Properties

Label 2850.2.d.q.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.q.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -8.00000 q^{41} +2.00000i q^{42} -6.00000i q^{43} -2.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +2.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +14.0000i q^{73} -8.00000 q^{74} -1.00000 q^{76} +4.00000i q^{77} +4.00000i q^{78} +1.00000 q^{81} +8.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -6.00000 q^{86} +2.00000i q^{88} -8.00000 q^{91} -4.00000i q^{92} -8.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +12.0000i q^{97} -3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{44} + 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 8 q^{69} - 16 q^{71} - 16 q^{74} - 2 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 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 \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 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
\(22\) − 2.00000i − 0.426401i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 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
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −4.00000 −0.640513
\(40\) 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 \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 4.00000i − 0.554700i
\(53\) − 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
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 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
\(66\) 2.00000 0.246183
\(67\) − 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
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.00000i 0.455842i
\(78\) 4.00000i 0.452911i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000i 0.883452i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 2.00000i 0.213201i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) − 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
\(99\) −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
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 2.00000i 0.188982i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) − 8.00000i − 0.721336i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) 2.00000i 0.173422i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000i 1.87959i 0.341743 + 0.939793i \(0.388983\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 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
\(142\) 8.00000i 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 3.00000i 0.247436i
\(148\) 8.00000i 0.657596i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 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
\(156\) 4.00000 0.320256
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 6.00000i 0.457496i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000i 0.592999i
\(183\) 2.00000i 0.147844i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 4.00000i 0.292509i
\(188\) − 12.0000i − 0.875190i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 2.00000i 0.142134i
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 2.00000i − 0.140720i
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 4.00000i − 0.278019i
\(208\) 4.00000i 0.277350i
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000i 0.412082i
\(213\) − 8.00000i − 0.548151i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 16.0000i − 1.08615i
\(218\) 10.0000i 0.677285i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 8.00000i − 0.536925i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 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 \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 4.00000i 0.259281i
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 4.00000i 0.254514i
\(248\) − 8.00000i − 0.508001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 8.00000i 0.502956i
\(254\) −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 \(-0.189739\pi\)
\(258\) − 6.00000i − 0.373544i
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 18.0000i 1.11204i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 8.00000i − 0.484182i
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) − 16.0000i − 0.944450i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) − 14.0000i − 0.819288i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) − 2.00000i − 0.116052i
\(298\) 10.0000i 0.579284i
\(299\) −16.0000 −0.925304
\(300\) 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
\(307\) 32.0000i 1.82634i 0.407583 + 0.913168i \(0.366372\pi\)
−0.407583 + 0.913168i \(0.633628\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 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
\(326\) 14.0000 0.775388
\(327\) − 10.0000i − 0.553001i
\(328\) − 8.00000i − 0.441726i
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 8.00000i 0.438397i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 28.0000i − 1.52526i −0.646837 0.762629i \(-0.723908\pi\)
0.646837 0.762629i \(-0.276092\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 1.00000i 0.0540738i
\(343\) 20.0000i 1.07990i
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) − 2.00000i − 0.106600i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 2.00000i − 0.105118i
\(363\) − 7.00000i − 0.367405i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000i 0.414781i
\(373\) − 36.0000i − 1.86401i −0.362446 0.932005i \(-0.618058\pi\)
0.362446 0.932005i \(-0.381942\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) − 2.00000i − 0.102869i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 18.0000i 0.920960i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 6.00000i 0.304997i
\(388\) − 12.0000i − 0.609208i
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000i 0.151523i
\(393\) − 18.0000i − 0.907980i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 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
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) − 32.0000i − 1.59403i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.0000i − 0.793091i
\(408\) − 2.00000i − 0.0990148i
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) − 4.00000i − 0.197066i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) − 2.00000i − 0.0978232i
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 12.0000i − 0.584151i
\(423\) − 12.0000i − 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 4.00000i 0.193574i
\(428\) − 12.0000i − 0.580042i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 16.0000i − 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 4.00000i 0.191346i
\(438\) 14.0000i 0.668946i
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000i 0.380521i
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) − 10.0000i − 0.472984i
\(448\) − 2.00000i − 0.0944911i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 6.00000i 0.282216i
\(453\) − 8.00000i − 0.375873i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 18.0000i − 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 6.00000i − 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) − 12.0000i − 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 6.00000i 0.274721i
\(478\) 10.0000i 0.457389i
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) − 2.00000i − 0.0910975i
\(483\) − 8.00000i − 0.364013i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 8.00000i 0.360668i
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 16.0000i − 0.717698i
\(498\) 4.00000i 0.179244i
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 2.00000i − 0.0892644i
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) − 3.00000i − 0.133235i
\(508\) 8.00000i 0.354943i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 24.0000i 1.05552i
\(518\) − 16.0000i − 0.703000i
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) − 16.0000i − 0.696971i
\(528\) 2.00000i 0.0870388i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.00000i − 0.0867110i
\(533\) − 32.0000i − 1.38607i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) − 12.0000i − 0.515444i
\(543\) 2.00000i 0.0858282i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) − 22.0000i − 0.939793i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) − 4.00000i − 0.170251i
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) 0 0
\(557\) 42.0000i 1.77960i 0.456354 + 0.889799i \(0.349155\pi\)
−0.456354 + 0.889799i \(0.650845\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) − 12.0000i − 0.506189i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 2.00000i 0.0839921i
\(568\) − 8.00000i − 0.335673i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) − 18.0000i − 0.751961i
\(574\) −16.0000 −0.667827
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 12.0000i 0.497416i
\(583\) − 12.0000i − 0.496989i
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) − 8.00000i − 0.330195i −0.986277 0.165098i \(-0.947206\pi\)
0.986277 0.165098i \(-0.0527939\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) − 8.00000i − 0.328798i
\(593\) − 46.0000i − 1.88899i −0.328521 0.944497i \(-0.606550\pi\)
0.328521 0.944497i \(-0.393450\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 20.0000i 0.818546i
\(598\) 16.0000i 0.654289i
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) 8.00000i 0.325785i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 12.0000i 0.487065i 0.969893 + 0.243532i \(0.0783062\pi\)
−0.969893 + 0.243532i \(0.921694\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 2.00000i 0.0808452i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 18.0000i 0.721734i
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 2.00000i 0.0798723i
\(628\) 18.0000i 0.718278i
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 32.0000 1.26392 0.631962 0.774999i \(-0.282250\pi\)
0.631962 + 0.774999i \(0.282250\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) − 14.0000i − 0.548282i
\(653\) − 46.0000i − 1.80012i −0.435767 0.900060i \(-0.643523\pi\)
0.435767 0.900060i \(-0.356477\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) − 14.0000i − 0.546192i
\(658\) 24.0000i 0.935617i
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) − 8.00000i − 0.310694i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) 8.00000i 0.309529i
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 2.00000i 0.0771517i
\(673\) 4.00000i 0.154189i 0.997024 + 0.0770943i \(0.0245643\pi\)
−0.997024 + 0.0770943i \(0.975436\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 16.0000i 0.612672i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) − 10.0000i − 0.381524i
\(688\) − 6.00000i − 0.228748i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) − 4.00000i − 0.151947i
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) − 10.0000i − 0.378506i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) − 8.00000i − 0.301726i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 32.0000i − 1.19841i
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 0 0
\(717\) − 10.0000i − 0.373457i
\(718\) − 10.0000i − 0.373197i
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) − 1.00000i − 0.0372161i
\(723\) 2.00000i 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 18.0000i − 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −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 \(-0.840576\pi\)
0.877178 0.480166i \(-0.159424\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) − 16.0000i − 0.589368i
\(738\) − 8.00000i − 0.294484i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) − 12.0000i − 0.440534i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −36.0000 −1.31805
\(747\) − 4.00000i − 0.146352i
\(748\) − 4.00000i − 0.146254i
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 2.00000i 0.0728841i
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) − 20.0000i − 0.724049i
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 24.0000i − 0.863779i
\(773\) 34.0000i 1.22290i 0.791285 + 0.611448i \(0.209412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 16.0000i 0.573997i
\(778\) 30.0000i 1.07555i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 8.00000i 0.286079i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 18.0000i 0.641223i
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) − 2.00000i − 0.0710669i
\(793\) 8.00000i 0.284088i
\(794\) 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
\(801\) 0 0
\(802\) − 12.0000i − 0.423735i
\(803\) 28.0000i 0.988099i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) 2.00000i 0.0703598i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 12.0000i 0.420858i
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 6.00000i − 0.209913i
\(818\) 30.0000i 1.04893i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 22.0000i 0.767338i
\(823\) 34.0000i 1.18517i 0.805510 + 0.592583i \(0.201892\pi\)
−0.805510 + 0.592583i \(0.798108\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) − 4.00000i − 0.138675i
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 8.00000i 0.276520i
\(838\) − 30.0000i − 1.03633i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 22.0000i − 0.758170i
\(843\) 12.0000i 0.413302i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) − 14.0000i − 0.481046i
\(848\) − 6.00000i − 0.206041i
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 8.00000i 0.274075i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 8.00000i 0.273115i
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) − 12.0000i − 0.408722i
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 13.0000i 0.441503i
\(868\) 16.0000i 0.543075i
\(869\) 0 0
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) − 10.0000i − 0.338643i
\(873\) − 12.0000i − 0.406138i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 12.0000i 0.405211i 0.979260 + 0.202606i \(0.0649409\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000i 0.101015i
\(883\) − 26.0000i − 0.874970i −0.899226 0.437485i \(-0.855869\pi\)
0.899226 0.437485i \(-0.144131\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 8.00000i 0.268462i
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) − 4.00000i − 0.133930i
\(893\) 12.0000i 0.401565i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) − 16.0000i − 0.534224i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 16.0000i 0.532742i
\(903\) 12.0000i 0.399335i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 8.00000i 0.264761i
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) − 36.0000i − 1.18882i
\(918\) − 2.00000i − 0.0660098i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) − 2.00000i − 0.0658665i
\(923\) − 32.0000i − 1.05329i
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 6.00000i 0.196537i
\(933\) − 18.0000i − 0.589294i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) − 32.0000i − 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 0 0
\(949\) −56.0000 −1.81784
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 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
\(956\) 10.0000 0.323423
\(957\) 0 0
\(958\) − 10.0000i − 0.323085i
\(959\) −44.0000 −1.42083
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 32.0000i − 1.03172i
\(963\) − 12.0000i − 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) − 58.0000i − 1.86515i −0.360971 0.932577i \(-0.617555\pi\)
0.360971 0.932577i \(-0.382445\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −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
\(973\) 0 0
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 38.0000i − 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 14.0000i 0.447671i
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) − 22.0000i − 0.702048i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 8.00000 0.255031
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) − 4.00000i − 0.127257i
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 12.0000i 0.380808i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 62.0000i 1.96356i 0.190022 + 0.981780i \(0.439144\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) − 40.0000i − 1.26618i
\(999\) −8.00000 −0.253109
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.q.799.1 2
5.2 odd 4 570.2.a.l.1.1 1
5.3 odd 4 2850.2.a.e.1.1 1
5.4 even 2 inner 2850.2.d.q.799.2 2
15.2 even 4 1710.2.a.b.1.1 1
15.8 even 4 8550.2.a.bf.1.1 1
20.7 even 4 4560.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.l.1.1 1 5.2 odd 4
1710.2.a.b.1.1 1 15.2 even 4
2850.2.a.e.1.1 1 5.3 odd 4
2850.2.d.q.799.1 2 1.1 even 1 trivial
2850.2.d.q.799.2 2 5.4 even 2 inner
4560.2.a.o.1.1 1 20.7 even 4
8550.2.a.bf.1.1 1 15.8 even 4