Properties

Label 950.2.a.c.1.1
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{11} +3.00000 q^{12} +1.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -6.00000 q^{18} +1.00000 q^{19} +15.0000 q^{21} +4.00000 q^{22} -7.00000 q^{23} -3.00000 q^{24} -1.00000 q^{26} +9.00000 q^{27} +5.00000 q^{28} -3.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -12.0000 q^{33} -3.00000 q^{34} +6.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} +3.00000 q^{39} -6.00000 q^{41} -15.0000 q^{42} -6.00000 q^{43} -4.00000 q^{44} +7.00000 q^{46} +3.00000 q^{48} +18.0000 q^{49} +9.00000 q^{51} +1.00000 q^{52} +13.0000 q^{53} -9.00000 q^{54} -5.00000 q^{56} +3.00000 q^{57} +3.00000 q^{58} -9.00000 q^{59} -12.0000 q^{61} +2.00000 q^{62} +30.0000 q^{63} +1.00000 q^{64} +12.0000 q^{66} +3.00000 q^{67} +3.00000 q^{68} -21.0000 q^{69} -6.00000 q^{72} -11.0000 q^{73} -2.00000 q^{74} +1.00000 q^{76} -20.0000 q^{77} -3.00000 q^{78} -2.00000 q^{79} +9.00000 q^{81} +6.00000 q^{82} +10.0000 q^{83} +15.0000 q^{84} +6.00000 q^{86} -9.00000 q^{87} +4.00000 q^{88} +2.00000 q^{89} +5.00000 q^{91} -7.00000 q^{92} -6.00000 q^{93} -3.00000 q^{96} +2.00000 q^{97} -18.0000 q^{98} -24.0000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 5.00000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 3.00000 0.866025
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −5.00000 −1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −6.00000 −1.41421
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 15.0000 3.27327
\(22\) 4.00000 0.852803
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 9.00000 1.73205
\(28\) 5.00000 0.944911
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 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
\(33\) −12.0000 −2.08893
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −15.0000 −2.31455
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 3.00000 0.433013
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 9.00000 1.26025
\(52\) 1.00000 0.138675
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −5.00000 −0.668153
\(57\) 3.00000 0.397360
\(58\) 3.00000 0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 2.00000 0.254000
\(63\) 30.0000 3.77964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 12.0000 1.47710
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 3.00000 0.363803
\(69\) −21.0000 −2.52810
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −6.00000 −0.707107
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −20.0000 −2.27921
\(78\) −3.00000 −0.339683
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 6.00000 0.662589
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 15.0000 1.63663
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −9.00000 −0.964901
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −7.00000 −0.729800
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −18.0000 −1.81827
\(99\) −24.0000 −2.41209
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −9.00000 −0.891133
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 9.00000 0.866025
\(109\) 19.0000 1.81987 0.909935 0.414751i \(-0.136131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 5.00000 0.472456
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 6.00000 0.554700
\(118\) 9.00000 0.828517
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 12.0000 1.08643
\(123\) −18.0000 −1.62301
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −30.0000 −2.67261
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −18.0000 −1.58481
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) −12.0000 −1.04447
\(133\) 5.00000 0.433555
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 21.0000 1.78764
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) 54.0000 4.45384
\(148\) 2.00000 0.164399
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 18.0000 1.45521
\(154\) 20.0000 1.61165
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 2.00000 0.159111
\(159\) 39.0000 3.09290
\(160\) 0 0
\(161\) −35.0000 −2.75839
\(162\) −9.00000 −0.707107
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −10.0000 −0.776151
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) −15.0000 −1.15728
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −6.00000 −0.457496
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −27.0000 −2.02944
\(178\) −2.00000 −0.149906
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) −5.00000 −0.370625
\(183\) −36.0000 −2.66120
\(184\) 7.00000 0.516047
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 45.0000 3.27327
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 3.00000 0.216506
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 24.0000 1.70561
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 8.00000 0.562878
\(203\) −15.0000 −1.05279
\(204\) 9.00000 0.630126
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −42.0000 −2.91920
\(208\) 1.00000 0.0693375
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 13.0000 0.892844
\(213\) 0 0
\(214\) −13.0000 −0.888662
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) −10.0000 −0.678844
\(218\) −19.0000 −1.28684
\(219\) −33.0000 −2.22993
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) −6.00000 −0.402694
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 0 0
\(227\) −5.00000 −0.331862 −0.165931 0.986137i \(-0.553063\pi\)
−0.165931 + 0.986137i \(0.553063\pi\)
\(228\) 3.00000 0.198680
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −60.0000 −3.94771
\(232\) 3.00000 0.196960
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) −6.00000 −0.389742
\(238\) −15.0000 −0.972306
\(239\) −11.0000 −0.711531 −0.355765 0.934575i \(-0.615780\pi\)
−0.355765 + 0.934575i \(0.615780\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) 18.0000 1.14764
\(247\) 1.00000 0.0636285
\(248\) 2.00000 0.127000
\(249\) 30.0000 1.90117
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 30.0000 1.88982
\(253\) 28.0000 1.76034
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 18.0000 1.12063
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) −16.0000 −0.988483
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) −5.00000 −0.306570
\(267\) 6.00000 0.367194
\(268\) 3.00000 0.183254
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −27.0000 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(272\) 3.00000 0.181902
\(273\) 15.0000 0.907841
\(274\) 9.00000 0.543710
\(275\) 0 0
\(276\) −21.0000 −1.26405
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −16.0000 −0.959616
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −30.0000 −1.77084
\(288\) −6.00000 −0.353553
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) −11.0000 −0.643726
\(293\) 27.0000 1.57736 0.788678 0.614806i \(-0.210766\pi\)
0.788678 + 0.614806i \(0.210766\pi\)
\(294\) −54.0000 −3.14934
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −36.0000 −2.08893
\(298\) 4.00000 0.231714
\(299\) −7.00000 −0.404820
\(300\) 0 0
\(301\) −30.0000 −1.72917
\(302\) 10.0000 0.575435
\(303\) −24.0000 −1.37876
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −18.0000 −1.02899
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −20.0000 −1.13961
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 25.0000 1.41762 0.708810 0.705399i \(-0.249232\pi\)
0.708810 + 0.705399i \(0.249232\pi\)
\(312\) −3.00000 −0.169842
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) −39.0000 −2.18701
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 39.0000 2.17677
\(322\) 35.0000 1.95047
\(323\) 3.00000 0.166924
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 57.0000 3.15211
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 10.0000 0.548821
\(333\) 12.0000 0.657596
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 15.0000 0.818317
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) −6.00000 −0.324443
\(343\) 55.0000 2.96972
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) −9.00000 −0.482451
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 4.00000 0.213201
\(353\) −7.00000 −0.372572 −0.186286 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) 27.0000 1.43503
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 45.0000 2.38165
\(358\) 8.00000 0.422813
\(359\) −5.00000 −0.263890 −0.131945 0.991257i \(-0.542122\pi\)
−0.131945 + 0.991257i \(0.542122\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −26.0000 −1.36653
\(363\) 15.0000 0.787296
\(364\) 5.00000 0.262071
\(365\) 0 0
\(366\) 36.0000 1.88175
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −7.00000 −0.364900
\(369\) −36.0000 −1.87409
\(370\) 0 0
\(371\) 65.0000 3.37463
\(372\) −6.00000 −0.311086
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) −45.0000 −2.31455
\(379\) −33.0000 −1.69510 −0.847548 0.530719i \(-0.821922\pi\)
−0.847548 + 0.530719i \(0.821922\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) −9.00000 −0.460480
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −36.0000 −1.82998
\(388\) 2.00000 0.101535
\(389\) −4.00000 −0.202808 −0.101404 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) −21.0000 −1.06202
\(392\) −18.0000 −0.909137
\(393\) 48.0000 2.42128
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −24.0000 −1.20605
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 15.0000 0.751882
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −9.00000 −0.448879
\(403\) −2.00000 −0.0996271
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) −8.00000 −0.396545
\(408\) −9.00000 −0.445566
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −27.0000 −1.33181
\(412\) −4.00000 −0.197066
\(413\) −45.0000 −2.21431
\(414\) 42.0000 2.06419
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 48.0000 2.35057
\(418\) 4.00000 0.195646
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −13.0000 −0.631336
\(425\) 0 0
\(426\) 0 0
\(427\) −60.0000 −2.90360
\(428\) 13.0000 0.628379
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 9.00000 0.433013
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 19.0000 0.909935
\(437\) −7.00000 −0.334855
\(438\) 33.0000 1.57680
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 108.000 5.14286
\(442\) −3.00000 −0.142695
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −12.0000 −0.567581
\(448\) 5.00000 0.236228
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) −30.0000 −1.40952
\(454\) 5.00000 0.234662
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 6.00000 0.280362
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 60.0000 2.79145
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 6.00000 0.277350
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 9.00000 0.414259
\(473\) 24.0000 1.10352
\(474\) 6.00000 0.275589
\(475\) 0 0
\(476\) 15.0000 0.687524
\(477\) 78.0000 3.57137
\(478\) 11.0000 0.503128
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 12.0000 0.546585
\(483\) −105.000 −4.77767
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 12.0000 0.543214
\(489\) −66.0000 −2.98462
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) −18.0000 −0.811503
\(493\) −9.00000 −0.405340
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −30.0000 −1.34433
\(499\) 42.0000 1.88018 0.940089 0.340929i \(-0.110742\pi\)
0.940089 + 0.340929i \(0.110742\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 12.0000 0.535586
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) −30.0000 −1.33631
\(505\) 0 0
\(506\) −28.0000 −1.24475
\(507\) −36.0000 −1.59882
\(508\) 6.00000 0.266207
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −55.0000 −2.43306
\(512\) −1.00000 −0.0441942
\(513\) 9.00000 0.397360
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −18.0000 −0.792406
\(517\) 0 0
\(518\) −10.0000 −0.439375
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 18.0000 0.787839
\(523\) 9.00000 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −6.00000 −0.261364
\(528\) −12.0000 −0.522233
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −54.0000 −2.34340
\(532\) 5.00000 0.216777
\(533\) −6.00000 −0.259889
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −3.00000 −0.129580
\(537\) −24.0000 −1.03568
\(538\) −2.00000 −0.0862261
\(539\) −72.0000 −3.10126
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 27.0000 1.15975
\(543\) 78.0000 3.34730
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) −15.0000 −0.641941
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −9.00000 −0.384461
\(549\) −72.0000 −3.07289
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 21.0000 0.893819
\(553\) −10.0000 −0.425243
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 12.0000 0.508001
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 18.0000 0.759284
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 45.0000 1.88982
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) −4.00000 −0.167248
\(573\) 27.0000 1.12794
\(574\) 30.0000 1.25218
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) 7.00000 0.291414 0.145707 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 8.00000 0.332756
\(579\) −30.0000 −1.24676
\(580\) 0 0
\(581\) 50.0000 2.07435
\(582\) −6.00000 −0.248708
\(583\) −52.0000 −2.15362
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) −27.0000 −1.11536
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 54.0000 2.22692
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 66.0000 2.71488
\(592\) 2.00000 0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 36.0000 1.47710
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) −45.0000 −1.84173
\(598\) 7.00000 0.286251
\(599\) −26.0000 −1.06233 −0.531166 0.847268i \(-0.678246\pi\)
−0.531166 + 0.847268i \(0.678246\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 30.0000 1.22271
\(603\) 18.0000 0.733017
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) 26.0000 1.05531 0.527654 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −45.0000 −1.82349
\(610\) 0 0
\(611\) 0 0
\(612\) 18.0000 0.727607
\(613\) 20.0000 0.807792 0.403896 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 20.0000 0.805823
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 12.0000 0.482711
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) −63.0000 −2.52810
\(622\) −25.0000 −1.00241
\(623\) 10.0000 0.400642
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) −1.00000 −0.0399680
\(627\) −12.0000 −0.479234
\(628\) −6.00000 −0.239426
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 2.00000 0.0795557
\(633\) −15.0000 −0.596196
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) 39.0000 1.54645
\(637\) 18.0000 0.713186
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) −39.0000 −1.53921
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −35.0000 −1.37919
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) −9.00000 −0.353553
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) −22.0000 −0.861586
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) −57.0000 −2.22888
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −66.0000 −2.57491
\(658\) 0 0
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) 15.0000 0.583432 0.291716 0.956505i \(-0.405774\pi\)
0.291716 + 0.956505i \(0.405774\pi\)
\(662\) 7.00000 0.272063
\(663\) 9.00000 0.349531
\(664\) −10.0000 −0.388075
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 21.0000 0.813123
\(668\) 2.00000 0.0773823
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) −15.0000 −0.578638
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) −8.00000 −0.306336
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −55.0000 −2.09991
\(687\) −18.0000 −0.686743
\(688\) −6.00000 −0.228748
\(689\) 13.0000 0.495261
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 14.0000 0.532200
\(693\) −120.000 −4.55842
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) −18.0000 −0.681799
\(698\) −14.0000 −0.529908
\(699\) −30.0000 −1.13470
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) −9.00000 −0.339683
\(703\) 2.00000 0.0754314
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 7.00000 0.263448
\(707\) −40.0000 −1.50435
\(708\) −27.0000 −1.01472
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −2.00000 −0.0749532
\(713\) 14.0000 0.524304
\(714\) −45.0000 −1.68408
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) −33.0000 −1.23241
\(718\) 5.00000 0.186598
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) −1.00000 −0.0372161
\(723\) −36.0000 −1.33885
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) −15.0000 −0.556702
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) −5.00000 −0.185312
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) −36.0000 −1.33060
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) −12.0000 −0.442026
\(738\) 36.0000 1.32518
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) −65.0000 −2.38623
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) 60.0000 2.19529
\(748\) −12.0000 −0.438763
\(749\) 65.0000 2.37505
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) 45.0000 1.63663
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 33.0000 1.19861
\(759\) 84.0000 3.04901
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) −18.0000 −0.652071
\(763\) 95.0000 3.43923
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −9.00000 −0.324971
\(768\) 3.00000 0.108253
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) −66.0000 −2.37693
\(772\) −10.0000 −0.359908
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 36.0000 1.29399
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 30.0000 1.07624
\(778\) 4.00000 0.143407
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 21.0000 0.750958
\(783\) −27.0000 −0.964901
\(784\) 18.0000 0.642857
\(785\) 0 0
\(786\) −48.0000 −1.71210
\(787\) 39.0000 1.39020 0.695100 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) 22.0000 0.783718
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 24.0000 0.852803
\(793\) −12.0000 −0.426132
\(794\) −16.0000 −0.567819
\(795\) 0 0
\(796\) −15.0000 −0.531661
\(797\) −31.0000 −1.09808 −0.549038 0.835797i \(-0.685006\pi\)
−0.549038 + 0.835797i \(0.685006\pi\)
\(798\) −15.0000 −0.530994
\(799\) 0 0
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 6.00000 0.211867
\(803\) 44.0000 1.55273
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 6.00000 0.211210
\(808\) 8.00000 0.281439
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) 0 0
\(811\) 37.0000 1.29925 0.649623 0.760257i \(-0.274927\pi\)
0.649623 + 0.760257i \(0.274927\pi\)
\(812\) −15.0000 −0.526397
\(813\) −81.0000 −2.84079
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 9.00000 0.315063
\(817\) −6.00000 −0.209913
\(818\) −22.0000 −0.769212
\(819\) 30.0000 1.04828
\(820\) 0 0
\(821\) −52.0000 −1.81481 −0.907406 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) 27.0000 0.941733
\(823\) 43.0000 1.49889 0.749443 0.662069i \(-0.230321\pi\)
0.749443 + 0.662069i \(0.230321\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −42.0000 −1.45960
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 1.00000 0.0346688
\(833\) 54.0000 1.87099
\(834\) −48.0000 −1.66210
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) −18.0000 −0.622171
\(838\) −14.0000 −0.483622
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −1.00000 −0.0344623
\(843\) −54.0000 −1.85986
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) 13.0000 0.446422
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) −14.0000 −0.479914
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 60.0000 2.05316
\(855\) 0 0
\(856\) −13.0000 −0.444331
\(857\) 40.0000 1.36637 0.683187 0.730243i \(-0.260593\pi\)
0.683187 + 0.730243i \(0.260593\pi\)
\(858\) 12.0000 0.409673
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) −90.0000 −3.06719
\(862\) 36.0000 1.22616
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) −24.0000 −0.815083
\(868\) −10.0000 −0.339422
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) −19.0000 −0.643421
\(873\) 12.0000 0.406138
\(874\) 7.00000 0.236779
\(875\) 0 0
\(876\) −33.0000 −1.11497
\(877\) −33.0000 −1.11433 −0.557165 0.830402i \(-0.688111\pi\)
−0.557165 + 0.830402i \(0.688111\pi\)
\(878\) −26.0000 −0.877457
\(879\) 81.0000 2.73206
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) −108.000 −3.63655
\(883\) −30.0000 −1.00958 −0.504790 0.863242i \(-0.668430\pi\)
−0.504790 + 0.863242i \(0.668430\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) −6.00000 −0.201347
\(889\) 30.0000 1.00617
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) −5.00000 −0.167038
\(897\) −21.0000 −0.701170
\(898\) 22.0000 0.734150
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) −24.0000 −0.799113
\(903\) −90.0000 −2.99501
\(904\) 0 0
\(905\) 0 0
\(906\) 30.0000 0.996683
\(907\) 1.00000 0.0332045 0.0166022 0.999862i \(-0.494715\pi\)
0.0166022 + 0.999862i \(0.494715\pi\)
\(908\) −5.00000 −0.165931
\(909\) −48.0000 −1.59206
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 3.00000 0.0993399
\(913\) −40.0000 −1.32381
\(914\) −29.0000 −0.959235
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 80.0000 2.64183
\(918\) −27.0000 −0.891133
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) −60.0000 −1.97386
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) −24.0000 −0.788263
\(928\) 3.00000 0.0984798
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −10.0000 −0.327561
\(933\) 75.0000 2.45539
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −47.0000 −1.53542 −0.767712 0.640796i \(-0.778605\pi\)
−0.767712 + 0.640796i \(0.778605\pi\)
\(938\) −15.0000 −0.489767
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) −51.0000 −1.66255 −0.831276 0.555860i \(-0.812389\pi\)
−0.831276 + 0.555860i \(0.812389\pi\)
\(942\) 18.0000 0.586472
\(943\) 42.0000 1.36771
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −6.00000 −0.194871
\(949\) −11.0000 −0.357075
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) −15.0000 −0.486153
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −78.0000 −2.52534
\(955\) 0 0
\(956\) −11.0000 −0.355765
\(957\) 36.0000 1.16371
\(958\) 0 0
\(959\) −45.0000 −1.45313
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −2.00000 −0.0644826
\(963\) 78.0000 2.51351
\(964\) −12.0000 −0.386494
\(965\) 0 0
\(966\) 105.000 3.37832
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −5.00000 −0.160706
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 80.0000 2.56468
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 62.0000 1.98356 0.991778 0.127971i \(-0.0408466\pi\)
0.991778 + 0.127971i \(0.0408466\pi\)
\(978\) 66.0000 2.11045
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 114.000 3.63974
\(982\) −18.0000 −0.574403
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) 30.0000 0.952981 0.476491 0.879180i \(-0.341909\pi\)
0.476491 + 0.879180i \(0.341909\pi\)
\(992\) 2.00000 0.0635001
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) 0 0
\(996\) 30.0000 0.950586
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) −42.0000 −1.32949
\(999\) 18.0000 0.569495
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 950.2.a.c.1.1 1
3.2 odd 2 8550.2.a.bm.1.1 1
4.3 odd 2 7600.2.a.a.1.1 1
5.2 odd 4 950.2.b.a.799.1 2
5.3 odd 4 950.2.b.a.799.2 2
5.4 even 2 190.2.a.b.1.1 1
15.14 odd 2 1710.2.a.g.1.1 1
20.19 odd 2 1520.2.a.j.1.1 1
35.34 odd 2 9310.2.a.u.1.1 1
40.19 odd 2 6080.2.a.b.1.1 1
40.29 even 2 6080.2.a.x.1.1 1
95.94 odd 2 3610.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 5.4 even 2
950.2.a.c.1.1 1 1.1 even 1 trivial
950.2.b.a.799.1 2 5.2 odd 4
950.2.b.a.799.2 2 5.3 odd 4
1520.2.a.j.1.1 1 20.19 odd 2
1710.2.a.g.1.1 1 15.14 odd 2
3610.2.a.e.1.1 1 95.94 odd 2
6080.2.a.b.1.1 1 40.19 odd 2
6080.2.a.x.1.1 1 40.29 even 2
7600.2.a.a.1.1 1 4.3 odd 2
8550.2.a.bm.1.1 1 3.2 odd 2
9310.2.a.u.1.1 1 35.34 odd 2