Properties

Label 1850.2.b.g.149.2
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.g.149.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000i 0.577350i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000i 0.639602i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 4.00000i − 0.769800i
\(28\) − 1.00000i − 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 6.00000i − 1.04447i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 1.00000i − 0.164399i
\(38\) − 2.00000i − 0.324443i
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 4.00000i 0.554700i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.00000i 0.529813i
\(58\) − 3.00000i − 0.393919i
\(59\) 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 \(0.520392\pi\)
\(62\) 5.00000i 0.635001i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 16.0000i − 1.87266i −0.351123 0.936329i \(-0.614200\pi\)
0.351123 0.936329i \(-0.385800\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000i 0.341882i
\(78\) − 8.00000i − 0.905822i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000i 0.331295i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 6.00000i 0.643268i
\(88\) − 3.00000i − 0.319801i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 6.00000i − 0.625543i
\(93\) − 10.0000i − 1.03695i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) − 17.0000i − 1.72609i −0.505128 0.863044i \(-0.668555\pi\)
0.505128 0.863044i \(-0.331445\pi\)
\(98\) 6.00000i 0.606092i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000i 0.0944911i
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 1.00000i − 0.0905357i
\(123\) − 6.00000i − 0.541002i
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 20.0000i − 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000i 0.522233i
\(133\) − 2.00000i − 0.173422i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 12.0000i 1.02151i
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 6.00000i 0.503509i
\(143\) − 12.0000i − 1.00349i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) − 12.0000i − 0.989743i
\(148\) 1.00000i 0.0821995i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000i 0.162221i
\(153\) 3.00000i 0.242536i
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) 13.0000i 1.03751i 0.854922 + 0.518756i \(0.173605\pi\)
−0.854922 + 0.518756i \(0.826395\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) − 11.0000i − 0.864242i
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 1.00000i 0.0762493i
\(173\) 15.0000i 1.14043i 0.821496 + 0.570214i \(0.193140\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 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 \(0.476318\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 2.00000i 0.147844i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) − 9.00000i − 0.658145i
\(188\) 12.0000i 0.875190i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 2.00000i 0.144338i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 17.0000 1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 6.00000i − 0.422159i
\(203\) − 3.00000i − 0.210559i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 6.00000i − 0.417029i
\(208\) − 4.00000i − 0.277350i
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) − 3.00000i − 0.206041i
\(213\) − 12.0000i − 0.822226i
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 5.00000i 0.339422i
\(218\) − 11.0000i − 0.745014i
\(219\) −32.0000 −2.16236
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) − 2.00000i − 0.134231i
\(223\) 17.0000i 1.13840i 0.822198 + 0.569202i \(0.192748\pi\)
−0.822198 + 0.569202i \(0.807252\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000i 1.79205i 0.444001 + 0.896026i \(0.353559\pi\)
−0.444001 + 0.896026i \(0.646441\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 3.00000i 0.196960i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 3.00000i 0.194461i
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 10.0000i 0.641500i
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 8.00000i 0.509028i
\(248\) − 5.00000i − 0.317500i
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) 18.0000i 1.13165i
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 2.00000i − 0.124515i
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 9.00000i 0.554964i 0.960731 + 0.277482i \(0.0894999\pi\)
−0.960731 + 0.277482i \(0.910500\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) − 12.0000i − 0.734388i
\(268\) − 4.00000i − 0.244339i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) − 8.00000i − 0.484182i
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 13.0000i 0.779688i
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) − 24.0000i − 1.42918i
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 3.00000i 0.177084i
\(288\) − 1.00000i − 0.0589256i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −34.0000 −1.99312
\(292\) 16.0000i 0.936329i
\(293\) 21.0000i 1.22683i 0.789760 + 0.613417i \(0.210205\pi\)
−0.789760 + 0.613417i \(0.789795\pi\)
\(294\) 12.0000 0.699854
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) − 12.0000i − 0.696311i
\(298\) − 6.00000i − 0.347571i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 8.00000i 0.460348i
\(303\) 12.0000i 0.689382i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 34.0000i 1.94048i 0.242140 + 0.970241i \(0.422151\pi\)
−0.242140 + 0.970241i \(0.577849\pi\)
\(308\) − 3.00000i − 0.170941i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 8.00000i 0.452911i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 21.0000i 1.17948i 0.807594 + 0.589739i \(0.200769\pi\)
−0.807594 + 0.589739i \(0.799231\pi\)
\(318\) 6.00000i 0.336463i
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 6.00000i − 0.334367i
\(323\) 6.00000i 0.333849i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −11.0000 −0.609234
\(327\) 22.0000i 1.21660i
\(328\) − 3.00000i − 0.165647i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 1.00000i 0.0547997i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 4.00000i 0.217894i 0.994048 + 0.108947i \(0.0347479\pi\)
−0.994048 + 0.108947i \(0.965252\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 2.00000i 0.108148i
\(343\) 13.0000i 0.701934i
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 3.00000i 0.159901i
\(353\) − 21.0000i − 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) − 6.00000i − 0.317554i
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 2.00000i 0.105118i
\(363\) 4.00000i 0.209946i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 35.0000i − 1.82699i −0.406855 0.913493i \(-0.633375\pi\)
0.406855 0.913493i \(-0.366625\pi\)
\(368\) 6.00000i 0.312772i
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 10.0000i 0.518476i
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 12.0000i 0.618031i
\(378\) 4.00000i 0.205738i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −40.0000 −2.04926
\(382\) 3.00000i 0.153493i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 1.00000i 0.0508329i
\(388\) 17.0000i 0.863044i
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) − 6.00000i − 0.303046i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 16.0000i 0.802008i
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 20.0000i − 0.996271i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) − 3.00000i − 0.148704i
\(408\) 6.00000i 0.297044i
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 4.00000i 0.197066i
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) − 26.0000i − 1.27323i
\(418\) − 6.00000i − 0.293470i
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 5.00000i 0.243396i
\(423\) 12.0000i 0.583460i
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) − 1.00000i − 0.0483934i
\(428\) 6.00000i 0.290021i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) − 12.0000i − 0.574038i
\(438\) − 32.0000i − 1.52902i
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) − 12.0000i − 0.570782i
\(443\) 30.0000i 1.42534i 0.701498 + 0.712672i \(0.252515\pi\)
−0.701498 + 0.712672i \(0.747485\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −17.0000 −0.804973
\(447\) 12.0000i 0.567581i
\(448\) − 1.00000i − 0.0472456i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) − 9.00000i − 0.423324i
\(453\) − 16.0000i − 0.751746i
\(454\) −27.0000 −1.26717
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 17.0000i − 0.795226i −0.917553 0.397613i \(-0.869839\pi\)
0.917553 0.397613i \(-0.130161\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 6.00000i 0.279145i
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 0 0
\(467\) − 27.0000i − 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 0 0
\(473\) − 3.00000i − 0.137940i
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) − 3.00000i − 0.137361i
\(478\) 9.00000i 0.411650i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) − 28.0000i − 1.27537i
\(483\) 12.0000i 0.546019i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 1.00000i 0.0452679i
\(489\) 22.0000 0.994874
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 9.00000i 0.405340i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 6.00000i 0.269137i
\(498\) − 24.0000i − 1.07547i
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 18.0000i 0.803379i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 6.00000i 0.266469i
\(508\) 20.0000i 0.887357i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000i 0.0441942i
\(513\) 8.00000i 0.353209i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 36.0000i − 1.58328i
\(518\) 1.00000i 0.0439375i
\(519\) 30.0000 1.31685
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 3.00000i 0.131306i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) − 15.0000i − 0.653410i
\(528\) − 6.00000i − 0.261116i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000i 0.0867110i
\(533\) − 12.0000i − 0.519778i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 18.0000i 0.776035i
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 2.00000i 0.0859074i
\(543\) − 4.00000i − 0.171656i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) 1.00000i 0.0427569i 0.999771 + 0.0213785i \(0.00680549\pi\)
−0.999771 + 0.0213785i \(0.993195\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) − 12.0000i − 0.510754i
\(553\) − 8.00000i − 0.340195i
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) − 5.00000i − 0.211667i
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 24.0000i 1.01238i
\(563\) − 3.00000i − 0.126435i −0.998000 0.0632175i \(-0.979864\pi\)
0.998000 0.0632175i \(-0.0201362\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) − 11.0000i − 0.461957i
\(568\) − 6.00000i − 0.251754i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 12.0000i 0.501745i
\(573\) − 6.00000i − 0.250654i
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 8.00000i 0.332756i
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) − 34.0000i − 1.40935i
\(583\) 9.00000i 0.372742i
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) − 27.0000i − 1.11441i −0.830375 0.557205i \(-0.811874\pi\)
0.830375 0.557205i \(-0.188126\pi\)
\(588\) 12.0000i 0.494872i
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) − 1.00000i − 0.0410997i
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 12.0000 0.492366
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) − 32.0000i − 1.30967i
\(598\) 24.0000i 0.981433i
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) − 4.00000i − 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) − 2.00000i − 0.0811107i
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) − 3.00000i − 0.121268i
\(613\) 29.0000i 1.17130i 0.810564 + 0.585649i \(0.199160\pi\)
−0.810564 + 0.585649i \(0.800840\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) − 9.00000i − 0.360867i
\(623\) 6.00000i 0.240385i
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 12.0000i 0.479234i
\(628\) − 13.0000i − 0.518756i
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 10.0000i − 0.397464i
\(634\) −21.0000 −0.834017
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) − 24.0000i − 0.950915i
\(638\) − 9.00000i − 0.356313i
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) − 13.0000i − 0.512670i −0.966588 0.256335i \(-0.917485\pi\)
0.966588 0.256335i \(-0.0825150\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 48.0000i 1.88707i 0.331266 + 0.943537i \(0.392524\pi\)
−0.331266 + 0.943537i \(0.607476\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) − 11.0000i − 0.430793i
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) −22.0000 −0.860268
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 16.0000i 0.624219i
\(658\) 12.0000i 0.467809i
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 26.0000i 1.01052i
\(663\) 24.0000i 0.932083i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) − 18.0000i − 0.696963i
\(668\) 12.0000i 0.464294i
\(669\) 34.0000 1.31452
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 2.00000i 0.0771517i
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 18.0000i 0.691286i
\(679\) 17.0000 0.652400
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) 15.0000i 0.574380i
\(683\) − 15.0000i − 0.573959i −0.957937 0.286980i \(-0.907349\pi\)
0.957937 0.286980i \(-0.0926512\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 28.0000i 1.06827i
\(688\) − 1.00000i − 0.0381246i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −19.0000 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) − 15.0000i − 0.570214i
\(693\) − 3.00000i − 0.113961i
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 9.00000i − 0.340899i
\(698\) − 26.0000i − 0.984115i
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) − 16.0000i − 0.603881i
\(703\) 2.00000i 0.0754314i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) − 6.00000i − 0.224860i
\(713\) 30.0000i 1.12351i
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) − 18.0000i − 0.672222i
\(718\) 36.0000i 1.34351i
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) − 15.0000i − 0.558242i
\(723\) 56.0000i 2.08266i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) − 4.00000i − 0.148250i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) − 2.00000i − 0.0739221i
\(733\) − 31.0000i − 1.14501i −0.819901 0.572506i \(-0.805971\pi\)
0.819901 0.572506i \(-0.194029\pi\)
\(734\) 35.0000 1.29187
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 12.0000i 0.442026i
\(738\) − 3.00000i − 0.110432i
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) − 3.00000i − 0.110133i
\(743\) − 51.0000i − 1.87101i −0.353315 0.935504i \(-0.614946\pi\)
0.353315 0.935504i \(-0.385054\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 12.0000i 0.439057i
\(748\) 9.00000i 0.329073i
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) − 36.0000i − 1.31191i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 34.0000i 1.23575i 0.786276 + 0.617876i \(0.212006\pi\)
−0.786276 + 0.617876i \(0.787994\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) − 40.0000i − 1.44905i
\(763\) − 11.0000i − 0.398227i
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) − 2.00000i − 0.0721688i
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) − 14.0000i − 0.503871i
\(773\) − 39.0000i − 1.40273i −0.712801 0.701366i \(-0.752574\pi\)
0.712801 0.701366i \(-0.247426\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −17.0000 −0.610264
\(777\) − 2.00000i − 0.0717496i
\(778\) − 9.00000i − 0.322666i
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 18.0000i 0.643679i
\(783\) 12.0000i 0.428845i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 3.00000i 0.106600i
\(793\) 4.00000i 0.142044i
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 6.00000i 0.211867i
\(803\) − 48.0000i − 1.69388i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) − 36.0000i − 1.26726i
\(808\) 6.00000i 0.211079i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 3.00000i 0.105279i
\(813\) − 4.00000i − 0.140286i
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 2.00000i 0.0699711i
\(818\) − 32.0000i − 1.11885i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) − 24.0000i − 0.837096i
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000i 0.104320i 0.998639 + 0.0521601i \(0.0166106\pi\)
−0.998639 + 0.0521601i \(0.983389\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −47.0000 −1.63238 −0.816189 0.577785i \(-0.803917\pi\)
−0.816189 + 0.577785i \(0.803917\pi\)
\(830\) 0 0
\(831\) 56.0000 1.94262
\(832\) 4.00000i 0.138675i
\(833\) − 18.0000i − 0.623663i
\(834\) 26.0000 0.900306
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) − 20.0000i − 0.691301i
\(838\) 36.0000i 1.24360i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) − 10.0000i − 0.344623i
\(843\) − 48.0000i − 1.65321i
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) − 2.00000i − 0.0687208i
\(848\) 3.00000i 0.103020i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 12.0000i 0.411113i
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) 15.0000i 0.512390i 0.966625 + 0.256195i \(0.0824690\pi\)
−0.966625 + 0.256195i \(0.917531\pi\)
\(858\) − 24.0000i − 0.819346i
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 15.0000i 0.510902i
\(863\) 39.0000i 1.32758i 0.747921 + 0.663788i \(0.231052\pi\)
−0.747921 + 0.663788i \(0.768948\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) − 16.0000i − 0.543388i
\(868\) − 5.00000i − 0.169711i
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 11.0000i 0.372507i
\(873\) 17.0000i 0.575363i
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 32.0000 1.08118
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) − 35.0000i − 1.18119i
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) − 6.00000i − 0.202031i
\(883\) 29.0000i 0.975928i 0.872864 + 0.487964i \(0.162260\pi\)
−0.872864 + 0.487964i \(0.837740\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −30.0000 −1.00787
\(887\) − 33.0000i − 1.10803i −0.832506 0.554016i \(-0.813095\pi\)
0.832506 0.554016i \(-0.186905\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) − 17.0000i − 0.569202i
\(893\) 24.0000i 0.803129i
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 48.0000i − 1.60267i
\(898\) − 6.00000i − 0.200223i
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 9.00000i 0.299667i
\(903\) − 2.00000i − 0.0665558i
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) − 27.0000i − 0.896026i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 4.00000i 0.132453i
\(913\) − 36.0000i − 1.19143i
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) − 12.0000i − 0.396059i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) 33.0000i 1.08680i
\(923\) − 24.0000i − 0.789970i
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 4.00000i 0.131377i
\(928\) − 3.00000i − 0.0984798i
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 18.0000i 0.589294i
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 52.0000 1.69696
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 26.0000i 0.847126i
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) − 33.0000i − 1.07236i −0.844105 0.536178i \(-0.819868\pi\)
0.844105 0.536178i \(-0.180132\pi\)
\(948\) − 16.0000i − 0.519656i
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) − 3.00000i − 0.0972306i
\(953\) − 60.0000i − 1.94359i −0.235826 0.971795i \(-0.575780\pi\)
0.235826 0.971795i \(-0.424220\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) 18.0000i 0.581857i
\(958\) − 24.0000i − 0.775405i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) − 4.00000i − 0.128965i
\(963\) 6.00000i 0.193347i
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 40.0000i 1.28631i 0.765735 + 0.643157i \(0.222376\pi\)
−0.765735 + 0.643157i \(0.777624\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 57.0000 1.82922 0.914609 0.404341i \(-0.132499\pi\)
0.914609 + 0.404341i \(0.132499\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 13.0000i 0.416761i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 45.0000i 1.43968i 0.694141 + 0.719839i \(0.255784\pi\)
−0.694141 + 0.719839i \(0.744216\pi\)
\(978\) 22.0000i 0.703482i
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) − 12.0000i − 0.382935i
\(983\) 51.0000i 1.62665i 0.581811 + 0.813324i \(0.302344\pi\)
−0.581811 + 0.813324i \(0.697656\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) − 24.0000i − 0.763928i
\(988\) − 8.00000i − 0.254514i
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 5.00000i 0.158750i
\(993\) − 52.0000i − 1.65017i
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) 24.0000 0.760469
\(997\) 28.0000i 0.886769i 0.896332 + 0.443384i \(0.146222\pi\)
−0.896332 + 0.443384i \(0.853778\pi\)
\(998\) − 14.0000i − 0.443162i
\(999\) −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.b.g.149.2 2
5.2 odd 4 370.2.a.a.1.1 1
5.3 odd 4 1850.2.a.o.1.1 1
5.4 even 2 inner 1850.2.b.g.149.1 2
15.2 even 4 3330.2.a.v.1.1 1
20.7 even 4 2960.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.a.1.1 1 5.2 odd 4
1850.2.a.o.1.1 1 5.3 odd 4
1850.2.b.g.149.1 2 5.4 even 2 inner
1850.2.b.g.149.2 2 1.1 even 1 trivial
2960.2.a.j.1.1 1 20.7 even 4
3330.2.a.v.1.1 1 15.2 even 4