Properties

Label 2850.2.d.c.799.1
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 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} +8.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{42} -8.00000i q^{43} +8.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} +1.00000i q^{57} +2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +12.0000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{84} -8.00000 q^{86} +2.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} -8.00000i q^{92} +8.00000i q^{93} -1.00000 q^{96} +10.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{14} + 2q^{16} - 2q^{19} - 8q^{21} + 2q^{24} - 12q^{26} - 4q^{29} - 16q^{31} - 4q^{34} + 2q^{36} - 12q^{39} + 12q^{41} + 16q^{46} - 18q^{49} - 4q^{51} + 2q^{54} + 8q^{56} + 24q^{59} - 4q^{61} - 2q^{64} + 16q^{69} - 16q^{71} - 20q^{74} + 2q^{76} + 32q^{79} + 2q^{81} + 8q^{84} - 16q^{86} - 12q^{89} - 48q^{91} - 2q^{96} + O(q^{100}) \)

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
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000i 0.617213i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 6.00000i 0.832050i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 1.00000i 0.132453i
\(58\) 2.00000i 0.262613i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) − 8.00000i − 0.834058i
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 20.0000i − 1.93347i −0.255774 0.966736i \(-0.582330\pi\)
0.255774 0.966736i \(-0.417670\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) − 4.00000i − 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 6.00000i 0.554700i
\(118\) − 12.0000i − 1.10469i
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000i 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 9.00000i 0.742307i
\(148\) 10.0000i 0.821995i
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 16.0000i − 1.27289i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 32.0000 2.52195
\(162\) − 1.00000i − 0.0785674i
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 8.00000i 0.609994i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 6.00000i 0.449719i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 24.0000i 1.77900i
\(183\) 2.00000i 0.147844i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 18.0000i 1.26648i
\(203\) 8.00000i 0.561490i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) − 8.00000i − 0.556038i
\(208\) − 6.00000i − 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 8.00000i 0.548151i
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 32.0000i 2.17230i
\(218\) 6.00000i 0.406371i
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 10.0000i 0.671156i
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 2.00000i − 0.131306i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 16.0000i − 1.03931i
\(238\) 8.00000i 0.518563i
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 11.0000i 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 6.00000i 0.381771i
\(248\) − 8.00000i − 0.508001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 6.00000i 0.367194i
\(268\) − 12.0000i − 0.733017i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 24.0000i 1.45255i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) − 24.0000i − 1.41668i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 2.00000i − 0.117041i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) − 2.00000i − 0.115857i
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 18.0000i 1.03407i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 26.0000i 1.46031i 0.683284 + 0.730153i \(0.260551\pi\)
−0.683284 + 0.730153i \(0.739449\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 0 0
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) − 32.0000i − 1.78329i
\(323\) 2.00000i 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) 6.00000i 0.331801i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 10.0000i 0.547997i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) − 1.00000i − 0.0540738i
\(343\) 8.00000i 0.431959i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 8.00000i 0.423405i
\(358\) − 20.0000i − 1.05703i
\(359\) −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
\(362\) 18.0000i 0.946059i
\(363\) 11.0000i 0.577350i
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 8.00000i 0.417029i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) − 8.00000i − 0.414781i
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) − 4.00000i − 0.205738i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 12.0000i 0.613973i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 8.00000i 0.406663i
\(388\) − 10.0000i − 0.507673i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) − 9.00000i − 0.454569i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) − 38.0000i − 1.90717i −0.301131 0.953583i \(-0.597364\pi\)
0.301131 0.953583i \(-0.402636\pi\)
\(398\) 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 \(0.452133\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) 48.0000i 2.39105i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) − 2.00000i − 0.0990148i
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) − 8.00000i − 0.394132i
\(413\) − 48.0000i − 2.36193i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) − 12.0000i − 0.587643i
\(418\) 0 0
\(419\) 40.0000 1.95413 0.977064 0.212946i \(-0.0683059\pi\)
0.977064 + 0.212946i \(0.0683059\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 8.00000i 0.387147i
\(428\) 20.0000i 0.966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) − 8.00000i − 0.382692i
\(438\) − 2.00000i − 0.0955637i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 12.0000i 0.570782i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 2.00000i − 0.0945968i
\(448\) 4.00000i 0.188982i
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 6.00000i 0.280362i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 12.0000i 0.552345i
\(473\) 0 0
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 2.00000i 0.0915737i
\(478\) − 20.0000i − 0.914779i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 14.0000i 0.637683i
\(483\) − 32.0000i − 1.45605i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 4.00000i 0.180151i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 32.0000i 1.43540i
\(498\) 4.00000i 0.179244i
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) − 8.00000i − 0.354943i
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 40.0000i 1.75750i
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 44.0000i − 1.92399i −0.273075 0.961993i \(-0.588041\pi\)
0.273075 0.961993i \(-0.411959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) − 4.00000i − 0.173422i
\(533\) − 36.0000i − 1.55933i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 20.0000i − 0.863064i
\(538\) 18.0000i 0.776035i
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 18.0000i 0.772454i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 8.00000i 0.340503i
\(553\) − 64.0000i − 2.72156i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000i 0.0843649i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) − 4.00000i − 0.167984i
\(568\) − 8.00000i − 0.335673i
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) − 10.0000i − 0.414513i
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 20.0000i 0.825488i 0.910847 + 0.412744i \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 10.0000i − 0.410997i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 16.0000i 0.654836i
\(598\) − 48.0000i − 1.96287i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 32.0000i 1.30422i
\(603\) − 12.0000i − 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) − 26.0000i − 1.04672i −0.852111 0.523360i \(-0.824678\pi\)
0.852111 0.523360i \(-0.175322\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) − 12.0000i − 0.481156i
\(623\) 24.0000i 0.961540i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 16.0000i 0.636446i
\(633\) − 20.0000i − 0.794929i
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 54.0000i 2.13956i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 20.0000i 0.789337i
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) −32.0000 −1.26098
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) − 32.0000i − 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 32.0000 1.25418
\(652\) − 8.00000i − 0.313304i
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) − 20.0000i − 0.777322i
\(663\) 12.0000i 0.466041i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) − 16.0000i − 0.619522i
\(668\) 8.00000i 0.309529i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000i 0.154303i
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) − 18.0000i − 0.691286i
\(679\) 40.0000 1.53506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 6.00000i 0.228914i
\(688\) − 8.00000i − 0.304997i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) − 12.0000i − 0.454532i
\(698\) 14.0000i 0.529908i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) 10.0000i 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 72.0000i 2.70784i
\(708\) 12.0000i 0.450988i
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) − 6.00000i − 0.224860i
\(713\) − 64.0000i − 2.39682i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) − 20.0000i − 0.746914i
\(718\) 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
\(721\) 32.0000 1.19174
\(722\) − 1.00000i − 0.0372161i
\(723\) 14.0000i 0.520666i
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) − 12.0000i − 0.445055i −0.974926 0.222528i \(-0.928569\pi\)
0.974926 0.222528i \(-0.0714308\pi\)
\(728\) − 24.0000i − 0.889499i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) − 2.00000i − 0.0739221i
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 6.00000i 0.220863i
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 8.00000i 0.293689i
\(743\) − 32.0000i − 1.17397i −0.809599 0.586983i \(-0.800316\pi\)
0.809599 0.586983i \(-0.199684\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 34.0000 1.24483
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) −80.0000 −2.92314
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) 24.0000i 0.868858i
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) − 72.0000i − 2.59977i
\(768\) − 1.00000i − 0.0360844i
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000i 0.0719816i
\(773\) 46.0000i 1.65451i 0.561830 + 0.827253i \(0.310097\pi\)
−0.561830 + 0.827253i \(0.689903\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 40.0000i 1.43499i
\(778\) 6.00000i 0.215110i
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) − 16.0000i − 0.572159i
\(783\) − 2.00000i − 0.0714742i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) −38.0000 −1.34857
\(795\) 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
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) − 6.00000i − 0.211867i
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) 18.0000i 0.633630i
\(808\) − 18.0000i − 0.633238i
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) − 8.00000i − 0.280745i
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 8.00000i 0.279885i
\(818\) 18.0000i 0.629355i
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) 44.0000i 1.53374i 0.641800 + 0.766872i \(0.278188\pi\)
−0.641800 + 0.766872i \(0.721812\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 52.0000i 1.80822i 0.427303 + 0.904109i \(0.359464\pi\)
−0.427303 + 0.904109i \(0.640536\pi\)
\(828\) 8.00000i 0.278019i
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 6.00000i 0.208013i
\(833\) 18.0000i 0.623663i
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) − 40.0000i − 1.38178i
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 22.0000i − 0.758170i
\(843\) 2.00000i 0.0688837i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) − 2.00000i − 0.0686803i
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) − 8.00000i − 0.274075i
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) − 24.0000i − 0.817443i
\(863\) 8.00000i 0.272323i 0.990687 + 0.136162i \(0.0434766\pi\)
−0.990687 + 0.136162i \(0.956523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) − 13.0000i − 0.441503i
\(868\) − 32.0000i − 1.08615i
\(869\) 0 0
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) − 6.00000i − 0.203186i
\(873\) − 10.0000i − 0.338449i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 6.00000i 0.202606i 0.994856 + 0.101303i \(0.0323011\pi\)
−0.994856 + 0.101303i \(0.967699\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 16.0000i 0.537227i 0.963248 + 0.268614i \(0.0865655\pi\)
−0.963248 + 0.268614i \(0.913434\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000i 0.535720i
\(893\) 0 0
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) − 48.0000i − 1.60267i
\(898\) 30.0000i 1.00111i
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 32.0000i 1.06489i
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 60.0000i 1.99227i 0.0878507 + 0.996134i \(0.472000\pi\)
−0.0878507 + 0.996134i \(0.528000\pi\)
\(908\) 20.0000i 0.663723i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) − 2.00000i − 0.0660098i
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) − 6.00000i − 0.197599i
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 8.00000i − 0.262754i
\(928\) 2.00000i 0.0656532i
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) 6.00000i 0.196537i
\(933\) − 12.0000i − 0.392862i
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) − 48.0000i − 1.56726i
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) − 10.0000i − 0.325818i
\(943\) 48.0000i 1.56310i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 44.0000i − 1.42981i −0.699223 0.714904i \(-0.746470\pi\)
0.699223 0.714904i \(-0.253530\pi\)
\(948\) 16.0000i 0.519656i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) − 8.00000i − 0.259281i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 36.0000i 1.16311i
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 60.0000i 1.93448i
\(963\) 20.0000i 0.644491i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) −32.0000 −1.02958
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 48.0000i − 1.53881i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) 0 0
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 40.0000i 1.27645i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) − 6.00000i − 0.190885i
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 20.0000i − 0.634681i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) − 14.0000i − 0.443384i −0.975117 0.221692i \(-0.928842\pi\)
0.975117 0.221692i \(-0.0711580\pi\)
\(998\) − 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.c.799.1 2
5.2 odd 4 570.2.a.i.1.1 1
5.3 odd 4 2850.2.a.h.1.1 1
5.4 even 2 inner 2850.2.d.c.799.2 2
15.2 even 4 1710.2.a.e.1.1 1
15.8 even 4 8550.2.a.s.1.1 1
20.7 even 4 4560.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.i.1.1 1 5.2 odd 4
1710.2.a.e.1.1 1 15.2 even 4
2850.2.a.h.1.1 1 5.3 odd 4
2850.2.d.c.799.1 2 1.1 even 1 trivial
2850.2.d.c.799.2 2 5.4 even 2 inner
4560.2.a.x.1.1 1 20.7 even 4
8550.2.a.s.1.1 1 15.8 even 4