Properties

Label 224.3.g.b.15.2
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.2
Root \(-1.67467 - 1.09337i\) of \(x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.b.15.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.56747 q^{3} +5.73252i q^{5} +2.64575i q^{7} +11.8618 q^{9} +O(q^{10})\) \(q-4.56747 q^{3} +5.73252i q^{5} +2.64575i q^{7} +11.8618 q^{9} +1.40065 q^{11} -19.0821i q^{13} -26.1831i q^{15} -32.2699 q^{17} -12.5675 q^{19} -12.0844i q^{21} -15.8893i q^{23} -7.86180 q^{25} -13.0712 q^{27} -3.29194i q^{29} -22.6705i q^{31} -6.39741 q^{33} -15.1668 q^{35} -54.1537i q^{37} +87.1569i q^{39} -7.59607 q^{41} +20.8478 q^{43} +67.9980i q^{45} -21.6384i q^{47} -7.00000 q^{49} +147.392 q^{51} +0.356667i q^{53} +8.02924i q^{55} +57.4016 q^{57} -26.8583 q^{59} +86.2287i q^{61} +31.3834i q^{63} +109.389 q^{65} -114.523 q^{67} +72.5739i q^{69} +104.792i q^{71} -24.3974 q^{73} +35.9085 q^{75} +3.70576i q^{77} -117.128i q^{79} -47.0539 q^{81} -79.2706 q^{83} -184.988i q^{85} +15.0359i q^{87} +2.66078 q^{89} +50.4865 q^{91} +103.547i q^{93} -72.0433i q^{95} -52.0930 q^{97} +16.6142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 48q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −4.56747 −1.52249 −0.761245 0.648464i \(-0.775412\pi\)
−0.761245 + 0.648464i \(0.775412\pi\)
\(4\) 0 0
\(5\) 5.73252i 1.14650i 0.819379 + 0.573252i \(0.194318\pi\)
−0.819379 + 0.573252i \(0.805682\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) 11.8618 1.31798
\(10\) 0 0
\(11\) 1.40065 0.127332 0.0636658 0.997971i \(-0.479721\pi\)
0.0636658 + 0.997971i \(0.479721\pi\)
\(12\) 0 0
\(13\) − 19.0821i − 1.46785i −0.679228 0.733927i \(-0.737685\pi\)
0.679228 0.733927i \(-0.262315\pi\)
\(14\) 0 0
\(15\) − 26.1831i − 1.74554i
\(16\) 0 0
\(17\) −32.2699 −1.89823 −0.949114 0.314932i \(-0.898018\pi\)
−0.949114 + 0.314932i \(0.898018\pi\)
\(18\) 0 0
\(19\) −12.5675 −0.661446 −0.330723 0.943728i \(-0.607293\pi\)
−0.330723 + 0.943728i \(0.607293\pi\)
\(20\) 0 0
\(21\) − 12.0844i − 0.575447i
\(22\) 0 0
\(23\) − 15.8893i − 0.690839i −0.938448 0.345419i \(-0.887737\pi\)
0.938448 0.345419i \(-0.112263\pi\)
\(24\) 0 0
\(25\) −7.86180 −0.314472
\(26\) 0 0
\(27\) −13.0712 −0.484118
\(28\) 0 0
\(29\) − 3.29194i − 0.113515i −0.998388 0.0567576i \(-0.981924\pi\)
0.998388 0.0567576i \(-0.0180762\pi\)
\(30\) 0 0
\(31\) − 22.6705i − 0.731306i −0.930751 0.365653i \(-0.880846\pi\)
0.930751 0.365653i \(-0.119154\pi\)
\(32\) 0 0
\(33\) −6.39741 −0.193861
\(34\) 0 0
\(35\) −15.1668 −0.433338
\(36\) 0 0
\(37\) − 54.1537i − 1.46361i −0.681512 0.731807i \(-0.738677\pi\)
0.681512 0.731807i \(-0.261323\pi\)
\(38\) 0 0
\(39\) 87.1569i 2.23479i
\(40\) 0 0
\(41\) −7.59607 −0.185270 −0.0926350 0.995700i \(-0.529529\pi\)
−0.0926350 + 0.995700i \(0.529529\pi\)
\(42\) 0 0
\(43\) 20.8478 0.484833 0.242417 0.970172i \(-0.422060\pi\)
0.242417 + 0.970172i \(0.422060\pi\)
\(44\) 0 0
\(45\) 67.9980i 1.51107i
\(46\) 0 0
\(47\) − 21.6384i − 0.460392i −0.973144 0.230196i \(-0.926063\pi\)
0.973144 0.230196i \(-0.0739367\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 147.392 2.89004
\(52\) 0 0
\(53\) 0.356667i 0.00672957i 0.999994 + 0.00336479i \(0.00107105\pi\)
−0.999994 + 0.00336479i \(0.998929\pi\)
\(54\) 0 0
\(55\) 8.02924i 0.145986i
\(56\) 0 0
\(57\) 57.4016 1.00705
\(58\) 0 0
\(59\) −26.8583 −0.455226 −0.227613 0.973752i \(-0.573092\pi\)
−0.227613 + 0.973752i \(0.573092\pi\)
\(60\) 0 0
\(61\) 86.2287i 1.41359i 0.707420 + 0.706793i \(0.249859\pi\)
−0.707420 + 0.706793i \(0.750141\pi\)
\(62\) 0 0
\(63\) 31.3834i 0.498149i
\(64\) 0 0
\(65\) 109.389 1.68290
\(66\) 0 0
\(67\) −114.523 −1.70929 −0.854646 0.519211i \(-0.826226\pi\)
−0.854646 + 0.519211i \(0.826226\pi\)
\(68\) 0 0
\(69\) 72.5739i 1.05180i
\(70\) 0 0
\(71\) 104.792i 1.47594i 0.674834 + 0.737969i \(0.264215\pi\)
−0.674834 + 0.737969i \(0.735785\pi\)
\(72\) 0 0
\(73\) −24.3974 −0.334211 −0.167106 0.985939i \(-0.553442\pi\)
−0.167106 + 0.985939i \(0.553442\pi\)
\(74\) 0 0
\(75\) 35.9085 0.478781
\(76\) 0 0
\(77\) 3.70576i 0.0481268i
\(78\) 0 0
\(79\) − 117.128i − 1.48263i −0.671157 0.741315i \(-0.734202\pi\)
0.671157 0.741315i \(-0.265798\pi\)
\(80\) 0 0
\(81\) −47.0539 −0.580913
\(82\) 0 0
\(83\) −79.2706 −0.955067 −0.477534 0.878614i \(-0.658469\pi\)
−0.477534 + 0.878614i \(0.658469\pi\)
\(84\) 0 0
\(85\) − 184.988i − 2.17633i
\(86\) 0 0
\(87\) 15.0359i 0.172826i
\(88\) 0 0
\(89\) 2.66078 0.0298964 0.0149482 0.999888i \(-0.495242\pi\)
0.0149482 + 0.999888i \(0.495242\pi\)
\(90\) 0 0
\(91\) 50.4865 0.554797
\(92\) 0 0
\(93\) 103.547i 1.11341i
\(94\) 0 0
\(95\) − 72.0433i − 0.758350i
\(96\) 0 0
\(97\) −52.0930 −0.537042 −0.268521 0.963274i \(-0.586535\pi\)
−0.268521 + 0.963274i \(0.586535\pi\)
\(98\) 0 0
\(99\) 16.6142 0.167820
\(100\) 0 0
\(101\) 91.4742i 0.905685i 0.891591 + 0.452842i \(0.149590\pi\)
−0.891591 + 0.452842i \(0.850410\pi\)
\(102\) 0 0
\(103\) − 39.7891i − 0.386302i −0.981169 0.193151i \(-0.938129\pi\)
0.981169 0.193151i \(-0.0618708\pi\)
\(104\) 0 0
\(105\) 69.2740 0.659753
\(106\) 0 0
\(107\) −82.6631 −0.772552 −0.386276 0.922383i \(-0.626239\pi\)
−0.386276 + 0.922383i \(0.626239\pi\)
\(108\) 0 0
\(109\) − 29.4719i − 0.270384i −0.990819 0.135192i \(-0.956835\pi\)
0.990819 0.135192i \(-0.0431652\pi\)
\(110\) 0 0
\(111\) 247.346i 2.22834i
\(112\) 0 0
\(113\) 159.133 1.40826 0.704130 0.710071i \(-0.251337\pi\)
0.704130 + 0.710071i \(0.251337\pi\)
\(114\) 0 0
\(115\) 91.0857 0.792049
\(116\) 0 0
\(117\) − 226.348i − 1.93460i
\(118\) 0 0
\(119\) − 85.3781i − 0.717463i
\(120\) 0 0
\(121\) −119.038 −0.983787
\(122\) 0 0
\(123\) 34.6948 0.282072
\(124\) 0 0
\(125\) 98.2451i 0.785961i
\(126\) 0 0
\(127\) − 16.0834i − 0.126641i −0.997993 0.0633205i \(-0.979831\pi\)
0.997993 0.0633205i \(-0.0201690\pi\)
\(128\) 0 0
\(129\) −95.2219 −0.738154
\(130\) 0 0
\(131\) 118.136 0.901799 0.450899 0.892575i \(-0.351103\pi\)
0.450899 + 0.892575i \(0.351103\pi\)
\(132\) 0 0
\(133\) − 33.2504i − 0.250003i
\(134\) 0 0
\(135\) − 74.9308i − 0.555043i
\(136\) 0 0
\(137\) −19.1708 −0.139933 −0.0699664 0.997549i \(-0.522289\pi\)
−0.0699664 + 0.997549i \(0.522289\pi\)
\(138\) 0 0
\(139\) −104.954 −0.755062 −0.377531 0.925997i \(-0.623227\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(140\) 0 0
\(141\) 98.8329i 0.700942i
\(142\) 0 0
\(143\) − 26.7273i − 0.186904i
\(144\) 0 0
\(145\) 18.8711 0.130146
\(146\) 0 0
\(147\) 31.9723 0.217499
\(148\) 0 0
\(149\) − 82.3906i − 0.552957i −0.961020 0.276478i \(-0.910833\pi\)
0.961020 0.276478i \(-0.0891674\pi\)
\(150\) 0 0
\(151\) 57.7395i 0.382381i 0.981553 + 0.191190i \(0.0612347\pi\)
−0.981553 + 0.191190i \(0.938765\pi\)
\(152\) 0 0
\(153\) −382.779 −2.50182
\(154\) 0 0
\(155\) 129.959 0.838445
\(156\) 0 0
\(157\) 3.72975i 0.0237564i 0.999929 + 0.0118782i \(0.00378104\pi\)
−0.999929 + 0.0118782i \(0.996219\pi\)
\(158\) 0 0
\(159\) − 1.62907i − 0.0102457i
\(160\) 0 0
\(161\) 42.0391 0.261112
\(162\) 0 0
\(163\) −77.7069 −0.476729 −0.238365 0.971176i \(-0.576611\pi\)
−0.238365 + 0.971176i \(0.576611\pi\)
\(164\) 0 0
\(165\) − 36.6733i − 0.222262i
\(166\) 0 0
\(167\) − 62.0837i − 0.371759i −0.982573 0.185879i \(-0.940487\pi\)
0.982573 0.185879i \(-0.0595133\pi\)
\(168\) 0 0
\(169\) −195.127 −1.15459
\(170\) 0 0
\(171\) −149.073 −0.871771
\(172\) 0 0
\(173\) − 195.614i − 1.13072i −0.824846 0.565358i \(-0.808738\pi\)
0.824846 0.565358i \(-0.191262\pi\)
\(174\) 0 0
\(175\) − 20.8004i − 0.118859i
\(176\) 0 0
\(177\) 122.675 0.693077
\(178\) 0 0
\(179\) −72.2099 −0.403407 −0.201704 0.979447i \(-0.564648\pi\)
−0.201704 + 0.979447i \(0.564648\pi\)
\(180\) 0 0
\(181\) − 140.980i − 0.778895i −0.921049 0.389448i \(-0.872666\pi\)
0.921049 0.389448i \(-0.127334\pi\)
\(182\) 0 0
\(183\) − 393.847i − 2.15217i
\(184\) 0 0
\(185\) 310.437 1.67804
\(186\) 0 0
\(187\) −45.1987 −0.241704
\(188\) 0 0
\(189\) − 34.5831i − 0.182979i
\(190\) 0 0
\(191\) 284.473i 1.48939i 0.667407 + 0.744693i \(0.267404\pi\)
−0.667407 + 0.744693i \(0.732596\pi\)
\(192\) 0 0
\(193\) −123.850 −0.641710 −0.320855 0.947128i \(-0.603970\pi\)
−0.320855 + 0.947128i \(0.603970\pi\)
\(194\) 0 0
\(195\) −499.629 −2.56220
\(196\) 0 0
\(197\) 108.098i 0.548721i 0.961627 + 0.274361i \(0.0884662\pi\)
−0.961627 + 0.274361i \(0.911534\pi\)
\(198\) 0 0
\(199\) − 331.854i − 1.66761i −0.552060 0.833804i \(-0.686158\pi\)
0.552060 0.833804i \(-0.313842\pi\)
\(200\) 0 0
\(201\) 523.079 2.60238
\(202\) 0 0
\(203\) 8.70966 0.0429047
\(204\) 0 0
\(205\) − 43.5446i − 0.212413i
\(206\) 0 0
\(207\) − 188.476i − 0.910510i
\(208\) 0 0
\(209\) −17.6026 −0.0842229
\(210\) 0 0
\(211\) −26.3950 −0.125095 −0.0625475 0.998042i \(-0.519922\pi\)
−0.0625475 + 0.998042i \(0.519922\pi\)
\(212\) 0 0
\(213\) − 478.633i − 2.24710i
\(214\) 0 0
\(215\) 119.511i 0.555864i
\(216\) 0 0
\(217\) 59.9804 0.276408
\(218\) 0 0
\(219\) 111.434 0.508833
\(220\) 0 0
\(221\) 615.777i 2.78632i
\(222\) 0 0
\(223\) 161.183i 0.722796i 0.932412 + 0.361398i \(0.117700\pi\)
−0.932412 + 0.361398i \(0.882300\pi\)
\(224\) 0 0
\(225\) −93.2551 −0.414467
\(226\) 0 0
\(227\) 171.279 0.754533 0.377266 0.926105i \(-0.376864\pi\)
0.377266 + 0.926105i \(0.376864\pi\)
\(228\) 0 0
\(229\) 229.251i 1.00110i 0.865709 + 0.500548i \(0.166868\pi\)
−0.865709 + 0.500548i \(0.833132\pi\)
\(230\) 0 0
\(231\) − 16.9260i − 0.0732726i
\(232\) 0 0
\(233\) −270.154 −1.15946 −0.579730 0.814808i \(-0.696842\pi\)
−0.579730 + 0.814808i \(0.696842\pi\)
\(234\) 0 0
\(235\) 124.043 0.527841
\(236\) 0 0
\(237\) 534.978i 2.25729i
\(238\) 0 0
\(239\) 157.155i 0.657551i 0.944408 + 0.328776i \(0.106636\pi\)
−0.944408 + 0.328776i \(0.893364\pi\)
\(240\) 0 0
\(241\) −97.7124 −0.405445 −0.202723 0.979236i \(-0.564979\pi\)
−0.202723 + 0.979236i \(0.564979\pi\)
\(242\) 0 0
\(243\) 332.558 1.36855
\(244\) 0 0
\(245\) − 40.1276i − 0.163786i
\(246\) 0 0
\(247\) 239.814i 0.970906i
\(248\) 0 0
\(249\) 362.066 1.45408
\(250\) 0 0
\(251\) 313.145 1.24759 0.623796 0.781587i \(-0.285590\pi\)
0.623796 + 0.781587i \(0.285590\pi\)
\(252\) 0 0
\(253\) − 22.2553i − 0.0879655i
\(254\) 0 0
\(255\) 844.927i 3.31344i
\(256\) 0 0
\(257\) −348.855 −1.35741 −0.678707 0.734409i \(-0.737459\pi\)
−0.678707 + 0.734409i \(0.737459\pi\)
\(258\) 0 0
\(259\) 143.277 0.553194
\(260\) 0 0
\(261\) − 39.0484i − 0.149611i
\(262\) 0 0
\(263\) 384.364i 1.46146i 0.682667 + 0.730729i \(0.260820\pi\)
−0.682667 + 0.730729i \(0.739180\pi\)
\(264\) 0 0
\(265\) −2.04460 −0.00771548
\(266\) 0 0
\(267\) −12.1530 −0.0455170
\(268\) 0 0
\(269\) − 37.7613i − 0.140376i −0.997534 0.0701882i \(-0.977640\pi\)
0.997534 0.0701882i \(-0.0223600\pi\)
\(270\) 0 0
\(271\) − 308.730i − 1.13922i −0.821914 0.569612i \(-0.807093\pi\)
0.821914 0.569612i \(-0.192907\pi\)
\(272\) 0 0
\(273\) −230.596 −0.844673
\(274\) 0 0
\(275\) −11.0116 −0.0400422
\(276\) 0 0
\(277\) − 244.210i − 0.881623i −0.897600 0.440812i \(-0.854691\pi\)
0.897600 0.440812i \(-0.145309\pi\)
\(278\) 0 0
\(279\) − 268.913i − 0.963845i
\(280\) 0 0
\(281\) 266.569 0.948646 0.474323 0.880351i \(-0.342693\pi\)
0.474323 + 0.880351i \(0.342693\pi\)
\(282\) 0 0
\(283\) −165.605 −0.585177 −0.292589 0.956238i \(-0.594517\pi\)
−0.292589 + 0.956238i \(0.594517\pi\)
\(284\) 0 0
\(285\) 329.056i 1.15458i
\(286\) 0 0
\(287\) − 20.0973i − 0.0700255i
\(288\) 0 0
\(289\) 752.346 2.60327
\(290\) 0 0
\(291\) 237.933 0.817641
\(292\) 0 0
\(293\) − 34.3652i − 0.117288i −0.998279 0.0586438i \(-0.981322\pi\)
0.998279 0.0586438i \(-0.0186776\pi\)
\(294\) 0 0
\(295\) − 153.966i − 0.521918i
\(296\) 0 0
\(297\) −18.3081 −0.0616434
\(298\) 0 0
\(299\) −303.201 −1.01405
\(300\) 0 0
\(301\) 55.1582i 0.183250i
\(302\) 0 0
\(303\) − 417.806i − 1.37890i
\(304\) 0 0
\(305\) −494.308 −1.62068
\(306\) 0 0
\(307\) 222.934 0.726170 0.363085 0.931756i \(-0.381724\pi\)
0.363085 + 0.931756i \(0.381724\pi\)
\(308\) 0 0
\(309\) 181.736i 0.588141i
\(310\) 0 0
\(311\) 419.934i 1.35027i 0.737694 + 0.675135i \(0.235915\pi\)
−0.737694 + 0.675135i \(0.764085\pi\)
\(312\) 0 0
\(313\) −293.869 −0.938878 −0.469439 0.882965i \(-0.655544\pi\)
−0.469439 + 0.882965i \(0.655544\pi\)
\(314\) 0 0
\(315\) −179.906 −0.571130
\(316\) 0 0
\(317\) − 423.461i − 1.33584i −0.744234 0.667919i \(-0.767185\pi\)
0.744234 0.667919i \(-0.232815\pi\)
\(318\) 0 0
\(319\) − 4.61085i − 0.0144541i
\(320\) 0 0
\(321\) 377.561 1.17620
\(322\) 0 0
\(323\) 405.551 1.25558
\(324\) 0 0
\(325\) 150.020i 0.461599i
\(326\) 0 0
\(327\) 134.612i 0.411658i
\(328\) 0 0
\(329\) 57.2499 0.174012
\(330\) 0 0
\(331\) 126.666 0.382678 0.191339 0.981524i \(-0.438717\pi\)
0.191339 + 0.981524i \(0.438717\pi\)
\(332\) 0 0
\(333\) − 642.361i − 1.92901i
\(334\) 0 0
\(335\) − 656.503i − 1.95971i
\(336\) 0 0
\(337\) 302.404 0.897341 0.448671 0.893697i \(-0.351898\pi\)
0.448671 + 0.893697i \(0.351898\pi\)
\(338\) 0 0
\(339\) −726.838 −2.14406
\(340\) 0 0
\(341\) − 31.7533i − 0.0931183i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −416.031 −1.20589
\(346\) 0 0
\(347\) −320.532 −0.923724 −0.461862 0.886952i \(-0.652818\pi\)
−0.461862 + 0.886952i \(0.652818\pi\)
\(348\) 0 0
\(349\) − 380.678i − 1.09077i −0.838186 0.545385i \(-0.816384\pi\)
0.838186 0.545385i \(-0.183616\pi\)
\(350\) 0 0
\(351\) 249.426i 0.710614i
\(352\) 0 0
\(353\) −364.369 −1.03221 −0.516104 0.856526i \(-0.672618\pi\)
−0.516104 + 0.856526i \(0.672618\pi\)
\(354\) 0 0
\(355\) −600.720 −1.69217
\(356\) 0 0
\(357\) 389.962i 1.09233i
\(358\) 0 0
\(359\) 111.995i 0.311965i 0.987760 + 0.155982i \(0.0498543\pi\)
−0.987760 + 0.155982i \(0.950146\pi\)
\(360\) 0 0
\(361\) −203.059 −0.562489
\(362\) 0 0
\(363\) 543.704 1.49781
\(364\) 0 0
\(365\) − 139.859i − 0.383174i
\(366\) 0 0
\(367\) − 439.042i − 1.19630i −0.801384 0.598150i \(-0.795903\pi\)
0.801384 0.598150i \(-0.204097\pi\)
\(368\) 0 0
\(369\) −90.1030 −0.244182
\(370\) 0 0
\(371\) −0.943653 −0.00254354
\(372\) 0 0
\(373\) 254.996i 0.683637i 0.939766 + 0.341818i \(0.111043\pi\)
−0.939766 + 0.341818i \(0.888957\pi\)
\(374\) 0 0
\(375\) − 448.732i − 1.19662i
\(376\) 0 0
\(377\) −62.8172 −0.166624
\(378\) 0 0
\(379\) −603.048 −1.59116 −0.795578 0.605852i \(-0.792833\pi\)
−0.795578 + 0.605852i \(0.792833\pi\)
\(380\) 0 0
\(381\) 73.4605i 0.192810i
\(382\) 0 0
\(383\) − 73.3855i − 0.191607i −0.995400 0.0958035i \(-0.969458\pi\)
0.995400 0.0958035i \(-0.0305420\pi\)
\(384\) 0 0
\(385\) −21.2434 −0.0551776
\(386\) 0 0
\(387\) 247.293 0.639000
\(388\) 0 0
\(389\) 340.800i 0.876092i 0.898953 + 0.438046i \(0.144329\pi\)
−0.898953 + 0.438046i \(0.855671\pi\)
\(390\) 0 0
\(391\) 512.745i 1.31137i
\(392\) 0 0
\(393\) −539.581 −1.37298
\(394\) 0 0
\(395\) 671.438 1.69984
\(396\) 0 0
\(397\) − 111.540i − 0.280957i −0.990084 0.140478i \(-0.955136\pi\)
0.990084 0.140478i \(-0.0448640\pi\)
\(398\) 0 0
\(399\) 151.870i 0.380627i
\(400\) 0 0
\(401\) 340.535 0.849215 0.424607 0.905378i \(-0.360412\pi\)
0.424607 + 0.905378i \(0.360412\pi\)
\(402\) 0 0
\(403\) −432.600 −1.07345
\(404\) 0 0
\(405\) − 269.738i − 0.666019i
\(406\) 0 0
\(407\) − 75.8502i − 0.186364i
\(408\) 0 0
\(409\) 666.959 1.63071 0.815354 0.578963i \(-0.196543\pi\)
0.815354 + 0.578963i \(0.196543\pi\)
\(410\) 0 0
\(411\) 87.5620 0.213046
\(412\) 0 0
\(413\) − 71.0604i − 0.172059i
\(414\) 0 0
\(415\) − 454.420i − 1.09499i
\(416\) 0 0
\(417\) 479.373 1.14958
\(418\) 0 0
\(419\) 200.191 0.477783 0.238891 0.971046i \(-0.423216\pi\)
0.238891 + 0.971046i \(0.423216\pi\)
\(420\) 0 0
\(421\) − 15.9136i − 0.0377996i −0.999821 0.0188998i \(-0.993984\pi\)
0.999821 0.0188998i \(-0.00601636\pi\)
\(422\) 0 0
\(423\) − 256.671i − 0.606786i
\(424\) 0 0
\(425\) 253.699 0.596940
\(426\) 0 0
\(427\) −228.140 −0.534285
\(428\) 0 0
\(429\) 122.076i 0.284560i
\(430\) 0 0
\(431\) − 628.013i − 1.45711i −0.684989 0.728553i \(-0.740193\pi\)
0.684989 0.728553i \(-0.259807\pi\)
\(432\) 0 0
\(433\) 789.232 1.82271 0.911353 0.411625i \(-0.135039\pi\)
0.911353 + 0.411625i \(0.135039\pi\)
\(434\) 0 0
\(435\) −86.1933 −0.198146
\(436\) 0 0
\(437\) 199.688i 0.456952i
\(438\) 0 0
\(439\) 665.570i 1.51610i 0.652194 + 0.758052i \(0.273849\pi\)
−0.652194 + 0.758052i \(0.726151\pi\)
\(440\) 0 0
\(441\) −83.0326 −0.188283
\(442\) 0 0
\(443\) −507.152 −1.14481 −0.572406 0.819970i \(-0.693990\pi\)
−0.572406 + 0.819970i \(0.693990\pi\)
\(444\) 0 0
\(445\) 15.2530i 0.0342764i
\(446\) 0 0
\(447\) 376.317i 0.841872i
\(448\) 0 0
\(449\) −279.029 −0.621446 −0.310723 0.950501i \(-0.600571\pi\)
−0.310723 + 0.950501i \(0.600571\pi\)
\(450\) 0 0
\(451\) −10.6394 −0.0235907
\(452\) 0 0
\(453\) − 263.723i − 0.582171i
\(454\) 0 0
\(455\) 289.415i 0.636077i
\(456\) 0 0
\(457\) −720.881 −1.57742 −0.788710 0.614765i \(-0.789251\pi\)
−0.788710 + 0.614765i \(0.789251\pi\)
\(458\) 0 0
\(459\) 421.805 0.918966
\(460\) 0 0
\(461\) − 483.262i − 1.04829i −0.851629 0.524145i \(-0.824385\pi\)
0.851629 0.524145i \(-0.175615\pi\)
\(462\) 0 0
\(463\) − 39.6326i − 0.0855995i −0.999084 0.0427997i \(-0.986372\pi\)
0.999084 0.0427997i \(-0.0136277\pi\)
\(464\) 0 0
\(465\) −593.584 −1.27652
\(466\) 0 0
\(467\) −17.7868 −0.0380874 −0.0190437 0.999819i \(-0.506062\pi\)
−0.0190437 + 0.999819i \(0.506062\pi\)
\(468\) 0 0
\(469\) − 302.998i − 0.646052i
\(470\) 0 0
\(471\) − 17.0355i − 0.0361689i
\(472\) 0 0
\(473\) 29.2005 0.0617346
\(474\) 0 0
\(475\) 98.8029 0.208006
\(476\) 0 0
\(477\) 4.23072i 0.00886943i
\(478\) 0 0
\(479\) − 668.616i − 1.39586i −0.716166 0.697930i \(-0.754105\pi\)
0.716166 0.697930i \(-0.245895\pi\)
\(480\) 0 0
\(481\) −1033.37 −2.14837
\(482\) 0 0
\(483\) −192.012 −0.397541
\(484\) 0 0
\(485\) − 298.624i − 0.615720i
\(486\) 0 0
\(487\) 418.484i 0.859311i 0.902993 + 0.429656i \(0.141365\pi\)
−0.902993 + 0.429656i \(0.858635\pi\)
\(488\) 0 0
\(489\) 354.924 0.725816
\(490\) 0 0
\(491\) −381.031 −0.776030 −0.388015 0.921653i \(-0.626839\pi\)
−0.388015 + 0.921653i \(0.626839\pi\)
\(492\) 0 0
\(493\) 106.231i 0.215478i
\(494\) 0 0
\(495\) 95.2412i 0.192406i
\(496\) 0 0
\(497\) −277.253 −0.557852
\(498\) 0 0
\(499\) 438.392 0.878541 0.439271 0.898355i \(-0.355237\pi\)
0.439271 + 0.898355i \(0.355237\pi\)
\(500\) 0 0
\(501\) 283.565i 0.565999i
\(502\) 0 0
\(503\) − 754.754i − 1.50050i −0.661151 0.750252i \(-0.729932\pi\)
0.661151 0.750252i \(-0.270068\pi\)
\(504\) 0 0
\(505\) −524.378 −1.03837
\(506\) 0 0
\(507\) 891.235 1.75786
\(508\) 0 0
\(509\) − 494.029i − 0.970588i −0.874351 0.485294i \(-0.838713\pi\)
0.874351 0.485294i \(-0.161287\pi\)
\(510\) 0 0
\(511\) − 64.5495i − 0.126320i
\(512\) 0 0
\(513\) 164.272 0.320218
\(514\) 0 0
\(515\) 228.092 0.442897
\(516\) 0 0
\(517\) − 30.3078i − 0.0586224i
\(518\) 0 0
\(519\) 893.460i 1.72150i
\(520\) 0 0
\(521\) −32.8747 −0.0630993 −0.0315496 0.999502i \(-0.510044\pi\)
−0.0315496 + 0.999502i \(0.510044\pi\)
\(522\) 0 0
\(523\) 28.2755 0.0540640 0.0270320 0.999635i \(-0.491394\pi\)
0.0270320 + 0.999635i \(0.491394\pi\)
\(524\) 0 0
\(525\) 95.0051i 0.180962i
\(526\) 0 0
\(527\) 731.574i 1.38819i
\(528\) 0 0
\(529\) 276.531 0.522742
\(530\) 0 0
\(531\) −318.588 −0.599977
\(532\) 0 0
\(533\) 144.949i 0.271949i
\(534\) 0 0
\(535\) − 473.868i − 0.885735i
\(536\) 0 0
\(537\) 329.817 0.614183
\(538\) 0 0
\(539\) −9.80453 −0.0181902
\(540\) 0 0
\(541\) 1071.59i 1.98077i 0.138352 + 0.990383i \(0.455820\pi\)
−0.138352 + 0.990383i \(0.544180\pi\)
\(542\) 0 0
\(543\) 643.922i 1.18586i
\(544\) 0 0
\(545\) 168.948 0.309997
\(546\) 0 0
\(547\) 986.888 1.80418 0.902091 0.431545i \(-0.142032\pi\)
0.902091 + 0.431545i \(0.142032\pi\)
\(548\) 0 0
\(549\) 1022.83i 1.86307i
\(550\) 0 0
\(551\) 41.3714i 0.0750842i
\(552\) 0 0
\(553\) 309.891 0.560382
\(554\) 0 0
\(555\) −1417.91 −2.55480
\(556\) 0 0
\(557\) − 483.550i − 0.868133i −0.900881 0.434067i \(-0.857078\pi\)
0.900881 0.434067i \(-0.142922\pi\)
\(558\) 0 0
\(559\) − 397.821i − 0.711665i
\(560\) 0 0
\(561\) 206.444 0.367993
\(562\) 0 0
\(563\) −520.893 −0.925210 −0.462605 0.886564i \(-0.653085\pi\)
−0.462605 + 0.886564i \(0.653085\pi\)
\(564\) 0 0
\(565\) 912.236i 1.61458i
\(566\) 0 0
\(567\) − 124.493i − 0.219564i
\(568\) 0 0
\(569\) −732.959 −1.28815 −0.644077 0.764961i \(-0.722758\pi\)
−0.644077 + 0.764961i \(0.722758\pi\)
\(570\) 0 0
\(571\) 999.584 1.75058 0.875292 0.483595i \(-0.160669\pi\)
0.875292 + 0.483595i \(0.160669\pi\)
\(572\) 0 0
\(573\) − 1299.32i − 2.26758i
\(574\) 0 0
\(575\) 124.918i 0.217249i
\(576\) 0 0
\(577\) 465.859 0.807381 0.403690 0.914896i \(-0.367727\pi\)
0.403690 + 0.914896i \(0.367727\pi\)
\(578\) 0 0
\(579\) 565.682 0.976998
\(580\) 0 0
\(581\) − 209.730i − 0.360981i
\(582\) 0 0
\(583\) 0.499565i 0 0.000856887i
\(584\) 0 0
\(585\) 1297.54 2.21803
\(586\) 0 0
\(587\) 574.851 0.979303 0.489651 0.871918i \(-0.337124\pi\)
0.489651 + 0.871918i \(0.337124\pi\)
\(588\) 0 0
\(589\) 284.911i 0.483719i
\(590\) 0 0
\(591\) − 493.735i − 0.835423i
\(592\) 0 0
\(593\) 943.055 1.59031 0.795156 0.606405i \(-0.207389\pi\)
0.795156 + 0.606405i \(0.207389\pi\)
\(594\) 0 0
\(595\) 489.432 0.822574
\(596\) 0 0
\(597\) 1515.73i 2.53892i
\(598\) 0 0
\(599\) − 9.26699i − 0.0154708i −0.999970 0.00773538i \(-0.997538\pi\)
0.999970 0.00773538i \(-0.00246227\pi\)
\(600\) 0 0
\(601\) 57.7003 0.0960072 0.0480036 0.998847i \(-0.484714\pi\)
0.0480036 + 0.998847i \(0.484714\pi\)
\(602\) 0 0
\(603\) −1358.44 −2.25281
\(604\) 0 0
\(605\) − 682.389i − 1.12792i
\(606\) 0 0
\(607\) 1024.68i 1.68810i 0.536264 + 0.844050i \(0.319835\pi\)
−0.536264 + 0.844050i \(0.680165\pi\)
\(608\) 0 0
\(609\) −39.7811 −0.0653221
\(610\) 0 0
\(611\) −412.906 −0.675788
\(612\) 0 0
\(613\) − 404.818i − 0.660389i −0.943913 0.330195i \(-0.892886\pi\)
0.943913 0.330195i \(-0.107114\pi\)
\(614\) 0 0
\(615\) 198.889i 0.323396i
\(616\) 0 0
\(617\) 894.209 1.44928 0.724642 0.689125i \(-0.242005\pi\)
0.724642 + 0.689125i \(0.242005\pi\)
\(618\) 0 0
\(619\) 779.388 1.25911 0.629554 0.776957i \(-0.283238\pi\)
0.629554 + 0.776957i \(0.283238\pi\)
\(620\) 0 0
\(621\) 207.692i 0.334447i
\(622\) 0 0
\(623\) 7.03977i 0.0112998i
\(624\) 0 0
\(625\) −759.737 −1.21558
\(626\) 0 0
\(627\) 80.3993 0.128229
\(628\) 0 0
\(629\) 1747.53i 2.77827i
\(630\) 0 0
\(631\) 780.191i 1.23644i 0.786007 + 0.618218i \(0.212145\pi\)
−0.786007 + 0.618218i \(0.787855\pi\)
\(632\) 0 0
\(633\) 120.559 0.190456
\(634\) 0 0
\(635\) 92.1985 0.145194
\(636\) 0 0
\(637\) 133.575i 0.209693i
\(638\) 0 0
\(639\) 1243.02i 1.94525i
\(640\) 0 0
\(641\) −23.3139 −0.0363712 −0.0181856 0.999835i \(-0.505789\pi\)
−0.0181856 + 0.999835i \(0.505789\pi\)
\(642\) 0 0
\(643\) −530.706 −0.825360 −0.412680 0.910876i \(-0.635407\pi\)
−0.412680 + 0.910876i \(0.635407\pi\)
\(644\) 0 0
\(645\) − 545.862i − 0.846297i
\(646\) 0 0
\(647\) 213.435i 0.329883i 0.986303 + 0.164942i \(0.0527436\pi\)
−0.986303 + 0.164942i \(0.947256\pi\)
\(648\) 0 0
\(649\) −37.6190 −0.0579646
\(650\) 0 0
\(651\) −273.959 −0.420828
\(652\) 0 0
\(653\) 274.874i 0.420941i 0.977600 + 0.210470i \(0.0674995\pi\)
−0.977600 + 0.210470i \(0.932500\pi\)
\(654\) 0 0
\(655\) 677.215i 1.03392i
\(656\) 0 0
\(657\) −289.397 −0.440483
\(658\) 0 0
\(659\) −1234.48 −1.87327 −0.936633 0.350313i \(-0.886075\pi\)
−0.936633 + 0.350313i \(0.886075\pi\)
\(660\) 0 0
\(661\) − 582.733i − 0.881593i −0.897607 0.440797i \(-0.854696\pi\)
0.897607 0.440797i \(-0.145304\pi\)
\(662\) 0 0
\(663\) − 2812.54i − 4.24215i
\(664\) 0 0
\(665\) 190.609 0.286630
\(666\) 0 0
\(667\) −52.3066 −0.0784207
\(668\) 0 0
\(669\) − 736.201i − 1.10045i
\(670\) 0 0
\(671\) 120.776i 0.179994i
\(672\) 0 0
\(673\) −399.145 −0.593083 −0.296542 0.955020i \(-0.595833\pi\)
−0.296542 + 0.955020i \(0.595833\pi\)
\(674\) 0 0
\(675\) 102.763 0.152241
\(676\) 0 0
\(677\) 754.467i 1.11443i 0.830369 + 0.557214i \(0.188130\pi\)
−0.830369 + 0.557214i \(0.811870\pi\)
\(678\) 0 0
\(679\) − 137.825i − 0.202983i
\(680\) 0 0
\(681\) −782.312 −1.14877
\(682\) 0 0
\(683\) −288.264 −0.422055 −0.211028 0.977480i \(-0.567681\pi\)
−0.211028 + 0.977480i \(0.567681\pi\)
\(684\) 0 0
\(685\) − 109.897i − 0.160433i
\(686\) 0 0
\(687\) − 1047.10i − 1.52416i
\(688\) 0 0
\(689\) 6.80596 0.00987803
\(690\) 0 0
\(691\) 156.692 0.226761 0.113380 0.993552i \(-0.463832\pi\)
0.113380 + 0.993552i \(0.463832\pi\)
\(692\) 0 0
\(693\) 43.9570i 0.0634300i
\(694\) 0 0
\(695\) − 601.649i − 0.865682i
\(696\) 0 0
\(697\) 245.124 0.351685
\(698\) 0 0
\(699\) 1233.92 1.76527
\(700\) 0 0
\(701\) − 1126.50i − 1.60700i −0.595307 0.803498i \(-0.702970\pi\)
0.595307 0.803498i \(-0.297030\pi\)
\(702\) 0 0
\(703\) 680.575i 0.968102i
\(704\) 0 0
\(705\) −566.562 −0.803633
\(706\) 0 0
\(707\) −242.018 −0.342317
\(708\) 0 0
\(709\) − 1096.17i − 1.54608i −0.634356 0.773041i \(-0.718734\pi\)
0.634356 0.773041i \(-0.281266\pi\)
\(710\) 0 0
\(711\) − 1389.35i − 1.95407i
\(712\) 0 0
\(713\) −360.218 −0.505214
\(714\) 0 0
\(715\) 153.215 0.214286
\(716\) 0 0
\(717\) − 717.800i − 1.00112i
\(718\) 0 0
\(719\) − 605.362i − 0.841949i −0.907072 0.420975i \(-0.861688\pi\)
0.907072 0.420975i \(-0.138312\pi\)
\(720\) 0 0
\(721\) 105.272 0.146009
\(722\) 0 0
\(723\) 446.298 0.617287
\(724\) 0 0
\(725\) 25.8806i 0.0356974i
\(726\) 0 0
\(727\) 443.659i 0.610260i 0.952311 + 0.305130i \(0.0986999\pi\)
−0.952311 + 0.305130i \(0.901300\pi\)
\(728\) 0 0
\(729\) −1095.46 −1.50269
\(730\) 0 0
\(731\) −672.757 −0.920325
\(732\) 0 0
\(733\) 750.026i 1.02323i 0.859216 + 0.511614i \(0.170952\pi\)
−0.859216 + 0.511614i \(0.829048\pi\)
\(734\) 0 0
\(735\) 183.282i 0.249363i
\(736\) 0 0
\(737\) −160.406 −0.217647
\(738\) 0 0
\(739\) −619.293 −0.838015 −0.419007 0.907983i \(-0.637622\pi\)
−0.419007 + 0.907983i \(0.637622\pi\)
\(740\) 0 0
\(741\) − 1095.34i − 1.47819i
\(742\) 0 0
\(743\) 30.5255i 0.0410842i 0.999789 + 0.0205421i \(0.00653921\pi\)
−0.999789 + 0.0205421i \(0.993461\pi\)
\(744\) 0 0
\(745\) 472.306 0.633967
\(746\) 0 0
\(747\) −940.291 −1.25876
\(748\) 0 0
\(749\) − 218.706i − 0.291997i
\(750\) 0 0
\(751\) − 968.214i − 1.28923i −0.764506 0.644616i \(-0.777017\pi\)
0.764506 0.644616i \(-0.222983\pi\)
\(752\) 0 0
\(753\) −1430.28 −1.89945
\(754\) 0 0
\(755\) −330.993 −0.438401
\(756\) 0 0
\(757\) 1171.15i 1.54710i 0.633736 + 0.773550i \(0.281521\pi\)
−0.633736 + 0.773550i \(0.718479\pi\)
\(758\) 0 0
\(759\) 101.650i 0.133927i
\(760\) 0 0
\(761\) 235.996 0.310113 0.155057 0.987906i \(-0.450444\pi\)
0.155057 + 0.987906i \(0.450444\pi\)
\(762\) 0 0
\(763\) 77.9753 0.102196
\(764\) 0 0
\(765\) − 2194.29i − 2.86835i
\(766\) 0 0
\(767\) 512.513i 0.668205i
\(768\) 0 0
\(769\) −124.257 −0.161582 −0.0807912 0.996731i \(-0.525745\pi\)
−0.0807912 + 0.996731i \(0.525745\pi\)
\(770\) 0 0
\(771\) 1593.39 2.06665
\(772\) 0 0
\(773\) − 178.223i − 0.230560i −0.993333 0.115280i \(-0.963224\pi\)
0.993333 0.115280i \(-0.0367765\pi\)
\(774\) 0 0
\(775\) 178.231i 0.229975i
\(776\) 0 0
\(777\) −654.415 −0.842233
\(778\) 0 0
\(779\) 95.4634 0.122546
\(780\) 0 0
\(781\) 146.776i 0.187933i
\(782\) 0 0
\(783\) 43.0296i 0.0549548i
\(784\) 0 0
\(785\) −21.3809 −0.0272368
\(786\) 0 0
\(787\) 1107.90 1.40775 0.703873 0.710326i \(-0.251453\pi\)
0.703873 + 0.710326i \(0.251453\pi\)
\(788\) 0 0
\(789\) − 1755.57i − 2.22506i
\(790\) 0 0
\(791\) 421.028i 0.532273i
\(792\) 0 0
\(793\) 1645.43 2.07494
\(794\) 0 0
\(795\) 9.33867 0.0117468
\(796\) 0 0
\(797\) 1094.69i 1.37351i 0.726889 + 0.686755i \(0.240966\pi\)
−0.726889 + 0.686755i \(0.759034\pi\)
\(798\) 0 0
\(799\) 698.269i 0.873929i
\(800\) 0 0
\(801\) 31.5617 0.0394028
\(802\) 0 0
\(803\) −34.1722 −0.0425556
\(804\) 0 0
\(805\) 240.990i 0.299367i
\(806\) 0 0
\(807\) 172.473i 0.213722i
\(808\) 0 0
\(809\) −1386.75 −1.71416 −0.857079 0.515185i \(-0.827723\pi\)
−0.857079 + 0.515185i \(0.827723\pi\)
\(810\) 0 0
\(811\) 312.204 0.384962 0.192481 0.981301i \(-0.438347\pi\)
0.192481 + 0.981301i \(0.438347\pi\)
\(812\) 0 0
\(813\) 1410.11i 1.73446i
\(814\) 0 0
\(815\) − 445.456i − 0.546572i
\(816\) 0 0
\(817\) −262.005 −0.320691
\(818\) 0 0
\(819\) 598.861 0.731209
\(820\) 0 0
\(821\) 1092.89i 1.33117i 0.746324 + 0.665583i \(0.231817\pi\)
−0.746324 + 0.665583i \(0.768183\pi\)
\(822\) 0 0
\(823\) − 907.162i − 1.10226i −0.834419 0.551131i \(-0.814196\pi\)
0.834419 0.551131i \(-0.185804\pi\)
\(824\) 0 0
\(825\) 50.2952 0.0609638
\(826\) 0 0
\(827\) 607.144 0.734152 0.367076 0.930191i \(-0.380359\pi\)
0.367076 + 0.930191i \(0.380359\pi\)
\(828\) 0 0
\(829\) 427.969i 0.516247i 0.966112 + 0.258124i \(0.0831042\pi\)
−0.966112 + 0.258124i \(0.916896\pi\)
\(830\) 0 0
\(831\) 1115.42i 1.34226i
\(832\) 0 0
\(833\) 225.889 0.271176
\(834\) 0 0
\(835\) 355.896 0.426223
\(836\) 0 0
\(837\) 296.330i 0.354038i
\(838\) 0 0
\(839\) − 1133.09i − 1.35053i −0.737575 0.675265i \(-0.764029\pi\)
0.737575 0.675265i \(-0.235971\pi\)
\(840\) 0 0
\(841\) 830.163 0.987114
\(842\) 0 0
\(843\) −1217.55 −1.44430
\(844\) 0 0
\(845\) − 1118.57i − 1.32375i
\(846\) 0 0
\(847\) − 314.945i − 0.371836i
\(848\) 0 0
\(849\) 756.397 0.890927
\(850\) 0 0
\(851\) −860.464 −1.01112
\(852\) 0 0
\(853\) 169.502i 0.198712i 0.995052 + 0.0993562i \(0.0316783\pi\)
−0.995052 + 0.0993562i \(0.968322\pi\)
\(854\) 0 0
\(855\) − 854.563i − 0.999489i
\(856\) 0 0
\(857\) −234.079 −0.273138 −0.136569 0.990631i \(-0.543607\pi\)
−0.136569 + 0.990631i \(0.543607\pi\)
\(858\) 0 0
\(859\) −894.342 −1.04114 −0.520571 0.853818i \(-0.674281\pi\)
−0.520571 + 0.853818i \(0.674281\pi\)
\(860\) 0 0
\(861\) 91.7939i 0.106613i
\(862\) 0 0
\(863\) 778.580i 0.902178i 0.892479 + 0.451089i \(0.148964\pi\)
−0.892479 + 0.451089i \(0.851036\pi\)
\(864\) 0 0
\(865\) 1121.36 1.29637
\(866\) 0 0
\(867\) −3436.32 −3.96346
\(868\) 0 0
\(869\) − 164.055i − 0.188786i
\(870\) 0 0
\(871\) 2185.33i 2.50899i
\(872\) 0 0
\(873\) −617.917 −0.707809
\(874\) 0 0
\(875\) −259.932 −0.297065
\(876\) 0 0
\(877\) − 17.2780i − 0.0197013i −0.999951 0.00985064i \(-0.996864\pi\)
0.999951 0.00985064i \(-0.00313561\pi\)
\(878\) 0 0
\(879\) 156.962i 0.178569i
\(880\) 0 0
\(881\) 770.918 0.875049 0.437524 0.899207i \(-0.355855\pi\)
0.437524 + 0.899207i \(0.355855\pi\)
\(882\) 0 0
\(883\) −776.362 −0.879232 −0.439616 0.898186i \(-0.644886\pi\)
−0.439616 + 0.898186i \(0.644886\pi\)
\(884\) 0 0
\(885\) 703.235i 0.794616i
\(886\) 0 0
\(887\) 1630.80i 1.83856i 0.393603 + 0.919280i \(0.371228\pi\)
−0.393603 + 0.919280i \(0.628772\pi\)
\(888\) 0 0
\(889\) 42.5527 0.0478658
\(890\) 0 0
\(891\) −65.9059 −0.0739685
\(892\) 0 0
\(893\) 271.940i 0.304524i
\(894\) 0 0
\(895\) − 413.945i − 0.462508i
\(896\) 0 0
\(897\) 1384.86 1.54388
\(898\) 0 0
\(899\) −74.6299 −0.0830144
\(900\) 0 0
\(901\) − 11.5096i − 0.0127743i
\(902\) 0 0
\(903\) − 251.933i − 0.278996i
\(904\) 0 0
\(905\) 808.171 0.893007
\(906\) 0 0
\(907\) 953.863 1.05167 0.525834 0.850587i \(-0.323753\pi\)
0.525834 + 0.850587i \(0.323753\pi\)
\(908\) 0 0
\(909\) 1085.05i 1.19367i
\(910\) 0 0
\(911\) − 1681.15i − 1.84539i −0.385534 0.922694i \(-0.625983\pi\)
0.385534 0.922694i \(-0.374017\pi\)
\(912\) 0 0
\(913\) −111.030 −0.121610
\(914\) 0 0
\(915\) 2257.74 2.46747
\(916\) 0 0
\(917\) 312.557i 0.340848i
\(918\) 0 0
\(919\) − 504.991i − 0.549500i −0.961516 0.274750i \(-0.911405\pi\)
0.961516 0.274750i \(-0.0885952\pi\)
\(920\) 0 0
\(921\) −1018.24 −1.10559
\(922\) 0 0
\(923\) 1999.64 2.16646
\(924\) 0 0
\(925\) 425.746i 0.460266i
\(926\) 0 0
\(927\) − 471.971i − 0.509138i
\(928\) 0 0
\(929\) 983.851 1.05904 0.529521 0.848297i \(-0.322372\pi\)
0.529521 + 0.848297i \(0.322372\pi\)
\(930\) 0 0
\(931\) 87.9723 0.0944923
\(932\) 0 0
\(933\) − 1918.04i − 2.05577i
\(934\) 0 0
\(935\) − 259.103i − 0.277115i
\(936\) 0 0
\(937\) −389.648 −0.415846 −0.207923 0.978145i \(-0.566670\pi\)
−0.207923 + 0.978145i \(0.566670\pi\)
\(938\) 0 0
\(939\) 1342.24 1.42943
\(940\) 0 0
\(941\) 875.465i 0.930356i 0.885217 + 0.465178i \(0.154010\pi\)
−0.885217 + 0.465178i \(0.845990\pi\)
\(942\) 0 0
\(943\) 120.696i 0.127992i
\(944\) 0 0
\(945\) 198.248 0.209787
\(946\) 0 0
\(947\) 1251.29 1.32132 0.660661 0.750685i \(-0.270276\pi\)
0.660661 + 0.750685i \(0.270276\pi\)
\(948\) 0 0
\(949\) 465.554i 0.490573i
\(950\) 0 0
\(951\) 1934.14i 2.03380i
\(952\) 0 0
\(953\) −882.129 −0.925633 −0.462817 0.886454i \(-0.653161\pi\)
−0.462817 + 0.886454i \(0.653161\pi\)
\(954\) 0 0
\(955\) −1630.75 −1.70759
\(956\) 0 0
\(957\) 21.0599i 0.0220062i
\(958\) 0 0
\(959\) − 50.7211i − 0.0528896i
\(960\) 0 0
\(961\) 447.049 0.465192
\(962\) 0 0
\(963\) −980.533 −1.01821
\(964\) 0 0
\(965\) − 709.973i − 0.735724i
\(966\) 0 0
\(967\) − 1410.24i − 1.45836i −0.684320 0.729182i \(-0.739901\pi\)
0.684320 0.729182i \(-0.260099\pi\)
\(968\) 0 0
\(969\) −1852.34 −1.91160
\(970\) 0 0
\(971\) −678.550 −0.698815 −0.349408 0.936971i \(-0.613617\pi\)
−0.349408 + 0.936971i \(0.613617\pi\)
\(972\) 0 0
\(973\) − 277.681i − 0.285387i
\(974\) 0 0
\(975\) − 685.210i − 0.702780i
\(976\) 0 0
\(977\) 111.815 0.114447 0.0572235 0.998361i \(-0.481775\pi\)
0.0572235 + 0.998361i \(0.481775\pi\)
\(978\) 0 0
\(979\) 3.72682 0.00380676
\(980\) 0 0
\(981\) − 349.590i − 0.356360i
\(982\) 0 0
\(983\) 202.226i 0.205723i 0.994696 + 0.102862i \(0.0327999\pi\)
−0.994696 + 0.102862i \(0.967200\pi\)
\(984\) 0 0
\(985\) −619.675 −0.629111
\(986\) 0 0
\(987\) −261.487 −0.264931
\(988\) 0 0
\(989\) − 331.257i − 0.334942i
\(990\) 0 0
\(991\) − 189.064i − 0.190781i −0.995440 0.0953907i \(-0.969590\pi\)
0.995440 0.0953907i \(-0.0304100\pi\)
\(992\) 0 0
\(993\) −578.546 −0.582624
\(994\) 0 0
\(995\) 1902.36 1.91192
\(996\) 0 0
\(997\) − 1632.91i − 1.63783i −0.573917 0.818914i \(-0.694577\pi\)
0.573917 0.818914i \(-0.305423\pi\)
\(998\) 0 0
\(999\) 707.853i 0.708562i
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 224.3.g.b.15.2 8
3.2 odd 2 2016.3.g.b.1135.2 8
4.3 odd 2 56.3.g.b.43.1 8
7.6 odd 2 1568.3.g.m.687.7 8
8.3 odd 2 inner 224.3.g.b.15.1 8
8.5 even 2 56.3.g.b.43.2 yes 8
12.11 even 2 504.3.g.b.379.8 8
16.3 odd 4 1792.3.d.j.1023.14 16
16.5 even 4 1792.3.d.j.1023.13 16
16.11 odd 4 1792.3.d.j.1023.3 16
16.13 even 4 1792.3.d.j.1023.4 16
24.5 odd 2 504.3.g.b.379.7 8
24.11 even 2 2016.3.g.b.1135.7 8
28.3 even 6 392.3.k.n.275.7 16
28.11 odd 6 392.3.k.o.275.7 16
28.19 even 6 392.3.k.n.67.5 16
28.23 odd 6 392.3.k.o.67.5 16
28.27 even 2 392.3.g.m.99.1 8
56.5 odd 6 392.3.k.n.67.7 16
56.13 odd 2 392.3.g.m.99.2 8
56.27 even 2 1568.3.g.m.687.8 8
56.37 even 6 392.3.k.o.67.7 16
56.45 odd 6 392.3.k.n.275.5 16
56.53 even 6 392.3.k.o.275.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.1 8 4.3 odd 2
56.3.g.b.43.2 yes 8 8.5 even 2
224.3.g.b.15.1 8 8.3 odd 2 inner
224.3.g.b.15.2 8 1.1 even 1 trivial
392.3.g.m.99.1 8 28.27 even 2
392.3.g.m.99.2 8 56.13 odd 2
392.3.k.n.67.5 16 28.19 even 6
392.3.k.n.67.7 16 56.5 odd 6
392.3.k.n.275.5 16 56.45 odd 6
392.3.k.n.275.7 16 28.3 even 6
392.3.k.o.67.5 16 28.23 odd 6
392.3.k.o.67.7 16 56.37 even 6
392.3.k.o.275.5 16 56.53 even 6
392.3.k.o.275.7 16 28.11 odd 6
504.3.g.b.379.7 8 24.5 odd 2
504.3.g.b.379.8 8 12.11 even 2
1568.3.g.m.687.7 8 7.6 odd 2
1568.3.g.m.687.8 8 56.27 even 2
1792.3.d.j.1023.3 16 16.11 odd 4
1792.3.d.j.1023.4 16 16.13 even 4
1792.3.d.j.1023.13 16 16.5 even 4
1792.3.d.j.1023.14 16 16.3 odd 4
2016.3.g.b.1135.2 8 3.2 odd 2
2016.3.g.b.1135.7 8 24.11 even 2