Properties

Label 2850.2.d.b.799.2
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
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 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.b.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} +1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +2.00000i q^{52} +10.0000i q^{53} +1.00000 q^{54} +1.00000i q^{57} +2.00000i q^{58} -12.0000 q^{59} +14.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} -10.0000 q^{74} -1.00000 q^{76} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} -4.00000i q^{92} +4.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} + 2q^{19} + 2q^{24} + 4q^{26} + 4q^{29} + 8q^{31} + 12q^{34} + 2q^{36} + 4q^{39} + 20q^{41} + 8q^{44} - 8q^{46} + 14q^{49} + 12q^{51} + 2q^{54} - 24q^{59} + 28q^{61} - 2q^{64} + 8q^{66} - 8q^{69} + 16q^{71} - 20q^{74} - 2q^{76} + 8q^{79} + 2q^{81} + 8q^{86} + 12q^{89} + 8q^{94} - 2q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 2.00000i 0.277350i
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 2.00000i 0.262613i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.00000i 0.226455i
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) 4.00000i 0.414781i
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000i 0.184900i
\(118\) − 12.0000i − 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 14.0000i 1.26750i
\(123\) 10.0000i 0.901670i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 8.00000i 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 7.00000i 0.577350i
\(148\) − 10.0000i − 0.821995i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 4.00000i 0.318223i
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 4.00000i 0.304997i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 12.0000i − 0.901975i
\(178\) 6.00000i 0.449719i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 24.0000i 1.75505i
\(188\) 4.00000i 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 2.00000i 0.140720i
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) − 4.00000i − 0.278019i
\(208\) − 2.00000i − 0.138675i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) 8.00000i 0.548151i
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000i 0.406371i
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) − 10.0000i − 0.671156i
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 28.0000i 1.85843i 0.369546 + 0.929213i \(0.379513\pi\)
−0.369546 + 0.929213i \(0.620487\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 2.00000i − 0.131306i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) − 2.00000i − 0.127257i
\(248\) − 4.00000i − 0.254000i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) − 16.0000i − 1.00591i
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 12.0000i 0.741362i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 12.0000i 0.733017i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 6.00000i − 0.351123i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 4.00000i 0.232104i
\(298\) 6.00000i 0.347571i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 20.0000i 1.15087i
\(303\) 2.00000i 0.114897i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 6.00000i 0.331801i
\(328\) − 10.0000i − 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) − 1.00000i − 0.0540738i
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 4.00000i − 0.213201i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 14.0000i − 0.735824i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) − 4.00000i − 0.207390i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) − 4.00000i − 0.206010i
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 4.00000i 0.204658i
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 4.00000i 0.203331i
\(388\) − 10.0000i − 0.507673i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) − 7.00000i − 0.353553i
\(393\) 12.0000i 0.605320i
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 8.00000i − 0.398508i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) − 40.0000i − 1.98273i
\(408\) − 6.00000i − 0.297044i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) − 12.0000i − 0.591198i
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 12.0000i − 0.587643i
\(418\) − 4.00000i − 0.195646i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 4.00000i 0.194487i
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 4.00000i 0.193347i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 4.00000i 0.191346i
\(438\) − 6.00000i − 0.286691i
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 12.0000i − 0.570782i
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 2.00000i 0.0940721i
\(453\) 20.0000i 0.939682i
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) 10.0000i 0.467269i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 20.0000i − 0.925490i −0.886492 0.462745i \(-0.846865\pi\)
0.886492 0.462745i \(-0.153135\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 12.0000i 0.552345i
\(473\) 16.0000i 0.735681i
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) − 10.0000i − 0.457869i
\(478\) − 12.0000i − 0.548867i
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) − 12.0000i − 0.540453i
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 28.0000i − 1.24970i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 9.00000i 0.399704i
\(508\) 12.0000i 0.532414i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) − 24.0000i − 1.04546i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 12.0000i − 0.517838i
\(538\) − 6.00000i − 0.258678i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) − 14.0000i − 0.600798i
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 14.0000i 0.598050i
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 34.0000i 1.44063i 0.693649 + 0.720313i \(0.256002\pi\)
−0.693649 + 0.720313i \(0.743998\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 10.0000i 0.421825i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) − 8.00000i − 0.335673i
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) 4.00000i 0.167102i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 0 0
\(582\) − 10.0000i − 0.414513i
\(583\) − 40.0000i − 1.65663i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 10.0000i 0.410997i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 16.0000i 0.654836i
\(598\) 8.00000i 0.327144i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) − 4.00000i − 0.162355i −0.996700 0.0811775i \(-0.974132\pi\)
0.996700 0.0811775i \(-0.0258681\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) − 6.00000i − 0.242536i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 4.00000i 0.160385i
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) − 4.00000i − 0.159745i
\(628\) − 22.0000i − 0.877896i
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) 12.0000i 0.476957i
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) − 14.0000i − 0.554700i
\(638\) − 8.00000i − 0.316723i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000i 0.157867i
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 4.00000i − 0.157256i −0.996904 0.0786281i \(-0.974946\pi\)
0.996904 0.0786281i \(-0.0250540\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000i 0.783260i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) − 12.0000i − 0.466041i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 8.00000i 0.309761i
\(668\) 0 0
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 26.0000i − 0.999261i −0.866239 0.499631i \(-0.833469\pi\)
0.866239 0.499631i \(-0.166531\pi\)
\(678\) 2.00000i 0.0768095i
\(679\) 0 0
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) − 16.0000i − 0.612672i
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) − 4.00000i − 0.152499i
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) − 60.0000i − 2.27266i
\(698\) 26.0000i 0.984115i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 10.0000i 0.377157i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) − 6.00000i − 0.224860i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 12.0000i − 0.448148i
\(718\) − 12.0000i − 0.447836i
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000i 0.0372161i
\(723\) 10.0000i 0.371904i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) − 14.0000i − 0.517455i
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000i 1.76810i
\(738\) − 10.0000i − 0.368105i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) − 32.0000i − 1.17397i −0.809599 0.586983i \(-0.800316\pi\)
0.809599 0.586983i \(-0.199684\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 12.0000i 0.439057i
\(748\) − 24.0000i − 0.877527i
\(749\) 0 0
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) − 4.00000i − 0.145865i
\(753\) − 28.0000i − 1.02038i
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 36.0000i 1.30758i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 12.0000i 0.434714i
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 24.0000i 0.866590i
\(768\) 1.00000i 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) − 6.00000i − 0.215945i
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) − 18.0000i − 0.645331i
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 24.0000i 0.858238i
\(783\) − 2.00000i − 0.0714742i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) 22.0000i 0.783718i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) − 4.00000i − 0.142134i
\(793\) − 28.0000i − 0.994309i
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 14.0000i − 0.494357i
\(803\) − 24.0000i − 0.846942i
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) − 6.00000i − 0.211210i
\(808\) − 2.00000i − 0.0703598i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) − 4.00000i − 0.139942i
\(818\) 14.0000i 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 14.0000i 0.488306i
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 2.00000i 0.0693375i
\(833\) − 42.0000i − 1.45521i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) − 4.00000i − 0.138260i
\(838\) 12.0000i 0.414533i
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000i 0.896019i
\(843\) 10.0000i 0.344418i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) 10.0000i 0.343401i
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) − 8.00000i − 0.274075i
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) − 6.00000i − 0.203186i
\(873\) − 10.0000i − 0.338449i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 4.00000i 0.134993i
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) − 32.0000i − 1.07445i −0.843437 0.537227i \(-0.819472\pi\)
0.843437 0.537227i \(-0.180528\pi\)
\(888\) 10.0000i 0.335578i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 28.0000i − 0.937509i
\(893\) − 4.00000i − 0.133855i
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000i 0.267112i
\(898\) − 34.0000i − 1.13459i
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) − 28.0000i − 0.929213i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 48.0000i 1.58857i
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) − 6.00000i − 0.198030i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 2.00000i 0.0658665i
\(923\) − 16.0000i − 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) − 12.0000i − 0.394132i
\(928\) 2.00000i 0.0656532i
\(929\) 22.0000 0.721797 0.360898 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) − 6.00000i − 0.196537i
\(933\) 4.00000i 0.130954i
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) − 22.0000i − 0.716799i
\(943\) 40.0000i 1.30258i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) − 4.00000i − 0.129914i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) − 8.00000i − 0.258603i
\(958\) − 4.00000i − 0.129234i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 20.0000i 0.644826i
\(963\) 4.00000i 0.128898i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) − 46.0000i − 1.47167i −0.677161 0.735835i \(-0.736790\pi\)
0.677161 0.735835i \(-0.263210\pi\)
\(978\) 20.0000i 0.639529i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 20.0000i 0.638226i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000i 0.127000i
\(993\) − 4.00000i − 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) 20.0000i 0.633089i
\(999\) 10.0000 0.316386
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 2850.2.d.b.799.2 2
5.2 odd 4 2850.2.a.j.1.1 1
5.3 odd 4 114.2.a.b.1.1 1
5.4 even 2 inner 2850.2.d.b.799.1 2
15.2 even 4 8550.2.a.ba.1.1 1
15.8 even 4 342.2.a.b.1.1 1
20.3 even 4 912.2.a.k.1.1 1
35.13 even 4 5586.2.a.y.1.1 1
40.3 even 4 3648.2.a.c.1.1 1
40.13 odd 4 3648.2.a.x.1.1 1
60.23 odd 4 2736.2.a.d.1.1 1
95.18 even 4 2166.2.a.d.1.1 1
285.113 odd 4 6498.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.b.1.1 1 5.3 odd 4
342.2.a.b.1.1 1 15.8 even 4
912.2.a.k.1.1 1 20.3 even 4
2166.2.a.d.1.1 1 95.18 even 4
2736.2.a.d.1.1 1 60.23 odd 4
2850.2.a.j.1.1 1 5.2 odd 4
2850.2.d.b.799.1 2 5.4 even 2 inner
2850.2.d.b.799.2 2 1.1 even 1 trivial
3648.2.a.c.1.1 1 40.3 even 4
3648.2.a.x.1.1 1 40.13 odd 4
5586.2.a.y.1.1 1 35.13 even 4
6498.2.a.p.1.1 1 285.113 odd 4
8550.2.a.ba.1.1 1 15.2 even 4