Properties

Label 546.4.a.b.1.1
Level $546$
Weight $4$
Character 546.1
Self dual yes
Analytic conductor $32.215$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2150428631\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -18.0000 q^{10} -18.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} +27.0000 q^{15} +16.0000 q^{16} +60.0000 q^{17} +18.0000 q^{18} -43.0000 q^{19} -36.0000 q^{20} -21.0000 q^{21} -36.0000 q^{22} +9.00000 q^{23} -24.0000 q^{24} -44.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -249.000 q^{29} +54.0000 q^{30} -79.0000 q^{31} +32.0000 q^{32} +54.0000 q^{33} +120.000 q^{34} -63.0000 q^{35} +36.0000 q^{36} -412.000 q^{37} -86.0000 q^{38} -39.0000 q^{39} -72.0000 q^{40} +222.000 q^{41} -42.0000 q^{42} -295.000 q^{43} -72.0000 q^{44} -81.0000 q^{45} +18.0000 q^{46} +411.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -88.0000 q^{50} -180.000 q^{51} +52.0000 q^{52} -237.000 q^{53} -54.0000 q^{54} +162.000 q^{55} +56.0000 q^{56} +129.000 q^{57} -498.000 q^{58} -384.000 q^{59} +108.000 q^{60} -466.000 q^{61} -158.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -117.000 q^{65} +108.000 q^{66} -1042.00 q^{67} +240.000 q^{68} -27.0000 q^{69} -126.000 q^{70} -288.000 q^{71} +72.0000 q^{72} -691.000 q^{73} -824.000 q^{74} +132.000 q^{75} -172.000 q^{76} -126.000 q^{77} -78.0000 q^{78} +1001.00 q^{79} -144.000 q^{80} +81.0000 q^{81} +444.000 q^{82} +39.0000 q^{83} -84.0000 q^{84} -540.000 q^{85} -590.000 q^{86} +747.000 q^{87} -144.000 q^{88} -339.000 q^{89} -162.000 q^{90} +91.0000 q^{91} +36.0000 q^{92} +237.000 q^{93} +822.000 q^{94} +387.000 q^{95} -96.0000 q^{96} +713.000 q^{97} +98.0000 q^{98} -162.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −9.00000 −0.804984 −0.402492 0.915423i \(-0.631856\pi\)
−0.402492 + 0.915423i \(0.631856\pi\)
\(6\) −6.00000 −0.408248
\(7\) 7.00000 0.377964
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −18.0000 −0.569210
\(11\) −18.0000 −0.493382 −0.246691 0.969094i \(-0.579343\pi\)
−0.246691 + 0.969094i \(0.579343\pi\)
\(12\) −12.0000 −0.288675
\(13\) 13.0000 0.277350
\(14\) 14.0000 0.267261
\(15\) 27.0000 0.464758
\(16\) 16.0000 0.250000
\(17\) 60.0000 0.856008 0.428004 0.903777i \(-0.359217\pi\)
0.428004 + 0.903777i \(0.359217\pi\)
\(18\) 18.0000 0.235702
\(19\) −43.0000 −0.519204 −0.259602 0.965716i \(-0.583591\pi\)
−0.259602 + 0.965716i \(0.583591\pi\)
\(20\) −36.0000 −0.402492
\(21\) −21.0000 −0.218218
\(22\) −36.0000 −0.348874
\(23\) 9.00000 0.0815926 0.0407963 0.999167i \(-0.487011\pi\)
0.0407963 + 0.999167i \(0.487011\pi\)
\(24\) −24.0000 −0.204124
\(25\) −44.0000 −0.352000
\(26\) 26.0000 0.196116
\(27\) −27.0000 −0.192450
\(28\) 28.0000 0.188982
\(29\) −249.000 −1.59442 −0.797209 0.603703i \(-0.793691\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(30\) 54.0000 0.328634
\(31\) −79.0000 −0.457704 −0.228852 0.973461i \(-0.573497\pi\)
−0.228852 + 0.973461i \(0.573497\pi\)
\(32\) 32.0000 0.176777
\(33\) 54.0000 0.284854
\(34\) 120.000 0.605289
\(35\) −63.0000 −0.304256
\(36\) 36.0000 0.166667
\(37\) −412.000 −1.83060 −0.915302 0.402767i \(-0.868048\pi\)
−0.915302 + 0.402767i \(0.868048\pi\)
\(38\) −86.0000 −0.367133
\(39\) −39.0000 −0.160128
\(40\) −72.0000 −0.284605
\(41\) 222.000 0.845624 0.422812 0.906217i \(-0.361043\pi\)
0.422812 + 0.906217i \(0.361043\pi\)
\(42\) −42.0000 −0.154303
\(43\) −295.000 −1.04621 −0.523106 0.852268i \(-0.675227\pi\)
−0.523106 + 0.852268i \(0.675227\pi\)
\(44\) −72.0000 −0.246691
\(45\) −81.0000 −0.268328
\(46\) 18.0000 0.0576947
\(47\) 411.000 1.27554 0.637771 0.770226i \(-0.279856\pi\)
0.637771 + 0.770226i \(0.279856\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) −88.0000 −0.248902
\(51\) −180.000 −0.494217
\(52\) 52.0000 0.138675
\(53\) −237.000 −0.614235 −0.307117 0.951672i \(-0.599364\pi\)
−0.307117 + 0.951672i \(0.599364\pi\)
\(54\) −54.0000 −0.136083
\(55\) 162.000 0.397165
\(56\) 56.0000 0.133631
\(57\) 129.000 0.299763
\(58\) −498.000 −1.12742
\(59\) −384.000 −0.847331 −0.423666 0.905819i \(-0.639257\pi\)
−0.423666 + 0.905819i \(0.639257\pi\)
\(60\) 108.000 0.232379
\(61\) −466.000 −0.978118 −0.489059 0.872251i \(-0.662660\pi\)
−0.489059 + 0.872251i \(0.662660\pi\)
\(62\) −158.000 −0.323645
\(63\) 63.0000 0.125988
\(64\) 64.0000 0.125000
\(65\) −117.000 −0.223263
\(66\) 108.000 0.201422
\(67\) −1042.00 −1.90001 −0.950004 0.312237i \(-0.898922\pi\)
−0.950004 + 0.312237i \(0.898922\pi\)
\(68\) 240.000 0.428004
\(69\) −27.0000 −0.0471075
\(70\) −126.000 −0.215141
\(71\) −288.000 −0.481399 −0.240699 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) 72.0000 0.117851
\(73\) −691.000 −1.10788 −0.553941 0.832556i \(-0.686877\pi\)
−0.553941 + 0.832556i \(0.686877\pi\)
\(74\) −824.000 −1.29443
\(75\) 132.000 0.203227
\(76\) −172.000 −0.259602
\(77\) −126.000 −0.186481
\(78\) −78.0000 −0.113228
\(79\) 1001.00 1.42559 0.712793 0.701374i \(-0.247430\pi\)
0.712793 + 0.701374i \(0.247430\pi\)
\(80\) −144.000 −0.201246
\(81\) 81.0000 0.111111
\(82\) 444.000 0.597946
\(83\) 39.0000 0.0515760 0.0257880 0.999667i \(-0.491791\pi\)
0.0257880 + 0.999667i \(0.491791\pi\)
\(84\) −84.0000 −0.109109
\(85\) −540.000 −0.689073
\(86\) −590.000 −0.739783
\(87\) 747.000 0.920538
\(88\) −144.000 −0.174437
\(89\) −339.000 −0.403752 −0.201876 0.979411i \(-0.564704\pi\)
−0.201876 + 0.979411i \(0.564704\pi\)
\(90\) −162.000 −0.189737
\(91\) 91.0000 0.104828
\(92\) 36.0000 0.0407963
\(93\) 237.000 0.264255
\(94\) 822.000 0.901945
\(95\) 387.000 0.417951
\(96\) −96.0000 −0.102062
\(97\) 713.000 0.746332 0.373166 0.927765i \(-0.378272\pi\)
0.373166 + 0.927765i \(0.378272\pi\)
\(98\) 98.0000 0.101015
\(99\) −162.000 −0.164461
\(100\) −176.000 −0.176000
\(101\) 810.000 0.798000 0.399000 0.916951i \(-0.369357\pi\)
0.399000 + 0.916951i \(0.369357\pi\)
\(102\) −360.000 −0.349464
\(103\) −1456.00 −1.39285 −0.696427 0.717628i \(-0.745228\pi\)
−0.696427 + 0.717628i \(0.745228\pi\)
\(104\) 104.000 0.0980581
\(105\) 189.000 0.175662
\(106\) −474.000 −0.434330
\(107\) −924.000 −0.834827 −0.417413 0.908717i \(-0.637063\pi\)
−0.417413 + 0.908717i \(0.637063\pi\)
\(108\) −108.000 −0.0962250
\(109\) 1154.00 1.01407 0.507033 0.861927i \(-0.330742\pi\)
0.507033 + 0.861927i \(0.330742\pi\)
\(110\) 324.000 0.280838
\(111\) 1236.00 1.05690
\(112\) 112.000 0.0944911
\(113\) 1725.00 1.43606 0.718028 0.696014i \(-0.245045\pi\)
0.718028 + 0.696014i \(0.245045\pi\)
\(114\) 258.000 0.211964
\(115\) −81.0000 −0.0656808
\(116\) −996.000 −0.797209
\(117\) 117.000 0.0924500
\(118\) −768.000 −0.599154
\(119\) 420.000 0.323541
\(120\) 216.000 0.164317
\(121\) −1007.00 −0.756574
\(122\) −932.000 −0.691634
\(123\) −666.000 −0.488221
\(124\) −316.000 −0.228852
\(125\) 1521.00 1.08834
\(126\) 126.000 0.0890871
\(127\) −1204.00 −0.841242 −0.420621 0.907236i \(-0.638188\pi\)
−0.420621 + 0.907236i \(0.638188\pi\)
\(128\) 128.000 0.0883883
\(129\) 885.000 0.604030
\(130\) −234.000 −0.157870
\(131\) 828.000 0.552234 0.276117 0.961124i \(-0.410952\pi\)
0.276117 + 0.961124i \(0.410952\pi\)
\(132\) 216.000 0.142427
\(133\) −301.000 −0.196241
\(134\) −2084.00 −1.34351
\(135\) 243.000 0.154919
\(136\) 480.000 0.302645
\(137\) −2664.00 −1.66132 −0.830660 0.556780i \(-0.812037\pi\)
−0.830660 + 0.556780i \(0.812037\pi\)
\(138\) −54.0000 −0.0333100
\(139\) −322.000 −0.196487 −0.0982435 0.995162i \(-0.531322\pi\)
−0.0982435 + 0.995162i \(0.531322\pi\)
\(140\) −252.000 −0.152128
\(141\) −1233.00 −0.736435
\(142\) −576.000 −0.340400
\(143\) −234.000 −0.136840
\(144\) 144.000 0.0833333
\(145\) 2241.00 1.28348
\(146\) −1382.00 −0.783391
\(147\) −147.000 −0.0824786
\(148\) −1648.00 −0.915302
\(149\) −1602.00 −0.880812 −0.440406 0.897799i \(-0.645165\pi\)
−0.440406 + 0.897799i \(0.645165\pi\)
\(150\) 264.000 0.143703
\(151\) 1928.00 1.03906 0.519531 0.854451i \(-0.326107\pi\)
0.519531 + 0.854451i \(0.326107\pi\)
\(152\) −344.000 −0.183566
\(153\) 540.000 0.285336
\(154\) −252.000 −0.131862
\(155\) 711.000 0.368444
\(156\) −156.000 −0.0800641
\(157\) 1748.00 0.888571 0.444285 0.895885i \(-0.353458\pi\)
0.444285 + 0.895885i \(0.353458\pi\)
\(158\) 2002.00 1.00804
\(159\) 711.000 0.354629
\(160\) −288.000 −0.142302
\(161\) 63.0000 0.0308391
\(162\) 162.000 0.0785674
\(163\) 3008.00 1.44543 0.722714 0.691147i \(-0.242894\pi\)
0.722714 + 0.691147i \(0.242894\pi\)
\(164\) 888.000 0.422812
\(165\) −486.000 −0.229303
\(166\) 78.0000 0.0364697
\(167\) −1767.00 −0.818770 −0.409385 0.912362i \(-0.634257\pi\)
−0.409385 + 0.912362i \(0.634257\pi\)
\(168\) −168.000 −0.0771517
\(169\) 169.000 0.0769231
\(170\) −1080.00 −0.487248
\(171\) −387.000 −0.173068
\(172\) −1180.00 −0.523106
\(173\) 96.0000 0.0421893 0.0210946 0.999777i \(-0.493285\pi\)
0.0210946 + 0.999777i \(0.493285\pi\)
\(174\) 1494.00 0.650919
\(175\) −308.000 −0.133043
\(176\) −288.000 −0.123346
\(177\) 1152.00 0.489207
\(178\) −678.000 −0.285496
\(179\) 3345.00 1.39674 0.698372 0.715735i \(-0.253908\pi\)
0.698372 + 0.715735i \(0.253908\pi\)
\(180\) −324.000 −0.134164
\(181\) 2378.00 0.976549 0.488274 0.872690i \(-0.337627\pi\)
0.488274 + 0.872690i \(0.337627\pi\)
\(182\) 182.000 0.0741249
\(183\) 1398.00 0.564717
\(184\) 72.0000 0.0288473
\(185\) 3708.00 1.47361
\(186\) 474.000 0.186857
\(187\) −1080.00 −0.422339
\(188\) 1644.00 0.637771
\(189\) −189.000 −0.0727393
\(190\) 774.000 0.295536
\(191\) −768.000 −0.290945 −0.145473 0.989362i \(-0.546470\pi\)
−0.145473 + 0.989362i \(0.546470\pi\)
\(192\) −192.000 −0.0721688
\(193\) −3202.00 −1.19422 −0.597111 0.802158i \(-0.703685\pi\)
−0.597111 + 0.802158i \(0.703685\pi\)
\(194\) 1426.00 0.527736
\(195\) 351.000 0.128901
\(196\) 196.000 0.0714286
\(197\) −774.000 −0.279925 −0.139962 0.990157i \(-0.544698\pi\)
−0.139962 + 0.990157i \(0.544698\pi\)
\(198\) −324.000 −0.116291
\(199\) −1060.00 −0.377595 −0.188798 0.982016i \(-0.560459\pi\)
−0.188798 + 0.982016i \(0.560459\pi\)
\(200\) −352.000 −0.124451
\(201\) 3126.00 1.09697
\(202\) 1620.00 0.564271
\(203\) −1743.00 −0.602634
\(204\) −720.000 −0.247108
\(205\) −1998.00 −0.680714
\(206\) −2912.00 −0.984896
\(207\) 81.0000 0.0271975
\(208\) 208.000 0.0693375
\(209\) 774.000 0.256166
\(210\) 378.000 0.124212
\(211\) 2189.00 0.714204 0.357102 0.934065i \(-0.383765\pi\)
0.357102 + 0.934065i \(0.383765\pi\)
\(212\) −948.000 −0.307117
\(213\) 864.000 0.277936
\(214\) −1848.00 −0.590312
\(215\) 2655.00 0.842184
\(216\) −216.000 −0.0680414
\(217\) −553.000 −0.172996
\(218\) 2308.00 0.717053
\(219\) 2073.00 0.639636
\(220\) 648.000 0.198583
\(221\) 780.000 0.237414
\(222\) 2472.00 0.747341
\(223\) 2891.00 0.868142 0.434071 0.900879i \(-0.357077\pi\)
0.434071 + 0.900879i \(0.357077\pi\)
\(224\) 224.000 0.0668153
\(225\) −396.000 −0.117333
\(226\) 3450.00 1.01545
\(227\) 3132.00 0.915763 0.457881 0.889013i \(-0.348608\pi\)
0.457881 + 0.889013i \(0.348608\pi\)
\(228\) 516.000 0.149881
\(229\) −1150.00 −0.331852 −0.165926 0.986138i \(-0.553061\pi\)
−0.165926 + 0.986138i \(0.553061\pi\)
\(230\) −162.000 −0.0464433
\(231\) 378.000 0.107665
\(232\) −1992.00 −0.563712
\(233\) 2847.00 0.800486 0.400243 0.916409i \(-0.368926\pi\)
0.400243 + 0.916409i \(0.368926\pi\)
\(234\) 234.000 0.0653720
\(235\) −3699.00 −1.02679
\(236\) −1536.00 −0.423666
\(237\) −3003.00 −0.823062
\(238\) 840.000 0.228778
\(239\) 2220.00 0.600836 0.300418 0.953808i \(-0.402874\pi\)
0.300418 + 0.953808i \(0.402874\pi\)
\(240\) 432.000 0.116190
\(241\) −6703.00 −1.79161 −0.895805 0.444447i \(-0.853400\pi\)
−0.895805 + 0.444447i \(0.853400\pi\)
\(242\) −2014.00 −0.534979
\(243\) −243.000 −0.0641500
\(244\) −1864.00 −0.489059
\(245\) −441.000 −0.114998
\(246\) −1332.00 −0.345224
\(247\) −559.000 −0.144001
\(248\) −632.000 −0.161823
\(249\) −117.000 −0.0297774
\(250\) 3042.00 0.769572
\(251\) 4770.00 1.19952 0.599760 0.800180i \(-0.295263\pi\)
0.599760 + 0.800180i \(0.295263\pi\)
\(252\) 252.000 0.0629941
\(253\) −162.000 −0.0402563
\(254\) −2408.00 −0.594848
\(255\) 1620.00 0.397837
\(256\) 256.000 0.0625000
\(257\) −1062.00 −0.257766 −0.128883 0.991660i \(-0.541139\pi\)
−0.128883 + 0.991660i \(0.541139\pi\)
\(258\) 1770.00 0.427114
\(259\) −2884.00 −0.691904
\(260\) −468.000 −0.111631
\(261\) −2241.00 −0.531473
\(262\) 1656.00 0.390489
\(263\) 5283.00 1.23865 0.619323 0.785137i \(-0.287407\pi\)
0.619323 + 0.785137i \(0.287407\pi\)
\(264\) 432.000 0.100711
\(265\) 2133.00 0.494450
\(266\) −602.000 −0.138763
\(267\) 1017.00 0.233106
\(268\) −4168.00 −0.950004
\(269\) −6672.00 −1.51226 −0.756132 0.654419i \(-0.772913\pi\)
−0.756132 + 0.654419i \(0.772913\pi\)
\(270\) 486.000 0.109545
\(271\) −5992.00 −1.34313 −0.671565 0.740946i \(-0.734377\pi\)
−0.671565 + 0.740946i \(0.734377\pi\)
\(272\) 960.000 0.214002
\(273\) −273.000 −0.0605228
\(274\) −5328.00 −1.17473
\(275\) 792.000 0.173671
\(276\) −108.000 −0.0235538
\(277\) −5731.00 −1.24311 −0.621557 0.783369i \(-0.713499\pi\)
−0.621557 + 0.783369i \(0.713499\pi\)
\(278\) −644.000 −0.138937
\(279\) −711.000 −0.152568
\(280\) −504.000 −0.107571
\(281\) 6402.00 1.35911 0.679557 0.733622i \(-0.262172\pi\)
0.679557 + 0.733622i \(0.262172\pi\)
\(282\) −2466.00 −0.520738
\(283\) 7796.00 1.63754 0.818770 0.574121i \(-0.194656\pi\)
0.818770 + 0.574121i \(0.194656\pi\)
\(284\) −1152.00 −0.240699
\(285\) −1161.00 −0.241304
\(286\) −468.000 −0.0967602
\(287\) 1554.00 0.319616
\(288\) 288.000 0.0589256
\(289\) −1313.00 −0.267250
\(290\) 4482.00 0.907559
\(291\) −2139.00 −0.430895
\(292\) −2764.00 −0.553941
\(293\) 6279.00 1.25196 0.625978 0.779841i \(-0.284700\pi\)
0.625978 + 0.779841i \(0.284700\pi\)
\(294\) −294.000 −0.0583212
\(295\) 3456.00 0.682088
\(296\) −3296.00 −0.647217
\(297\) 486.000 0.0949514
\(298\) −3204.00 −0.622828
\(299\) 117.000 0.0226297
\(300\) 528.000 0.101614
\(301\) −2065.00 −0.395431
\(302\) 3856.00 0.734728
\(303\) −2430.00 −0.460726
\(304\) −688.000 −0.129801
\(305\) 4194.00 0.787370
\(306\) 1080.00 0.201763
\(307\) −1681.00 −0.312507 −0.156254 0.987717i \(-0.549942\pi\)
−0.156254 + 0.987717i \(0.549942\pi\)
\(308\) −504.000 −0.0932405
\(309\) 4368.00 0.804165
\(310\) 1422.00 0.260530
\(311\) −1770.00 −0.322725 −0.161363 0.986895i \(-0.551589\pi\)
−0.161363 + 0.986895i \(0.551589\pi\)
\(312\) −312.000 −0.0566139
\(313\) −4426.00 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(314\) 3496.00 0.628314
\(315\) −567.000 −0.101419
\(316\) 4004.00 0.712793
\(317\) −5592.00 −0.990782 −0.495391 0.868670i \(-0.664975\pi\)
−0.495391 + 0.868670i \(0.664975\pi\)
\(318\) 1422.00 0.250760
\(319\) 4482.00 0.786658
\(320\) −576.000 −0.100623
\(321\) 2772.00 0.481987
\(322\) 126.000 0.0218065
\(323\) −2580.00 −0.444443
\(324\) 324.000 0.0555556
\(325\) −572.000 −0.0976272
\(326\) 6016.00 1.02207
\(327\) −3462.00 −0.585471
\(328\) 1776.00 0.298973
\(329\) 2877.00 0.482110
\(330\) −972.000 −0.162142
\(331\) −5938.00 −0.986048 −0.493024 0.870016i \(-0.664109\pi\)
−0.493024 + 0.870016i \(0.664109\pi\)
\(332\) 156.000 0.0257880
\(333\) −3708.00 −0.610202
\(334\) −3534.00 −0.578958
\(335\) 9378.00 1.52948
\(336\) −336.000 −0.0545545
\(337\) 641.000 0.103613 0.0518064 0.998657i \(-0.483502\pi\)
0.0518064 + 0.998657i \(0.483502\pi\)
\(338\) 338.000 0.0543928
\(339\) −5175.00 −0.829107
\(340\) −2160.00 −0.344537
\(341\) 1422.00 0.225823
\(342\) −774.000 −0.122378
\(343\) 343.000 0.0539949
\(344\) −2360.00 −0.369891
\(345\) 243.000 0.0379208
\(346\) 192.000 0.0298323
\(347\) 5736.00 0.887391 0.443695 0.896178i \(-0.353667\pi\)
0.443695 + 0.896178i \(0.353667\pi\)
\(348\) 2988.00 0.460269
\(349\) −3535.00 −0.542190 −0.271095 0.962553i \(-0.587386\pi\)
−0.271095 + 0.962553i \(0.587386\pi\)
\(350\) −616.000 −0.0940760
\(351\) −351.000 −0.0533761
\(352\) −576.000 −0.0872185
\(353\) −3870.00 −0.583511 −0.291755 0.956493i \(-0.594239\pi\)
−0.291755 + 0.956493i \(0.594239\pi\)
\(354\) 2304.00 0.345922
\(355\) 2592.00 0.387519
\(356\) −1356.00 −0.201876
\(357\) −1260.00 −0.186796
\(358\) 6690.00 0.987647
\(359\) 7572.00 1.11319 0.556595 0.830784i \(-0.312108\pi\)
0.556595 + 0.830784i \(0.312108\pi\)
\(360\) −648.000 −0.0948683
\(361\) −5010.00 −0.730427
\(362\) 4756.00 0.690524
\(363\) 3021.00 0.436808
\(364\) 364.000 0.0524142
\(365\) 6219.00 0.891828
\(366\) 2796.00 0.399315
\(367\) 3206.00 0.456000 0.228000 0.973661i \(-0.426781\pi\)
0.228000 + 0.973661i \(0.426781\pi\)
\(368\) 144.000 0.0203981
\(369\) 1998.00 0.281875
\(370\) 7416.00 1.04200
\(371\) −1659.00 −0.232159
\(372\) 948.000 0.132128
\(373\) −9034.00 −1.25406 −0.627028 0.778997i \(-0.715729\pi\)
−0.627028 + 0.778997i \(0.715729\pi\)
\(374\) −2160.00 −0.298639
\(375\) −4563.00 −0.628353
\(376\) 3288.00 0.450972
\(377\) −3237.00 −0.442212
\(378\) −378.000 −0.0514344
\(379\) 8570.00 1.16151 0.580754 0.814079i \(-0.302758\pi\)
0.580754 + 0.814079i \(0.302758\pi\)
\(380\) 1548.00 0.208976
\(381\) 3612.00 0.485691
\(382\) −1536.00 −0.205729
\(383\) 10836.0 1.44568 0.722838 0.691018i \(-0.242837\pi\)
0.722838 + 0.691018i \(0.242837\pi\)
\(384\) −384.000 −0.0510310
\(385\) 1134.00 0.150114
\(386\) −6404.00 −0.844443
\(387\) −2655.00 −0.348737
\(388\) 2852.00 0.373166
\(389\) −6282.00 −0.818792 −0.409396 0.912357i \(-0.634261\pi\)
−0.409396 + 0.912357i \(0.634261\pi\)
\(390\) 702.000 0.0911465
\(391\) 540.000 0.0698439
\(392\) 392.000 0.0505076
\(393\) −2484.00 −0.318833
\(394\) −1548.00 −0.197937
\(395\) −9009.00 −1.14757
\(396\) −648.000 −0.0822304
\(397\) 4745.00 0.599861 0.299930 0.953961i \(-0.403037\pi\)
0.299930 + 0.953961i \(0.403037\pi\)
\(398\) −2120.00 −0.267000
\(399\) 903.000 0.113300
\(400\) −704.000 −0.0880000
\(401\) −5328.00 −0.663510 −0.331755 0.943366i \(-0.607641\pi\)
−0.331755 + 0.943366i \(0.607641\pi\)
\(402\) 6252.00 0.775675
\(403\) −1027.00 −0.126944
\(404\) 3240.00 0.399000
\(405\) −729.000 −0.0894427
\(406\) −3486.00 −0.426126
\(407\) 7416.00 0.903188
\(408\) −1440.00 −0.174732
\(409\) −13795.0 −1.66777 −0.833886 0.551937i \(-0.813889\pi\)
−0.833886 + 0.551937i \(0.813889\pi\)
\(410\) −3996.00 −0.481337
\(411\) 7992.00 0.959164
\(412\) −5824.00 −0.696427
\(413\) −2688.00 −0.320261
\(414\) 162.000 0.0192316
\(415\) −351.000 −0.0415179
\(416\) 416.000 0.0490290
\(417\) 966.000 0.113442
\(418\) 1548.00 0.181137
\(419\) 4398.00 0.512784 0.256392 0.966573i \(-0.417466\pi\)
0.256392 + 0.966573i \(0.417466\pi\)
\(420\) 756.000 0.0878310
\(421\) −484.000 −0.0560302 −0.0280151 0.999607i \(-0.508919\pi\)
−0.0280151 + 0.999607i \(0.508919\pi\)
\(422\) 4378.00 0.505018
\(423\) 3699.00 0.425181
\(424\) −1896.00 −0.217165
\(425\) −2640.00 −0.301315
\(426\) 1728.00 0.196530
\(427\) −3262.00 −0.369694
\(428\) −3696.00 −0.417413
\(429\) 702.000 0.0790044
\(430\) 5310.00 0.595514
\(431\) −3414.00 −0.381547 −0.190773 0.981634i \(-0.561100\pi\)
−0.190773 + 0.981634i \(0.561100\pi\)
\(432\) −432.000 −0.0481125
\(433\) 812.000 0.0901206 0.0450603 0.998984i \(-0.485652\pi\)
0.0450603 + 0.998984i \(0.485652\pi\)
\(434\) −1106.00 −0.122326
\(435\) −6723.00 −0.741019
\(436\) 4616.00 0.507033
\(437\) −387.000 −0.0423632
\(438\) 4146.00 0.452291
\(439\) 7778.00 0.845612 0.422806 0.906220i \(-0.361045\pi\)
0.422806 + 0.906220i \(0.361045\pi\)
\(440\) 1296.00 0.140419
\(441\) 441.000 0.0476190
\(442\) 1560.00 0.167877
\(443\) −6987.00 −0.749351 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(444\) 4944.00 0.528450
\(445\) 3051.00 0.325014
\(446\) 5782.00 0.613869
\(447\) 4806.00 0.508537
\(448\) 448.000 0.0472456
\(449\) 16344.0 1.71786 0.858932 0.512089i \(-0.171128\pi\)
0.858932 + 0.512089i \(0.171128\pi\)
\(450\) −792.000 −0.0829672
\(451\) −3996.00 −0.417216
\(452\) 6900.00 0.718028
\(453\) −5784.00 −0.599903
\(454\) 6264.00 0.647542
\(455\) −819.000 −0.0843853
\(456\) 1032.00 0.105982
\(457\) −17116.0 −1.75198 −0.875988 0.482334i \(-0.839789\pi\)
−0.875988 + 0.482334i \(0.839789\pi\)
\(458\) −2300.00 −0.234655
\(459\) −1620.00 −0.164739
\(460\) −324.000 −0.0328404
\(461\) −6882.00 −0.695286 −0.347643 0.937627i \(-0.613018\pi\)
−0.347643 + 0.937627i \(0.613018\pi\)
\(462\) 756.000 0.0761305
\(463\) 3854.00 0.386848 0.193424 0.981115i \(-0.438041\pi\)
0.193424 + 0.981115i \(0.438041\pi\)
\(464\) −3984.00 −0.398605
\(465\) −2133.00 −0.212722
\(466\) 5694.00 0.566029
\(467\) 1830.00 0.181333 0.0906663 0.995881i \(-0.471100\pi\)
0.0906663 + 0.995881i \(0.471100\pi\)
\(468\) 468.000 0.0462250
\(469\) −7294.00 −0.718136
\(470\) −7398.00 −0.726052
\(471\) −5244.00 −0.513016
\(472\) −3072.00 −0.299577
\(473\) 5310.00 0.516182
\(474\) −6006.00 −0.581993
\(475\) 1892.00 0.182760
\(476\) 1680.00 0.161770
\(477\) −2133.00 −0.204745
\(478\) 4440.00 0.424855
\(479\) −12663.0 −1.20791 −0.603953 0.797020i \(-0.706409\pi\)
−0.603953 + 0.797020i \(0.706409\pi\)
\(480\) 864.000 0.0821584
\(481\) −5356.00 −0.507718
\(482\) −13406.0 −1.26686
\(483\) −189.000 −0.0178050
\(484\) −4028.00 −0.378287
\(485\) −6417.00 −0.600785
\(486\) −486.000 −0.0453609
\(487\) −5578.00 −0.519021 −0.259511 0.965740i \(-0.583561\pi\)
−0.259511 + 0.965740i \(0.583561\pi\)
\(488\) −3728.00 −0.345817
\(489\) −9024.00 −0.834518
\(490\) −882.000 −0.0813157
\(491\) −17724.0 −1.62907 −0.814535 0.580115i \(-0.803008\pi\)
−0.814535 + 0.580115i \(0.803008\pi\)
\(492\) −2664.00 −0.244111
\(493\) −14940.0 −1.36484
\(494\) −1118.00 −0.101824
\(495\) 1458.00 0.132388
\(496\) −1264.00 −0.114426
\(497\) −2016.00 −0.181952
\(498\) −234.000 −0.0210558
\(499\) −15820.0 −1.41924 −0.709620 0.704585i \(-0.751133\pi\)
−0.709620 + 0.704585i \(0.751133\pi\)
\(500\) 6084.00 0.544170
\(501\) 5301.00 0.472717
\(502\) 9540.00 0.848189
\(503\) 414.000 0.0366985 0.0183493 0.999832i \(-0.494159\pi\)
0.0183493 + 0.999832i \(0.494159\pi\)
\(504\) 504.000 0.0445435
\(505\) −7290.00 −0.642378
\(506\) −324.000 −0.0284655
\(507\) −507.000 −0.0444116
\(508\) −4816.00 −0.420621
\(509\) −1521.00 −0.132450 −0.0662251 0.997805i \(-0.521096\pi\)
−0.0662251 + 0.997805i \(0.521096\pi\)
\(510\) 3240.00 0.281313
\(511\) −4837.00 −0.418740
\(512\) 512.000 0.0441942
\(513\) 1161.00 0.0999209
\(514\) −2124.00 −0.182268
\(515\) 13104.0 1.12123
\(516\) 3540.00 0.302015
\(517\) −7398.00 −0.629330
\(518\) −5768.00 −0.489250
\(519\) −288.000 −0.0243580
\(520\) −936.000 −0.0789352
\(521\) 2436.00 0.204843 0.102421 0.994741i \(-0.467341\pi\)
0.102421 + 0.994741i \(0.467341\pi\)
\(522\) −4482.00 −0.375808
\(523\) 13178.0 1.10179 0.550893 0.834576i \(-0.314287\pi\)
0.550893 + 0.834576i \(0.314287\pi\)
\(524\) 3312.00 0.276117
\(525\) 924.000 0.0768127
\(526\) 10566.0 0.875855
\(527\) −4740.00 −0.391798
\(528\) 864.000 0.0712136
\(529\) −12086.0 −0.993343
\(530\) 4266.00 0.349629
\(531\) −3456.00 −0.282444
\(532\) −1204.00 −0.0981203
\(533\) 2886.00 0.234534
\(534\) 2034.00 0.164831
\(535\) 8316.00 0.672022
\(536\) −8336.00 −0.671754
\(537\) −10035.0 −0.806410
\(538\) −13344.0 −1.06933
\(539\) −882.000 −0.0704832
\(540\) 972.000 0.0774597
\(541\) 13700.0 1.08874 0.544371 0.838845i \(-0.316769\pi\)
0.544371 + 0.838845i \(0.316769\pi\)
\(542\) −11984.0 −0.949736
\(543\) −7134.00 −0.563811
\(544\) 1920.00 0.151322
\(545\) −10386.0 −0.816307
\(546\) −546.000 −0.0427960
\(547\) −6649.00 −0.519727 −0.259864 0.965645i \(-0.583678\pi\)
−0.259864 + 0.965645i \(0.583678\pi\)
\(548\) −10656.0 −0.830660
\(549\) −4194.00 −0.326039
\(550\) 1584.00 0.122804
\(551\) 10707.0 0.827829
\(552\) −216.000 −0.0166550
\(553\) 7007.00 0.538821
\(554\) −11462.0 −0.879014
\(555\) −11124.0 −0.850788
\(556\) −1288.00 −0.0982435
\(557\) −1272.00 −0.0967619 −0.0483809 0.998829i \(-0.515406\pi\)
−0.0483809 + 0.998829i \(0.515406\pi\)
\(558\) −1422.00 −0.107882
\(559\) −3835.00 −0.290167
\(560\) −1008.00 −0.0760639
\(561\) 3240.00 0.243838
\(562\) 12804.0 0.961039
\(563\) −17052.0 −1.27648 −0.638238 0.769839i \(-0.720336\pi\)
−0.638238 + 0.769839i \(0.720336\pi\)
\(564\) −4932.00 −0.368217
\(565\) −15525.0 −1.15600
\(566\) 15592.0 1.15792
\(567\) 567.000 0.0419961
\(568\) −2304.00 −0.170200
\(569\) −19713.0 −1.45239 −0.726197 0.687487i \(-0.758714\pi\)
−0.726197 + 0.687487i \(0.758714\pi\)
\(570\) −2322.00 −0.170628
\(571\) 16625.0 1.21845 0.609225 0.792998i \(-0.291481\pi\)
0.609225 + 0.792998i \(0.291481\pi\)
\(572\) −936.000 −0.0684198
\(573\) 2304.00 0.167977
\(574\) 3108.00 0.226002
\(575\) −396.000 −0.0287206
\(576\) 576.000 0.0416667
\(577\) −7234.00 −0.521933 −0.260967 0.965348i \(-0.584041\pi\)
−0.260967 + 0.965348i \(0.584041\pi\)
\(578\) −2626.00 −0.188974
\(579\) 9606.00 0.689485
\(580\) 8964.00 0.641741
\(581\) 273.000 0.0194939
\(582\) −4278.00 −0.304689
\(583\) 4266.00 0.303053
\(584\) −5528.00 −0.391696
\(585\) −1053.00 −0.0744208
\(586\) 12558.0 0.885267
\(587\) −7737.00 −0.544021 −0.272010 0.962294i \(-0.587689\pi\)
−0.272010 + 0.962294i \(0.587689\pi\)
\(588\) −588.000 −0.0412393
\(589\) 3397.00 0.237642
\(590\) 6912.00 0.482309
\(591\) 2322.00 0.161615
\(592\) −6592.00 −0.457651
\(593\) 18723.0 1.29656 0.648281 0.761401i \(-0.275488\pi\)
0.648281 + 0.761401i \(0.275488\pi\)
\(594\) 972.000 0.0671408
\(595\) −3780.00 −0.260445
\(596\) −6408.00 −0.440406
\(597\) 3180.00 0.218005
\(598\) 234.000 0.0160016
\(599\) −24855.0 −1.69541 −0.847703 0.530472i \(-0.822015\pi\)
−0.847703 + 0.530472i \(0.822015\pi\)
\(600\) 1056.00 0.0718517
\(601\) 5114.00 0.347096 0.173548 0.984825i \(-0.444477\pi\)
0.173548 + 0.984825i \(0.444477\pi\)
\(602\) −4130.00 −0.279612
\(603\) −9378.00 −0.633336
\(604\) 7712.00 0.519531
\(605\) 9063.00 0.609030
\(606\) −4860.00 −0.325782
\(607\) 16886.0 1.12913 0.564565 0.825389i \(-0.309044\pi\)
0.564565 + 0.825389i \(0.309044\pi\)
\(608\) −1376.00 −0.0917832
\(609\) 5229.00 0.347931
\(610\) 8388.00 0.556754
\(611\) 5343.00 0.353772
\(612\) 2160.00 0.142668
\(613\) 812.000 0.0535014 0.0267507 0.999642i \(-0.491484\pi\)
0.0267507 + 0.999642i \(0.491484\pi\)
\(614\) −3362.00 −0.220976
\(615\) 5994.00 0.393010
\(616\) −1008.00 −0.0659310
\(617\) 11094.0 0.723870 0.361935 0.932203i \(-0.382116\pi\)
0.361935 + 0.932203i \(0.382116\pi\)
\(618\) 8736.00 0.568630
\(619\) −21508.0 −1.39657 −0.698287 0.715818i \(-0.746054\pi\)
−0.698287 + 0.715818i \(0.746054\pi\)
\(620\) 2844.00 0.184222
\(621\) −243.000 −0.0157025
\(622\) −3540.00 −0.228201
\(623\) −2373.00 −0.152604
\(624\) −624.000 −0.0400320
\(625\) −8189.00 −0.524096
\(626\) −8852.00 −0.565171
\(627\) −2322.00 −0.147898
\(628\) 6992.00 0.444285
\(629\) −24720.0 −1.56701
\(630\) −1134.00 −0.0717137
\(631\) 15014.0 0.947223 0.473612 0.880734i \(-0.342950\pi\)
0.473612 + 0.880734i \(0.342950\pi\)
\(632\) 8008.00 0.504021
\(633\) −6567.00 −0.412346
\(634\) −11184.0 −0.700589
\(635\) 10836.0 0.677187
\(636\) 2844.00 0.177314
\(637\) 637.000 0.0396214
\(638\) 8964.00 0.556251
\(639\) −2592.00 −0.160466
\(640\) −1152.00 −0.0711512
\(641\) 12321.0 0.759205 0.379602 0.925150i \(-0.376061\pi\)
0.379602 + 0.925150i \(0.376061\pi\)
\(642\) 5544.00 0.340817
\(643\) 13376.0 0.820370 0.410185 0.912002i \(-0.365464\pi\)
0.410185 + 0.912002i \(0.365464\pi\)
\(644\) 252.000 0.0154196
\(645\) −7965.00 −0.486235
\(646\) −5160.00 −0.314269
\(647\) −15150.0 −0.920569 −0.460284 0.887772i \(-0.652253\pi\)
−0.460284 + 0.887772i \(0.652253\pi\)
\(648\) 648.000 0.0392837
\(649\) 6912.00 0.418058
\(650\) −1144.00 −0.0690329
\(651\) 1659.00 0.0998792
\(652\) 12032.0 0.722714
\(653\) −414.000 −0.0248102 −0.0124051 0.999923i \(-0.503949\pi\)
−0.0124051 + 0.999923i \(0.503949\pi\)
\(654\) −6924.00 −0.413991
\(655\) −7452.00 −0.444540
\(656\) 3552.00 0.211406
\(657\) −6219.00 −0.369294
\(658\) 5754.00 0.340903
\(659\) 20631.0 1.21953 0.609765 0.792583i \(-0.291264\pi\)
0.609765 + 0.792583i \(0.291264\pi\)
\(660\) −1944.00 −0.114652
\(661\) 25589.0 1.50574 0.752872 0.658167i \(-0.228668\pi\)
0.752872 + 0.658167i \(0.228668\pi\)
\(662\) −11876.0 −0.697241
\(663\) −2340.00 −0.137071
\(664\) 312.000 0.0182349
\(665\) 2709.00 0.157971
\(666\) −7416.00 −0.431478
\(667\) −2241.00 −0.130093
\(668\) −7068.00 −0.409385
\(669\) −8673.00 −0.501222
\(670\) 18756.0 1.08150
\(671\) 8388.00 0.482586
\(672\) −672.000 −0.0385758
\(673\) 32105.0 1.83887 0.919433 0.393247i \(-0.128648\pi\)
0.919433 + 0.393247i \(0.128648\pi\)
\(674\) 1282.00 0.0732653
\(675\) 1188.00 0.0677424
\(676\) 676.000 0.0384615
\(677\) 29022.0 1.64757 0.823786 0.566901i \(-0.191858\pi\)
0.823786 + 0.566901i \(0.191858\pi\)
\(678\) −10350.0 −0.586267
\(679\) 4991.00 0.282087
\(680\) −4320.00 −0.243624
\(681\) −9396.00 −0.528716
\(682\) 2844.00 0.159681
\(683\) 18960.0 1.06220 0.531101 0.847308i \(-0.321778\pi\)
0.531101 + 0.847308i \(0.321778\pi\)
\(684\) −1548.00 −0.0865340
\(685\) 23976.0 1.33734
\(686\) 686.000 0.0381802
\(687\) 3450.00 0.191595
\(688\) −4720.00 −0.261553
\(689\) −3081.00 −0.170358
\(690\) 486.000 0.0268141
\(691\) −20653.0 −1.13701 −0.568507 0.822678i \(-0.692479\pi\)
−0.568507 + 0.822678i \(0.692479\pi\)
\(692\) 384.000 0.0210946
\(693\) −1134.00 −0.0621603
\(694\) 11472.0 0.627480
\(695\) 2898.00 0.158169
\(696\) 5976.00 0.325459
\(697\) 13320.0 0.723861
\(698\) −7070.00 −0.383386
\(699\) −8541.00 −0.462161
\(700\) −1232.00 −0.0665217
\(701\) −33543.0 −1.80728 −0.903639 0.428295i \(-0.859114\pi\)
−0.903639 + 0.428295i \(0.859114\pi\)
\(702\) −702.000 −0.0377426
\(703\) 17716.0 0.950457
\(704\) −1152.00 −0.0616728
\(705\) 11097.0 0.592819
\(706\) −7740.00 −0.412604
\(707\) 5670.00 0.301616
\(708\) 4608.00 0.244603
\(709\) 2630.00 0.139311 0.0696557 0.997571i \(-0.477810\pi\)
0.0696557 + 0.997571i \(0.477810\pi\)
\(710\) 5184.00 0.274017
\(711\) 9009.00 0.475195
\(712\) −2712.00 −0.142748
\(713\) −711.000 −0.0373452
\(714\) −2520.00 −0.132085
\(715\) 2106.00 0.110154
\(716\) 13380.0 0.698372
\(717\) −6660.00 −0.346893
\(718\) 15144.0 0.787144
\(719\) 21414.0 1.11072 0.555360 0.831610i \(-0.312581\pi\)
0.555360 + 0.831610i \(0.312581\pi\)
\(720\) −1296.00 −0.0670820
\(721\) −10192.0 −0.526449
\(722\) −10020.0 −0.516490
\(723\) 20109.0 1.03439
\(724\) 9512.00 0.488274
\(725\) 10956.0 0.561235
\(726\) 6042.00 0.308870
\(727\) 13466.0 0.686969 0.343484 0.939158i \(-0.388393\pi\)
0.343484 + 0.939158i \(0.388393\pi\)
\(728\) 728.000 0.0370625
\(729\) 729.000 0.0370370
\(730\) 12438.0 0.630618
\(731\) −17700.0 −0.895565
\(732\) 5592.00 0.282358
\(733\) −3895.00 −0.196269 −0.0981345 0.995173i \(-0.531288\pi\)
−0.0981345 + 0.995173i \(0.531288\pi\)
\(734\) 6412.00 0.322440
\(735\) 1323.00 0.0663940
\(736\) 288.000 0.0144237
\(737\) 18756.0 0.937430
\(738\) 3996.00 0.199315
\(739\) −27250.0 −1.35644 −0.678219 0.734860i \(-0.737248\pi\)
−0.678219 + 0.734860i \(0.737248\pi\)
\(740\) 14832.0 0.736804
\(741\) 1677.00 0.0831392
\(742\) −3318.00 −0.164161
\(743\) −19632.0 −0.969352 −0.484676 0.874694i \(-0.661062\pi\)
−0.484676 + 0.874694i \(0.661062\pi\)
\(744\) 1896.00 0.0934284
\(745\) 14418.0 0.709040
\(746\) −18068.0 −0.886751
\(747\) 351.000 0.0171920
\(748\) −4320.00 −0.211170
\(749\) −6468.00 −0.315535
\(750\) −9126.00 −0.444313
\(751\) −3499.00 −0.170014 −0.0850069 0.996380i \(-0.527091\pi\)
−0.0850069 + 0.996380i \(0.527091\pi\)
\(752\) 6576.00 0.318886
\(753\) −14310.0 −0.692544
\(754\) −6474.00 −0.312691
\(755\) −17352.0 −0.836429
\(756\) −756.000 −0.0363696
\(757\) 10469.0 0.502645 0.251323 0.967903i \(-0.419135\pi\)
0.251323 + 0.967903i \(0.419135\pi\)
\(758\) 17140.0 0.821310
\(759\) 486.000 0.0232420
\(760\) 3096.00 0.147768
\(761\) 5607.00 0.267088 0.133544 0.991043i \(-0.457364\pi\)
0.133544 + 0.991043i \(0.457364\pi\)
\(762\) 7224.00 0.343436
\(763\) 8078.00 0.383281
\(764\) −3072.00 −0.145473
\(765\) −4860.00 −0.229691
\(766\) 21672.0 1.02225
\(767\) −4992.00 −0.235007
\(768\) −768.000 −0.0360844
\(769\) 14843.0 0.696037 0.348018 0.937488i \(-0.386855\pi\)
0.348018 + 0.937488i \(0.386855\pi\)
\(770\) 2268.00 0.106147
\(771\) 3186.00 0.148821
\(772\) −12808.0 −0.597111
\(773\) −7518.00 −0.349811 −0.174905 0.984585i \(-0.555962\pi\)
−0.174905 + 0.984585i \(0.555962\pi\)
\(774\) −5310.00 −0.246594
\(775\) 3476.00 0.161112
\(776\) 5704.00 0.263868
\(777\) 8652.00 0.399471
\(778\) −12564.0 −0.578973
\(779\) −9546.00 −0.439051
\(780\) 1404.00 0.0644503
\(781\) 5184.00 0.237514
\(782\) 1080.00 0.0493871
\(783\) 6723.00 0.306846
\(784\) 784.000 0.0357143
\(785\) −15732.0 −0.715286
\(786\) −4968.00 −0.225449
\(787\) 35813.0 1.62210 0.811052 0.584974i \(-0.198895\pi\)
0.811052 + 0.584974i \(0.198895\pi\)
\(788\) −3096.00 −0.139962
\(789\) −15849.0 −0.715132
\(790\) −18018.0 −0.811458
\(791\) 12075.0 0.542778
\(792\) −1296.00 −0.0581456
\(793\) −6058.00 −0.271281
\(794\) 9490.00 0.424166
\(795\) −6399.00 −0.285471
\(796\) −4240.00 −0.188798
\(797\) 35718.0 1.58745 0.793724 0.608278i \(-0.208139\pi\)
0.793724 + 0.608278i \(0.208139\pi\)
\(798\) 1806.00 0.0801149
\(799\) 24660.0 1.09187
\(800\) −1408.00 −0.0622254
\(801\) −3051.00 −0.134584
\(802\) −10656.0 −0.469173
\(803\) 12438.0 0.546610
\(804\) 12504.0 0.548485
\(805\) −567.000 −0.0248250
\(806\) −2054.00 −0.0897631
\(807\) 20016.0 0.873106
\(808\) 6480.00 0.282136
\(809\) 10401.0 0.452014 0.226007 0.974126i \(-0.427433\pi\)
0.226007 + 0.974126i \(0.427433\pi\)
\(810\) −1458.00 −0.0632456
\(811\) 21548.0 0.932987 0.466494 0.884525i \(-0.345517\pi\)
0.466494 + 0.884525i \(0.345517\pi\)
\(812\) −6972.00 −0.301317
\(813\) 17976.0 0.775456
\(814\) 14832.0 0.638650
\(815\) −27072.0 −1.16355
\(816\) −2880.00 −0.123554
\(817\) 12685.0 0.543197
\(818\) −27590.0 −1.17929
\(819\) 819.000 0.0349428
\(820\) −7992.00 −0.340357
\(821\) −7554.00 −0.321116 −0.160558 0.987026i \(-0.551329\pi\)
−0.160558 + 0.987026i \(0.551329\pi\)
\(822\) 15984.0 0.678231
\(823\) 13448.0 0.569584 0.284792 0.958589i \(-0.408075\pi\)
0.284792 + 0.958589i \(0.408075\pi\)
\(824\) −11648.0 −0.492448
\(825\) −2376.00 −0.100269
\(826\) −5376.00 −0.226459
\(827\) −10728.0 −0.451087 −0.225544 0.974233i \(-0.572416\pi\)
−0.225544 + 0.974233i \(0.572416\pi\)
\(828\) 324.000 0.0135988
\(829\) −42190.0 −1.76757 −0.883787 0.467889i \(-0.845015\pi\)
−0.883787 + 0.467889i \(0.845015\pi\)
\(830\) −702.000 −0.0293576
\(831\) 17193.0 0.717712
\(832\) 832.000 0.0346688
\(833\) 2940.00 0.122287
\(834\) 1932.00 0.0802155
\(835\) 15903.0 0.659097
\(836\) 3096.00 0.128083
\(837\) 2133.00 0.0880851
\(838\) 8796.00 0.362593
\(839\) −14856.0 −0.611306 −0.305653 0.952143i \(-0.598875\pi\)
−0.305653 + 0.952143i \(0.598875\pi\)
\(840\) 1512.00 0.0621059
\(841\) 37612.0 1.54217
\(842\) −968.000 −0.0396193
\(843\) −19206.0 −0.784685
\(844\) 8756.00 0.357102
\(845\) −1521.00 −0.0619219
\(846\) 7398.00 0.300648
\(847\) −7049.00 −0.285958
\(848\) −3792.00 −0.153559
\(849\) −23388.0 −0.945435
\(850\) −5280.00 −0.213062
\(851\) −3708.00 −0.149364
\(852\) 3456.00 0.138968
\(853\) −29185.0 −1.17148 −0.585742 0.810498i \(-0.699197\pi\)
−0.585742 + 0.810498i \(0.699197\pi\)
\(854\) −6524.00 −0.261413
\(855\) 3483.00 0.139317
\(856\) −7392.00 −0.295156
\(857\) −2178.00 −0.0868134 −0.0434067 0.999057i \(-0.513821\pi\)
−0.0434067 + 0.999057i \(0.513821\pi\)
\(858\) 1404.00 0.0558645
\(859\) 11702.0 0.464805 0.232402 0.972620i \(-0.425341\pi\)
0.232402 + 0.972620i \(0.425341\pi\)
\(860\) 10620.0 0.421092
\(861\) −4662.00 −0.184530
\(862\) −6828.00 −0.269794
\(863\) 31044.0 1.22451 0.612254 0.790661i \(-0.290263\pi\)
0.612254 + 0.790661i \(0.290263\pi\)
\(864\) −864.000 −0.0340207
\(865\) −864.000 −0.0339617
\(866\) 1624.00 0.0637249
\(867\) 3939.00 0.154297
\(868\) −2212.00 −0.0864979
\(869\) −18018.0 −0.703359
\(870\) −13446.0 −0.523979
\(871\) −13546.0 −0.526968
\(872\) 9232.00 0.358526
\(873\) 6417.00 0.248777
\(874\) −774.000 −0.0299553
\(875\) 10647.0 0.411353
\(876\) 8292.00 0.319818
\(877\) 13700.0 0.527498 0.263749 0.964591i \(-0.415041\pi\)
0.263749 + 0.964591i \(0.415041\pi\)
\(878\) 15556.0 0.597938
\(879\) −18837.0 −0.722817
\(880\) 2592.00 0.0992913
\(881\) −19710.0 −0.753742 −0.376871 0.926266i \(-0.623000\pi\)
−0.376871 + 0.926266i \(0.623000\pi\)
\(882\) 882.000 0.0336718
\(883\) 12476.0 0.475482 0.237741 0.971329i \(-0.423593\pi\)
0.237741 + 0.971329i \(0.423593\pi\)
\(884\) 3120.00 0.118707
\(885\) −10368.0 −0.393804
\(886\) −13974.0 −0.529871
\(887\) 14940.0 0.565542 0.282771 0.959187i \(-0.408746\pi\)
0.282771 + 0.959187i \(0.408746\pi\)
\(888\) 9888.00 0.373671
\(889\) −8428.00 −0.317960
\(890\) 6102.00 0.229820
\(891\) −1458.00 −0.0548202
\(892\) 11564.0 0.434071
\(893\) −17673.0 −0.662267
\(894\) 9612.00 0.359590
\(895\) −30105.0 −1.12436
\(896\) 896.000 0.0334077
\(897\) −351.000 −0.0130653
\(898\) 32688.0 1.21471
\(899\) 19671.0 0.729772
\(900\) −1584.00 −0.0586667
\(901\) −14220.0 −0.525790
\(902\) −7992.00 −0.295016
\(903\) 6195.00 0.228302
\(904\) 13800.0 0.507723
\(905\) −21402.0 −0.786107
\(906\) −11568.0 −0.424195
\(907\) 12953.0 0.474198 0.237099 0.971486i \(-0.423803\pi\)
0.237099 + 0.971486i \(0.423803\pi\)
\(908\) 12528.0 0.457881
\(909\) 7290.00 0.266000
\(910\) −1638.00 −0.0596694
\(911\) 15411.0 0.560471 0.280236 0.959931i \(-0.409587\pi\)
0.280236 + 0.959931i \(0.409587\pi\)
\(912\) 2064.00 0.0749406
\(913\) −702.000 −0.0254467
\(914\) −34232.0 −1.23883
\(915\) −12582.0 −0.454588
\(916\) −4600.00 −0.165926
\(917\) 5796.00 0.208725
\(918\) −3240.00 −0.116488
\(919\) 22520.0 0.808342 0.404171 0.914683i \(-0.367560\pi\)
0.404171 + 0.914683i \(0.367560\pi\)
\(920\) −648.000 −0.0232217
\(921\) 5043.00 0.180426
\(922\) −13764.0 −0.491641
\(923\) −3744.00 −0.133516
\(924\) 1512.00 0.0538324
\(925\) 18128.0 0.644373
\(926\) 7708.00 0.273543
\(927\) −13104.0 −0.464285
\(928\) −7968.00 −0.281856
\(929\) 31803.0 1.12317 0.561584 0.827420i \(-0.310192\pi\)
0.561584 + 0.827420i \(0.310192\pi\)
\(930\) −4266.00 −0.150417
\(931\) −2107.00 −0.0741720
\(932\) 11388.0 0.400243
\(933\) 5310.00 0.186325
\(934\) 3660.00 0.128221
\(935\) 9720.00 0.339976
\(936\) 936.000 0.0326860
\(937\) −30148.0 −1.05111 −0.525556 0.850759i \(-0.676143\pi\)
−0.525556 + 0.850759i \(0.676143\pi\)
\(938\) −14588.0 −0.507799
\(939\) 13278.0 0.461460
\(940\) −14796.0 −0.513396
\(941\) 5709.00 0.197777 0.0988885 0.995099i \(-0.468471\pi\)
0.0988885 + 0.995099i \(0.468471\pi\)
\(942\) −10488.0 −0.362757
\(943\) 1998.00 0.0689966
\(944\) −6144.00 −0.211833
\(945\) 1701.00 0.0585540
\(946\) 10620.0 0.364996
\(947\) −42474.0 −1.45747 −0.728733 0.684798i \(-0.759890\pi\)
−0.728733 + 0.684798i \(0.759890\pi\)
\(948\) −12012.0 −0.411531
\(949\) −8983.00 −0.307271
\(950\) 3784.00 0.129231
\(951\) 16776.0 0.572028
\(952\) 3360.00 0.114389
\(953\) 10953.0 0.372301 0.186150 0.982521i \(-0.440399\pi\)
0.186150 + 0.982521i \(0.440399\pi\)
\(954\) −4266.00 −0.144777
\(955\) 6912.00 0.234206
\(956\) 8880.00 0.300418
\(957\) −13446.0 −0.454177
\(958\) −25326.0 −0.854119
\(959\) −18648.0 −0.627920
\(960\) 1728.00 0.0580948
\(961\) −23550.0 −0.790507
\(962\) −10712.0 −0.359011
\(963\) −8316.00 −0.278276
\(964\) −26812.0 −0.895805
\(965\) 28818.0 0.961331
\(966\) −378.000 −0.0125900
\(967\) −4498.00 −0.149582 −0.0747911 0.997199i \(-0.523829\pi\)
−0.0747911 + 0.997199i \(0.523829\pi\)
\(968\) −8056.00 −0.267489
\(969\) 7740.00 0.256599
\(970\) −12834.0 −0.424819
\(971\) 42978.0 1.42042 0.710211 0.703989i \(-0.248600\pi\)
0.710211 + 0.703989i \(0.248600\pi\)
\(972\) −972.000 −0.0320750
\(973\) −2254.00 −0.0742651
\(974\) −11156.0 −0.367003
\(975\) 1716.00 0.0563651
\(976\) −7456.00 −0.244529
\(977\) −1554.00 −0.0508873 −0.0254436 0.999676i \(-0.508100\pi\)
−0.0254436 + 0.999676i \(0.508100\pi\)
\(978\) −18048.0 −0.590093
\(979\) 6102.00 0.199204
\(980\) −1764.00 −0.0574989
\(981\) 10386.0 0.338022
\(982\) −35448.0 −1.15193
\(983\) 15597.0 0.506070 0.253035 0.967457i \(-0.418571\pi\)
0.253035 + 0.967457i \(0.418571\pi\)
\(984\) −5328.00 −0.172612
\(985\) 6966.00 0.225335
\(986\) −29880.0 −0.965084
\(987\) −8631.00 −0.278346
\(988\) −2236.00 −0.0720006
\(989\) −2655.00 −0.0853631
\(990\) 2916.00 0.0936127
\(991\) 52004.0 1.66696 0.833482 0.552546i \(-0.186344\pi\)
0.833482 + 0.552546i \(0.186344\pi\)
\(992\) −2528.00 −0.0809114
\(993\) 17814.0 0.569295
\(994\) −4032.00 −0.128659
\(995\) 9540.00 0.303958
\(996\) −468.000 −0.0148887
\(997\) 16184.0 0.514095 0.257047 0.966399i \(-0.417250\pi\)
0.257047 + 0.966399i \(0.417250\pi\)
\(998\) −31640.0 −1.00355
\(999\) 11124.0 0.352300
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 546.4.a.b.1.1 1
3.2 odd 2 1638.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.a.b.1.1 1 1.1 even 1 trivial
1638.4.a.f.1.1 1 3.2 odd 2