Properties

Label 2898.2.a.bc.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.70156 q^{10} -0.701562 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.40312 q^{19} -2.70156 q^{20} +1.00000 q^{23} +2.29844 q^{25} -0.701562 q^{26} -1.00000 q^{28} -6.70156 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.70156 q^{35} -2.70156 q^{37} +7.40312 q^{38} -2.70156 q^{40} -1.29844 q^{41} -0.701562 q^{43} +1.00000 q^{46} +2.70156 q^{47} +1.00000 q^{49} +2.29844 q^{50} -0.701562 q^{52} +3.40312 q^{53} -1.00000 q^{56} -6.70156 q^{58} +13.4031 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +1.89531 q^{65} +4.00000 q^{67} +2.70156 q^{70} +5.40312 q^{71} +7.40312 q^{73} -2.70156 q^{74} +7.40312 q^{76} -2.70156 q^{80} -1.29844 q^{82} +15.4031 q^{83} -0.701562 q^{86} +2.59688 q^{89} +0.701562 q^{91} +1.00000 q^{92} +2.70156 q^{94} -20.0000 q^{95} +18.1047 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + q^{5} - 2q^{7} + 2q^{8} + q^{10} + 5q^{13} - 2q^{14} + 2q^{16} + 2q^{19} + q^{20} + 2q^{23} + 11q^{25} + 5q^{26} - 2q^{28} - 7q^{29} + 12q^{31} + 2q^{32} - q^{35} + q^{37} + 2q^{38} + q^{40} - 9q^{41} + 5q^{43} + 2q^{46} - q^{47} + 2q^{49} + 11q^{50} + 5q^{52} - 6q^{53} - 2q^{56} - 7q^{58} + 14q^{59} + 20q^{61} + 12q^{62} + 2q^{64} + 23q^{65} + 8q^{67} - q^{70} - 2q^{71} + 2q^{73} + q^{74} + 2q^{76} + q^{80} - 9q^{82} + 18q^{83} + 5q^{86} + 18q^{89} - 5q^{91} + 2q^{92} - q^{94} - 40q^{95} + 17q^{97} + 2q^{98} + O(q^{100}) \)

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 \(-0.706462\pi\)
−0.604088 + 0.796918i \(0.706462\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.70156 −0.854309
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 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
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(22\) 0 0
\(23\) 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
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.70156 0.456647
\(36\) 0 0
\(37\) −2.70156 −0.444134 −0.222067 0.975031i \(-0.571280\pi\)
−0.222067 + 0.975031i \(0.571280\pi\)
\(38\) 7.40312 1.20095
\(39\) 0 0
\(40\) −2.70156 −0.427154
\(41\) −1.29844 −0.202782 −0.101391 0.994847i \(-0.532329\pi\)
−0.101391 + 0.994847i \(0.532329\pi\)
\(42\) 0 0
\(43\) −0.701562 −0.106987 −0.0534936 0.998568i \(-0.517036\pi\)
−0.0534936 + 0.998568i \(0.517036\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 2.70156 0.394063 0.197032 0.980397i \(-0.436870\pi\)
0.197032 + 0.980397i \(0.436870\pi\)
\(48\) 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 \(-0.424908\pi\)
0.233728 + 0.972302i \(0.424908\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.70156 −0.879958
\(59\) 13.4031 1.74494 0.872469 0.488669i \(-0.162518\pi\)
0.872469 + 0.488669i \(0.162518\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.89531 0.235085
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 2.70156 0.322898
\(71\) 5.40312 0.641233 0.320616 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(72\) 0 0
\(73\) 7.40312 0.866470 0.433235 0.901281i \(-0.357372\pi\)
0.433235 + 0.901281i \(0.357372\pi\)
\(74\) −2.70156 −0.314050
\(75\) 0 0
\(76\) 7.40312 0.849197
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.70156 −0.302044
\(81\) 0 0
\(82\) −1.29844 −0.143388
\(83\) 15.4031 1.69071 0.845356 0.534203i \(-0.179388\pi\)
0.845356 + 0.534203i \(0.179388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.701562 −0.0756514
\(87\) 0 0
\(88\) 0 0
\(89\) 2.59688 0.275268 0.137634 0.990483i \(-0.456050\pi\)
0.137634 + 0.990483i \(0.456050\pi\)
\(90\) 0 0
\(91\) 0.701562 0.0735437
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 2.70156 0.278645
\(95\) −20.0000 −2.05196
\(96\) 0 0
\(97\) 18.1047 1.83825 0.919126 0.393963i \(-0.128896\pi\)
0.919126 + 0.393963i \(0.128896\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 2.29844 0.229844
\(101\) −17.4031 −1.73168 −0.865838 0.500325i \(-0.833214\pi\)
−0.865838 + 0.500325i \(0.833214\pi\)
\(102\) 0 0
\(103\) 2.10469 0.207381 0.103690 0.994610i \(-0.466935\pi\)
0.103690 + 0.994610i \(0.466935\pi\)
\(104\) −0.701562 −0.0687938
\(105\) 0 0
\(106\) 3.40312 0.330541
\(107\) 10.8062 1.04468 0.522340 0.852737i \(-0.325059\pi\)
0.522340 + 0.852737i \(0.325059\pi\)
\(108\) 0 0
\(109\) 9.29844 0.890629 0.445314 0.895374i \(-0.353092\pi\)
0.445314 + 0.895374i \(0.353092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 10.7016 1.00672 0.503359 0.864077i \(-0.332097\pi\)
0.503359 + 0.864077i \(0.332097\pi\)
\(114\) 0 0
\(115\) −2.70156 −0.251922
\(116\) −6.70156 −0.622224
\(117\) 0 0
\(118\) 13.4031 1.23386
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 7.29844 0.652792
\(126\) 0 0
\(127\) −2.10469 −0.186761 −0.0933804 0.995631i \(-0.529767\pi\)
−0.0933804 + 0.995631i \(0.529767\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.89531 0.166230
\(131\) −2.59688 −0.226890 −0.113445 0.993544i \(-0.536189\pi\)
−0.113445 + 0.993544i \(0.536189\pi\)
\(132\) 0 0
\(133\) −7.40312 −0.641932
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −20.1047 −1.71766 −0.858830 0.512261i \(-0.828808\pi\)
−0.858830 + 0.512261i \(0.828808\pi\)
\(138\) 0 0
\(139\) −1.89531 −0.160758 −0.0803792 0.996764i \(-0.525613\pi\)
−0.0803792 + 0.996764i \(0.525613\pi\)
\(140\) 2.70156 0.228324
\(141\) 0 0
\(142\) 5.40312 0.453420
\(143\) 0 0
\(144\) 0 0
\(145\) 18.1047 1.50351
\(146\) 7.40312 0.612687
\(147\) 0 0
\(148\) −2.70156 −0.222067
\(149\) 4.59688 0.376591 0.188295 0.982112i \(-0.439704\pi\)
0.188295 + 0.982112i \(0.439704\pi\)
\(150\) 0 0
\(151\) −10.1047 −0.822308 −0.411154 0.911566i \(-0.634874\pi\)
−0.411154 + 0.911566i \(0.634874\pi\)
\(152\) 7.40312 0.600473
\(153\) 0 0
\(154\) 0 0
\(155\) −16.2094 −1.30197
\(156\) 0 0
\(157\) −0.596876 −0.0476359 −0.0238179 0.999716i \(-0.507582\pi\)
−0.0238179 + 0.999716i \(0.507582\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.70156 −0.213577
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −1.29844 −0.101391
\(165\) 0 0
\(166\) 15.4031 1.19551
\(167\) 8.59688 0.665246 0.332623 0.943060i \(-0.392066\pi\)
0.332623 + 0.943060i \(0.392066\pi\)
\(168\) 0 0
\(169\) −12.5078 −0.962139
\(170\) 0 0
\(171\) 0 0
\(172\) −0.701562 −0.0534936
\(173\) −9.40312 −0.714906 −0.357453 0.933931i \(-0.616355\pi\)
−0.357453 + 0.933931i \(0.616355\pi\)
\(174\) 0 0
\(175\) −2.29844 −0.173746
\(176\) 0 0
\(177\) 0 0
\(178\) 2.59688 0.194644
\(179\) 0.701562 0.0524372 0.0262186 0.999656i \(-0.491653\pi\)
0.0262186 + 0.999656i \(0.491653\pi\)
\(180\) 0 0
\(181\) −8.80625 −0.654563 −0.327282 0.944927i \(-0.606133\pi\)
−0.327282 + 0.944927i \(0.606133\pi\)
\(182\) 0.701562 0.0520032
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 7.29844 0.536592
\(186\) 0 0
\(187\) 0 0
\(188\) 2.70156 0.197032
\(189\) 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
\(193\) 12.1047 0.871314 0.435657 0.900113i \(-0.356516\pi\)
0.435657 + 0.900113i \(0.356516\pi\)
\(194\) 18.1047 1.29984
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 25.5078 1.81736 0.908678 0.417497i \(-0.137093\pi\)
0.908678 + 0.417497i \(0.137093\pi\)
\(198\) 0 0
\(199\) −18.1047 −1.28341 −0.641704 0.766953i \(-0.721772\pi\)
−0.641704 + 0.766953i \(0.721772\pi\)
\(200\) 2.29844 0.162524
\(201\) 0 0
\(202\) −17.4031 −1.22448
\(203\) 6.70156 0.470357
\(204\) 0 0
\(205\) 3.50781 0.244996
\(206\) 2.10469 0.146640
\(207\) 0 0
\(208\) −0.701562 −0.0486446
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 3.40312 0.233728
\(213\) 0 0
\(214\) 10.8062 0.738700
\(215\) 1.89531 0.129259
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 9.29844 0.629770
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.80625 0.321850 0.160925 0.986967i \(-0.448552\pi\)
0.160925 + 0.986967i \(0.448552\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.7016 0.711857
\(227\) −5.50781 −0.365566 −0.182783 0.983153i \(-0.558511\pi\)
−0.182783 + 0.983153i \(0.558511\pi\)
\(228\) 0 0
\(229\) −10.2094 −0.674654 −0.337327 0.941387i \(-0.609523\pi\)
−0.337327 + 0.941387i \(0.609523\pi\)
\(230\) −2.70156 −0.178136
\(231\) 0 0
\(232\) −6.70156 −0.439979
\(233\) −3.19375 −0.209230 −0.104615 0.994513i \(-0.533361\pi\)
−0.104615 + 0.994513i \(0.533361\pi\)
\(234\) 0 0
\(235\) −7.29844 −0.476098
\(236\) 13.4031 0.872469
\(237\) 0 0
\(238\) 0 0
\(239\) −6.80625 −0.440260 −0.220130 0.975471i \(-0.570648\pi\)
−0.220130 + 0.975471i \(0.570648\pi\)
\(240\) 0 0
\(241\) −14.1047 −0.908563 −0.454281 0.890858i \(-0.650104\pi\)
−0.454281 + 0.890858i \(0.650104\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −2.70156 −0.172596
\(246\) 0 0
\(247\) −5.19375 −0.330470
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 7.29844 0.461594
\(251\) 30.9109 1.95108 0.975540 0.219820i \(-0.0705470\pi\)
0.975540 + 0.219820i \(0.0705470\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.10469 −0.132060
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.6125 1.72242 0.861210 0.508249i \(-0.169707\pi\)
0.861210 + 0.508249i \(0.169707\pi\)
\(258\) 0 0
\(259\) 2.70156 0.167867
\(260\) 1.89531 0.117542
\(261\) 0 0
\(262\) −2.59688 −0.160436
\(263\) −15.5078 −0.956253 −0.478126 0.878291i \(-0.658684\pi\)
−0.478126 + 0.878291i \(0.658684\pi\)
\(264\) 0 0
\(265\) −9.19375 −0.564768
\(266\) −7.40312 −0.453915
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) −9.40312 −0.573319 −0.286659 0.958033i \(-0.592545\pi\)
−0.286659 + 0.958033i \(0.592545\pi\)
\(270\) 0 0
\(271\) 4.59688 0.279240 0.139620 0.990205i \(-0.455412\pi\)
0.139620 + 0.990205i \(0.455412\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −20.1047 −1.21457
\(275\) 0 0
\(276\) 0 0
\(277\) −0.596876 −0.0358628 −0.0179314 0.999839i \(-0.505708\pi\)
−0.0179314 + 0.999839i \(0.505708\pi\)
\(278\) −1.89531 −0.113673
\(279\) 0 0
\(280\) 2.70156 0.161449
\(281\) −10.7016 −0.638402 −0.319201 0.947687i \(-0.603414\pi\)
−0.319201 + 0.947687i \(0.603414\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 5.40312 0.320616
\(285\) 0 0
\(286\) 0 0
\(287\) 1.29844 0.0766444
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 18.1047 1.06314
\(291\) 0 0
\(292\) 7.40312 0.433235
\(293\) 28.8062 1.68288 0.841440 0.540351i \(-0.181709\pi\)
0.841440 + 0.540351i \(0.181709\pi\)
\(294\) 0 0
\(295\) −36.2094 −2.10819
\(296\) −2.70156 −0.157025
\(297\) 0 0
\(298\) 4.59688 0.266290
\(299\) −0.701562 −0.0405724
\(300\) 0 0
\(301\) 0.701562 0.0404374
\(302\) −10.1047 −0.581459
\(303\) 0 0
\(304\) 7.40312 0.424598
\(305\) −27.0156 −1.54691
\(306\) 0 0
\(307\) −14.1047 −0.804997 −0.402498 0.915421i \(-0.631858\pi\)
−0.402498 + 0.915421i \(0.631858\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −16.2094 −0.920631
\(311\) −22.2094 −1.25938 −0.629689 0.776847i \(-0.716818\pi\)
−0.629689 + 0.776847i \(0.716818\pi\)
\(312\) 0 0
\(313\) −17.4031 −0.983683 −0.491841 0.870685i \(-0.663676\pi\)
−0.491841 + 0.870685i \(0.663676\pi\)
\(314\) −0.596876 −0.0336836
\(315\) 0 0
\(316\) 0 0
\(317\) −1.29844 −0.0729275 −0.0364638 0.999335i \(-0.511609\pi\)
−0.0364638 + 0.999335i \(0.511609\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.70156 −0.151022
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) −1.61250 −0.0894452
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −1.29844 −0.0716942
\(329\) −2.70156 −0.148942
\(330\) 0 0
\(331\) 33.6125 1.84751 0.923755 0.382984i \(-0.125104\pi\)
0.923755 + 0.382984i \(0.125104\pi\)
\(332\) 15.4031 0.845356
\(333\) 0 0
\(334\) 8.59688 0.470400
\(335\) −10.8062 −0.590408
\(336\) 0 0
\(337\) 4.59688 0.250408 0.125204 0.992131i \(-0.460041\pi\)
0.125204 + 0.992131i \(0.460041\pi\)
\(338\) −12.5078 −0.680335
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −0.701562 −0.0378257
\(345\) 0 0
\(346\) −9.40312 −0.505515
\(347\) 11.5078 0.617772 0.308886 0.951099i \(-0.400044\pi\)
0.308886 + 0.951099i \(0.400044\pi\)
\(348\) 0 0
\(349\) −4.20937 −0.225323 −0.112661 0.993633i \(-0.535937\pi\)
−0.112661 + 0.993633i \(0.535937\pi\)
\(350\) −2.29844 −0.122857
\(351\) 0 0
\(352\) 0 0
\(353\) −24.3141 −1.29411 −0.647053 0.762445i \(-0.723999\pi\)
−0.647053 + 0.762445i \(0.723999\pi\)
\(354\) 0 0
\(355\) −14.5969 −0.774722
\(356\) 2.59688 0.137634
\(357\) 0 0
\(358\) 0.701562 0.0370787
\(359\) −3.29844 −0.174085 −0.0870424 0.996205i \(-0.527742\pi\)
−0.0870424 + 0.996205i \(0.527742\pi\)
\(360\) 0 0
\(361\) 35.8062 1.88454
\(362\) −8.80625 −0.462846
\(363\) 0 0
\(364\) 0.701562 0.0367718
\(365\) −20.0000 −1.04685
\(366\) 0 0
\(367\) −28.9109 −1.50914 −0.754569 0.656220i \(-0.772154\pi\)
−0.754569 + 0.656220i \(0.772154\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 7.29844 0.379428
\(371\) −3.40312 −0.176681
\(372\) 0 0
\(373\) 20.8062 1.07731 0.538653 0.842527i \(-0.318933\pi\)
0.538653 + 0.842527i \(0.318933\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.70156 0.139322
\(377\) 4.70156 0.242143
\(378\) 0 0
\(379\) −11.2984 −0.580362 −0.290181 0.956972i \(-0.593715\pi\)
−0.290181 + 0.956972i \(0.593715\pi\)
\(380\) −20.0000 −1.02598
\(381\) 0 0
\(382\) 2.80625 0.143580
\(383\) 18.8062 0.960954 0.480477 0.877007i \(-0.340463\pi\)
0.480477 + 0.877007i \(0.340463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.1047 0.616112
\(387\) 0 0
\(388\) 18.1047 0.919126
\(389\) −11.1938 −0.567546 −0.283773 0.958892i \(-0.591586\pi\)
−0.283773 + 0.958892i \(0.591586\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 25.5078 1.28506
\(395\) 0 0
\(396\) 0 0
\(397\) −32.2094 −1.61654 −0.808271 0.588811i \(-0.799596\pi\)
−0.808271 + 0.588811i \(0.799596\pi\)
\(398\) −18.1047 −0.907506
\(399\) 0 0
\(400\) 2.29844 0.114922
\(401\) −3.19375 −0.159488 −0.0797442 0.996815i \(-0.525410\pi\)
−0.0797442 + 0.996815i \(0.525410\pi\)
\(402\) 0 0
\(403\) −4.20937 −0.209684
\(404\) −17.4031 −0.865838
\(405\) 0 0
\(406\) 6.70156 0.332593
\(407\) 0 0
\(408\) 0 0
\(409\) 6.20937 0.307034 0.153517 0.988146i \(-0.450940\pi\)
0.153517 + 0.988146i \(0.450940\pi\)
\(410\) 3.50781 0.173238
\(411\) 0 0
\(412\) 2.10469 0.103690
\(413\) −13.4031 −0.659525
\(414\) 0 0
\(415\) −41.6125 −2.04268
\(416\) −0.701562 −0.0343969
\(417\) 0 0
\(418\) 0 0
\(419\) 15.4031 0.752492 0.376246 0.926520i \(-0.377215\pi\)
0.376246 + 0.926520i \(0.377215\pi\)
\(420\) 0 0
\(421\) 9.29844 0.453178 0.226589 0.973990i \(-0.427243\pi\)
0.226589 + 0.973990i \(0.427243\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 3.40312 0.165270
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 10.8062 0.522340
\(429\) 0 0
\(430\) 1.89531 0.0914001
\(431\) 14.1047 0.679399 0.339699 0.940534i \(-0.389675\pi\)
0.339699 + 0.940534i \(0.389675\pi\)
\(432\) 0 0
\(433\) −2.10469 −0.101145 −0.0505724 0.998720i \(-0.516105\pi\)
−0.0505724 + 0.998720i \(0.516105\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 9.29844 0.445314
\(437\) 7.40312 0.354139
\(438\) 0 0
\(439\) 28.8062 1.37485 0.687424 0.726257i \(-0.258742\pi\)
0.687424 + 0.726257i \(0.258742\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.10469 0.0999967 0.0499983 0.998749i \(-0.484078\pi\)
0.0499983 + 0.998749i \(0.484078\pi\)
\(444\) 0 0
\(445\) −7.01562 −0.332572
\(446\) 4.80625 0.227582
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 23.6125 1.11434 0.557171 0.830398i \(-0.311887\pi\)
0.557171 + 0.830398i \(0.311887\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.7016 0.503359
\(453\) 0 0
\(454\) −5.50781 −0.258494
\(455\) −1.89531 −0.0888537
\(456\) 0 0
\(457\) 22.2094 1.03891 0.519455 0.854498i \(-0.326135\pi\)
0.519455 + 0.854498i \(0.326135\pi\)
\(458\) −10.2094 −0.477053
\(459\) 0 0
\(460\) −2.70156 −0.125961
\(461\) −14.5969 −0.679844 −0.339922 0.940454i \(-0.610401\pi\)
−0.339922 + 0.940454i \(0.610401\pi\)
\(462\) 0 0
\(463\) 6.10469 0.283709 0.141854 0.989888i \(-0.454694\pi\)
0.141854 + 0.989888i \(0.454694\pi\)
\(464\) −6.70156 −0.311112
\(465\) 0 0
\(466\) −3.19375 −0.147948
\(467\) 14.7016 0.680307 0.340154 0.940370i \(-0.389521\pi\)
0.340154 + 0.940370i \(0.389521\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −7.29844 −0.336652
\(471\) 0 0
\(472\) 13.4031 0.616929
\(473\) 0 0
\(474\) 0 0
\(475\) 17.0156 0.780730
\(476\) 0 0
\(477\) 0 0
\(478\) −6.80625 −0.311311
\(479\) 5.40312 0.246875 0.123438 0.992352i \(-0.460608\pi\)
0.123438 + 0.992352i \(0.460608\pi\)
\(480\) 0 0
\(481\) 1.89531 0.0864189
\(482\) −14.1047 −0.642451
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −48.9109 −2.22093
\(486\) 0 0
\(487\) 6.10469 0.276630 0.138315 0.990388i \(-0.455831\pi\)
0.138315 + 0.990388i \(0.455831\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) −2.70156 −0.122044
\(491\) −33.6125 −1.51691 −0.758455 0.651725i \(-0.774046\pi\)
−0.758455 + 0.651725i \(0.774046\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −5.19375 −0.233678
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −5.40312 −0.242363
\(498\) 0 0
\(499\) −27.0156 −1.20939 −0.604693 0.796459i \(-0.706704\pi\)
−0.604693 + 0.796459i \(0.706704\pi\)
\(500\) 7.29844 0.326396
\(501\) 0 0
\(502\) 30.9109 1.37962
\(503\) −1.40312 −0.0625622 −0.0312811 0.999511i \(-0.509959\pi\)
−0.0312811 + 0.999511i \(0.509959\pi\)
\(504\) 0 0
\(505\) 47.0156 2.09217
\(506\) 0 0
\(507\) 0 0
\(508\) −2.10469 −0.0933804
\(509\) −10.8062 −0.478979 −0.239489 0.970899i \(-0.576980\pi\)
−0.239489 + 0.970899i \(0.576980\pi\)
\(510\) 0 0
\(511\) −7.40312 −0.327495
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 27.6125 1.21794
\(515\) −5.68594 −0.250552
\(516\) 0 0
\(517\) 0 0
\(518\) 2.70156 0.118700
\(519\) 0 0
\(520\) 1.89531 0.0831150
\(521\) −2.59688 −0.113771 −0.0568856 0.998381i \(-0.518117\pi\)
−0.0568856 + 0.998381i \(0.518117\pi\)
\(522\) 0 0
\(523\) 38.4187 1.67993 0.839967 0.542637i \(-0.182574\pi\)
0.839967 + 0.542637i \(0.182574\pi\)
\(524\) −2.59688 −0.113445
\(525\) 0 0
\(526\) −15.5078 −0.676173
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −9.19375 −0.399351
\(531\) 0 0
\(532\) −7.40312 −0.320966
\(533\) 0.910935 0.0394570
\(534\) 0 0
\(535\) −29.1938 −1.26216
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −9.40312 −0.405397
\(539\) 0 0
\(540\) 0 0
\(541\) 18.2094 0.782882 0.391441 0.920203i \(-0.371977\pi\)
0.391441 + 0.920203i \(0.371977\pi\)
\(542\) 4.59688 0.197453
\(543\) 0 0
\(544\) 0 0
\(545\) −25.1203 −1.07604
\(546\) 0 0
\(547\) 24.2094 1.03512 0.517559 0.855648i \(-0.326841\pi\)
0.517559 + 0.855648i \(0.326841\pi\)
\(548\) −20.1047 −0.858830
\(549\) 0 0
\(550\) 0 0
\(551\) −49.6125 −2.11356
\(552\) 0 0
\(553\) 0 0
\(554\) −0.596876 −0.0253588
\(555\) 0 0
\(556\) −1.89531 −0.0803792
\(557\) −8.59688 −0.364261 −0.182131 0.983274i \(-0.558299\pi\)
−0.182131 + 0.983274i \(0.558299\pi\)
\(558\) 0 0
\(559\) 0.492189 0.0208174
\(560\) 2.70156 0.114162
\(561\) 0 0
\(562\) −10.7016 −0.451418
\(563\) 25.7172 1.08385 0.541925 0.840427i \(-0.317696\pi\)
0.541925 + 0.840427i \(0.317696\pi\)
\(564\) 0 0
\(565\) −28.9109 −1.21629
\(566\) −10.0000 −0.420331
\(567\) 0 0
\(568\) 5.40312 0.226710
\(569\) 8.10469 0.339766 0.169883 0.985464i \(-0.445661\pi\)
0.169883 + 0.985464i \(0.445661\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.29844 0.0541958
\(575\) 2.29844 0.0958515
\(576\) 0 0
\(577\) 27.6125 1.14952 0.574762 0.818321i \(-0.305095\pi\)
0.574762 + 0.818321i \(0.305095\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 18.1047 0.751756
\(581\) −15.4031 −0.639029
\(582\) 0 0
\(583\) 0 0
\(584\) 7.40312 0.306343
\(585\) 0 0
\(586\) 28.8062 1.18998
\(587\) 26.8062 1.10641 0.553206 0.833044i \(-0.313404\pi\)
0.553206 + 0.833044i \(0.313404\pi\)
\(588\) 0 0
\(589\) 44.4187 1.83024
\(590\) −36.2094 −1.49072
\(591\) 0 0
\(592\) −2.70156 −0.111034
\(593\) −2.91093 −0.119538 −0.0597689 0.998212i \(-0.519036\pi\)
−0.0597689 + 0.998212i \(0.519036\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.59688 0.188295
\(597\) 0 0
\(598\) −0.701562 −0.0286890
\(599\) −21.6125 −0.883063 −0.441531 0.897246i \(-0.645565\pi\)
−0.441531 + 0.897246i \(0.645565\pi\)
\(600\) 0 0
\(601\) 12.5969 0.513837 0.256919 0.966433i \(-0.417293\pi\)
0.256919 + 0.966433i \(0.417293\pi\)
\(602\) 0.701562 0.0285935
\(603\) 0 0
\(604\) −10.1047 −0.411154
\(605\) 29.7172 1.20818
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 7.40312 0.300236
\(609\) 0 0
\(610\) −27.0156 −1.09383
\(611\) −1.89531 −0.0766762
\(612\) 0 0
\(613\) 32.3141 1.30515 0.652576 0.757723i \(-0.273688\pi\)
0.652576 + 0.757723i \(0.273688\pi\)
\(614\) −14.1047 −0.569219
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) −16.2094 −0.650984
\(621\) 0 0
\(622\) −22.2094 −0.890515
\(623\) −2.59688 −0.104042
\(624\) 0 0
\(625\) −31.2094 −1.24837
\(626\) −17.4031 −0.695569
\(627\) 0 0
\(628\) −0.596876 −0.0238179
\(629\) 0 0
\(630\) 0 0
\(631\) 34.8062 1.38561 0.692807 0.721123i \(-0.256374\pi\)
0.692807 + 0.721123i \(0.256374\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −1.29844 −0.0515676
\(635\) 5.68594 0.225640
\(636\) 0 0
\(637\) −0.701562 −0.0277969
\(638\) 0 0
\(639\) 0 0
\(640\) −2.70156 −0.106789
\(641\) 5.29844 0.209276 0.104638 0.994510i \(-0.466632\pi\)
0.104638 + 0.994510i \(0.466632\pi\)
\(642\) 0 0
\(643\) 28.8062 1.13601 0.568004 0.823026i \(-0.307716\pi\)
0.568004 + 0.823026i \(0.307716\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −34.2094 −1.34491 −0.672455 0.740138i \(-0.734760\pi\)
−0.672455 + 0.740138i \(0.734760\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.61250 −0.0632473
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −27.8953 −1.09163 −0.545814 0.837906i \(-0.683779\pi\)
−0.545814 + 0.837906i \(0.683779\pi\)
\(654\) 0 0
\(655\) 7.01562 0.274123
\(656\) −1.29844 −0.0506955
\(657\) 0 0
\(658\) −2.70156 −0.105318
\(659\) 25.6125 0.997721 0.498861 0.866682i \(-0.333752\pi\)
0.498861 + 0.866682i \(0.333752\pi\)
\(660\) 0 0
\(661\) 4.80625 0.186941 0.0934707 0.995622i \(-0.470204\pi\)
0.0934707 + 0.995622i \(0.470204\pi\)
\(662\) 33.6125 1.30639
\(663\) 0 0
\(664\) 15.4031 0.597757
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) −6.70156 −0.259486
\(668\) 8.59688 0.332623
\(669\) 0 0
\(670\) −10.8062 −0.417482
\(671\) 0 0
\(672\) 0 0
\(673\) −14.7016 −0.566704 −0.283352 0.959016i \(-0.591446\pi\)
−0.283352 + 0.959016i \(0.591446\pi\)
\(674\) 4.59688 0.177065
\(675\) 0 0
\(676\) −12.5078 −0.481070
\(677\) 28.8062 1.10711 0.553557 0.832811i \(-0.313270\pi\)
0.553557 + 0.832811i \(0.313270\pi\)
\(678\) 0 0
\(679\) −18.1047 −0.694794
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 54.3141 2.07523
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −0.701562 −0.0267468
\(689\) −2.38750 −0.0909566
\(690\) 0 0
\(691\) −35.2984 −1.34282 −0.671408 0.741088i \(-0.734310\pi\)
−0.671408 + 0.741088i \(0.734310\pi\)
\(692\) −9.40312 −0.357453
\(693\) 0 0
\(694\) 11.5078 0.436831
\(695\) 5.12031 0.194224
\(696\) 0 0
\(697\) 0 0
\(698\) −4.20937 −0.159327
\(699\) 0 0
\(700\) −2.29844 −0.0868728
\(701\) −27.4031 −1.03500 −0.517501 0.855683i \(-0.673138\pi\)
−0.517501 + 0.855683i \(0.673138\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) −24.3141 −0.915072
\(707\) 17.4031 0.654512
\(708\) 0 0
\(709\) −40.8062 −1.53251 −0.766255 0.642536i \(-0.777882\pi\)
−0.766255 + 0.642536i \(0.777882\pi\)
\(710\) −14.5969 −0.547811
\(711\) 0 0
\(712\) 2.59688 0.0973220
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 0.701562 0.0262186
\(717\) 0 0
\(718\) −3.29844 −0.123097
\(719\) 17.5078 0.652931 0.326466 0.945209i \(-0.394142\pi\)
0.326466 + 0.945209i \(0.394142\pi\)
\(720\) 0 0
\(721\) −2.10469 −0.0783826
\(722\) 35.8062 1.33257
\(723\) 0 0
\(724\) −8.80625 −0.327282
\(725\) −15.4031 −0.572058
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0.701562 0.0260016
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) −33.0156 −1.21946 −0.609730 0.792609i \(-0.708722\pi\)
−0.609730 + 0.792609i \(0.708722\pi\)
\(734\) −28.9109 −1.06712
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −0.209373 −0.00770190 −0.00385095 0.999993i \(-0.501226\pi\)
−0.00385095 + 0.999993i \(0.501226\pi\)
\(740\) 7.29844 0.268296
\(741\) 0 0
\(742\) −3.40312 −0.124933
\(743\) −29.6125 −1.08638 −0.543189 0.839611i \(-0.682783\pi\)
−0.543189 + 0.839611i \(0.682783\pi\)
\(744\) 0 0
\(745\) −12.4187 −0.454988
\(746\) 20.8062 0.761771
\(747\) 0 0
\(748\) 0 0
\(749\) −10.8062 −0.394852
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 2.70156 0.0985158
\(753\) 0 0
\(754\) 4.70156 0.171221
\(755\) 27.2984 0.993492
\(756\) 0 0
\(757\) 32.8062 1.19236 0.596182 0.802850i \(-0.296684\pi\)
0.596182 + 0.802850i \(0.296684\pi\)
\(758\) −11.2984 −0.410378
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) 35.6125 1.29095 0.645476 0.763781i \(-0.276659\pi\)
0.645476 + 0.763781i \(0.276659\pi\)
\(762\) 0 0
\(763\) −9.29844 −0.336626
\(764\) 2.80625 0.101527
\(765\) 0 0
\(766\) 18.8062 0.679497
\(767\) −9.40312 −0.339527
\(768\) 0 0
\(769\) −9.89531 −0.356834 −0.178417 0.983955i \(-0.557098\pi\)
−0.178417 + 0.983955i \(0.557098\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12.1047 0.435657
\(773\) 25.7172 0.924983 0.462491 0.886624i \(-0.346956\pi\)
0.462491 + 0.886624i \(0.346956\pi\)
\(774\) 0 0
\(775\) 13.7906 0.495374
\(776\) 18.1047 0.649920
\(777\) 0 0
\(778\) −11.1938 −0.401315
\(779\) −9.61250 −0.344403
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 1.61250 0.0575525
\(786\) 0 0
\(787\) −39.8219 −1.41950 −0.709748 0.704455i \(-0.751191\pi\)
−0.709748 + 0.704455i \(0.751191\pi\)
\(788\) 25.5078 0.908678
\(789\) 0 0
\(790\) 0 0
\(791\) −10.7016 −0.380504
\(792\) 0 0
\(793\) −7.01562 −0.249132
\(794\) −32.2094 −1.14307
\(795\) 0 0
\(796\) −18.1047 −0.641704
\(797\) 25.2984 0.896117 0.448058 0.894004i \(-0.352116\pi\)
0.448058 + 0.894004i \(0.352116\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 2.29844 0.0812621
\(801\) 0 0
\(802\) −3.19375 −0.112775
\(803\) 0 0
\(804\) 0 0
\(805\) 2.70156 0.0952176
\(806\) −4.20937 −0.148269
\(807\) 0 0
\(808\) −17.4031 −0.612240
\(809\) 8.59688 0.302250 0.151125 0.988515i \(-0.451710\pi\)
0.151125 + 0.988515i \(0.451710\pi\)
\(810\) 0 0
\(811\) −46.3141 −1.62631 −0.813153 0.582050i \(-0.802251\pi\)
−0.813153 + 0.582050i \(0.802251\pi\)
\(812\) 6.70156 0.235179
\(813\) 0 0
\(814\) 0 0
\(815\) 32.4187 1.13558
\(816\) 0 0
\(817\) −5.19375 −0.181706
\(818\) 6.20937 0.217106
\(819\) 0 0
\(820\) 3.50781 0.122498
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) −23.7172 −0.826729 −0.413365 0.910566i \(-0.635646\pi\)
−0.413365 + 0.910566i \(0.635646\pi\)
\(824\) 2.10469 0.0733202
\(825\) 0 0
\(826\) −13.4031 −0.466354
\(827\) −31.0156 −1.07852 −0.539259 0.842140i \(-0.681296\pi\)
−0.539259 + 0.842140i \(0.681296\pi\)
\(828\) 0 0
\(829\) 8.20937 0.285123 0.142562 0.989786i \(-0.454466\pi\)
0.142562 + 0.989786i \(0.454466\pi\)
\(830\) −41.6125 −1.44439
\(831\) 0 0
\(832\) −0.701562 −0.0243223
\(833\) 0 0
\(834\) 0 0
\(835\) −23.2250 −0.803734
\(836\) 0 0
\(837\) 0 0
\(838\) 15.4031 0.532092
\(839\) 6.59688 0.227749 0.113875 0.993495i \(-0.463674\pi\)
0.113875 + 0.993495i \(0.463674\pi\)
\(840\) 0 0
\(841\) 15.9109 0.548653
\(842\) 9.29844 0.320445
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 33.7906 1.16243
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 3.40312 0.116864
\(849\) 0 0
\(850\) 0 0
\(851\) −2.70156 −0.0926084
\(852\) 0 0
\(853\) 21.8953 0.749681 0.374841 0.927089i \(-0.377697\pi\)
0.374841 + 0.927089i \(0.377697\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 10.8062 0.369350
\(857\) −46.9109 −1.60245 −0.801224 0.598365i \(-0.795817\pi\)
−0.801224 + 0.598365i \(0.795817\pi\)
\(858\) 0 0
\(859\) 40.9109 1.39586 0.697932 0.716164i \(-0.254104\pi\)
0.697932 + 0.716164i \(0.254104\pi\)
\(860\) 1.89531 0.0646297
\(861\) 0 0
\(862\) 14.1047 0.480408
\(863\) −14.8062 −0.504011 −0.252005 0.967726i \(-0.581090\pi\)
−0.252005 + 0.967726i \(0.581090\pi\)
\(864\) 0 0
\(865\) 25.4031 0.863732
\(866\) −2.10469 −0.0715202
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) −2.80625 −0.0950861
\(872\) 9.29844 0.314885
\(873\) 0 0
\(874\) 7.40312 0.250414
\(875\) −7.29844 −0.246732
\(876\) 0 0
\(877\) −9.79063 −0.330606 −0.165303 0.986243i \(-0.552860\pi\)
−0.165303 + 0.986243i \(0.552860\pi\)
\(878\) 28.8062 0.972164
\(879\) 0 0
\(880\) 0 0
\(881\) −28.2094 −0.950398 −0.475199 0.879878i \(-0.657624\pi\)
−0.475199 + 0.879878i \(0.657624\pi\)
\(882\) 0 0
\(883\) 29.4031 0.989494 0.494747 0.869037i \(-0.335261\pi\)
0.494747 + 0.869037i \(0.335261\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.10469 0.0707083
\(887\) 13.7906 0.463044 0.231522 0.972830i \(-0.425629\pi\)
0.231522 + 0.972830i \(0.425629\pi\)
\(888\) 0 0
\(889\) 2.10469 0.0705889
\(890\) −7.01562 −0.235164
\(891\) 0 0
\(892\) 4.80625 0.160925
\(893\) 20.0000 0.669274
\(894\) 0 0
\(895\) −1.89531 −0.0633533
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 23.6125 0.787959
\(899\) −40.2094 −1.34106
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 10.7016 0.355929
\(905\) 23.7906 0.790827
\(906\) 0 0
\(907\) −22.1047 −0.733974 −0.366987 0.930226i \(-0.619611\pi\)
−0.366987 + 0.930226i \(0.619611\pi\)
\(908\) −5.50781 −0.182783
\(909\) 0 0
\(910\) −1.89531 −0.0628290
\(911\) 4.70156 0.155770 0.0778849 0.996962i \(-0.475183\pi\)
0.0778849 + 0.996962i \(0.475183\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 22.2094 0.734621
\(915\) 0 0
\(916\) −10.2094 −0.337327
\(917\) 2.59688 0.0857564
\(918\) 0 0
\(919\) 26.8062 0.884257 0.442128 0.896952i \(-0.354224\pi\)
0.442128 + 0.896952i \(0.354224\pi\)
\(920\) −2.70156 −0.0890679
\(921\) 0 0
\(922\) −14.5969 −0.480723
\(923\) −3.79063 −0.124770
\(924\) 0 0
\(925\) −6.20937 −0.204163
\(926\) 6.10469 0.200612
\(927\) 0 0
\(928\) −6.70156 −0.219990
\(929\) −17.2984 −0.567543 −0.283772 0.958892i \(-0.591586\pi\)
−0.283772 + 0.958892i \(0.591586\pi\)
\(930\) 0 0
\(931\) 7.40312 0.242628
\(932\) −3.19375 −0.104615
\(933\) 0 0
\(934\) 14.7016 0.481050
\(935\) 0 0
\(936\) 0 0
\(937\) −10.3141 −0.336946 −0.168473 0.985706i \(-0.553884\pi\)
−0.168473 + 0.985706i \(0.553884\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −7.29844 −0.238049
\(941\) −48.3141 −1.57499 −0.787497 0.616319i \(-0.788623\pi\)
−0.787497 + 0.616319i \(0.788623\pi\)
\(942\) 0 0
\(943\) −1.29844 −0.0422830
\(944\) 13.4031 0.436235
\(945\) 0 0
\(946\) 0 0
\(947\) −35.5078 −1.15385 −0.576924 0.816798i \(-0.695747\pi\)
−0.576924 + 0.816798i \(0.695747\pi\)
\(948\) 0 0
\(949\) −5.19375 −0.168596
\(950\) 17.0156 0.552060
\(951\) 0 0
\(952\) 0 0
\(953\) −4.80625 −0.155690 −0.0778448 0.996965i \(-0.524804\pi\)
−0.0778448 + 0.996965i \(0.524804\pi\)
\(954\) 0 0
\(955\) −7.58125 −0.245324
\(956\) −6.80625 −0.220130
\(957\) 0 0
\(958\) 5.40312 0.174567
\(959\) 20.1047 0.649214
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 1.89531 0.0611074
\(963\) 0 0
\(964\) −14.1047 −0.454281
\(965\) −32.7016 −1.05270
\(966\) 0 0
\(967\) −56.4187 −1.81430 −0.907152 0.420803i \(-0.861749\pi\)
−0.907152 + 0.420803i \(0.861749\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) −48.9109 −1.57044
\(971\) 20.5969 0.660985 0.330493 0.943809i \(-0.392785\pi\)
0.330493 + 0.943809i \(0.392785\pi\)
\(972\) 0 0
\(973\) 1.89531 0.0607610
\(974\) 6.10469 0.195607
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 46.7016 1.49412 0.747058 0.664759i \(-0.231466\pi\)
0.747058 + 0.664759i \(0.231466\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.70156 −0.0862982
\(981\) 0 0
\(982\) −33.6125 −1.07262
\(983\) −33.8219 −1.07875 −0.539375 0.842066i \(-0.681339\pi\)
−0.539375 + 0.842066i \(0.681339\pi\)
\(984\) 0 0
\(985\) −68.9109 −2.19568
\(986\) 0 0
\(987\) 0 0
\(988\) −5.19375 −0.165235
\(989\) −0.701562 −0.0223084
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −5.40312 −0.171377
\(995\) 48.9109 1.55058
\(996\) 0 0
\(997\) 61.4031 1.94466 0.972328 0.233619i \(-0.0750568\pi\)
0.972328 + 0.233619i \(0.0750568\pi\)
\(998\) −27.0156 −0.855165
\(999\) 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.bc.1.1 2
3.2 odd 2 966.2.a.m.1.2 2
12.11 even 2 7728.2.a.z.1.2 2
21.20 even 2 6762.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.m.1.2 2 3.2 odd 2
2898.2.a.bc.1.1 2 1.1 even 1 trivial
6762.2.a.bq.1.1 2 21.20 even 2
7728.2.a.z.1.2 2 12.11 even 2