Properties

Label 2850.2.d.j.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: \(x^{2} + 1\)
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.j.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -1.00000i q^{12} +2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} +6.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -4.00000 q^{41} +2.00000i q^{42} +6.00000i q^{43} +6.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} -8.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} +4.00000 q^{69} -1.00000i q^{72} -6.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} +6.00000 q^{86} +8.00000i q^{87} -6.00000i q^{88} +4.00000 q^{89} +4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} +12.0000i q^{97} -3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 12q^{11} + 4q^{14} + 2q^{16} + 2q^{19} - 4q^{21} - 2q^{24} + 16q^{29} - 16q^{31} + 4q^{34} + 2q^{36} - 8q^{41} + 12q^{44} - 8q^{46} + 6q^{49} - 4q^{51} - 2q^{54} - 4q^{56} + 8q^{59} + 4q^{61} - 2q^{64} - 12q^{66} + 8q^{69} - 8q^{74} - 2q^{76} - 16q^{79} + 2q^{81} + 4q^{84} + 12q^{86} + 8q^{89} - 24q^{94} + 2q^{96} + 12q^{99} + O(q^{100}) \)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) − 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 \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 6.00000i 1.27920i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 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
\(33\) − 6.00000i − 1.04447i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 1.00000i 0.132453i
\(58\) − 8.00000i − 1.05045i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000i 0.441726i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 8.00000i 0.857690i
\(88\) − 6.00000i − 0.639602i
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) − 8.00000i − 0.829561i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 2.00000i 0.188982i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) − 4.00000i − 0.368230i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) − 2.00000i − 0.181071i
\(123\) − 4.00000i − 0.360668i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 22.0000 1.92215 0.961074 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 2.00000i 0.173422i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 3.00000i 0.247436i
\(148\) 4.00000i 0.328798i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 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
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 6.00000i − 0.457496i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 4.00000i 0.300658i
\(178\) − 4.00000i − 0.299813i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) − 12.0000i − 0.877527i
\(188\) 12.0000i 0.875190i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 8.00000i 0.575853i 0.957653 + 0.287926i \(0.0929658\pi\)
−0.957653 + 0.287926i \(0.907034\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 10.0000i 0.703598i
\(203\) 16.0000i 1.12298i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 16.0000i − 1.08615i
\(218\) − 14.0000i − 0.948200i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) − 4.00000i − 0.268462i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 8.00000i 0.525226i
\(233\) − 14.0000i − 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) − 8.00000i − 0.519656i
\(238\) 4.00000i 0.259281i
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) − 8.00000i − 0.508001i
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 24.0000i 1.50887i
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 6.00000i 0.373544i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) − 22.0000i − 1.35916i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 4.00000i 0.244796i
\(268\) 8.00000i 0.488678i
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.00000i − 0.472225i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 6.00000i 0.351123i
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 6.00000i 0.348155i
\(298\) 22.0000i 1.27443i
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) − 10.0000i − 0.574485i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 12.0000i 0.683763i
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 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 \(0.946109\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) −48.0000 −2.68748
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) − 8.00000i − 0.445823i
\(323\) 2.00000i 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 14.0000i 0.774202i
\(328\) − 4.00000i − 0.220863i
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 4.00000i 0.219199i
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 20.0000i − 1.08947i −0.838608 0.544735i \(-0.816630\pi\)
0.838608 0.544735i \(-0.183370\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) 1.00000i 0.0540738i
\(343\) 20.0000i 1.07990i
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 36.0000i 1.93258i 0.257454 + 0.966291i \(0.417117\pi\)
−0.257454 + 0.966291i \(0.582883\pi\)
\(348\) − 8.00000i − 0.428845i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.00000i 0.319801i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) − 4.00000i − 0.211702i
\(358\) 12.0000i 0.634220i
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 2.00000i − 0.105118i
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000i 0.414781i
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) − 2.00000i − 0.102869i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 22.0000i 1.12562i
\(383\) − 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
\(386\) 8.00000 0.407189
\(387\) − 6.00000i − 0.304997i
\(388\) − 12.0000i − 0.609208i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 3.00000i 0.151523i
\(393\) 22.0000i 1.10975i
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 20.0000i 1.00251i
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) 24.0000i 1.18964i
\(408\) − 2.00000i − 0.0990148i
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000i 0.293470i
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 28.0000i 1.36302i
\(423\) 12.0000i 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 4.00000i − 0.191346i
\(438\) − 6.00000i − 0.286691i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 22.0000i − 1.04056i
\(448\) − 2.00000i − 0.0944911i
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 14.0000i 0.658505i
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 14.0000i − 0.654892i −0.944870 0.327446i \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 18.0000i 0.841085i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) − 12.0000i − 0.558291i
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 4.00000i 0.184115i
\(473\) − 36.0000i − 1.65528i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 6.00000i 0.274721i
\(478\) − 18.0000i − 0.823301i
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 22.0000i 1.00207i
\(483\) 8.00000i 0.364013i
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 4.00000i 0.180334i
\(493\) 16.0000i 0.720604i
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) − 4.00000i − 0.179244i
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 14.0000i 0.624851i
\(503\) − 40.0000i − 1.78351i −0.452517 0.891756i \(-0.649474\pi\)
0.452517 0.891756i \(-0.350526\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 13.0000i 0.577350i
\(508\) 12.0000i 0.532414i
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) 72.0000i 3.16656i
\(518\) − 8.00000i − 0.351500i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 8.00000i 0.350150i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −22.0000 −0.961074
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) − 16.0000i − 0.696971i
\(528\) − 6.00000i − 0.261116i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) − 2.00000i − 0.0867110i
\(533\) 0 0
\(534\) 4.00000 0.173097
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) − 12.0000i − 0.517838i
\(538\) 8.00000i 0.344904i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 2.00000i 0.0858282i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 40.0000i 1.71028i 0.518400 + 0.855138i \(0.326528\pi\)
−0.518400 + 0.855138i \(0.673472\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 4.00000i 0.170251i
\(553\) − 16.0000i − 0.680389i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 0 0
\(557\) − 22.0000i − 0.932170i −0.884740 0.466085i \(-0.845664\pi\)
0.884740 0.466085i \(-0.154336\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) − 16.0000i − 0.674919i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) − 22.0000i − 0.919063i
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 12.0000i 0.497416i
\(583\) 36.0000i 1.49097i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) − 4.00000i − 0.164399i
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) − 20.0000i − 0.818546i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 12.0000i 0.489083i
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000i 0.0808452i
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) − 38.0000i − 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 6.00000i 0.240578i
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) 14.0000 0.559553
\(627\) − 6.00000i − 0.239617i
\(628\) − 6.00000i − 0.239426i
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) − 28.0000i − 1.11290i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 48.0000i 1.90034i
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 26.0000i 1.02534i 0.858586 + 0.512670i \(0.171344\pi\)
−0.858586 + 0.512670i \(0.828656\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 6.00000i 0.234978i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 6.00000i 0.234082i
\(658\) − 24.0000i − 0.935617i
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) − 32.0000i − 1.23904i
\(668\) 16.0000i 0.619059i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 2.00000i 0.0771517i
\(673\) 28.0000i 1.07932i 0.841883 + 0.539660i \(0.181447\pi\)
−0.841883 + 0.539660i \(0.818553\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) − 48.0000i − 1.83801i
\(683\) 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
\(687\) − 18.0000i − 0.686743i
\(688\) 6.00000i 0.228748i
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 12.0000i 0.455842i
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) − 8.00000i − 0.303022i
\(698\) − 2.00000i − 0.0757011i
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) − 20.0000i − 0.752177i
\(708\) − 4.00000i − 0.150329i
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 4.00000i 0.149906i
\(713\) 32.0000i 1.19841i
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 18.0000i 0.672222i
\(718\) − 6.00000i − 0.223918i
\(719\) −14.0000 −0.522112 −0.261056 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 0.0372161i
\(723\) − 22.0000i − 0.818189i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 14.0000i 0.519231i 0.965712 + 0.259616i \(0.0835959\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) − 2.00000i − 0.0739221i
\(733\) − 18.0000i − 0.664845i −0.943131 0.332423i \(-0.892134\pi\)
0.943131 0.332423i \(-0.107866\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000i 1.76810i
\(738\) − 4.00000i − 0.147242i
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 12.0000i − 0.440534i
\(743\) 8.00000i 0.293492i 0.989174 + 0.146746i \(0.0468799\pi\)
−0.989174 + 0.146746i \(0.953120\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) 4.00000i 0.146352i
\(748\) 12.0000i 0.438763i
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) − 14.0000i − 0.510188i
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 20.0000i 0.726433i
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 28.0000i 1.01367i
\(764\) 22.0000 0.795932
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 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
\(771\) −6.00000 −0.216085
\(772\) − 8.00000i − 0.287926i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 8.00000i 0.286998i
\(778\) − 30.0000i − 1.07555i
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) − 8.00000i − 0.286079i
\(783\) − 8.00000i − 0.285897i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 22.0000 0.784714
\(787\) 44.0000i 1.56843i 0.620489 + 0.784215i \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 14.0000i − 0.498729i
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) 6.00000i 0.213201i
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 2.00000i 0.0708436i 0.999372 + 0.0354218i \(0.0112775\pi\)
−0.999372 + 0.0354218i \(0.988723\pi\)
\(798\) 2.00000i 0.0707992i
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) 36.0000i 1.27041i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.00000i − 0.281613i
\(808\) − 10.0000i − 0.351799i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) − 16.0000i − 0.561490i
\(813\) 20.0000i 0.701431i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 6.00000i 0.209913i
\(818\) 14.0000i 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 6.00000i 0.209274i
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 8.00000i 0.276520i
\(838\) − 14.0000i − 0.483622i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 2.00000i 0.0689246i
\(843\) 16.0000i 0.551069i
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 50.0000i 1.71802i
\(848\) − 6.00000i − 0.206041i
\(849\) 22.0000 0.755038
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) − 36.0000i − 1.22616i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 16.0000i 0.543075i
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 14.0000i 0.474100i
\(873\) − 12.0000i − 0.406138i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 3.00000i 0.101015i
\(883\) − 14.0000i − 0.471138i −0.971858 0.235569i \(-0.924305\pi\)
0.971858 0.235569i \(-0.0756953\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) − 16.0000i − 0.535720i
\(893\) − 12.0000i − 0.401565i
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) − 4.00000i − 0.133482i
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 24.0000i − 0.799113i
\(903\) − 12.0000i − 0.399335i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 28.0000i 0.929213i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 24.0000i 0.794284i
\(914\) −14.0000 −0.463079
\(915\) 0 0
\(916\) 18.0000 0.594737
\(917\) 44.0000i 1.45301i
\(918\) − 2.00000i − 0.0660098i
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 34.0000i 1.11973i
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) − 8.00000i − 0.262613i
\(929\) 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
\(932\) 14.0000i 0.458585i
\(933\) − 6.00000i − 0.196431i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) − 34.0000i − 1.11073i −0.831606 0.555366i \(-0.812578\pi\)
0.831606 0.555366i \(-0.187422\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 6.00000i 0.195491i
\(943\) 16.0000i 0.521032i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −36.0000 −1.17046
\(947\) 44.0000i 1.42981i 0.699223 + 0.714904i \(0.253530\pi\)
−0.699223 + 0.714904i \(0.746470\pi\)
\(948\) 8.00000i 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) − 4.00000i − 0.129641i
\(953\) 34.0000i 1.10137i 0.834714 + 0.550684i \(0.185633\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −18.0000 −0.582162
\(957\) − 48.0000i − 1.55162i
\(958\) 18.0000i 0.581554i
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) − 18.0000i − 0.578841i −0.957202 0.289420i \(-0.906537\pi\)
0.957202 0.289420i \(-0.0934626\pi\)
\(968\) 25.0000i 0.803530i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 6.00000i − 0.191859i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 34.0000i 1.08498i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) − 58.0000i − 1.83688i −0.395562 0.918439i \(-0.629450\pi\)
0.395562 0.918439i \(-0.370550\pi\)
\(998\) 24.0000i 0.759707i
\(999\) −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.j.799.1 2
5.2 odd 4 2850.2.a.y.1.1 1
5.3 odd 4 570.2.a.b.1.1 1
5.4 even 2 inner 2850.2.d.j.799.2 2
15.2 even 4 8550.2.a.g.1.1 1
15.8 even 4 1710.2.a.s.1.1 1
20.3 even 4 4560.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.b.1.1 1 5.3 odd 4
1710.2.a.s.1.1 1 15.8 even 4
2850.2.a.y.1.1 1 5.2 odd 4
2850.2.d.j.799.1 2 1.1 even 1 trivial
2850.2.d.j.799.2 2 5.4 even 2 inner
4560.2.a.r.1.1 1 20.3 even 4
8550.2.a.g.1.1 1 15.2 even 4