Properties

Label 1176.4.a.bb.1.3
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.25358\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-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 \(-0.399251\pi\)
0.311255 + 0.950326i \(0.399251\pi\)
\(6\) 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 \(-0.294497\pi\)
0.601684 + 0.798734i \(0.294497\pi\)
\(12\) 0 0
\(13\) 83.5699 1.78293 0.891466 0.453088i \(-0.149678\pi\)
0.891466 + 0.453088i \(0.149678\pi\)
\(14\) 0 0
\(15\) −20.8796 −0.359406
\(16\) 0 0
\(17\) −10.4354 −0.148880 −0.0744401 0.997225i \(-0.523717\pi\)
−0.0744401 + 0.997225i \(0.523717\pi\)
\(18\) 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 \(-0.241555\pi\)
0.725616 + 0.688100i \(0.241555\pi\)
\(24\) 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 \(-0.422128\pi\)
0.242210 + 0.970224i \(0.422128\pi\)
\(42\) 0 0
\(43\) 343.250 1.21733 0.608664 0.793428i \(-0.291706\pi\)
0.608664 + 0.793428i \(0.291706\pi\)
\(44\) 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 \(-0.482825\pi\)
0.0539302 + 0.998545i \(0.482825\pi\)
\(62\) 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 \(-0.596387\pi\)
−0.298204 + 0.954502i \(0.596387\pi\)
\(68\) 0 0
\(69\) −480.231 −0.837869
\(70\) 0 0
\(71\) −571.877 −0.955905 −0.477953 0.878386i \(-0.658621\pi\)
−0.477953 + 0.878386i \(0.658621\pi\)
\(72\) 0 0
\(73\) −206.313 −0.330783 −0.165391 0.986228i \(-0.552889\pi\)
−0.165391 + 0.986228i \(0.552889\pi\)
\(74\) 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 \(-0.271443\pi\)
0.657904 + 0.753102i \(0.271443\pi\)
\(80\) 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 \(-0.134029\pi\)
0.912655 + 0.408731i \(0.134029\pi\)
\(90\) 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 \(-0.516268\pi\)
−0.0510844 + 0.998694i \(0.516268\pi\)
\(98\) 0 0
\(99\) 395.121 0.401123
\(100\) 0 0
\(101\) 1326.39 1.30674 0.653368 0.757040i \(-0.273355\pi\)
0.653368 + 0.757040i \(0.273355\pi\)
\(102\) 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 \(-0.563146\pi\)
−0.197080 + 0.980387i \(0.563146\pi\)
\(108\) 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 \(-0.456207\pi\)
0.137145 + 0.990551i \(0.456207\pi\)
\(128\) 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 \(-0.544100\pi\)
−0.138102 + 0.990418i \(0.544100\pi\)
\(152\) 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 \(-0.546586\pi\)
−0.145832 + 0.989309i \(0.546586\pi\)
\(158\) 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 \(-0.115782\pi\)
0.934573 + 0.355771i \(0.115782\pi\)
\(164\) 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 \(-0.309896\pi\)
0.562355 + 0.826896i \(0.309896\pi\)
\(174\) 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 \(-0.581603\pi\)
−0.253565 + 0.967318i \(0.581603\pi\)
\(180\) 0 0
\(181\) 2555.03 1.04925 0.524625 0.851334i \(-0.324206\pi\)
0.524625 + 0.851334i \(0.324206\pi\)
\(182\) 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 \(-0.288427\pi\)
0.616804 + 0.787117i \(0.288427\pi\)
\(192\) 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 \(-0.462421\pi\)
0.117783 + 0.993039i \(0.462421\pi\)
\(212\) 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 \(-0.250484\pi\)
0.706030 + 0.708182i \(0.250484\pi\)
\(230\) 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 \(-0.690615\pi\)
−0.563680 + 0.825993i \(0.690615\pi\)
\(240\) 0 0
\(241\) 6240.57 1.66801 0.834006 0.551756i \(-0.186042\pi\)
0.834006 + 0.551756i \(0.186042\pi\)
\(242\) 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 \(-0.683842\pi\)
−0.545979 + 0.837799i \(0.683842\pi\)
\(258\) 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 \(-0.793145\pi\)
−0.796171 + 0.605071i \(0.793145\pi\)
\(264\) 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 \(-0.918683\pi\)
−0.967546 + 0.252694i \(0.918683\pi\)
\(270\) 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 \(-0.593467\pi\)
−0.289434 + 0.957198i \(0.593467\pi\)
\(294\) 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 \(-0.850317\pi\)
−0.891459 + 0.453102i \(0.850317\pi\)
\(314\) 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 \(-0.287671\pi\)
0.618671 + 0.785650i \(0.287671\pi\)
\(332\) 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 \(-0.153183\pi\)
0.886423 + 0.462876i \(0.153183\pi\)
\(348\) 0 0
\(349\) 1293.03 0.198322 0.0991610 0.995071i \(-0.468384\pi\)
0.0991610 + 0.995071i \(0.468384\pi\)
\(350\) 0 0
\(351\) −2256.39 −0.343125
\(352\) 0 0
\(353\) −7265.01 −1.09540 −0.547702 0.836674i \(-0.684497\pi\)
−0.547702 + 0.836674i \(0.684497\pi\)
\(354\) 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 \(-0.532387\pi\)
−0.101572 + 0.994828i \(0.532387\pi\)
\(360\) 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 \(-0.845892\pi\)
−0.885073 + 0.465453i \(0.845892\pi\)
\(380\) 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 \(-0.842431\pi\)
−0.879960 + 0.475047i \(0.842431\pi\)
\(398\) 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 \(-0.389938\pi\)
0.338920 + 0.940815i \(0.389938\pi\)
\(410\) 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 \(-0.523260\pi\)
−0.0730092 + 0.997331i \(0.523260\pi\)
\(432\) 0 0
\(433\) 4039.33 0.448309 0.224154 0.974554i \(-0.428038\pi\)
0.224154 + 0.974554i \(0.428038\pi\)
\(434\) 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 \(-0.840994\pi\)
−0.877806 + 0.479016i \(0.840994\pi\)
\(444\) 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 \(-0.541826\pi\)
−0.131024 + 0.991379i \(0.541826\pi\)
\(462\) 0 0
\(463\) 15296.9 1.53543 0.767717 0.640789i \(-0.221393\pi\)
0.767717 + 0.640789i \(0.221393\pi\)
\(464\) 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 \(-0.383943\pi\)
0.356579 + 0.934265i \(0.383943\pi\)
\(488\) 0 0
\(489\) −11669.3 −1.07915
\(490\) 0 0
\(491\) −14761.3 −1.35676 −0.678380 0.734711i \(-0.737318\pi\)
−0.678380 + 0.734711i \(0.737318\pi\)
\(492\) 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 \(-0.325707\pi\)
0.520604 + 0.853798i \(0.325707\pi\)
\(500\) 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 \(-0.448101\pi\)
0.162324 + 0.986737i \(0.448101\pi\)
\(510\) 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 \(-0.731344\pi\)
−0.664473 + 0.747312i \(0.731344\pi\)
\(522\) 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 \(-0.244496\pi\)
0.719228 + 0.694774i \(0.244496\pi\)
\(548\) 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 \(-0.186824\pi\)
0.832647 + 0.553804i \(0.186824\pi\)
\(572\) 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 \(-0.743811\pi\)
−0.693226 + 0.720721i \(0.743811\pi\)
\(578\) 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 \(-0.365158\pi\)
0.411061 + 0.911608i \(0.365158\pi\)
\(594\) 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 \(-0.302849\pi\)
0.580520 + 0.814246i \(0.302849\pi\)
\(600\) 0 0
\(601\) 5518.23 0.374531 0.187266 0.982309i \(-0.440037\pi\)
0.187266 + 0.982309i \(0.440037\pi\)
\(602\) 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 \(-0.451029\pi\)
0.153241 + 0.988189i \(0.451029\pi\)
\(632\) 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 \(-0.555790\pi\)
−0.174373 + 0.984680i \(0.555790\pi\)
\(660\) 0 0
\(661\) 25442.5 1.49713 0.748563 0.663064i \(-0.230744\pi\)
0.748563 + 0.663064i \(0.230744\pi\)
\(662\) 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 \(-0.489250\pi\)
0.0337654 + 0.999430i \(0.489250\pi\)
\(678\) 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 \(-0.688626\pi\)
−0.558509 + 0.829499i \(0.688626\pi\)
\(684\) 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 \(-0.564382\pi\)
−0.200887 + 0.979614i \(0.564382\pi\)
\(734\) 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 \(-0.401730\pi\)
0.303843 + 0.952722i \(0.401730\pi\)
\(740\) 0 0
\(741\) 1072.14 0.0531526
\(742\) 0 0
\(743\) 13351.2 0.659232 0.329616 0.944115i \(-0.393081\pi\)
0.329616 + 0.944115i \(0.393081\pi\)
\(744\) 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 \(-0.262168\pi\)
0.679567 + 0.733614i \(0.262168\pi\)
\(752\) 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 \(-0.304725\pi\)
0.575713 + 0.817652i \(0.304725\pi\)
\(762\) 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 \(-0.509860\pi\)
−0.0309705 + 0.999520i \(0.509860\pi\)
\(770\) 0 0
\(771\) 13496.7 0.630443
\(772\) 0 0
\(773\) −28790.1 −1.33960 −0.669798 0.742544i \(-0.733619\pi\)
−0.669798 + 0.742544i \(0.733619\pi\)
\(774\) 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 \(-0.271850\pi\)
0.656941 + 0.753942i \(0.271850\pi\)
\(798\) 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 \(-0.791595\pi\)
−0.793217 + 0.608939i \(0.791595\pi\)
\(824\) 0 0
\(825\) 10083.5 0.425531
\(826\) 0 0
\(827\) 18855.9 0.792847 0.396423 0.918068i \(-0.370251\pi\)
0.396423 + 0.918068i \(0.370251\pi\)
\(828\) 0 0
\(829\) 34027.1 1.42559 0.712793 0.701375i \(-0.247430\pi\)
0.712793 + 0.701375i \(0.247430\pi\)
\(830\) 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 \(-0.298480\pi\)
0.591641 + 0.806201i \(0.298480\pi\)
\(854\) 0 0
\(855\) −267.870 −0.0107146
\(856\) 0 0
\(857\) 18935.8 0.754765 0.377383 0.926057i \(-0.376824\pi\)
0.377383 + 0.926057i \(0.376824\pi\)
\(858\) 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 \(-0.593140\pi\)
−0.288450 + 0.957495i \(0.593140\pi\)
\(864\) 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 \(-0.605805\pi\)
−0.326308 + 0.945264i \(0.605805\pi\)
\(882\) 0 0
\(883\) −20428.3 −0.778557 −0.389279 0.921120i \(-0.627276\pi\)
−0.389279 + 0.921120i \(0.627276\pi\)
\(884\) 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 \(-0.653755\pi\)
−0.464469 + 0.885589i \(0.653755\pi\)
\(908\) 0 0
\(909\) 11937.5 0.435579
\(910\) 0 0
\(911\) −4890.79 −0.177869 −0.0889347 0.996037i \(-0.528346\pi\)
−0.0889347 + 0.996037i \(0.528346\pi\)
\(912\) 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 \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(920\) 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 \(-0.870263\pi\)
−0.918083 + 0.396388i \(0.870263\pi\)
\(930\) 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 \(-0.652183\pi\)
−0.460091 + 0.887872i \(0.652183\pi\)
\(938\) 0 0
\(939\) 29618.9 1.02937
\(940\) 0 0
\(941\) 40403.8 1.39971 0.699854 0.714286i \(-0.253248\pi\)
0.699854 + 0.714286i \(0.253248\pi\)
\(942\) 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 \(-0.603142\pi\)
−0.318391 + 0.947959i \(0.603142\pi\)
\(948\) 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 \(-0.384874\pi\)
0.353844 + 0.935304i \(0.384874\pi\)
\(968\) 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 \(-0.482543\pi\)
0.0548155 + 0.998497i \(0.482543\pi\)
\(992\) 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 \(-0.241757\pi\)
0.725180 + 0.688560i \(0.241757\pi\)
\(998\) 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 1176.4.a.bb.1.3 4
4.3 odd 2 2352.4.a.cr.1.3 4
7.6 odd 2 1176.4.a.bc.1.2 yes 4
28.27 even 2 2352.4.a.ck.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.3 4 1.1 even 1 trivial
1176.4.a.bc.1.2 yes 4 7.6 odd 2
2352.4.a.ck.1.2 4 28.27 even 2
2352.4.a.cr.1.3 4 4.3 odd 2