Properties

Label 7728.2.a.bc.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
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 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -0.701562 q^{13} +2.70156 q^{15} +4.00000 q^{17} -7.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.29844 q^{25} -1.00000 q^{27} +6.70156 q^{29} +2.00000 q^{31} -4.00000 q^{33} +2.70156 q^{35} +10.7016 q^{37} +0.701562 q^{39} -6.70156 q^{41} -4.70156 q^{43} -2.70156 q^{45} -8.10469 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.40312 q^{53} -10.8062 q^{55} +7.40312 q^{57} -5.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +1.89531 q^{65} +10.8062 q^{67} +1.00000 q^{69} -5.40312 q^{71} -11.4031 q^{73} -2.29844 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -0.596876 q^{83} -10.8062 q^{85} -6.70156 q^{87} +1.40312 q^{89} +0.701562 q^{91} -2.00000 q^{93} +20.0000 q^{95} +12.7016 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + 8q^{11} + 5q^{13} - q^{15} + 8q^{17} - 2q^{19} + 2q^{21} - 2q^{23} + 11q^{25} - 2q^{27} + 7q^{29} + 4q^{31} - 8q^{33} - q^{35} + 15q^{37} - 5q^{39} - 7q^{41} - 3q^{43} + q^{45} + 3q^{47} + 2q^{49} - 8q^{51} - 6q^{53} + 4q^{55} + 2q^{57} + 2q^{59} - 4q^{61} - 2q^{63} + 23q^{65} - 4q^{67} + 2q^{69} + 2q^{71} - 10q^{73} - 11q^{75} - 8q^{77} - 16q^{79} + 2q^{81} - 14q^{83} + 4q^{85} - 7q^{87} - 10q^{89} - 5q^{91} - 4q^{93} + 40q^{95} + 19q^{97} + 8q^{99} + O(q^{100}) \)

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 \(-0.706462\pi\)
−0.604088 + 0.796918i \(0.706462\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −0.701562 −0.194578 −0.0972892 0.995256i \(-0.531017\pi\)
−0.0972892 + 0.995256i \(0.531017\pi\)
\(14\) 0 0
\(15\) 2.70156 0.697540
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −7.40312 −1.69839 −0.849197 0.528077i \(-0.822913\pi\)
−0.849197 + 0.528077i \(0.822913\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.29844 0.459688
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.70156 1.24445 0.622224 0.782839i \(-0.286229\pi\)
0.622224 + 0.782839i \(0.286229\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 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
\(39\) 0.701562 0.112340
\(40\) 0 0
\(41\) −6.70156 −1.04661 −0.523304 0.852146i \(-0.675301\pi\)
−0.523304 + 0.852146i \(0.675301\pi\)
\(42\) 0 0
\(43\) −4.70156 −0.716982 −0.358491 0.933533i \(-0.616709\pi\)
−0.358491 + 0.933533i \(0.616709\pi\)
\(44\) 0 0
\(45\) −2.70156 −0.402725
\(46\) 0 0
\(47\) −8.10469 −1.18219 −0.591095 0.806602i \(-0.701304\pi\)
−0.591095 + 0.806602i \(0.701304\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 3.40312 0.467455 0.233728 0.972302i \(-0.424908\pi\)
0.233728 + 0.972302i \(0.424908\pi\)
\(54\) 0 0
\(55\) −10.8062 −1.45711
\(56\) 0 0
\(57\) 7.40312 0.980568
\(58\) 0 0
\(59\) −5.40312 −0.703427 −0.351713 0.936108i \(-0.614401\pi\)
−0.351713 + 0.936108i \(0.614401\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.89531 0.235085
\(66\) 0 0
\(67\) 10.8062 1.32019 0.660097 0.751181i \(-0.270515\pi\)
0.660097 + 0.751181i \(0.270515\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.40312 −0.641233 −0.320616 0.947209i \(-0.603890\pi\)
−0.320616 + 0.947209i \(0.603890\pi\)
\(72\) 0 0
\(73\) −11.4031 −1.33463 −0.667317 0.744773i \(-0.732558\pi\)
−0.667317 + 0.744773i \(0.732558\pi\)
\(74\) 0 0
\(75\) −2.29844 −0.265401
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.596876 −0.0655156 −0.0327578 0.999463i \(-0.510429\pi\)
−0.0327578 + 0.999463i \(0.510429\pi\)
\(84\) 0 0
\(85\) −10.8062 −1.17210
\(86\) 0 0
\(87\) −6.70156 −0.718483
\(88\) 0 0
\(89\) 1.40312 0.148731 0.0743654 0.997231i \(-0.476307\pi\)
0.0743654 + 0.997231i \(0.476307\pi\)
\(90\) 0 0
\(91\) 0.701562 0.0735437
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 12.7016 1.28965 0.644824 0.764331i \(-0.276931\pi\)
0.644824 + 0.764331i \(0.276931\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 9.40312 0.935646 0.467823 0.883822i \(-0.345039\pi\)
0.467823 + 0.883822i \(0.345039\pi\)
\(102\) 0 0
\(103\) −8.70156 −0.857390 −0.428695 0.903449i \(-0.641027\pi\)
−0.428695 + 0.903449i \(0.641027\pi\)
\(104\) 0 0
\(105\) −2.70156 −0.263645
\(106\) 0 0
\(107\) 6.80625 0.657985 0.328992 0.944333i \(-0.393291\pi\)
0.328992 + 0.944333i \(0.393291\pi\)
\(108\) 0 0
\(109\) −1.29844 −0.124368 −0.0621839 0.998065i \(-0.519807\pi\)
−0.0621839 + 0.998065i \(0.519807\pi\)
\(110\) 0 0
\(111\) −10.7016 −1.01575
\(112\) 0 0
\(113\) −12.1047 −1.13871 −0.569357 0.822091i \(-0.692808\pi\)
−0.569357 + 0.822091i \(0.692808\pi\)
\(114\) 0 0
\(115\) 2.70156 0.251922
\(116\) 0 0
\(117\) −0.701562 −0.0648594
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 6.70156 0.604260
\(124\) 0 0
\(125\) 7.29844 0.652792
\(126\) 0 0
\(127\) −0.701562 −0.0622536 −0.0311268 0.999515i \(-0.509910\pi\)
−0.0311268 + 0.999515i \(0.509910\pi\)
\(128\) 0 0
\(129\) 4.70156 0.413949
\(130\) 0 0
\(131\) −5.40312 −0.472073 −0.236037 0.971744i \(-0.575849\pi\)
−0.236037 + 0.971744i \(0.575849\pi\)
\(132\) 0 0
\(133\) 7.40312 0.641932
\(134\) 0 0
\(135\) 2.70156 0.232513
\(136\) 0 0
\(137\) −13.2984 −1.13616 −0.568081 0.822973i \(-0.692314\pi\)
−0.568081 + 0.822973i \(0.692314\pi\)
\(138\) 0 0
\(139\) −11.2984 −0.958321 −0.479160 0.877727i \(-0.659059\pi\)
−0.479160 + 0.877727i \(0.659059\pi\)
\(140\) 0 0
\(141\) 8.10469 0.682538
\(142\) 0 0
\(143\) −2.80625 −0.234670
\(144\) 0 0
\(145\) −18.1047 −1.50351
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 18.2094 1.49177 0.745885 0.666075i \(-0.232027\pi\)
0.745885 + 0.666075i \(0.232027\pi\)
\(150\) 0 0
\(151\) 20.9109 1.70171 0.850854 0.525402i \(-0.176085\pi\)
0.850854 + 0.525402i \(0.176085\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −5.40312 −0.433989
\(156\) 0 0
\(157\) 11.4031 0.910068 0.455034 0.890474i \(-0.349627\pi\)
0.455034 + 0.890474i \(0.349627\pi\)
\(158\) 0 0
\(159\) −3.40312 −0.269885
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −14.8062 −1.15971 −0.579857 0.814718i \(-0.696892\pi\)
−0.579857 + 0.814718i \(0.696892\pi\)
\(164\) 0 0
\(165\) 10.8062 0.841265
\(166\) 0 0
\(167\) 0.596876 0.0461876 0.0230938 0.999733i \(-0.492648\pi\)
0.0230938 + 0.999733i \(0.492648\pi\)
\(168\) 0 0
\(169\) −12.5078 −0.962139
\(170\) 0 0
\(171\) −7.40312 −0.566131
\(172\) 0 0
\(173\) 6.59688 0.501551 0.250776 0.968045i \(-0.419314\pi\)
0.250776 + 0.968045i \(0.419314\pi\)
\(174\) 0 0
\(175\) −2.29844 −0.173746
\(176\) 0 0
\(177\) 5.40312 0.406124
\(178\) 0 0
\(179\) 5.89531 0.440636 0.220318 0.975428i \(-0.429290\pi\)
0.220318 + 0.975428i \(0.429290\pi\)
\(180\) 0 0
\(181\) 8.80625 0.654563 0.327282 0.944927i \(-0.393867\pi\)
0.327282 + 0.944927i \(0.393867\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −28.9109 −2.12557
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 2.80625 0.203053 0.101527 0.994833i \(-0.467627\pi\)
0.101527 + 0.994833i \(0.467627\pi\)
\(192\) 0 0
\(193\) −3.89531 −0.280391 −0.140195 0.990124i \(-0.544773\pi\)
−0.140195 + 0.990124i \(0.544773\pi\)
\(194\) 0 0
\(195\) −1.89531 −0.135726
\(196\) 0 0
\(197\) −9.50781 −0.677403 −0.338702 0.940894i \(-0.609988\pi\)
−0.338702 + 0.940894i \(0.609988\pi\)
\(198\) 0 0
\(199\) 8.70156 0.616837 0.308419 0.951251i \(-0.400200\pi\)
0.308419 + 0.951251i \(0.400200\pi\)
\(200\) 0 0
\(201\) −10.8062 −0.762214
\(202\) 0 0
\(203\) −6.70156 −0.470357
\(204\) 0 0
\(205\) 18.1047 1.26449
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −29.6125 −2.04834
\(210\) 0 0
\(211\) 14.8062 1.01930 0.509652 0.860381i \(-0.329774\pi\)
0.509652 + 0.860381i \(0.329774\pi\)
\(212\) 0 0
\(213\) 5.40312 0.370216
\(214\) 0 0
\(215\) 12.7016 0.866239
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 11.4031 0.770552
\(220\) 0 0
\(221\) −2.80625 −0.188769
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 2.29844 0.153229
\(226\) 0 0
\(227\) 3.89531 0.258541 0.129271 0.991609i \(-0.458736\pi\)
0.129271 + 0.991609i \(0.458736\pi\)
\(228\) 0 0
\(229\) −27.4031 −1.81085 −0.905425 0.424507i \(-0.860447\pi\)
−0.905425 + 0.424507i \(0.860447\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 21.8953 1.42829
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 14.8062 0.957737 0.478868 0.877887i \(-0.341047\pi\)
0.478868 + 0.877887i \(0.341047\pi\)
\(240\) 0 0
\(241\) 28.9109 1.86232 0.931159 0.364615i \(-0.118799\pi\)
0.931159 + 0.364615i \(0.118799\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.70156 −0.172596
\(246\) 0 0
\(247\) 5.19375 0.330470
\(248\) 0 0
\(249\) 0.596876 0.0378255
\(250\) 0 0
\(251\) −13.2984 −0.839390 −0.419695 0.907665i \(-0.637863\pi\)
−0.419695 + 0.907665i \(0.637863\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 10.8062 0.676714
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −10.7016 −0.664963
\(260\) 0 0
\(261\) 6.70156 0.414816
\(262\) 0 0
\(263\) −3.29844 −0.203390 −0.101695 0.994816i \(-0.532427\pi\)
−0.101695 + 0.994816i \(0.532427\pi\)
\(264\) 0 0
\(265\) −9.19375 −0.564768
\(266\) 0 0
\(267\) −1.40312 −0.0858698
\(268\) 0 0
\(269\) 28.2094 1.71996 0.859978 0.510331i \(-0.170477\pi\)
0.859978 + 0.510331i \(0.170477\pi\)
\(270\) 0 0
\(271\) 11.4031 0.692690 0.346345 0.938107i \(-0.387423\pi\)
0.346345 + 0.938107i \(0.387423\pi\)
\(272\) 0 0
\(273\) −0.701562 −0.0424605
\(274\) 0 0
\(275\) 9.19375 0.554404
\(276\) 0 0
\(277\) −8.59688 −0.516536 −0.258268 0.966073i \(-0.583152\pi\)
−0.258268 + 0.966073i \(0.583152\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −17.5078 −1.04443 −0.522214 0.852814i \(-0.674894\pi\)
−0.522214 + 0.852814i \(0.674894\pi\)
\(282\) 0 0
\(283\) 20.8062 1.23680 0.618402 0.785862i \(-0.287781\pi\)
0.618402 + 0.785862i \(0.287781\pi\)
\(284\) 0 0
\(285\) −20.0000 −1.18470
\(286\) 0 0
\(287\) 6.70156 0.395581
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −12.7016 −0.744579
\(292\) 0 0
\(293\) 4.80625 0.280784 0.140392 0.990096i \(-0.455164\pi\)
0.140392 + 0.990096i \(0.455164\pi\)
\(294\) 0 0
\(295\) 14.5969 0.849863
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0.701562 0.0405724
\(300\) 0 0
\(301\) 4.70156 0.270994
\(302\) 0 0
\(303\) −9.40312 −0.540195
\(304\) 0 0
\(305\) 5.40312 0.309382
\(306\) 0 0
\(307\) 32.9109 1.87833 0.939163 0.343471i \(-0.111603\pi\)
0.939163 + 0.343471i \(0.111603\pi\)
\(308\) 0 0
\(309\) 8.70156 0.495015
\(310\) 0 0
\(311\) −0.596876 −0.0338457 −0.0169229 0.999857i \(-0.505387\pi\)
−0.0169229 + 0.999857i \(0.505387\pi\)
\(312\) 0 0
\(313\) 32.2094 1.82058 0.910291 0.413970i \(-0.135858\pi\)
0.910291 + 0.413970i \(0.135858\pi\)
\(314\) 0 0
\(315\) 2.70156 0.152216
\(316\) 0 0
\(317\) −14.7016 −0.825722 −0.412861 0.910794i \(-0.635470\pi\)
−0.412861 + 0.910794i \(0.635470\pi\)
\(318\) 0 0
\(319\) 26.8062 1.50086
\(320\) 0 0
\(321\) −6.80625 −0.379888
\(322\) 0 0
\(323\) −29.6125 −1.64768
\(324\) 0 0
\(325\) −1.61250 −0.0894452
\(326\) 0 0
\(327\) 1.29844 0.0718038
\(328\) 0 0
\(329\) 8.10469 0.446826
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 10.7016 0.586442
\(334\) 0 0
\(335\) −29.1938 −1.59503
\(336\) 0 0
\(337\) 34.2094 1.86350 0.931752 0.363096i \(-0.118280\pi\)
0.931752 + 0.363096i \(0.118280\pi\)
\(338\) 0 0
\(339\) 12.1047 0.657436
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.70156 −0.145447
\(346\) 0 0
\(347\) 16.7016 0.896587 0.448293 0.893886i \(-0.352032\pi\)
0.448293 + 0.893886i \(0.352032\pi\)
\(348\) 0 0
\(349\) 25.4031 1.35980 0.679899 0.733306i \(-0.262024\pi\)
0.679899 + 0.733306i \(0.262024\pi\)
\(350\) 0 0
\(351\) 0.701562 0.0374466
\(352\) 0 0
\(353\) 10.7016 0.569587 0.284793 0.958589i \(-0.408075\pi\)
0.284793 + 0.958589i \(0.408075\pi\)
\(354\) 0 0
\(355\) 14.5969 0.774722
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 22.1047 1.16664 0.583320 0.812242i \(-0.301753\pi\)
0.583320 + 0.812242i \(0.301753\pi\)
\(360\) 0 0
\(361\) 35.8062 1.88454
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 30.8062 1.61247
\(366\) 0 0
\(367\) −2.10469 −0.109864 −0.0549319 0.998490i \(-0.517494\pi\)
−0.0549319 + 0.998490i \(0.517494\pi\)
\(368\) 0 0
\(369\) −6.70156 −0.348869
\(370\) 0 0
\(371\) −3.40312 −0.176681
\(372\) 0 0
\(373\) 24.8062 1.28442 0.642209 0.766529i \(-0.278018\pi\)
0.642209 + 0.766529i \(0.278018\pi\)
\(374\) 0 0
\(375\) −7.29844 −0.376890
\(376\) 0 0
\(377\) −4.70156 −0.242143
\(378\) 0 0
\(379\) −7.29844 −0.374896 −0.187448 0.982275i \(-0.560022\pi\)
−0.187448 + 0.982275i \(0.560022\pi\)
\(380\) 0 0
\(381\) 0.701562 0.0359421
\(382\) 0 0
\(383\) 21.6125 1.10435 0.552174 0.833729i \(-0.313799\pi\)
0.552174 + 0.833729i \(0.313799\pi\)
\(384\) 0 0
\(385\) 10.8062 0.550737
\(386\) 0 0
\(387\) −4.70156 −0.238994
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 5.40312 0.272552
\(394\) 0 0
\(395\) 21.6125 1.08744
\(396\) 0 0
\(397\) 21.4031 1.07419 0.537096 0.843521i \(-0.319521\pi\)
0.537096 + 0.843521i \(0.319521\pi\)
\(398\) 0 0
\(399\) −7.40312 −0.370620
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −1.40312 −0.0698946
\(404\) 0 0
\(405\) −2.70156 −0.134242
\(406\) 0 0
\(407\) 42.8062 2.12183
\(408\) 0 0
\(409\) −7.40312 −0.366061 −0.183030 0.983107i \(-0.558591\pi\)
−0.183030 + 0.983107i \(0.558591\pi\)
\(410\) 0 0
\(411\) 13.2984 0.655964
\(412\) 0 0
\(413\) 5.40312 0.265870
\(414\) 0 0
\(415\) 1.61250 0.0791544
\(416\) 0 0
\(417\) 11.2984 0.553287
\(418\) 0 0
\(419\) −38.2094 −1.86665 −0.933325 0.359033i \(-0.883107\pi\)
−0.933325 + 0.359033i \(0.883107\pi\)
\(420\) 0 0
\(421\) −9.29844 −0.453178 −0.226589 0.973990i \(-0.572757\pi\)
−0.226589 + 0.973990i \(0.572757\pi\)
\(422\) 0 0
\(423\) −8.10469 −0.394063
\(424\) 0 0
\(425\) 9.19375 0.445962
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 2.80625 0.135487
\(430\) 0 0
\(431\) −40.9109 −1.97061 −0.985305 0.170803i \(-0.945364\pi\)
−0.985305 + 0.170803i \(0.945364\pi\)
\(432\) 0 0
\(433\) −26.3141 −1.26457 −0.632286 0.774735i \(-0.717883\pi\)
−0.632286 + 0.774735i \(0.717883\pi\)
\(434\) 0 0
\(435\) 18.1047 0.868053
\(436\) 0 0
\(437\) 7.40312 0.354139
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.29844 0.346759 0.173380 0.984855i \(-0.444531\pi\)
0.173380 + 0.984855i \(0.444531\pi\)
\(444\) 0 0
\(445\) −3.79063 −0.179693
\(446\) 0 0
\(447\) −18.2094 −0.861274
\(448\) 0 0
\(449\) 38.4187 1.81309 0.906546 0.422106i \(-0.138709\pi\)
0.906546 + 0.422106i \(0.138709\pi\)
\(450\) 0 0
\(451\) −26.8062 −1.26226
\(452\) 0 0
\(453\) −20.9109 −0.982481
\(454\) 0 0
\(455\) −1.89531 −0.0888537
\(456\) 0 0
\(457\) −10.2094 −0.477574 −0.238787 0.971072i \(-0.576750\pi\)
−0.238787 + 0.971072i \(0.576750\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 36.2094 1.68644 0.843219 0.537570i \(-0.180658\pi\)
0.843219 + 0.537570i \(0.180658\pi\)
\(462\) 0 0
\(463\) 28.7016 1.33387 0.666937 0.745114i \(-0.267605\pi\)
0.666937 + 0.745114i \(0.267605\pi\)
\(464\) 0 0
\(465\) 5.40312 0.250564
\(466\) 0 0
\(467\) −2.70156 −0.125013 −0.0625067 0.998045i \(-0.519909\pi\)
−0.0625067 + 0.998045i \(0.519909\pi\)
\(468\) 0 0
\(469\) −10.8062 −0.498986
\(470\) 0 0
\(471\) −11.4031 −0.525428
\(472\) 0 0
\(473\) −18.8062 −0.864712
\(474\) 0 0
\(475\) −17.0156 −0.780730
\(476\) 0 0
\(477\) 3.40312 0.155818
\(478\) 0 0
\(479\) −29.4031 −1.34346 −0.671732 0.740795i \(-0.734449\pi\)
−0.671732 + 0.740795i \(0.734449\pi\)
\(480\) 0 0
\(481\) −7.50781 −0.342327
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −34.3141 −1.55812
\(486\) 0 0
\(487\) 12.7016 0.575563 0.287781 0.957696i \(-0.407082\pi\)
0.287781 + 0.957696i \(0.407082\pi\)
\(488\) 0 0
\(489\) 14.8062 0.669562
\(490\) 0 0
\(491\) −41.6125 −1.87795 −0.938973 0.343991i \(-0.888221\pi\)
−0.938973 + 0.343991i \(0.888221\pi\)
\(492\) 0 0
\(493\) 26.8062 1.20729
\(494\) 0 0
\(495\) −10.8062 −0.485705
\(496\) 0 0
\(497\) 5.40312 0.242363
\(498\) 0 0
\(499\) 10.5969 0.474381 0.237191 0.971463i \(-0.423773\pi\)
0.237191 + 0.971463i \(0.423773\pi\)
\(500\) 0 0
\(501\) −0.596876 −0.0266664
\(502\) 0 0
\(503\) −17.4031 −0.775967 −0.387983 0.921666i \(-0.626828\pi\)
−0.387983 + 0.921666i \(0.626828\pi\)
\(504\) 0 0
\(505\) −25.4031 −1.13042
\(506\) 0 0
\(507\) 12.5078 0.555491
\(508\) 0 0
\(509\) −13.6125 −0.603363 −0.301682 0.953409i \(-0.597548\pi\)
−0.301682 + 0.953409i \(0.597548\pi\)
\(510\) 0 0
\(511\) 11.4031 0.504445
\(512\) 0 0
\(513\) 7.40312 0.326856
\(514\) 0 0
\(515\) 23.5078 1.03588
\(516\) 0 0
\(517\) −32.4187 −1.42577
\(518\) 0 0
\(519\) −6.59688 −0.289571
\(520\) 0 0
\(521\) −33.4031 −1.46342 −0.731709 0.681617i \(-0.761277\pi\)
−0.731709 + 0.681617i \(0.761277\pi\)
\(522\) 0 0
\(523\) 4.80625 0.210163 0.105081 0.994464i \(-0.466490\pi\)
0.105081 + 0.994464i \(0.466490\pi\)
\(524\) 0 0
\(525\) 2.29844 0.100312
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.40312 −0.234476
\(532\) 0 0
\(533\) 4.70156 0.203647
\(534\) 0 0
\(535\) −18.3875 −0.794961
\(536\) 0 0
\(537\) −5.89531 −0.254402
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 20.5969 0.885529 0.442764 0.896638i \(-0.353998\pi\)
0.442764 + 0.896638i \(0.353998\pi\)
\(542\) 0 0
\(543\) −8.80625 −0.377912
\(544\) 0 0
\(545\) 3.50781 0.150258
\(546\) 0 0
\(547\) 35.0156 1.49716 0.748580 0.663045i \(-0.230736\pi\)
0.748580 + 0.663045i \(0.230736\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −49.6125 −2.11356
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 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
\(559\) 3.29844 0.139509
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) −16.1047 −0.678732 −0.339366 0.940654i \(-0.610212\pi\)
−0.339366 + 0.940654i \(0.610212\pi\)
\(564\) 0 0
\(565\) 32.7016 1.37577
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 25.2984 1.06057 0.530283 0.847821i \(-0.322086\pi\)
0.530283 + 0.847821i \(0.322086\pi\)
\(570\) 0 0
\(571\) −10.8062 −0.452227 −0.226114 0.974101i \(-0.572602\pi\)
−0.226114 + 0.974101i \(0.572602\pi\)
\(572\) 0 0
\(573\) −2.80625 −0.117233
\(574\) 0 0
\(575\) −2.29844 −0.0958515
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 3.89531 0.161884
\(580\) 0 0
\(581\) 0.596876 0.0247626
\(582\) 0 0
\(583\) 13.6125 0.563772
\(584\) 0 0
\(585\) 1.89531 0.0783616
\(586\) 0 0
\(587\) 10.8062 0.446022 0.223011 0.974816i \(-0.428411\pi\)
0.223011 + 0.974816i \(0.428411\pi\)
\(588\) 0 0
\(589\) −14.8062 −0.610081
\(590\) 0 0
\(591\) 9.50781 0.391099
\(592\) 0 0
\(593\) 37.2984 1.53166 0.765832 0.643041i \(-0.222328\pi\)
0.765832 + 0.643041i \(0.222328\pi\)
\(594\) 0 0
\(595\) 10.8062 0.443013
\(596\) 0 0
\(597\) −8.70156 −0.356131
\(598\) 0 0
\(599\) 37.6125 1.53680 0.768402 0.639967i \(-0.221052\pi\)
0.768402 + 0.639967i \(0.221052\pi\)
\(600\) 0 0
\(601\) −0.596876 −0.0243471 −0.0121735 0.999926i \(-0.503875\pi\)
−0.0121735 + 0.999926i \(0.503875\pi\)
\(602\) 0 0
\(603\) 10.8062 0.440064
\(604\) 0 0
\(605\) −13.5078 −0.549171
\(606\) 0 0
\(607\) −11.6125 −0.471337 −0.235668 0.971834i \(-0.575728\pi\)
−0.235668 + 0.971834i \(0.575728\pi\)
\(608\) 0 0
\(609\) 6.70156 0.271561
\(610\) 0 0
\(611\) 5.68594 0.230029
\(612\) 0 0
\(613\) −2.70156 −0.109115 −0.0545575 0.998511i \(-0.517375\pi\)
−0.0545575 + 0.998511i \(0.517375\pi\)
\(614\) 0 0
\(615\) −18.1047 −0.730051
\(616\) 0 0
\(617\) −44.8062 −1.80383 −0.901916 0.431912i \(-0.857839\pi\)
−0.901916 + 0.431912i \(0.857839\pi\)
\(618\) 0 0
\(619\) 32.8062 1.31859 0.659297 0.751882i \(-0.270854\pi\)
0.659297 + 0.751882i \(0.270854\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −1.40312 −0.0562150
\(624\) 0 0
\(625\) −31.2094 −1.24837
\(626\) 0 0
\(627\) 29.6125 1.18261
\(628\) 0 0
\(629\) 42.8062 1.70680
\(630\) 0 0
\(631\) −21.6125 −0.860380 −0.430190 0.902738i \(-0.641553\pi\)
−0.430190 + 0.902738i \(0.641553\pi\)
\(632\) 0 0
\(633\) −14.8062 −0.588496
\(634\) 0 0
\(635\) 1.89531 0.0752132
\(636\) 0 0
\(637\) −0.701562 −0.0277969
\(638\) 0 0
\(639\) −5.40312 −0.213744
\(640\) 0 0
\(641\) 49.7172 1.96371 0.981855 0.189631i \(-0.0607293\pi\)
0.981855 + 0.189631i \(0.0607293\pi\)
\(642\) 0 0
\(643\) 24.8062 0.978263 0.489131 0.872210i \(-0.337314\pi\)
0.489131 + 0.872210i \(0.337314\pi\)
\(644\) 0 0
\(645\) −12.7016 −0.500124
\(646\) 0 0
\(647\) 19.4031 0.762816 0.381408 0.924407i \(-0.375439\pi\)
0.381408 + 0.924407i \(0.375439\pi\)
\(648\) 0 0
\(649\) −21.6125 −0.848365
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) −12.1047 −0.473693 −0.236846 0.971547i \(-0.576114\pi\)
−0.236846 + 0.971547i \(0.576114\pi\)
\(654\) 0 0
\(655\) 14.5969 0.570347
\(656\) 0 0
\(657\) −11.4031 −0.444878
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −34.4187 −1.33873 −0.669367 0.742932i \(-0.733435\pi\)
−0.669367 + 0.742932i \(0.733435\pi\)
\(662\) 0 0
\(663\) 2.80625 0.108986
\(664\) 0 0
\(665\) −20.0000 −0.775567
\(666\) 0 0
\(667\) −6.70156 −0.259486
\(668\) 0 0
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 1.29844 0.0500511 0.0250256 0.999687i \(-0.492033\pi\)
0.0250256 + 0.999687i \(0.492033\pi\)
\(674\) 0 0
\(675\) −2.29844 −0.0884669
\(676\) 0 0
\(677\) 26.4187 1.01535 0.507677 0.861547i \(-0.330504\pi\)
0.507677 + 0.861547i \(0.330504\pi\)
\(678\) 0 0
\(679\) −12.7016 −0.487441
\(680\) 0 0
\(681\) −3.89531 −0.149269
\(682\) 0 0
\(683\) 17.6125 0.673923 0.336962 0.941518i \(-0.390601\pi\)
0.336962 + 0.941518i \(0.390601\pi\)
\(684\) 0 0
\(685\) 35.9266 1.37268
\(686\) 0 0
\(687\) 27.4031 1.04549
\(688\) 0 0
\(689\) −2.38750 −0.0909566
\(690\) 0 0
\(691\) −31.5078 −1.19861 −0.599307 0.800519i \(-0.704557\pi\)
−0.599307 + 0.800519i \(0.704557\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) 30.5234 1.15782
\(696\) 0 0
\(697\) −26.8062 −1.01536
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −17.0156 −0.642671 −0.321336 0.946965i \(-0.604132\pi\)
−0.321336 + 0.946965i \(0.604132\pi\)
\(702\) 0 0
\(703\) −79.2250 −2.98803
\(704\) 0 0
\(705\) −21.8953 −0.824625
\(706\) 0 0
\(707\) −9.40312 −0.353641
\(708\) 0 0
\(709\) 48.8062 1.83296 0.916479 0.400084i \(-0.131019\pi\)
0.916479 + 0.400084i \(0.131019\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 7.58125 0.283523
\(716\) 0 0
\(717\) −14.8062 −0.552949
\(718\) 0 0
\(719\) −38.9109 −1.45113 −0.725567 0.688152i \(-0.758422\pi\)
−0.725567 + 0.688152i \(0.758422\pi\)
\(720\) 0 0
\(721\) 8.70156 0.324063
\(722\) 0 0
\(723\) −28.9109 −1.07521
\(724\) 0 0
\(725\) 15.4031 0.572058
\(726\) 0 0
\(727\) 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
\(731\) −18.8062 −0.695574
\(732\) 0 0
\(733\) 13.7906 0.509368 0.254684 0.967024i \(-0.418028\pi\)
0.254684 + 0.967024i \(0.418028\pi\)
\(734\) 0 0
\(735\) 2.70156 0.0996486
\(736\) 0 0
\(737\) 43.2250 1.59221
\(738\) 0 0
\(739\) −35.0156 −1.28807 −0.644035 0.764996i \(-0.722741\pi\)
−0.644035 + 0.764996i \(0.722741\pi\)
\(740\) 0 0
\(741\) −5.19375 −0.190797
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −49.1938 −1.80232
\(746\) 0 0
\(747\) −0.596876 −0.0218385
\(748\) 0 0
\(749\) −6.80625 −0.248695
\(750\) 0 0
\(751\) 21.6125 0.788651 0.394326 0.918971i \(-0.370978\pi\)
0.394326 + 0.918971i \(0.370978\pi\)
\(752\) 0 0
\(753\) 13.2984 0.484622
\(754\) 0 0
\(755\) −56.4922 −2.05596
\(756\) 0 0
\(757\) 50.4187 1.83250 0.916250 0.400606i \(-0.131201\pi\)
0.916250 + 0.400606i \(0.131201\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 15.6125 0.565953 0.282976 0.959127i \(-0.408678\pi\)
0.282976 + 0.959127i \(0.408678\pi\)
\(762\) 0 0
\(763\) 1.29844 0.0470066
\(764\) 0 0
\(765\) −10.8062 −0.390701
\(766\) 0 0
\(767\) 3.79063 0.136872
\(768\) 0 0
\(769\) −28.9109 −1.04255 −0.521277 0.853387i \(-0.674544\pi\)
−0.521277 + 0.853387i \(0.674544\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) 12.1047 0.435375 0.217688 0.976018i \(-0.430149\pi\)
0.217688 + 0.976018i \(0.430149\pi\)
\(774\) 0 0
\(775\) 4.59688 0.165125
\(776\) 0 0
\(777\) 10.7016 0.383916
\(778\) 0 0
\(779\) 49.6125 1.77755
\(780\) 0 0
\(781\) −21.6125 −0.773356
\(782\) 0 0
\(783\) −6.70156 −0.239494
\(784\) 0 0
\(785\) −30.8062 −1.09952
\(786\) 0 0
\(787\) 31.4031 1.11940 0.559700 0.828695i \(-0.310916\pi\)
0.559700 + 0.828695i \(0.310916\pi\)
\(788\) 0 0
\(789\) 3.29844 0.117427
\(790\) 0 0
\(791\) 12.1047 0.430393
\(792\) 0 0
\(793\) 1.40312 0.0498264
\(794\) 0 0
\(795\) 9.19375 0.326069
\(796\) 0 0
\(797\) 9.29844 0.329368 0.164684 0.986346i \(-0.447340\pi\)
0.164684 + 0.986346i \(0.447340\pi\)
\(798\) 0 0
\(799\) −32.4187 −1.14689
\(800\) 0 0
\(801\) 1.40312 0.0495770
\(802\) 0 0
\(803\) −45.6125 −1.60963
\(804\) 0 0
\(805\) −2.70156 −0.0952176
\(806\) 0 0
\(807\) −28.2094 −0.993017
\(808\) 0 0
\(809\) −33.0156 −1.16077 −0.580384 0.814343i \(-0.697097\pi\)
−0.580384 + 0.814343i \(0.697097\pi\)
\(810\) 0 0
\(811\) −20.4922 −0.719578 −0.359789 0.933034i \(-0.617151\pi\)
−0.359789 + 0.933034i \(0.617151\pi\)
\(812\) 0 0
\(813\) −11.4031 −0.399925
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 34.8062 1.21772
\(818\) 0 0
\(819\) 0.701562 0.0245146
\(820\) 0 0
\(821\) −23.6125 −0.824082 −0.412041 0.911165i \(-0.635184\pi\)
−0.412041 + 0.911165i \(0.635184\pi\)
\(822\) 0 0
\(823\) 7.29844 0.254408 0.127204 0.991877i \(-0.459400\pi\)
0.127204 + 0.991877i \(0.459400\pi\)
\(824\) 0 0
\(825\) −9.19375 −0.320085
\(826\) 0 0
\(827\) 34.5969 1.20305 0.601526 0.798854i \(-0.294560\pi\)
0.601526 + 0.798854i \(0.294560\pi\)
\(828\) 0 0
\(829\) 42.5969 1.47945 0.739725 0.672909i \(-0.234955\pi\)
0.739725 + 0.672909i \(0.234955\pi\)
\(830\) 0 0
\(831\) 8.59688 0.298222
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −1.61250 −0.0558028
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −3.79063 −0.130867 −0.0654335 0.997857i \(-0.520843\pi\)
−0.0654335 + 0.997857i \(0.520843\pi\)
\(840\) 0 0
\(841\) 15.9109 0.548653
\(842\) 0 0
\(843\) 17.5078 0.603001
\(844\) 0 0
\(845\) 33.7906 1.16243
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −20.8062 −0.714069
\(850\) 0 0
\(851\) −10.7016 −0.366845
\(852\) 0 0
\(853\) −10.1047 −0.345978 −0.172989 0.984924i \(-0.555342\pi\)
−0.172989 + 0.984924i \(0.555342\pi\)
\(854\) 0 0
\(855\) 20.0000 0.683986
\(856\) 0 0
\(857\) −6.70156 −0.228921 −0.114461 0.993428i \(-0.536514\pi\)
−0.114461 + 0.993428i \(0.536514\pi\)
\(858\) 0 0
\(859\) 47.5078 1.62095 0.810473 0.585776i \(-0.199210\pi\)
0.810473 + 0.585776i \(0.199210\pi\)
\(860\) 0 0
\(861\) −6.70156 −0.228389
\(862\) 0 0
\(863\) 57.6125 1.96115 0.980576 0.196139i \(-0.0628404\pi\)
0.980576 + 0.196139i \(0.0628404\pi\)
\(864\) 0 0
\(865\) −17.8219 −0.605962
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −7.58125 −0.256881
\(872\) 0 0
\(873\) 12.7016 0.429883
\(874\) 0 0
\(875\) −7.29844 −0.246732
\(876\) 0 0
\(877\) −4.59688 −0.155225 −0.0776127 0.996984i \(-0.524730\pi\)
−0.0776127 + 0.996984i \(0.524730\pi\)
\(878\) 0 0
\(879\) −4.80625 −0.162111
\(880\) 0 0
\(881\) −32.2094 −1.08516 −0.542581 0.840004i \(-0.682553\pi\)
−0.542581 + 0.840004i \(0.682553\pi\)
\(882\) 0 0
\(883\) 37.4031 1.25872 0.629358 0.777116i \(-0.283318\pi\)
0.629358 + 0.777116i \(0.283318\pi\)
\(884\) 0 0
\(885\) −14.5969 −0.490669
\(886\) 0 0
\(887\) −10.2094 −0.342797 −0.171399 0.985202i \(-0.554829\pi\)
−0.171399 + 0.985202i \(0.554829\pi\)
\(888\) 0 0
\(889\) 0.701562 0.0235296
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 60.0000 2.00782
\(894\) 0 0
\(895\) −15.9266 −0.532366
\(896\) 0 0
\(897\) −0.701562 −0.0234245
\(898\) 0 0
\(899\) 13.4031 0.447019
\(900\) 0 0
\(901\) 13.6125 0.453498
\(902\) 0 0
\(903\) −4.70156 −0.156458
\(904\) 0 0
\(905\) −23.7906 −0.790827
\(906\) 0 0
\(907\) 11.5078 0.382111 0.191055 0.981579i \(-0.438809\pi\)
0.191055 + 0.981579i \(0.438809\pi\)
\(908\) 0 0
\(909\) 9.40312 0.311882
\(910\) 0 0
\(911\) −33.8953 −1.12300 −0.561501 0.827476i \(-0.689776\pi\)
−0.561501 + 0.827476i \(0.689776\pi\)
\(912\) 0 0
\(913\) −2.38750 −0.0790148
\(914\) 0 0
\(915\) −5.40312 −0.178622
\(916\) 0 0
\(917\) 5.40312 0.178427
\(918\) 0 0
\(919\) −24.4187 −0.805500 −0.402750 0.915310i \(-0.631946\pi\)
−0.402750 + 0.915310i \(0.631946\pi\)
\(920\) 0 0
\(921\) −32.9109 −1.08445
\(922\) 0 0
\(923\) 3.79063 0.124770
\(924\) 0 0
\(925\) 24.5969 0.808740
\(926\) 0 0
\(927\) −8.70156 −0.285797
\(928\) 0 0
\(929\) −1.08907 −0.0357311 −0.0178655 0.999840i \(-0.505687\pi\)
−0.0178655 + 0.999840i \(0.505687\pi\)
\(930\) 0 0
\(931\) −7.40312 −0.242628
\(932\) 0 0
\(933\) 0.596876 0.0195408
\(934\) 0 0
\(935\) −43.2250 −1.41361
\(936\) 0 0
\(937\) 5.89531 0.192592 0.0962958 0.995353i \(-0.469301\pi\)
0.0962958 + 0.995353i \(0.469301\pi\)
\(938\) 0 0
\(939\) −32.2094 −1.05111
\(940\) 0 0
\(941\) −32.3141 −1.05341 −0.526704 0.850049i \(-0.676572\pi\)
−0.526704 + 0.850049i \(0.676572\pi\)
\(942\) 0 0
\(943\) 6.70156 0.218233
\(944\) 0 0
\(945\) −2.70156 −0.0878818
\(946\) 0 0
\(947\) 28.9109 0.939479 0.469740 0.882805i \(-0.344348\pi\)
0.469740 + 0.882805i \(0.344348\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 14.7016 0.476731
\(952\) 0 0
\(953\) −41.2250 −1.33541 −0.667704 0.744427i \(-0.732723\pi\)
−0.667704 + 0.744427i \(0.732723\pi\)
\(954\) 0 0
\(955\) −7.58125 −0.245324
\(956\) 0 0
\(957\) −26.8062 −0.866523
\(958\) 0 0
\(959\) 13.2984 0.429429
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.80625 0.219328
\(964\) 0 0
\(965\) 10.5234 0.338761
\(966\) 0 0
\(967\) −10.8062 −0.347506 −0.173753 0.984789i \(-0.555589\pi\)
−0.173753 + 0.984789i \(0.555589\pi\)
\(968\) 0 0
\(969\) 29.6125 0.951290
\(970\) 0 0
\(971\) 39.8219 1.27794 0.638972 0.769230i \(-0.279360\pi\)
0.638972 + 0.769230i \(0.279360\pi\)
\(972\) 0 0
\(973\) 11.2984 0.362211
\(974\) 0 0
\(975\) 1.61250 0.0516412
\(976\) 0 0
\(977\) −53.7172 −1.71856 −0.859282 0.511501i \(-0.829090\pi\)
−0.859282 + 0.511501i \(0.829090\pi\)
\(978\) 0 0
\(979\) 5.61250 0.179376
\(980\) 0 0
\(981\) −1.29844 −0.0414559
\(982\) 0 0
\(983\) −3.79063 −0.120902 −0.0604511 0.998171i \(-0.519254\pi\)
−0.0604511 + 0.998171i \(0.519254\pi\)
\(984\) 0 0
\(985\) 25.6859 0.818422
\(986\) 0 0
\(987\) −8.10469 −0.257975
\(988\) 0 0
\(989\) 4.70156 0.149501
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −23.5078 −0.745248
\(996\) 0 0
\(997\) −10.5969 −0.335606 −0.167803 0.985821i \(-0.553667\pi\)
−0.167803 + 0.985821i \(0.553667\pi\)
\(998\) 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 7728.2.a.bc.1.1 2
4.3 odd 2 966.2.a.n.1.1 2
12.11 even 2 2898.2.a.bb.1.2 2
28.27 even 2 6762.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.1 2 4.3 odd 2
2898.2.a.bb.1.2 2 12.11 even 2
6762.2.a.bo.1.2 2 28.27 even 2
7728.2.a.bc.1.1 2 1.1 even 1 trivial