Properties

Label 1850.2.a.o.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} +2.00000 q^{24} +4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} +3.00000 q^{29} +5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +8.00000 q^{39} +3.00000 q^{41} +2.00000 q^{42} +1.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -6.00000 q^{51} +4.00000 q^{52} -3.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +4.00000 q^{57} +3.00000 q^{58} -1.00000 q^{61} +5.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} -3.00000 q^{68} -12.0000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +16.0000 q^{73} -1.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} +3.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +1.00000 q^{86} +6.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} +10.0000 q^{93} -12.0000 q^{94} +2.00000 q^{96} -17.0000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 2.00000 0.308607
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 2.00000 0.288675
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 3.00000 0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 5.00000 0.635001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −3.00000 −0.363803
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 16.0000 1.87266 0.936329 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000 0.331295
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 6.00000 0.643268
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 10.0000 1.03695
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) −6.00000 −0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −6.00000 −0.594089
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −4.00000 −0.384900
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) 6.00000 0.541002
\(124\) 5.00000 0.449013
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000 0.522233
\(133\) 2.00000 0.173422
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −12.0000 −1.02151
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 6.00000 0.503509
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) −12.0000 −0.989743
\(148\) −1.00000 −0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) −3.00000 −0.242536
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) −11.0000 −0.864242
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 1.00000 0.0762493
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.00000 0.296500
\(183\) −2.00000 −0.147844
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) −9.00000 −0.658145
\(188\) −12.0000 −0.875190
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 2.00000 0.144338
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 3.00000 0.213201
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −6.00000 −0.422159
\(203\) 3.00000 0.210559
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −6.00000 −0.417029
\(208\) 4.00000 0.277350
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −3.00000 −0.206041
\(213\) 12.0000 0.822226
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 5.00000 0.339422
\(218\) 11.0000 0.745014
\(219\) 32.0000 2.16236
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −17.0000 −1.13840 −0.569202 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 4.00000 0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 3.00000 0.196960
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) −3.00000 −0.194461
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −2.00000 −0.128565
\(243\) −10.0000 −0.641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 8.00000 0.509028
\(248\) 5.00000 0.317500
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 1.00000 0.0629941
\(253\) −18.0000 −1.13165
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 2.00000 0.124515
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) −12.0000 −0.734388
\(268\) 4.00000 0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −3.00000 −0.181902
\(273\) 8.00000 0.484182
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −13.0000 −0.779688
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) −24.0000 −1.42918
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 3.00000 0.177084
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −34.0000 −1.99312
\(292\) 16.0000 0.936329
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) −12.0000 −0.699854
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −12.0000 −0.696311
\(298\) 6.00000 0.347571
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 8.00000 0.460348
\(303\) −12.0000 −0.689382
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) 3.00000 0.170941
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 8.00000 0.452911
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) −6.00000 −0.336463
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −6.00000 −0.334367
\(323\) −6.00000 −0.333849
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −11.0000 −0.609234
\(327\) 22.0000 1.21660
\(328\) 3.00000 0.165647
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 12.0000 0.658586
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 3.00000 0.163178
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 2.00000 0.108148
\(343\) −13.0000 −0.701934
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 3.00000 0.159901
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) −4.00000 −0.209946
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −35.0000 −1.82699 −0.913493 0.406855i \(-0.866625\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) −6.00000 −0.312772
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 10.0000 0.518476
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 12.0000 0.618031
\(378\) −4.00000 −0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −40.0000 −2.04926
\(382\) 3.00000 0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 1.00000 0.0508329
\(388\) −17.0000 −0.863044
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −16.0000 −0.802008
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 8.00000 0.399004
\(403\) 20.0000 0.996271
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −3.00000 −0.148704
\(408\) −6.00000 −0.297044
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −26.0000 −1.27323
\(418\) 6.00000 0.293470
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 5.00000 0.243396
\(423\) −12.0000 −0.583460
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) −1.00000 −0.0483934
\(428\) −6.00000 −0.290021
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) −4.00000 −0.192450
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 5.00000 0.240008
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) −12.0000 −0.574038
\(438\) 32.0000 1.52902
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −12.0000 −0.570782
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −17.0000 −0.804973
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) −9.00000 −0.423324
\(453\) 16.0000 0.751746
\(454\) 27.0000 1.26717
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 14.0000 0.654177
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 6.00000 0.279145
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 0 0
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 4.00000 0.184900
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) −3.00000 −0.137361
\(478\) −9.00000 −0.411650
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −28.0000 −1.27537
\(483\) −12.0000 −0.546019
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) −9.00000 −0.405340
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 6.00000 0.269137
\(498\) 24.0000 1.07547
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 18.0000 0.803379
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 6.00000 0.266469
\(508\) −20.0000 −0.887357
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) −36.0000 −1.58328
\(518\) −1.00000 −0.0439375
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 3.00000 0.131306
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) −15.0000 −0.653410
\(528\) 6.00000 0.261116
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) 12.0000 0.519778
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 2.00000 0.0859074
\(543\) 4.00000 0.171656
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) −12.0000 −0.512615
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) −12.0000 −0.510754
\(553\) 8.00000 0.340195
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 5.00000 0.211667
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 24.0000 1.01238
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) −11.0000 −0.461957
\(568\) 6.00000 0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 12.0000 0.501745
\(573\) 6.00000 0.250654
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −8.00000 −0.332756
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) −34.0000 −1.40935
\(583\) −9.00000 −0.372742
\(584\) 16.0000 0.662085
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) −12.0000 −0.494872
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −1.00000 −0.0410997
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −32.0000 −1.30967
\(598\) −24.0000 −0.981433
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 1.00000 0.0407570
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 2.00000 0.0811107
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) −3.00000 −0.121268
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 8.00000 0.321807
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) −9.00000 −0.360867
\(623\) −6.00000 −0.240385
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 12.0000 0.479234
\(628\) 13.0000 0.518756
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 8.00000 0.318223
\(633\) 10.0000 0.397464
\(634\) 21.0000 0.834017
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −24.0000 −0.950915
\(638\) 9.00000 0.356313
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) −12.0000 −0.473602
\(643\) 13.0000 0.512670 0.256335 0.966588i \(-0.417485\pi\)
0.256335 + 0.966588i \(0.417485\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) −11.0000 −0.430793
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 22.0000 0.860268
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 16.0000 0.624219
\(658\) −12.0000 −0.467809
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 26.0000 1.01052
\(663\) −24.0000 −0.932083
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −18.0000 −0.696963
\(668\) −12.0000 −0.464294
\(669\) −34.0000 −1.31452
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 2.00000 0.0771517
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 4.00000 0.154074
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −18.0000 −0.691286
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) 15.0000 0.574380
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 28.0000 1.06827
\(688\) 1.00000 0.0381246
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −19.0000 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) −15.0000 −0.570214
\(693\) 3.00000 0.113961
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −9.00000 −0.340899
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −16.0000 −0.603881
\(703\) −2.00000 −0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) −30.0000 −1.12351
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) −36.0000 −1.34351
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −15.0000 −0.558242
\(723\) −56.0000 −2.08266
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) −2.00000 −0.0739221
\(733\) 31.0000 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) −35.0000 −1.29187
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 12.0000 0.442026
\(738\) 3.00000 0.110432
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) −3.00000 −0.110133
\(743\) 51.0000 1.87101 0.935504 0.353315i \(-0.114946\pi\)
0.935504 + 0.353315i \(0.114946\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 12.0000 0.439057
\(748\) −9.00000 −0.329073
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −12.0000 −0.437595
\(753\) 36.0000 1.31191
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −16.0000 −0.581146
\(759\) −36.0000 −1.30672
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) −40.0000 −1.44905
\(763\) 11.0000 0.398227
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) −14.0000 −0.503871
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) −17.0000 −0.610264
\(777\) −2.00000 −0.0717496
\(778\) 9.00000 0.322666
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 18.0000 0.643679
\(783\) −12.0000 −0.428845
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 6.00000 0.213741
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 3.00000 0.106600
\(793\) −4.00000 −0.142044
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 4.00000 0.141598
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 6.00000 0.211867
\(803\) 48.0000 1.69388
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) −36.0000 −1.26726
\(808\) −6.00000 −0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 3.00000 0.105279
\(813\) 4.00000 0.140286
\(814\) −3.00000 −0.105150
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 2.00000 0.0699711
\(818\) 32.0000 1.11885
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) −24.0000 −0.837096
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −6.00000 −0.208514
\(829\) 47.0000 1.63238 0.816189 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) 0 0
\(831\) 56.0000 1.94262
\(832\) 4.00000 0.138675
\(833\) 18.0000 0.623663
\(834\) −26.0000 −0.900306
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) −20.0000 −0.691301
\(838\) −36.0000 −1.24360
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) 48.0000 1.65321
\(844\) 5.00000 0.172107
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) −2.00000 −0.0687208
\(848\) −3.00000 −0.103020
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 12.0000 0.411113
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −1.00000 −0.0342193
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 24.0000 0.819346
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 15.0000 0.510902
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −16.0000 −0.543388
\(868\) 5.00000 0.169711
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 11.0000 0.372507
\(873\) −17.0000 −0.575363
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 32.0000 1.08118
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 35.0000 1.18119
\(879\) −42.0000 −1.41662
\(880\) 0 0
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) −6.00000 −0.202031
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −30.0000 −1.00787
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) −17.0000 −0.569202
\(893\) −24.0000 −0.803129
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −48.0000 −1.60267
\(898\) 6.00000 0.200223
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 9.00000 0.299667
\(903\) 2.00000 0.0665558
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 27.0000 0.896026
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 4.00000 0.132453
\(913\) 36.0000 1.19143
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 12.0000 0.396059
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) 33.0000 1.08680
\(923\) 24.0000 0.789970
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 4.00000 0.131377
\(928\) 3.00000 0.0984798
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 4.00000 0.130605
\(939\) −52.0000 −1.69696
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 26.0000 0.847126
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 16.0000 0.519656
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) −3.00000 −0.0972306
\(953\) 60.0000 1.94359 0.971795 0.235826i \(-0.0757795\pi\)
0.971795 + 0.235826i \(0.0757795\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) 18.0000 0.581857
\(958\) 24.0000 0.775405
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −4.00000 −0.128965
\(963\) −6.00000 −0.193347
\(964\) −28.0000 −0.901819
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 57.0000 1.82922 0.914609 0.404341i \(-0.132499\pi\)
0.914609 + 0.404341i \(0.132499\pi\)
\(972\) −10.0000 −0.320750
\(973\) −13.0000 −0.416761
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) −22.0000 −0.703482
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) −12.0000 −0.382935
\(983\) −51.0000 −1.62665 −0.813324 0.581811i \(-0.802344\pi\)
−0.813324 + 0.581811i \(0.802344\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) −24.0000 −0.763928
\(988\) 8.00000 0.254514
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 5.00000 0.158750
\(993\) 52.0000 1.65017
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 24.0000 0.760469
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 14.0000 0.443162
\(999\) 4.00000 0.126554
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 1850.2.a.o.1.1 1
5.2 odd 4 1850.2.b.g.149.2 2
5.3 odd 4 1850.2.b.g.149.1 2
5.4 even 2 370.2.a.a.1.1 1
15.14 odd 2 3330.2.a.v.1.1 1
20.19 odd 2 2960.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.a.1.1 1 5.4 even 2
1850.2.a.o.1.1 1 1.1 even 1 trivial
1850.2.b.g.149.1 2 5.3 odd 4
1850.2.b.g.149.2 2 5.2 odd 4
2960.2.a.j.1.1 1 20.19 odd 2
3330.2.a.v.1.1 1 15.14 odd 2