Properties

Label 1216.3.e.i.1025.2
Level $1216$
Weight $3$
Character 1216.1025
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1025
Dual form 1216.3.e.i.1025.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{3} +1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.82843i q^{3} +1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +16.9706i q^{13} +2.82843i q^{15} -25.0000 q^{17} +19.0000 q^{19} -14.1421i q^{21} +10.0000 q^{23} -24.0000 q^{25} +28.2843i q^{27} -42.4264i q^{29} +42.4264i q^{31} +14.1421i q^{33} -5.00000 q^{35} +25.4558i q^{37} -48.0000 q^{39} -42.4264i q^{41} +5.00000 q^{43} +1.00000 q^{45} -5.00000 q^{47} -24.0000 q^{49} -70.7107i q^{51} +25.4558i q^{53} +5.00000 q^{55} +53.7401i q^{57} +84.8528i q^{59} -95.0000 q^{61} -5.00000 q^{63} +16.9706i q^{65} +110.309i q^{67} +28.2843i q^{69} -25.0000 q^{73} -67.8823i q^{75} -25.0000 q^{77} -42.4264i q^{79} -71.0000 q^{81} -130.000 q^{83} -25.0000 q^{85} +120.000 q^{87} -127.279i q^{89} -84.8528i q^{91} -120.000 q^{93} +19.0000 q^{95} +16.9706i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 10q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 10q^{7} + 2q^{9} + 10q^{11} - 50q^{17} + 38q^{19} + 20q^{23} - 48q^{25} - 10q^{35} - 96q^{39} + 10q^{43} + 2q^{45} - 10q^{47} - 48q^{49} + 10q^{55} - 190q^{61} - 10q^{63} - 50q^{73} - 50q^{77} - 142q^{81} - 260q^{83} - 50q^{85} + 240q^{87} - 240q^{93} + 38q^{95} + 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.82843i 0.942809i 0.881917 + 0.471405i \(0.156253\pi\)
−0.881917 + 0.471405i \(0.843747\pi\)
\(4\) 0 0
\(5\) 1.00000 0.200000 0.100000 0.994987i \(-0.468116\pi\)
0.100000 + 0.994987i \(0.468116\pi\)
\(6\) 0 0
\(7\) −5.00000 −0.714286 −0.357143 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) 0 0
\(9\) 1.00000 0.111111
\(10\) 0 0
\(11\) 5.00000 0.454545 0.227273 0.973831i \(-0.427019\pi\)
0.227273 + 0.973831i \(0.427019\pi\)
\(12\) 0 0
\(13\) 16.9706i 1.30543i 0.757604 + 0.652714i \(0.226370\pi\)
−0.757604 + 0.652714i \(0.773630\pi\)
\(14\) 0 0
\(15\) 2.82843i 0.188562i
\(16\) 0 0
\(17\) −25.0000 −1.47059 −0.735294 0.677748i \(-0.762956\pi\)
−0.735294 + 0.677748i \(0.762956\pi\)
\(18\) 0 0
\(19\) 19.0000 1.00000
\(20\) 0 0
\(21\) − 14.1421i − 0.673435i
\(22\) 0 0
\(23\) 10.0000 0.434783 0.217391 0.976085i \(-0.430245\pi\)
0.217391 + 0.976085i \(0.430245\pi\)
\(24\) 0 0
\(25\) −24.0000 −0.960000
\(26\) 0 0
\(27\) 28.2843i 1.04757i
\(28\) 0 0
\(29\) − 42.4264i − 1.46298i −0.681852 0.731490i \(-0.738825\pi\)
0.681852 0.731490i \(-0.261175\pi\)
\(30\) 0 0
\(31\) 42.4264i 1.36859i 0.729204 + 0.684297i \(0.239891\pi\)
−0.729204 + 0.684297i \(0.760109\pi\)
\(32\) 0 0
\(33\) 14.1421i 0.428550i
\(34\) 0 0
\(35\) −5.00000 −0.142857
\(36\) 0 0
\(37\) 25.4558i 0.687996i 0.938970 + 0.343998i \(0.111781\pi\)
−0.938970 + 0.343998i \(0.888219\pi\)
\(38\) 0 0
\(39\) −48.0000 −1.23077
\(40\) 0 0
\(41\) − 42.4264i − 1.03479i −0.855747 0.517395i \(-0.826902\pi\)
0.855747 0.517395i \(-0.173098\pi\)
\(42\) 0 0
\(43\) 5.00000 0.116279 0.0581395 0.998308i \(-0.481483\pi\)
0.0581395 + 0.998308i \(0.481483\pi\)
\(44\) 0 0
\(45\) 1.00000 0.0222222
\(46\) 0 0
\(47\) −5.00000 −0.106383 −0.0531915 0.998584i \(-0.516939\pi\)
−0.0531915 + 0.998584i \(0.516939\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) − 70.7107i − 1.38648i
\(52\) 0 0
\(53\) 25.4558i 0.480299i 0.970736 + 0.240149i \(0.0771965\pi\)
−0.970736 + 0.240149i \(0.922804\pi\)
\(54\) 0 0
\(55\) 5.00000 0.0909091
\(56\) 0 0
\(57\) 53.7401i 0.942809i
\(58\) 0 0
\(59\) 84.8528i 1.43818i 0.694915 + 0.719092i \(0.255442\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 0 0
\(61\) −95.0000 −1.55738 −0.778689 0.627411i \(-0.784115\pi\)
−0.778689 + 0.627411i \(0.784115\pi\)
\(62\) 0 0
\(63\) −5.00000 −0.0793651
\(64\) 0 0
\(65\) 16.9706i 0.261086i
\(66\) 0 0
\(67\) 110.309i 1.64640i 0.567753 + 0.823199i \(0.307813\pi\)
−0.567753 + 0.823199i \(0.692187\pi\)
\(68\) 0 0
\(69\) 28.2843i 0.409917i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −25.0000 −0.342466 −0.171233 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(74\) 0 0
\(75\) − 67.8823i − 0.905097i
\(76\) 0 0
\(77\) −25.0000 −0.324675
\(78\) 0 0
\(79\) − 42.4264i − 0.537043i −0.963274 0.268522i \(-0.913465\pi\)
0.963274 0.268522i \(-0.0865351\pi\)
\(80\) 0 0
\(81\) −71.0000 −0.876543
\(82\) 0 0
\(83\) −130.000 −1.56627 −0.783133 0.621855i \(-0.786379\pi\)
−0.783133 + 0.621855i \(0.786379\pi\)
\(84\) 0 0
\(85\) −25.0000 −0.294118
\(86\) 0 0
\(87\) 120.000 1.37931
\(88\) 0 0
\(89\) − 127.279i − 1.43010i −0.699071 0.715052i \(-0.746403\pi\)
0.699071 0.715052i \(-0.253597\pi\)
\(90\) 0 0
\(91\) − 84.8528i − 0.932449i
\(92\) 0 0
\(93\) −120.000 −1.29032
\(94\) 0 0
\(95\) 19.0000 0.200000
\(96\) 0 0
\(97\) 16.9706i 0.174954i 0.996167 + 0.0874771i \(0.0278805\pi\)
−0.996167 + 0.0874771i \(0.972120\pi\)
\(98\) 0 0
\(99\) 5.00000 0.0505051
\(100\) 0 0
\(101\) −50.0000 −0.495050 −0.247525 0.968882i \(-0.579617\pi\)
−0.247525 + 0.968882i \(0.579617\pi\)
\(102\) 0 0
\(103\) 16.9706i 0.164763i 0.996601 + 0.0823814i \(0.0262526\pi\)
−0.996601 + 0.0823814i \(0.973747\pi\)
\(104\) 0 0
\(105\) − 14.1421i − 0.134687i
\(106\) 0 0
\(107\) − 101.823i − 0.951620i −0.879548 0.475810i \(-0.842155\pi\)
0.879548 0.475810i \(-0.157845\pi\)
\(108\) 0 0
\(109\) − 127.279i − 1.16770i −0.811862 0.583850i \(-0.801546\pi\)
0.811862 0.583850i \(-0.198454\pi\)
\(110\) 0 0
\(111\) −72.0000 −0.648649
\(112\) 0 0
\(113\) 110.309i 0.976183i 0.872793 + 0.488091i \(0.162307\pi\)
−0.872793 + 0.488091i \(0.837693\pi\)
\(114\) 0 0
\(115\) 10.0000 0.0869565
\(116\) 0 0
\(117\) 16.9706i 0.145048i
\(118\) 0 0
\(119\) 125.000 1.05042
\(120\) 0 0
\(121\) −96.0000 −0.793388
\(122\) 0 0
\(123\) 120.000 0.975610
\(124\) 0 0
\(125\) −49.0000 −0.392000
\(126\) 0 0
\(127\) − 229.103i − 1.80396i −0.431780 0.901979i \(-0.642114\pi\)
0.431780 0.901979i \(-0.357886\pi\)
\(128\) 0 0
\(129\) 14.1421i 0.109629i
\(130\) 0 0
\(131\) −163.000 −1.24427 −0.622137 0.782908i \(-0.713735\pi\)
−0.622137 + 0.782908i \(0.713735\pi\)
\(132\) 0 0
\(133\) −95.0000 −0.714286
\(134\) 0 0
\(135\) 28.2843i 0.209513i
\(136\) 0 0
\(137\) 95.0000 0.693431 0.346715 0.937970i \(-0.387297\pi\)
0.346715 + 0.937970i \(0.387297\pi\)
\(138\) 0 0
\(139\) 125.000 0.899281 0.449640 0.893210i \(-0.351552\pi\)
0.449640 + 0.893210i \(0.351552\pi\)
\(140\) 0 0
\(141\) − 14.1421i − 0.100299i
\(142\) 0 0
\(143\) 84.8528i 0.593376i
\(144\) 0 0
\(145\) − 42.4264i − 0.292596i
\(146\) 0 0
\(147\) − 67.8823i − 0.461784i
\(148\) 0 0
\(149\) −215.000 −1.44295 −0.721477 0.692439i \(-0.756536\pi\)
−0.721477 + 0.692439i \(0.756536\pi\)
\(150\) 0 0
\(151\) 84.8528i 0.561939i 0.959717 + 0.280970i \(0.0906560\pi\)
−0.959717 + 0.280970i \(0.909344\pi\)
\(152\) 0 0
\(153\) −25.0000 −0.163399
\(154\) 0 0
\(155\) 42.4264i 0.273719i
\(156\) 0 0
\(157\) 190.000 1.21019 0.605096 0.796153i \(-0.293135\pi\)
0.605096 + 0.796153i \(0.293135\pi\)
\(158\) 0 0
\(159\) −72.0000 −0.452830
\(160\) 0 0
\(161\) −50.0000 −0.310559
\(162\) 0 0
\(163\) 110.000 0.674847 0.337423 0.941353i \(-0.390445\pi\)
0.337423 + 0.941353i \(0.390445\pi\)
\(164\) 0 0
\(165\) 14.1421i 0.0857099i
\(166\) 0 0
\(167\) − 59.3970i − 0.355670i −0.984060 0.177835i \(-0.943091\pi\)
0.984060 0.177835i \(-0.0569094\pi\)
\(168\) 0 0
\(169\) −119.000 −0.704142
\(170\) 0 0
\(171\) 19.0000 0.111111
\(172\) 0 0
\(173\) 186.676i 1.07905i 0.841969 + 0.539527i \(0.181397\pi\)
−0.841969 + 0.539527i \(0.818603\pi\)
\(174\) 0 0
\(175\) 120.000 0.685714
\(176\) 0 0
\(177\) −240.000 −1.35593
\(178\) 0 0
\(179\) − 127.279i − 0.711057i −0.934665 0.355529i \(-0.884301\pi\)
0.934665 0.355529i \(-0.115699\pi\)
\(180\) 0 0
\(181\) 254.558i 1.40640i 0.710992 + 0.703200i \(0.248246\pi\)
−0.710992 + 0.703200i \(0.751754\pi\)
\(182\) 0 0
\(183\) − 268.701i − 1.46831i
\(184\) 0 0
\(185\) 25.4558i 0.137599i
\(186\) 0 0
\(187\) −125.000 −0.668449
\(188\) 0 0
\(189\) − 141.421i − 0.748261i
\(190\) 0 0
\(191\) −293.000 −1.53403 −0.767016 0.641628i \(-0.778259\pi\)
−0.767016 + 0.641628i \(0.778259\pi\)
\(192\) 0 0
\(193\) 59.3970i 0.307756i 0.988090 + 0.153878i \(0.0491763\pi\)
−0.988090 + 0.153878i \(0.950824\pi\)
\(194\) 0 0
\(195\) −48.0000 −0.246154
\(196\) 0 0
\(197\) 70.0000 0.355330 0.177665 0.984091i \(-0.443146\pi\)
0.177665 + 0.984091i \(0.443146\pi\)
\(198\) 0 0
\(199\) −173.000 −0.869347 −0.434673 0.900588i \(-0.643136\pi\)
−0.434673 + 0.900588i \(0.643136\pi\)
\(200\) 0 0
\(201\) −312.000 −1.55224
\(202\) 0 0
\(203\) 212.132i 1.04499i
\(204\) 0 0
\(205\) − 42.4264i − 0.206958i
\(206\) 0 0
\(207\) 10.0000 0.0483092
\(208\) 0 0
\(209\) 95.0000 0.454545
\(210\) 0 0
\(211\) 84.8528i 0.402146i 0.979576 + 0.201073i \(0.0644429\pi\)
−0.979576 + 0.201073i \(0.935557\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.00000 0.0232558
\(216\) 0 0
\(217\) − 212.132i − 0.977567i
\(218\) 0 0
\(219\) − 70.7107i − 0.322880i
\(220\) 0 0
\(221\) − 424.264i − 1.91975i
\(222\) 0 0
\(223\) 364.867i 1.63618i 0.575094 + 0.818088i \(0.304966\pi\)
−0.575094 + 0.818088i \(0.695034\pi\)
\(224\) 0 0
\(225\) −24.0000 −0.106667
\(226\) 0 0
\(227\) 67.8823i 0.299041i 0.988759 + 0.149520i \(0.0477730\pi\)
−0.988759 + 0.149520i \(0.952227\pi\)
\(228\) 0 0
\(229\) 145.000 0.633188 0.316594 0.948561i \(-0.397461\pi\)
0.316594 + 0.948561i \(0.397461\pi\)
\(230\) 0 0
\(231\) − 70.7107i − 0.306107i
\(232\) 0 0
\(233\) 335.000 1.43777 0.718884 0.695130i \(-0.244653\pi\)
0.718884 + 0.695130i \(0.244653\pi\)
\(234\) 0 0
\(235\) −5.00000 −0.0212766
\(236\) 0 0
\(237\) 120.000 0.506329
\(238\) 0 0
\(239\) −197.000 −0.824268 −0.412134 0.911123i \(-0.635216\pi\)
−0.412134 + 0.911123i \(0.635216\pi\)
\(240\) 0 0
\(241\) 296.985i 1.23230i 0.787628 + 0.616151i \(0.211309\pi\)
−0.787628 + 0.616151i \(0.788691\pi\)
\(242\) 0 0
\(243\) 53.7401i 0.221153i
\(244\) 0 0
\(245\) −24.0000 −0.0979592
\(246\) 0 0
\(247\) 322.441i 1.30543i
\(248\) 0 0
\(249\) − 367.696i − 1.47669i
\(250\) 0 0
\(251\) 173.000 0.689243 0.344622 0.938742i \(-0.388007\pi\)
0.344622 + 0.938742i \(0.388007\pi\)
\(252\) 0 0
\(253\) 50.0000 0.197628
\(254\) 0 0
\(255\) − 70.7107i − 0.277297i
\(256\) 0 0
\(257\) − 67.8823i − 0.264133i −0.991241 0.132067i \(-0.957839\pi\)
0.991241 0.132067i \(-0.0421613\pi\)
\(258\) 0 0
\(259\) − 127.279i − 0.491426i
\(260\) 0 0
\(261\) − 42.4264i − 0.162553i
\(262\) 0 0
\(263\) 355.000 1.34981 0.674905 0.737905i \(-0.264185\pi\)
0.674905 + 0.737905i \(0.264185\pi\)
\(264\) 0 0
\(265\) 25.4558i 0.0960598i
\(266\) 0 0
\(267\) 360.000 1.34831
\(268\) 0 0
\(269\) 381.838i 1.41947i 0.704468 + 0.709735i \(0.251186\pi\)
−0.704468 + 0.709735i \(0.748814\pi\)
\(270\) 0 0
\(271\) −110.000 −0.405904 −0.202952 0.979189i \(-0.565054\pi\)
−0.202952 + 0.979189i \(0.565054\pi\)
\(272\) 0 0
\(273\) 240.000 0.879121
\(274\) 0 0
\(275\) −120.000 −0.436364
\(276\) 0 0
\(277\) 265.000 0.956679 0.478339 0.878175i \(-0.341239\pi\)
0.478339 + 0.878175i \(0.341239\pi\)
\(278\) 0 0
\(279\) 42.4264i 0.152066i
\(280\) 0 0
\(281\) 424.264i 1.50984i 0.655819 + 0.754918i \(0.272323\pi\)
−0.655819 + 0.754918i \(0.727677\pi\)
\(282\) 0 0
\(283\) 125.000 0.441696 0.220848 0.975308i \(-0.429118\pi\)
0.220848 + 0.975308i \(0.429118\pi\)
\(284\) 0 0
\(285\) 53.7401i 0.188562i
\(286\) 0 0
\(287\) 212.132i 0.739136i
\(288\) 0 0
\(289\) 336.000 1.16263
\(290\) 0 0
\(291\) −48.0000 −0.164948
\(292\) 0 0
\(293\) − 186.676i − 0.637120i −0.947903 0.318560i \(-0.896801\pi\)
0.947903 0.318560i \(-0.103199\pi\)
\(294\) 0 0
\(295\) 84.8528i 0.287637i
\(296\) 0 0
\(297\) 141.421i 0.476166i
\(298\) 0 0
\(299\) 169.706i 0.567577i
\(300\) 0 0
\(301\) −25.0000 −0.0830565
\(302\) 0 0
\(303\) − 141.421i − 0.466737i
\(304\) 0 0
\(305\) −95.0000 −0.311475
\(306\) 0 0
\(307\) 280.014i 0.912099i 0.889955 + 0.456049i \(0.150736\pi\)
−0.889955 + 0.456049i \(0.849264\pi\)
\(308\) 0 0
\(309\) −48.0000 −0.155340
\(310\) 0 0
\(311\) 235.000 0.755627 0.377814 0.925882i \(-0.376676\pi\)
0.377814 + 0.925882i \(0.376676\pi\)
\(312\) 0 0
\(313\) −310.000 −0.990415 −0.495208 0.868775i \(-0.664908\pi\)
−0.495208 + 0.868775i \(0.664908\pi\)
\(314\) 0 0
\(315\) −5.00000 −0.0158730
\(316\) 0 0
\(317\) 186.676i 0.588884i 0.955669 + 0.294442i \(0.0951338\pi\)
−0.955669 + 0.294442i \(0.904866\pi\)
\(318\) 0 0
\(319\) − 212.132i − 0.664991i
\(320\) 0 0
\(321\) 288.000 0.897196
\(322\) 0 0
\(323\) −475.000 −1.47059
\(324\) 0 0
\(325\) − 407.294i − 1.25321i
\(326\) 0 0
\(327\) 360.000 1.10092
\(328\) 0 0
\(329\) 25.0000 0.0759878
\(330\) 0 0
\(331\) 296.985i 0.897235i 0.893724 + 0.448618i \(0.148083\pi\)
−0.893724 + 0.448618i \(0.851917\pi\)
\(332\) 0 0
\(333\) 25.4558i 0.0764440i
\(334\) 0 0
\(335\) 110.309i 0.329280i
\(336\) 0 0
\(337\) − 526.087i − 1.56109i −0.625099 0.780545i \(-0.714942\pi\)
0.625099 0.780545i \(-0.285058\pi\)
\(338\) 0 0
\(339\) −312.000 −0.920354
\(340\) 0 0
\(341\) 212.132i 0.622088i
\(342\) 0 0
\(343\) 365.000 1.06414
\(344\) 0 0
\(345\) 28.2843i 0.0819834i
\(346\) 0 0
\(347\) 125.000 0.360231 0.180115 0.983646i \(-0.442353\pi\)
0.180115 + 0.983646i \(0.442353\pi\)
\(348\) 0 0
\(349\) −23.0000 −0.0659026 −0.0329513 0.999457i \(-0.510491\pi\)
−0.0329513 + 0.999457i \(0.510491\pi\)
\(350\) 0 0
\(351\) −480.000 −1.36752
\(352\) 0 0
\(353\) 410.000 1.16147 0.580737 0.814092i \(-0.302765\pi\)
0.580737 + 0.814092i \(0.302765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 353.553i 0.990346i
\(358\) 0 0
\(359\) 475.000 1.32312 0.661560 0.749892i \(-0.269895\pi\)
0.661560 + 0.749892i \(0.269895\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) − 271.529i − 0.748014i
\(364\) 0 0
\(365\) −25.0000 −0.0684932
\(366\) 0 0
\(367\) −230.000 −0.626703 −0.313351 0.949637i \(-0.601452\pi\)
−0.313351 + 0.949637i \(0.601452\pi\)
\(368\) 0 0
\(369\) − 42.4264i − 0.114977i
\(370\) 0 0
\(371\) − 127.279i − 0.343071i
\(372\) 0 0
\(373\) 67.8823i 0.181990i 0.995851 + 0.0909950i \(0.0290047\pi\)
−0.995851 + 0.0909950i \(0.970995\pi\)
\(374\) 0 0
\(375\) − 138.593i − 0.369581i
\(376\) 0 0
\(377\) 720.000 1.90981
\(378\) 0 0
\(379\) − 254.558i − 0.671658i −0.941923 0.335829i \(-0.890984\pi\)
0.941923 0.335829i \(-0.109016\pi\)
\(380\) 0 0
\(381\) 648.000 1.70079
\(382\) 0 0
\(383\) 144.250i 0.376631i 0.982109 + 0.188316i \(0.0603028\pi\)
−0.982109 + 0.188316i \(0.939697\pi\)
\(384\) 0 0
\(385\) −25.0000 −0.0649351
\(386\) 0 0
\(387\) 5.00000 0.0129199
\(388\) 0 0
\(389\) 553.000 1.42159 0.710797 0.703397i \(-0.248334\pi\)
0.710797 + 0.703397i \(0.248334\pi\)
\(390\) 0 0
\(391\) −250.000 −0.639386
\(392\) 0 0
\(393\) − 461.034i − 1.17311i
\(394\) 0 0
\(395\) − 42.4264i − 0.107409i
\(396\) 0 0
\(397\) −335.000 −0.843829 −0.421914 0.906636i \(-0.638642\pi\)
−0.421914 + 0.906636i \(0.638642\pi\)
\(398\) 0 0
\(399\) − 268.701i − 0.673435i
\(400\) 0 0
\(401\) 212.132i 0.529008i 0.964385 + 0.264504i \(0.0852082\pi\)
−0.964385 + 0.264504i \(0.914792\pi\)
\(402\) 0 0
\(403\) −720.000 −1.78660
\(404\) 0 0
\(405\) −71.0000 −0.175309
\(406\) 0 0
\(407\) 127.279i 0.312725i
\(408\) 0 0
\(409\) 721.249i 1.76344i 0.471769 + 0.881722i \(0.343616\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(410\) 0 0
\(411\) 268.701i 0.653773i
\(412\) 0 0
\(413\) − 424.264i − 1.02727i
\(414\) 0 0
\(415\) −130.000 −0.313253
\(416\) 0 0
\(417\) 353.553i 0.847850i
\(418\) 0 0
\(419\) 62.0000 0.147971 0.0739857 0.997259i \(-0.476428\pi\)
0.0739857 + 0.997259i \(0.476428\pi\)
\(420\) 0 0
\(421\) − 296.985i − 0.705427i −0.935731 0.352714i \(-0.885259\pi\)
0.935731 0.352714i \(-0.114741\pi\)
\(422\) 0 0
\(423\) −5.00000 −0.0118203
\(424\) 0 0
\(425\) 600.000 1.41176
\(426\) 0 0
\(427\) 475.000 1.11241
\(428\) 0 0
\(429\) −240.000 −0.559441
\(430\) 0 0
\(431\) 509.117i 1.18125i 0.806948 + 0.590623i \(0.201118\pi\)
−0.806948 + 0.590623i \(0.798882\pi\)
\(432\) 0 0
\(433\) − 229.103i − 0.529105i −0.964371 0.264553i \(-0.914776\pi\)
0.964371 0.264553i \(-0.0852243\pi\)
\(434\) 0 0
\(435\) 120.000 0.275862
\(436\) 0 0
\(437\) 190.000 0.434783
\(438\) 0 0
\(439\) − 806.102i − 1.83622i −0.396323 0.918111i \(-0.629714\pi\)
0.396323 0.918111i \(-0.370286\pi\)
\(440\) 0 0
\(441\) −24.0000 −0.0544218
\(442\) 0 0
\(443\) 365.000 0.823928 0.411964 0.911200i \(-0.364843\pi\)
0.411964 + 0.911200i \(0.364843\pi\)
\(444\) 0 0
\(445\) − 127.279i − 0.286021i
\(446\) 0 0
\(447\) − 608.112i − 1.36043i
\(448\) 0 0
\(449\) − 763.675i − 1.70084i −0.526108 0.850418i \(-0.676349\pi\)
0.526108 0.850418i \(-0.323651\pi\)
\(450\) 0 0
\(451\) − 212.132i − 0.470359i
\(452\) 0 0
\(453\) −240.000 −0.529801
\(454\) 0 0
\(455\) − 84.8528i − 0.186490i
\(456\) 0 0
\(457\) −265.000 −0.579869 −0.289934 0.957047i \(-0.593633\pi\)
−0.289934 + 0.957047i \(0.593633\pi\)
\(458\) 0 0
\(459\) − 707.107i − 1.54054i
\(460\) 0 0
\(461\) 553.000 1.19957 0.599783 0.800163i \(-0.295254\pi\)
0.599783 + 0.800163i \(0.295254\pi\)
\(462\) 0 0
\(463\) −485.000 −1.04752 −0.523758 0.851867i \(-0.675470\pi\)
−0.523758 + 0.851867i \(0.675470\pi\)
\(464\) 0 0
\(465\) −120.000 −0.258065
\(466\) 0 0
\(467\) −115.000 −0.246253 −0.123126 0.992391i \(-0.539292\pi\)
−0.123126 + 0.992391i \(0.539292\pi\)
\(468\) 0 0
\(469\) − 551.543i − 1.17600i
\(470\) 0 0
\(471\) 537.401i 1.14098i
\(472\) 0 0
\(473\) 25.0000 0.0528541
\(474\) 0 0
\(475\) −456.000 −0.960000
\(476\) 0 0
\(477\) 25.4558i 0.0533665i
\(478\) 0 0
\(479\) 490.000 1.02296 0.511482 0.859294i \(-0.329097\pi\)
0.511482 + 0.859294i \(0.329097\pi\)
\(480\) 0 0
\(481\) −432.000 −0.898129
\(482\) 0 0
\(483\) − 141.421i − 0.292798i
\(484\) 0 0
\(485\) 16.9706i 0.0349909i
\(486\) 0 0
\(487\) 610.940i 1.25450i 0.778819 + 0.627249i \(0.215819\pi\)
−0.778819 + 0.627249i \(0.784181\pi\)
\(488\) 0 0
\(489\) 311.127i 0.636252i
\(490\) 0 0
\(491\) −82.0000 −0.167006 −0.0835031 0.996508i \(-0.526611\pi\)
−0.0835031 + 0.996508i \(0.526611\pi\)
\(492\) 0 0
\(493\) 1060.66i 2.15144i
\(494\) 0 0
\(495\) 5.00000 0.0101010
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 485.000 0.971944 0.485972 0.873974i \(-0.338466\pi\)
0.485972 + 0.873974i \(0.338466\pi\)
\(500\) 0 0
\(501\) 168.000 0.335329
\(502\) 0 0
\(503\) 250.000 0.497018 0.248509 0.968630i \(-0.420059\pi\)
0.248509 + 0.968630i \(0.420059\pi\)
\(504\) 0 0
\(505\) −50.0000 −0.0990099
\(506\) 0 0
\(507\) − 336.583i − 0.663871i
\(508\) 0 0
\(509\) 169.706i 0.333410i 0.986007 + 0.166705i \(0.0533127\pi\)
−0.986007 + 0.166705i \(0.946687\pi\)
\(510\) 0 0
\(511\) 125.000 0.244618
\(512\) 0 0
\(513\) 537.401i 1.04757i
\(514\) 0 0
\(515\) 16.9706i 0.0329525i
\(516\) 0 0
\(517\) −25.0000 −0.0483559
\(518\) 0 0
\(519\) −528.000 −1.01734
\(520\) 0 0
\(521\) − 127.279i − 0.244298i −0.992512 0.122149i \(-0.961021\pi\)
0.992512 0.122149i \(-0.0389786\pi\)
\(522\) 0 0
\(523\) 356.382i 0.681418i 0.940169 + 0.340709i \(0.110667\pi\)
−0.940169 + 0.340709i \(0.889333\pi\)
\(524\) 0 0
\(525\) 339.411i 0.646498i
\(526\) 0 0
\(527\) − 1060.66i − 2.01264i
\(528\) 0 0
\(529\) −429.000 −0.810964
\(530\) 0 0
\(531\) 84.8528i 0.159798i
\(532\) 0 0
\(533\) 720.000 1.35084
\(534\) 0 0
\(535\) − 101.823i − 0.190324i
\(536\) 0 0
\(537\) 360.000 0.670391
\(538\) 0 0
\(539\) −120.000 −0.222635
\(540\) 0 0
\(541\) 25.0000 0.0462107 0.0231054 0.999733i \(-0.492645\pi\)
0.0231054 + 0.999733i \(0.492645\pi\)
\(542\) 0 0
\(543\) −720.000 −1.32597
\(544\) 0 0
\(545\) − 127.279i − 0.233540i
\(546\) 0 0
\(547\) 16.9706i 0.0310248i 0.999880 + 0.0155124i \(0.00493795\pi\)
−0.999880 + 0.0155124i \(0.995062\pi\)
\(548\) 0 0
\(549\) −95.0000 −0.173042
\(550\) 0 0
\(551\) − 806.102i − 1.46298i
\(552\) 0 0
\(553\) 212.132i 0.383602i
\(554\) 0 0
\(555\) −72.0000 −0.129730
\(556\) 0 0
\(557\) 745.000 1.33752 0.668761 0.743477i \(-0.266825\pi\)
0.668761 + 0.743477i \(0.266825\pi\)
\(558\) 0 0
\(559\) 84.8528i 0.151794i
\(560\) 0 0
\(561\) − 353.553i − 0.630220i
\(562\) 0 0
\(563\) 313.955i 0.557647i 0.960342 + 0.278824i \(0.0899445\pi\)
−0.960342 + 0.278824i \(0.910056\pi\)
\(564\) 0 0
\(565\) 110.309i 0.195237i
\(566\) 0 0
\(567\) 355.000 0.626102
\(568\) 0 0
\(569\) 424.264i 0.745631i 0.927905 + 0.372816i \(0.121608\pi\)
−0.927905 + 0.372816i \(0.878392\pi\)
\(570\) 0 0
\(571\) 1070.00 1.87391 0.936953 0.349456i \(-0.113634\pi\)
0.936953 + 0.349456i \(0.113634\pi\)
\(572\) 0 0
\(573\) − 828.729i − 1.44630i
\(574\) 0 0
\(575\) −240.000 −0.417391
\(576\) 0 0
\(577\) −25.0000 −0.0433276 −0.0216638 0.999765i \(-0.506896\pi\)
−0.0216638 + 0.999765i \(0.506896\pi\)
\(578\) 0 0
\(579\) −168.000 −0.290155
\(580\) 0 0
\(581\) 650.000 1.11876
\(582\) 0 0
\(583\) 127.279i 0.218318i
\(584\) 0 0
\(585\) 16.9706i 0.0290095i
\(586\) 0 0
\(587\) 725.000 1.23509 0.617547 0.786534i \(-0.288127\pi\)
0.617547 + 0.786534i \(0.288127\pi\)
\(588\) 0 0
\(589\) 806.102i 1.36859i
\(590\) 0 0
\(591\) 197.990i 0.335008i
\(592\) 0 0
\(593\) 650.000 1.09612 0.548061 0.836439i \(-0.315366\pi\)
0.548061 + 0.836439i \(0.315366\pi\)
\(594\) 0 0
\(595\) 125.000 0.210084
\(596\) 0 0
\(597\) − 489.318i − 0.819628i
\(598\) 0 0
\(599\) 296.985i 0.495801i 0.968785 + 0.247901i \(0.0797406\pi\)
−0.968785 + 0.247901i \(0.920259\pi\)
\(600\) 0 0
\(601\) − 848.528i − 1.41186i −0.708281 0.705930i \(-0.750529\pi\)
0.708281 0.705930i \(-0.249471\pi\)
\(602\) 0 0
\(603\) 110.309i 0.182933i
\(604\) 0 0
\(605\) −96.0000 −0.158678
\(606\) 0 0
\(607\) 271.529i 0.447329i 0.974666 + 0.223665i \(0.0718021\pi\)
−0.974666 + 0.223665i \(0.928198\pi\)
\(608\) 0 0
\(609\) −600.000 −0.985222
\(610\) 0 0
\(611\) − 84.8528i − 0.138875i
\(612\) 0 0
\(613\) −1055.00 −1.72104 −0.860522 0.509413i \(-0.829862\pi\)
−0.860522 + 0.509413i \(0.829862\pi\)
\(614\) 0 0
\(615\) 120.000 0.195122
\(616\) 0 0
\(617\) −505.000 −0.818476 −0.409238 0.912428i \(-0.634206\pi\)
−0.409238 + 0.912428i \(0.634206\pi\)
\(618\) 0 0
\(619\) −130.000 −0.210016 −0.105008 0.994471i \(-0.533487\pi\)
−0.105008 + 0.994471i \(0.533487\pi\)
\(620\) 0 0
\(621\) 282.843i 0.455463i
\(622\) 0 0
\(623\) 636.396i 1.02150i
\(624\) 0 0
\(625\) 551.000 0.881600
\(626\) 0 0
\(627\) 268.701i 0.428550i
\(628\) 0 0
\(629\) − 636.396i − 1.01176i
\(630\) 0 0
\(631\) 475.000 0.752773 0.376387 0.926463i \(-0.377166\pi\)
0.376387 + 0.926463i \(0.377166\pi\)
\(632\) 0 0
\(633\) −240.000 −0.379147
\(634\) 0 0
\(635\) − 229.103i − 0.360791i
\(636\) 0 0
\(637\) − 407.294i − 0.639393i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 848.528i − 1.32376i −0.749611 0.661878i \(-0.769760\pi\)
0.749611 0.661878i \(-0.230240\pi\)
\(642\) 0 0
\(643\) −955.000 −1.48523 −0.742613 0.669721i \(-0.766414\pi\)
−0.742613 + 0.669721i \(0.766414\pi\)
\(644\) 0 0
\(645\) 14.1421i 0.0219258i
\(646\) 0 0
\(647\) −965.000 −1.49150 −0.745750 0.666226i \(-0.767908\pi\)
−0.745750 + 0.666226i \(0.767908\pi\)
\(648\) 0 0
\(649\) 424.264i 0.653720i
\(650\) 0 0
\(651\) 600.000 0.921659
\(652\) 0 0
\(653\) −935.000 −1.43185 −0.715926 0.698176i \(-0.753995\pi\)
−0.715926 + 0.698176i \(0.753995\pi\)
\(654\) 0 0
\(655\) −163.000 −0.248855
\(656\) 0 0
\(657\) −25.0000 −0.0380518
\(658\) 0 0
\(659\) − 84.8528i − 0.128760i −0.997925 0.0643800i \(-0.979493\pi\)
0.997925 0.0643800i \(-0.0205070\pi\)
\(660\) 0 0
\(661\) 678.823i 1.02696i 0.858101 + 0.513481i \(0.171644\pi\)
−0.858101 + 0.513481i \(0.828356\pi\)
\(662\) 0 0
\(663\) 1200.00 1.80995
\(664\) 0 0
\(665\) −95.0000 −0.142857
\(666\) 0 0
\(667\) − 424.264i − 0.636078i
\(668\) 0 0
\(669\) −1032.00 −1.54260
\(670\) 0 0
\(671\) −475.000 −0.707899
\(672\) 0 0
\(673\) − 186.676i − 0.277379i −0.990336 0.138690i \(-0.955711\pi\)
0.990336 0.138690i \(-0.0442890\pi\)
\(674\) 0 0
\(675\) − 678.823i − 1.00566i
\(676\) 0 0
\(677\) − 907.925i − 1.34110i −0.741864 0.670550i \(-0.766058\pi\)
0.741864 0.670550i \(-0.233942\pi\)
\(678\) 0 0
\(679\) − 84.8528i − 0.124967i
\(680\) 0 0
\(681\) −192.000 −0.281938
\(682\) 0 0
\(683\) 1120.06i 1.63991i 0.572429 + 0.819954i \(0.306001\pi\)
−0.572429 + 0.819954i \(0.693999\pi\)
\(684\) 0 0
\(685\) 95.0000 0.138686
\(686\) 0 0
\(687\) 410.122i 0.596975i
\(688\) 0 0
\(689\) −432.000 −0.626996
\(690\) 0 0
\(691\) −715.000 −1.03473 −0.517366 0.855764i \(-0.673087\pi\)
−0.517366 + 0.855764i \(0.673087\pi\)
\(692\) 0 0
\(693\) −25.0000 −0.0360750
\(694\) 0 0
\(695\) 125.000 0.179856
\(696\) 0 0
\(697\) 1060.66i 1.52175i
\(698\) 0 0
\(699\) 947.523i 1.35554i
\(700\) 0 0
\(701\) 430.000 0.613409 0.306705 0.951805i \(-0.400774\pi\)
0.306705 + 0.951805i \(0.400774\pi\)
\(702\) 0 0
\(703\) 483.661i 0.687996i
\(704\) 0 0
\(705\) − 14.1421i − 0.0200598i
\(706\) 0 0
\(707\) 250.000 0.353607
\(708\) 0 0
\(709\) 382.000 0.538787 0.269394 0.963030i \(-0.413177\pi\)
0.269394 + 0.963030i \(0.413177\pi\)
\(710\) 0 0
\(711\) − 42.4264i − 0.0596715i
\(712\) 0 0
\(713\) 424.264i 0.595041i
\(714\) 0 0
\(715\) 84.8528i 0.118675i
\(716\) 0 0
\(717\) − 557.200i − 0.777127i
\(718\) 0 0
\(719\) 115.000 0.159944 0.0799722 0.996797i \(-0.474517\pi\)
0.0799722 + 0.996797i \(0.474517\pi\)
\(720\) 0 0
\(721\) − 84.8528i − 0.117688i
\(722\) 0 0
\(723\) −840.000 −1.16183
\(724\) 0 0
\(725\) 1018.23i 1.40446i
\(726\) 0 0
\(727\) 1075.00 1.47868 0.739340 0.673333i \(-0.235138\pi\)
0.739340 + 0.673333i \(0.235138\pi\)
\(728\) 0 0
\(729\) −791.000 −1.08505
\(730\) 0 0
\(731\) −125.000 −0.170999
\(732\) 0 0
\(733\) −530.000 −0.723056 −0.361528 0.932361i \(-0.617745\pi\)
−0.361528 + 0.932361i \(0.617745\pi\)
\(734\) 0 0
\(735\) − 67.8823i − 0.0923568i
\(736\) 0 0
\(737\) 551.543i 0.748363i
\(738\) 0 0
\(739\) −547.000 −0.740189 −0.370095 0.928994i \(-0.620675\pi\)
−0.370095 + 0.928994i \(0.620675\pi\)
\(740\) 0 0
\(741\) −912.000 −1.23077
\(742\) 0 0
\(743\) 958.837i 1.29049i 0.763974 + 0.645247i \(0.223245\pi\)
−0.763974 + 0.645247i \(0.776755\pi\)
\(744\) 0 0
\(745\) −215.000 −0.288591
\(746\) 0 0
\(747\) −130.000 −0.174029
\(748\) 0 0
\(749\) 509.117i 0.679729i
\(750\) 0 0
\(751\) 169.706i 0.225973i 0.993597 + 0.112986i \(0.0360417\pi\)
−0.993597 + 0.112986i \(0.963958\pi\)
\(752\) 0 0
\(753\) 489.318i 0.649825i
\(754\) 0 0
\(755\) 84.8528i 0.112388i
\(756\) 0 0
\(757\) −1055.00 −1.39366 −0.696830 0.717237i \(-0.745407\pi\)
−0.696830 + 0.717237i \(0.745407\pi\)
\(758\) 0 0
\(759\) 141.421i 0.186326i
\(760\) 0 0
\(761\) 215.000 0.282523 0.141261 0.989972i \(-0.454884\pi\)
0.141261 + 0.989972i \(0.454884\pi\)
\(762\) 0 0
\(763\) 636.396i 0.834071i
\(764\) 0 0
\(765\) −25.0000 −0.0326797
\(766\) 0 0
\(767\) −1440.00 −1.87744
\(768\) 0 0
\(769\) −145.000 −0.188557 −0.0942783 0.995546i \(-0.530054\pi\)
−0.0942783 + 0.995546i \(0.530054\pi\)
\(770\) 0 0
\(771\) 192.000 0.249027
\(772\) 0 0
\(773\) − 407.294i − 0.526900i −0.964673 0.263450i \(-0.915140\pi\)
0.964673 0.263450i \(-0.0848604\pi\)
\(774\) 0 0
\(775\) − 1018.23i − 1.31385i
\(776\) 0 0
\(777\) 360.000 0.463320
\(778\) 0 0
\(779\) − 806.102i − 1.03479i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1200.00 1.53257
\(784\) 0 0
\(785\) 190.000 0.242038
\(786\) 0 0
\(787\) 186.676i 0.237200i 0.992942 + 0.118600i \(0.0378406\pi\)
−0.992942 + 0.118600i \(0.962159\pi\)
\(788\) 0 0
\(789\) 1004.09i 1.27261i
\(790\) 0 0
\(791\) − 551.543i − 0.697273i
\(792\) 0 0
\(793\) − 1612.20i − 2.03304i
\(794\) 0 0
\(795\) −72.0000 −0.0905660
\(796\) 0 0
\(797\) 704.278i 0.883662i 0.897098 + 0.441831i \(0.145671\pi\)
−0.897098 + 0.441831i \(0.854329\pi\)
\(798\) 0 0
\(799\) 125.000 0.156446
\(800\) 0 0
\(801\) − 127.279i − 0.158900i
\(802\) 0 0
\(803\) −125.000 −0.155666
\(804\) 0 0
\(805\) −50.0000 −0.0621118
\(806\) 0 0
\(807\) −1080.00 −1.33829
\(808\) 0 0
\(809\) −457.000 −0.564895 −0.282447 0.959283i \(-0.591146\pi\)
−0.282447 + 0.959283i \(0.591146\pi\)
\(810\) 0 0
\(811\) 509.117i 0.627764i 0.949462 + 0.313882i \(0.101630\pi\)
−0.949462 + 0.313882i \(0.898370\pi\)
\(812\) 0 0
\(813\) − 311.127i − 0.382690i
\(814\) 0 0
\(815\) 110.000 0.134969
\(816\) 0 0
\(817\) 95.0000 0.116279
\(818\) 0 0
\(819\) − 84.8528i − 0.103605i
\(820\) 0 0
\(821\) −167.000 −0.203410 −0.101705 0.994815i \(-0.532430\pi\)
−0.101705 + 0.994815i \(0.532430\pi\)
\(822\) 0 0
\(823\) 1315.00 1.59781 0.798906 0.601455i \(-0.205412\pi\)
0.798906 + 0.601455i \(0.205412\pi\)
\(824\) 0 0
\(825\) − 339.411i − 0.411408i
\(826\) 0 0
\(827\) − 534.573i − 0.646400i −0.946331 0.323200i \(-0.895241\pi\)
0.946331 0.323200i \(-0.104759\pi\)
\(828\) 0 0
\(829\) − 763.675i − 0.921201i −0.887608 0.460600i \(-0.847634\pi\)
0.887608 0.460600i \(-0.152366\pi\)
\(830\) 0 0
\(831\) 749.533i 0.901965i
\(832\) 0 0
\(833\) 600.000 0.720288
\(834\) 0 0
\(835\) − 59.3970i − 0.0711341i
\(836\) 0 0
\(837\) −1200.00 −1.43369
\(838\) 0 0
\(839\) 339.411i 0.404543i 0.979330 + 0.202271i \(0.0648323\pi\)
−0.979330 + 0.202271i \(0.935168\pi\)
\(840\) 0 0
\(841\) −959.000 −1.14031
\(842\) 0 0
\(843\) −1200.00 −1.42349
\(844\) 0 0
\(845\) −119.000 −0.140828
\(846\) 0 0
\(847\) 480.000 0.566706
\(848\) 0 0
\(849\) 353.553i 0.416435i
\(850\) 0 0
\(851\) 254.558i 0.299129i
\(852\) 0 0
\(853\) −770.000 −0.902696 −0.451348 0.892348i \(-0.649057\pi\)
−0.451348 + 0.892348i \(0.649057\pi\)
\(854\) 0 0
\(855\) 19.0000 0.0222222
\(856\) 0 0
\(857\) − 1255.82i − 1.46537i −0.680568 0.732685i \(-0.738267\pi\)
0.680568 0.732685i \(-0.261733\pi\)
\(858\) 0 0
\(859\) 557.000 0.648428 0.324214 0.945984i \(-0.394900\pi\)
0.324214 + 0.945984i \(0.394900\pi\)
\(860\) 0 0
\(861\) −600.000 −0.696864
\(862\) 0 0
\(863\) − 992.778i − 1.15038i −0.818020 0.575190i \(-0.804928\pi\)
0.818020 0.575190i \(-0.195072\pi\)
\(864\) 0 0
\(865\) 186.676i 0.215811i
\(866\) 0 0
\(867\) 950.352i 1.09614i
\(868\) 0 0
\(869\) − 212.132i − 0.244111i
\(870\) 0 0
\(871\) −1872.00 −2.14925
\(872\) 0 0
\(873\) 16.9706i 0.0194394i
\(874\) 0 0
\(875\) 245.000 0.280000
\(876\) 0 0
\(877\) − 186.676i − 0.212858i −0.994320 0.106429i \(-0.966058\pi\)
0.994320 0.106429i \(-0.0339416\pi\)
\(878\) 0 0
\(879\) 528.000 0.600683
\(880\) 0 0
\(881\) −25.0000 −0.0283768 −0.0141884 0.999899i \(-0.504516\pi\)
−0.0141884 + 0.999899i \(0.504516\pi\)
\(882\) 0 0
\(883\) 965.000 1.09287 0.546433 0.837503i \(-0.315985\pi\)
0.546433 + 0.837503i \(0.315985\pi\)
\(884\) 0 0
\(885\) −240.000 −0.271186
\(886\) 0 0
\(887\) 780.646i 0.880097i 0.897974 + 0.440048i \(0.145039\pi\)
−0.897974 + 0.440048i \(0.854961\pi\)
\(888\) 0 0
\(889\) 1145.51i 1.28854i
\(890\) 0 0
\(891\) −355.000 −0.398429
\(892\) 0 0
\(893\) −95.0000 −0.106383
\(894\) 0 0
\(895\) − 127.279i − 0.142211i
\(896\) 0 0
\(897\) −480.000 −0.535117
\(898\) 0 0
\(899\) 1800.00 2.00222
\(900\) 0 0
\(901\) − 636.396i − 0.706322i
\(902\) 0 0
\(903\) − 70.7107i − 0.0783064i
\(904\) 0 0
\(905\) 254.558i 0.281280i
\(906\) 0 0
\(907\) − 313.955i − 0.346147i −0.984909 0.173074i \(-0.944630\pi\)
0.984909 0.173074i \(-0.0553698\pi\)
\(908\) 0 0
\(909\) −50.0000 −0.0550055
\(910\) 0 0
\(911\) − 933.381i − 1.02457i −0.858816 0.512284i \(-0.828800\pi\)
0.858816 0.512284i \(-0.171200\pi\)
\(912\) 0 0
\(913\) −650.000 −0.711939
\(914\) 0 0
\(915\) − 268.701i − 0.293662i
\(916\) 0 0
\(917\) 815.000 0.888768
\(918\) 0 0
\(919\) 538.000 0.585419 0.292709 0.956201i \(-0.405443\pi\)
0.292709 + 0.956201i \(0.405443\pi\)
\(920\) 0 0
\(921\) −792.000 −0.859935
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 610.940i − 0.660476i
\(926\) 0 0
\(927\) 16.9706i 0.0183070i
\(928\) 0 0
\(929\) −742.000 −0.798708 −0.399354 0.916797i \(-0.630766\pi\)
−0.399354 + 0.916797i \(0.630766\pi\)
\(930\) 0 0
\(931\) −456.000 −0.489796
\(932\) 0 0
\(933\) 664.680i 0.712412i
\(934\) 0 0
\(935\) −125.000 −0.133690
\(936\) 0 0
\(937\) 335.000 0.357524 0.178762 0.983892i \(-0.442791\pi\)
0.178762 + 0.983892i \(0.442791\pi\)
\(938\) 0 0
\(939\) − 876.812i − 0.933773i
\(940\) 0 0
\(941\) 424.264i 0.450865i 0.974259 + 0.225433i \(0.0723795\pi\)
−0.974259 + 0.225433i \(0.927620\pi\)
\(942\) 0 0
\(943\) − 424.264i − 0.449909i
\(944\) 0 0
\(945\) − 141.421i − 0.149652i
\(946\) 0 0
\(947\) −1210.00 −1.27772 −0.638860 0.769323i \(-0.720594\pi\)
−0.638860 + 0.769323i \(0.720594\pi\)
\(948\) 0 0
\(949\) − 424.264i − 0.447064i
\(950\) 0 0
\(951\) −528.000 −0.555205
\(952\) 0 0
\(953\) 992.778i 1.04174i 0.853636 + 0.520870i \(0.174392\pi\)
−0.853636 + 0.520870i \(0.825608\pi\)
\(954\) 0 0
\(955\) −293.000 −0.306806
\(956\) 0 0
\(957\) 600.000 0.626959
\(958\) 0 0
\(959\) −475.000 −0.495308
\(960\) 0 0
\(961\) −839.000 −0.873049
\(962\) 0 0
\(963\) − 101.823i − 0.105736i
\(964\) 0 0
\(965\) 59.3970i 0.0615513i
\(966\) 0 0
\(967\) −350.000 −0.361944 −0.180972 0.983488i \(-0.557924\pi\)
−0.180972 + 0.983488i \(0.557924\pi\)
\(968\) 0 0
\(969\) − 1343.50i − 1.38648i
\(970\) 0 0
\(971\) 254.558i 0.262161i 0.991372 + 0.131081i \(0.0418447\pi\)
−0.991372 + 0.131081i \(0.958155\pi\)
\(972\) 0 0
\(973\) −625.000 −0.642343
\(974\) 0 0
\(975\) 1152.00 1.18154
\(976\) 0 0
\(977\) − 398.808i − 0.408197i −0.978950 0.204098i \(-0.934574\pi\)
0.978950 0.204098i \(-0.0654262\pi\)
\(978\) 0 0
\(979\) − 636.396i − 0.650047i
\(980\) 0 0
\(981\) − 127.279i − 0.129744i
\(982\) 0 0
\(983\) 695.793i 0.707826i 0.935278 + 0.353913i \(0.115149\pi\)
−0.935278 + 0.353913i \(0.884851\pi\)
\(984\) 0 0
\(985\) 70.0000 0.0710660
\(986\) 0 0
\(987\) 70.7107i 0.0716420i
\(988\) 0 0
\(989\) 50.0000 0.0505561
\(990\) 0 0
\(991\) − 381.838i − 0.385305i −0.981267 0.192653i \(-0.938291\pi\)
0.981267 0.192653i \(-0.0617091\pi\)
\(992\) 0 0
\(993\) −840.000 −0.845921
\(994\) 0 0
\(995\) −173.000 −0.173869
\(996\) 0 0
\(997\) 265.000 0.265797 0.132899 0.991130i \(-0.457572\pi\)
0.132899 + 0.991130i \(0.457572\pi\)
\(998\) 0 0
\(999\) −720.000 −0.720721
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 1216.3.e.i.1025.2 2
4.3 odd 2 1216.3.e.j.1025.1 2
8.3 odd 2 38.3.b.a.37.2 yes 2
8.5 even 2 304.3.e.c.113.1 2
19.18 odd 2 inner 1216.3.e.i.1025.1 2
24.5 odd 2 2736.3.o.h.721.1 2
24.11 even 2 342.3.d.a.37.1 2
40.3 even 4 950.3.d.a.949.3 4
40.19 odd 2 950.3.c.a.151.1 2
40.27 even 4 950.3.d.a.949.2 4
76.75 even 2 1216.3.e.j.1025.2 2
152.37 odd 2 304.3.e.c.113.2 2
152.75 even 2 38.3.b.a.37.1 2
456.227 odd 2 342.3.d.a.37.2 2
456.341 even 2 2736.3.o.h.721.2 2
760.227 odd 4 950.3.d.a.949.4 4
760.379 even 2 950.3.c.a.151.2 2
760.683 odd 4 950.3.d.a.949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 152.75 even 2
38.3.b.a.37.2 yes 2 8.3 odd 2
304.3.e.c.113.1 2 8.5 even 2
304.3.e.c.113.2 2 152.37 odd 2
342.3.d.a.37.1 2 24.11 even 2
342.3.d.a.37.2 2 456.227 odd 2
950.3.c.a.151.1 2 40.19 odd 2
950.3.c.a.151.2 2 760.379 even 2
950.3.d.a.949.1 4 760.683 odd 4
950.3.d.a.949.2 4 40.27 even 4
950.3.d.a.949.3 4 40.3 even 4
950.3.d.a.949.4 4 760.227 odd 4
1216.3.e.i.1025.1 2 19.18 odd 2 inner
1216.3.e.i.1025.2 2 1.1 even 1 trivial
1216.3.e.j.1025.1 2 4.3 odd 2
1216.3.e.j.1025.2 2 76.75 even 2
2736.3.o.h.721.1 2 24.5 odd 2
2736.3.o.h.721.2 2 456.341 even 2