Properties

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

$q$-expansion

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

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 \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 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 \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 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 \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) −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 \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 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 \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) 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 \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 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 \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 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 \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −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 \(0.331445\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) − 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 \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) − 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 \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) −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 \(0.347461\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 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 \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 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 \(-0.826395\pi\)
0.854922 0.518756i \(-0.173605\pi\)
\(158\) 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 \(-0.858234\pi\)
0.902451 0.430793i \(-0.141766\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 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 \(-0.806860\pi\)
0.821496 0.570214i \(-0.193140\pi\)
\(174\) −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 \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 17.0000 1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 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 \(-0.807252\pi\)
0.822198 0.569202i \(-0.192748\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) − 27.0000i − 1.79205i −0.444001 0.896026i \(-0.646441\pi\)
0.444001 0.896026i \(-0.353559\pi\)
\(228\) 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 \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 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 \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) −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 \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) − 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 \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) −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 \(-0.789795\pi\)
0.789760 0.613417i \(-0.210205\pi\)
\(294\) 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 \(-0.577849\pi\)
0.242140 0.970241i \(-0.422151\pi\)
\(308\) 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 \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 21.0000i − 1.17948i −0.807594 0.589739i \(-0.799231\pi\)
0.807594 0.589739i \(-0.200769\pi\)
\(318\) − 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 \(-0.965252\pi\)
0.994048 0.108947i \(-0.0347479\pi\)
\(338\) 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 \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 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 \(0.188761\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(354\) 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 \(0.366625\pi\)
−0.406855 + 0.913493i \(0.633375\pi\)
\(368\) − 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 \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 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 \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) −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 \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) − 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 \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) −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 \(-0.747485\pi\)
0.701498 0.712672i \(-0.252515\pi\)
\(444\) 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 \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) 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 \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 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 \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) − 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 \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) −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 \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 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 \(-0.993195\pi\)
0.999771 0.0213785i \(-0.00680549\pi\)
\(548\) − 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 \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 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 \(0.0201362\pi\)
−0.998000 + 0.0632175i \(0.979864\pi\)
\(564\) 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 \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) − 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 \(0.188126\pi\)
−0.830375 + 0.557205i \(0.811874\pi\)
\(588\) − 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 \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) 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 \(-0.852679\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(608\) 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 \(-0.800840\pi\)
0.810564 0.585649i \(-0.199160\pi\)
\(614\) −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 \(0.0825150\pi\)
−0.966588 + 0.256335i \(0.917485\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) − 48.0000i − 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) − 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 \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) −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 \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 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 \(0.0926512\pi\)
−0.957937 + 0.286980i \(0.907349\pi\)
\(684\) −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 \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 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 \(0.194029\pi\)
−0.819901 + 0.572506i \(0.805971\pi\)
\(734\) 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 \(0.385054\pi\)
−0.353315 + 0.935504i \(0.614946\pi\)
\(744\) −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 \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) − 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 \(0.247426\pi\)
−0.712801 + 0.701366i \(0.752574\pi\)
\(774\) −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 \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 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 \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 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 \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.00000i − 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) − 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 \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) − 15.0000i − 0.512390i −0.966625 0.256195i \(-0.917531\pi\)
0.966625 0.256195i \(-0.0824690\pi\)
\(858\) 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 \(-0.768948\pi\)
0.747921 0.663788i \(-0.231052\pi\)
\(864\) 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 \(-0.929561\pi\)
0.975615 0.219489i \(-0.0704391\pi\)
\(878\) 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 \(-0.837740\pi\)
0.872864 0.487964i \(-0.162260\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −30.0000 −1.00787
\(887\) 33.0000i 1.10803i 0.832506 + 0.554016i \(0.186905\pi\)
−0.832506 + 0.554016i \(0.813095\pi\)
\(888\) − 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 \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 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 \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 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 \(0.180132\pi\)
−0.844105 + 0.536178i \(0.819868\pi\)
\(948\) 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 \(0.424220\pi\)
−0.235826 + 0.971795i \(0.575780\pi\)
\(954\) 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 \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) − 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 \(-0.744216\pi\)
0.694141 0.719839i \(-0.255784\pi\)
\(978\) − 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 \(-0.697656\pi\)
0.581811 0.813324i \(-0.302344\pi\)
\(984\) −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 \(-0.853778\pi\)
0.896332 0.443384i \(-0.146222\pi\)
\(998\) 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.149.1 2
5.2 odd 4 1850.2.a.o.1.1 1
5.3 odd 4 370.2.a.a.1.1 1
5.4 even 2 inner 1850.2.b.g.149.2 2
15.8 even 4 3330.2.a.v.1.1 1
20.3 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 5.3 odd 4
1850.2.a.o.1.1 1 5.2 odd 4
1850.2.b.g.149.1 2 1.1 even 1 trivial
1850.2.b.g.149.2 2 5.4 even 2 inner
2960.2.a.j.1.1 1 20.3 even 4
3330.2.a.v.1.1 1 15.8 even 4