Properties

Label 2352.4.a.ck.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.145408.2
Defining polynomial: \(x^{4} - 24 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.25358\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} -43.9023 q^{11} -83.5699 q^{13} +20.8796 q^{15} +10.4354 q^{17} -4.27642 q^{19} -160.077 q^{23} -76.5601 q^{25} -27.0000 q^{27} +9.93138 q^{29} +133.715 q^{31} +131.707 q^{33} -357.606 q^{37} +250.710 q^{39} -127.174 q^{41} -343.250 q^{43} -62.6389 q^{45} -77.4628 q^{47} -31.3063 q^{51} -460.812 q^{53} +305.555 q^{55} +12.8293 q^{57} -272.735 q^{59} -51.3874 q^{61} +581.636 q^{65} +327.081 q^{67} +480.231 q^{69} +571.877 q^{71} +206.313 q^{73} +229.680 q^{75} -923.918 q^{79} +81.0000 q^{81} +1105.73 q^{83} -72.6293 q^{85} -29.7941 q^{87} -1532.57 q^{89} -401.146 q^{93} +29.7634 q^{95} +97.6059 q^{97} -395.121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + 40q^{11} - 48q^{13} + 24q^{15} + 152q^{17} + 224q^{19} + 8q^{23} - 28q^{25} - 108q^{27} - 144q^{29} + 400q^{31} - 120q^{33} - 304q^{37} + 144q^{39} + 152q^{41} - 160q^{43} - 72q^{45} + 544q^{47} - 456q^{51} - 1320q^{53} + 16q^{55} - 672q^{57} + 1040q^{59} - 896q^{61} - 648q^{65} + 416q^{67} - 24q^{69} - 248q^{71} + 752q^{73} + 84q^{75} - 864q^{79} + 324q^{81} + 1456q^{83} - 1608q^{85} + 432q^{87} - 2936q^{89} - 1200q^{93} + 80q^{95} - 144q^{97} + 360q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −6.95987 −0.622510 −0.311255 0.950326i \(-0.600749\pi\)
−0.311255 + 0.950326i \(0.600749\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −43.9023 −1.20337 −0.601684 0.798734i \(-0.705503\pi\)
−0.601684 + 0.798734i \(0.705503\pi\)
\(12\) 0 0
\(13\) −83.5699 −1.78293 −0.891466 0.453088i \(-0.850322\pi\)
−0.891466 + 0.453088i \(0.850322\pi\)
\(14\) 0 0
\(15\) 20.8796 0.359406
\(16\) 0 0
\(17\) 10.4354 0.148880 0.0744401 0.997225i \(-0.476283\pi\)
0.0744401 + 0.997225i \(0.476283\pi\)
\(18\) 0 0
\(19\) −4.27642 −0.0516357 −0.0258179 0.999667i \(-0.508219\pi\)
−0.0258179 + 0.999667i \(0.508219\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −160.077 −1.45123 −0.725616 0.688100i \(-0.758445\pi\)
−0.725616 + 0.688100i \(0.758445\pi\)
\(24\) 0 0
\(25\) −76.5601 −0.612481
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 9.93138 0.0635935 0.0317967 0.999494i \(-0.489877\pi\)
0.0317967 + 0.999494i \(0.489877\pi\)
\(30\) 0 0
\(31\) 133.715 0.774710 0.387355 0.921931i \(-0.373389\pi\)
0.387355 + 0.921931i \(0.373389\pi\)
\(32\) 0 0
\(33\) 131.707 0.694765
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −357.606 −1.58892 −0.794460 0.607316i \(-0.792246\pi\)
−0.794460 + 0.607316i \(0.792246\pi\)
\(38\) 0 0
\(39\) 250.710 1.02938
\(40\) 0 0
\(41\) −127.174 −0.484420 −0.242210 0.970224i \(-0.577872\pi\)
−0.242210 + 0.970224i \(0.577872\pi\)
\(42\) 0 0
\(43\) −343.250 −1.21733 −0.608664 0.793428i \(-0.708294\pi\)
−0.608664 + 0.793428i \(0.708294\pi\)
\(44\) 0 0
\(45\) −62.6389 −0.207503
\(46\) 0 0
\(47\) −77.4628 −0.240406 −0.120203 0.992749i \(-0.538355\pi\)
−0.120203 + 0.992749i \(0.538355\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −31.3063 −0.0859560
\(52\) 0 0
\(53\) −460.812 −1.19429 −0.597145 0.802133i \(-0.703698\pi\)
−0.597145 + 0.802133i \(0.703698\pi\)
\(54\) 0 0
\(55\) 305.555 0.749109
\(56\) 0 0
\(57\) 12.8293 0.0298119
\(58\) 0 0
\(59\) −272.735 −0.601816 −0.300908 0.953653i \(-0.597290\pi\)
−0.300908 + 0.953653i \(0.597290\pi\)
\(60\) 0 0
\(61\) −51.3874 −0.107860 −0.0539302 0.998545i \(-0.517175\pi\)
−0.0539302 + 0.998545i \(0.517175\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 581.636 1.10989
\(66\) 0 0
\(67\) 327.081 0.596407 0.298204 0.954502i \(-0.403613\pi\)
0.298204 + 0.954502i \(0.403613\pi\)
\(68\) 0 0
\(69\) 480.231 0.837869
\(70\) 0 0
\(71\) 571.877 0.955905 0.477953 0.878386i \(-0.341379\pi\)
0.477953 + 0.878386i \(0.341379\pi\)
\(72\) 0 0
\(73\) 206.313 0.330783 0.165391 0.986228i \(-0.447111\pi\)
0.165391 + 0.986228i \(0.447111\pi\)
\(74\) 0 0
\(75\) 229.680 0.353616
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −923.918 −1.31581 −0.657904 0.753102i \(-0.728557\pi\)
−0.657904 + 0.753102i \(0.728557\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1105.73 1.46229 0.731143 0.682224i \(-0.238987\pi\)
0.731143 + 0.682224i \(0.238987\pi\)
\(84\) 0 0
\(85\) −72.6293 −0.0926794
\(86\) 0 0
\(87\) −29.7941 −0.0367157
\(88\) 0 0
\(89\) −1532.57 −1.82531 −0.912655 0.408731i \(-0.865971\pi\)
−0.912655 + 0.408731i \(0.865971\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −401.146 −0.447279
\(94\) 0 0
\(95\) 29.7634 0.0321438
\(96\) 0 0
\(97\) 97.6059 0.102169 0.0510844 0.998694i \(-0.483732\pi\)
0.0510844 + 0.998694i \(0.483732\pi\)
\(98\) 0 0
\(99\) −395.121 −0.401123
\(100\) 0 0
\(101\) −1326.39 −1.30674 −0.653368 0.757040i \(-0.726645\pi\)
−0.653368 + 0.757040i \(0.726645\pi\)
\(102\) 0 0
\(103\) 780.354 0.746511 0.373255 0.927729i \(-0.378242\pi\)
0.373255 + 0.927729i \(0.378242\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 436.263 0.394160 0.197080 0.980387i \(-0.436854\pi\)
0.197080 + 0.980387i \(0.436854\pi\)
\(108\) 0 0
\(109\) −2092.32 −1.83861 −0.919304 0.393547i \(-0.871248\pi\)
−0.919304 + 0.393547i \(0.871248\pi\)
\(110\) 0 0
\(111\) 1072.82 0.917363
\(112\) 0 0
\(113\) 129.323 0.107661 0.0538303 0.998550i \(-0.482857\pi\)
0.0538303 + 0.998550i \(0.482857\pi\)
\(114\) 0 0
\(115\) 1114.12 0.903407
\(116\) 0 0
\(117\) −752.129 −0.594311
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 596.415 0.448096
\(122\) 0 0
\(123\) 381.522 0.279680
\(124\) 0 0
\(125\) 1402.83 1.00379
\(126\) 0 0
\(127\) −392.568 −0.274289 −0.137145 0.990551i \(-0.543793\pi\)
−0.137145 + 0.990551i \(0.543793\pi\)
\(128\) 0 0
\(129\) 1029.75 0.702824
\(130\) 0 0
\(131\) 952.085 0.634993 0.317496 0.948259i \(-0.397158\pi\)
0.317496 + 0.948259i \(0.397158\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 187.917 0.119802
\(136\) 0 0
\(137\) −1850.66 −1.15411 −0.577053 0.816706i \(-0.695798\pi\)
−0.577053 + 0.816706i \(0.695798\pi\)
\(138\) 0 0
\(139\) −1874.96 −1.14412 −0.572059 0.820213i \(-0.693855\pi\)
−0.572059 + 0.820213i \(0.693855\pi\)
\(140\) 0 0
\(141\) 232.388 0.138799
\(142\) 0 0
\(143\) 3668.91 2.14552
\(144\) 0 0
\(145\) −69.1211 −0.0395876
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −956.162 −0.525717 −0.262859 0.964834i \(-0.584665\pi\)
−0.262859 + 0.964834i \(0.584665\pi\)
\(150\) 0 0
\(151\) 512.503 0.276205 0.138102 0.990418i \(-0.455900\pi\)
0.138102 + 0.990418i \(0.455900\pi\)
\(152\) 0 0
\(153\) 93.9189 0.0496267
\(154\) 0 0
\(155\) −930.643 −0.482265
\(156\) 0 0
\(157\) 573.762 0.291663 0.145832 0.989309i \(-0.453414\pi\)
0.145832 + 0.989309i \(0.453414\pi\)
\(158\) 0 0
\(159\) 1382.44 0.689524
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3889.78 −1.86915 −0.934573 0.355771i \(-0.884218\pi\)
−0.934573 + 0.355771i \(0.884218\pi\)
\(164\) 0 0
\(165\) −916.664 −0.432498
\(166\) 0 0
\(167\) −167.287 −0.0775154 −0.0387577 0.999249i \(-0.512340\pi\)
−0.0387577 + 0.999249i \(0.512340\pi\)
\(168\) 0 0
\(169\) 4786.92 2.17885
\(170\) 0 0
\(171\) −38.4878 −0.0172119
\(172\) 0 0
\(173\) −2559.23 −1.12471 −0.562355 0.826896i \(-0.690104\pi\)
−0.562355 + 0.826896i \(0.690104\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 818.206 0.347458
\(178\) 0 0
\(179\) 1214.50 0.507130 0.253565 0.967318i \(-0.418397\pi\)
0.253565 + 0.967318i \(0.418397\pi\)
\(180\) 0 0
\(181\) −2555.03 −1.04925 −0.524625 0.851334i \(-0.675794\pi\)
−0.524625 + 0.851334i \(0.675794\pi\)
\(182\) 0 0
\(183\) 154.162 0.0622732
\(184\) 0 0
\(185\) 2488.89 0.989119
\(186\) 0 0
\(187\) −458.140 −0.179158
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3256.32 −1.23361 −0.616804 0.787117i \(-0.711573\pi\)
−0.616804 + 0.787117i \(0.711573\pi\)
\(192\) 0 0
\(193\) 96.0749 0.0358323 0.0179161 0.999839i \(-0.494297\pi\)
0.0179161 + 0.999839i \(0.494297\pi\)
\(194\) 0 0
\(195\) −1744.91 −0.640797
\(196\) 0 0
\(197\) 1335.35 0.482941 0.241471 0.970408i \(-0.422370\pi\)
0.241471 + 0.970408i \(0.422370\pi\)
\(198\) 0 0
\(199\) 2907.06 1.03556 0.517778 0.855515i \(-0.326759\pi\)
0.517778 + 0.855515i \(0.326759\pi\)
\(200\) 0 0
\(201\) −981.242 −0.344336
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 885.115 0.301557
\(206\) 0 0
\(207\) −1440.69 −0.483744
\(208\) 0 0
\(209\) 187.745 0.0621368
\(210\) 0 0
\(211\) −721.998 −0.235566 −0.117783 0.993039i \(-0.537579\pi\)
−0.117783 + 0.993039i \(0.537579\pi\)
\(212\) 0 0
\(213\) −1715.63 −0.551892
\(214\) 0 0
\(215\) 2388.98 0.757799
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −618.940 −0.190978
\(220\) 0 0
\(221\) −872.088 −0.265443
\(222\) 0 0
\(223\) 5906.68 1.77373 0.886863 0.462033i \(-0.152880\pi\)
0.886863 + 0.462033i \(0.152880\pi\)
\(224\) 0 0
\(225\) −689.041 −0.204160
\(226\) 0 0
\(227\) 4877.87 1.42624 0.713118 0.701044i \(-0.247283\pi\)
0.713118 + 0.701044i \(0.247283\pi\)
\(228\) 0 0
\(229\) −4893.35 −1.41206 −0.706030 0.708182i \(-0.749516\pi\)
−0.706030 + 0.708182i \(0.749516\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3155.44 0.887208 0.443604 0.896223i \(-0.353700\pi\)
0.443604 + 0.896223i \(0.353700\pi\)
\(234\) 0 0
\(235\) 539.131 0.149655
\(236\) 0 0
\(237\) 2771.75 0.759682
\(238\) 0 0
\(239\) 4165.43 1.12736 0.563680 0.825993i \(-0.309385\pi\)
0.563680 + 0.825993i \(0.309385\pi\)
\(240\) 0 0
\(241\) −6240.57 −1.66801 −0.834006 0.551756i \(-0.813958\pi\)
−0.834006 + 0.551756i \(0.813958\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 357.380 0.0920630
\(248\) 0 0
\(249\) −3317.19 −0.844252
\(250\) 0 0
\(251\) 387.084 0.0973407 0.0486703 0.998815i \(-0.484502\pi\)
0.0486703 + 0.998815i \(0.484502\pi\)
\(252\) 0 0
\(253\) 7027.75 1.74637
\(254\) 0 0
\(255\) 217.888 0.0535085
\(256\) 0 0
\(257\) 4498.89 1.09196 0.545979 0.837799i \(-0.316158\pi\)
0.545979 + 0.837799i \(0.316158\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 89.3824 0.0211978
\(262\) 0 0
\(263\) 6791.57 1.59234 0.796171 0.605071i \(-0.206855\pi\)
0.796171 + 0.605071i \(0.206855\pi\)
\(264\) 0 0
\(265\) 3207.19 0.743458
\(266\) 0 0
\(267\) 4597.72 1.05384
\(268\) 0 0
\(269\) 8537.49 1.93509 0.967546 0.252694i \(-0.0813167\pi\)
0.967546 + 0.252694i \(0.0813167\pi\)
\(270\) 0 0
\(271\) −4743.77 −1.06333 −0.531667 0.846953i \(-0.678434\pi\)
−0.531667 + 0.846953i \(0.678434\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3361.17 0.737040
\(276\) 0 0
\(277\) −1004.32 −0.217847 −0.108924 0.994050i \(-0.534740\pi\)
−0.108924 + 0.994050i \(0.534740\pi\)
\(278\) 0 0
\(279\) 1203.44 0.258237
\(280\) 0 0
\(281\) 4563.78 0.968869 0.484435 0.874828i \(-0.339025\pi\)
0.484435 + 0.874828i \(0.339025\pi\)
\(282\) 0 0
\(283\) 5558.25 1.16750 0.583752 0.811932i \(-0.301584\pi\)
0.583752 + 0.811932i \(0.301584\pi\)
\(284\) 0 0
\(285\) −89.2901 −0.0185582
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4804.10 −0.977835
\(290\) 0 0
\(291\) −292.818 −0.0589872
\(292\) 0 0
\(293\) 2903.23 0.578868 0.289434 0.957198i \(-0.406533\pi\)
0.289434 + 0.957198i \(0.406533\pi\)
\(294\) 0 0
\(295\) 1898.20 0.374636
\(296\) 0 0
\(297\) 1185.36 0.231588
\(298\) 0 0
\(299\) 13377.6 2.58745
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3979.16 0.754444
\(304\) 0 0
\(305\) 357.650 0.0671442
\(306\) 0 0
\(307\) 7974.96 1.48259 0.741295 0.671179i \(-0.234212\pi\)
0.741295 + 0.671179i \(0.234212\pi\)
\(308\) 0 0
\(309\) −2341.06 −0.430998
\(310\) 0 0
\(311\) −4744.75 −0.865114 −0.432557 0.901607i \(-0.642389\pi\)
−0.432557 + 0.901607i \(0.642389\pi\)
\(312\) 0 0
\(313\) 9872.97 1.78292 0.891459 0.453102i \(-0.149683\pi\)
0.891459 + 0.453102i \(0.149683\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −311.027 −0.0551073 −0.0275537 0.999620i \(-0.508772\pi\)
−0.0275537 + 0.999620i \(0.508772\pi\)
\(318\) 0 0
\(319\) −436.011 −0.0765263
\(320\) 0 0
\(321\) −1308.79 −0.227568
\(322\) 0 0
\(323\) −44.6263 −0.00768754
\(324\) 0 0
\(325\) 6398.12 1.09201
\(326\) 0 0
\(327\) 6276.97 1.06152
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7451.30 −1.23734 −0.618671 0.785650i \(-0.712329\pi\)
−0.618671 + 0.785650i \(0.712329\pi\)
\(332\) 0 0
\(333\) −3218.45 −0.529640
\(334\) 0 0
\(335\) −2276.44 −0.371269
\(336\) 0 0
\(337\) 4112.84 0.664808 0.332404 0.943137i \(-0.392140\pi\)
0.332404 + 0.943137i \(0.392140\pi\)
\(338\) 0 0
\(339\) −387.968 −0.0621578
\(340\) 0 0
\(341\) −5870.42 −0.932262
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3342.35 −0.521582
\(346\) 0 0
\(347\) −11459.5 −1.77285 −0.886423 0.462876i \(-0.846817\pi\)
−0.886423 + 0.462876i \(0.846817\pi\)
\(348\) 0 0
\(349\) −1293.03 −0.198322 −0.0991610 0.995071i \(-0.531616\pi\)
−0.0991610 + 0.995071i \(0.531616\pi\)
\(350\) 0 0
\(351\) 2256.39 0.343125
\(352\) 0 0
\(353\) 7265.01 1.09540 0.547702 0.836674i \(-0.315503\pi\)
0.547702 + 0.836674i \(0.315503\pi\)
\(354\) 0 0
\(355\) −3980.19 −0.595061
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1381.80 0.203143 0.101572 0.994828i \(-0.467613\pi\)
0.101572 + 0.994828i \(0.467613\pi\)
\(360\) 0 0
\(361\) −6840.71 −0.997334
\(362\) 0 0
\(363\) −1789.25 −0.258708
\(364\) 0 0
\(365\) −1435.91 −0.205916
\(366\) 0 0
\(367\) −4502.44 −0.640397 −0.320198 0.947351i \(-0.603750\pi\)
−0.320198 + 0.947351i \(0.603750\pi\)
\(368\) 0 0
\(369\) −1144.57 −0.161473
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11140.1 1.54642 0.773208 0.634152i \(-0.218651\pi\)
0.773208 + 0.634152i \(0.218651\pi\)
\(374\) 0 0
\(375\) −4208.50 −0.579536
\(376\) 0 0
\(377\) −829.964 −0.113383
\(378\) 0 0
\(379\) 13060.7 1.77015 0.885073 0.465453i \(-0.154108\pi\)
0.885073 + 0.465453i \(0.154108\pi\)
\(380\) 0 0
\(381\) 1177.70 0.158361
\(382\) 0 0
\(383\) −11248.6 −1.50072 −0.750362 0.661027i \(-0.770121\pi\)
−0.750362 + 0.661027i \(0.770121\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3089.25 −0.405776
\(388\) 0 0
\(389\) −11873.8 −1.54763 −0.773813 0.633414i \(-0.781653\pi\)
−0.773813 + 0.633414i \(0.781653\pi\)
\(390\) 0 0
\(391\) −1670.47 −0.216060
\(392\) 0 0
\(393\) −2856.26 −0.366613
\(394\) 0 0
\(395\) 6430.35 0.819104
\(396\) 0 0
\(397\) 13921.3 1.75992 0.879960 0.475047i \(-0.157569\pi\)
0.879960 + 0.475047i \(0.157569\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12830.2 −1.59778 −0.798891 0.601475i \(-0.794580\pi\)
−0.798891 + 0.601475i \(0.794580\pi\)
\(402\) 0 0
\(403\) −11174.6 −1.38126
\(404\) 0 0
\(405\) −563.750 −0.0691678
\(406\) 0 0
\(407\) 15699.7 1.91206
\(408\) 0 0
\(409\) −5606.76 −0.677840 −0.338920 0.940815i \(-0.610062\pi\)
−0.338920 + 0.940815i \(0.610062\pi\)
\(410\) 0 0
\(411\) 5551.98 0.666324
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −7695.75 −0.910288
\(416\) 0 0
\(417\) 5624.89 0.660556
\(418\) 0 0
\(419\) −77.6276 −0.00905097 −0.00452549 0.999990i \(-0.501441\pi\)
−0.00452549 + 0.999990i \(0.501441\pi\)
\(420\) 0 0
\(421\) 9742.15 1.12780 0.563900 0.825843i \(-0.309301\pi\)
0.563900 + 0.825843i \(0.309301\pi\)
\(422\) 0 0
\(423\) −697.165 −0.0801355
\(424\) 0 0
\(425\) −798.938 −0.0911863
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −11006.7 −1.23872
\(430\) 0 0
\(431\) 1306.54 0.146018 0.0730092 0.997331i \(-0.476740\pi\)
0.0730092 + 0.997331i \(0.476740\pi\)
\(432\) 0 0
\(433\) −4039.33 −0.448309 −0.224154 0.974554i \(-0.571962\pi\)
−0.224154 + 0.974554i \(0.571962\pi\)
\(434\) 0 0
\(435\) 207.363 0.0228559
\(436\) 0 0
\(437\) 684.557 0.0749355
\(438\) 0 0
\(439\) 58.3195 0.00634040 0.00317020 0.999995i \(-0.498991\pi\)
0.00317020 + 0.999995i \(0.498991\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16369.5 1.75561 0.877806 0.479016i \(-0.159006\pi\)
0.877806 + 0.479016i \(0.159006\pi\)
\(444\) 0 0
\(445\) 10666.5 1.13627
\(446\) 0 0
\(447\) 2868.49 0.303523
\(448\) 0 0
\(449\) 1584.33 0.166523 0.0832617 0.996528i \(-0.473466\pi\)
0.0832617 + 0.996528i \(0.473466\pi\)
\(450\) 0 0
\(451\) 5583.23 0.582936
\(452\) 0 0
\(453\) −1537.51 −0.159467
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8530.39 −0.873162 −0.436581 0.899665i \(-0.643811\pi\)
−0.436581 + 0.899665i \(0.643811\pi\)
\(458\) 0 0
\(459\) −281.757 −0.0286520
\(460\) 0 0
\(461\) 2593.77 0.262048 0.131024 0.991379i \(-0.458174\pi\)
0.131024 + 0.991379i \(0.458174\pi\)
\(462\) 0 0
\(463\) −15296.9 −1.53543 −0.767717 0.640789i \(-0.778607\pi\)
−0.767717 + 0.640789i \(0.778607\pi\)
\(464\) 0 0
\(465\) 2791.93 0.278436
\(466\) 0 0
\(467\) −13942.1 −1.38151 −0.690754 0.723090i \(-0.742721\pi\)
−0.690754 + 0.723090i \(0.742721\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1721.28 −0.168392
\(472\) 0 0
\(473\) 15069.5 1.46489
\(474\) 0 0
\(475\) 327.404 0.0316259
\(476\) 0 0
\(477\) −4147.31 −0.398097
\(478\) 0 0
\(479\) 1431.14 0.136515 0.0682574 0.997668i \(-0.478256\pi\)
0.0682574 + 0.997668i \(0.478256\pi\)
\(480\) 0 0
\(481\) 29885.1 2.83294
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −679.325 −0.0636012
\(486\) 0 0
\(487\) −7664.42 −0.713158 −0.356579 0.934265i \(-0.616057\pi\)
−0.356579 + 0.934265i \(0.616057\pi\)
\(488\) 0 0
\(489\) 11669.3 1.07915
\(490\) 0 0
\(491\) 14761.3 1.35676 0.678380 0.734711i \(-0.262682\pi\)
0.678380 + 0.734711i \(0.262682\pi\)
\(492\) 0 0
\(493\) 103.638 0.00946781
\(494\) 0 0
\(495\) 2749.99 0.249703
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −11606.1 −1.04121 −0.520604 0.853798i \(-0.674293\pi\)
−0.520604 + 0.853798i \(0.674293\pi\)
\(500\) 0 0
\(501\) 501.862 0.0447535
\(502\) 0 0
\(503\) −1650.23 −0.146283 −0.0731414 0.997322i \(-0.523302\pi\)
−0.0731414 + 0.997322i \(0.523302\pi\)
\(504\) 0 0
\(505\) 9231.48 0.813456
\(506\) 0 0
\(507\) −14360.8 −1.25796
\(508\) 0 0
\(509\) −3728.12 −0.324649 −0.162324 0.986737i \(-0.551899\pi\)
−0.162324 + 0.986737i \(0.551899\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 115.463 0.00993730
\(514\) 0 0
\(515\) −5431.17 −0.464710
\(516\) 0 0
\(517\) 3400.80 0.289298
\(518\) 0 0
\(519\) 7677.69 0.649351
\(520\) 0 0
\(521\) 15803.9 1.32895 0.664473 0.747312i \(-0.268656\pi\)
0.664473 + 0.747312i \(0.268656\pi\)
\(522\) 0 0
\(523\) −9235.54 −0.772164 −0.386082 0.922464i \(-0.626172\pi\)
−0.386082 + 0.922464i \(0.626172\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1395.38 0.115339
\(528\) 0 0
\(529\) 13457.6 1.10607
\(530\) 0 0
\(531\) −2454.62 −0.200605
\(532\) 0 0
\(533\) 10627.9 0.863688
\(534\) 0 0
\(535\) −3036.34 −0.245369
\(536\) 0 0
\(537\) −3643.51 −0.292792
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8399.16 −0.667483 −0.333741 0.942665i \(-0.608311\pi\)
−0.333741 + 0.942665i \(0.608311\pi\)
\(542\) 0 0
\(543\) 7665.10 0.605784
\(544\) 0 0
\(545\) 14562.3 1.14455
\(546\) 0 0
\(547\) −18402.5 −1.43846 −0.719228 0.694774i \(-0.755504\pi\)
−0.719228 + 0.694774i \(0.755504\pi\)
\(548\) 0 0
\(549\) −462.487 −0.0359535
\(550\) 0 0
\(551\) −42.4708 −0.00328370
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −7466.68 −0.571068
\(556\) 0 0
\(557\) 7803.37 0.593607 0.296804 0.954938i \(-0.404079\pi\)
0.296804 + 0.954938i \(0.404079\pi\)
\(558\) 0 0
\(559\) 28685.3 2.17041
\(560\) 0 0
\(561\) 1374.42 0.103437
\(562\) 0 0
\(563\) 708.972 0.0530721 0.0265361 0.999648i \(-0.491552\pi\)
0.0265361 + 0.999648i \(0.491552\pi\)
\(564\) 0 0
\(565\) −900.069 −0.0670198
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15799.2 1.16403 0.582017 0.813176i \(-0.302264\pi\)
0.582017 + 0.813176i \(0.302264\pi\)
\(570\) 0 0
\(571\) −22721.9 −1.66529 −0.832647 0.553804i \(-0.813176\pi\)
−0.832647 + 0.553804i \(0.813176\pi\)
\(572\) 0 0
\(573\) 9768.97 0.712224
\(574\) 0 0
\(575\) 12255.5 0.888852
\(576\) 0 0
\(577\) 19216.2 1.38645 0.693226 0.720721i \(-0.256189\pi\)
0.693226 + 0.720721i \(0.256189\pi\)
\(578\) 0 0
\(579\) −288.225 −0.0206878
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20230.7 1.43717
\(584\) 0 0
\(585\) 5234.72 0.369964
\(586\) 0 0
\(587\) −19262.1 −1.35440 −0.677201 0.735798i \(-0.736807\pi\)
−0.677201 + 0.735798i \(0.736807\pi\)
\(588\) 0 0
\(589\) −571.824 −0.0400027
\(590\) 0 0
\(591\) −4006.04 −0.278826
\(592\) 0 0
\(593\) −11871.8 −0.822122 −0.411061 0.911608i \(-0.634842\pi\)
−0.411061 + 0.911608i \(0.634842\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8721.17 −0.597879
\(598\) 0 0
\(599\) −17021.1 −1.16104 −0.580520 0.814246i \(-0.697151\pi\)
−0.580520 + 0.814246i \(0.697151\pi\)
\(600\) 0 0
\(601\) −5518.23 −0.374531 −0.187266 0.982309i \(-0.559963\pi\)
−0.187266 + 0.982309i \(0.559963\pi\)
\(602\) 0 0
\(603\) 2943.73 0.198802
\(604\) 0 0
\(605\) −4150.97 −0.278944
\(606\) 0 0
\(607\) −20388.4 −1.36333 −0.681663 0.731667i \(-0.738743\pi\)
−0.681663 + 0.731667i \(0.738743\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6473.55 0.428628
\(612\) 0 0
\(613\) −14237.0 −0.938053 −0.469026 0.883184i \(-0.655395\pi\)
−0.469026 + 0.883184i \(0.655395\pi\)
\(614\) 0 0
\(615\) −2655.34 −0.174104
\(616\) 0 0
\(617\) −20383.4 −1.32999 −0.664997 0.746846i \(-0.731567\pi\)
−0.664997 + 0.746846i \(0.731567\pi\)
\(618\) 0 0
\(619\) 15997.5 1.03876 0.519380 0.854543i \(-0.326163\pi\)
0.519380 + 0.854543i \(0.326163\pi\)
\(620\) 0 0
\(621\) 4322.08 0.279290
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −193.527 −0.0123857
\(626\) 0 0
\(627\) −563.235 −0.0358747
\(628\) 0 0
\(629\) −3731.77 −0.236559
\(630\) 0 0
\(631\) −4857.90 −0.306481 −0.153241 0.988189i \(-0.548971\pi\)
−0.153241 + 0.988189i \(0.548971\pi\)
\(632\) 0 0
\(633\) 2165.99 0.136004
\(634\) 0 0
\(635\) 2732.22 0.170748
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 5146.89 0.318635
\(640\) 0 0
\(641\) 24563.9 1.51360 0.756798 0.653649i \(-0.226763\pi\)
0.756798 + 0.653649i \(0.226763\pi\)
\(642\) 0 0
\(643\) −6010.79 −0.368650 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(644\) 0 0
\(645\) −7166.93 −0.437515
\(646\) 0 0
\(647\) 16927.4 1.02857 0.514285 0.857619i \(-0.328057\pi\)
0.514285 + 0.857619i \(0.328057\pi\)
\(648\) 0 0
\(649\) 11973.7 0.724206
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7702.96 0.461623 0.230812 0.972998i \(-0.425862\pi\)
0.230812 + 0.972998i \(0.425862\pi\)
\(654\) 0 0
\(655\) −6626.39 −0.395290
\(656\) 0 0
\(657\) 1856.82 0.110261
\(658\) 0 0
\(659\) 5899.80 0.348746 0.174373 0.984680i \(-0.444210\pi\)
0.174373 + 0.984680i \(0.444210\pi\)
\(660\) 0 0
\(661\) −25442.5 −1.49713 −0.748563 0.663064i \(-0.769256\pi\)
−0.748563 + 0.663064i \(0.769256\pi\)
\(662\) 0 0
\(663\) 2616.26 0.153254
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1589.78 −0.0922889
\(668\) 0 0
\(669\) −17720.1 −1.02406
\(670\) 0 0
\(671\) 2256.03 0.129796
\(672\) 0 0
\(673\) −18424.7 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(674\) 0 0
\(675\) 2067.12 0.117872
\(676\) 0 0
\(677\) −1189.55 −0.0675307 −0.0337654 0.999430i \(-0.510750\pi\)
−0.0337654 + 0.999430i \(0.510750\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −14633.6 −0.823437
\(682\) 0 0
\(683\) 19938.4 1.11702 0.558509 0.829499i \(-0.311374\pi\)
0.558509 + 0.829499i \(0.311374\pi\)
\(684\) 0 0
\(685\) 12880.4 0.718443
\(686\) 0 0
\(687\) 14680.1 0.815253
\(688\) 0 0
\(689\) 38510.0 2.12934
\(690\) 0 0
\(691\) −5625.32 −0.309692 −0.154846 0.987939i \(-0.549488\pi\)
−0.154846 + 0.987939i \(0.549488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13049.5 0.712225
\(696\) 0 0
\(697\) −1327.11 −0.0721206
\(698\) 0 0
\(699\) −9466.31 −0.512230
\(700\) 0 0
\(701\) −29961.0 −1.61428 −0.807140 0.590360i \(-0.798986\pi\)
−0.807140 + 0.590360i \(0.798986\pi\)
\(702\) 0 0
\(703\) 1529.27 0.0820451
\(704\) 0 0
\(705\) −1617.39 −0.0864036
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14177.2 0.750968 0.375484 0.926829i \(-0.377476\pi\)
0.375484 + 0.926829i \(0.377476\pi\)
\(710\) 0 0
\(711\) −8315.26 −0.438603
\(712\) 0 0
\(713\) −21404.8 −1.12428
\(714\) 0 0
\(715\) −25535.2 −1.33561
\(716\) 0 0
\(717\) −12496.3 −0.650882
\(718\) 0 0
\(719\) −425.151 −0.0220521 −0.0110260 0.999939i \(-0.503510\pi\)
−0.0110260 + 0.999939i \(0.503510\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 18721.7 0.963027
\(724\) 0 0
\(725\) −760.348 −0.0389498
\(726\) 0 0
\(727\) −12678.5 −0.646793 −0.323397 0.946264i \(-0.604825\pi\)
−0.323397 + 0.946264i \(0.604825\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −3581.96 −0.181236
\(732\) 0 0
\(733\) 7973.30 0.401774 0.200887 0.979614i \(-0.435618\pi\)
0.200887 + 0.979614i \(0.435618\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14359.6 −0.717698
\(738\) 0 0
\(739\) −12208.1 −0.607687 −0.303843 0.952722i \(-0.598270\pi\)
−0.303843 + 0.952722i \(0.598270\pi\)
\(740\) 0 0
\(741\) −1072.14 −0.0531526
\(742\) 0 0
\(743\) −13351.2 −0.659232 −0.329616 0.944115i \(-0.606919\pi\)
−0.329616 + 0.944115i \(0.606919\pi\)
\(744\) 0 0
\(745\) 6654.77 0.327264
\(746\) 0 0
\(747\) 9951.58 0.487429
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −27971.9 −1.35913 −0.679567 0.733614i \(-0.737832\pi\)
−0.679567 + 0.733614i \(0.737832\pi\)
\(752\) 0 0
\(753\) −1161.25 −0.0561997
\(754\) 0 0
\(755\) −3566.96 −0.171940
\(756\) 0 0
\(757\) 20382.6 0.978623 0.489312 0.872109i \(-0.337248\pi\)
0.489312 + 0.872109i \(0.337248\pi\)
\(758\) 0 0
\(759\) −21083.2 −1.00827
\(760\) 0 0
\(761\) −24172.0 −1.15143 −0.575713 0.817652i \(-0.695275\pi\)
−0.575713 + 0.817652i \(0.695275\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −653.664 −0.0308931
\(766\) 0 0
\(767\) 22792.5 1.07300
\(768\) 0 0
\(769\) 1320.89 0.0619409 0.0309705 0.999520i \(-0.490140\pi\)
0.0309705 + 0.999520i \(0.490140\pi\)
\(770\) 0 0
\(771\) −13496.7 −0.630443
\(772\) 0 0
\(773\) 28790.1 1.33960 0.669798 0.742544i \(-0.266381\pi\)
0.669798 + 0.742544i \(0.266381\pi\)
\(774\) 0 0
\(775\) −10237.3 −0.474495
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 543.850 0.0250134
\(780\) 0 0
\(781\) −25106.7 −1.15031
\(782\) 0 0
\(783\) −268.147 −0.0122386
\(784\) 0 0
\(785\) −3993.31 −0.181563
\(786\) 0 0
\(787\) −38138.2 −1.72742 −0.863710 0.503989i \(-0.831865\pi\)
−0.863710 + 0.503989i \(0.831865\pi\)
\(788\) 0 0
\(789\) −20374.7 −0.919339
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4294.44 0.192308
\(794\) 0 0
\(795\) −9621.58 −0.429235
\(796\) 0 0
\(797\) −29562.7 −1.31388 −0.656941 0.753942i \(-0.728150\pi\)
−0.656941 + 0.753942i \(0.728150\pi\)
\(798\) 0 0
\(799\) −808.357 −0.0357918
\(800\) 0 0
\(801\) −13793.2 −0.608437
\(802\) 0 0
\(803\) −9057.63 −0.398054
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25612.5 −1.11723
\(808\) 0 0
\(809\) 28800.4 1.25163 0.625814 0.779972i \(-0.284767\pi\)
0.625814 + 0.779972i \(0.284767\pi\)
\(810\) 0 0
\(811\) 23960.3 1.03743 0.518717 0.854946i \(-0.326410\pi\)
0.518717 + 0.854946i \(0.326410\pi\)
\(812\) 0 0
\(813\) 14231.3 0.613917
\(814\) 0 0
\(815\) 27072.4 1.16356
\(816\) 0 0
\(817\) 1467.88 0.0628576
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7480.81 0.318005 0.159003 0.987278i \(-0.449172\pi\)
0.159003 + 0.987278i \(0.449172\pi\)
\(822\) 0 0
\(823\) 37456.0 1.58643 0.793217 0.608939i \(-0.208405\pi\)
0.793217 + 0.608939i \(0.208405\pi\)
\(824\) 0 0
\(825\) −10083.5 −0.425531
\(826\) 0 0
\(827\) −18855.9 −0.792847 −0.396423 0.918068i \(-0.629749\pi\)
−0.396423 + 0.918068i \(0.629749\pi\)
\(828\) 0 0
\(829\) −34027.1 −1.42559 −0.712793 0.701375i \(-0.752570\pi\)
−0.712793 + 0.701375i \(0.752570\pi\)
\(830\) 0 0
\(831\) 3012.95 0.125774
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1164.30 0.0482541
\(836\) 0 0
\(837\) −3610.32 −0.149093
\(838\) 0 0
\(839\) 9279.70 0.381848 0.190924 0.981605i \(-0.438852\pi\)
0.190924 + 0.981605i \(0.438852\pi\)
\(840\) 0 0
\(841\) −24290.4 −0.995956
\(842\) 0 0
\(843\) −13691.3 −0.559377
\(844\) 0 0
\(845\) −33316.4 −1.35635
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16674.7 −0.674059
\(850\) 0 0
\(851\) 57244.4 2.30589
\(852\) 0 0
\(853\) −29478.9 −1.18328 −0.591641 0.806201i \(-0.701520\pi\)
−0.591641 + 0.806201i \(0.701520\pi\)
\(854\) 0 0
\(855\) 267.870 0.0107146
\(856\) 0 0
\(857\) −18935.8 −0.754765 −0.377383 0.926057i \(-0.623176\pi\)
−0.377383 + 0.926057i \(0.623176\pi\)
\(858\) 0 0
\(859\) −11678.8 −0.463883 −0.231941 0.972730i \(-0.574508\pi\)
−0.231941 + 0.972730i \(0.574508\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14625.7 0.576899 0.288450 0.957495i \(-0.406860\pi\)
0.288450 + 0.957495i \(0.406860\pi\)
\(864\) 0 0
\(865\) 17811.9 0.700143
\(866\) 0 0
\(867\) 14412.3 0.564553
\(868\) 0 0
\(869\) 40562.1 1.58340
\(870\) 0 0
\(871\) −27334.1 −1.06335
\(872\) 0 0
\(873\) 878.454 0.0340563
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26195.9 1.00863 0.504317 0.863519i \(-0.331744\pi\)
0.504317 + 0.863519i \(0.331744\pi\)
\(878\) 0 0
\(879\) −8709.68 −0.334210
\(880\) 0 0
\(881\) 17065.6 0.652616 0.326308 0.945264i \(-0.394195\pi\)
0.326308 + 0.945264i \(0.394195\pi\)
\(882\) 0 0
\(883\) 20428.3 0.778557 0.389279 0.921120i \(-0.372724\pi\)
0.389279 + 0.921120i \(0.372724\pi\)
\(884\) 0 0
\(885\) −5694.61 −0.216296
\(886\) 0 0
\(887\) −3679.59 −0.139288 −0.0696441 0.997572i \(-0.522186\pi\)
−0.0696441 + 0.997572i \(0.522186\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3556.09 −0.133708
\(892\) 0 0
\(893\) 331.264 0.0124136
\(894\) 0 0
\(895\) −8452.80 −0.315694
\(896\) 0 0
\(897\) −40132.8 −1.49386
\(898\) 0 0
\(899\) 1327.98 0.0492665
\(900\) 0 0
\(901\) −4808.77 −0.177806
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17782.7 0.653168
\(906\) 0 0
\(907\) 25374.5 0.928938 0.464469 0.885589i \(-0.346245\pi\)
0.464469 + 0.885589i \(0.346245\pi\)
\(908\) 0 0
\(909\) −11937.5 −0.435579
\(910\) 0 0
\(911\) 4890.79 0.177869 0.0889347 0.996037i \(-0.471654\pi\)
0.0889347 + 0.996037i \(0.471654\pi\)
\(912\) 0 0
\(913\) −48544.2 −1.75967
\(914\) 0 0
\(915\) −1072.95 −0.0387657
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16590.3 −0.595501 −0.297750 0.954644i \(-0.596236\pi\)
−0.297750 + 0.954644i \(0.596236\pi\)
\(920\) 0 0
\(921\) −23924.9 −0.855974
\(922\) 0 0
\(923\) −47791.7 −1.70431
\(924\) 0 0
\(925\) 27378.4 0.973184
\(926\) 0 0
\(927\) 7023.19 0.248837
\(928\) 0 0
\(929\) 51991.9 1.83617 0.918083 0.396388i \(-0.129737\pi\)
0.918083 + 0.396388i \(0.129737\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 14234.3 0.499474
\(934\) 0 0
\(935\) 3188.60 0.111528
\(936\) 0 0
\(937\) 26392.7 0.920182 0.460091 0.887872i \(-0.347817\pi\)
0.460091 + 0.887872i \(0.347817\pi\)
\(938\) 0 0
\(939\) −29618.9 −1.02937
\(940\) 0 0
\(941\) −40403.8 −1.39971 −0.699854 0.714286i \(-0.746752\pi\)
−0.699854 + 0.714286i \(0.746752\pi\)
\(942\) 0 0
\(943\) 20357.6 0.703006
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18557.3 0.636782 0.318391 0.947959i \(-0.396858\pi\)
0.318391 + 0.947959i \(0.396858\pi\)
\(948\) 0 0
\(949\) −17241.6 −0.589763
\(950\) 0 0
\(951\) 933.081 0.0318162
\(952\) 0 0
\(953\) −30432.6 −1.03443 −0.517213 0.855857i \(-0.673031\pi\)
−0.517213 + 0.855857i \(0.673031\pi\)
\(954\) 0 0
\(955\) 22663.6 0.767934
\(956\) 0 0
\(957\) 1308.03 0.0441825
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11911.2 −0.399824
\(962\) 0 0
\(963\) 3926.37 0.131387
\(964\) 0 0
\(965\) −668.670 −0.0223059
\(966\) 0 0
\(967\) −21280.5 −0.707688 −0.353844 0.935304i \(-0.615126\pi\)
−0.353844 + 0.935304i \(0.615126\pi\)
\(968\) 0 0
\(969\) 133.879 0.00443840
\(970\) 0 0
\(971\) 10383.6 0.343179 0.171589 0.985169i \(-0.445110\pi\)
0.171589 + 0.985169i \(0.445110\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −19194.4 −0.630473
\(976\) 0 0
\(977\) −50807.3 −1.66374 −0.831868 0.554973i \(-0.812728\pi\)
−0.831868 + 0.554973i \(0.812728\pi\)
\(978\) 0 0
\(979\) 67283.6 2.19652
\(980\) 0 0
\(981\) −18830.9 −0.612870
\(982\) 0 0
\(983\) 21417.1 0.694913 0.347456 0.937696i \(-0.387045\pi\)
0.347456 + 0.937696i \(0.387045\pi\)
\(984\) 0 0
\(985\) −9293.84 −0.300636
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 54946.3 1.76663
\(990\) 0 0
\(991\) −3420.14 −0.109631 −0.0548155 0.998497i \(-0.517457\pi\)
−0.0548155 + 0.998497i \(0.517457\pi\)
\(992\) 0 0
\(993\) 22353.9 0.714380
\(994\) 0 0
\(995\) −20232.7 −0.644644
\(996\) 0 0
\(997\) −45658.1 −1.45036 −0.725180 0.688560i \(-0.758243\pi\)
−0.725180 + 0.688560i \(0.758243\pi\)
\(998\) 0 0
\(999\) 9655.36 0.305788
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 2352.4.a.ck.1.2 4
4.3 odd 2 1176.4.a.bc.1.2 yes 4
7.6 odd 2 2352.4.a.cr.1.3 4
28.27 even 2 1176.4.a.bb.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.3 4 28.27 even 2
1176.4.a.bc.1.2 yes 4 4.3 odd 2
2352.4.a.ck.1.2 4 1.1 even 1 trivial
2352.4.a.cr.1.3 4 7.6 odd 2