Properties

Label 1950.2.e.l.1249.2
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.l.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +1.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +10.0000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -6.00000 q^{74} -1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +6.00000 q^{89} +4.00000i q^{92} +1.00000 q^{96} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 2q^{16} - 2q^{24} + 2q^{26} + 20q^{29} - 12q^{34} + 2q^{36} - 2q^{39} + 4q^{41} + 8q^{46} + 14q^{49} + 12q^{51} - 2q^{54} + 12q^{61} - 2q^{64} - 8q^{69} + 32q^{71} - 12q^{74} + 2q^{81} + 8q^{86} + 12q^{89} + 2q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 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
\(26\) 1.00000 0.196116
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 1.00000i 0.138675i
\(53\) − 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
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) − 1.00000i − 0.113228i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 10.0000i − 1.07211i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) − 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
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 6.00000i 0.543214i
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000i 1.34269i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) − 7.00000i − 0.577350i
\(148\) − 6.00000i − 0.493197i
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) − 6.00000i − 0.443533i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) 4.00000i 0.278019i
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000i 0.412082i
\(213\) − 16.0000i − 1.09630i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000i 0.948200i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 6.00000i 0.402694i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 10.0000i − 0.656532i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −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 \(0.940080\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 12.0000i 0.741362i
\(263\) − 28.0000i − 1.72655i −0.504730 0.863277i \(-0.668408\pi\)
0.504730 0.863277i \(-0.331592\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 4.00000i 0.244339i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 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 \(-0.0191369\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 2.00000i 0.117041i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 14.0000i 0.810998i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) − 16.0000i − 0.920697i
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 14.0000i − 0.774202i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 6.00000i − 0.328798i
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) − 18.0000i − 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 10.0000i 0.536056i
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 20.0000i 1.05703i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 10.0000i − 0.525588i
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) − 20.0000i − 1.04399i −0.852948 0.521996i \(-0.825188\pi\)
0.852948 0.521996i \(-0.174812\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 10.0000i − 0.515026i
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 24.0000i 1.22795i
\(383\) − 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
\(386\) −14.0000 −0.712581
\(387\) 4.00000i 0.203331i
\(388\) 14.0000i 0.710742i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) − 7.00000i − 0.353553i
\(393\) − 12.0000i − 0.605320i
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) − 6.00000i − 0.297044i
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 12.0000i − 0.591198i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 4.00000i 0.195881i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) − 12.0000i − 0.584151i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) − 4.00000i − 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) − 2.00000i − 0.0955637i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 6.00000i 0.285391i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 14.0000i − 0.662177i
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 10.0000i − 0.470360i
\(453\) 16.0000i 0.751746i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.00000i − 0.280668i −0.990104 0.140334i \(-0.955182\pi\)
0.990104 0.140334i \(-0.0448177\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) − 16.0000i − 0.731823i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) − 14.0000i − 0.637683i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 60.0000i 2.70226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 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
\(501\) −24.0000 −1.07224
\(502\) − 4.00000i − 0.178529i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) − 12.0000i − 0.532414i
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) − 10.0000i − 0.437688i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 28.0000 1.22086
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 2.00000i − 0.0866296i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) − 20.0000i − 0.863064i
\(538\) − 14.0000i − 0.603583i
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) − 24.0000i − 1.03089i
\(543\) 10.0000i 0.429141i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) − 6.00000i − 0.253095i
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) − 16.0000i − 0.671345i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) − 24.0000i − 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 22.0000i − 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) − 14.0000i − 0.580319i
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 7.00000i 0.288675i
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 6.00000i 0.246598i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) − 16.0000i − 0.654836i
\(598\) − 4.00000i − 0.163572i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 36.0000i 1.46119i 0.682808 + 0.730597i \(0.260758\pi\)
−0.682808 + 0.730597i \(0.739242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) − 22.0000i − 0.888572i −0.895885 0.444286i \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 12.0000i 0.482711i
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 16.0000i − 0.641542i
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −48.0000 −1.91085 −0.955425 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 7.00000i − 0.277350i
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 4.00000i 0.157867i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 8.00000i 0.310929i
\(663\) − 6.00000i − 0.233021i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) − 40.0000i − 1.54881i
\(668\) 24.0000i 0.928588i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 42.0000i 1.61898i 0.587133 + 0.809491i \(0.300257\pi\)
−0.587133 + 0.809491i \(0.699743\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 30.0000i 1.15299i 0.817099 + 0.576497i \(0.195581\pi\)
−0.817099 + 0.576497i \(0.804419\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) − 4.00000i − 0.152499i
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 12.0000i 0.454532i
\(698\) − 34.0000i − 1.28692i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 16.0000i 0.597531i
\(718\) − 24.0000i − 0.895672i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 19.0000i − 0.707107i
\(723\) 14.0000i 0.520666i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) − 12.0000i − 0.445055i −0.974926 0.222528i \(-0.928569\pi\)
0.974926 0.222528i \(-0.0714308\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 6.00000i 0.221766i
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) − 2.00000i − 0.0736210i
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 4.00000i 0.145768i
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) − 24.0000i − 0.871719i
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 12.0000i 0.434714i
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 14.0000i − 0.503871i
\(773\) 26.0000i 0.935155i 0.883952 + 0.467578i \(0.154873\pi\)
−0.883952 + 0.467578i \(0.845127\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 18.0000i 0.645331i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 24.0000i 0.858238i
\(783\) 10.0000i 0.357371i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) −28.0000 −0.996826
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 6.00000i − 0.213066i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000i 0.492823i
\(808\) − 6.00000i − 0.211079i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) − 2.00000i − 0.0699284i
\(819\) 0 0
\(820\) 0 0
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 44.0000i − 1.53374i −0.641800 0.766872i \(-0.721812\pi\)
0.641800 0.766872i \(-0.278188\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 1.00000i 0.0346688i
\(833\) 42.0000i 1.45521i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 12.0000i − 0.414533i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) − 22.0000i − 0.758170i
\(843\) 6.00000i 0.206651i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 16.0000i 0.548151i
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 38.0000i 1.29806i 0.760765 + 0.649028i \(0.224824\pi\)
−0.760765 + 0.649028i \(0.775176\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) − 14.0000i − 0.474100i
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000i 0.133556i
\(898\) 14.0000i 0.467186i
\(899\) 0 0
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) − 6.00000i − 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) − 14.0000i − 0.461065i
\(923\) − 16.0000i − 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) − 12.0000i − 0.394132i
\(928\) 10.0000i 0.328266i
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 18.0000i − 0.589610i
\(933\) 16.0000i 0.523816i
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 14.0000i 0.456145i
\(943\) − 8.00000i − 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 10.0000i 0.323932i 0.986796 + 0.161966i \(0.0517835\pi\)
−0.986796 + 0.161966i \(0.948217\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 6.00000i 0.193448i
\(963\) − 4.00000i − 0.128898i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 0 0
\(970\) 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
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 20.0000i 0.638226i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −60.0000 −1.91079
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 22.0000i 0.696747i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(998\) 40.0000i 1.26618i
\(999\) −6.00000 −0.189832
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 1950.2.e.l.1249.2 2
3.2 odd 2 5850.2.e.p.5149.1 2
5.2 odd 4 390.2.a.a.1.1 1
5.3 odd 4 1950.2.a.y.1.1 1
5.4 even 2 inner 1950.2.e.l.1249.1 2
15.2 even 4 1170.2.a.m.1.1 1
15.8 even 4 5850.2.a.m.1.1 1
15.14 odd 2 5850.2.e.p.5149.2 2
20.7 even 4 3120.2.a.q.1.1 1
60.47 odd 4 9360.2.a.bn.1.1 1
65.12 odd 4 5070.2.a.s.1.1 1
65.47 even 4 5070.2.b.c.1351.1 2
65.57 even 4 5070.2.b.c.1351.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.a.1.1 1 5.2 odd 4
1170.2.a.m.1.1 1 15.2 even 4
1950.2.a.y.1.1 1 5.3 odd 4
1950.2.e.l.1249.1 2 5.4 even 2 inner
1950.2.e.l.1249.2 2 1.1 even 1 trivial
3120.2.a.q.1.1 1 20.7 even 4
5070.2.a.s.1.1 1 65.12 odd 4
5070.2.b.c.1351.1 2 65.47 even 4
5070.2.b.c.1351.2 2 65.57 even 4
5850.2.a.m.1.1 1 15.8 even 4
5850.2.e.p.5149.1 2 3.2 odd 2
5850.2.e.p.5149.2 2 15.14 odd 2
9360.2.a.bn.1.1 1 60.47 odd 4