Properties

Label 2850.2.d.n.799.1
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \( x^{2} + 1 \) Copy content Toggle raw display
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.n.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -1.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +8.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +2.00000i q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{82} +16.0000i q^{83} -2.00000 q^{84} +6.00000 q^{86} -2.00000i q^{88} -16.0000 q^{89} -8.00000i q^{92} -8.00000 q^{94} +1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} - 4 q^{34} + 2 q^{36} - 16 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} - 2 q^{54} + 4 q^{56} + 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} + 16 q^{71} + 8 q^{74} + 2 q^{76} + 16 q^{79} + 2 q^{81} - 4 q^{84} + 12 q^{86} - 32 q^{89} - 16 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −2.00000 −0.534522
\(15\) 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
\(21\) 2.00000 0.436436
\(22\) 2.00000i 0.426401i
\(23\) 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
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 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
\(56\) 2.00000 0.267261
\(57\) − 1.00000i − 0.132453i
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −8.00000 −0.963087
\(70\) 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
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000i 0.883452i
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) − 2.00000i − 0.213201i
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 8.00000i − 0.834058i
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) − 12.0000i − 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 20.0000i 1.93347i 0.255774 + 0.966736i \(0.417670\pi\)
−0.255774 + 0.966736i \(0.582330\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) − 2.00000i − 0.188982i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) − 8.00000i − 0.736460i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) − 8.00000i − 0.671345i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 3.00000i 0.247436i
\(148\) − 4.00000i − 0.328798i
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 2.00000i 0.161690i
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) − 1.00000i − 0.0785674i
\(163\) 10.0000i 0.783260i 0.920123 + 0.391630i \(0.128089\pi\)
−0.920123 + 0.391630i \(0.871911\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 6.00000i − 0.457496i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 8.00000i 0.601317i
\(178\) 16.0000i 1.19925i
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 8.00000i 0.583460i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 2.00000i − 0.140720i
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) − 8.00000i − 0.556038i
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) 8.00000i 0.548151i
\(214\) 20.0000 1.36717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000i 0.268462i
\(223\) − 12.0000i − 0.803579i −0.915732 0.401790i \(-0.868388\pi\)
0.915732 0.401790i \(-0.131612\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 8.00000i 0.519656i
\(238\) 4.00000i 0.259281i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 0 0
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) − 16.0000i − 1.00591i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.0000i 1.87135i 0.352865 + 0.935674i \(0.385208\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 6.00000i 0.373544i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000i 0.370681i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) − 16.0000i − 0.979184i
\(268\) 0 0
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 14.0000i − 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) 2.00000i 0.118888i 0.998232 + 0.0594438i \(0.0189327\pi\)
−0.998232 + 0.0594438i \(0.981067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 16.0000i 0.944450i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) − 2.00000i − 0.117041i
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 2.00000i 0.116052i
\(298\) 2.00000i 0.115857i
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 8.00000i 0.460348i
\(303\) 2.00000i 0.114897i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 24.0000i − 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) − 18.0000i − 1.01742i −0.860938 0.508710i \(-0.830123\pi\)
0.860938 0.508710i \(-0.169877\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 10.0000i 0.560772i
\(319\) 0 0
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) − 16.0000i − 0.891645i
\(323\) 2.00000i 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) − 2.00000i − 0.110600i
\(328\) − 8.00000i − 0.441726i
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) − 4.00000i − 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 16.0000i − 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) − 1.00000i − 0.0540738i
\(343\) − 20.0000i − 1.07990i
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 8.00000i − 0.429463i −0.976673 0.214731i \(-0.931112\pi\)
0.976673 0.214731i \(-0.0688876\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000i 0.106600i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 16.0000 0.847998
\(357\) − 4.00000i − 0.211702i
\(358\) − 8.00000i − 0.422813i
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.00000i 0.315353i
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 26.0000i 1.35719i 0.734513 + 0.678594i \(0.237411\pi\)
−0.734513 + 0.678594i \(0.762589\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 2.00000i 0.102869i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) − 10.0000i − 0.511645i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) − 6.00000i − 0.304997i
\(388\) − 8.00000i − 0.406138i
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 3.00000i 0.151523i
\(393\) − 6.00000i − 0.302660i
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.00000i − 0.396545i
\(408\) 2.00000i 0.0990148i
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 12.0000i 0.591198i
\(413\) − 16.0000i − 0.787309i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) − 8.00000i − 0.391762i
\(418\) − 2.00000i − 0.0978232i
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 8.00000i 0.388973i
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) − 4.00000i − 0.193574i
\(428\) − 20.0000i − 0.966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 12.0000i 0.576683i 0.957528 + 0.288342i \(0.0931039\pi\)
−0.957528 + 0.288342i \(0.906896\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 8.00000i − 0.382692i
\(438\) 2.00000i 0.0955637i
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) − 2.00000i − 0.0945968i
\(448\) 2.00000i 0.0944911i
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) − 2.00000i − 0.0940721i
\(453\) − 8.00000i − 0.375873i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 10.0000i 0.467269i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 18.0000i − 0.836531i −0.908325 0.418265i \(-0.862638\pi\)
0.908325 0.418265i \(-0.137362\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 20.0000i 0.925490i 0.886492 + 0.462745i \(0.153135\pi\)
−0.886492 + 0.462745i \(0.846865\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 8.00000i 0.368230i
\(473\) − 12.0000i − 0.551761i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) − 10.0000i − 0.457869i
\(478\) 6.00000i 0.274434i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 10.0000i − 0.455488i
\(483\) 16.0000i 0.728025i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 8.00000i 0.360668i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 16.0000i − 0.717698i
\(498\) 16.0000i 0.716977i
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 6.00000i − 0.267793i
\(503\) − 28.0000i − 1.24846i −0.781241 0.624229i \(-0.785413\pi\)
0.781241 0.624229i \(-0.214587\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 13.0000i 0.577350i
\(508\) − 16.0000i − 0.709885i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) 16.0000i 0.703679i
\(518\) − 8.00000i − 0.351500i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) − 2.00000i − 0.0867110i
\(533\) 0 0
\(534\) −16.0000 −0.692388
\(535\) 0 0
\(536\) 0 0
\(537\) 8.00000i 0.345225i
\(538\) 8.00000i 0.344904i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 4.00000i 0.171815i
\(543\) − 6.00000i − 0.257485i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 40.0000i − 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) − 8.00000i − 0.340503i
\(553\) − 16.0000i − 0.680389i
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 20.0000i 0.843649i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) − 2.00000i − 0.0839921i
\(568\) 8.00000i 0.335673i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 16.0000 0.667827
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 8.00000i 0.331611i
\(583\) − 20.0000i − 0.828315i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) − 44.0000i − 1.81607i −0.418890 0.908037i \(-0.637581\pi\)
0.418890 0.908037i \(-0.362419\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 4.00000i 0.164399i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 44.0000i 1.78590i 0.450151 + 0.892952i \(0.351370\pi\)
−0.450151 + 0.892952i \(0.648630\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 46.0000i 1.85792i 0.370177 + 0.928961i \(0.379297\pi\)
−0.370177 + 0.928961i \(0.620703\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 14.0000i 0.561349i
\(623\) 32.0000i 1.28205i
\(624\) 0 0
\(625\) 0 0
\(626\) −18.0000 −0.719425
\(627\) 2.00000i 0.0798723i
\(628\) 6.00000i 0.239426i
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 4.00000i 0.158986i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 20.0000i 0.789337i
\(643\) − 46.0000i − 1.81406i −0.421063 0.907031i \(-0.638343\pi\)
0.421063 0.907031i \(-0.361657\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) − 10.0000i − 0.391630i
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) − 2.00000i − 0.0780274i
\(658\) 16.0000i 0.623745i
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) − 2.00000i − 0.0771517i
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 2.00000i 0.0768095i
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 10.0000i − 0.381524i
\(688\) 6.00000i 0.228748i
\(689\) 0 0
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) − 4.00000i − 0.151947i
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) − 26.0000i − 0.984115i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) − 4.00000i − 0.150435i
\(708\) − 8.00000i − 0.300658i
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) − 16.0000i − 0.599625i
\(713\) 0 0
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) − 6.00000i − 0.224074i
\(718\) − 30.0000i − 1.11959i
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) − 1.00000i − 0.0372161i
\(723\) 10.0000i 0.371904i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 22.0000i − 0.815935i −0.912996 0.407967i \(-0.866238\pi\)
0.912996 0.407967i \(-0.133762\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) − 2.00000i − 0.0739221i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) − 8.00000i − 0.294484i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 20.0000i − 0.734223i
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) − 16.0000i − 0.585409i
\(748\) − 4.00000i − 0.146254i
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 6.00000i 0.218652i
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) − 28.0000i − 1.01701i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 4.00000i 0.144810i
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) − 4.00000i − 0.143963i
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) 8.00000i 0.286998i
\(778\) 14.0000i 0.501924i
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) − 16.0000i − 0.572159i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 52.0000i 1.85360i 0.375555 + 0.926800i \(0.377452\pi\)
−0.375555 + 0.926800i \(0.622548\pi\)
\(788\) − 10.0000i − 0.356235i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 2.00000i 0.0710669i
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) − 22.0000i − 0.779280i −0.920967 0.389640i \(-0.872599\pi\)
0.920967 0.389640i \(-0.127401\pi\)
\(798\) 2.00000i 0.0707992i
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 16.0000 0.565332
\(802\) 12.0000i 0.423735i
\(803\) − 4.00000i − 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.00000i − 0.281613i
\(808\) 2.00000i 0.0703598i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) − 4.00000i − 0.140286i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) − 6.00000i − 0.209913i
\(818\) − 18.0000i − 0.629355i
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 18.0000i 0.627822i
\(823\) − 34.0000i − 1.18517i −0.805510 0.592583i \(-0.798108\pi\)
0.805510 0.592583i \(-0.201892\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 8.00000i 0.278019i
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 0 0
\(833\) − 6.00000i − 0.207888i
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) 30.0000i 1.03633i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 26.0000i 0.896019i
\(843\) − 20.0000i − 0.688837i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 14.0000i 0.481046i
\(848\) 10.0000i 0.343401i
\(849\) −2.00000 −0.0686398
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) − 8.00000i − 0.274075i
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) − 30.0000i − 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) − 20.0000i − 0.681203i
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) − 2.00000i − 0.0677285i
\(873\) − 8.00000i − 0.270759i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 8.00000i 0.270141i 0.990836 + 0.135070i \(0.0431261\pi\)
−0.990836 + 0.135070i \(0.956874\pi\)
\(878\) 16.0000i 0.539974i
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 18.0000i 0.605748i 0.953031 + 0.302874i \(0.0979462\pi\)
−0.953031 + 0.302874i \(0.902054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) − 4.00000i − 0.134231i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 12.0000i 0.401790i
\(893\) 8.00000i 0.267710i
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 24.0000i 0.800890i
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) − 16.0000i − 0.532742i
\(903\) 12.0000i 0.399335i
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 4.00000i 0.132745i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 32.0000i − 1.05905i
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 12.0000i 0.396275i
\(918\) 2.00000i 0.0660098i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 6.00000i 0.197599i
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −18.0000 −0.591517
\(927\) 12.0000i 0.394132i
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) − 6.00000i − 0.196537i
\(933\) − 14.0000i − 0.458339i
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) − 58.0000i − 1.89478i −0.320085 0.947389i \(-0.603712\pi\)
0.320085 0.947389i \(-0.396288\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) − 6.00000i − 0.195491i
\(943\) − 64.0000i − 2.08413i
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) − 4.00000i − 0.129641i
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) − 30.0000i − 0.969256i
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 20.0000i − 0.644491i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) − 6.00000i − 0.192947i −0.995336 0.0964735i \(-0.969244\pi\)
0.995336 0.0964735i \(-0.0307563\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 16.0000i 0.512936i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 10.0000i 0.319765i
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 30.0000i 0.957338i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 8.00000 0.255031
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) − 12.0000i − 0.380808i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) 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 2850.2.d.n.799.1 2
5.2 odd 4 2850.2.a.ba.1.1 1
5.3 odd 4 570.2.a.c.1.1 1
5.4 even 2 inner 2850.2.d.n.799.2 2
15.2 even 4 8550.2.a.o.1.1 1
15.8 even 4 1710.2.a.n.1.1 1
20.3 even 4 4560.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.c.1.1 1 5.3 odd 4
1710.2.a.n.1.1 1 15.8 even 4
2850.2.a.ba.1.1 1 5.2 odd 4
2850.2.d.n.799.1 2 1.1 even 1 trivial
2850.2.d.n.799.2 2 5.4 even 2 inner
4560.2.a.bd.1.1 1 20.3 even 4
8550.2.a.o.1.1 1 15.2 even 4