\(266\) 0 0
\(267\) −8.80625 −0.538934
\(268\) 0 0
\(269\) −4.80625 −0.293042 −0.146521 0.989208i \(-0.546808\pi\)
−0.146521 + 0.989208i \(0.546808\pi\)
\(270\) 0 0
\(271\) −12.2094 −0.741667 −0.370833 0.928699i \(-0.620928\pi\)
−0.370833 + 0.928699i \(0.620928\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 1.61250 0.0972372
\(276\) 0 0
\(277\) 27.6125 1.65907 0.829537 0.558452i \(-0.188604\pi\)
0.829537 + 0.558452i \(0.188604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.8062 0.763957 0.381978 0.924171i \(-0.375243\pi\)
0.381978 + 0.924171i \(0.375243\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 1.89531 0.112269
\(286\) 0 0
\(287\) −3.40312 −0.200880
\(288\) 0 0
\(289\) −9.70156 −0.570680
\(290\) 0 0
\(291\) −8.80625 −0.516231
\(292\) 0 0
\(293\) −12.8062 −0.748149 −0.374075 0.927399i \(-0.622040\pi\)
−0.374075 + 0.927399i \(0.622040\pi\)
\(294\) 0 0
\(295\) 40.0000 2.32889
\(296\) 0 0
\(297\) 0.701562 0.0407088
\(298\) 0 0
\(299\) −4.70156 −0.271898
\(300\) 0 0
\(301\) −10.1047 −0.582424
\(302\) 0 0
\(303\) −3.40312 −0.195504
\(304\) 0 0
\(305\) 3.50781 0.200857
\(306\) 0 0
\(307\) 6.80625 0.388453 0.194227 0.980957i \(-0.437780\pi\)
0.194227 + 0.980957i \(0.437780\pi\)
\(308\) 0 0
\(309\) 3.29844 0.187642
\(310\) 0 0
\(311\) −14.5969 −0.827713 −0.413856 0.910342i \(-0.635818\pi\)
−0.413856 + 0.910342i \(0.635818\pi\)
\(312\) 0 0
\(313\) 22.2094 1.25535 0.627674 0.778476i \(-0.284007\pi\)
0.627674 + 0.778476i \(0.284007\pi\)
\(314\) 0 0
\(315\) −2.70156 −0.152216
\(316\) 0 0
\(317\) 7.40312 0.415801 0.207900 0.978150i \(-0.433337\pi\)
0.207900 + 0.978150i \(0.433337\pi\)
\(318\) 0 0
\(319\) 1.89531 0.106117
\(320\) 0 0
\(321\) 5.40312 0.301573
\(322\) 0 0
\(323\) −1.89531 −0.105458
\(324\) 0 0
\(325\) 2.29844 0.127494
\(326\) 0 0
\(327\) 9.29844 0.514205
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −32.2094 −1.77039 −0.885194 0.465223i \(-0.845974\pi\)
−0.885194 + 0.465223i \(0.845974\pi\)
\(332\) 0 0
\(333\) 10.7016 0.586442
\(334\) 0 0
\(335\) −14.5969 −0.797513
\(336\) 0 0
\(337\) −29.5078 −1.60739 −0.803696 0.595040i \(-0.797136\pi\)
−0.803696 + 0.595040i \(0.797136\pi\)
\(338\) 0 0
\(339\) 4.80625 0.261040
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −12.7016 −0.683829
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 20.8062 1.10740 0.553702 0.832715i \(-0.313215\pi\)
0.553702 + 0.832715i \(0.313215\pi\)
\(354\) 0 0
\(355\) 21.6125 1.14707
\(356\) 0 0
\(357\) 2.70156 0.142982
\(358\) 0 0
\(359\) −26.8062 −1.41478 −0.707390 0.706824i \(-0.750127\pi\)
−0.707390 + 0.706824i \(0.750127\pi\)
\(360\) 0 0
\(361\) −18.5078 −0.974095
\(362\) 0 0
\(363\) −10.5078 −0.551517
\(364\) 0 0
\(365\) −3.50781 −0.183607
\(366\) 0 0
\(367\) 29.6125 1.54576 0.772880 0.634552i \(-0.218816\pi\)
0.772880 + 0.634552i \(0.218816\pi\)
\(368\) 0 0
\(369\) 3.40312 0.177160
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −19.4031 −1.00466 −0.502328 0.864677i \(-0.667523\pi\)
−0.502328 + 0.864677i \(0.667523\pi\)
\(374\) 0 0
\(375\) −7.29844 −0.376890
\(376\) 0 0
\(377\) 2.70156 0.139138
\(378\) 0 0
\(379\) 17.6125 0.904693 0.452347 0.891842i \(-0.350587\pi\)
0.452347 + 0.891842i \(0.350587\pi\)
\(380\) 0 0
\(381\) 6.59688 0.337968
\(382\) 0 0
\(383\) −16.9109 −0.864108 −0.432054 0.901848i \(-0.642211\pi\)
−0.432054 + 0.901848i \(0.642211\pi\)
\(384\) 0 0
\(385\) −1.89531 −0.0965941
\(386\) 0 0
\(387\) 10.1047 0.513650
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 12.7016 0.642346
\(392\) 0 0
\(393\) 7.29844 0.368157
\(394\) 0 0
\(395\) −25.4031 −1.27817
\(396\) 0 0
\(397\) 0.806248 0.0404645 0.0202322 0.999795i \(-0.493559\pi\)
0.0202322 + 0.999795i \(0.493559\pi\)
\(398\) 0 0
\(399\) −0.701562 −0.0351220
\(400\) 0 0
\(401\) 4.80625 0.240013 0.120006 0.992773i \(-0.461709\pi\)
0.120006 + 0.992773i \(0.461709\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.70156 0.134242
\(406\) 0 0
\(407\) 7.50781 0.372148
\(408\) 0 0
\(409\) 5.29844 0.261991 0.130995 0.991383i \(-0.458183\pi\)
0.130995 + 0.991383i \(0.458183\pi\)
\(410\) 0 0
\(411\) −18.7016 −0.922480
\(412\) 0 0
\(413\) −14.8062 −0.728568
\(414\) 0 0
\(415\) 36.2094 1.77745
\(416\) 0 0
\(417\) −6.80625 −0.333303
\(418\) 0 0
\(419\) −34.1047 −1.66612 −0.833061 0.553180i \(-0.813414\pi\)
−0.833061 + 0.553180i \(0.813414\pi\)
\(420\) 0 0
\(421\) 19.6125 0.955855 0.477927 0.878399i \(-0.341388\pi\)
0.477927 + 0.878399i \(0.341388\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −6.20937 −0.301199
\(426\) 0 0
\(427\) −1.29844 −0.0628358
\(428\) 0 0
\(429\) 0.701562 0.0338717
\(430\) 0 0
\(431\) 12.2094 0.588105 0.294052 0.955789i \(-0.404996\pi\)
0.294052 + 0.955789i \(0.404996\pi\)
\(432\) 0 0
\(433\) 14.2094 0.682859 0.341429 0.939907i \(-0.389089\pi\)
0.341429 + 0.939907i \(0.389089\pi\)
\(434\) 0 0
\(435\) 7.29844 0.349933
\(436\) 0 0
\(437\) −3.29844 −0.157786
\(438\) 0 0
\(439\) 0.492189 0.0234909 0.0117455 0.999931i \(-0.496261\pi\)
0.0117455 + 0.999931i \(0.496261\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −0.209373 −0.00994760 −0.00497380 0.999988i \(-0.501583\pi\)
−0.00497380 + 0.999988i \(0.501583\pi\)
\(444\) 0 0
\(445\) −23.7906 −1.12778
\(446\) 0 0
\(447\) 15.4031 0.728543
\(448\) 0 0
\(449\) −7.89531 −0.372603 −0.186301 0.982493i \(-0.559650\pi\)
−0.186301 + 0.982493i \(0.559650\pi\)
\(450\) 0 0
\(451\) 2.38750 0.112423
\(452\) 0 0
\(453\) 4.70156 0.220899
\(454\) 0 0
\(455\) −2.70156 −0.126651
\(456\) 0 0
\(457\) 0.596876 0.0279207 0.0139603 0.999903i \(-0.495556\pi\)
0.0139603 + 0.999903i \(0.495556\pi\)
\(458\) 0 0
\(459\) −2.70156 −0.126098
\(460\) 0 0
\(461\) −20.3141 −0.946120 −0.473060 0.881030i \(-0.656851\pi\)
−0.473060 + 0.881030i \(0.656851\pi\)
\(462\) 0 0
\(463\) 34.3141 1.59471 0.797355 0.603511i \(-0.206232\pi\)
0.797355 + 0.603511i \(0.206232\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.49219 0.207874 0.103937 0.994584i \(-0.466856\pi\)
0.103937 + 0.994584i \(0.466856\pi\)
\(468\) 0 0
\(469\) 5.40312 0.249493
\(470\) 0 0
\(471\) 20.1047 0.926375
\(472\) 0 0
\(473\) 7.08907 0.325956
\(474\) 0 0
\(475\) 1.61250 0.0739864
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −7.50781 −0.343041 −0.171520 0.985181i \(-0.554868\pi\)
−0.171520 + 0.985181i \(0.554868\pi\)
\(480\) 0 0
\(481\) 10.7016 0.487949
\(482\) 0 0
\(483\) 4.70156 0.213928
\(484\) 0 0
\(485\) −23.7906 −1.08028
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −5.40312 −0.244338
\(490\) 0 0
\(491\) −26.5969 −1.20030 −0.600150 0.799887i \(-0.704892\pi\)
−0.600150 + 0.799887i \(0.704892\pi\)
\(492\) 0 0
\(493\) −7.29844 −0.328705
\(494\) 0 0
\(495\) 1.89531 0.0851880
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 1.19375 0.0534397 0.0267198 0.999643i \(-0.491494\pi\)
0.0267198 + 0.999643i \(0.491494\pi\)
\(500\) 0 0
\(501\) −3.29844 −0.147363
\(502\) 0 0
\(503\) 33.4031 1.48937 0.744686 0.667415i \(-0.232599\pi\)
0.744686 + 0.667415i \(0.232599\pi\)
\(504\) 0 0
\(505\) −9.19375 −0.409117
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 38.9109 1.72470 0.862348 0.506315i \(-0.168993\pi\)
0.862348 + 0.506315i \(0.168993\pi\)
\(510\) 0 0
\(511\) 1.29844 0.0574395
\(512\) 0 0
\(513\) 0.701562 0.0309747
\(514\) 0 0
\(515\) 8.91093 0.392663
\(516\) 0 0
\(517\) 5.61250 0.246837
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −1.29844 −0.0568856 −0.0284428 0.999595i \(-0.509055\pi\)
−0.0284428 + 0.999595i \(0.509055\pi\)
\(522\) 0 0
\(523\) 33.6125 1.46977 0.734886 0.678191i \(-0.237236\pi\)
0.734886 + 0.678191i \(0.237236\pi\)
\(524\) 0 0
\(525\) −2.29844 −0.100312
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.895314 −0.0389267
\(530\) 0 0
\(531\) 14.8062 0.642536
\(532\) 0 0
\(533\) 3.40312 0.147406
\(534\) 0 0
\(535\) 14.5969 0.631078
\(536\) 0 0
\(537\) −14.8062 −0.638937
\(538\) 0 0
\(539\) 0.701562 0.0302184
\(540\) 0 0
\(541\) −6.70156 −0.288123 −0.144061 0.989569i \(-0.546016\pi\)
−0.144061 + 0.989569i \(0.546016\pi\)
\(542\) 0 0
\(543\) 8.80625 0.377912
\(544\) 0 0
\(545\) 25.1203 1.07604
\(546\) 0 0
\(547\) −9.61250 −0.411001 −0.205500 0.978657i \(-0.565882\pi\)
−0.205500 + 0.978657i \(0.565882\pi\)
\(548\) 0 0
\(549\) 1.29844 0.0554160
\(550\) 0 0
\(551\) 1.89531 0.0807431
\(552\) 0 0
\(553\) 9.40312 0.399862
\(554\) 0 0
\(555\) 28.9109 1.22720
\(556\) 0 0
\(557\) −24.5969 −1.04220 −0.521102 0.853495i \(-0.674479\pi\)
−0.521102 + 0.853495i \(0.674479\pi\)
\(558\) 0 0
\(559\) 10.1047 0.427383
\(560\) 0 0
\(561\) −1.89531 −0.0800202
\(562\) 0 0
\(563\) −8.70156 −0.366727 −0.183364 0.983045i \(-0.558699\pi\)
−0.183364 + 0.983045i \(0.558699\pi\)
\(564\) 0 0
\(565\) 12.9844 0.546257
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 1.19375 0.0499569 0.0249785 0.999688i \(-0.492048\pi\)
0.0249785 + 0.999688i \(0.492048\pi\)
\(572\) 0 0
\(573\) −12.7016 −0.530615
\(574\) 0 0
\(575\) −10.8062 −0.450652
\(576\) 0 0
\(577\) −8.80625 −0.366609 −0.183304 0.983056i \(-0.558679\pi\)
−0.183304 + 0.983056i \(0.558679\pi\)
\(578\) 0 0
\(579\) 11.4031 0.473898
\(580\) 0 0
\(581\) −13.4031 −0.556055
\(582\) 0 0
\(583\) −1.40312 −0.0581115
\(584\) 0 0
\(585\) 2.70156 0.111696
\(586\) 0 0
\(587\) −35.0156 −1.44525 −0.722625 0.691241i \(-0.757064\pi\)
−0.722625 + 0.691241i \(0.757064\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −3.40312 −0.139986
\(592\) 0 0
\(593\) −16.8062 −0.690150 −0.345075 0.938575i \(-0.612147\pi\)
−0.345075 + 0.938575i \(0.612147\pi\)
\(594\) 0 0
\(595\) 7.29844 0.299207
\(596\) 0 0
\(597\) −22.1047 −0.904685
\(598\) 0 0
\(599\) −11.2984 −0.461642 −0.230821 0.972996i \(-0.574141\pi\)
−0.230821 + 0.972996i \(0.574141\pi\)
\(600\) 0 0
\(601\) −15.4031 −0.628307 −0.314153 0.949372i \(-0.601721\pi\)
−0.314153 + 0.949372i \(0.601721\pi\)
\(602\) 0 0
\(603\) −5.40312 −0.220032
\(604\) 0 0
\(605\) −28.3875 −1.15412
\(606\) 0 0
\(607\) −7.50781 −0.304733 −0.152366 0.988324i \(-0.548689\pi\)
−0.152366 + 0.988324i \(0.548689\pi\)
\(608\) 0 0
\(609\) −2.70156 −0.109473
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 38.9109 1.57160 0.785799 0.618482i \(-0.212252\pi\)
0.785799 + 0.618482i \(0.212252\pi\)
\(614\) 0 0
\(615\) 9.19375 0.370728
\(616\) 0 0
\(617\) −30.9109 −1.24443 −0.622214 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(618\) 0 0
\(619\) −7.29844 −0.293349 −0.146674 0.989185i \(-0.546857\pi\)
−0.146674 + 0.989185i \(0.546857\pi\)
\(620\) 0 0
\(621\) −4.70156 −0.188667
\(622\) 0 0
\(623\) 8.80625 0.352815
\(624\) 0 0
\(625\) −31.2094 −1.24837
\(626\) 0 0
\(627\) 0.492189 0.0196562
\(628\) 0 0
\(629\) −28.9109 −1.15275
\(630\) 0 0
\(631\) 7.50781 0.298881 0.149441 0.988771i \(-0.452253\pi\)
0.149441 + 0.988771i \(0.452253\pi\)
\(632\) 0 0
\(633\) 24.7016 0.981799
\(634\) 0 0
\(635\) 17.8219 0.707239
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −48.8062 −1.92773 −0.963865 0.266390i \(-0.914169\pi\)
−0.963865 + 0.266390i \(0.914169\pi\)
\(642\) 0 0
\(643\) −46.3141 −1.82645 −0.913224 0.407458i \(-0.866415\pi\)
−0.913224 + 0.407458i \(0.866415\pi\)
\(644\) 0 0
\(645\) 27.2984 1.07487
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 10.3875 0.407745
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.50781 −0.372069 −0.186035 0.982543i \(-0.559564\pi\)
−0.186035 + 0.982543i \(0.559564\pi\)
\(654\) 0 0
\(655\) 19.7172 0.770414
\(656\) 0 0
\(657\) −1.29844 −0.0506569
\(658\) 0 0
\(659\) 35.0156 1.36401 0.682007 0.731345i \(-0.261107\pi\)
0.682007 + 0.731345i \(0.261107\pi\)
\(660\) 0 0
\(661\) −50.4187 −1.96106 −0.980531 0.196365i \(-0.937086\pi\)
−0.980531 + 0.196365i \(0.937086\pi\)
\(662\) 0 0
\(663\) −2.70156 −0.104920
\(664\) 0 0
\(665\) −1.89531 −0.0734971
\(666\) 0 0
\(667\) −12.7016 −0.491806
\(668\) 0 0
\(669\) −9.40312 −0.363546
\(670\) 0 0
\(671\) 0.910935 0.0351662
\(672\) 0 0
\(673\) 42.9109 1.65409 0.827047 0.562132i \(-0.190019\pi\)
0.827047 + 0.562132i \(0.190019\pi\)
\(674\) 0 0
\(675\) 2.29844 0.0884669
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 8.80625 0.337953
\(680\) 0 0
\(681\) −21.4031 −0.820170
\(682\) 0 0
\(683\) −11.5078 −0.440334 −0.220167 0.975462i \(-0.570660\pi\)
−0.220167 + 0.975462i \(0.570660\pi\)
\(684\) 0 0
\(685\) −50.5234 −1.93040
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −49.6125 −1.88735 −0.943674 0.330876i \(-0.892656\pi\)
−0.943674 + 0.330876i \(0.892656\pi\)
\(692\) 0 0
\(693\) −0.701562 −0.0266501
\(694\) 0 0
\(695\) −18.3875 −0.697478
\(696\) 0 0
\(697\) −9.19375 −0.348238
\(698\) 0 0
\(699\) −18.2094 −0.688742
\(700\) 0 0
\(701\) 3.19375 0.120626 0.0603132 0.998180i \(-0.480790\pi\)
0.0603132 + 0.998180i \(0.480790\pi\)
\(702\) 0 0
\(703\) 7.50781 0.283162
\(704\) 0 0
\(705\) 21.6125 0.813974
\(706\) 0 0
\(707\) 3.40312 0.127988
\(708\) 0 0
\(709\) 51.6125 1.93835 0.969174 0.246377i \(-0.0792402\pi\)
0.969174 + 0.246377i \(0.0792402\pi\)
\(710\) 0 0
\(711\) −9.40312 −0.352645
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.89531 0.0708807
\(716\) 0 0
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) 31.0156 1.15669 0.578344 0.815793i \(-0.303699\pi\)
0.578344 + 0.815793i \(0.303699\pi\)
\(720\) 0 0
\(721\) −3.29844 −0.122840
\(722\) 0 0
\(723\) −24.8062 −0.922554
\(724\) 0 0
\(725\) 6.20937 0.230610
\(726\) 0 0
\(727\) 22.1047 0.819817 0.409909 0.912127i \(-0.365561\pi\)
0.409909 + 0.912127i \(0.365561\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −27.2984 −1.00967
\(732\) 0 0
\(733\) 20.5969 0.760763 0.380381 0.924830i \(-0.375793\pi\)
0.380381 + 0.924830i \(0.375793\pi\)
\(734\) 0 0
\(735\) 2.70156 0.0996486
\(736\) 0 0
\(737\) −3.79063 −0.139630
\(738\) 0 0
\(739\) −40.2094 −1.47913 −0.739563 0.673088i \(-0.764968\pi\)
−0.739563 + 0.673088i \(0.764968\pi\)
\(740\) 0 0
\(741\) 0.701562 0.0257725
\(742\) 0 0
\(743\) −23.0156 −0.844361 −0.422181 0.906512i \(-0.638735\pi\)
−0.422181 + 0.906512i \(0.638735\pi\)
\(744\) 0 0
\(745\) 41.6125 1.52456
\(746\) 0 0
\(747\) 13.4031 0.490395
\(748\) 0 0
\(749\) −5.40312 −0.197426
\(750\) 0 0
\(751\) −34.8062 −1.27010 −0.635049 0.772472i \(-0.719020\pi\)
−0.635049 + 0.772472i \(0.719020\pi\)
\(752\) 0 0
\(753\) −3.50781 −0.127832
\(754\) 0 0
\(755\) 12.7016 0.462257
\(756\) 0 0
\(757\) 30.4187 1.10559 0.552794 0.833318i \(-0.313562\pi\)
0.552794 + 0.833318i \(0.313562\pi\)
\(758\) 0 0
\(759\) −3.29844 −0.119726
\(760\) 0 0
\(761\) 32.5969 1.18164 0.590818 0.806805i \(-0.298805\pi\)
0.590818 + 0.806805i \(0.298805\pi\)
\(762\) 0 0
\(763\) −9.29844 −0.336626
\(764\) 0 0
\(765\) −7.29844 −0.263876
\(766\) 0 0
\(767\) 14.8062 0.534623
\(768\) 0 0
\(769\) 50.9109 1.83590 0.917948 0.396702i \(-0.129845\pi\)
0.917948 + 0.396702i \(0.129845\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) −28.3141 −1.01839 −0.509193 0.860652i \(-0.670056\pi\)
−0.509193 + 0.860652i \(0.670056\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −10.7016 −0.383916
\(778\) 0 0
\(779\) 2.38750 0.0855412
\(780\) 0 0
\(781\) 5.61250 0.200831
\(782\) 0 0
\(783\) 2.70156 0.0965460
\(784\) 0 0
\(785\) 54.3141 1.93855
\(786\) 0 0
\(787\) −20.9109 −0.745394 −0.372697 0.927953i \(-0.621567\pi\)
−0.372697 + 0.927953i \(0.621567\pi\)
\(788\) 0 0
\(789\) 26.8062 0.954328
\(790\) 0 0
\(791\) −4.80625 −0.170891
\(792\) 0 0
\(793\) 1.29844 0.0461089
\(794\) 0 0
\(795\) −5.40312 −0.191629
\(796\) 0 0
\(797\) −40.5969 −1.43802 −0.719008 0.695002i \(-0.755403\pi\)
−0.719008 + 0.695002i \(0.755403\pi\)
\(798\) 0 0
\(799\) −21.6125 −0.764595
\(800\) 0 0
\(801\) −8.80625 −0.311153
\(802\) 0 0
\(803\) −0.910935 −0.0321462
\(804\) 0 0
\(805\) 12.7016 0.447671
\(806\) 0 0
\(807\) −4.80625 −0.169188
\(808\) 0 0
\(809\) −11.6125 −0.408274 −0.204137 0.978942i \(-0.565439\pi\)
−0.204137 + 0.978942i \(0.565439\pi\)
\(810\) 0 0
\(811\) 21.8953 0.768848 0.384424 0.923157i \(-0.374400\pi\)
0.384424 + 0.923157i \(0.374400\pi\)
\(812\) 0 0
\(813\) −12.2094 −0.428201
\(814\) 0 0
\(815\) −14.5969 −0.511306
\(816\) 0 0
\(817\) 7.08907 0.248015
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 28.5969 0.998038 0.499019 0.866591i \(-0.333694\pi\)
0.499019 + 0.866591i \(0.333694\pi\)
\(822\) 0 0
\(823\) −5.19375 −0.181043 −0.0905214 0.995895i \(-0.528853\pi\)
−0.0905214 + 0.995895i \(0.528853\pi\)
\(824\) 0 0
\(825\) 1.61250 0.0561399
\(826\) 0 0
\(827\) −52.9109 −1.83989 −0.919947 0.392043i \(-0.871768\pi\)
−0.919947 + 0.392043i \(0.871768\pi\)
\(828\) 0 0
\(829\) −36.3141 −1.26124 −0.630620 0.776092i \(-0.717199\pi\)
−0.630620 + 0.776092i \(0.717199\pi\)
\(830\) 0 0
\(831\) 27.6125 0.957867
\(832\) 0 0
\(833\) −2.70156 −0.0936036
\(834\) 0 0
\(835\) −8.91093 −0.308376
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.8062 −1.20165 −0.600823 0.799382i \(-0.705160\pi\)
−0.600823 + 0.799382i \(0.705160\pi\)
\(840\) 0 0
\(841\) −21.7016 −0.748330
\(842\) 0 0
\(843\) 12.8062 0.441071
\(844\) 0 0
\(845\) 2.70156 0.0929366
\(846\) 0 0
\(847\) 10.5078 0.361053
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −50.3141 −1.72474
\(852\) 0 0
\(853\) −22.2094 −0.760434 −0.380217 0.924897i \(-0.624151\pi\)
−0.380217 + 0.924897i \(0.624151\pi\)
\(854\) 0 0
\(855\) 1.89531 0.0648184
\(856\) 0 0
\(857\) −16.8062 −0.574091 −0.287045 0.957917i \(-0.592673\pi\)
−0.287045 + 0.957917i \(0.592673\pi\)
\(858\) 0 0
\(859\) −38.8062 −1.32405 −0.662026 0.749481i \(-0.730303\pi\)
−0.662026 + 0.749481i \(0.730303\pi\)
\(860\) 0 0
\(861\) −3.40312 −0.115978
\(862\) 0 0
\(863\) 22.5969 0.769207 0.384603 0.923082i \(-0.374338\pi\)
0.384603 + 0.923082i \(0.374338\pi\)
\(864\) 0 0
\(865\) −48.6281 −1.65341
\(866\) 0 0
\(867\) −9.70156 −0.329482
\(868\) 0 0
\(869\) −6.59688 −0.223784
\(870\) 0 0
\(871\) −5.40312 −0.183078
\(872\) 0 0
\(873\) −8.80625 −0.298046
\(874\) 0 0
\(875\) 7.29844 0.246732
\(876\) 0 0
\(877\) 0.387503 0.0130850 0.00654252 0.999979i \(-0.497917\pi\)
0.00654252 + 0.999979i \(0.497917\pi\)
\(878\) 0 0
\(879\) −12.8062 −0.431944
\(880\) 0 0
\(881\) −8.31406 −0.280108 −0.140054 0.990144i \(-0.544728\pi\)
−0.140054 + 0.990144i \(0.544728\pi\)
\(882\) 0 0
\(883\) −13.8953 −0.467615 −0.233807 0.972283i \(-0.575118\pi\)
−0.233807 + 0.972283i \(0.575118\pi\)
\(884\) 0 0
\(885\) 40.0000 1.34459
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −6.59688 −0.221252
\(890\) 0 0
\(891\) 0.701562 0.0235032
\(892\) 0 0
\(893\) 5.61250 0.187815
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) 0 0
\(897\) −4.70156 −0.156981
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 5.40312 0.180004
\(902\) 0 0
\(903\) −10.1047 −0.336263
\(904\) 0 0
\(905\) 23.7906 0.790827
\(906\) 0 0
\(907\) 25.6125 0.850449 0.425225 0.905088i \(-0.360195\pi\)
0.425225 + 0.905088i \(0.360195\pi\)
\(908\) 0 0
\(909\) −3.40312 −0.112875
\(910\) 0 0
\(911\) −40.9109 −1.35544 −0.677720 0.735320i \(-0.737032\pi\)
−0.677720 + 0.735320i \(0.737032\pi\)
\(912\) 0 0
\(913\) 9.40312 0.311198
\(914\) 0 0
\(915\) 3.50781 0.115965
\(916\) 0 0
\(917\) −7.29844 −0.241016
\(918\) 0 0
\(919\) 14.5969 0.481507 0.240753 0.970586i \(-0.422606\pi\)
0.240753 + 0.970586i \(0.422606\pi\)
\(920\) 0 0
\(921\) 6.80625 0.224274
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 24.5969 0.808740
\(926\) 0 0
\(927\) 3.29844 0.108335
\(928\) 0 0
\(929\) 33.0156 1.08321 0.541604 0.840634i \(-0.317817\pi\)
0.541604 + 0.840634i \(0.317817\pi\)
\(930\) 0 0
\(931\) 0.701562 0.0229928
\(932\) 0 0
\(933\) −14.5969 −0.477880
\(934\) 0 0
\(935\) −5.12031 −0.167452
\(936\) 0 0
\(937\) 28.8062 0.941059 0.470530 0.882384i \(-0.344063\pi\)
0.470530 + 0.882384i \(0.344063\pi\)
\(938\) 0 0
\(939\) 22.2094 0.724775
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) −2.70156 −0.0878818
\(946\) 0 0
\(947\) −4.49219 −0.145977 −0.0729883 0.997333i \(-0.523254\pi\)
−0.0729883 + 0.997333i \(0.523254\pi\)
\(948\) 0 0
\(949\) −1.29844 −0.0421491
\(950\) 0 0
\(951\) 7.40312 0.240063
\(952\) 0 0
\(953\) −39.8219 −1.28996 −0.644978 0.764201i \(-0.723134\pi\)
−0.644978 + 0.764201i \(0.723134\pi\)
\(954\) 0 0
\(955\) −34.3141 −1.11038
\(956\) 0 0
\(957\) 1.89531 0.0612668
\(958\) 0 0
\(959\) 18.7016 0.603905
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 5.40312 0.174113
\(964\) 0 0
\(965\) 30.8062 0.991688
\(966\) 0 0
\(967\) 51.7172 1.66311 0.831556 0.555441i \(-0.187450\pi\)
0.831556 + 0.555441i \(0.187450\pi\)
\(968\) 0 0
\(969\) −1.89531 −0.0608862
\(970\) 0 0
\(971\) −54.8062 −1.75882 −0.879408 0.476069i \(-0.842061\pi\)
−0.879408 + 0.476069i \(0.842061\pi\)
\(972\) 0 0
\(973\) 6.80625 0.218198
\(974\) 0 0
\(975\) 2.29844 0.0736089
\(976\) 0 0
\(977\) −41.7172 −1.33465 −0.667325 0.744766i \(-0.732561\pi\)
−0.667325 + 0.744766i \(0.732561\pi\)
\(978\) 0 0
\(979\) −6.17813 −0.197454
\(980\) 0 0
\(981\) 9.29844 0.296876
\(982\) 0 0
\(983\) 38.1047 1.21535 0.607675 0.794186i \(-0.292102\pi\)
0.607675 + 0.794186i \(0.292102\pi\)
\(984\) 0 0
\(985\) −9.19375 −0.292937
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −47.5078 −1.51066
\(990\) 0 0
\(991\) 38.5969 1.22607 0.613035 0.790056i \(-0.289948\pi\)
0.613035 + 0.790056i \(0.289948\pi\)
\(992\) 0 0
\(993\) −32.2094 −1.02213
\(994\) 0 0
\(995\) −59.7172 −1.89316
\(996\) 0 0
\(997\) −12.8062 −0.405578 −0.202789 0.979222i \(-0.565001\pi\)
−0.202789 + 0.979222i \(0.565001\pi\)
\(998\) 0 0
\(999\) 10.7016 0.338582
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 4368.2.a.bg.1.2 2
4.3 odd 2 546.2.a.i.1.2 2
12.11 even 2 1638.2.a.w.1.1 2
28.27 even 2 3822.2.a.bt.1.1 2
52.51 odd 2 7098.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.i.1.2 2 4.3 odd 2
1638.2.a.w.1.1 2 12.11 even 2
3822.2.a.bt.1.1 2 28.27 even 2
4368.2.a.bg.1.2 2 1.1 even 1 trivial
7098.2.a.bh.1.1 2 52.51 odd 2