Properties

Label 45.4.a.b.1.1
Level $45$
Weight $4$
Character 45.1
Self dual yes
Analytic conductor $2.655$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 45.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +20.0000 q^{7} +21.0000 q^{8} +O(q^{10})\) \(q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +20.0000 q^{7} +21.0000 q^{8} -15.0000 q^{10} +24.0000 q^{11} +74.0000 q^{13} -60.0000 q^{14} -71.0000 q^{16} -54.0000 q^{17} -124.000 q^{19} +5.00000 q^{20} -72.0000 q^{22} +120.000 q^{23} +25.0000 q^{25} -222.000 q^{26} +20.0000 q^{28} +78.0000 q^{29} +200.000 q^{31} +45.0000 q^{32} +162.000 q^{34} +100.000 q^{35} -70.0000 q^{37} +372.000 q^{38} +105.000 q^{40} -330.000 q^{41} +92.0000 q^{43} +24.0000 q^{44} -360.000 q^{46} +24.0000 q^{47} +57.0000 q^{49} -75.0000 q^{50} +74.0000 q^{52} -450.000 q^{53} +120.000 q^{55} +420.000 q^{56} -234.000 q^{58} -24.0000 q^{59} -322.000 q^{61} -600.000 q^{62} +433.000 q^{64} +370.000 q^{65} -196.000 q^{67} -54.0000 q^{68} -300.000 q^{70} +288.000 q^{71} -430.000 q^{73} +210.000 q^{74} -124.000 q^{76} +480.000 q^{77} -520.000 q^{79} -355.000 q^{80} +990.000 q^{82} -156.000 q^{83} -270.000 q^{85} -276.000 q^{86} +504.000 q^{88} -1026.00 q^{89} +1480.00 q^{91} +120.000 q^{92} -72.0000 q^{94} -620.000 q^{95} -286.000 q^{97} -171.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) −15.0000 −0.474342
\(11\) 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 \(-0.210402\pi\)
0.789381 + 0.613904i \(0.210402\pi\)
\(14\) −60.0000 −1.14541
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −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 \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) 5.00000 0.0559017
\(21\) 0 0
\(22\) −72.0000 −0.697748
\(23\) 120.000 1.08790 0.543951 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −222.000 −1.67453
\(27\) 0 0
\(28\) 20.0000 0.134987
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) 162.000 0.817140
\(35\) 100.000 0.482945
\(36\) 0 0
\(37\) −70.0000 −0.311025 −0.155513 0.987834i \(-0.549703\pi\)
−0.155513 + 0.987834i \(0.549703\pi\)
\(38\) 372.000 1.58806
\(39\) 0 0
\(40\) 105.000 0.415049
\(41\) −330.000 −1.25701 −0.628504 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 0 0
\(43\) 92.0000 0.326276 0.163138 0.986603i \(-0.447838\pi\)
0.163138 + 0.986603i \(0.447838\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) 24.0000 0.0744843 0.0372421 0.999306i \(-0.488143\pi\)
0.0372421 + 0.999306i \(0.488143\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) −75.0000 −0.212132
\(51\) 0 0
\(52\) 74.0000 0.197345
\(53\) −450.000 −1.16627 −0.583134 0.812376i \(-0.698174\pi\)
−0.583134 + 0.812376i \(0.698174\pi\)
\(54\) 0 0
\(55\) 120.000 0.294196
\(56\) 420.000 1.00223
\(57\) 0 0
\(58\) −234.000 −0.529754
\(59\) −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 \(0.609728\pi\)
\(62\) −600.000 −1.22903
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 370.000 0.706044
\(66\) 0 0
\(67\) −196.000 −0.357391 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(68\) −54.0000 −0.0963009
\(69\) 0 0
\(70\) −300.000 −0.512241
\(71\) 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 \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(74\) 210.000 0.329892
\(75\) 0 0
\(76\) −124.000 −0.187155
\(77\) 480.000 0.710404
\(78\) 0 0
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) −355.000 −0.496128
\(81\) 0 0
\(82\) 990.000 1.33326
\(83\) −156.000 −0.206304 −0.103152 0.994666i \(-0.532893\pi\)
−0.103152 + 0.994666i \(0.532893\pi\)
\(84\) 0 0
\(85\) −270.000 −0.344537
\(86\) −276.000 −0.346068
\(87\) 0 0
\(88\) 504.000 0.610529
\(89\) −1026.00 −1.22198 −0.610988 0.791640i \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 1480.00 1.70490
\(92\) 120.000 0.135988
\(93\) 0 0
\(94\) −72.0000 −0.0790025
\(95\) −620.000 −0.669586
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) −171.000 −0.176261
\(99\) 0 0
\(100\) 25.0000 0.0250000
\(101\) 1734.00 1.70831 0.854156 0.520017i \(-0.174075\pi\)
0.854156 + 0.520017i \(0.174075\pi\)
\(102\) 0 0
\(103\) 452.000 0.432397 0.216198 0.976349i \(-0.430634\pi\)
0.216198 + 0.976349i \(0.430634\pi\)
\(104\) 1554.00 1.46521
\(105\) 0 0
\(106\) 1350.00 1.23702
\(107\) 1404.00 1.26850 0.634251 0.773127i \(-0.281308\pi\)
0.634251 + 0.773127i \(0.281308\pi\)
\(108\) 0 0
\(109\) −1474.00 −1.29526 −0.647631 0.761954i \(-0.724240\pi\)
−0.647631 + 0.761954i \(0.724240\pi\)
\(110\) −360.000 −0.312042
\(111\) 0 0
\(112\) −1420.00 −1.19801
\(113\) −1086.00 −0.904091 −0.452046 0.891995i \(-0.649306\pi\)
−0.452046 + 0.891995i \(0.649306\pi\)
\(114\) 0 0
\(115\) 600.000 0.486524
\(116\) 78.0000 0.0624321
\(117\) 0 0
\(118\) 72.0000 0.0561707
\(119\) −1080.00 −0.831962
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 966.000 0.716865
\(123\) 0 0
\(124\) 200.000 0.144843
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1244.00 0.869190 0.434595 0.900626i \(-0.356891\pi\)
0.434595 + 0.900626i \(0.356891\pi\)
\(128\) −1659.00 −1.14560
\(129\) 0 0
\(130\) −1110.00 −0.748873
\(131\) −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
\(134\) 588.000 0.379071
\(135\) 0 0
\(136\) −1134.00 −0.714998
\(137\) −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 \(-0.249119\pi\)
0.709062 + 0.705147i \(0.249119\pi\)
\(140\) 100.000 0.0603682
\(141\) 0 0
\(142\) −864.000 −0.510600
\(143\) 1776.00 1.03858
\(144\) 0 0
\(145\) 390.000 0.223364
\(146\) 1290.00 0.731241
\(147\) 0 0
\(148\) −70.0000 −0.0388781
\(149\) −258.000 −0.141854 −0.0709268 0.997482i \(-0.522596\pi\)
−0.0709268 + 0.997482i \(0.522596\pi\)
\(150\) 0 0
\(151\) −808.000 −0.435458 −0.217729 0.976009i \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(152\) −2604.00 −1.38955
\(153\) 0 0
\(154\) −1440.00 −0.753497
\(155\) 1000.00 0.518206
\(156\) 0 0
\(157\) 2378.00 1.20882 0.604411 0.796673i \(-0.293408\pi\)
0.604411 + 0.796673i \(0.293408\pi\)
\(158\) 1560.00 0.785487
\(159\) 0 0
\(160\) 225.000 0.111174
\(161\) 2400.00 1.17482
\(162\) 0 0
\(163\) −52.0000 −0.0249874 −0.0124937 0.999922i \(-0.503977\pi\)
−0.0124937 + 0.999922i \(0.503977\pi\)
\(164\) −330.000 −0.157126
\(165\) 0 0
\(166\) 468.000 0.218818
\(167\) 3720.00 1.72373 0.861863 0.507141i \(-0.169298\pi\)
0.861863 + 0.507141i \(0.169298\pi\)
\(168\) 0 0
\(169\) 3279.00 1.49249
\(170\) 810.000 0.365436
\(171\) 0 0
\(172\) 92.0000 0.0407845
\(173\) −426.000 −0.187215 −0.0936075 0.995609i \(-0.529840\pi\)
−0.0936075 + 0.995609i \(0.529840\pi\)
\(174\) 0 0
\(175\) 500.000 0.215980
\(176\) −1704.00 −0.729795
\(177\) 0 0
\(178\) 3078.00 1.29610
\(179\) 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 \(0.722179\pi\)
\(182\) −4440.00 −1.80832
\(183\) 0 0
\(184\) 2520.00 1.00966
\(185\) −350.000 −0.139095
\(186\) 0 0
\(187\) −1296.00 −0.506807
\(188\) 24.0000 0.00931053
\(189\) 0 0
\(190\) 1860.00 0.710203
\(191\) −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 \(-0.334377\pi\)
0.497158 + 0.867660i \(0.334377\pi\)
\(194\) 858.000 0.317530
\(195\) 0 0
\(196\) 57.0000 0.0207726
\(197\) 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 \(-0.739116\pi\)
−0.682521 + 0.730866i \(0.739116\pi\)
\(200\) 525.000 0.185616
\(201\) 0 0
\(202\) −5202.00 −1.81194
\(203\) 1560.00 0.539362
\(204\) 0 0
\(205\) −1650.00 −0.562151
\(206\) −1356.00 −0.458626
\(207\) 0 0
\(208\) −5254.00 −1.75144
\(209\) −2976.00 −0.984948
\(210\) 0 0
\(211\) 1100.00 0.358896 0.179448 0.983767i \(-0.442569\pi\)
0.179448 + 0.983767i \(0.442569\pi\)
\(212\) −450.000 −0.145784
\(213\) 0 0
\(214\) −4212.00 −1.34545
\(215\) 460.000 0.145915
\(216\) 0 0
\(217\) 4000.00 1.25133
\(218\) 4422.00 1.37383
\(219\) 0 0
\(220\) 120.000 0.0367745
\(221\) −3996.00 −1.21629
\(222\) 0 0
\(223\) 1964.00 0.589772 0.294886 0.955532i \(-0.404718\pi\)
0.294886 + 0.955532i \(0.404718\pi\)
\(224\) 900.000 0.268454
\(225\) 0 0
\(226\) 3258.00 0.958933
\(227\) −660.000 −0.192977 −0.0964884 0.995334i \(-0.530761\pi\)
−0.0964884 + 0.995334i \(0.530761\pi\)
\(228\) 0 0
\(229\) −1906.00 −0.550009 −0.275004 0.961443i \(-0.588679\pi\)
−0.275004 + 0.961443i \(0.588679\pi\)
\(230\) −1800.00 −0.516037
\(231\) 0 0
\(232\) 1638.00 0.463534
\(233\) 1458.00 0.409943 0.204972 0.978768i \(-0.434290\pi\)
0.204972 + 0.978768i \(0.434290\pi\)
\(234\) 0 0
\(235\) 120.000 0.0333104
\(236\) −24.0000 −0.00661978
\(237\) 0 0
\(238\) 3240.00 0.882429
\(239\) −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 \(0.463078\pi\)
\(242\) 2265.00 0.601652
\(243\) 0 0
\(244\) −322.000 −0.0844834
\(245\) 285.000 0.0743183
\(246\) 0 0
\(247\) −9176.00 −2.36379
\(248\) 4200.00 1.07540
\(249\) 0 0
\(250\) −375.000 −0.0948683
\(251\) −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
\(254\) −3732.00 −0.921915
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −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
\(260\) 370.000 0.0882555
\(261\) 0 0
\(262\) 6984.00 1.64684
\(263\) −5448.00 −1.27733 −0.638666 0.769484i \(-0.720513\pi\)
−0.638666 + 0.769484i \(0.720513\pi\)
\(264\) 0 0
\(265\) −2250.00 −0.521571
\(266\) 7440.00 1.71495
\(267\) 0 0
\(268\) −196.000 −0.0446739
\(269\) 2574.00 0.583418 0.291709 0.956507i \(-0.405776\pi\)
0.291709 + 0.956507i \(0.405776\pi\)
\(270\) 0 0
\(271\) −3184.00 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(272\) 3834.00 0.854671
\(273\) 0 0
\(274\) 6354.00 1.40095
\(275\) 600.000 0.131569
\(276\) 0 0
\(277\) 3962.00 0.859399 0.429699 0.902972i \(-0.358620\pi\)
0.429699 + 0.902972i \(0.358620\pi\)
\(278\) −6972.00 −1.50415
\(279\) 0 0
\(280\) 2100.00 0.448211
\(281\) 8286.00 1.75908 0.879540 0.475825i \(-0.157851\pi\)
0.879540 + 0.475825i \(0.157851\pi\)
\(282\) 0 0
\(283\) −2716.00 −0.570493 −0.285246 0.958454i \(-0.592075\pi\)
−0.285246 + 0.958454i \(0.592075\pi\)
\(284\) 288.000 0.0601748
\(285\) 0 0
\(286\) −5328.00 −1.10158
\(287\) −6600.00 −1.35744
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) −1170.00 −0.236913
\(291\) 0 0
\(292\) −430.000 −0.0861776
\(293\) −6018.00 −1.19992 −0.599958 0.800032i \(-0.704816\pi\)
−0.599958 + 0.800032i \(0.704816\pi\)
\(294\) 0 0
\(295\) −120.000 −0.0236836
\(296\) −1470.00 −0.288655
\(297\) 0 0
\(298\) 774.000 0.150458
\(299\) 8880.00 1.71754
\(300\) 0 0
\(301\) 1840.00 0.352345
\(302\) 2424.00 0.461873
\(303\) 0 0
\(304\) 8804.00 1.66100
\(305\) −1610.00 −0.302257
\(306\) 0 0
\(307\) 9236.00 1.71702 0.858512 0.512793i \(-0.171389\pi\)
0.858512 + 0.512793i \(0.171389\pi\)
\(308\) 480.000 0.0888004
\(309\) 0 0
\(310\) −3000.00 −0.549640
\(311\) −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 \(-0.730687\pi\)
−0.662930 + 0.748681i \(0.730687\pi\)
\(314\) −7134.00 −1.28215
\(315\) 0 0
\(316\) −520.000 −0.0925705
\(317\) 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
\(320\) 2165.00 0.378210
\(321\) 0 0
\(322\) −7200.00 −1.24609
\(323\) 6696.00 1.15348
\(324\) 0 0
\(325\) 1850.00 0.315752
\(326\) 156.000 0.0265032
\(327\) 0 0
\(328\) −6930.00 −1.16660
\(329\) 480.000 0.0804354
\(330\) 0 0
\(331\) 3692.00 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(332\) −156.000 −0.0257880
\(333\) 0 0
\(334\) −11160.0 −1.82829
\(335\) −980.000 −0.159830
\(336\) 0 0
\(337\) −8998.00 −1.45446 −0.727229 0.686395i \(-0.759192\pi\)
−0.727229 + 0.686395i \(0.759192\pi\)
\(338\) −9837.00 −1.58302
\(339\) 0 0
\(340\) −270.000 −0.0430671
\(341\) 4800.00 0.762271
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 1932.00 0.302809
\(345\) 0 0
\(346\) 1278.00 0.198571
\(347\) −5244.00 −0.811276 −0.405638 0.914034i \(-0.632951\pi\)
−0.405638 + 0.914034i \(0.632951\pi\)
\(348\) 0 0
\(349\) 6302.00 0.966585 0.483293 0.875459i \(-0.339441\pi\)
0.483293 + 0.875459i \(0.339441\pi\)
\(350\) −1500.00 −0.229081
\(351\) 0 0
\(352\) 1080.00 0.163535
\(353\) −3414.00 −0.514756 −0.257378 0.966311i \(-0.582859\pi\)
−0.257378 + 0.966311i \(0.582859\pi\)
\(354\) 0 0
\(355\) 1440.00 0.215288
\(356\) −1026.00 −0.152747
\(357\) 0 0
\(358\) −4320.00 −0.637763
\(359\) −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
\(362\) 9390.00 1.36334
\(363\) 0 0
\(364\) 1480.00 0.213113
\(365\) −2150.00 −0.308318
\(366\) 0 0
\(367\) −3508.00 −0.498954 −0.249477 0.968381i \(-0.580259\pi\)
−0.249477 + 0.968381i \(0.580259\pi\)
\(368\) −8520.00 −1.20689
\(369\) 0 0
\(370\) 1050.00 0.147532
\(371\) −9000.00 −1.25945
\(372\) 0 0
\(373\) 10802.0 1.49948 0.749740 0.661732i \(-0.230178\pi\)
0.749740 + 0.661732i \(0.230178\pi\)
\(374\) 3888.00 0.537550
\(375\) 0 0
\(376\) 504.000 0.0691272
\(377\) 5772.00 0.788523
\(378\) 0 0
\(379\) 1460.00 0.197876 0.0989382 0.995094i \(-0.468455\pi\)
0.0989382 + 0.995094i \(0.468455\pi\)
\(380\) −620.000 −0.0836982
\(381\) 0 0
\(382\) 10728.0 1.43689
\(383\) 4872.00 0.649994 0.324997 0.945715i \(-0.394637\pi\)
0.324997 + 0.945715i \(0.394637\pi\)
\(384\) 0 0
\(385\) 2400.00 0.317702
\(386\) −7998.00 −1.05463
\(387\) 0 0
\(388\) −286.000 −0.0374213
\(389\) 14046.0 1.83075 0.915373 0.402606i \(-0.131896\pi\)
0.915373 + 0.402606i \(0.131896\pi\)
\(390\) 0 0
\(391\) −6480.00 −0.838127
\(392\) 1197.00 0.154229
\(393\) 0 0
\(394\) −8154.00 −1.04262
\(395\) −2600.00 −0.331190
\(396\) 0 0
\(397\) −2734.00 −0.345631 −0.172816 0.984954i \(-0.555286\pi\)
−0.172816 + 0.984954i \(0.555286\pi\)
\(398\) 11496.0 1.44785
\(399\) 0 0
\(400\) −1775.00 −0.221875
\(401\) 15942.0 1.98530 0.992650 0.121019i \(-0.0386161\pi\)
0.992650 + 0.121019i \(0.0386161\pi\)
\(402\) 0 0
\(403\) 14800.0 1.82938
\(404\) 1734.00 0.213539
\(405\) 0 0
\(406\) −4680.00 −0.572080
\(407\) −1680.00 −0.204606
\(408\) 0 0
\(409\) 8714.00 1.05350 0.526748 0.850022i \(-0.323411\pi\)
0.526748 + 0.850022i \(0.323411\pi\)
\(410\) 4950.00 0.596251
\(411\) 0 0
\(412\) 452.000 0.0540496
\(413\) −480.000 −0.0571895
\(414\) 0 0
\(415\) −780.000 −0.0922619
\(416\) 3330.00 0.392468
\(417\) 0 0
\(418\) 8928.00 1.04470
\(419\) −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 \(0.279004\pi\)
\(422\) −3300.00 −0.380667
\(423\) 0 0
\(424\) −9450.00 −1.08239
\(425\) −1350.00 −0.154081
\(426\) 0 0
\(427\) −6440.00 −0.729868
\(428\) 1404.00 0.158563
\(429\) 0 0
\(430\) −1380.00 −0.154766
\(431\) −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 \(-0.833898\pi\)
−0.866912 + 0.498462i \(0.833898\pi\)
\(434\) −12000.0 −1.32723
\(435\) 0 0
\(436\) −1474.00 −0.161908
\(437\) −14880.0 −1.62885
\(438\) 0 0
\(439\) −9880.00 −1.07414 −0.537069 0.843538i \(-0.680469\pi\)
−0.537069 + 0.843538i \(0.680469\pi\)
\(440\) 2520.00 0.273037
\(441\) 0 0
\(442\) 11988.0 1.29007
\(443\) 16116.0 1.72843 0.864215 0.503123i \(-0.167816\pi\)
0.864215 + 0.503123i \(0.167816\pi\)
\(444\) 0 0
\(445\) −5130.00 −0.546484
\(446\) −5892.00 −0.625548
\(447\) 0 0
\(448\) 8660.00 0.913274
\(449\) −9018.00 −0.947852 −0.473926 0.880565i \(-0.657164\pi\)
−0.473926 + 0.880565i \(0.657164\pi\)
\(450\) 0 0
\(451\) −7920.00 −0.826914
\(452\) −1086.00 −0.113011
\(453\) 0 0
\(454\) 1980.00 0.204683
\(455\) 7400.00 0.762456
\(456\) 0 0
\(457\) −3670.00 −0.375657 −0.187829 0.982202i \(-0.560145\pi\)
−0.187829 + 0.982202i \(0.560145\pi\)
\(458\) 5718.00 0.583372
\(459\) 0 0
\(460\) 600.000 0.0608155
\(461\) −17562.0 −1.77428 −0.887141 0.461499i \(-0.847312\pi\)
−0.887141 + 0.461499i \(0.847312\pi\)
\(462\) 0 0
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) −5538.00 −0.554084
\(465\) 0 0
\(466\) −4374.00 −0.434810
\(467\) −6876.00 −0.681335 −0.340667 0.940184i \(-0.610653\pi\)
−0.340667 + 0.940184i \(0.610653\pi\)
\(468\) 0 0
\(469\) −3920.00 −0.385946
\(470\) −360.000 −0.0353310
\(471\) 0 0
\(472\) −504.000 −0.0491493
\(473\) 2208.00 0.214638
\(474\) 0 0
\(475\) −3100.00 −0.299448
\(476\) −1080.00 −0.103995
\(477\) 0 0
\(478\) 3528.00 0.337588
\(479\) −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
\(482\) −2598.00 −0.245510
\(483\) 0 0
\(484\) −755.000 −0.0709053
\(485\) −1430.00 −0.133882
\(486\) 0 0
\(487\) −3076.00 −0.286215 −0.143108 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(488\) −6762.00 −0.627257
\(489\) 0 0
\(490\) −855.000 −0.0788265
\(491\) 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
\(494\) 27528.0 2.50717
\(495\) 0 0
\(496\) −14200.0 −1.28548
\(497\) 5760.00 0.519862
\(498\) 0 0
\(499\) 9956.00 0.893170 0.446585 0.894741i \(-0.352640\pi\)
0.446585 + 0.894741i \(0.352640\pi\)
\(500\) 125.000 0.0111803
\(501\) 0 0
\(502\) 1296.00 0.115226
\(503\) 10656.0 0.944588 0.472294 0.881441i \(-0.343426\pi\)
0.472294 + 0.881441i \(0.343426\pi\)
\(504\) 0 0
\(505\) 8670.00 0.763980
\(506\) −8640.00 −0.759081
\(507\) 0 0
\(508\) 1244.00 0.108649
\(509\) 2766.00 0.240866 0.120433 0.992721i \(-0.461572\pi\)
0.120433 + 0.992721i \(0.461572\pi\)
\(510\) 0 0
\(511\) −8600.00 −0.744504
\(512\) 8733.00 0.753804
\(513\) 0 0
\(514\) 7578.00 0.650294
\(515\) 2260.00 0.193374
\(516\) 0 0
\(517\) 576.000 0.0489989
\(518\) 4200.00 0.356250
\(519\) 0 0
\(520\) 7770.00 0.655264
\(521\) −10530.0 −0.885466 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(522\) 0 0
\(523\) 12692.0 1.06115 0.530576 0.847637i \(-0.321976\pi\)
0.530576 + 0.847637i \(0.321976\pi\)
\(524\) −2328.00 −0.194082
\(525\) 0 0
\(526\) 16344.0 1.35481
\(527\) −10800.0 −0.892705
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 6750.00 0.553210
\(531\) 0 0
\(532\) −2480.00 −0.202108
\(533\) −24420.0 −1.98452
\(534\) 0 0
\(535\) 7020.00 0.567292
\(536\) −4116.00 −0.331687
\(537\) 0 0
\(538\) −7722.00 −0.618809
\(539\) 1368.00 0.109321
\(540\) 0 0
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) 9552.00 0.756999
\(543\) 0 0
\(544\) −2430.00 −0.191517
\(545\) −7370.00 −0.579259
\(546\) 0 0
\(547\) 3620.00 0.282962 0.141481 0.989941i \(-0.454814\pi\)
0.141481 + 0.989941i \(0.454814\pi\)
\(548\) −2118.00 −0.165103
\(549\) 0 0
\(550\) −1800.00 −0.139550
\(551\) −9672.00 −0.747806
\(552\) 0 0
\(553\) −10400.0 −0.799734
\(554\) −11886.0 −0.911530
\(555\) 0 0
\(556\) 2324.00 0.177265
\(557\) 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
\(560\) −7100.00 −0.535767
\(561\) 0 0
\(562\) −24858.0 −1.86579
\(563\) 13404.0 1.00339 0.501697 0.865043i \(-0.332709\pi\)
0.501697 + 0.865043i \(0.332709\pi\)
\(564\) 0 0
\(565\) −5430.00 −0.404322
\(566\) 8148.00 0.605099
\(567\) 0 0
\(568\) 6048.00 0.446775
\(569\) 18654.0 1.37437 0.687185 0.726483i \(-0.258846\pi\)
0.687185 + 0.726483i \(0.258846\pi\)
\(570\) 0 0
\(571\) −7684.00 −0.563162 −0.281581 0.959537i \(-0.590859\pi\)
−0.281581 + 0.959537i \(0.590859\pi\)
\(572\) 1776.00 0.129822
\(573\) 0 0
\(574\) 19800.0 1.43978
\(575\) 3000.00 0.217580
\(576\) 0 0
\(577\) −1726.00 −0.124531 −0.0622654 0.998060i \(-0.519833\pi\)
−0.0622654 + 0.998060i \(0.519833\pi\)
\(578\) 5991.00 0.431129
\(579\) 0 0
\(580\) 390.000 0.0279205
\(581\) −3120.00 −0.222787
\(582\) 0 0
\(583\) −10800.0 −0.767222
\(584\) −9030.00 −0.639836
\(585\) 0 0
\(586\) 18054.0 1.27270
\(587\) −10596.0 −0.745049 −0.372524 0.928022i \(-0.621508\pi\)
−0.372524 + 0.928022i \(0.621508\pi\)
\(588\) 0 0
\(589\) −24800.0 −1.73492
\(590\) 360.000 0.0251203
\(591\) 0 0
\(592\) 4970.00 0.345043
\(593\) −2862.00 −0.198193 −0.0990963 0.995078i \(-0.531595\pi\)
−0.0990963 + 0.995078i \(0.531595\pi\)
\(594\) 0 0
\(595\) −5400.00 −0.372065
\(596\) −258.000 −0.0177317
\(597\) 0 0
\(598\) −26640.0 −1.82172
\(599\) 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 \(0.605331\pi\)
\(602\) −5520.00 −0.373718
\(603\) 0 0
\(604\) −808.000 −0.0544322
\(605\) −3775.00 −0.253679
\(606\) 0 0
\(607\) 17444.0 1.16644 0.583221 0.812314i \(-0.301792\pi\)
0.583221 + 0.812314i \(0.301792\pi\)
\(608\) −5580.00 −0.372202
\(609\) 0 0
\(610\) 4830.00 0.320592
\(611\) 1776.00 0.117593
\(612\) 0 0
\(613\) −2374.00 −0.156419 −0.0782096 0.996937i \(-0.524920\pi\)
−0.0782096 + 0.996937i \(0.524920\pi\)
\(614\) −27708.0 −1.82118
\(615\) 0 0
\(616\) 10080.0 0.659310
\(617\) 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 \(-0.407729\pi\)
0.285834 + 0.958279i \(0.407729\pi\)
\(620\) 1000.00 0.0647758
\(621\) 0 0
\(622\) 4608.00 0.297048
\(623\) −20520.0 −1.31961
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 22026.0 1.40629
\(627\) 0 0
\(628\) 2378.00 0.151103
\(629\) 3780.00 0.239616
\(630\) 0 0
\(631\) −12688.0 −0.800478 −0.400239 0.916411i \(-0.631073\pi\)
−0.400239 + 0.916411i \(0.631073\pi\)
\(632\) −10920.0 −0.687301
\(633\) 0 0
\(634\) −11682.0 −0.731785
\(635\) 6220.00 0.388714
\(636\) 0 0
\(637\) 4218.00 0.262360
\(638\) −5616.00 −0.348495
\(639\) 0 0
\(640\) −8295.00 −0.512326
\(641\) 9150.00 0.563812 0.281906 0.959442i \(-0.409033\pi\)
0.281906 + 0.959442i \(0.409033\pi\)
\(642\) 0 0
\(643\) 25292.0 1.55120 0.775598 0.631227i \(-0.217448\pi\)
0.775598 + 0.631227i \(0.217448\pi\)
\(644\) 2400.00 0.146853
\(645\) 0 0
\(646\) −20088.0 −1.22345
\(647\) 2736.00 0.166249 0.0831246 0.996539i \(-0.473510\pi\)
0.0831246 + 0.996539i \(0.473510\pi\)
\(648\) 0 0
\(649\) −576.000 −0.0348382
\(650\) −5550.00 −0.334906
\(651\) 0 0
\(652\) −52.0000 −0.00312343
\(653\) −22218.0 −1.33148 −0.665741 0.746183i \(-0.731884\pi\)
−0.665741 + 0.746183i \(0.731884\pi\)
\(654\) 0 0
\(655\) −11640.0 −0.694370
\(656\) 23430.0 1.39449
\(657\) 0 0
\(658\) −1440.00 −0.0853147
\(659\) −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 \(0.601133\pi\)
\(662\) −11076.0 −0.650273
\(663\) 0 0
\(664\) −3276.00 −0.191466
\(665\) −12400.0 −0.723085
\(666\) 0 0
\(667\) 9360.00 0.543359
\(668\) 3720.00 0.215466
\(669\) 0 0
\(670\) 2940.00 0.169526
\(671\) −7728.00 −0.444614
\(672\) 0 0
\(673\) 1370.00 0.0784690 0.0392345 0.999230i \(-0.487508\pi\)
0.0392345 + 0.999230i \(0.487508\pi\)
\(674\) 26994.0 1.54269
\(675\) 0 0
\(676\) 3279.00 0.186561
\(677\) 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
\(680\) −5670.00 −0.319757
\(681\) 0 0
\(682\) −14400.0 −0.808511
\(683\) −11988.0 −0.671608 −0.335804 0.941932i \(-0.609008\pi\)
−0.335804 + 0.941932i \(0.609008\pi\)
\(684\) 0 0
\(685\) −10590.0 −0.590691
\(686\) 17160.0 0.955061
\(687\) 0 0
\(688\) −6532.00 −0.361962
\(689\) −33300.0 −1.84126
\(690\) 0 0
\(691\) 32996.0 1.81654 0.908268 0.418388i \(-0.137405\pi\)
0.908268 + 0.418388i \(0.137405\pi\)
\(692\) −426.000 −0.0234019
\(693\) 0 0
\(694\) 15732.0 0.860488
\(695\) 11620.0 0.634204
\(696\) 0 0
\(697\) 17820.0 0.968408
\(698\) −18906.0 −1.02522
\(699\) 0 0
\(700\) 500.000 0.0269975
\(701\) 25902.0 1.39558 0.697792 0.716300i \(-0.254166\pi\)
0.697792 + 0.716300i \(0.254166\pi\)
\(702\) 0 0
\(703\) 8680.00 0.465679
\(704\) 10392.0 0.556340
\(705\) 0 0
\(706\) 10242.0 0.545981
\(707\) 34680.0 1.84480
\(708\) 0 0
\(709\) −27394.0 −1.45106 −0.725531 0.688189i \(-0.758406\pi\)
−0.725531 + 0.688189i \(0.758406\pi\)
\(710\) −4320.00 −0.228347
\(711\) 0 0
\(712\) −21546.0 −1.13409
\(713\) 24000.0 1.26060
\(714\) 0 0
\(715\) 8880.00 0.464466
\(716\) 1440.00 0.0751611
\(717\) 0 0
\(718\) 14472.0 0.752215
\(719\) −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
\(722\) −25551.0 −1.31705
\(723\) 0 0
\(724\) −3130.00 −0.160671
\(725\) 1950.00 0.0998913
\(726\) 0 0
\(727\) 28028.0 1.42985 0.714925 0.699201i \(-0.246461\pi\)
0.714925 + 0.699201i \(0.246461\pi\)
\(728\) 31080.0 1.58228
\(729\) 0 0
\(730\) 6450.00 0.327021
\(731\) −4968.00 −0.251365
\(732\) 0 0
\(733\) 18002.0 0.907120 0.453560 0.891226i \(-0.350154\pi\)
0.453560 + 0.891226i \(0.350154\pi\)
\(734\) 10524.0 0.529221
\(735\) 0 0
\(736\) 5400.00 0.270444
\(737\) −4704.00 −0.235107
\(738\) 0 0
\(739\) 15284.0 0.760800 0.380400 0.924822i \(-0.375786\pi\)
0.380400 + 0.924822i \(0.375786\pi\)
\(740\) −350.000 −0.0173868
\(741\) 0 0
\(742\) 27000.0 1.33585
\(743\) 18768.0 0.926691 0.463345 0.886178i \(-0.346649\pi\)
0.463345 + 0.886178i \(0.346649\pi\)
\(744\) 0 0
\(745\) −1290.00 −0.0634388
\(746\) −32406.0 −1.59044
\(747\) 0 0
\(748\) −1296.00 −0.0633509
\(749\) 28080.0 1.36985
\(750\) 0 0
\(751\) 8696.00 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(752\) −1704.00 −0.0826310
\(753\) 0 0
\(754\) −17316.0 −0.836355
\(755\) −4040.00 −0.194743
\(756\) 0 0
\(757\) −38662.0 −1.85627 −0.928134 0.372247i \(-0.878587\pi\)
−0.928134 + 0.372247i \(0.878587\pi\)
\(758\) −4380.00 −0.209880
\(759\) 0 0
\(760\) −13020.0 −0.621428
\(761\) −23874.0 −1.13723 −0.568615 0.822604i \(-0.692521\pi\)
−0.568615 + 0.822604i \(0.692521\pi\)
\(762\) 0 0
\(763\) −29480.0 −1.39875
\(764\) −3576.00 −0.169339
\(765\) 0 0
\(766\) −14616.0 −0.689422
\(767\) −1776.00 −0.0836084
\(768\) 0 0
\(769\) 23618.0 1.10753 0.553763 0.832675i \(-0.313192\pi\)
0.553763 + 0.832675i \(0.313192\pi\)
\(770\) −7200.00 −0.336974
\(771\) 0 0
\(772\) 2666.00 0.124289
\(773\) −11538.0 −0.536860 −0.268430 0.963299i \(-0.586505\pi\)
−0.268430 + 0.963299i \(0.586505\pi\)
\(774\) 0 0
\(775\) 5000.00 0.231749
\(776\) −6006.00 −0.277839
\(777\) 0 0
\(778\) −42138.0 −1.94180
\(779\) 40920.0 1.88204
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 19440.0 0.888968
\(783\) 0 0
\(784\) −4047.00 −0.184357
\(785\) 11890.0 0.540602
\(786\) 0 0
\(787\) −14884.0 −0.674152 −0.337076 0.941478i \(-0.609438\pi\)
−0.337076 + 0.941478i \(0.609438\pi\)
\(788\) 2718.00 0.122874
\(789\) 0 0
\(790\) 7800.00 0.351280
\(791\) −21720.0 −0.976327
\(792\) 0 0
\(793\) −23828.0 −1.06703
\(794\) 8202.00 0.366597
\(795\) 0 0
\(796\) −3832.00 −0.170630
\(797\) 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
\(800\) 1125.00 0.0497184
\(801\) 0 0
\(802\) −47826.0 −2.10573
\(803\) −10320.0 −0.453530
\(804\) 0 0
\(805\) 12000.0 0.525397
\(806\) −44400.0 −1.94035
\(807\) 0 0
\(808\) 36414.0 1.58545
\(809\) −44730.0 −1.94391 −0.971955 0.235167i \(-0.924436\pi\)
−0.971955 + 0.235167i \(0.924436\pi\)
\(810\) 0 0
\(811\) −42748.0 −1.85091 −0.925453 0.378862i \(-0.876316\pi\)
−0.925453 + 0.378862i \(0.876316\pi\)
\(812\) 1560.00 0.0674203
\(813\) 0 0
\(814\) 5040.00 0.217017
\(815\) −260.000 −0.0111747
\(816\) 0 0
\(817\) −11408.0 −0.488513
\(818\) −26142.0 −1.11740
\(819\) 0 0
\(820\) −1650.00 −0.0702689
\(821\) 31686.0 1.34695 0.673477 0.739208i \(-0.264800\pi\)
0.673477 + 0.739208i \(0.264800\pi\)
\(822\) 0 0
\(823\) 11036.0 0.467425 0.233713 0.972306i \(-0.424913\pi\)
0.233713 + 0.972306i \(0.424913\pi\)
\(824\) 9492.00 0.401298
\(825\) 0 0
\(826\) 1440.00 0.0606586
\(827\) −25884.0 −1.08836 −0.544181 0.838968i \(-0.683159\pi\)
−0.544181 + 0.838968i \(0.683159\pi\)
\(828\) 0 0
\(829\) 15950.0 0.668234 0.334117 0.942532i \(-0.391562\pi\)
0.334117 + 0.942532i \(0.391562\pi\)
\(830\) 2340.00 0.0978585
\(831\) 0 0
\(832\) 32042.0 1.33516
\(833\) −3078.00 −0.128027
\(834\) 0 0
\(835\) 18600.0 0.770874
\(836\) −2976.00 −0.123119
\(837\) 0 0
\(838\) 35928.0 1.48104
\(839\) −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
\(842\) −33162.0 −1.35729
\(843\) 0 0
\(844\) 1100.00 0.0448620
\(845\) 16395.0 0.667462
\(846\) 0 0
\(847\) −15100.0 −0.612565
\(848\) 31950.0 1.29383
\(849\) 0 0
\(850\) 4050.00 0.163428
\(851\) −8400.00 −0.338365
\(852\) 0 0
\(853\) −27862.0 −1.11838 −0.559189 0.829040i \(-0.688887\pi\)
−0.559189 + 0.829040i \(0.688887\pi\)
\(854\) 19320.0 0.774141
\(855\) 0 0
\(856\) 29484.0 1.17727
\(857\) 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 \(-0.693666\pi\)
−0.571572 + 0.820552i \(0.693666\pi\)
\(860\) 460.000 0.0182394
\(861\) 0 0
\(862\) 2160.00 0.0853479
\(863\) 32688.0 1.28935 0.644677 0.764455i \(-0.276992\pi\)
0.644677 + 0.764455i \(0.276992\pi\)
\(864\) 0 0
\(865\) −2130.00 −0.0837251
\(866\) 46866.0 1.83900
\(867\) 0 0
\(868\) 4000.00 0.156416
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) −14504.0 −0.564236
\(872\) −30954.0 −1.20210
\(873\) 0 0
\(874\) 44640.0 1.72766
\(875\) 2500.00 0.0965891
\(876\) 0 0
\(877\) 36650.0 1.41115 0.705577 0.708633i \(-0.250688\pi\)
0.705577 + 0.708633i \(0.250688\pi\)
\(878\) 29640.0 1.13930
\(879\) 0 0
\(880\) −8520.00 −0.326374
\(881\) 2646.00 0.101187 0.0505936 0.998719i \(-0.483889\pi\)
0.0505936 + 0.998719i \(0.483889\pi\)
\(882\) 0 0
\(883\) 10892.0 0.415113 0.207557 0.978223i \(-0.433449\pi\)
0.207557 + 0.978223i \(0.433449\pi\)
\(884\) −3996.00 −0.152036
\(885\) 0 0
\(886\) −48348.0 −1.83328
\(887\) 43464.0 1.64530 0.822648 0.568550i \(-0.192496\pi\)
0.822648 + 0.568550i \(0.192496\pi\)
\(888\) 0 0
\(889\) 24880.0 0.938637
\(890\) 15390.0 0.579634
\(891\) 0 0
\(892\) 1964.00 0.0737215
\(893\) −2976.00 −0.111521
\(894\) 0 0
\(895\) 7200.00 0.268904
\(896\) −33180.0 −1.23713
\(897\) 0 0
\(898\) 27054.0 1.00535
\(899\) 15600.0 0.578742
\(900\) 0 0
\(901\) 24300.0 0.898502
\(902\) 23760.0 0.877075
\(903\) 0 0
\(904\) −22806.0 −0.839067
\(905\) −15650.0 −0.574833
\(906\) 0 0
\(907\) −14884.0 −0.544890 −0.272445 0.962171i \(-0.587832\pi\)
−0.272445 + 0.962171i \(0.587832\pi\)
\(908\) −660.000 −0.0241221
\(909\) 0 0
\(910\) −22200.0 −0.808706
\(911\) 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
\(914\) 11010.0 0.398445
\(915\) 0 0
\(916\) −1906.00 −0.0687511
\(917\) −46560.0 −1.67671
\(918\) 0 0
\(919\) −6640.00 −0.238339 −0.119169 0.992874i \(-0.538023\pi\)
−0.119169 + 0.992874i \(0.538023\pi\)
\(920\) 12600.0 0.451532
\(921\) 0 0
\(922\) 52686.0 1.88191
\(923\) 21312.0 0.760014
\(924\) 0 0
\(925\) −1750.00 −0.0622050
\(926\) −3516.00 −0.124776
\(927\) 0 0
\(928\) 3510.00 0.124161
\(929\) −29946.0 −1.05758 −0.528792 0.848751i \(-0.677355\pi\)
−0.528792 + 0.848751i \(0.677355\pi\)
\(930\) 0 0
\(931\) −7068.00 −0.248812
\(932\) 1458.00 0.0512429
\(933\) 0 0
\(934\) 20628.0 0.722665
\(935\) −6480.00 −0.226651
\(936\) 0 0
\(937\) 45002.0 1.56900 0.784499 0.620130i \(-0.212920\pi\)
0.784499 + 0.620130i \(0.212920\pi\)
\(938\) 11760.0 0.409358
\(939\) 0 0
\(940\) 120.000 0.00416380
\(941\) −6090.00 −0.210976 −0.105488 0.994421i \(-0.533640\pi\)
−0.105488 + 0.994421i \(0.533640\pi\)
\(942\) 0 0
\(943\) −39600.0 −1.36750
\(944\) 1704.00 0.0587505
\(945\) 0 0
\(946\) −6624.00 −0.227658
\(947\) −56388.0 −1.93491 −0.967457 0.253035i \(-0.918571\pi\)
−0.967457 + 0.253035i \(0.918571\pi\)
\(948\) 0 0
\(949\) −31820.0 −1.08843
\(950\) 9300.00 0.317612
\(951\) 0 0
\(952\) −22680.0 −0.772125
\(953\) −10854.0 −0.368936 −0.184468 0.982839i \(-0.559056\pi\)
−0.184468 + 0.982839i \(0.559056\pi\)
\(954\) 0 0
\(955\) −17880.0 −0.605846
\(956\) −1176.00 −0.0397851
\(957\) 0 0
\(958\) 6840.00 0.230679
\(959\) −42360.0 −1.42636
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 15540.0 0.520821
\(963\) 0 0
\(964\) 866.000 0.0289336
\(965\) 13330.0 0.444671
\(966\) 0 0
\(967\) −42316.0 −1.40723 −0.703615 0.710582i \(-0.748432\pi\)
−0.703615 + 0.710582i \(0.748432\pi\)
\(968\) −15855.0 −0.526445
\(969\) 0 0
\(970\) 4290.00 0.142004
\(971\) −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
\(974\) 9228.00 0.303577
\(975\) 0 0
\(976\) 22862.0 0.749790
\(977\) 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
\(980\) 285.000 0.00928979
\(981\) 0 0
\(982\) −56736.0 −1.84371
\(983\) −6960.00 −0.225829 −0.112914 0.993605i \(-0.536019\pi\)
−0.112914 + 0.993605i \(0.536019\pi\)
\(984\) 0 0
\(985\) 13590.0 0.439608
\(986\) 12636.0 0.408126
\(987\) 0 0
\(988\) −9176.00 −0.295473
\(989\) 11040.0 0.354956
\(990\) 0 0
\(991\) 47792.0 1.53195 0.765975 0.642870i \(-0.222256\pi\)
0.765975 + 0.642870i \(0.222256\pi\)
\(992\) 9000.00 0.288055
\(993\) 0 0
\(994\) −17280.0 −0.551397
\(995\) −19160.0 −0.610465
\(996\) 0 0
\(997\) 9938.00 0.315687 0.157843 0.987464i \(-0.449546\pi\)
0.157843 + 0.987464i \(0.449546\pi\)
\(998\) −29868.0 −0.947350
\(999\) 0 0
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 45.4.a.b.1.1 1
3.2 odd 2 15.4.a.b.1.1 1
4.3 odd 2 720.4.a.r.1.1 1
5.2 odd 4 225.4.b.d.199.1 2
5.3 odd 4 225.4.b.d.199.2 2
5.4 even 2 225.4.a.g.1.1 1
7.6 odd 2 2205.4.a.c.1.1 1
9.2 odd 6 405.4.e.d.271.1 2
9.4 even 3 405.4.e.k.136.1 2
9.5 odd 6 405.4.e.d.136.1 2
9.7 even 3 405.4.e.k.271.1 2
12.11 even 2 240.4.a.f.1.1 1
15.2 even 4 75.4.b.a.49.2 2
15.8 even 4 75.4.b.a.49.1 2
15.14 odd 2 75.4.a.a.1.1 1
21.20 even 2 735.4.a.i.1.1 1
24.5 odd 2 960.4.a.bi.1.1 1
24.11 even 2 960.4.a.l.1.1 1
33.32 even 2 1815.4.a.a.1.1 1
60.23 odd 4 1200.4.f.m.49.2 2
60.47 odd 4 1200.4.f.m.49.1 2
60.59 even 2 1200.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 3.2 odd 2
45.4.a.b.1.1 1 1.1 even 1 trivial
75.4.a.a.1.1 1 15.14 odd 2
75.4.b.a.49.1 2 15.8 even 4
75.4.b.a.49.2 2 15.2 even 4
225.4.a.g.1.1 1 5.4 even 2
225.4.b.d.199.1 2 5.2 odd 4
225.4.b.d.199.2 2 5.3 odd 4
240.4.a.f.1.1 1 12.11 even 2
405.4.e.d.136.1 2 9.5 odd 6
405.4.e.d.271.1 2 9.2 odd 6
405.4.e.k.136.1 2 9.4 even 3
405.4.e.k.271.1 2 9.7 even 3
720.4.a.r.1.1 1 4.3 odd 2
735.4.a.i.1.1 1 21.20 even 2
960.4.a.l.1.1 1 24.11 even 2
960.4.a.bi.1.1 1 24.5 odd 2
1200.4.a.o.1.1 1 60.59 even 2
1200.4.f.m.49.1 2 60.47 odd 4
1200.4.f.m.49.2 2 60.23 odd 4
1815.4.a.a.1.1 1 33.32 even 2
2205.4.a.c.1.1 1 7.6 odd 2