Properties

Label 2898.2.a.bb.1.2
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
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 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.70156 q^{10} +4.00000 q^{11} -0.701562 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +7.40312 q^{19} +2.70156 q^{20} +4.00000 q^{22} -1.00000 q^{23} +2.29844 q^{25} -0.701562 q^{26} +1.00000 q^{28} -6.70156 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} +2.70156 q^{35} +10.7016 q^{37} +7.40312 q^{38} +2.70156 q^{40} +6.70156 q^{41} +4.70156 q^{43} +4.00000 q^{44} -1.00000 q^{46} -8.10469 q^{47} +1.00000 q^{49} +2.29844 q^{50} -0.701562 q^{52} -3.40312 q^{53} +10.8062 q^{55} +1.00000 q^{56} -6.70156 q^{58} -5.40312 q^{59} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.89531 q^{65} -10.8062 q^{67} -4.00000 q^{68} +2.70156 q^{70} -5.40312 q^{71} -11.4031 q^{73} +10.7016 q^{74} +7.40312 q^{76} +4.00000 q^{77} +8.00000 q^{79} +2.70156 q^{80} +6.70156 q^{82} -0.596876 q^{83} -10.8062 q^{85} +4.70156 q^{86} +4.00000 q^{88} -1.40312 q^{89} -0.701562 q^{91} -1.00000 q^{92} -8.10469 q^{94} +20.0000 q^{95} +12.7016 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} + 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} + 2q^{7} + 2q^{8} - q^{10} + 8q^{11} + 5q^{13} + 2q^{14} + 2q^{16} - 8q^{17} + 2q^{19} - q^{20} + 8q^{22} - 2q^{23} + 11q^{25} + 5q^{26} + 2q^{28} - 7q^{29} - 4q^{31} + 2q^{32} - 8q^{34} - q^{35} + 15q^{37} + 2q^{38} - q^{40} + 7q^{41} + 3q^{43} + 8q^{44} - 2q^{46} + 3q^{47} + 2q^{49} + 11q^{50} + 5q^{52} + 6q^{53} - 4q^{55} + 2q^{56} - 7q^{58} + 2q^{59} - 4q^{61} - 4q^{62} + 2q^{64} - 23q^{65} + 4q^{67} - 8q^{68} - q^{70} + 2q^{71} - 10q^{73} + 15q^{74} + 2q^{76} + 8q^{77} + 16q^{79} - q^{80} + 7q^{82} - 14q^{83} + 4q^{85} + 3q^{86} + 8q^{88} + 10q^{89} + 5q^{91} - 2q^{92} + 3q^{94} + 40q^{95} + 19q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 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
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.70156 0.854309
\(11\) 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 \(0.531017\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 7.40312 1.69839 0.849197 0.528077i \(-0.177087\pi\)
0.849197 + 0.528077i \(0.177087\pi\)
\(20\) 2.70156 0.604088
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.29844 0.459688
\(26\) −0.701562 −0.137588
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.70156 −1.24445 −0.622224 0.782839i \(-0.713771\pi\)
−0.622224 + 0.782839i \(0.713771\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 2.70156 0.456647
\(36\) 0 0
\(37\) 10.7016 1.75933 0.879663 0.475598i \(-0.157768\pi\)
0.879663 + 0.475598i \(0.157768\pi\)
\(38\) 7.40312 1.20095
\(39\) 0 0
\(40\) 2.70156 0.427154
\(41\) 6.70156 1.04661 0.523304 0.852146i \(-0.324699\pi\)
0.523304 + 0.852146i \(0.324699\pi\)
\(42\) 0 0
\(43\) 4.70156 0.716982 0.358491 0.933533i \(-0.383291\pi\)
0.358491 + 0.933533i \(0.383291\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −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
\(50\) 2.29844 0.325048
\(51\) 0 0
\(52\) −0.701562 −0.0972892
\(53\) −3.40312 −0.467455 −0.233728 0.972302i \(-0.575092\pi\)
−0.233728 + 0.972302i \(0.575092\pi\)
\(54\) 0 0
\(55\) 10.8062 1.45711
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.70156 −0.879958
\(59\) −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 \(0.540868\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.89531 −0.235085
\(66\) 0 0
\(67\) −10.8062 −1.32019 −0.660097 0.751181i \(-0.729485\pi\)
−0.660097 + 0.751181i \(0.729485\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 2.70156 0.322898
\(71\) −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 \(0.732558\pi\)
\(74\) 10.7016 1.24403
\(75\) 0 0
\(76\) 7.40312 0.849197
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.70156 0.302044
\(81\) 0 0
\(82\) 6.70156 0.740064
\(83\) −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
\(86\) 4.70156 0.506982
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −1.40312 −0.148731 −0.0743654 0.997231i \(-0.523693\pi\)
−0.0743654 + 0.997231i \(0.523693\pi\)
\(90\) 0 0
\(91\) −0.701562 −0.0735437
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −8.10469 −0.835935
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 12.7016 1.28965 0.644824 0.764331i \(-0.276931\pi\)
0.644824 + 0.764331i \(0.276931\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 2.29844 0.229844
\(101\) −9.40312 −0.935646 −0.467823 0.883822i \(-0.654961\pi\)
−0.467823 + 0.883822i \(0.654961\pi\)
\(102\) 0 0
\(103\) 8.70156 0.857390 0.428695 0.903449i \(-0.358973\pi\)
0.428695 + 0.903449i \(0.358973\pi\)
\(104\) −0.701562 −0.0687938
\(105\) 0 0
\(106\) −3.40312 −0.330541
\(107\) 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 \(0.519807\pi\)
\(110\) 10.8062 1.03034
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 12.1047 1.13871 0.569357 0.822091i \(-0.307192\pi\)
0.569357 + 0.822091i \(0.307192\pi\)
\(114\) 0 0
\(115\) −2.70156 −0.251922
\(116\) −6.70156 −0.622224
\(117\) 0 0
\(118\) −5.40312 −0.497398
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −7.29844 −0.652792
\(126\) 0 0
\(127\) 0.701562 0.0622536 0.0311268 0.999515i \(-0.490090\pi\)
0.0311268 + 0.999515i \(0.490090\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.89531 −0.166230
\(131\) −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
\(134\) −10.8062 −0.933518
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 13.2984 1.13616 0.568081 0.822973i \(-0.307686\pi\)
0.568081 + 0.822973i \(0.307686\pi\)
\(138\) 0 0
\(139\) 11.2984 0.958321 0.479160 0.877727i \(-0.340941\pi\)
0.479160 + 0.877727i \(0.340941\pi\)
\(140\) 2.70156 0.228324
\(141\) 0 0
\(142\) −5.40312 −0.453420
\(143\) −2.80625 −0.234670
\(144\) 0 0
\(145\) −18.1047 −1.50351
\(146\) −11.4031 −0.943729
\(147\) 0 0
\(148\) 10.7016 0.879663
\(149\) −18.2094 −1.49177 −0.745885 0.666075i \(-0.767973\pi\)
−0.745885 + 0.666075i \(0.767973\pi\)
\(150\) 0 0
\(151\) −20.9109 −1.70171 −0.850854 0.525402i \(-0.823915\pi\)
−0.850854 + 0.525402i \(0.823915\pi\)
\(152\) 7.40312 0.600473
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) −5.40312 −0.433989
\(156\) 0 0
\(157\) 11.4031 0.910068 0.455034 0.890474i \(-0.349627\pi\)
0.455034 + 0.890474i \(0.349627\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 2.70156 0.213577
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 14.8062 1.15971 0.579857 0.814718i \(-0.303108\pi\)
0.579857 + 0.814718i \(0.303108\pi\)
\(164\) 6.70156 0.523304
\(165\) 0 0
\(166\) −0.596876 −0.0463265
\(167\) 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
\(170\) −10.8062 −0.828801
\(171\) 0 0
\(172\) 4.70156 0.358491
\(173\) −6.59688 −0.501551 −0.250776 0.968045i \(-0.580686\pi\)
−0.250776 + 0.968045i \(0.580686\pi\)
\(174\) 0 0
\(175\) 2.29844 0.173746
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −1.40312 −0.105169
\(179\) 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 \(0.393867\pi\)
\(182\) −0.701562 −0.0520032
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 28.9109 2.12557
\(186\) 0 0
\(187\) −16.0000 −1.17004
\(188\) −8.10469 −0.591095
\(189\) 0 0
\(190\) 20.0000 1.45095
\(191\) 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 \(0.544773\pi\)
\(194\) 12.7016 0.911919
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 9.50781 0.677403 0.338702 0.940894i \(-0.390012\pi\)
0.338702 + 0.940894i \(0.390012\pi\)
\(198\) 0 0
\(199\) −8.70156 −0.616837 −0.308419 0.951251i \(-0.599800\pi\)
−0.308419 + 0.951251i \(0.599800\pi\)
\(200\) 2.29844 0.162524
\(201\) 0 0
\(202\) −9.40312 −0.661602
\(203\) −6.70156 −0.470357
\(204\) 0 0
\(205\) 18.1047 1.26449
\(206\) 8.70156 0.606267
\(207\) 0 0
\(208\) −0.701562 −0.0486446
\(209\) 29.6125 2.04834
\(210\) 0 0
\(211\) −14.8062 −1.01930 −0.509652 0.860381i \(-0.670226\pi\)
−0.509652 + 0.860381i \(0.670226\pi\)
\(212\) −3.40312 −0.233728
\(213\) 0 0
\(214\) 6.80625 0.465266
\(215\) 12.7016 0.866239
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −1.29844 −0.0879413
\(219\) 0 0
\(220\) 10.8062 0.728557
\(221\) 2.80625 0.188769
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 12.1047 0.805192
\(227\) 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 \(0.860447\pi\)
\(230\) −2.70156 −0.178136
\(231\) 0 0
\(232\) −6.70156 −0.439979
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −21.8953 −1.42829
\(236\) −5.40312 −0.351713
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 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 \(0.118799\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 2.70156 0.172596
\(246\) 0 0
\(247\) −5.19375 −0.330470
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −7.29844 −0.461594
\(251\) −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
\(254\) 0.701562 0.0440199
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 10.7016 0.664963
\(260\) −1.89531 −0.117542
\(261\) 0 0
\(262\) −5.40312 −0.333806
\(263\) −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
\(266\) 7.40312 0.453915
\(267\) 0 0
\(268\) −10.8062 −0.660097
\(269\) −28.2094 −1.71996 −0.859978 0.510331i \(-0.829523\pi\)
−0.859978 + 0.510331i \(0.829523\pi\)
\(270\) 0 0
\(271\) −11.4031 −0.692690 −0.346345 0.938107i \(-0.612577\pi\)
−0.346345 + 0.938107i \(0.612577\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 13.2984 0.803388
\(275\) 9.19375 0.554404
\(276\) 0 0
\(277\) −8.59688 −0.516536 −0.258268 0.966073i \(-0.583152\pi\)
−0.258268 + 0.966073i \(0.583152\pi\)
\(278\) 11.2984 0.677635
\(279\) 0 0
\(280\) 2.70156 0.161449
\(281\) 17.5078 1.04443 0.522214 0.852814i \(-0.325106\pi\)
0.522214 + 0.852814i \(0.325106\pi\)
\(282\) 0 0
\(283\) −20.8062 −1.23680 −0.618402 0.785862i \(-0.712219\pi\)
−0.618402 + 0.785862i \(0.712219\pi\)
\(284\) −5.40312 −0.320616
\(285\) 0 0
\(286\) −2.80625 −0.165937
\(287\) 6.70156 0.395581
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −18.1047 −1.06314
\(291\) 0 0
\(292\) −11.4031 −0.667317
\(293\) −4.80625 −0.280784 −0.140392 0.990096i \(-0.544836\pi\)
−0.140392 + 0.990096i \(0.544836\pi\)
\(294\) 0 0
\(295\) −14.5969 −0.849863
\(296\) 10.7016 0.622016
\(297\) 0 0
\(298\) −18.2094 −1.05484
\(299\) 0.701562 0.0405724
\(300\) 0 0
\(301\) 4.70156 0.270994
\(302\) −20.9109 −1.20329
\(303\) 0 0
\(304\) 7.40312 0.424598
\(305\) −5.40312 −0.309382
\(306\) 0 0
\(307\) −32.9109 −1.87833 −0.939163 0.343471i \(-0.888397\pi\)
−0.939163 + 0.343471i \(0.888397\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −5.40312 −0.306877
\(311\) −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 \(0.135858\pi\)
\(314\) 11.4031 0.643516
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.7016 0.825722 0.412861 0.910794i \(-0.364530\pi\)
0.412861 + 0.910794i \(0.364530\pi\)
\(318\) 0 0
\(319\) −26.8062 −1.50086
\(320\) 2.70156 0.151022
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −29.6125 −1.64768
\(324\) 0 0
\(325\) −1.61250 −0.0894452
\(326\) 14.8062 0.820042
\(327\) 0 0
\(328\) 6.70156 0.370032
\(329\) −8.10469 −0.446826
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −0.596876 −0.0327578
\(333\) 0 0
\(334\) 0.596876 0.0326596
\(335\) −29.1938 −1.59503
\(336\) 0 0
\(337\) 34.2094 1.86350 0.931752 0.363096i \(-0.118280\pi\)
0.931752 + 0.363096i \(0.118280\pi\)
\(338\) −12.5078 −0.680335
\(339\) 0 0
\(340\) −10.8062 −0.586051
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.70156 0.253491
\(345\) 0 0
\(346\) −6.59688 −0.354650
\(347\) 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 \(0.262024\pi\)
\(350\) 2.29844 0.122857
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −10.7016 −0.569587 −0.284793 0.958589i \(-0.591925\pi\)
−0.284793 + 0.958589i \(0.591925\pi\)
\(354\) 0 0
\(355\) −14.5969 −0.774722
\(356\) −1.40312 −0.0743654
\(357\) 0 0
\(358\) 5.89531 0.311577
\(359\) 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
\(362\) 8.80625 0.462846
\(363\) 0 0
\(364\) −0.701562 −0.0367718
\(365\) −30.8062 −1.61247
\(366\) 0 0
\(367\) 2.10469 0.109864 0.0549319 0.998490i \(-0.482506\pi\)
0.0549319 + 0.998490i \(0.482506\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 28.9109 1.50301
\(371\) −3.40312 −0.176681
\(372\) 0 0
\(373\) 24.8062 1.28442 0.642209 0.766529i \(-0.278018\pi\)
0.642209 + 0.766529i \(0.278018\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) −8.10469 −0.417967
\(377\) 4.70156 0.242143
\(378\) 0 0
\(379\) 7.29844 0.374896 0.187448 0.982275i \(-0.439978\pi\)
0.187448 + 0.982275i \(0.439978\pi\)
\(380\) 20.0000 1.02598
\(381\) 0 0
\(382\) 2.80625 0.143580
\(383\) 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
\(386\) −3.89531 −0.198266
\(387\) 0 0
\(388\) 12.7016 0.644824
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 9.50781 0.478997
\(395\) 21.6125 1.08744
\(396\) 0 0
\(397\) 21.4031 1.07419 0.537096 0.843521i \(-0.319521\pi\)
0.537096 + 0.843521i \(0.319521\pi\)
\(398\) −8.70156 −0.436170
\(399\) 0 0
\(400\) 2.29844 0.114922
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 1.40312 0.0698946
\(404\) −9.40312 −0.467823
\(405\) 0 0
\(406\) −6.70156 −0.332593
\(407\) 42.8062 2.12183
\(408\) 0 0
\(409\) −7.40312 −0.366061 −0.183030 0.983107i \(-0.558591\pi\)
−0.183030 + 0.983107i \(0.558591\pi\)
\(410\) 18.1047 0.894127
\(411\) 0 0
\(412\) 8.70156 0.428695
\(413\) −5.40312 −0.265870
\(414\) 0 0
\(415\) −1.61250 −0.0791544
\(416\) −0.701562 −0.0343969
\(417\) 0 0
\(418\) 29.6125 1.44839
\(419\) −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 \(0.572757\pi\)
\(422\) −14.8062 −0.720757
\(423\) 0 0
\(424\) −3.40312 −0.165270
\(425\) −9.19375 −0.445962
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 6.80625 0.328992
\(429\) 0 0
\(430\) 12.7016 0.612524
\(431\) −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 \(0.717883\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −1.29844 −0.0621839
\(437\) −7.40312 −0.354139
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 10.8062 0.515168
\(441\) 0 0
\(442\) 2.80625 0.133480
\(443\) 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
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −38.4187 −1.81309 −0.906546 0.422106i \(-0.861291\pi\)
−0.906546 + 0.422106i \(0.861291\pi\)
\(450\) 0 0
\(451\) 26.8062 1.26226
\(452\) 12.1047 0.569357
\(453\) 0 0
\(454\) 3.89531 0.182816
\(455\) −1.89531 −0.0888537
\(456\) 0 0
\(457\) −10.2094 −0.477574 −0.238787 0.971072i \(-0.576750\pi\)
−0.238787 + 0.971072i \(0.576750\pi\)
\(458\) −27.4031 −1.28046
\(459\) 0 0
\(460\) −2.70156 −0.125961
\(461\) −36.2094 −1.68644 −0.843219 0.537570i \(-0.819342\pi\)
−0.843219 + 0.537570i \(0.819342\pi\)
\(462\) 0 0
\(463\) −28.7016 −1.33387 −0.666937 0.745114i \(-0.732395\pi\)
−0.666937 + 0.745114i \(0.732395\pi\)
\(464\) −6.70156 −0.311112
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −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
\(470\) −21.8953 −1.00996
\(471\) 0 0
\(472\) −5.40312 −0.248699
\(473\) 18.8062 0.864712
\(474\) 0 0
\(475\) 17.0156 0.780730
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 14.8062 0.677222
\(479\) −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
\(482\) 28.9109 1.31686
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 34.3141 1.55812
\(486\) 0 0
\(487\) −12.7016 −0.575563 −0.287781 0.957696i \(-0.592918\pi\)
−0.287781 + 0.957696i \(0.592918\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 2.70156 0.122044
\(491\) −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
\(494\) −5.19375 −0.233678
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −5.40312 −0.242363
\(498\) 0 0
\(499\) −10.5969 −0.474381 −0.237191 0.971463i \(-0.576227\pi\)
−0.237191 + 0.971463i \(0.576227\pi\)
\(500\) −7.29844 −0.326396
\(501\) 0 0
\(502\) −13.2984 −0.593538
\(503\) −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
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 0.701562 0.0311268
\(509\) 13.6125 0.603363 0.301682 0.953409i \(-0.402452\pi\)
0.301682 + 0.953409i \(0.402452\pi\)
\(510\) 0 0
\(511\) −11.4031 −0.504445
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 23.5078 1.03588
\(516\) 0 0
\(517\) −32.4187 −1.42577
\(518\) 10.7016 0.470200
\(519\) 0 0
\(520\) −1.89531 −0.0831150
\(521\) 33.4031 1.46342 0.731709 0.681617i \(-0.238723\pi\)
0.731709 + 0.681617i \(0.238723\pi\)
\(522\) 0 0
\(523\) −4.80625 −0.210163 −0.105081 0.994464i \(-0.533510\pi\)
−0.105081 + 0.994464i \(0.533510\pi\)
\(524\) −5.40312 −0.236037
\(525\) 0 0
\(526\) −3.29844 −0.143819
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −9.19375 −0.399351
\(531\) 0 0
\(532\) 7.40312 0.320966
\(533\) −4.70156 −0.203647
\(534\) 0 0
\(535\) 18.3875 0.794961
\(536\) −10.8062 −0.466759
\(537\) 0 0
\(538\) −28.2094 −1.21619
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 20.5969 0.885529 0.442764 0.896638i \(-0.353998\pi\)
0.442764 + 0.896638i \(0.353998\pi\)
\(542\) −11.4031 −0.489806
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −3.50781 −0.150258
\(546\) 0 0
\(547\) −35.0156 −1.49716 −0.748580 0.663045i \(-0.769264\pi\)
−0.748580 + 0.663045i \(0.769264\pi\)
\(548\) 13.2984 0.568081
\(549\) 0 0
\(550\) 9.19375 0.392023
\(551\) −49.6125 −2.11356
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −8.59688 −0.365246
\(555\) 0 0
\(556\) 11.2984 0.479160
\(557\) 24.5969 1.04220 0.521102 0.853495i \(-0.325521\pi\)
0.521102 + 0.853495i \(0.325521\pi\)
\(558\) 0 0
\(559\) −3.29844 −0.139509
\(560\) 2.70156 0.114162
\(561\) 0 0
\(562\) 17.5078 0.738522
\(563\) −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
\(566\) −20.8062 −0.874552
\(567\) 0 0
\(568\) −5.40312 −0.226710
\(569\) −25.2984 −1.06057 −0.530283 0.847821i \(-0.677914\pi\)
−0.530283 + 0.847821i \(0.677914\pi\)
\(570\) 0 0
\(571\) 10.8062 0.452227 0.226114 0.974101i \(-0.427398\pi\)
0.226114 + 0.974101i \(0.427398\pi\)
\(572\) −2.80625 −0.117335
\(573\) 0 0
\(574\) 6.70156 0.279718
\(575\) −2.29844 −0.0958515
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −18.1047 −0.751756
\(581\) −0.596876 −0.0247626
\(582\) 0 0
\(583\) −13.6125 −0.563772
\(584\) −11.4031 −0.471865
\(585\) 0 0
\(586\) −4.80625 −0.198544
\(587\) 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
\(590\) −14.5969 −0.600944
\(591\) 0 0
\(592\) 10.7016 0.439831
\(593\) −37.2984 −1.53166 −0.765832 0.643041i \(-0.777672\pi\)
−0.765832 + 0.643041i \(0.777672\pi\)
\(594\) 0 0
\(595\) −10.8062 −0.443013
\(596\) −18.2094 −0.745885
\(597\) 0 0
\(598\) 0.701562 0.0286890
\(599\) 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 \(0.503875\pi\)
\(602\) 4.70156 0.191621
\(603\) 0 0
\(604\) −20.9109 −0.850854
\(605\) 13.5078 0.549171
\(606\) 0 0
\(607\) 11.6125 0.471337 0.235668 0.971834i \(-0.424272\pi\)
0.235668 + 0.971834i \(0.424272\pi\)
\(608\) 7.40312 0.300236
\(609\) 0 0
\(610\) −5.40312 −0.218766
\(611\) 5.68594 0.230029
\(612\) 0 0
\(613\) −2.70156 −0.109115 −0.0545575 0.998511i \(-0.517375\pi\)
−0.0545575 + 0.998511i \(0.517375\pi\)
\(614\) −32.9109 −1.32818
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 44.8062 1.80383 0.901916 0.431912i \(-0.142161\pi\)
0.901916 + 0.431912i \(0.142161\pi\)
\(618\) 0 0
\(619\) −32.8062 −1.31859 −0.659297 0.751882i \(-0.729146\pi\)
−0.659297 + 0.751882i \(0.729146\pi\)
\(620\) −5.40312 −0.216995
\(621\) 0 0
\(622\) −0.596876 −0.0239325
\(623\) −1.40312 −0.0562150
\(624\) 0 0
\(625\) −31.2094 −1.24837
\(626\) 32.2094 1.28735
\(627\) 0 0
\(628\) 11.4031 0.455034
\(629\) −42.8062 −1.70680
\(630\) 0 0
\(631\) 21.6125 0.860380 0.430190 0.902738i \(-0.358447\pi\)
0.430190 + 0.902738i \(0.358447\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 14.7016 0.583874
\(635\) 1.89531 0.0752132
\(636\) 0 0
\(637\) −0.701562 −0.0277969
\(638\) −26.8062 −1.06127
\(639\) 0 0
\(640\) 2.70156 0.106789
\(641\) −49.7172 −1.96371 −0.981855 0.189631i \(-0.939271\pi\)
−0.981855 + 0.189631i \(0.939271\pi\)
\(642\) 0 0
\(643\) −24.8062 −0.978263 −0.489131 0.872210i \(-0.662686\pi\)
−0.489131 + 0.872210i \(0.662686\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −29.6125 −1.16509
\(647\) 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
\(650\) −1.61250 −0.0632473
\(651\) 0 0
\(652\) 14.8062 0.579857
\(653\) 12.1047 0.473693 0.236846 0.971547i \(-0.423886\pi\)
0.236846 + 0.971547i \(0.423886\pi\)
\(654\) 0 0
\(655\) −14.5969 −0.570347
\(656\) 6.70156 0.261652
\(657\) 0 0
\(658\) −8.10469 −0.315954
\(659\) −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 \(0.733435\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −0.596876 −0.0231633
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) 6.70156 0.259486
\(668\) 0.596876 0.0230938
\(669\) 0 0
\(670\) −29.1938 −1.12785
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 1.29844 0.0500511 0.0250256 0.999687i \(-0.492033\pi\)
0.0250256 + 0.999687i \(0.492033\pi\)
\(674\) 34.2094 1.31770
\(675\) 0 0
\(676\) −12.5078 −0.481070
\(677\) −26.4187 −1.01535 −0.507677 0.861547i \(-0.669496\pi\)
−0.507677 + 0.861547i \(0.669496\pi\)
\(678\) 0 0
\(679\) 12.7016 0.487441
\(680\) −10.8062 −0.414401
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 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
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.70156 0.179245
\(689\) 2.38750 0.0909566
\(690\) 0 0
\(691\) 31.5078 1.19861 0.599307 0.800519i \(-0.295443\pi\)
0.599307 + 0.800519i \(0.295443\pi\)
\(692\) −6.59688 −0.250776
\(693\) 0 0
\(694\) 16.7016 0.633983
\(695\) 30.5234 1.15782
\(696\) 0 0
\(697\) −26.8062 −1.01536
\(698\) 25.4031 0.961522
\(699\) 0 0
\(700\) 2.29844 0.0868728
\(701\) 17.0156 0.642671 0.321336 0.946965i \(-0.395868\pi\)
0.321336 + 0.946965i \(0.395868\pi\)
\(702\) 0 0
\(703\) 79.2250 2.98803
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −10.7016 −0.402759
\(707\) −9.40312 −0.353641
\(708\) 0 0
\(709\) 48.8062 1.83296 0.916479 0.400084i \(-0.131019\pi\)
0.916479 + 0.400084i \(0.131019\pi\)
\(710\) −14.5969 −0.547811
\(711\) 0 0
\(712\) −1.40312 −0.0525843
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −7.58125 −0.283523
\(716\) 5.89531 0.220318
\(717\) 0 0
\(718\) 22.1047 0.824940
\(719\) −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
\(722\) 35.8062 1.33257
\(723\) 0 0
\(724\) 8.80625 0.327282
\(725\) −15.4031 −0.572058
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) −0.701562 −0.0260016
\(729\) 0 0
\(730\) −30.8062 −1.14019
\(731\) −18.8062 −0.695574
\(732\) 0 0
\(733\) 13.7906 0.509368 0.254684 0.967024i \(-0.418028\pi\)
0.254684 + 0.967024i \(0.418028\pi\)
\(734\) 2.10469 0.0776854
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −43.2250 −1.59221
\(738\) 0 0
\(739\) 35.0156 1.28807 0.644035 0.764996i \(-0.277259\pi\)
0.644035 + 0.764996i \(0.277259\pi\)
\(740\) 28.9109 1.06279
\(741\) 0 0
\(742\) −3.40312 −0.124933
\(743\) −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
\(746\) 24.8062 0.908221
\(747\) 0 0
\(748\) −16.0000 −0.585018
\(749\) 6.80625 0.248695
\(750\) 0 0
\(751\) −21.6125 −0.788651 −0.394326 0.918971i \(-0.629022\pi\)
−0.394326 + 0.918971i \(0.629022\pi\)
\(752\) −8.10469 −0.295548
\(753\) 0 0
\(754\) 4.70156 0.171221
\(755\) −56.4922 −2.05596
\(756\) 0 0
\(757\) 50.4187 1.83250 0.916250 0.400606i \(-0.131201\pi\)
0.916250 + 0.400606i \(0.131201\pi\)
\(758\) 7.29844 0.265091
\(759\) 0 0
\(760\) 20.0000 0.725476
\(761\) −15.6125 −0.565953 −0.282976 0.959127i \(-0.591322\pi\)
−0.282976 + 0.959127i \(0.591322\pi\)
\(762\) 0 0
\(763\) −1.29844 −0.0470066
\(764\) 2.80625 0.101527
\(765\) 0 0
\(766\) 21.6125 0.780891
\(767\) 3.79063 0.136872
\(768\) 0 0
\(769\) −28.9109 −1.04255 −0.521277 0.853387i \(-0.674544\pi\)
−0.521277 + 0.853387i \(0.674544\pi\)
\(770\) 10.8062 0.389430
\(771\) 0 0
\(772\) −3.89531 −0.140195
\(773\) −12.1047 −0.435375 −0.217688 0.976018i \(-0.569851\pi\)
−0.217688 + 0.976018i \(0.569851\pi\)
\(774\) 0 0
\(775\) −4.59688 −0.165125
\(776\) 12.7016 0.455960
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 49.6125 1.77755
\(780\) 0 0
\(781\) −21.6125 −0.773356
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 30.8062 1.09952
\(786\) 0 0
\(787\) −31.4031 −1.11940 −0.559700 0.828695i \(-0.689084\pi\)
−0.559700 + 0.828695i \(0.689084\pi\)
\(788\) 9.50781 0.338702
\(789\) 0 0
\(790\) 21.6125 0.768938
\(791\) 12.1047 0.430393
\(792\) 0 0
\(793\) 1.40312 0.0498264
\(794\) 21.4031 0.759568
\(795\) 0 0
\(796\) −8.70156 −0.308419
\(797\) −9.29844 −0.329368 −0.164684 0.986346i \(-0.552660\pi\)
−0.164684 + 0.986346i \(0.552660\pi\)
\(798\) 0 0
\(799\) 32.4187 1.14689
\(800\) 2.29844 0.0812621
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −45.6125 −1.60963
\(804\) 0 0
\(805\) −2.70156 −0.0952176
\(806\) 1.40312 0.0494229
\(807\) 0 0
\(808\) −9.40312 −0.330801
\(809\) 33.0156 1.16077 0.580384 0.814343i \(-0.302903\pi\)
0.580384 + 0.814343i \(0.302903\pi\)
\(810\) 0 0
\(811\) 20.4922 0.719578 0.359789 0.933034i \(-0.382849\pi\)
0.359789 + 0.933034i \(0.382849\pi\)
\(812\) −6.70156 −0.235179
\(813\) 0 0
\(814\) 42.8062 1.50036
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 34.8062 1.21772
\(818\) −7.40312 −0.258844
\(819\) 0 0
\(820\) 18.1047 0.632243
\(821\) 23.6125 0.824082 0.412041 0.911165i \(-0.364816\pi\)
0.412041 + 0.911165i \(0.364816\pi\)
\(822\) 0 0
\(823\) −7.29844 −0.254408 −0.127204 0.991877i \(-0.540600\pi\)
−0.127204 + 0.991877i \(0.540600\pi\)
\(824\) 8.70156 0.303133
\(825\) 0 0
\(826\) −5.40312 −0.187999
\(827\) 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 \(0.234955\pi\)
\(830\) −1.61250 −0.0559706
\(831\) 0 0
\(832\) −0.701562 −0.0243223
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) 1.61250 0.0558028
\(836\) 29.6125 1.02417
\(837\) 0 0
\(838\) −38.2094 −1.31992
\(839\) −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
\(842\) −9.29844 −0.320445
\(843\) 0 0
\(844\) −14.8062 −0.509652
\(845\) −33.7906 −1.16243
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −3.40312 −0.116864
\(849\) 0 0
\(850\) −9.19375 −0.315343
\(851\) −10.7016 −0.366845
\(852\) 0 0
\(853\) −10.1047 −0.345978 −0.172989 0.984924i \(-0.555342\pi\)
−0.172989 + 0.984924i \(0.555342\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 6.80625 0.232633
\(857\) 6.70156 0.228921 0.114461 0.993428i \(-0.463486\pi\)
0.114461 + 0.993428i \(0.463486\pi\)
\(858\) 0 0
\(859\) −47.5078 −1.62095 −0.810473 0.585776i \(-0.800790\pi\)
−0.810473 + 0.585776i \(0.800790\pi\)
\(860\) 12.7016 0.433120
\(861\) 0 0
\(862\) −40.9109 −1.39343
\(863\) 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
\(866\) −26.3141 −0.894188
\(867\) 0 0
\(868\) −2.00000 −0.0678844
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 7.58125 0.256881
\(872\) −1.29844 −0.0439707
\(873\) 0 0
\(874\) −7.40312 −0.250414
\(875\) −7.29844 −0.246732
\(876\) 0 0
\(877\) −4.59688 −0.155225 −0.0776127 0.996984i \(-0.524730\pi\)
−0.0776127 + 0.996984i \(0.524730\pi\)
\(878\) −22.0000 −0.742464
\(879\) 0 0
\(880\) 10.8062 0.364279
\(881\) 32.2094 1.08516 0.542581 0.840004i \(-0.317447\pi\)
0.542581 + 0.840004i \(0.317447\pi\)
\(882\) 0 0
\(883\) −37.4031 −1.25872 −0.629358 0.777116i \(-0.716682\pi\)
−0.629358 + 0.777116i \(0.716682\pi\)
\(884\) 2.80625 0.0943844
\(885\) 0 0
\(886\) 7.29844 0.245196
\(887\) −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
\(890\) −3.79063 −0.127062
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) 15.9266 0.532366
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −38.4187 −1.28205
\(899\) 13.4031 0.447019
\(900\) 0 0
\(901\) 13.6125 0.453498
\(902\) 26.8062 0.892550
\(903\) 0 0
\(904\) 12.1047 0.402596
\(905\) 23.7906 0.790827
\(906\) 0 0
\(907\) −11.5078 −0.382111 −0.191055 0.981579i \(-0.561191\pi\)
−0.191055 + 0.981579i \(0.561191\pi\)
\(908\) 3.89531 0.129271
\(909\) 0 0
\(910\) −1.89531 −0.0628290
\(911\) −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
\(914\) −10.2094 −0.337696
\(915\) 0 0
\(916\) −27.4031 −0.905425
\(917\) −5.40312 −0.178427
\(918\) 0 0
\(919\) 24.4187 0.805500 0.402750 0.915310i \(-0.368054\pi\)
0.402750 + 0.915310i \(0.368054\pi\)
\(920\) −2.70156 −0.0890679
\(921\) 0 0
\(922\) −36.2094 −1.19249
\(923\) 3.79063 0.124770
\(924\) 0 0
\(925\) 24.5969 0.808740
\(926\) −28.7016 −0.943192
\(927\) 0 0
\(928\) −6.70156 −0.219990
\(929\) 1.08907 0.0357311 0.0178655 0.999840i \(-0.494313\pi\)
0.0178655 + 0.999840i \(0.494313\pi\)
\(930\) 0 0
\(931\) 7.40312 0.242628
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) −2.70156 −0.0883978
\(935\) −43.2250 −1.41361
\(936\) 0 0
\(937\) 5.89531 0.192592 0.0962958 0.995353i \(-0.469301\pi\)
0.0962958 + 0.995353i \(0.469301\pi\)
\(938\) −10.8062 −0.352837
\(939\) 0 0
\(940\) −21.8953 −0.714146
\(941\) 32.3141 1.05341 0.526704 0.850049i \(-0.323428\pi\)
0.526704 + 0.850049i \(0.323428\pi\)
\(942\) 0 0
\(943\) −6.70156 −0.218233
\(944\) −5.40312 −0.175857
\(945\) 0 0
\(946\) 18.8062 0.611444
\(947\) 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
\(950\) 17.0156 0.552060
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) 41.2250 1.33541 0.667704 0.744427i \(-0.267277\pi\)
0.667704 + 0.744427i \(0.267277\pi\)
\(954\) 0 0
\(955\) 7.58125 0.245324
\(956\) 14.8062 0.478868
\(957\) 0 0
\(958\) −29.4031 −0.949972
\(959\) 13.2984 0.429429
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −7.50781 −0.242062
\(963\) 0 0
\(964\) 28.9109 0.931159
\(965\) −10.5234 −0.338761
\(966\) 0 0
\(967\) 10.8062 0.347506 0.173753 0.984789i \(-0.444411\pi\)
0.173753 + 0.984789i \(0.444411\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 34.3141 1.10176
\(971\) 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
\(974\) −12.7016 −0.406984
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 53.7172 1.71856 0.859282 0.511501i \(-0.170910\pi\)
0.859282 + 0.511501i \(0.170910\pi\)
\(978\) 0 0
\(979\) −5.61250 −0.179376
\(980\) 2.70156 0.0862982
\(981\) 0 0
\(982\) −41.6125 −1.32791
\(983\) −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
\(986\) 26.8062 0.853685
\(987\) 0 0
\(988\) −5.19375 −0.165235
\(989\) −4.70156 −0.149501
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) −5.40312 −0.171377
\(995\) −23.5078 −0.745248
\(996\) 0 0
\(997\) −10.5969 −0.335606 −0.167803 0.985821i \(-0.553667\pi\)
−0.167803 + 0.985821i \(0.553667\pi\)
\(998\) −10.5969 −0.335438
\(999\) 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 2898.2.a.bb.1.2 2
3.2 odd 2 966.2.a.n.1.1 2
12.11 even 2 7728.2.a.bc.1.1 2
21.20 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 3.2 odd 2
2898.2.a.bb.1.2 2 1.1 even 1 trivial
6762.2.a.bo.1.2 2 21.20 even 2
7728.2.a.bc.1.1 2 12.11 even 2