Properties

Label 1200.4.a.o.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +20.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +20.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} -74.0000 q^{13} -54.0000 q^{17} +124.000 q^{19} -60.0000 q^{21} -120.000 q^{23} -27.0000 q^{27} -78.0000 q^{29} -200.000 q^{31} -72.0000 q^{33} +70.0000 q^{37} +222.000 q^{39} +330.000 q^{41} +92.0000 q^{43} -24.0000 q^{47} +57.0000 q^{49} +162.000 q^{51} -450.000 q^{53} -372.000 q^{57} -24.0000 q^{59} -322.000 q^{61} +180.000 q^{63} -196.000 q^{67} +360.000 q^{69} +288.000 q^{71} +430.000 q^{73} +480.000 q^{77} +520.000 q^{79} +81.0000 q^{81} +156.000 q^{83} +234.000 q^{87} +1026.00 q^{89} -1480.00 q^{91} +600.000 q^{93} +286.000 q^{97} +216.000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) −74.0000 −1.57876 −0.789381 0.613904i \(-0.789598\pi\)
−0.789381 + 0.613904i \(0.789598\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) −60.0000 −0.623480
\(22\) 0 0
\(23\) −120.000 −1.08790 −0.543951 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −78.0000 −0.499456 −0.249728 0.968316i \(-0.580341\pi\)
−0.249728 + 0.968316i \(0.580341\pi\)
\(30\) 0 0
\(31\) −200.000 −1.15874 −0.579372 0.815063i \(-0.696702\pi\)
−0.579372 + 0.815063i \(0.696702\pi\)
\(32\) 0 0
\(33\) −72.0000 −0.379806
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 70.0000 0.311025 0.155513 0.987834i \(-0.450297\pi\)
0.155513 + 0.987834i \(0.450297\pi\)
\(38\) 0 0
\(39\) 222.000 0.911499
\(40\) 0 0
\(41\) 330.000 1.25701 0.628504 0.777806i \(-0.283668\pi\)
0.628504 + 0.777806i \(0.283668\pi\)
\(42\) 0 0
\(43\) 92.0000 0.326276 0.163138 0.986603i \(-0.447838\pi\)
0.163138 + 0.986603i \(0.447838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −24.0000 −0.0744843 −0.0372421 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 162.000 0.444795
\(52\) 0 0
\(53\) −450.000 −1.16627 −0.583134 0.812376i \(-0.698174\pi\)
−0.583134 + 0.812376i \(0.698174\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −372.000 −0.864432
\(58\) 0 0
\(59\) −24.0000 −0.0529582 −0.0264791 0.999649i \(-0.508430\pi\)
−0.0264791 + 0.999649i \(0.508430\pi\)
\(60\) 0 0
\(61\) −322.000 −0.675867 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) 0 0
\(63\) 180.000 0.359966
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −196.000 −0.357391 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(68\) 0 0
\(69\) 360.000 0.628100
\(70\) 0 0
\(71\) 288.000 0.481399 0.240699 0.970600i \(-0.422623\pi\)
0.240699 + 0.970600i \(0.422623\pi\)
\(72\) 0 0
\(73\) 430.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 480.000 0.710404
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 156.000 0.206304 0.103152 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 234.000 0.288361
\(88\) 0 0
\(89\) 1026.00 1.22198 0.610988 0.791640i \(-0.290773\pi\)
0.610988 + 0.791640i \(0.290773\pi\)
\(90\) 0 0
\(91\) −1480.00 −1.70490
\(92\) 0 0
\(93\) 600.000 0.669001
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 286.000 0.299370 0.149685 0.988734i \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 0 0
\(99\) 216.000 0.219281
\(100\) 0 0
\(101\) −1734.00 −1.70831 −0.854156 0.520017i \(-0.825925\pi\)
−0.854156 + 0.520017i \(0.825925\pi\)
\(102\) 0 0
\(103\) 452.000 0.432397 0.216198 0.976349i \(-0.430634\pi\)
0.216198 + 0.976349i \(0.430634\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1404.00 −1.26850 −0.634251 0.773127i \(-0.718692\pi\)
−0.634251 + 0.773127i \(0.718692\pi\)
\(108\) 0 0
\(109\) −1474.00 −1.29526 −0.647631 0.761954i \(-0.724240\pi\)
−0.647631 + 0.761954i \(0.724240\pi\)
\(110\) 0 0
\(111\) −210.000 −0.179570
\(112\) 0 0
\(113\) −1086.00 −0.904091 −0.452046 0.891995i \(-0.649306\pi\)
−0.452046 + 0.891995i \(0.649306\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −666.000 −0.526254
\(118\) 0 0
\(119\) −1080.00 −0.831962
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) −990.000 −0.725734
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1244.00 0.869190 0.434595 0.900626i \(-0.356891\pi\)
0.434595 + 0.900626i \(0.356891\pi\)
\(128\) 0 0
\(129\) −276.000 −0.188376
\(130\) 0 0
\(131\) −2328.00 −1.55266 −0.776329 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) 2480.00 1.61687
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2118.00 −1.32082 −0.660412 0.750903i \(-0.729618\pi\)
−0.660412 + 0.750903i \(0.729618\pi\)
\(138\) 0 0
\(139\) −2324.00 −1.41812 −0.709062 0.705147i \(-0.750881\pi\)
−0.709062 + 0.705147i \(0.750881\pi\)
\(140\) 0 0
\(141\) 72.0000 0.0430035
\(142\) 0 0
\(143\) −1776.00 −1.03858
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −171.000 −0.0959445
\(148\) 0 0
\(149\) 258.000 0.141854 0.0709268 0.997482i \(-0.477404\pi\)
0.0709268 + 0.997482i \(0.477404\pi\)
\(150\) 0 0
\(151\) 808.000 0.435458 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(152\) 0 0
\(153\) −486.000 −0.256802
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2378.00 −1.20882 −0.604411 0.796673i \(-0.706592\pi\)
−0.604411 + 0.796673i \(0.706592\pi\)
\(158\) 0 0
\(159\) 1350.00 0.673346
\(160\) 0 0
\(161\) −2400.00 −1.17482
\(162\) 0 0
\(163\) −52.0000 −0.0249874 −0.0124937 0.999922i \(-0.503977\pi\)
−0.0124937 + 0.999922i \(0.503977\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3720.00 −1.72373 −0.861863 0.507141i \(-0.830702\pi\)
−0.861863 + 0.507141i \(0.830702\pi\)
\(168\) 0 0
\(169\) 3279.00 1.49249
\(170\) 0 0
\(171\) 1116.00 0.499080
\(172\) 0 0
\(173\) −426.000 −0.187215 −0.0936075 0.995609i \(-0.529840\pi\)
−0.0936075 + 0.995609i \(0.529840\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 72.0000 0.0305754
\(178\) 0 0
\(179\) 1440.00 0.601289 0.300644 0.953736i \(-0.402798\pi\)
0.300644 + 0.953736i \(0.402798\pi\)
\(180\) 0 0
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) 0 0
\(183\) 966.000 0.390212
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1296.00 −0.506807
\(188\) 0 0
\(189\) −540.000 −0.207827
\(190\) 0 0
\(191\) −3576.00 −1.35471 −0.677357 0.735655i \(-0.736875\pi\)
−0.677357 + 0.735655i \(0.736875\pi\)
\(192\) 0 0
\(193\) −2666.00 −0.994315 −0.497158 0.867660i \(-0.665623\pi\)
−0.497158 + 0.867660i \(0.665623\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2718.00 0.982992 0.491496 0.870880i \(-0.336450\pi\)
0.491496 + 0.870880i \(0.336450\pi\)
\(198\) 0 0
\(199\) 3832.00 1.36504 0.682521 0.730866i \(-0.260884\pi\)
0.682521 + 0.730866i \(0.260884\pi\)
\(200\) 0 0
\(201\) 588.000 0.206340
\(202\) 0 0
\(203\) −1560.00 −0.539362
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1080.00 −0.362634
\(208\) 0 0
\(209\) 2976.00 0.984948
\(210\) 0 0
\(211\) −1100.00 −0.358896 −0.179448 0.983767i \(-0.557431\pi\)
−0.179448 + 0.983767i \(0.557431\pi\)
\(212\) 0 0
\(213\) −864.000 −0.277936
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4000.00 −1.25133
\(218\) 0 0
\(219\) −1290.00 −0.398037
\(220\) 0 0
\(221\) 3996.00 1.21629
\(222\) 0 0
\(223\) 1964.00 0.589772 0.294886 0.955532i \(-0.404718\pi\)
0.294886 + 0.955532i \(0.404718\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 660.000 0.192977 0.0964884 0.995334i \(-0.469239\pi\)
0.0964884 + 0.995334i \(0.469239\pi\)
\(228\) 0 0
\(229\) −1906.00 −0.550009 −0.275004 0.961443i \(-0.588679\pi\)
−0.275004 + 0.961443i \(0.588679\pi\)
\(230\) 0 0
\(231\) −1440.00 −0.410152
\(232\) 0 0
\(233\) 1458.00 0.409943 0.204972 0.978768i \(-0.434290\pi\)
0.204972 + 0.978768i \(0.434290\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1560.00 −0.427565
\(238\) 0 0
\(239\) −1176.00 −0.318281 −0.159140 0.987256i \(-0.550872\pi\)
−0.159140 + 0.987256i \(0.550872\pi\)
\(240\) 0 0
\(241\) 866.000 0.231469 0.115734 0.993280i \(-0.463078\pi\)
0.115734 + 0.993280i \(0.463078\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9176.00 −2.36379
\(248\) 0 0
\(249\) −468.000 −0.119110
\(250\) 0 0
\(251\) −432.000 −0.108636 −0.0543179 0.998524i \(-0.517298\pi\)
−0.0543179 + 0.998524i \(0.517298\pi\)
\(252\) 0 0
\(253\) −2880.00 −0.715668
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2526.00 −0.613103 −0.306552 0.951854i \(-0.599175\pi\)
−0.306552 + 0.951854i \(0.599175\pi\)
\(258\) 0 0
\(259\) 1400.00 0.335876
\(260\) 0 0
\(261\) −702.000 −0.166485
\(262\) 0 0
\(263\) 5448.00 1.27733 0.638666 0.769484i \(-0.279487\pi\)
0.638666 + 0.769484i \(0.279487\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3078.00 −0.705508
\(268\) 0 0
\(269\) −2574.00 −0.583418 −0.291709 0.956507i \(-0.594224\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(270\) 0 0
\(271\) 3184.00 0.713706 0.356853 0.934161i \(-0.383850\pi\)
0.356853 + 0.934161i \(0.383850\pi\)
\(272\) 0 0
\(273\) 4440.00 0.984326
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3962.00 −0.859399 −0.429699 0.902972i \(-0.641380\pi\)
−0.429699 + 0.902972i \(0.641380\pi\)
\(278\) 0 0
\(279\) −1800.00 −0.386248
\(280\) 0 0
\(281\) −8286.00 −1.75908 −0.879540 0.475825i \(-0.842149\pi\)
−0.879540 + 0.475825i \(0.842149\pi\)
\(282\) 0 0
\(283\) −2716.00 −0.570493 −0.285246 0.958454i \(-0.592075\pi\)
−0.285246 + 0.958454i \(0.592075\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6600.00 1.35744
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) −858.000 −0.172841
\(292\) 0 0
\(293\) −6018.00 −1.19992 −0.599958 0.800032i \(-0.704816\pi\)
−0.599958 + 0.800032i \(0.704816\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −648.000 −0.126602
\(298\) 0 0
\(299\) 8880.00 1.71754
\(300\) 0 0
\(301\) 1840.00 0.352345
\(302\) 0 0
\(303\) 5202.00 0.986294
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9236.00 1.71702 0.858512 0.512793i \(-0.171389\pi\)
0.858512 + 0.512793i \(0.171389\pi\)
\(308\) 0 0
\(309\) −1356.00 −0.249644
\(310\) 0 0
\(311\) −1536.00 −0.280060 −0.140030 0.990147i \(-0.544720\pi\)
−0.140030 + 0.990147i \(0.544720\pi\)
\(312\) 0 0
\(313\) 7342.00 1.32586 0.662930 0.748681i \(-0.269313\pi\)
0.662930 + 0.748681i \(0.269313\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3894.00 0.689933 0.344967 0.938615i \(-0.387890\pi\)
0.344967 + 0.938615i \(0.387890\pi\)
\(318\) 0 0
\(319\) −1872.00 −0.328564
\(320\) 0 0
\(321\) 4212.00 0.732370
\(322\) 0 0
\(323\) −6696.00 −1.15348
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4422.00 0.747820
\(328\) 0 0
\(329\) −480.000 −0.0804354
\(330\) 0 0
\(331\) −3692.00 −0.613084 −0.306542 0.951857i \(-0.599172\pi\)
−0.306542 + 0.951857i \(0.599172\pi\)
\(332\) 0 0
\(333\) 630.000 0.103675
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8998.00 1.45446 0.727229 0.686395i \(-0.240808\pi\)
0.727229 + 0.686395i \(0.240808\pi\)
\(338\) 0 0
\(339\) 3258.00 0.521977
\(340\) 0 0
\(341\) −4800.00 −0.762271
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5244.00 0.811276 0.405638 0.914034i \(-0.367049\pi\)
0.405638 + 0.914034i \(0.367049\pi\)
\(348\) 0 0
\(349\) 6302.00 0.966585 0.483293 0.875459i \(-0.339441\pi\)
0.483293 + 0.875459i \(0.339441\pi\)
\(350\) 0 0
\(351\) 1998.00 0.303833
\(352\) 0 0
\(353\) −3414.00 −0.514756 −0.257378 0.966311i \(-0.582859\pi\)
−0.257378 + 0.966311i \(0.582859\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3240.00 0.480333
\(358\) 0 0
\(359\) −4824.00 −0.709195 −0.354597 0.935019i \(-0.615382\pi\)
−0.354597 + 0.935019i \(0.615382\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) 2265.00 0.327498
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3508.00 −0.498954 −0.249477 0.968381i \(-0.580259\pi\)
−0.249477 + 0.968381i \(0.580259\pi\)
\(368\) 0 0
\(369\) 2970.00 0.419003
\(370\) 0 0
\(371\) −9000.00 −1.25945
\(372\) 0 0
\(373\) −10802.0 −1.49948 −0.749740 0.661732i \(-0.769822\pi\)
−0.749740 + 0.661732i \(0.769822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5772.00 0.788523
\(378\) 0 0
\(379\) −1460.00 −0.197876 −0.0989382 0.995094i \(-0.531545\pi\)
−0.0989382 + 0.995094i \(0.531545\pi\)
\(380\) 0 0
\(381\) −3732.00 −0.501827
\(382\) 0 0
\(383\) −4872.00 −0.649994 −0.324997 0.945715i \(-0.605363\pi\)
−0.324997 + 0.945715i \(0.605363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 828.000 0.108759
\(388\) 0 0
\(389\) −14046.0 −1.83075 −0.915373 0.402606i \(-0.868104\pi\)
−0.915373 + 0.402606i \(0.868104\pi\)
\(390\) 0 0
\(391\) 6480.00 0.838127
\(392\) 0 0
\(393\) 6984.00 0.896428
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2734.00 0.345631 0.172816 0.984954i \(-0.444714\pi\)
0.172816 + 0.984954i \(0.444714\pi\)
\(398\) 0 0
\(399\) −7440.00 −0.933498
\(400\) 0 0
\(401\) −15942.0 −1.98530 −0.992650 0.121019i \(-0.961384\pi\)
−0.992650 + 0.121019i \(0.961384\pi\)
\(402\) 0 0
\(403\) 14800.0 1.82938
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1680.00 0.204606
\(408\) 0 0
\(409\) 8714.00 1.05350 0.526748 0.850022i \(-0.323411\pi\)
0.526748 + 0.850022i \(0.323411\pi\)
\(410\) 0 0
\(411\) 6354.00 0.762578
\(412\) 0 0
\(413\) −480.000 −0.0571895
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6972.00 0.818754
\(418\) 0 0
\(419\) −11976.0 −1.39634 −0.698169 0.715933i \(-0.746002\pi\)
−0.698169 + 0.715933i \(0.746002\pi\)
\(420\) 0 0
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) 0 0
\(423\) −216.000 −0.0248281
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6440.00 −0.729868
\(428\) 0 0
\(429\) 5328.00 0.599623
\(430\) 0 0
\(431\) −720.000 −0.0804668 −0.0402334 0.999190i \(-0.512810\pi\)
−0.0402334 + 0.999190i \(0.512810\pi\)
\(432\) 0 0
\(433\) 15622.0 1.73382 0.866912 0.498462i \(-0.166102\pi\)
0.866912 + 0.498462i \(0.166102\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14880.0 −1.62885
\(438\) 0 0
\(439\) 9880.00 1.07414 0.537069 0.843538i \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 0 0
\(443\) −16116.0 −1.72843 −0.864215 0.503123i \(-0.832184\pi\)
−0.864215 + 0.503123i \(0.832184\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −774.000 −0.0818992
\(448\) 0 0
\(449\) 9018.00 0.947852 0.473926 0.880565i \(-0.342836\pi\)
0.473926 + 0.880565i \(0.342836\pi\)
\(450\) 0 0
\(451\) 7920.00 0.826914
\(452\) 0 0
\(453\) −2424.00 −0.251412
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3670.00 0.375657 0.187829 0.982202i \(-0.439855\pi\)
0.187829 + 0.982202i \(0.439855\pi\)
\(458\) 0 0
\(459\) 1458.00 0.148265
\(460\) 0 0
\(461\) 17562.0 1.77428 0.887141 0.461499i \(-0.152688\pi\)
0.887141 + 0.461499i \(0.152688\pi\)
\(462\) 0 0
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6876.00 0.681335 0.340667 0.940184i \(-0.389347\pi\)
0.340667 + 0.940184i \(0.389347\pi\)
\(468\) 0 0
\(469\) −3920.00 −0.385946
\(470\) 0 0
\(471\) 7134.00 0.697914
\(472\) 0 0
\(473\) 2208.00 0.214638
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4050.00 −0.388756
\(478\) 0 0
\(479\) −2280.00 −0.217486 −0.108743 0.994070i \(-0.534683\pi\)
−0.108743 + 0.994070i \(0.534683\pi\)
\(480\) 0 0
\(481\) −5180.00 −0.491035
\(482\) 0 0
\(483\) 7200.00 0.678284
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3076.00 −0.286215 −0.143108 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(488\) 0 0
\(489\) 156.000 0.0144265
\(490\) 0 0
\(491\) 18912.0 1.73826 0.869131 0.494582i \(-0.164679\pi\)
0.869131 + 0.494582i \(0.164679\pi\)
\(492\) 0 0
\(493\) 4212.00 0.384785
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5760.00 0.519862
\(498\) 0 0
\(499\) −9956.00 −0.893170 −0.446585 0.894741i \(-0.647360\pi\)
−0.446585 + 0.894741i \(0.647360\pi\)
\(500\) 0 0
\(501\) 11160.0 0.995194
\(502\) 0 0
\(503\) −10656.0 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9837.00 −0.861689
\(508\) 0 0
\(509\) −2766.00 −0.240866 −0.120433 0.992721i \(-0.538428\pi\)
−0.120433 + 0.992721i \(0.538428\pi\)
\(510\) 0 0
\(511\) 8600.00 0.744504
\(512\) 0 0
\(513\) −3348.00 −0.288144
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −576.000 −0.0489989
\(518\) 0 0
\(519\) 1278.00 0.108089
\(520\) 0 0
\(521\) 10530.0 0.885466 0.442733 0.896654i \(-0.354009\pi\)
0.442733 + 0.896654i \(0.354009\pi\)
\(522\) 0 0
\(523\) 12692.0 1.06115 0.530576 0.847637i \(-0.321976\pi\)
0.530576 + 0.847637i \(0.321976\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10800.0 0.892705
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) −216.000 −0.0176527
\(532\) 0 0
\(533\) −24420.0 −1.98452
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4320.00 −0.347154
\(538\) 0 0
\(539\) 1368.00 0.109321
\(540\) 0 0
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) 0 0
\(543\) 9390.00 0.742106
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3620.00 0.282962 0.141481 0.989941i \(-0.454814\pi\)
0.141481 + 0.989941i \(0.454814\pi\)
\(548\) 0 0
\(549\) −2898.00 −0.225289
\(550\) 0 0
\(551\) −9672.00 −0.747806
\(552\) 0 0
\(553\) 10400.0 0.799734
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14166.0 1.07762 0.538809 0.842428i \(-0.318875\pi\)
0.538809 + 0.842428i \(0.318875\pi\)
\(558\) 0 0
\(559\) −6808.00 −0.515112
\(560\) 0 0
\(561\) 3888.00 0.292605
\(562\) 0 0
\(563\) −13404.0 −1.00339 −0.501697 0.865043i \(-0.667291\pi\)
−0.501697 + 0.865043i \(0.667291\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1620.00 0.119989
\(568\) 0 0
\(569\) −18654.0 −1.37437 −0.687185 0.726483i \(-0.741154\pi\)
−0.687185 + 0.726483i \(0.741154\pi\)
\(570\) 0 0
\(571\) 7684.00 0.563162 0.281581 0.959537i \(-0.409141\pi\)
0.281581 + 0.959537i \(0.409141\pi\)
\(572\) 0 0
\(573\) 10728.0 0.782144
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1726.00 0.124531 0.0622654 0.998060i \(-0.480167\pi\)
0.0622654 + 0.998060i \(0.480167\pi\)
\(578\) 0 0
\(579\) 7998.00 0.574068
\(580\) 0 0
\(581\) 3120.00 0.222787
\(582\) 0 0
\(583\) −10800.0 −0.767222
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10596.0 0.745049 0.372524 0.928022i \(-0.378492\pi\)
0.372524 + 0.928022i \(0.378492\pi\)
\(588\) 0 0
\(589\) −24800.0 −1.73492
\(590\) 0 0
\(591\) −8154.00 −0.567531
\(592\) 0 0
\(593\) −2862.00 −0.198193 −0.0990963 0.995078i \(-0.531595\pi\)
−0.0990963 + 0.995078i \(0.531595\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11496.0 −0.788107
\(598\) 0 0
\(599\) 23592.0 1.60925 0.804627 0.593781i \(-0.202365\pi\)
0.804627 + 0.593781i \(0.202365\pi\)
\(600\) 0 0
\(601\) −9574.00 −0.649803 −0.324902 0.945748i \(-0.605331\pi\)
−0.324902 + 0.945748i \(0.605331\pi\)
\(602\) 0 0
\(603\) −1764.00 −0.119130
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17444.0 1.16644 0.583221 0.812314i \(-0.301792\pi\)
0.583221 + 0.812314i \(0.301792\pi\)
\(608\) 0 0
\(609\) 4680.00 0.311401
\(610\) 0 0
\(611\) 1776.00 0.117593
\(612\) 0 0
\(613\) 2374.00 0.156419 0.0782096 0.996937i \(-0.475080\pi\)
0.0782096 + 0.996937i \(0.475080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12162.0 0.793555 0.396778 0.917915i \(-0.370128\pi\)
0.396778 + 0.917915i \(0.370128\pi\)
\(618\) 0 0
\(619\) −8804.00 −0.571668 −0.285834 0.958279i \(-0.592271\pi\)
−0.285834 + 0.958279i \(0.592271\pi\)
\(620\) 0 0
\(621\) 3240.00 0.209367
\(622\) 0 0
\(623\) 20520.0 1.31961
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −8928.00 −0.568660
\(628\) 0 0
\(629\) −3780.00 −0.239616
\(630\) 0 0
\(631\) 12688.0 0.800478 0.400239 0.916411i \(-0.368927\pi\)
0.400239 + 0.916411i \(0.368927\pi\)
\(632\) 0 0
\(633\) 3300.00 0.207209
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4218.00 −0.262360
\(638\) 0 0
\(639\) 2592.00 0.160466
\(640\) 0 0
\(641\) −9150.00 −0.563812 −0.281906 0.959442i \(-0.590967\pi\)
−0.281906 + 0.959442i \(0.590967\pi\)
\(642\) 0 0
\(643\) 25292.0 1.55120 0.775598 0.631227i \(-0.217448\pi\)
0.775598 + 0.631227i \(0.217448\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2736.00 −0.166249 −0.0831246 0.996539i \(-0.526490\pi\)
−0.0831246 + 0.996539i \(0.526490\pi\)
\(648\) 0 0
\(649\) −576.000 −0.0348382
\(650\) 0 0
\(651\) 12000.0 0.722453
\(652\) 0 0
\(653\) −22218.0 −1.33148 −0.665741 0.746183i \(-0.731884\pi\)
−0.665741 + 0.746183i \(0.731884\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3870.00 0.229807
\(658\) 0 0
\(659\) −14520.0 −0.858299 −0.429149 0.903234i \(-0.641187\pi\)
−0.429149 + 0.903234i \(0.641187\pi\)
\(660\) 0 0
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) 0 0
\(663\) −11988.0 −0.702225
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9360.00 0.543359
\(668\) 0 0
\(669\) −5892.00 −0.340505
\(670\) 0 0
\(671\) −7728.00 −0.444614
\(672\) 0 0
\(673\) −1370.00 −0.0784690 −0.0392345 0.999230i \(-0.512492\pi\)
−0.0392345 + 0.999230i \(0.512492\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13758.0 0.781038 0.390519 0.920595i \(-0.372296\pi\)
0.390519 + 0.920595i \(0.372296\pi\)
\(678\) 0 0
\(679\) 5720.00 0.323289
\(680\) 0 0
\(681\) −1980.00 −0.111415
\(682\) 0 0
\(683\) 11988.0 0.671608 0.335804 0.941932i \(-0.390992\pi\)
0.335804 + 0.941932i \(0.390992\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5718.00 0.317548
\(688\) 0 0
\(689\) 33300.0 1.84126
\(690\) 0 0
\(691\) −32996.0 −1.81654 −0.908268 0.418388i \(-0.862595\pi\)
−0.908268 + 0.418388i \(0.862595\pi\)
\(692\) 0 0
\(693\) 4320.00 0.236801
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −17820.0 −0.968408
\(698\) 0 0
\(699\) −4374.00 −0.236681
\(700\) 0 0
\(701\) −25902.0 −1.39558 −0.697792 0.716300i \(-0.745834\pi\)
−0.697792 + 0.716300i \(0.745834\pi\)
\(702\) 0 0
\(703\) 8680.00 0.465679
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −34680.0 −1.84480
\(708\) 0 0
\(709\) −27394.0 −1.45106 −0.725531 0.688189i \(-0.758406\pi\)
−0.725531 + 0.688189i \(0.758406\pi\)
\(710\) 0 0
\(711\) 4680.00 0.246855
\(712\) 0 0
\(713\) 24000.0 1.26060
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3528.00 0.183760
\(718\) 0 0
\(719\) −34848.0 −1.80753 −0.903763 0.428033i \(-0.859207\pi\)
−0.903763 + 0.428033i \(0.859207\pi\)
\(720\) 0 0
\(721\) 9040.00 0.466945
\(722\) 0 0
\(723\) −2598.00 −0.133639
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28028.0 1.42985 0.714925 0.699201i \(-0.246461\pi\)
0.714925 + 0.699201i \(0.246461\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4968.00 −0.251365
\(732\) 0 0
\(733\) −18002.0 −0.907120 −0.453560 0.891226i \(-0.649846\pi\)
−0.453560 + 0.891226i \(0.649846\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4704.00 −0.235107
\(738\) 0 0
\(739\) −15284.0 −0.760800 −0.380400 0.924822i \(-0.624214\pi\)
−0.380400 + 0.924822i \(0.624214\pi\)
\(740\) 0 0
\(741\) 27528.0 1.36473
\(742\) 0 0
\(743\) −18768.0 −0.926691 −0.463345 0.886178i \(-0.653351\pi\)
−0.463345 + 0.886178i \(0.653351\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1404.00 0.0687680
\(748\) 0 0
\(749\) −28080.0 −1.36985
\(750\) 0 0
\(751\) −8696.00 −0.422532 −0.211266 0.977429i \(-0.567759\pi\)
−0.211266 + 0.977429i \(0.567759\pi\)
\(752\) 0 0
\(753\) 1296.00 0.0627209
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38662.0 1.85627 0.928134 0.372247i \(-0.121413\pi\)
0.928134 + 0.372247i \(0.121413\pi\)
\(758\) 0 0
\(759\) 8640.00 0.413191
\(760\) 0 0
\(761\) 23874.0 1.13723 0.568615 0.822604i \(-0.307479\pi\)
0.568615 + 0.822604i \(0.307479\pi\)
\(762\) 0 0
\(763\) −29480.0 −1.39875
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1776.00 0.0836084
\(768\) 0 0
\(769\) 23618.0 1.10753 0.553763 0.832675i \(-0.313192\pi\)
0.553763 + 0.832675i \(0.313192\pi\)
\(770\) 0 0
\(771\) 7578.00 0.353975
\(772\) 0 0
\(773\) −11538.0 −0.536860 −0.268430 0.963299i \(-0.586505\pi\)
−0.268430 + 0.963299i \(0.586505\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4200.00 −0.193918
\(778\) 0 0
\(779\) 40920.0 1.88204
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 0 0
\(783\) 2106.00 0.0961204
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14884.0 −0.674152 −0.337076 0.941478i \(-0.609438\pi\)
−0.337076 + 0.941478i \(0.609438\pi\)
\(788\) 0 0
\(789\) −16344.0 −0.737467
\(790\) 0 0
\(791\) −21720.0 −0.976327
\(792\) 0 0
\(793\) 23828.0 1.06703
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11334.0 0.503728 0.251864 0.967763i \(-0.418957\pi\)
0.251864 + 0.967763i \(0.418957\pi\)
\(798\) 0 0
\(799\) 1296.00 0.0573832
\(800\) 0 0
\(801\) 9234.00 0.407325
\(802\) 0 0
\(803\) 10320.0 0.453530
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 7722.00 0.336837
\(808\) 0 0
\(809\) 44730.0 1.94391 0.971955 0.235167i \(-0.0755638\pi\)
0.971955 + 0.235167i \(0.0755638\pi\)
\(810\) 0 0
\(811\) 42748.0 1.85091 0.925453 0.378862i \(-0.123684\pi\)
0.925453 + 0.378862i \(0.123684\pi\)
\(812\) 0 0
\(813\) −9552.00 −0.412058
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 11408.0 0.488513
\(818\) 0 0
\(819\) −13320.0 −0.568301
\(820\) 0 0
\(821\) −31686.0 −1.34695 −0.673477 0.739208i \(-0.735200\pi\)
−0.673477 + 0.739208i \(0.735200\pi\)
\(822\) 0 0
\(823\) 11036.0 0.467425 0.233713 0.972306i \(-0.424913\pi\)
0.233713 + 0.972306i \(0.424913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25884.0 1.08836 0.544181 0.838968i \(-0.316841\pi\)
0.544181 + 0.838968i \(0.316841\pi\)
\(828\) 0 0
\(829\) 15950.0 0.668234 0.334117 0.942532i \(-0.391562\pi\)
0.334117 + 0.942532i \(0.391562\pi\)
\(830\) 0 0
\(831\) 11886.0 0.496174
\(832\) 0 0
\(833\) −3078.00 −0.128027
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5400.00 0.223000
\(838\) 0 0
\(839\) −13800.0 −0.567853 −0.283927 0.958846i \(-0.591637\pi\)
−0.283927 + 0.958846i \(0.591637\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) 0 0
\(843\) 24858.0 1.01560
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −15100.0 −0.612565
\(848\) 0 0
\(849\) 8148.00 0.329374
\(850\) 0 0
\(851\) −8400.00 −0.338365
\(852\) 0 0
\(853\) 27862.0 1.11838 0.559189 0.829040i \(-0.311113\pi\)
0.559189 + 0.829040i \(0.311113\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7314.00 0.291530 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(858\) 0 0
\(859\) 28780.0 1.14314 0.571572 0.820552i \(-0.306334\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(860\) 0 0
\(861\) −19800.0 −0.783719
\(862\) 0 0
\(863\) −32688.0 −1.28935 −0.644677 0.764455i \(-0.723008\pi\)
−0.644677 + 0.764455i \(0.723008\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5991.00 0.234677
\(868\) 0 0
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) 14504.0 0.564236
\(872\) 0 0
\(873\) 2574.00 0.0997900
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36650.0 −1.41115 −0.705577 0.708633i \(-0.749312\pi\)
−0.705577 + 0.708633i \(0.749312\pi\)
\(878\) 0 0
\(879\) 18054.0 0.692772
\(880\) 0 0
\(881\) −2646.00 −0.101187 −0.0505936 0.998719i \(-0.516111\pi\)
−0.0505936 + 0.998719i \(0.516111\pi\)
\(882\) 0 0
\(883\) 10892.0 0.415113 0.207557 0.978223i \(-0.433449\pi\)
0.207557 + 0.978223i \(0.433449\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43464.0 −1.64530 −0.822648 0.568550i \(-0.807504\pi\)
−0.822648 + 0.568550i \(0.807504\pi\)
\(888\) 0 0
\(889\) 24880.0 0.938637
\(890\) 0 0
\(891\) 1944.00 0.0730937
\(892\) 0 0
\(893\) −2976.00 −0.111521
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −26640.0 −0.991621
\(898\) 0 0
\(899\) 15600.0 0.578742
\(900\) 0 0
\(901\) 24300.0 0.898502
\(902\) 0 0
\(903\) −5520.00 −0.203426
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −14884.0 −0.544890 −0.272445 0.962171i \(-0.587832\pi\)
−0.272445 + 0.962171i \(0.587832\pi\)
\(908\) 0 0
\(909\) −15606.0 −0.569437
\(910\) 0 0
\(911\) 1248.00 0.0453876 0.0226938 0.999742i \(-0.492776\pi\)
0.0226938 + 0.999742i \(0.492776\pi\)
\(912\) 0 0
\(913\) 3744.00 0.135716
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −46560.0 −1.67671
\(918\) 0 0
\(919\) 6640.00 0.238339 0.119169 0.992874i \(-0.461977\pi\)
0.119169 + 0.992874i \(0.461977\pi\)
\(920\) 0 0
\(921\) −27708.0 −0.991324
\(922\) 0 0
\(923\) −21312.0 −0.760014
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4068.00 0.144132
\(928\) 0 0
\(929\) 29946.0 1.05758 0.528792 0.848751i \(-0.322645\pi\)
0.528792 + 0.848751i \(0.322645\pi\)
\(930\) 0 0
\(931\) 7068.00 0.248812
\(932\) 0 0
\(933\) 4608.00 0.161693
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −45002.0 −1.56900 −0.784499 0.620130i \(-0.787080\pi\)
−0.784499 + 0.620130i \(0.787080\pi\)
\(938\) 0 0
\(939\) −22026.0 −0.765486
\(940\) 0 0
\(941\) 6090.00 0.210976 0.105488 0.994421i \(-0.466360\pi\)
0.105488 + 0.994421i \(0.466360\pi\)
\(942\) 0 0
\(943\) −39600.0 −1.36750
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56388.0 1.93491 0.967457 0.253035i \(-0.0814288\pi\)
0.967457 + 0.253035i \(0.0814288\pi\)
\(948\) 0 0
\(949\) −31820.0 −1.08843
\(950\) 0 0
\(951\) −11682.0 −0.398333
\(952\) 0 0
\(953\) −10854.0 −0.368936 −0.184468 0.982839i \(-0.559056\pi\)
−0.184468 + 0.982839i \(0.559056\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5616.00 0.189696
\(958\) 0 0
\(959\) −42360.0 −1.42636
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 0 0
\(963\) −12636.0 −0.422834
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −42316.0 −1.40723 −0.703615 0.710582i \(-0.748432\pi\)
−0.703615 + 0.710582i \(0.748432\pi\)
\(968\) 0 0
\(969\) 20088.0 0.665964
\(970\) 0 0
\(971\) −24480.0 −0.809063 −0.404532 0.914524i \(-0.632565\pi\)
−0.404532 + 0.914524i \(0.632565\pi\)
\(972\) 0 0
\(973\) −46480.0 −1.53143
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6906.00 0.226144 0.113072 0.993587i \(-0.463931\pi\)
0.113072 + 0.993587i \(0.463931\pi\)
\(978\) 0 0
\(979\) 24624.0 0.803868
\(980\) 0 0
\(981\) −13266.0 −0.431754
\(982\) 0 0
\(983\) 6960.00 0.225829 0.112914 0.993605i \(-0.463981\pi\)
0.112914 + 0.993605i \(0.463981\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1440.00 0.0464394
\(988\) 0 0
\(989\) −11040.0 −0.354956
\(990\) 0 0
\(991\) −47792.0 −1.53195 −0.765975 0.642870i \(-0.777744\pi\)
−0.765975 + 0.642870i \(0.777744\pi\)
\(992\) 0 0
\(993\) 11076.0 0.353964
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9938.00 −0.315687 −0.157843 0.987464i \(-0.550454\pi\)
−0.157843 + 0.987464i \(0.550454\pi\)
\(998\) 0 0
\(999\) −1890.00 −0.0598568
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 1200.4.a.o.1.1 1
4.3 odd 2 75.4.a.a.1.1 1
5.2 odd 4 1200.4.f.m.49.2 2
5.3 odd 4 1200.4.f.m.49.1 2
5.4 even 2 240.4.a.f.1.1 1
12.11 even 2 225.4.a.g.1.1 1
15.14 odd 2 720.4.a.r.1.1 1
20.3 even 4 75.4.b.a.49.2 2
20.7 even 4 75.4.b.a.49.1 2
20.19 odd 2 15.4.a.b.1.1 1
40.19 odd 2 960.4.a.bi.1.1 1
40.29 even 2 960.4.a.l.1.1 1
60.23 odd 4 225.4.b.d.199.1 2
60.47 odd 4 225.4.b.d.199.2 2
60.59 even 2 45.4.a.b.1.1 1
140.139 even 2 735.4.a.i.1.1 1
180.59 even 6 405.4.e.k.136.1 2
180.79 odd 6 405.4.e.d.271.1 2
180.119 even 6 405.4.e.k.271.1 2
180.139 odd 6 405.4.e.d.136.1 2
220.219 even 2 1815.4.a.a.1.1 1
420.419 odd 2 2205.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 20.19 odd 2
45.4.a.b.1.1 1 60.59 even 2
75.4.a.a.1.1 1 4.3 odd 2
75.4.b.a.49.1 2 20.7 even 4
75.4.b.a.49.2 2 20.3 even 4
225.4.a.g.1.1 1 12.11 even 2
225.4.b.d.199.1 2 60.23 odd 4
225.4.b.d.199.2 2 60.47 odd 4
240.4.a.f.1.1 1 5.4 even 2
405.4.e.d.136.1 2 180.139 odd 6
405.4.e.d.271.1 2 180.79 odd 6
405.4.e.k.136.1 2 180.59 even 6
405.4.e.k.271.1 2 180.119 even 6
720.4.a.r.1.1 1 15.14 odd 2
735.4.a.i.1.1 1 140.139 even 2
960.4.a.l.1.1 1 40.29 even 2
960.4.a.bi.1.1 1 40.19 odd 2
1200.4.a.o.1.1 1 1.1 even 1 trivial
1200.4.f.m.49.1 2 5.3 odd 4
1200.4.f.m.49.2 2 5.2 odd 4
1815.4.a.a.1.1 1 220.219 even 2
2205.4.a.c.1.1 1 420.419 odd 2