Properties

Label 6498.2.a.y.1.1
Level $6498$
Weight $2$
Character 6498.1
Self dual yes
Analytic conductor $51.887$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.8867912334\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6498.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} +4.00000 q^{10} -2.00000 q^{11} +1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +4.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} +3.00000 q^{28} -5.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +12.0000 q^{35} +2.00000 q^{37} +4.00000 q^{40} -8.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{46} -8.00000 q^{47} +2.00000 q^{49} +11.0000 q^{50} +1.00000 q^{52} -1.00000 q^{53} -8.00000 q^{55} +3.00000 q^{56} -5.00000 q^{58} +15.0000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -3.00000 q^{67} -3.00000 q^{68} +12.0000 q^{70} +2.00000 q^{71} +9.00000 q^{73} +2.00000 q^{74} -6.00000 q^{77} +10.0000 q^{79} +4.00000 q^{80} -8.00000 q^{82} +6.00000 q^{83} -12.0000 q^{85} +4.00000 q^{86} -2.00000 q^{88} +3.00000 q^{91} +1.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 4.00000 0.632456
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 12.0000 1.43427
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) −8.00000 −0.883452
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −7.00000 −0.676716 −0.338358 0.941018i \(-0.609871\pi\)
−0.338358 + 0.941018i \(0.609871\pi\)
\(108\) 0 0
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) 15.0000 1.38086
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 17.0000 1.45241 0.726204 0.687479i \(-0.241283\pi\)
0.726204 + 0.687479i \(0.241283\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 12.0000 1.01419
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −20.0000 −1.66091
\(146\) 9.00000 0.744845
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 32.0000 2.57030
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 4.00000 0.316228
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 33.0000 2.49457
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 3.00000 0.222375
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 11.0000 0.777817
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −15.0000 −1.05279
\(204\) 0 0
\(205\) −32.0000 −2.23498
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 0 0
\(214\) −7.00000 −0.478510
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 15.0000 1.01593
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −17.0000 −1.12833 −0.564165 0.825662i \(-0.690802\pi\)
−0.564165 + 0.825662i \(0.690802\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −32.0000 −2.08745
\(236\) 15.0000 0.976417
\(237\) 0 0
\(238\) −9.00000 −0.583383
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 8.00000 0.511101
\(246\) 0 0
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) −22.0000 −1.32665
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −20.0000 −1.17444
\(291\) 0 0
\(292\) 9.00000 0.526685
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 60.0000 3.49334
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) 32.0000 1.81748
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) 0 0
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) −8.00000 −0.441726
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 33.0000 1.76392
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 3.00000 0.157243
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −3.00000 −0.155752
\(372\) 0 0
\(373\) −29.0000 −1.50156 −0.750782 0.660551i \(-0.770323\pi\)
−0.750782 + 0.660551i \(0.770323\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −7.00000 −0.358151
\(383\) −26.0000 −1.32854 −0.664269 0.747494i \(-0.731257\pi\)
−0.664269 + 0.747494i \(0.731257\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) −8.00000 −0.403034
\(395\) 40.0000 2.01262
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −25.0000 −1.25314
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −15.0000 −0.744438
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) −32.0000 −1.58037
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) 45.0000 2.21431
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) −27.0000 −1.31434
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) −33.0000 −1.60074
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) −7.00000 −0.338358
\(429\) 0 0
\(430\) 16.0000 0.771589
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 24.0000 1.15204
\(435\) 0 0
\(436\) 15.0000 0.718370
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) 26.0000 1.23530 0.617649 0.786454i \(-0.288085\pi\)
0.617649 + 0.786454i \(0.288085\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −17.0000 −0.797850
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 2.00000 0.0925490 0.0462745 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) −32.0000 −1.47605
\(471\) 0 0
\(472\) 15.0000 0.690431
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) −9.00000 −0.412514
\(477\) 0 0
\(478\) −15.0000 −0.686084
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 8.00000 0.361403
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −2.00000 −0.0892644
\(503\) −39.0000 −1.73892 −0.869462 0.494000i \(-0.835534\pi\)
−0.869462 + 0.494000i \(0.835534\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) −18.0000 −0.798621
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 27.0000 1.19441
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) 24.0000 1.05757
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −4.00000 −0.173749
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) −3.00000 −0.129580
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 7.00000 0.300676
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 60.0000 2.57012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 17.0000 0.726204
\(549\) 0 0
\(550\) −22.0000 −0.938083
\(551\) 0 0
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 0 0
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 56.0000 2.35594
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 11.0000 0.458732
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −20.0000 −0.830455
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 60.0000 2.47016
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 0 0
\(598\) 1.00000 0.0408930
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 32.0000 1.28515
\(621\) 0 0
\(622\) −7.00000 −0.280674
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 29.0000 1.15907
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) −27.0000 −1.07231
\(635\) −72.0000 −2.85723
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 10.0000 0.395904
\(639\) 0 0
\(640\) 4.00000 0.158114
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) 0 0
\(647\) −23.0000 −0.904223 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 11.0000 0.431455
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) −48.0000 −1.87552
\(656\) −8.00000 −0.312348
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) −17.0000 −0.660724
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 68.0000 2.59815
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 33.0000 1.24728
\(701\) 28.0000 1.05755 0.528773 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −9.00000 −0.338719
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 15.0000 0.559795
\(719\) 5.00000 0.186469 0.0932343 0.995644i \(-0.470279\pi\)
0.0932343 + 0.995644i \(0.470279\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −55.0000 −2.04265
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 3.00000 0.111187
\(729\) 0 0
\(730\) 36.0000 1.33242
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −3.00000 −0.110133
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.0000 −1.06177
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) −21.0000 −0.767323
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −5.00000 −0.182089
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −15.0000 −0.544825
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 45.0000 1.62911
\(764\) −7.00000 −0.253251
\(765\) 0 0
\(766\) −26.0000 −0.939418
\(767\) 15.0000 0.541619
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) −24.0000 −0.864900
\(771\) 0 0
\(772\) 6.00000 0.215945
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 88.0000 3.16105
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −3.00000 −0.107280
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) −8.00000 −0.284988
\(789\) 0 0
\(790\) 40.0000 1.42314
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −25.0000 −0.886102
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 11.0000 0.388909
\(801\) 0 0
\(802\) −8.00000 −0.282490
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 3.00000 0.105344 0.0526721 0.998612i \(-0.483226\pi\)
0.0526721 + 0.998612i \(0.483226\pi\)
\(812\) −15.0000 −0.526397
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) −64.0000 −2.24182
\(816\) 0 0
\(817\) 0 0
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) −32.0000 −1.11749
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) 29.0000 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) 23.0000 0.799788 0.399894 0.916561i \(-0.369047\pi\)
0.399894 + 0.916561i \(0.369047\pi\)
\(828\) 0 0
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 13.0000 0.448010
\(843\) 0 0
\(844\) −27.0000 −0.929378
\(845\) −48.0000 −1.65125
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) −1.00000 −0.0343401
\(849\) 0 0
\(850\) −33.0000 −1.13189
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −7.00000 −0.239255
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 24.0000 0.814613
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 15.0000 0.507964
\(873\) 0 0
\(874\) 0 0
\(875\) 72.0000 2.43404
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) 0 0
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 3.00000 0.0999445
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −88.0000 −2.92522
\(906\) 0 0
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) −17.0000 −0.564165
\(909\) 0 0
\(910\) 12.0000 0.397796
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −7.00000 −0.231539
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 28.0000 0.922131
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) 55.0000 1.80449 0.902246 0.431222i \(-0.141918\pi\)
0.902246 + 0.431222i \(0.141918\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 2.00000 0.0654420
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) −9.00000 −0.293860
\(939\) 0 0
\(940\) −32.0000 −1.04372
\(941\) 7.00000 0.228193 0.114097 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 15.0000 0.488208
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 0 0
\(952\) −9.00000 −0.291692
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 0 0
\(955\) −28.0000 −0.906059
\(956\) −15.0000 −0.485135
\(957\) 0 0
\(958\) 20.0000 0.646171
\(959\) 51.0000 1.64688
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 8.00000 0.255551
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 15.0000 0.477697
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) −100.000 −3.17021
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 40.0000 1.26618
\(999\) 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 6498.2.a.y.1.1 1
3.2 odd 2 722.2.a.b.1.1 1
12.11 even 2 5776.2.a.d.1.1 1
19.18 odd 2 342.2.a.d.1.1 1
57.2 even 18 722.2.e.c.99.1 6
57.5 odd 18 722.2.e.d.595.1 6
57.8 even 6 722.2.c.d.653.1 2
57.11 odd 6 722.2.c.f.653.1 2
57.14 even 18 722.2.e.c.595.1 6
57.17 odd 18 722.2.e.d.99.1 6
57.23 odd 18 722.2.e.d.415.1 6
57.26 odd 6 722.2.c.f.429.1 2
57.29 even 18 722.2.e.c.423.1 6
57.32 even 18 722.2.e.c.245.1 6
57.35 odd 18 722.2.e.d.389.1 6
57.41 even 18 722.2.e.c.389.1 6
57.44 odd 18 722.2.e.d.245.1 6
57.47 odd 18 722.2.e.d.423.1 6
57.50 even 6 722.2.c.d.429.1 2
57.53 even 18 722.2.e.c.415.1 6
57.56 even 2 38.2.a.b.1.1 1
76.75 even 2 2736.2.a.w.1.1 1
95.94 odd 2 8550.2.a.u.1.1 1
228.227 odd 2 304.2.a.d.1.1 1
285.113 odd 4 950.2.b.c.799.1 2
285.227 odd 4 950.2.b.c.799.2 2
285.284 even 2 950.2.a.b.1.1 1
399.398 odd 2 1862.2.a.f.1.1 1
456.227 odd 2 1216.2.a.g.1.1 1
456.341 even 2 1216.2.a.n.1.1 1
627.626 odd 2 4598.2.a.a.1.1 1
741.740 even 2 6422.2.a.b.1.1 1
1140.1139 odd 2 7600.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 57.56 even 2
304.2.a.d.1.1 1 228.227 odd 2
342.2.a.d.1.1 1 19.18 odd 2
722.2.a.b.1.1 1 3.2 odd 2
722.2.c.d.429.1 2 57.50 even 6
722.2.c.d.653.1 2 57.8 even 6
722.2.c.f.429.1 2 57.26 odd 6
722.2.c.f.653.1 2 57.11 odd 6
722.2.e.c.99.1 6 57.2 even 18
722.2.e.c.245.1 6 57.32 even 18
722.2.e.c.389.1 6 57.41 even 18
722.2.e.c.415.1 6 57.53 even 18
722.2.e.c.423.1 6 57.29 even 18
722.2.e.c.595.1 6 57.14 even 18
722.2.e.d.99.1 6 57.17 odd 18
722.2.e.d.245.1 6 57.44 odd 18
722.2.e.d.389.1 6 57.35 odd 18
722.2.e.d.415.1 6 57.23 odd 18
722.2.e.d.423.1 6 57.47 odd 18
722.2.e.d.595.1 6 57.5 odd 18
950.2.a.b.1.1 1 285.284 even 2
950.2.b.c.799.1 2 285.113 odd 4
950.2.b.c.799.2 2 285.227 odd 4
1216.2.a.g.1.1 1 456.227 odd 2
1216.2.a.n.1.1 1 456.341 even 2
1862.2.a.f.1.1 1 399.398 odd 2
2736.2.a.w.1.1 1 76.75 even 2
4598.2.a.a.1.1 1 627.626 odd 2
5776.2.a.d.1.1 1 12.11 even 2
6422.2.a.b.1.1 1 741.740 even 2
6498.2.a.y.1.1 1 1.1 even 1 trivial
7600.2.a.h.1.1 1 1140.1139 odd 2
8550.2.a.u.1.1 1 95.94 odd 2