Properties

Label 192.4.a.f.1.1
Level $192$
Weight $4$
Character 192.1
Self dual yes
Analytic conductor $11.328$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 192.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +18.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +18.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +10.0000 q^{13} -54.0000 q^{15} +18.0000 q^{17} +100.000 q^{19} -24.0000 q^{21} +72.0000 q^{23} +199.000 q^{25} -27.0000 q^{27} +234.000 q^{29} -16.0000 q^{31} +108.000 q^{33} +144.000 q^{35} +226.000 q^{37} -30.0000 q^{39} +90.0000 q^{41} -452.000 q^{43} +162.000 q^{45} +432.000 q^{47} -279.000 q^{49} -54.0000 q^{51} -414.000 q^{53} -648.000 q^{55} -300.000 q^{57} +684.000 q^{59} -422.000 q^{61} +72.0000 q^{63} +180.000 q^{65} -332.000 q^{67} -216.000 q^{69} -360.000 q^{71} +26.0000 q^{73} -597.000 q^{75} -288.000 q^{77} +512.000 q^{79} +81.0000 q^{81} +1188.00 q^{83} +324.000 q^{85} -702.000 q^{87} -630.000 q^{89} +80.0000 q^{91} +48.0000 q^{93} +1800.00 q^{95} -1054.00 q^{97} -324.000 q^{99} +O(q^{100})\)

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
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 10.0000 0.213346 0.106673 0.994294i \(-0.465980\pi\)
0.106673 + 0.994294i \(0.465980\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) 0 0
\(17\) 18.0000 0.256802 0.128401 0.991722i \(-0.459015\pi\)
0.128401 + 0.991722i \(0.459015\pi\)
\(18\) 0 0
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) −24.0000 −0.249392
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 234.000 1.49837 0.749185 0.662361i \(-0.230446\pi\)
0.749185 + 0.662361i \(0.230446\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) 108.000 0.569709
\(34\) 0 0
\(35\) 144.000 0.695441
\(36\) 0 0
\(37\) 226.000 1.00417 0.502083 0.864819i \(-0.332567\pi\)
0.502083 + 0.864819i \(0.332567\pi\)
\(38\) 0 0
\(39\) −30.0000 −0.123176
\(40\) 0 0
\(41\) 90.0000 0.342820 0.171410 0.985200i \(-0.445168\pi\)
0.171410 + 0.985200i \(0.445168\pi\)
\(42\) 0 0
\(43\) −452.000 −1.60301 −0.801504 0.597989i \(-0.795967\pi\)
−0.801504 + 0.597989i \(0.795967\pi\)
\(44\) 0 0
\(45\) 162.000 0.536656
\(46\) 0 0
\(47\) 432.000 1.34072 0.670358 0.742038i \(-0.266140\pi\)
0.670358 + 0.742038i \(0.266140\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −54.0000 −0.148265
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) −300.000 −0.697122
\(58\) 0 0
\(59\) 684.000 1.50931 0.754654 0.656123i \(-0.227805\pi\)
0.754654 + 0.656123i \(0.227805\pi\)
\(60\) 0 0
\(61\) −422.000 −0.885763 −0.442882 0.896580i \(-0.646044\pi\)
−0.442882 + 0.896580i \(0.646044\pi\)
\(62\) 0 0
\(63\) 72.0000 0.143986
\(64\) 0 0
\(65\) 180.000 0.343481
\(66\) 0 0
\(67\) −332.000 −0.605377 −0.302688 0.953090i \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) −360.000 −0.601748 −0.300874 0.953664i \(-0.597278\pi\)
−0.300874 + 0.953664i \(0.597278\pi\)
\(72\) 0 0
\(73\) 26.0000 0.0416859 0.0208429 0.999783i \(-0.493365\pi\)
0.0208429 + 0.999783i \(0.493365\pi\)
\(74\) 0 0
\(75\) −597.000 −0.919142
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) 512.000 0.729171 0.364585 0.931170i \(-0.381211\pi\)
0.364585 + 0.931170i \(0.381211\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1188.00 1.57108 0.785542 0.618809i \(-0.212384\pi\)
0.785542 + 0.618809i \(0.212384\pi\)
\(84\) 0 0
\(85\) 324.000 0.413444
\(86\) 0 0
\(87\) −702.000 −0.865084
\(88\) 0 0
\(89\) −630.000 −0.750336 −0.375168 0.926957i \(-0.622415\pi\)
−0.375168 + 0.926957i \(0.622415\pi\)
\(90\) 0 0
\(91\) 80.0000 0.0921569
\(92\) 0 0
\(93\) 48.0000 0.0535201
\(94\) 0 0
\(95\) 1800.00 1.94396
\(96\) 0 0
\(97\) −1054.00 −1.10327 −0.551637 0.834085i \(-0.685996\pi\)
−0.551637 + 0.834085i \(0.685996\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) −558.000 −0.549733 −0.274867 0.961482i \(-0.588634\pi\)
−0.274867 + 0.961482i \(0.588634\pi\)
\(102\) 0 0
\(103\) 8.00000 0.00765304 0.00382652 0.999993i \(-0.498782\pi\)
0.00382652 + 0.999993i \(0.498782\pi\)
\(104\) 0 0
\(105\) −432.000 −0.401513
\(106\) 0 0
\(107\) −1764.00 −1.59376 −0.796880 0.604138i \(-0.793518\pi\)
−0.796880 + 0.604138i \(0.793518\pi\)
\(108\) 0 0
\(109\) −1622.00 −1.42532 −0.712658 0.701512i \(-0.752509\pi\)
−0.712658 + 0.701512i \(0.752509\pi\)
\(110\) 0 0
\(111\) −678.000 −0.579756
\(112\) 0 0
\(113\) −1134.00 −0.944051 −0.472025 0.881585i \(-0.656477\pi\)
−0.472025 + 0.881585i \(0.656477\pi\)
\(114\) 0 0
\(115\) 1296.00 1.05089
\(116\) 0 0
\(117\) 90.0000 0.0711154
\(118\) 0 0
\(119\) 144.000 0.110928
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) −270.000 −0.197927
\(124\) 0 0
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) −592.000 −0.413634 −0.206817 0.978380i \(-0.566310\pi\)
−0.206817 + 0.978380i \(0.566310\pi\)
\(128\) 0 0
\(129\) 1356.00 0.925497
\(130\) 0 0
\(131\) 1908.00 1.27254 0.636270 0.771466i \(-0.280476\pi\)
0.636270 + 0.771466i \(0.280476\pi\)
\(132\) 0 0
\(133\) 800.000 0.521570
\(134\) 0 0
\(135\) −486.000 −0.309839
\(136\) 0 0
\(137\) 954.000 0.594932 0.297466 0.954732i \(-0.403858\pi\)
0.297466 + 0.954732i \(0.403858\pi\)
\(138\) 0 0
\(139\) −2564.00 −1.56457 −0.782286 0.622919i \(-0.785947\pi\)
−0.782286 + 0.622919i \(0.785947\pi\)
\(140\) 0 0
\(141\) −1296.00 −0.774063
\(142\) 0 0
\(143\) −360.000 −0.210522
\(144\) 0 0
\(145\) 4212.00 2.41233
\(146\) 0 0
\(147\) 837.000 0.469623
\(148\) 0 0
\(149\) 738.000 0.405767 0.202884 0.979203i \(-0.434969\pi\)
0.202884 + 0.979203i \(0.434969\pi\)
\(150\) 0 0
\(151\) −2440.00 −1.31500 −0.657498 0.753456i \(-0.728385\pi\)
−0.657498 + 0.753456i \(0.728385\pi\)
\(152\) 0 0
\(153\) 162.000 0.0856008
\(154\) 0 0
\(155\) −288.000 −0.149243
\(156\) 0 0
\(157\) 2554.00 1.29829 0.649145 0.760665i \(-0.275127\pi\)
0.649145 + 0.760665i \(0.275127\pi\)
\(158\) 0 0
\(159\) 1242.00 0.619478
\(160\) 0 0
\(161\) 576.000 0.281958
\(162\) 0 0
\(163\) 820.000 0.394033 0.197016 0.980400i \(-0.436875\pi\)
0.197016 + 0.980400i \(0.436875\pi\)
\(164\) 0 0
\(165\) 1944.00 0.917213
\(166\) 0 0
\(167\) 1944.00 0.900786 0.450393 0.892830i \(-0.351284\pi\)
0.450393 + 0.892830i \(0.351284\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) 900.000 0.402484
\(172\) 0 0
\(173\) 1242.00 0.545824 0.272912 0.962039i \(-0.412013\pi\)
0.272912 + 0.962039i \(0.412013\pi\)
\(174\) 0 0
\(175\) 1592.00 0.687679
\(176\) 0 0
\(177\) −2052.00 −0.871400
\(178\) 0 0
\(179\) −1116.00 −0.465999 −0.232999 0.972477i \(-0.574854\pi\)
−0.232999 + 0.972477i \(0.574854\pi\)
\(180\) 0 0
\(181\) −1070.00 −0.439406 −0.219703 0.975567i \(-0.570509\pi\)
−0.219703 + 0.975567i \(0.570509\pi\)
\(182\) 0 0
\(183\) 1266.00 0.511396
\(184\) 0 0
\(185\) 4068.00 1.61668
\(186\) 0 0
\(187\) −648.000 −0.253403
\(188\) 0 0
\(189\) −216.000 −0.0831306
\(190\) 0 0
\(191\) −576.000 −0.218209 −0.109104 0.994030i \(-0.534798\pi\)
−0.109104 + 0.994030i \(0.534798\pi\)
\(192\) 0 0
\(193\) −1342.00 −0.500514 −0.250257 0.968179i \(-0.580515\pi\)
−0.250257 + 0.968179i \(0.580515\pi\)
\(194\) 0 0
\(195\) −540.000 −0.198309
\(196\) 0 0
\(197\) −1422.00 −0.514281 −0.257140 0.966374i \(-0.582780\pi\)
−0.257140 + 0.966374i \(0.582780\pi\)
\(198\) 0 0
\(199\) 872.000 0.310625 0.155313 0.987865i \(-0.450361\pi\)
0.155313 + 0.987865i \(0.450361\pi\)
\(200\) 0 0
\(201\) 996.000 0.349515
\(202\) 0 0
\(203\) 1872.00 0.647235
\(204\) 0 0
\(205\) 1620.00 0.551930
\(206\) 0 0
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) −1340.00 −0.437201 −0.218600 0.975814i \(-0.570149\pi\)
−0.218600 + 0.975814i \(0.570149\pi\)
\(212\) 0 0
\(213\) 1080.00 0.347420
\(214\) 0 0
\(215\) −8136.00 −2.58079
\(216\) 0 0
\(217\) −128.000 −0.0400424
\(218\) 0 0
\(219\) −78.0000 −0.0240674
\(220\) 0 0
\(221\) 180.000 0.0547878
\(222\) 0 0
\(223\) 4880.00 1.46542 0.732711 0.680540i \(-0.238255\pi\)
0.732711 + 0.680540i \(0.238255\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) −2700.00 −0.789451 −0.394725 0.918799i \(-0.629160\pi\)
−0.394725 + 0.918799i \(0.629160\pi\)
\(228\) 0 0
\(229\) −254.000 −0.0732960 −0.0366480 0.999328i \(-0.511668\pi\)
−0.0366480 + 0.999328i \(0.511668\pi\)
\(230\) 0 0
\(231\) 864.000 0.246091
\(232\) 0 0
\(233\) 4410.00 1.23995 0.619976 0.784621i \(-0.287142\pi\)
0.619976 + 0.784621i \(0.287142\pi\)
\(234\) 0 0
\(235\) 7776.00 2.15851
\(236\) 0 0
\(237\) −1536.00 −0.420987
\(238\) 0 0
\(239\) −3888.00 −1.05228 −0.526138 0.850399i \(-0.676360\pi\)
−0.526138 + 0.850399i \(0.676360\pi\)
\(240\) 0 0
\(241\) 5138.00 1.37331 0.686655 0.726984i \(-0.259078\pi\)
0.686655 + 0.726984i \(0.259078\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −5022.00 −1.30957
\(246\) 0 0
\(247\) 1000.00 0.257605
\(248\) 0 0
\(249\) −3564.00 −0.907066
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) −972.000 −0.238702
\(256\) 0 0
\(257\) −5886.00 −1.42863 −0.714316 0.699823i \(-0.753262\pi\)
−0.714316 + 0.699823i \(0.753262\pi\)
\(258\) 0 0
\(259\) 1808.00 0.433759
\(260\) 0 0
\(261\) 2106.00 0.499456
\(262\) 0 0
\(263\) 2232.00 0.523312 0.261656 0.965161i \(-0.415731\pi\)
0.261656 + 0.965161i \(0.415731\pi\)
\(264\) 0 0
\(265\) −7452.00 −1.72744
\(266\) 0 0
\(267\) 1890.00 0.433206
\(268\) 0 0
\(269\) 666.000 0.150954 0.0754772 0.997148i \(-0.475952\pi\)
0.0754772 + 0.997148i \(0.475952\pi\)
\(270\) 0 0
\(271\) −5536.00 −1.24092 −0.620458 0.784240i \(-0.713053\pi\)
−0.620458 + 0.784240i \(0.713053\pi\)
\(272\) 0 0
\(273\) −240.000 −0.0532068
\(274\) 0 0
\(275\) −7164.00 −1.57093
\(276\) 0 0
\(277\) −2126.00 −0.461151 −0.230576 0.973054i \(-0.574061\pi\)
−0.230576 + 0.973054i \(0.574061\pi\)
\(278\) 0 0
\(279\) −144.000 −0.0308998
\(280\) 0 0
\(281\) −2934.00 −0.622875 −0.311437 0.950267i \(-0.600810\pi\)
−0.311437 + 0.950267i \(0.600810\pi\)
\(282\) 0 0
\(283\) −2036.00 −0.427659 −0.213830 0.976871i \(-0.568594\pi\)
−0.213830 + 0.976871i \(0.568594\pi\)
\(284\) 0 0
\(285\) −5400.00 −1.12235
\(286\) 0 0
\(287\) 720.000 0.148085
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 3162.00 0.636975
\(292\) 0 0
\(293\) −2286.00 −0.455800 −0.227900 0.973684i \(-0.573186\pi\)
−0.227900 + 0.973684i \(0.573186\pi\)
\(294\) 0 0
\(295\) 12312.0 2.42994
\(296\) 0 0
\(297\) 972.000 0.189903
\(298\) 0 0
\(299\) 720.000 0.139260
\(300\) 0 0
\(301\) −3616.00 −0.692434
\(302\) 0 0
\(303\) 1674.00 0.317389
\(304\) 0 0
\(305\) −7596.00 −1.42605
\(306\) 0 0
\(307\) −1244.00 −0.231267 −0.115633 0.993292i \(-0.536890\pi\)
−0.115633 + 0.993292i \(0.536890\pi\)
\(308\) 0 0
\(309\) −24.0000 −0.00441849
\(310\) 0 0
\(311\) 1224.00 0.223173 0.111586 0.993755i \(-0.464407\pi\)
0.111586 + 0.993755i \(0.464407\pi\)
\(312\) 0 0
\(313\) 1898.00 0.342752 0.171376 0.985206i \(-0.445179\pi\)
0.171376 + 0.985206i \(0.445179\pi\)
\(314\) 0 0
\(315\) 1296.00 0.231814
\(316\) 0 0
\(317\) 9162.00 1.62331 0.811655 0.584137i \(-0.198567\pi\)
0.811655 + 0.584137i \(0.198567\pi\)
\(318\) 0 0
\(319\) −8424.00 −1.47854
\(320\) 0 0
\(321\) 5292.00 0.920158
\(322\) 0 0
\(323\) 1800.00 0.310076
\(324\) 0 0
\(325\) 1990.00 0.339647
\(326\) 0 0
\(327\) 4866.00 0.822906
\(328\) 0 0
\(329\) 3456.00 0.579135
\(330\) 0 0
\(331\) 4348.00 0.722017 0.361009 0.932562i \(-0.382432\pi\)
0.361009 + 0.932562i \(0.382432\pi\)
\(332\) 0 0
\(333\) 2034.00 0.334722
\(334\) 0 0
\(335\) −5976.00 −0.974638
\(336\) 0 0
\(337\) 7154.00 1.15639 0.578195 0.815899i \(-0.303757\pi\)
0.578195 + 0.815899i \(0.303757\pi\)
\(338\) 0 0
\(339\) 3402.00 0.545048
\(340\) 0 0
\(341\) 576.000 0.0914726
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) −3888.00 −0.606733
\(346\) 0 0
\(347\) 1836.00 0.284039 0.142020 0.989864i \(-0.454640\pi\)
0.142020 + 0.989864i \(0.454640\pi\)
\(348\) 0 0
\(349\) −5894.00 −0.904007 −0.452004 0.892016i \(-0.649291\pi\)
−0.452004 + 0.892016i \(0.649291\pi\)
\(350\) 0 0
\(351\) −270.000 −0.0410585
\(352\) 0 0
\(353\) 11106.0 1.67454 0.837270 0.546789i \(-0.184150\pi\)
0.837270 + 0.546789i \(0.184150\pi\)
\(354\) 0 0
\(355\) −6480.00 −0.968796
\(356\) 0 0
\(357\) −432.000 −0.0640444
\(358\) 0 0
\(359\) 13176.0 1.93705 0.968527 0.248907i \(-0.0800713\pi\)
0.968527 + 0.248907i \(0.0800713\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 0 0
\(363\) 105.000 0.0151820
\(364\) 0 0
\(365\) 468.000 0.0671130
\(366\) 0 0
\(367\) −6112.00 −0.869329 −0.434665 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(368\) 0 0
\(369\) 810.000 0.114273
\(370\) 0 0
\(371\) −3312.00 −0.463478
\(372\) 0 0
\(373\) 13618.0 1.89038 0.945192 0.326515i \(-0.105874\pi\)
0.945192 + 0.326515i \(0.105874\pi\)
\(374\) 0 0
\(375\) −3996.00 −0.550273
\(376\) 0 0
\(377\) 2340.00 0.319671
\(378\) 0 0
\(379\) −692.000 −0.0937880 −0.0468940 0.998900i \(-0.514932\pi\)
−0.0468940 + 0.998900i \(0.514932\pi\)
\(380\) 0 0
\(381\) 1776.00 0.238812
\(382\) 0 0
\(383\) −8064.00 −1.07585 −0.537926 0.842992i \(-0.680792\pi\)
−0.537926 + 0.842992i \(0.680792\pi\)
\(384\) 0 0
\(385\) −5184.00 −0.686237
\(386\) 0 0
\(387\) −4068.00 −0.534336
\(388\) 0 0
\(389\) −12654.0 −1.64931 −0.824657 0.565633i \(-0.808632\pi\)
−0.824657 + 0.565633i \(0.808632\pi\)
\(390\) 0 0
\(391\) 1296.00 0.167625
\(392\) 0 0
\(393\) −5724.00 −0.734701
\(394\) 0 0
\(395\) 9216.00 1.17394
\(396\) 0 0
\(397\) 106.000 0.0134005 0.00670024 0.999978i \(-0.497867\pi\)
0.00670024 + 0.999978i \(0.497867\pi\)
\(398\) 0 0
\(399\) −2400.00 −0.301129
\(400\) 0 0
\(401\) −4014.00 −0.499874 −0.249937 0.968262i \(-0.580410\pi\)
−0.249937 + 0.968262i \(0.580410\pi\)
\(402\) 0 0
\(403\) −160.000 −0.0197771
\(404\) 0 0
\(405\) 1458.00 0.178885
\(406\) 0 0
\(407\) −8136.00 −0.990876
\(408\) 0 0
\(409\) 3914.00 0.473190 0.236595 0.971608i \(-0.423968\pi\)
0.236595 + 0.971608i \(0.423968\pi\)
\(410\) 0 0
\(411\) −2862.00 −0.343484
\(412\) 0 0
\(413\) 5472.00 0.651960
\(414\) 0 0
\(415\) 21384.0 2.52940
\(416\) 0 0
\(417\) 7692.00 0.903307
\(418\) 0 0
\(419\) −4428.00 −0.516282 −0.258141 0.966107i \(-0.583110\pi\)
−0.258141 + 0.966107i \(0.583110\pi\)
\(420\) 0 0
\(421\) 15490.0 1.79320 0.896599 0.442843i \(-0.146030\pi\)
0.896599 + 0.442843i \(0.146030\pi\)
\(422\) 0 0
\(423\) 3888.00 0.446906
\(424\) 0 0
\(425\) 3582.00 0.408829
\(426\) 0 0
\(427\) −3376.00 −0.382614
\(428\) 0 0
\(429\) 1080.00 0.121545
\(430\) 0 0
\(431\) 6768.00 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(432\) 0 0
\(433\) 1298.00 0.144060 0.0720299 0.997402i \(-0.477052\pi\)
0.0720299 + 0.997402i \(0.477052\pi\)
\(434\) 0 0
\(435\) −12636.0 −1.39276
\(436\) 0 0
\(437\) 7200.00 0.788153
\(438\) 0 0
\(439\) −2248.00 −0.244399 −0.122200 0.992506i \(-0.538995\pi\)
−0.122200 + 0.992506i \(0.538995\pi\)
\(440\) 0 0
\(441\) −2511.00 −0.271137
\(442\) 0 0
\(443\) 9612.00 1.03088 0.515440 0.856926i \(-0.327628\pi\)
0.515440 + 0.856926i \(0.327628\pi\)
\(444\) 0 0
\(445\) −11340.0 −1.20802
\(446\) 0 0
\(447\) −2214.00 −0.234270
\(448\) 0 0
\(449\) 162.000 0.0170273 0.00851364 0.999964i \(-0.497290\pi\)
0.00851364 + 0.999964i \(0.497290\pi\)
\(450\) 0 0
\(451\) −3240.00 −0.338283
\(452\) 0 0
\(453\) 7320.00 0.759213
\(454\) 0 0
\(455\) 1440.00 0.148370
\(456\) 0 0
\(457\) 1370.00 0.140232 0.0701159 0.997539i \(-0.477663\pi\)
0.0701159 + 0.997539i \(0.477663\pi\)
\(458\) 0 0
\(459\) −486.000 −0.0494217
\(460\) 0 0
\(461\) 15354.0 1.55121 0.775604 0.631220i \(-0.217445\pi\)
0.775604 + 0.631220i \(0.217445\pi\)
\(462\) 0 0
\(463\) −13024.0 −1.30729 −0.653646 0.756800i \(-0.726762\pi\)
−0.653646 + 0.756800i \(0.726762\pi\)
\(464\) 0 0
\(465\) 864.000 0.0861657
\(466\) 0 0
\(467\) 14436.0 1.43045 0.715223 0.698896i \(-0.246325\pi\)
0.715223 + 0.698896i \(0.246325\pi\)
\(468\) 0 0
\(469\) −2656.00 −0.261498
\(470\) 0 0
\(471\) −7662.00 −0.749568
\(472\) 0 0
\(473\) 16272.0 1.58179
\(474\) 0 0
\(475\) 19900.0 1.92226
\(476\) 0 0
\(477\) −3726.00 −0.357656
\(478\) 0 0
\(479\) 12096.0 1.15382 0.576911 0.816807i \(-0.304258\pi\)
0.576911 + 0.816807i \(0.304258\pi\)
\(480\) 0 0
\(481\) 2260.00 0.214235
\(482\) 0 0
\(483\) −1728.00 −0.162788
\(484\) 0 0
\(485\) −18972.0 −1.77624
\(486\) 0 0
\(487\) 6056.00 0.563498 0.281749 0.959488i \(-0.409085\pi\)
0.281749 + 0.959488i \(0.409085\pi\)
\(488\) 0 0
\(489\) −2460.00 −0.227495
\(490\) 0 0
\(491\) −7524.00 −0.691555 −0.345777 0.938317i \(-0.612385\pi\)
−0.345777 + 0.938317i \(0.612385\pi\)
\(492\) 0 0
\(493\) 4212.00 0.384785
\(494\) 0 0
\(495\) −5832.00 −0.529553
\(496\) 0 0
\(497\) −2880.00 −0.259931
\(498\) 0 0
\(499\) −5276.00 −0.473319 −0.236660 0.971593i \(-0.576053\pi\)
−0.236660 + 0.971593i \(0.576053\pi\)
\(500\) 0 0
\(501\) −5832.00 −0.520069
\(502\) 0 0
\(503\) 4968.00 0.440382 0.220191 0.975457i \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(504\) 0 0
\(505\) −10044.0 −0.885054
\(506\) 0 0
\(507\) 6291.00 0.551071
\(508\) 0 0
\(509\) −10998.0 −0.957717 −0.478858 0.877892i \(-0.658949\pi\)
−0.478858 + 0.877892i \(0.658949\pi\)
\(510\) 0 0
\(511\) 208.000 0.0180066
\(512\) 0 0
\(513\) −2700.00 −0.232374
\(514\) 0 0
\(515\) 144.000 0.0123212
\(516\) 0 0
\(517\) −15552.0 −1.32297
\(518\) 0 0
\(519\) −3726.00 −0.315131
\(520\) 0 0
\(521\) −8838.00 −0.743186 −0.371593 0.928396i \(-0.621188\pi\)
−0.371593 + 0.928396i \(0.621188\pi\)
\(522\) 0 0
\(523\) −22436.0 −1.87583 −0.937914 0.346869i \(-0.887245\pi\)
−0.937914 + 0.346869i \(0.887245\pi\)
\(524\) 0 0
\(525\) −4776.00 −0.397032
\(526\) 0 0
\(527\) −288.000 −0.0238055
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 6156.00 0.503103
\(532\) 0 0
\(533\) 900.000 0.0731395
\(534\) 0 0
\(535\) −31752.0 −2.56590
\(536\) 0 0
\(537\) 3348.00 0.269044
\(538\) 0 0
\(539\) 10044.0 0.802645
\(540\) 0 0
\(541\) 4762.00 0.378437 0.189218 0.981935i \(-0.439405\pi\)
0.189218 + 0.981935i \(0.439405\pi\)
\(542\) 0 0
\(543\) 3210.00 0.253691
\(544\) 0 0
\(545\) −29196.0 −2.29471
\(546\) 0 0
\(547\) 6004.00 0.469310 0.234655 0.972079i \(-0.424604\pi\)
0.234655 + 0.972079i \(0.424604\pi\)
\(548\) 0 0
\(549\) −3798.00 −0.295254
\(550\) 0 0
\(551\) 23400.0 1.80921
\(552\) 0 0
\(553\) 4096.00 0.314972
\(554\) 0 0
\(555\) −12204.0 −0.933389
\(556\) 0 0
\(557\) 5274.00 0.401197 0.200598 0.979674i \(-0.435711\pi\)
0.200598 + 0.979674i \(0.435711\pi\)
\(558\) 0 0
\(559\) −4520.00 −0.341996
\(560\) 0 0
\(561\) 1944.00 0.146303
\(562\) 0 0
\(563\) 12420.0 0.929735 0.464867 0.885380i \(-0.346102\pi\)
0.464867 + 0.885380i \(0.346102\pi\)
\(564\) 0 0
\(565\) −20412.0 −1.51989
\(566\) 0 0
\(567\) 648.000 0.0479955
\(568\) 0 0
\(569\) −21366.0 −1.57418 −0.787091 0.616837i \(-0.788414\pi\)
−0.787091 + 0.616837i \(0.788414\pi\)
\(570\) 0 0
\(571\) −21140.0 −1.54935 −0.774677 0.632357i \(-0.782088\pi\)
−0.774677 + 0.632357i \(0.782088\pi\)
\(572\) 0 0
\(573\) 1728.00 0.125983
\(574\) 0 0
\(575\) 14328.0 1.03916
\(576\) 0 0
\(577\) 3266.00 0.235642 0.117821 0.993035i \(-0.462409\pi\)
0.117821 + 0.993035i \(0.462409\pi\)
\(578\) 0 0
\(579\) 4026.00 0.288972
\(580\) 0 0
\(581\) 9504.00 0.678644
\(582\) 0 0
\(583\) 14904.0 1.05877
\(584\) 0 0
\(585\) 1620.00 0.114494
\(586\) 0 0
\(587\) −17028.0 −1.19731 −0.598655 0.801007i \(-0.704298\pi\)
−0.598655 + 0.801007i \(0.704298\pi\)
\(588\) 0 0
\(589\) −1600.00 −0.111930
\(590\) 0 0
\(591\) 4266.00 0.296920
\(592\) 0 0
\(593\) 9522.00 0.659396 0.329698 0.944086i \(-0.393053\pi\)
0.329698 + 0.944086i \(0.393053\pi\)
\(594\) 0 0
\(595\) 2592.00 0.178591
\(596\) 0 0
\(597\) −2616.00 −0.179340
\(598\) 0 0
\(599\) −10296.0 −0.702309 −0.351155 0.936318i \(-0.614211\pi\)
−0.351155 + 0.936318i \(0.614211\pi\)
\(600\) 0 0
\(601\) −3382.00 −0.229542 −0.114771 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(602\) 0 0
\(603\) −2988.00 −0.201792
\(604\) 0 0
\(605\) −630.000 −0.0423358
\(606\) 0 0
\(607\) −20656.0 −1.38122 −0.690611 0.723227i \(-0.742658\pi\)
−0.690611 + 0.723227i \(0.742658\pi\)
\(608\) 0 0
\(609\) −5616.00 −0.373681
\(610\) 0 0
\(611\) 4320.00 0.286037
\(612\) 0 0
\(613\) 22114.0 1.45706 0.728529 0.685015i \(-0.240205\pi\)
0.728529 + 0.685015i \(0.240205\pi\)
\(614\) 0 0
\(615\) −4860.00 −0.318657
\(616\) 0 0
\(617\) 19962.0 1.30250 0.651248 0.758865i \(-0.274246\pi\)
0.651248 + 0.758865i \(0.274246\pi\)
\(618\) 0 0
\(619\) 604.000 0.0392194 0.0196097 0.999808i \(-0.493758\pi\)
0.0196097 + 0.999808i \(0.493758\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) −5040.00 −0.324115
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) 10800.0 0.687895
\(628\) 0 0
\(629\) 4068.00 0.257872
\(630\) 0 0
\(631\) 152.000 0.00958958 0.00479479 0.999989i \(-0.498474\pi\)
0.00479479 + 0.999989i \(0.498474\pi\)
\(632\) 0 0
\(633\) 4020.00 0.252418
\(634\) 0 0
\(635\) −10656.0 −0.665938
\(636\) 0 0
\(637\) −2790.00 −0.173538
\(638\) 0 0
\(639\) −3240.00 −0.200583
\(640\) 0 0
\(641\) 4194.00 0.258429 0.129215 0.991617i \(-0.458754\pi\)
0.129215 + 0.991617i \(0.458754\pi\)
\(642\) 0 0
\(643\) 7252.00 0.444776 0.222388 0.974958i \(-0.428615\pi\)
0.222388 + 0.974958i \(0.428615\pi\)
\(644\) 0 0
\(645\) 24408.0 1.49002
\(646\) 0 0
\(647\) −6696.00 −0.406873 −0.203437 0.979088i \(-0.565211\pi\)
−0.203437 + 0.979088i \(0.565211\pi\)
\(648\) 0 0
\(649\) −24624.0 −1.48933
\(650\) 0 0
\(651\) 384.000 0.0231185
\(652\) 0 0
\(653\) −28422.0 −1.70328 −0.851638 0.524131i \(-0.824390\pi\)
−0.851638 + 0.524131i \(0.824390\pi\)
\(654\) 0 0
\(655\) 34344.0 2.04875
\(656\) 0 0
\(657\) 234.000 0.0138953
\(658\) 0 0
\(659\) 19908.0 1.17679 0.588396 0.808573i \(-0.299760\pi\)
0.588396 + 0.808573i \(0.299760\pi\)
\(660\) 0 0
\(661\) −14318.0 −0.842520 −0.421260 0.906940i \(-0.638412\pi\)
−0.421260 + 0.906940i \(0.638412\pi\)
\(662\) 0 0
\(663\) −540.000 −0.0316318
\(664\) 0 0
\(665\) 14400.0 0.839711
\(666\) 0 0
\(667\) 16848.0 0.978047
\(668\) 0 0
\(669\) −14640.0 −0.846061
\(670\) 0 0
\(671\) 15192.0 0.874040
\(672\) 0 0
\(673\) 30050.0 1.72116 0.860581 0.509313i \(-0.170101\pi\)
0.860581 + 0.509313i \(0.170101\pi\)
\(674\) 0 0
\(675\) −5373.00 −0.306381
\(676\) 0 0
\(677\) −22158.0 −1.25790 −0.628952 0.777444i \(-0.716516\pi\)
−0.628952 + 0.777444i \(0.716516\pi\)
\(678\) 0 0
\(679\) −8432.00 −0.476569
\(680\) 0 0
\(681\) 8100.00 0.455790
\(682\) 0 0
\(683\) 3132.00 0.175465 0.0877325 0.996144i \(-0.472038\pi\)
0.0877325 + 0.996144i \(0.472038\pi\)
\(684\) 0 0
\(685\) 17172.0 0.957822
\(686\) 0 0
\(687\) 762.000 0.0423175
\(688\) 0 0
\(689\) −4140.00 −0.228914
\(690\) 0 0
\(691\) 20932.0 1.15237 0.576187 0.817318i \(-0.304540\pi\)
0.576187 + 0.817318i \(0.304540\pi\)
\(692\) 0 0
\(693\) −2592.00 −0.142081
\(694\) 0 0
\(695\) −46152.0 −2.51891
\(696\) 0 0
\(697\) 1620.00 0.0880371
\(698\) 0 0
\(699\) −13230.0 −0.715886
\(700\) 0 0
\(701\) 21834.0 1.17640 0.588202 0.808714i \(-0.299836\pi\)
0.588202 + 0.808714i \(0.299836\pi\)
\(702\) 0 0
\(703\) 22600.0 1.21248
\(704\) 0 0
\(705\) −23328.0 −1.24622
\(706\) 0 0
\(707\) −4464.00 −0.237463
\(708\) 0 0
\(709\) −12446.0 −0.659266 −0.329633 0.944109i \(-0.606925\pi\)
−0.329633 + 0.944109i \(0.606925\pi\)
\(710\) 0 0
\(711\) 4608.00 0.243057
\(712\) 0 0
\(713\) −1152.00 −0.0605088
\(714\) 0 0
\(715\) −6480.00 −0.338935
\(716\) 0 0
\(717\) 11664.0 0.607531
\(718\) 0 0
\(719\) −12528.0 −0.649813 −0.324907 0.945746i \(-0.605333\pi\)
−0.324907 + 0.945746i \(0.605333\pi\)
\(720\) 0 0
\(721\) 64.0000 0.00330580
\(722\) 0 0
\(723\) −15414.0 −0.792881
\(724\) 0 0
\(725\) 46566.0 2.38540
\(726\) 0 0
\(727\) 11576.0 0.590550 0.295275 0.955412i \(-0.404589\pi\)
0.295275 + 0.955412i \(0.404589\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −8136.00 −0.411656
\(732\) 0 0
\(733\) 29338.0 1.47834 0.739170 0.673519i \(-0.235218\pi\)
0.739170 + 0.673519i \(0.235218\pi\)
\(734\) 0 0
\(735\) 15066.0 0.756079
\(736\) 0 0
\(737\) 11952.0 0.597364
\(738\) 0 0
\(739\) −2540.00 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(740\) 0 0
\(741\) −3000.00 −0.148728
\(742\) 0 0
\(743\) −18792.0 −0.927876 −0.463938 0.885868i \(-0.653564\pi\)
−0.463938 + 0.885868i \(0.653564\pi\)
\(744\) 0 0
\(745\) 13284.0 0.653273
\(746\) 0 0
\(747\) 10692.0 0.523695
\(748\) 0 0
\(749\) −14112.0 −0.688440
\(750\) 0 0
\(751\) 4832.00 0.234783 0.117392 0.993086i \(-0.462547\pi\)
0.117392 + 0.993086i \(0.462547\pi\)
\(752\) 0 0
\(753\) 14364.0 0.695157
\(754\) 0 0
\(755\) −43920.0 −2.11710
\(756\) 0 0
\(757\) 20818.0 0.999529 0.499764 0.866161i \(-0.333420\pi\)
0.499764 + 0.866161i \(0.333420\pi\)
\(758\) 0 0
\(759\) 7776.00 0.371872
\(760\) 0 0
\(761\) 12042.0 0.573617 0.286808 0.957988i \(-0.407406\pi\)
0.286808 + 0.957988i \(0.407406\pi\)
\(762\) 0 0
\(763\) −12976.0 −0.615679
\(764\) 0 0
\(765\) 2916.00 0.137815
\(766\) 0 0
\(767\) 6840.00 0.322005
\(768\) 0 0
\(769\) 13058.0 0.612332 0.306166 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(770\) 0 0
\(771\) 17658.0 0.824821
\(772\) 0 0
\(773\) 11826.0 0.550261 0.275130 0.961407i \(-0.411279\pi\)
0.275130 + 0.961407i \(0.411279\pi\)
\(774\) 0 0
\(775\) −3184.00 −0.147578
\(776\) 0 0
\(777\) −5424.00 −0.250431
\(778\) 0 0
\(779\) 9000.00 0.413939
\(780\) 0 0
\(781\) 12960.0 0.593784
\(782\) 0 0
\(783\) −6318.00 −0.288361
\(784\) 0 0
\(785\) 45972.0 2.09021
\(786\) 0 0
\(787\) −11996.0 −0.543343 −0.271672 0.962390i \(-0.587576\pi\)
−0.271672 + 0.962390i \(0.587576\pi\)
\(788\) 0 0
\(789\) −6696.00 −0.302134
\(790\) 0 0
\(791\) −9072.00 −0.407792
\(792\) 0 0
\(793\) −4220.00 −0.188974
\(794\) 0 0
\(795\) 22356.0 0.997340
\(796\) 0 0
\(797\) −6966.00 −0.309596 −0.154798 0.987946i \(-0.549473\pi\)
−0.154798 + 0.987946i \(0.549473\pi\)
\(798\) 0 0
\(799\) 7776.00 0.344299
\(800\) 0 0
\(801\) −5670.00 −0.250112
\(802\) 0 0
\(803\) −936.000 −0.0411342
\(804\) 0 0
\(805\) 10368.0 0.453943
\(806\) 0 0
\(807\) −1998.00 −0.0871536
\(808\) 0 0
\(809\) −40806.0 −1.77338 −0.886689 0.462367i \(-0.847000\pi\)
−0.886689 + 0.462367i \(0.847000\pi\)
\(810\) 0 0
\(811\) 17980.0 0.778500 0.389250 0.921132i \(-0.372734\pi\)
0.389250 + 0.921132i \(0.372734\pi\)
\(812\) 0 0
\(813\) 16608.0 0.716443
\(814\) 0 0
\(815\) 14760.0 0.634381
\(816\) 0 0
\(817\) −45200.0 −1.93555
\(818\) 0 0
\(819\) 720.000 0.0307190
\(820\) 0 0
\(821\) 12834.0 0.545566 0.272783 0.962076i \(-0.412056\pi\)
0.272783 + 0.962076i \(0.412056\pi\)
\(822\) 0 0
\(823\) −37864.0 −1.60371 −0.801857 0.597516i \(-0.796154\pi\)
−0.801857 + 0.597516i \(0.796154\pi\)
\(824\) 0 0
\(825\) 21492.0 0.906976
\(826\) 0 0
\(827\) −42516.0 −1.78770 −0.893849 0.448368i \(-0.852005\pi\)
−0.893849 + 0.448368i \(0.852005\pi\)
\(828\) 0 0
\(829\) −45638.0 −1.91203 −0.956015 0.293317i \(-0.905241\pi\)
−0.956015 + 0.293317i \(0.905241\pi\)
\(830\) 0 0
\(831\) 6378.00 0.266246
\(832\) 0 0
\(833\) −5022.00 −0.208886
\(834\) 0 0
\(835\) 34992.0 1.45024
\(836\) 0 0
\(837\) 432.000 0.0178400
\(838\) 0 0
\(839\) 17496.0 0.719939 0.359970 0.932964i \(-0.382787\pi\)
0.359970 + 0.932964i \(0.382787\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) 8802.00 0.359617
\(844\) 0 0
\(845\) −37746.0 −1.53669
\(846\) 0 0
\(847\) −280.000 −0.0113588
\(848\) 0 0
\(849\) 6108.00 0.246909
\(850\) 0 0
\(851\) 16272.0 0.655461
\(852\) 0 0
\(853\) −32174.0 −1.29146 −0.645731 0.763565i \(-0.723447\pi\)
−0.645731 + 0.763565i \(0.723447\pi\)
\(854\) 0 0
\(855\) 16200.0 0.647986
\(856\) 0 0
\(857\) −38934.0 −1.55188 −0.775939 0.630807i \(-0.782724\pi\)
−0.775939 + 0.630807i \(0.782724\pi\)
\(858\) 0 0
\(859\) −29780.0 −1.18286 −0.591432 0.806355i \(-0.701437\pi\)
−0.591432 + 0.806355i \(0.701437\pi\)
\(860\) 0 0
\(861\) −2160.00 −0.0854966
\(862\) 0 0
\(863\) −48096.0 −1.89711 −0.948556 0.316611i \(-0.897455\pi\)
−0.948556 + 0.316611i \(0.897455\pi\)
\(864\) 0 0
\(865\) 22356.0 0.878759
\(866\) 0 0
\(867\) 13767.0 0.539275
\(868\) 0 0
\(869\) −18432.0 −0.719520
\(870\) 0 0
\(871\) −3320.00 −0.129155
\(872\) 0 0
\(873\) −9486.00 −0.367758
\(874\) 0 0
\(875\) 10656.0 0.411701
\(876\) 0 0
\(877\) −21302.0 −0.820202 −0.410101 0.912040i \(-0.634507\pi\)
−0.410101 + 0.912040i \(0.634507\pi\)
\(878\) 0 0
\(879\) 6858.00 0.263157
\(880\) 0 0
\(881\) −7470.00 −0.285665 −0.142832 0.989747i \(-0.545621\pi\)
−0.142832 + 0.989747i \(0.545621\pi\)
\(882\) 0 0
\(883\) −764.000 −0.0291174 −0.0145587 0.999894i \(-0.504634\pi\)
−0.0145587 + 0.999894i \(0.504634\pi\)
\(884\) 0 0
\(885\) −36936.0 −1.40293
\(886\) 0 0
\(887\) 32328.0 1.22375 0.611876 0.790954i \(-0.290415\pi\)
0.611876 + 0.790954i \(0.290415\pi\)
\(888\) 0 0
\(889\) −4736.00 −0.178673
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 0 0
\(893\) 43200.0 1.61885
\(894\) 0 0
\(895\) −20088.0 −0.750243
\(896\) 0 0
\(897\) −2160.00 −0.0804017
\(898\) 0 0
\(899\) −3744.00 −0.138898
\(900\) 0 0
\(901\) −7452.00 −0.275541
\(902\) 0 0
\(903\) 10848.0 0.399777
\(904\) 0 0
\(905\) −19260.0 −0.707430
\(906\) 0 0
\(907\) 36316.0 1.32950 0.664748 0.747068i \(-0.268539\pi\)
0.664748 + 0.747068i \(0.268539\pi\)
\(908\) 0 0
\(909\) −5022.00 −0.183244
\(910\) 0 0
\(911\) −13392.0 −0.487044 −0.243522 0.969895i \(-0.578303\pi\)
−0.243522 + 0.969895i \(0.578303\pi\)
\(912\) 0 0
\(913\) −42768.0 −1.55029
\(914\) 0 0
\(915\) 22788.0 0.823331
\(916\) 0 0
\(917\) 15264.0 0.549686
\(918\) 0 0
\(919\) 38072.0 1.36657 0.683286 0.730151i \(-0.260550\pi\)
0.683286 + 0.730151i \(0.260550\pi\)
\(920\) 0 0
\(921\) 3732.00 0.133522
\(922\) 0 0
\(923\) −3600.00 −0.128381
\(924\) 0 0
\(925\) 44974.0 1.59863
\(926\) 0 0
\(927\) 72.0000 0.00255101
\(928\) 0 0
\(929\) −12798.0 −0.451979 −0.225990 0.974130i \(-0.572562\pi\)
−0.225990 + 0.974130i \(0.572562\pi\)
\(930\) 0 0
\(931\) −27900.0 −0.982154
\(932\) 0 0
\(933\) −3672.00 −0.128849
\(934\) 0 0
\(935\) −11664.0 −0.407972
\(936\) 0 0
\(937\) 34874.0 1.21588 0.607942 0.793981i \(-0.291995\pi\)
0.607942 + 0.793981i \(0.291995\pi\)
\(938\) 0 0
\(939\) −5694.00 −0.197888
\(940\) 0 0
\(941\) −17190.0 −0.595513 −0.297757 0.954642i \(-0.596238\pi\)
−0.297757 + 0.954642i \(0.596238\pi\)
\(942\) 0 0
\(943\) 6480.00 0.223773
\(944\) 0 0
\(945\) −3888.00 −0.133838
\(946\) 0 0
\(947\) −40284.0 −1.38232 −0.691158 0.722703i \(-0.742899\pi\)
−0.691158 + 0.722703i \(0.742899\pi\)
\(948\) 0 0
\(949\) 260.000 0.00889353
\(950\) 0 0
\(951\) −27486.0 −0.937218
\(952\) 0 0
\(953\) 15498.0 0.526789 0.263394 0.964688i \(-0.415158\pi\)
0.263394 + 0.964688i \(0.415158\pi\)
\(954\) 0 0
\(955\) −10368.0 −0.351310
\(956\) 0 0
\(957\) 25272.0 0.853634
\(958\) 0 0
\(959\) 7632.00 0.256987
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) −15876.0 −0.531253
\(964\) 0 0
\(965\) −24156.0 −0.805813
\(966\) 0 0
\(967\) 37160.0 1.23577 0.617883 0.786270i \(-0.287991\pi\)
0.617883 + 0.786270i \(0.287991\pi\)
\(968\) 0 0
\(969\) −5400.00 −0.179023
\(970\) 0 0
\(971\) −18468.0 −0.610367 −0.305183 0.952294i \(-0.598718\pi\)
−0.305183 + 0.952294i \(0.598718\pi\)
\(972\) 0 0
\(973\) −20512.0 −0.675832
\(974\) 0 0
\(975\) −5970.00 −0.196095
\(976\) 0 0
\(977\) 10386.0 0.340100 0.170050 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(978\) 0 0
\(979\) 22680.0 0.740404
\(980\) 0 0
\(981\) −14598.0 −0.475105
\(982\) 0 0
\(983\) 44136.0 1.43206 0.716032 0.698067i \(-0.245956\pi\)
0.716032 + 0.698067i \(0.245956\pi\)
\(984\) 0 0
\(985\) −25596.0 −0.827976
\(986\) 0 0
\(987\) −10368.0 −0.334364
\(988\) 0 0
\(989\) −32544.0 −1.04635
\(990\) 0 0
\(991\) −28432.0 −0.911375 −0.455687 0.890140i \(-0.650606\pi\)
−0.455687 + 0.890140i \(0.650606\pi\)
\(992\) 0 0
\(993\) −13044.0 −0.416857
\(994\) 0 0
\(995\) 15696.0 0.500097
\(996\) 0 0
\(997\) 39778.0 1.26357 0.631786 0.775143i \(-0.282322\pi\)
0.631786 + 0.775143i \(0.282322\pi\)
\(998\) 0 0
\(999\) −6102.00 −0.193252
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 192.4.a.f.1.1 1
3.2 odd 2 576.4.a.b.1.1 1
4.3 odd 2 192.4.a.l.1.1 1
8.3 odd 2 48.4.a.a.1.1 1
8.5 even 2 12.4.a.a.1.1 1
12.11 even 2 576.4.a.a.1.1 1
16.3 odd 4 768.4.d.j.385.2 2
16.5 even 4 768.4.d.g.385.2 2
16.11 odd 4 768.4.d.j.385.1 2
16.13 even 4 768.4.d.g.385.1 2
24.5 odd 2 36.4.a.a.1.1 1
24.11 even 2 144.4.a.g.1.1 1
40.3 even 4 1200.4.f.d.49.1 2
40.13 odd 4 300.4.d.e.49.2 2
40.19 odd 2 1200.4.a.be.1.1 1
40.27 even 4 1200.4.f.d.49.2 2
40.29 even 2 300.4.a.b.1.1 1
40.37 odd 4 300.4.d.e.49.1 2
56.5 odd 6 588.4.i.e.361.1 2
56.13 odd 2 588.4.a.c.1.1 1
56.27 even 2 2352.4.a.bk.1.1 1
56.37 even 6 588.4.i.d.361.1 2
56.45 odd 6 588.4.i.e.373.1 2
56.53 even 6 588.4.i.d.373.1 2
72.5 odd 6 324.4.e.a.217.1 2
72.13 even 6 324.4.e.h.217.1 2
72.29 odd 6 324.4.e.a.109.1 2
72.61 even 6 324.4.e.h.109.1 2
88.21 odd 2 1452.4.a.d.1.1 1
104.5 odd 4 2028.4.b.c.337.2 2
104.21 odd 4 2028.4.b.c.337.1 2
104.77 even 2 2028.4.a.c.1.1 1
120.29 odd 2 900.4.a.g.1.1 1
120.53 even 4 900.4.d.c.649.1 2
120.77 even 4 900.4.d.c.649.2 2
168.5 even 6 1764.4.k.o.361.1 2
168.53 odd 6 1764.4.k.b.1549.1 2
168.101 even 6 1764.4.k.o.1549.1 2
168.125 even 2 1764.4.a.b.1.1 1
168.149 odd 6 1764.4.k.b.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 1 8.5 even 2
36.4.a.a.1.1 1 24.5 odd 2
48.4.a.a.1.1 1 8.3 odd 2
144.4.a.g.1.1 1 24.11 even 2
192.4.a.f.1.1 1 1.1 even 1 trivial
192.4.a.l.1.1 1 4.3 odd 2
300.4.a.b.1.1 1 40.29 even 2
300.4.d.e.49.1 2 40.37 odd 4
300.4.d.e.49.2 2 40.13 odd 4
324.4.e.a.109.1 2 72.29 odd 6
324.4.e.a.217.1 2 72.5 odd 6
324.4.e.h.109.1 2 72.61 even 6
324.4.e.h.217.1 2 72.13 even 6
576.4.a.a.1.1 1 12.11 even 2
576.4.a.b.1.1 1 3.2 odd 2
588.4.a.c.1.1 1 56.13 odd 2
588.4.i.d.361.1 2 56.37 even 6
588.4.i.d.373.1 2 56.53 even 6
588.4.i.e.361.1 2 56.5 odd 6
588.4.i.e.373.1 2 56.45 odd 6
768.4.d.g.385.1 2 16.13 even 4
768.4.d.g.385.2 2 16.5 even 4
768.4.d.j.385.1 2 16.11 odd 4
768.4.d.j.385.2 2 16.3 odd 4
900.4.a.g.1.1 1 120.29 odd 2
900.4.d.c.649.1 2 120.53 even 4
900.4.d.c.649.2 2 120.77 even 4
1200.4.a.be.1.1 1 40.19 odd 2
1200.4.f.d.49.1 2 40.3 even 4
1200.4.f.d.49.2 2 40.27 even 4
1452.4.a.d.1.1 1 88.21 odd 2
1764.4.a.b.1.1 1 168.125 even 2
1764.4.k.b.361.1 2 168.149 odd 6
1764.4.k.b.1549.1 2 168.53 odd 6
1764.4.k.o.361.1 2 168.5 even 6
1764.4.k.o.1549.1 2 168.101 even 6
2028.4.a.c.1.1 1 104.77 even 2
2028.4.b.c.337.1 2 104.21 odd 4
2028.4.b.c.337.2 2 104.5 odd 4
2352.4.a.bk.1.1 1 56.27 even 2