Properties

Label 10.4.a.a.1.1
Level $10$
Weight $4$
Character 10.1
Self dual yes
Analytic conductor $0.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.590019100057\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 10.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -8.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -16.0000 q^{6} -4.00000 q^{7} +8.00000 q^{8} +37.0000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -8.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -16.0000 q^{6} -4.00000 q^{7} +8.00000 q^{8} +37.0000 q^{9} +10.0000 q^{10} +12.0000 q^{11} -32.0000 q^{12} -58.0000 q^{13} -8.00000 q^{14} -40.0000 q^{15} +16.0000 q^{16} +66.0000 q^{17} +74.0000 q^{18} -100.000 q^{19} +20.0000 q^{20} +32.0000 q^{21} +24.0000 q^{22} +132.000 q^{23} -64.0000 q^{24} +25.0000 q^{25} -116.000 q^{26} -80.0000 q^{27} -16.0000 q^{28} -90.0000 q^{29} -80.0000 q^{30} +152.000 q^{31} +32.0000 q^{32} -96.0000 q^{33} +132.000 q^{34} -20.0000 q^{35} +148.000 q^{36} -34.0000 q^{37} -200.000 q^{38} +464.000 q^{39} +40.0000 q^{40} -438.000 q^{41} +64.0000 q^{42} +32.0000 q^{43} +48.0000 q^{44} +185.000 q^{45} +264.000 q^{46} -204.000 q^{47} -128.000 q^{48} -327.000 q^{49} +50.0000 q^{50} -528.000 q^{51} -232.000 q^{52} +222.000 q^{53} -160.000 q^{54} +60.0000 q^{55} -32.0000 q^{56} +800.000 q^{57} -180.000 q^{58} +420.000 q^{59} -160.000 q^{60} +902.000 q^{61} +304.000 q^{62} -148.000 q^{63} +64.0000 q^{64} -290.000 q^{65} -192.000 q^{66} -1024.00 q^{67} +264.000 q^{68} -1056.00 q^{69} -40.0000 q^{70} +432.000 q^{71} +296.000 q^{72} +362.000 q^{73} -68.0000 q^{74} -200.000 q^{75} -400.000 q^{76} -48.0000 q^{77} +928.000 q^{78} -160.000 q^{79} +80.0000 q^{80} -359.000 q^{81} -876.000 q^{82} +72.0000 q^{83} +128.000 q^{84} +330.000 q^{85} +64.0000 q^{86} +720.000 q^{87} +96.0000 q^{88} +810.000 q^{89} +370.000 q^{90} +232.000 q^{91} +528.000 q^{92} -1216.00 q^{93} -408.000 q^{94} -500.000 q^{95} -256.000 q^{96} +1106.00 q^{97} -654.000 q^{98} +444.000 q^{99} +O(q^{100})\)

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
\(3\) −8.00000 −1.53960 −0.769800 0.638285i \(-0.779644\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) −16.0000 −1.08866
\(7\) −4.00000 −0.215980 −0.107990 0.994152i \(-0.534441\pi\)
−0.107990 + 0.994152i \(0.534441\pi\)
\(8\) 8.00000 0.353553
\(9\) 37.0000 1.37037
\(10\) 10.0000 0.316228
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) −32.0000 −0.769800
\(13\) −58.0000 −1.23741 −0.618704 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) −8.00000 −0.152721
\(15\) −40.0000 −0.688530
\(16\) 16.0000 0.250000
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 74.0000 0.968998
\(19\) −100.000 −1.20745 −0.603726 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 20.0000 0.223607
\(21\) 32.0000 0.332522
\(22\) 24.0000 0.232583
\(23\) 132.000 1.19669 0.598346 0.801238i \(-0.295825\pi\)
0.598346 + 0.801238i \(0.295825\pi\)
\(24\) −64.0000 −0.544331
\(25\) 25.0000 0.200000
\(26\) −116.000 −0.874980
\(27\) −80.0000 −0.570222
\(28\) −16.0000 −0.107990
\(29\) −90.0000 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) −80.0000 −0.486864
\(31\) 152.000 0.880645 0.440323 0.897840i \(-0.354864\pi\)
0.440323 + 0.897840i \(0.354864\pi\)
\(32\) 32.0000 0.176777
\(33\) −96.0000 −0.506408
\(34\) 132.000 0.665818
\(35\) −20.0000 −0.0965891
\(36\) 148.000 0.685185
\(37\) −34.0000 −0.151069 −0.0755347 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) −200.000 −0.853797
\(39\) 464.000 1.90511
\(40\) 40.0000 0.158114
\(41\) −438.000 −1.66839 −0.834196 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) 64.0000 0.235129
\(43\) 32.0000 0.113487 0.0567437 0.998389i \(-0.481928\pi\)
0.0567437 + 0.998389i \(0.481928\pi\)
\(44\) 48.0000 0.164461
\(45\) 185.000 0.612848
\(46\) 264.000 0.846189
\(47\) −204.000 −0.633116 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(48\) −128.000 −0.384900
\(49\) −327.000 −0.953353
\(50\) 50.0000 0.141421
\(51\) −528.000 −1.44970
\(52\) −232.000 −0.618704
\(53\) 222.000 0.575359 0.287680 0.957727i \(-0.407116\pi\)
0.287680 + 0.957727i \(0.407116\pi\)
\(54\) −160.000 −0.403208
\(55\) 60.0000 0.147098
\(56\) −32.0000 −0.0763604
\(57\) 800.000 1.85899
\(58\) −180.000 −0.407503
\(59\) 420.000 0.926769 0.463384 0.886157i \(-0.346635\pi\)
0.463384 + 0.886157i \(0.346635\pi\)
\(60\) −160.000 −0.344265
\(61\) 902.000 1.89327 0.946633 0.322312i \(-0.104460\pi\)
0.946633 + 0.322312i \(0.104460\pi\)
\(62\) 304.000 0.622710
\(63\) −148.000 −0.295972
\(64\) 64.0000 0.125000
\(65\) −290.000 −0.553386
\(66\) −192.000 −0.358084
\(67\) −1024.00 −1.86719 −0.933593 0.358334i \(-0.883345\pi\)
−0.933593 + 0.358334i \(0.883345\pi\)
\(68\) 264.000 0.470804
\(69\) −1056.00 −1.84243
\(70\) −40.0000 −0.0682988
\(71\) 432.000 0.722098 0.361049 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) 296.000 0.484499
\(73\) 362.000 0.580396 0.290198 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) −68.0000 −0.106822
\(75\) −200.000 −0.307920
\(76\) −400.000 −0.603726
\(77\) −48.0000 −0.0710404
\(78\) 928.000 1.34712
\(79\) −160.000 −0.227866 −0.113933 0.993488i \(-0.536345\pi\)
−0.113933 + 0.993488i \(0.536345\pi\)
\(80\) 80.0000 0.111803
\(81\) −359.000 −0.492455
\(82\) −876.000 −1.17973
\(83\) 72.0000 0.0952172 0.0476086 0.998866i \(-0.484840\pi\)
0.0476086 + 0.998866i \(0.484840\pi\)
\(84\) 128.000 0.166261
\(85\) 330.000 0.421100
\(86\) 64.0000 0.0802476
\(87\) 720.000 0.887266
\(88\) 96.0000 0.116291
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 370.000 0.433349
\(91\) 232.000 0.267255
\(92\) 528.000 0.598346
\(93\) −1216.00 −1.35584
\(94\) −408.000 −0.447681
\(95\) −500.000 −0.539989
\(96\) −256.000 −0.272166
\(97\) 1106.00 1.15770 0.578852 0.815433i \(-0.303501\pi\)
0.578852 + 0.815433i \(0.303501\pi\)
\(98\) −654.000 −0.674122
\(99\) 444.000 0.450744
\(100\) 100.000 0.100000
\(101\) −258.000 −0.254178 −0.127089 0.991891i \(-0.540563\pi\)
−0.127089 + 0.991891i \(0.540563\pi\)
\(102\) −1056.00 −1.02509
\(103\) −988.000 −0.945151 −0.472575 0.881290i \(-0.656676\pi\)
−0.472575 + 0.881290i \(0.656676\pi\)
\(104\) −464.000 −0.437490
\(105\) 160.000 0.148709
\(106\) 444.000 0.406840
\(107\) −24.0000 −0.0216838 −0.0108419 0.999941i \(-0.503451\pi\)
−0.0108419 + 0.999941i \(0.503451\pi\)
\(108\) −320.000 −0.285111
\(109\) 950.000 0.834803 0.417401 0.908722i \(-0.362941\pi\)
0.417401 + 0.908722i \(0.362941\pi\)
\(110\) 120.000 0.104014
\(111\) 272.000 0.232586
\(112\) −64.0000 −0.0539949
\(113\) −1038.00 −0.864131 −0.432066 0.901842i \(-0.642215\pi\)
−0.432066 + 0.901842i \(0.642215\pi\)
\(114\) 1600.00 1.31451
\(115\) 660.000 0.535177
\(116\) −360.000 −0.288148
\(117\) −2146.00 −1.69571
\(118\) 840.000 0.655324
\(119\) −264.000 −0.203368
\(120\) −320.000 −0.243432
\(121\) −1187.00 −0.891811
\(122\) 1804.00 1.33874
\(123\) 3504.00 2.56866
\(124\) 608.000 0.440323
\(125\) 125.000 0.0894427
\(126\) −296.000 −0.209284
\(127\) −124.000 −0.0866395 −0.0433198 0.999061i \(-0.513793\pi\)
−0.0433198 + 0.999061i \(0.513793\pi\)
\(128\) 128.000 0.0883883
\(129\) −256.000 −0.174725
\(130\) −580.000 −0.391303
\(131\) 132.000 0.0880374 0.0440187 0.999031i \(-0.485984\pi\)
0.0440187 + 0.999031i \(0.485984\pi\)
\(132\) −384.000 −0.253204
\(133\) 400.000 0.260785
\(134\) −2048.00 −1.32030
\(135\) −400.000 −0.255011
\(136\) 528.000 0.332909
\(137\) −1254.00 −0.782018 −0.391009 0.920387i \(-0.627874\pi\)
−0.391009 + 0.920387i \(0.627874\pi\)
\(138\) −2112.00 −1.30279
\(139\) −2860.00 −1.74519 −0.872597 0.488440i \(-0.837566\pi\)
−0.872597 + 0.488440i \(0.837566\pi\)
\(140\) −80.0000 −0.0482945
\(141\) 1632.00 0.974746
\(142\) 864.000 0.510600
\(143\) −696.000 −0.407010
\(144\) 592.000 0.342593
\(145\) −450.000 −0.257727
\(146\) 724.000 0.410402
\(147\) 2616.00 1.46778
\(148\) −136.000 −0.0755347
\(149\) 750.000 0.412365 0.206183 0.978514i \(-0.433896\pi\)
0.206183 + 0.978514i \(0.433896\pi\)
\(150\) −400.000 −0.217732
\(151\) −448.000 −0.241442 −0.120721 0.992686i \(-0.538521\pi\)
−0.120721 + 0.992686i \(0.538521\pi\)
\(152\) −800.000 −0.426898
\(153\) 2442.00 1.29035
\(154\) −96.0000 −0.0502331
\(155\) 760.000 0.393837
\(156\) 1856.00 0.952557
\(157\) 2246.00 1.14172 0.570861 0.821047i \(-0.306610\pi\)
0.570861 + 0.821047i \(0.306610\pi\)
\(158\) −320.000 −0.161126
\(159\) −1776.00 −0.885824
\(160\) 160.000 0.0790569
\(161\) −528.000 −0.258461
\(162\) −718.000 −0.348219
\(163\) −568.000 −0.272940 −0.136470 0.990644i \(-0.543576\pi\)
−0.136470 + 0.990644i \(0.543576\pi\)
\(164\) −1752.00 −0.834196
\(165\) −480.000 −0.226472
\(166\) 144.000 0.0673287
\(167\) −1524.00 −0.706172 −0.353086 0.935591i \(-0.614868\pi\)
−0.353086 + 0.935591i \(0.614868\pi\)
\(168\) 256.000 0.117564
\(169\) 1167.00 0.531179
\(170\) 660.000 0.297763
\(171\) −3700.00 −1.65466
\(172\) 128.000 0.0567437
\(173\) 3702.00 1.62692 0.813462 0.581618i \(-0.197580\pi\)
0.813462 + 0.581618i \(0.197580\pi\)
\(174\) 1440.00 0.627391
\(175\) −100.000 −0.0431959
\(176\) 192.000 0.0822304
\(177\) −3360.00 −1.42685
\(178\) 1620.00 0.682158
\(179\) 3180.00 1.32785 0.663923 0.747801i \(-0.268890\pi\)
0.663923 + 0.747801i \(0.268890\pi\)
\(180\) 740.000 0.306424
\(181\) −2098.00 −0.861564 −0.430782 0.902456i \(-0.641762\pi\)
−0.430782 + 0.902456i \(0.641762\pi\)
\(182\) 464.000 0.188978
\(183\) −7216.00 −2.91487
\(184\) 1056.00 0.423094
\(185\) −170.000 −0.0675603
\(186\) −2432.00 −0.958725
\(187\) 792.000 0.309715
\(188\) −816.000 −0.316558
\(189\) 320.000 0.123156
\(190\) −1000.00 −0.381830
\(191\) 4392.00 1.66384 0.831921 0.554894i \(-0.187241\pi\)
0.831921 + 0.554894i \(0.187241\pi\)
\(192\) −512.000 −0.192450
\(193\) −2158.00 −0.804851 −0.402425 0.915453i \(-0.631833\pi\)
−0.402425 + 0.915453i \(0.631833\pi\)
\(194\) 2212.00 0.818620
\(195\) 2320.00 0.851993
\(196\) −1308.00 −0.476676
\(197\) −1074.00 −0.388423 −0.194212 0.980960i \(-0.562215\pi\)
−0.194212 + 0.980960i \(0.562215\pi\)
\(198\) 888.000 0.318724
\(199\) 2840.00 1.01167 0.505835 0.862630i \(-0.331185\pi\)
0.505835 + 0.862630i \(0.331185\pi\)
\(200\) 200.000 0.0707107
\(201\) 8192.00 2.87472
\(202\) −516.000 −0.179731
\(203\) 360.000 0.124468
\(204\) −2112.00 −0.724851
\(205\) −2190.00 −0.746128
\(206\) −1976.00 −0.668323
\(207\) 4884.00 1.63991
\(208\) −928.000 −0.309352
\(209\) −1200.00 −0.397157
\(210\) 320.000 0.105153
\(211\) −2668.00 −0.870487 −0.435243 0.900313i \(-0.643338\pi\)
−0.435243 + 0.900313i \(0.643338\pi\)
\(212\) 888.000 0.287680
\(213\) −3456.00 −1.11174
\(214\) −48.0000 −0.0153328
\(215\) 160.000 0.0507531
\(216\) −640.000 −0.201604
\(217\) −608.000 −0.190202
\(218\) 1900.00 0.590295
\(219\) −2896.00 −0.893578
\(220\) 240.000 0.0735491
\(221\) −3828.00 −1.16515
\(222\) 544.000 0.164463
\(223\) 1772.00 0.532116 0.266058 0.963957i \(-0.414279\pi\)
0.266058 + 0.963957i \(0.414279\pi\)
\(224\) −128.000 −0.0381802
\(225\) 925.000 0.274074
\(226\) −2076.00 −0.611033
\(227\) −2784.00 −0.814011 −0.407006 0.913426i \(-0.633427\pi\)
−0.407006 + 0.913426i \(0.633427\pi\)
\(228\) 3200.00 0.929496
\(229\) 350.000 0.100998 0.0504992 0.998724i \(-0.483919\pi\)
0.0504992 + 0.998724i \(0.483919\pi\)
\(230\) 1320.00 0.378427
\(231\) 384.000 0.109374
\(232\) −720.000 −0.203751
\(233\) 1962.00 0.551652 0.275826 0.961208i \(-0.411049\pi\)
0.275826 + 0.961208i \(0.411049\pi\)
\(234\) −4292.00 −1.19905
\(235\) −1020.00 −0.283138
\(236\) 1680.00 0.463384
\(237\) 1280.00 0.350823
\(238\) −528.000 −0.143803
\(239\) −4320.00 −1.16919 −0.584597 0.811324i \(-0.698748\pi\)
−0.584597 + 0.811324i \(0.698748\pi\)
\(240\) −640.000 −0.172133
\(241\) −478.000 −0.127762 −0.0638811 0.997958i \(-0.520348\pi\)
−0.0638811 + 0.997958i \(0.520348\pi\)
\(242\) −2374.00 −0.630605
\(243\) 5032.00 1.32841
\(244\) 3608.00 0.946633
\(245\) −1635.00 −0.426352
\(246\) 7008.00 1.81632
\(247\) 5800.00 1.49411
\(248\) 1216.00 0.311355
\(249\) −576.000 −0.146596
\(250\) 250.000 0.0632456
\(251\) 2652.00 0.666903 0.333452 0.942767i \(-0.391787\pi\)
0.333452 + 0.942767i \(0.391787\pi\)
\(252\) −592.000 −0.147986
\(253\) 1584.00 0.393617
\(254\) −248.000 −0.0612634
\(255\) −2640.00 −0.648326
\(256\) 256.000 0.0625000
\(257\) −2334.00 −0.566502 −0.283251 0.959046i \(-0.591413\pi\)
−0.283251 + 0.959046i \(0.591413\pi\)
\(258\) −512.000 −0.123549
\(259\) 136.000 0.0326279
\(260\) −1160.00 −0.276693
\(261\) −3330.00 −0.789739
\(262\) 264.000 0.0622518
\(263\) −3948.00 −0.925643 −0.462822 0.886451i \(-0.653163\pi\)
−0.462822 + 0.886451i \(0.653163\pi\)
\(264\) −768.000 −0.179042
\(265\) 1110.00 0.257309
\(266\) 800.000 0.184403
\(267\) −6480.00 −1.48528
\(268\) −4096.00 −0.933593
\(269\) 1590.00 0.360387 0.180193 0.983631i \(-0.442328\pi\)
0.180193 + 0.983631i \(0.442328\pi\)
\(270\) −800.000 −0.180320
\(271\) 4952.00 1.11001 0.555005 0.831847i \(-0.312716\pi\)
0.555005 + 0.831847i \(0.312716\pi\)
\(272\) 1056.00 0.235402
\(273\) −1856.00 −0.411466
\(274\) −2508.00 −0.552970
\(275\) 300.000 0.0657843
\(276\) −4224.00 −0.921213
\(277\) 1646.00 0.357034 0.178517 0.983937i \(-0.442870\pi\)
0.178517 + 0.983937i \(0.442870\pi\)
\(278\) −5720.00 −1.23404
\(279\) 5624.00 1.20681
\(280\) −160.000 −0.0341494
\(281\) −1158.00 −0.245838 −0.122919 0.992417i \(-0.539226\pi\)
−0.122919 + 0.992417i \(0.539226\pi\)
\(282\) 3264.00 0.689250
\(283\) 6992.00 1.46866 0.734331 0.678792i \(-0.237496\pi\)
0.734331 + 0.678792i \(0.237496\pi\)
\(284\) 1728.00 0.361049
\(285\) 4000.00 0.831367
\(286\) −1392.00 −0.287800
\(287\) 1752.00 0.360339
\(288\) 1184.00 0.242250
\(289\) −557.000 −0.113373
\(290\) −900.000 −0.182241
\(291\) −8848.00 −1.78240
\(292\) 1448.00 0.290198
\(293\) −258.000 −0.0514421 −0.0257210 0.999669i \(-0.508188\pi\)
−0.0257210 + 0.999669i \(0.508188\pi\)
\(294\) 5232.00 1.03788
\(295\) 2100.00 0.414463
\(296\) −272.000 −0.0534111
\(297\) −960.000 −0.187558
\(298\) 1500.00 0.291586
\(299\) −7656.00 −1.48080
\(300\) −800.000 −0.153960
\(301\) −128.000 −0.0245110
\(302\) −896.000 −0.170725
\(303\) 2064.00 0.391332
\(304\) −1600.00 −0.301863
\(305\) 4510.00 0.846695
\(306\) 4884.00 0.912417
\(307\) −8944.00 −1.66274 −0.831370 0.555720i \(-0.812443\pi\)
−0.831370 + 0.555720i \(0.812443\pi\)
\(308\) −192.000 −0.0355202
\(309\) 7904.00 1.45515
\(310\) 1520.00 0.278485
\(311\) 1392.00 0.253804 0.126902 0.991915i \(-0.459497\pi\)
0.126902 + 0.991915i \(0.459497\pi\)
\(312\) 3712.00 0.673560
\(313\) −5878.00 −1.06148 −0.530742 0.847534i \(-0.678087\pi\)
−0.530742 + 0.847534i \(0.678087\pi\)
\(314\) 4492.00 0.807319
\(315\) −740.000 −0.132363
\(316\) −640.000 −0.113933
\(317\) 10326.0 1.82955 0.914773 0.403969i \(-0.132370\pi\)
0.914773 + 0.403969i \(0.132370\pi\)
\(318\) −3552.00 −0.626372
\(319\) −1080.00 −0.189556
\(320\) 320.000 0.0559017
\(321\) 192.000 0.0333844
\(322\) −1056.00 −0.182760
\(323\) −6600.00 −1.13695
\(324\) −1436.00 −0.246228
\(325\) −1450.00 −0.247482
\(326\) −1136.00 −0.192998
\(327\) −7600.00 −1.28526
\(328\) −3504.00 −0.589866
\(329\) 816.000 0.136740
\(330\) −960.000 −0.160140
\(331\) −4228.00 −0.702090 −0.351045 0.936359i \(-0.614174\pi\)
−0.351045 + 0.936359i \(0.614174\pi\)
\(332\) 288.000 0.0476086
\(333\) −1258.00 −0.207021
\(334\) −3048.00 −0.499339
\(335\) −5120.00 −0.835031
\(336\) 512.000 0.0831306
\(337\) 1106.00 0.178776 0.0893882 0.995997i \(-0.471509\pi\)
0.0893882 + 0.995997i \(0.471509\pi\)
\(338\) 2334.00 0.375600
\(339\) 8304.00 1.33042
\(340\) 1320.00 0.210550
\(341\) 1824.00 0.289663
\(342\) −7400.00 −1.17002
\(343\) 2680.00 0.421885
\(344\) 256.000 0.0401238
\(345\) −5280.00 −0.823958
\(346\) 7404.00 1.15041
\(347\) 9336.00 1.44433 0.722165 0.691720i \(-0.243147\pi\)
0.722165 + 0.691720i \(0.243147\pi\)
\(348\) 2880.00 0.443633
\(349\) −11770.0 −1.80525 −0.902627 0.430424i \(-0.858364\pi\)
−0.902627 + 0.430424i \(0.858364\pi\)
\(350\) −200.000 −0.0305441
\(351\) 4640.00 0.705598
\(352\) 384.000 0.0581456
\(353\) 8322.00 1.25477 0.627387 0.778707i \(-0.284124\pi\)
0.627387 + 0.778707i \(0.284124\pi\)
\(354\) −6720.00 −1.00894
\(355\) 2160.00 0.322932
\(356\) 3240.00 0.482359
\(357\) 2112.00 0.313106
\(358\) 6360.00 0.938929
\(359\) 10680.0 1.57011 0.785054 0.619427i \(-0.212635\pi\)
0.785054 + 0.619427i \(0.212635\pi\)
\(360\) 1480.00 0.216675
\(361\) 3141.00 0.457938
\(362\) −4196.00 −0.609218
\(363\) 9496.00 1.37303
\(364\) 928.000 0.133628
\(365\) 1810.00 0.259561
\(366\) −14432.0 −2.06113
\(367\) −5884.00 −0.836900 −0.418450 0.908240i \(-0.637426\pi\)
−0.418450 + 0.908240i \(0.637426\pi\)
\(368\) 2112.00 0.299173
\(369\) −16206.0 −2.28632
\(370\) −340.000 −0.0477723
\(371\) −888.000 −0.124266
\(372\) −4864.00 −0.677921
\(373\) −2098.00 −0.291234 −0.145617 0.989341i \(-0.546517\pi\)
−0.145617 + 0.989341i \(0.546517\pi\)
\(374\) 1584.00 0.219002
\(375\) −1000.00 −0.137706
\(376\) −1632.00 −0.223840
\(377\) 5220.00 0.713113
\(378\) 640.000 0.0870848
\(379\) 3860.00 0.523153 0.261576 0.965183i \(-0.415758\pi\)
0.261576 + 0.965183i \(0.415758\pi\)
\(380\) −2000.00 −0.269994
\(381\) 992.000 0.133390
\(382\) 8784.00 1.17651
\(383\) −9588.00 −1.27917 −0.639587 0.768718i \(-0.720895\pi\)
−0.639587 + 0.768718i \(0.720895\pi\)
\(384\) −1024.00 −0.136083
\(385\) −240.000 −0.0317702
\(386\) −4316.00 −0.569116
\(387\) 1184.00 0.155520
\(388\) 4424.00 0.578852
\(389\) −13410.0 −1.74785 −0.873925 0.486060i \(-0.838434\pi\)
−0.873925 + 0.486060i \(0.838434\pi\)
\(390\) 4640.00 0.602450
\(391\) 8712.00 1.12682
\(392\) −2616.00 −0.337061
\(393\) −1056.00 −0.135542
\(394\) −2148.00 −0.274657
\(395\) −800.000 −0.101905
\(396\) 1776.00 0.225372
\(397\) −13114.0 −1.65787 −0.828933 0.559348i \(-0.811052\pi\)
−0.828933 + 0.559348i \(0.811052\pi\)
\(398\) 5680.00 0.715358
\(399\) −3200.00 −0.401505
\(400\) 400.000 0.0500000
\(401\) −5838.00 −0.727022 −0.363511 0.931590i \(-0.618422\pi\)
−0.363511 + 0.931590i \(0.618422\pi\)
\(402\) 16384.0 2.03274
\(403\) −8816.00 −1.08972
\(404\) −1032.00 −0.127089
\(405\) −1795.00 −0.220233
\(406\) 720.000 0.0880123
\(407\) −408.000 −0.0496899
\(408\) −4224.00 −0.512547
\(409\) 9530.00 1.15215 0.576074 0.817398i \(-0.304584\pi\)
0.576074 + 0.817398i \(0.304584\pi\)
\(410\) −4380.00 −0.527592
\(411\) 10032.0 1.20400
\(412\) −3952.00 −0.472575
\(413\) −1680.00 −0.200163
\(414\) 9768.00 1.15959
\(415\) 360.000 0.0425824
\(416\) −1856.00 −0.218745
\(417\) 22880.0 2.68690
\(418\) −2400.00 −0.280832
\(419\) 7260.00 0.846478 0.423239 0.906018i \(-0.360893\pi\)
0.423239 + 0.906018i \(0.360893\pi\)
\(420\) 640.000 0.0743543
\(421\) 12062.0 1.39636 0.698178 0.715924i \(-0.253994\pi\)
0.698178 + 0.715924i \(0.253994\pi\)
\(422\) −5336.00 −0.615527
\(423\) −7548.00 −0.867604
\(424\) 1776.00 0.203420
\(425\) 1650.00 0.188322
\(426\) −6912.00 −0.786121
\(427\) −3608.00 −0.408907
\(428\) −96.0000 −0.0108419
\(429\) 5568.00 0.626633
\(430\) 320.000 0.0358878
\(431\) −13608.0 −1.52082 −0.760411 0.649442i \(-0.775002\pi\)
−0.760411 + 0.649442i \(0.775002\pi\)
\(432\) −1280.00 −0.142556
\(433\) −3838.00 −0.425964 −0.212982 0.977056i \(-0.568318\pi\)
−0.212982 + 0.977056i \(0.568318\pi\)
\(434\) −1216.00 −0.134493
\(435\) 3600.00 0.396797
\(436\) 3800.00 0.417401
\(437\) −13200.0 −1.44495
\(438\) −5792.00 −0.631855
\(439\) 7400.00 0.804516 0.402258 0.915526i \(-0.368225\pi\)
0.402258 + 0.915526i \(0.368225\pi\)
\(440\) 480.000 0.0520071
\(441\) −12099.0 −1.30645
\(442\) −7656.00 −0.823889
\(443\) 8352.00 0.895746 0.447873 0.894097i \(-0.352182\pi\)
0.447873 + 0.894097i \(0.352182\pi\)
\(444\) 1088.00 0.116293
\(445\) 4050.00 0.431435
\(446\) 3544.00 0.376263
\(447\) −6000.00 −0.634878
\(448\) −256.000 −0.0269975
\(449\) 10770.0 1.13200 0.566000 0.824405i \(-0.308490\pi\)
0.566000 + 0.824405i \(0.308490\pi\)
\(450\) 1850.00 0.193800
\(451\) −5256.00 −0.548770
\(452\) −4152.00 −0.432066
\(453\) 3584.00 0.371724
\(454\) −5568.00 −0.575593
\(455\) 1160.00 0.119520
\(456\) 6400.00 0.657253
\(457\) −6694.00 −0.685191 −0.342595 0.939483i \(-0.611306\pi\)
−0.342595 + 0.939483i \(0.611306\pi\)
\(458\) 700.000 0.0714167
\(459\) −5280.00 −0.536927
\(460\) 2640.00 0.267588
\(461\) −3018.00 −0.304907 −0.152454 0.988311i \(-0.548717\pi\)
−0.152454 + 0.988311i \(0.548717\pi\)
\(462\) 768.000 0.0773389
\(463\) 14492.0 1.45464 0.727322 0.686296i \(-0.240765\pi\)
0.727322 + 0.686296i \(0.240765\pi\)
\(464\) −1440.00 −0.144074
\(465\) −6080.00 −0.606351
\(466\) 3924.00 0.390077
\(467\) 7776.00 0.770515 0.385257 0.922809i \(-0.374113\pi\)
0.385257 + 0.922809i \(0.374113\pi\)
\(468\) −8584.00 −0.847854
\(469\) 4096.00 0.403274
\(470\) −2040.00 −0.200209
\(471\) −17968.0 −1.75780
\(472\) 3360.00 0.327662
\(473\) 384.000 0.0373284
\(474\) 2560.00 0.248069
\(475\) −2500.00 −0.241490
\(476\) −1056.00 −0.101684
\(477\) 8214.00 0.788455
\(478\) −8640.00 −0.826746
\(479\) −13680.0 −1.30492 −0.652458 0.757825i \(-0.726262\pi\)
−0.652458 + 0.757825i \(0.726262\pi\)
\(480\) −1280.00 −0.121716
\(481\) 1972.00 0.186934
\(482\) −956.000 −0.0903415
\(483\) 4224.00 0.397927
\(484\) −4748.00 −0.445905
\(485\) 5530.00 0.517741
\(486\) 10064.0 0.939326
\(487\) 7916.00 0.736567 0.368284 0.929714i \(-0.379946\pi\)
0.368284 + 0.929714i \(0.379946\pi\)
\(488\) 7216.00 0.669371
\(489\) 4544.00 0.420218
\(490\) −3270.00 −0.301477
\(491\) 13932.0 1.28053 0.640267 0.768152i \(-0.278824\pi\)
0.640267 + 0.768152i \(0.278824\pi\)
\(492\) 14016.0 1.28433
\(493\) −5940.00 −0.542645
\(494\) 11600.0 1.05650
\(495\) 2220.00 0.201579
\(496\) 2432.00 0.220161
\(497\) −1728.00 −0.155959
\(498\) −1152.00 −0.103659
\(499\) −8260.00 −0.741019 −0.370509 0.928829i \(-0.620817\pi\)
−0.370509 + 0.928829i \(0.620817\pi\)
\(500\) 500.000 0.0447214
\(501\) 12192.0 1.08722
\(502\) 5304.00 0.471572
\(503\) −11148.0 −0.988200 −0.494100 0.869405i \(-0.664502\pi\)
−0.494100 + 0.869405i \(0.664502\pi\)
\(504\) −1184.00 −0.104642
\(505\) −1290.00 −0.113672
\(506\) 3168.00 0.278330
\(507\) −9336.00 −0.817803
\(508\) −496.000 −0.0433198
\(509\) −9690.00 −0.843815 −0.421907 0.906639i \(-0.638639\pi\)
−0.421907 + 0.906639i \(0.638639\pi\)
\(510\) −5280.00 −0.458436
\(511\) −1448.00 −0.125354
\(512\) 512.000 0.0441942
\(513\) 8000.00 0.688516
\(514\) −4668.00 −0.400577
\(515\) −4940.00 −0.422684
\(516\) −1024.00 −0.0873626
\(517\) −2448.00 −0.208245
\(518\) 272.000 0.0230714
\(519\) −29616.0 −2.50481
\(520\) −2320.00 −0.195651
\(521\) −16038.0 −1.34863 −0.674316 0.738443i \(-0.735562\pi\)
−0.674316 + 0.738443i \(0.735562\pi\)
\(522\) −6660.00 −0.558430
\(523\) 992.000 0.0829391 0.0414695 0.999140i \(-0.486796\pi\)
0.0414695 + 0.999140i \(0.486796\pi\)
\(524\) 528.000 0.0440187
\(525\) 800.000 0.0665045
\(526\) −7896.00 −0.654528
\(527\) 10032.0 0.829223
\(528\) −1536.00 −0.126602
\(529\) 5257.00 0.432070
\(530\) 2220.00 0.181945
\(531\) 15540.0 1.27002
\(532\) 1600.00 0.130392
\(533\) 25404.0 2.06448
\(534\) −12960.0 −1.05025
\(535\) −120.000 −0.00969729
\(536\) −8192.00 −0.660150
\(537\) −25440.0 −2.04435
\(538\) 3180.00 0.254832
\(539\) −3924.00 −0.313578
\(540\) −1600.00 −0.127506
\(541\) 7142.00 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(542\) 9904.00 0.784895
\(543\) 16784.0 1.32646
\(544\) 2112.00 0.166455
\(545\) 4750.00 0.373335
\(546\) −3712.00 −0.290950
\(547\) 7616.00 0.595314 0.297657 0.954673i \(-0.403795\pi\)
0.297657 + 0.954673i \(0.403795\pi\)
\(548\) −5016.00 −0.391009
\(549\) 33374.0 2.59448
\(550\) 600.000 0.0465165
\(551\) 9000.00 0.695849
\(552\) −8448.00 −0.651396
\(553\) 640.000 0.0492144
\(554\) 3292.00 0.252462
\(555\) 1360.00 0.104016
\(556\) −11440.0 −0.872597
\(557\) −10314.0 −0.784593 −0.392296 0.919839i \(-0.628319\pi\)
−0.392296 + 0.919839i \(0.628319\pi\)
\(558\) 11248.0 0.853344
\(559\) −1856.00 −0.140430
\(560\) −320.000 −0.0241473
\(561\) −6336.00 −0.476838
\(562\) −2316.00 −0.173834
\(563\) −7128.00 −0.533587 −0.266793 0.963754i \(-0.585964\pi\)
−0.266793 + 0.963754i \(0.585964\pi\)
\(564\) 6528.00 0.487373
\(565\) −5190.00 −0.386451
\(566\) 13984.0 1.03850
\(567\) 1436.00 0.106360
\(568\) 3456.00 0.255300
\(569\) 2010.00 0.148091 0.0740453 0.997255i \(-0.476409\pi\)
0.0740453 + 0.997255i \(0.476409\pi\)
\(570\) 8000.00 0.587865
\(571\) −23188.0 −1.69945 −0.849726 0.527224i \(-0.823233\pi\)
−0.849726 + 0.527224i \(0.823233\pi\)
\(572\) −2784.00 −0.203505
\(573\) −35136.0 −2.56165
\(574\) 3504.00 0.254798
\(575\) 3300.00 0.239338
\(576\) 2368.00 0.171296
\(577\) 22466.0 1.62092 0.810461 0.585793i \(-0.199217\pi\)
0.810461 + 0.585793i \(0.199217\pi\)
\(578\) −1114.00 −0.0801666
\(579\) 17264.0 1.23915
\(580\) −1800.00 −0.128864
\(581\) −288.000 −0.0205650
\(582\) −17696.0 −1.26035
\(583\) 2664.00 0.189248
\(584\) 2896.00 0.205201
\(585\) −10730.0 −0.758343
\(586\) −516.000 −0.0363750
\(587\) 22776.0 1.60148 0.800738 0.599015i \(-0.204441\pi\)
0.800738 + 0.599015i \(0.204441\pi\)
\(588\) 10464.0 0.733891
\(589\) −15200.0 −1.06334
\(590\) 4200.00 0.293070
\(591\) 8592.00 0.598016
\(592\) −544.000 −0.0377673
\(593\) −21198.0 −1.46796 −0.733978 0.679174i \(-0.762338\pi\)
−0.733978 + 0.679174i \(0.762338\pi\)
\(594\) −1920.00 −0.132624
\(595\) −1320.00 −0.0909491
\(596\) 3000.00 0.206183
\(597\) −22720.0 −1.55757
\(598\) −15312.0 −1.04708
\(599\) 15960.0 1.08866 0.544330 0.838871i \(-0.316784\pi\)
0.544330 + 0.838871i \(0.316784\pi\)
\(600\) −1600.00 −0.108866
\(601\) 5882.00 0.399221 0.199610 0.979875i \(-0.436032\pi\)
0.199610 + 0.979875i \(0.436032\pi\)
\(602\) −256.000 −0.0173319
\(603\) −37888.0 −2.55874
\(604\) −1792.00 −0.120721
\(605\) −5935.00 −0.398830
\(606\) 4128.00 0.276714
\(607\) 8516.00 0.569446 0.284723 0.958610i \(-0.408098\pi\)
0.284723 + 0.958610i \(0.408098\pi\)
\(608\) −3200.00 −0.213449
\(609\) −2880.00 −0.191631
\(610\) 9020.00 0.598703
\(611\) 11832.0 0.783423
\(612\) 9768.00 0.645176
\(613\) 8462.00 0.557548 0.278774 0.960357i \(-0.410072\pi\)
0.278774 + 0.960357i \(0.410072\pi\)
\(614\) −17888.0 −1.17573
\(615\) 17520.0 1.14874
\(616\) −384.000 −0.0251166
\(617\) −11094.0 −0.723870 −0.361935 0.932203i \(-0.617884\pi\)
−0.361935 + 0.932203i \(0.617884\pi\)
\(618\) 15808.0 1.02895
\(619\) 2180.00 0.141553 0.0707767 0.997492i \(-0.477452\pi\)
0.0707767 + 0.997492i \(0.477452\pi\)
\(620\) 3040.00 0.196918
\(621\) −10560.0 −0.682380
\(622\) 2784.00 0.179467
\(623\) −3240.00 −0.208359
\(624\) 7424.00 0.476279
\(625\) 625.000 0.0400000
\(626\) −11756.0 −0.750582
\(627\) 9600.00 0.611463
\(628\) 8984.00 0.570861
\(629\) −2244.00 −0.142248
\(630\) −1480.00 −0.0935946
\(631\) −26848.0 −1.69382 −0.846911 0.531734i \(-0.821541\pi\)
−0.846911 + 0.531734i \(0.821541\pi\)
\(632\) −1280.00 −0.0805628
\(633\) 21344.0 1.34020
\(634\) 20652.0 1.29368
\(635\) −620.000 −0.0387464
\(636\) −7104.00 −0.442912
\(637\) 18966.0 1.17969
\(638\) −2160.00 −0.134036
\(639\) 15984.0 0.989542
\(640\) 640.000 0.0395285
\(641\) 26322.0 1.62193 0.810965 0.585095i \(-0.198943\pi\)
0.810965 + 0.585095i \(0.198943\pi\)
\(642\) 384.000 0.0236063
\(643\) −10168.0 −0.623619 −0.311809 0.950145i \(-0.600935\pi\)
−0.311809 + 0.950145i \(0.600935\pi\)
\(644\) −2112.00 −0.129231
\(645\) −1280.00 −0.0781395
\(646\) −13200.0 −0.803943
\(647\) −23604.0 −1.43426 −0.717132 0.696937i \(-0.754546\pi\)
−0.717132 + 0.696937i \(0.754546\pi\)
\(648\) −2872.00 −0.174109
\(649\) 5040.00 0.304834
\(650\) −2900.00 −0.174996
\(651\) 4864.00 0.292834
\(652\) −2272.00 −0.136470
\(653\) 16422.0 0.984139 0.492069 0.870556i \(-0.336241\pi\)
0.492069 + 0.870556i \(0.336241\pi\)
\(654\) −15200.0 −0.908818
\(655\) 660.000 0.0393715
\(656\) −7008.00 −0.417098
\(657\) 13394.0 0.795357
\(658\) 1632.00 0.0966899
\(659\) −26100.0 −1.54281 −0.771405 0.636345i \(-0.780446\pi\)
−0.771405 + 0.636345i \(0.780446\pi\)
\(660\) −1920.00 −0.113236
\(661\) −3058.00 −0.179943 −0.0899716 0.995944i \(-0.528678\pi\)
−0.0899716 + 0.995944i \(0.528678\pi\)
\(662\) −8456.00 −0.496453
\(663\) 30624.0 1.79387
\(664\) 576.000 0.0336644
\(665\) 2000.00 0.116627
\(666\) −2516.00 −0.146386
\(667\) −11880.0 −0.689648
\(668\) −6096.00 −0.353086
\(669\) −14176.0 −0.819246
\(670\) −10240.0 −0.590456
\(671\) 10824.0 0.622736
\(672\) 1024.00 0.0587822
\(673\) 10802.0 0.618702 0.309351 0.950948i \(-0.399888\pi\)
0.309351 + 0.950948i \(0.399888\pi\)
\(674\) 2212.00 0.126414
\(675\) −2000.00 −0.114044
\(676\) 4668.00 0.265589
\(677\) −10674.0 −0.605960 −0.302980 0.952997i \(-0.597982\pi\)
−0.302980 + 0.952997i \(0.597982\pi\)
\(678\) 16608.0 0.940747
\(679\) −4424.00 −0.250041
\(680\) 2640.00 0.148881
\(681\) 22272.0 1.25325
\(682\) 3648.00 0.204823
\(683\) −28608.0 −1.60272 −0.801358 0.598185i \(-0.795889\pi\)
−0.801358 + 0.598185i \(0.795889\pi\)
\(684\) −14800.0 −0.827328
\(685\) −6270.00 −0.349729
\(686\) 5360.00 0.298317
\(687\) −2800.00 −0.155497
\(688\) 512.000 0.0283718
\(689\) −12876.0 −0.711954
\(690\) −10560.0 −0.582627
\(691\) −2428.00 −0.133669 −0.0668346 0.997764i \(-0.521290\pi\)
−0.0668346 + 0.997764i \(0.521290\pi\)
\(692\) 14808.0 0.813462
\(693\) −1776.00 −0.0973516
\(694\) 18672.0 1.02130
\(695\) −14300.0 −0.780475
\(696\) 5760.00 0.313696
\(697\) −28908.0 −1.57097
\(698\) −23540.0 −1.27651
\(699\) −15696.0 −0.849324
\(700\) −400.000 −0.0215980
\(701\) −6618.00 −0.356574 −0.178287 0.983979i \(-0.557056\pi\)
−0.178287 + 0.983979i \(0.557056\pi\)
\(702\) 9280.00 0.498933
\(703\) 3400.00 0.182409
\(704\) 768.000 0.0411152
\(705\) 8160.00 0.435920
\(706\) 16644.0 0.887259
\(707\) 1032.00 0.0548972
\(708\) −13440.0 −0.713427
\(709\) 20510.0 1.08642 0.543208 0.839598i \(-0.317209\pi\)
0.543208 + 0.839598i \(0.317209\pi\)
\(710\) 4320.00 0.228347
\(711\) −5920.00 −0.312261
\(712\) 6480.00 0.341079
\(713\) 20064.0 1.05386
\(714\) 4224.00 0.221399
\(715\) −3480.00 −0.182020
\(716\) 12720.0 0.663923
\(717\) 34560.0 1.80009
\(718\) 21360.0 1.11023
\(719\) 31680.0 1.64321 0.821603 0.570061i \(-0.193080\pi\)
0.821603 + 0.570061i \(0.193080\pi\)
\(720\) 2960.00 0.153212
\(721\) 3952.00 0.204133
\(722\) 6282.00 0.323811
\(723\) 3824.00 0.196703
\(724\) −8392.00 −0.430782
\(725\) −2250.00 −0.115259
\(726\) 18992.0 0.970880
\(727\) 13196.0 0.673195 0.336597 0.941649i \(-0.390724\pi\)
0.336597 + 0.941649i \(0.390724\pi\)
\(728\) 1856.00 0.0944889
\(729\) −30563.0 −1.55276
\(730\) 3620.00 0.183537
\(731\) 2112.00 0.106861
\(732\) −28864.0 −1.45744
\(733\) 8102.00 0.408259 0.204130 0.978944i \(-0.434564\pi\)
0.204130 + 0.978944i \(0.434564\pi\)
\(734\) −11768.0 −0.591778
\(735\) 13080.0 0.656412
\(736\) 4224.00 0.211547
\(737\) −12288.0 −0.614158
\(738\) −32412.0 −1.61667
\(739\) −12580.0 −0.626201 −0.313101 0.949720i \(-0.601368\pi\)
−0.313101 + 0.949720i \(0.601368\pi\)
\(740\) −680.000 −0.0337801
\(741\) −46400.0 −2.30033
\(742\) −1776.00 −0.0878693
\(743\) 29892.0 1.47595 0.737975 0.674828i \(-0.235782\pi\)
0.737975 + 0.674828i \(0.235782\pi\)
\(744\) −9728.00 −0.479363
\(745\) 3750.00 0.184415
\(746\) −4196.00 −0.205934
\(747\) 2664.00 0.130483
\(748\) 3168.00 0.154858
\(749\) 96.0000 0.00468326
\(750\) −2000.00 −0.0973729
\(751\) −40408.0 −1.96339 −0.981697 0.190450i \(-0.939005\pi\)
−0.981697 + 0.190450i \(0.939005\pi\)
\(752\) −3264.00 −0.158279
\(753\) −21216.0 −1.02676
\(754\) 10440.0 0.504247
\(755\) −2240.00 −0.107976
\(756\) 1280.00 0.0615782
\(757\) 32366.0 1.55398 0.776990 0.629513i \(-0.216746\pi\)
0.776990 + 0.629513i \(0.216746\pi\)
\(758\) 7720.00 0.369925
\(759\) −12672.0 −0.606014
\(760\) −4000.00 −0.190915
\(761\) −17238.0 −0.821126 −0.410563 0.911832i \(-0.634668\pi\)
−0.410563 + 0.911832i \(0.634668\pi\)
\(762\) 1984.00 0.0943212
\(763\) −3800.00 −0.180300
\(764\) 17568.0 0.831921
\(765\) 12210.0 0.577063
\(766\) −19176.0 −0.904513
\(767\) −24360.0 −1.14679
\(768\) −2048.00 −0.0962250
\(769\) 10850.0 0.508792 0.254396 0.967100i \(-0.418123\pi\)
0.254396 + 0.967100i \(0.418123\pi\)
\(770\) −480.000 −0.0224649
\(771\) 18672.0 0.872186
\(772\) −8632.00 −0.402425
\(773\) 9102.00 0.423514 0.211757 0.977322i \(-0.432081\pi\)
0.211757 + 0.977322i \(0.432081\pi\)
\(774\) 2368.00 0.109969
\(775\) 3800.00 0.176129
\(776\) 8848.00 0.409310
\(777\) −1088.00 −0.0502340
\(778\) −26820.0 −1.23592
\(779\) 43800.0 2.01450
\(780\) 9280.00 0.425997
\(781\) 5184.00 0.237514
\(782\) 17424.0 0.796779
\(783\) 7200.00 0.328617
\(784\) −5232.00 −0.238338
\(785\) 11230.0 0.510593
\(786\) −2112.00 −0.0958429
\(787\) −25504.0 −1.15517 −0.577585 0.816330i \(-0.696005\pi\)
−0.577585 + 0.816330i \(0.696005\pi\)
\(788\) −4296.00 −0.194212
\(789\) 31584.0 1.42512
\(790\) −1600.00 −0.0720575
\(791\) 4152.00 0.186635
\(792\) 3552.00 0.159362
\(793\) −52316.0 −2.34274
\(794\) −26228.0 −1.17229
\(795\) −8880.00 −0.396152
\(796\) 11360.0 0.505835
\(797\) 14166.0 0.629593 0.314796 0.949159i \(-0.398064\pi\)
0.314796 + 0.949159i \(0.398064\pi\)
\(798\) −6400.00 −0.283907
\(799\) −13464.0 −0.596148
\(800\) 800.000 0.0353553
\(801\) 29970.0 1.32202
\(802\) −11676.0 −0.514082
\(803\) 4344.00 0.190905
\(804\) 32768.0 1.43736
\(805\) −2640.00 −0.115587
\(806\) −17632.0 −0.770547
\(807\) −12720.0 −0.554852
\(808\) −2064.00 −0.0898654
\(809\) 33210.0 1.44327 0.721633 0.692276i \(-0.243392\pi\)
0.721633 + 0.692276i \(0.243392\pi\)
\(810\) −3590.00 −0.155728
\(811\) 39212.0 1.69780 0.848902 0.528550i \(-0.177264\pi\)
0.848902 + 0.528550i \(0.177264\pi\)
\(812\) 1440.00 0.0622341
\(813\) −39616.0 −1.70897
\(814\) −816.000 −0.0351361
\(815\) −2840.00 −0.122062
\(816\) −8448.00 −0.362425
\(817\) −3200.00 −0.137030
\(818\) 19060.0 0.814691
\(819\) 8584.00 0.366238
\(820\) −8760.00 −0.373064
\(821\) 6222.00 0.264494 0.132247 0.991217i \(-0.457781\pi\)
0.132247 + 0.991217i \(0.457781\pi\)
\(822\) 20064.0 0.851353
\(823\) 31172.0 1.32028 0.660138 0.751144i \(-0.270498\pi\)
0.660138 + 0.751144i \(0.270498\pi\)
\(824\) −7904.00 −0.334161
\(825\) −2400.00 −0.101282
\(826\) −3360.00 −0.141537
\(827\) −264.000 −0.0111006 −0.00555029 0.999985i \(-0.501767\pi\)
−0.00555029 + 0.999985i \(0.501767\pi\)
\(828\) 19536.0 0.819955
\(829\) −29050.0 −1.21707 −0.608533 0.793528i \(-0.708242\pi\)
−0.608533 + 0.793528i \(0.708242\pi\)
\(830\) 720.000 0.0301103
\(831\) −13168.0 −0.549691
\(832\) −3712.00 −0.154676
\(833\) −21582.0 −0.897685
\(834\) 45760.0 1.89993
\(835\) −7620.00 −0.315810
\(836\) −4800.00 −0.198578
\(837\) −12160.0 −0.502164
\(838\) 14520.0 0.598550
\(839\) −21720.0 −0.893752 −0.446876 0.894596i \(-0.647463\pi\)
−0.446876 + 0.894596i \(0.647463\pi\)
\(840\) 1280.00 0.0525764
\(841\) −16289.0 −0.667883
\(842\) 24124.0 0.987373
\(843\) 9264.00 0.378492
\(844\) −10672.0 −0.435243
\(845\) 5835.00 0.237550
\(846\) −15096.0 −0.613488
\(847\) 4748.00 0.192613
\(848\) 3552.00 0.143840
\(849\) −55936.0 −2.26115
\(850\) 3300.00 0.133164
\(851\) −4488.00 −0.180783
\(852\) −13824.0 −0.555871
\(853\) −6658.00 −0.267252 −0.133626 0.991032i \(-0.542662\pi\)
−0.133626 + 0.991032i \(0.542662\pi\)
\(854\) −7216.00 −0.289141
\(855\) −18500.0 −0.739984
\(856\) −192.000 −0.00766638
\(857\) −13974.0 −0.556993 −0.278496 0.960437i \(-0.589836\pi\)
−0.278496 + 0.960437i \(0.589836\pi\)
\(858\) 11136.0 0.443096
\(859\) 23780.0 0.944544 0.472272 0.881453i \(-0.343434\pi\)
0.472272 + 0.881453i \(0.343434\pi\)
\(860\) 640.000 0.0253765
\(861\) −14016.0 −0.554778
\(862\) −27216.0 −1.07538
\(863\) −12228.0 −0.482324 −0.241162 0.970485i \(-0.577529\pi\)
−0.241162 + 0.970485i \(0.577529\pi\)
\(864\) −2560.00 −0.100802
\(865\) 18510.0 0.727583
\(866\) −7676.00 −0.301202
\(867\) 4456.00 0.174549
\(868\) −2432.00 −0.0951008
\(869\) −1920.00 −0.0749500
\(870\) 7200.00 0.280578
\(871\) 59392.0 2.31047
\(872\) 7600.00 0.295147
\(873\) 40922.0 1.58648
\(874\) −26400.0 −1.02173
\(875\) −500.000 −0.0193178
\(876\) −11584.0 −0.446789
\(877\) 11606.0 0.446872 0.223436 0.974719i \(-0.428273\pi\)
0.223436 + 0.974719i \(0.428273\pi\)
\(878\) 14800.0 0.568879
\(879\) 2064.00 0.0792002
\(880\) 960.000 0.0367745
\(881\) −32958.0 −1.26037 −0.630183 0.776446i \(-0.717020\pi\)
−0.630183 + 0.776446i \(0.717020\pi\)
\(882\) −24198.0 −0.923797
\(883\) 8072.00 0.307638 0.153819 0.988099i \(-0.450843\pi\)
0.153819 + 0.988099i \(0.450843\pi\)
\(884\) −15312.0 −0.582577
\(885\) −16800.0 −0.638108
\(886\) 16704.0 0.633388
\(887\) 15756.0 0.596431 0.298216 0.954498i \(-0.403609\pi\)
0.298216 + 0.954498i \(0.403609\pi\)
\(888\) 2176.00 0.0822317
\(889\) 496.000 0.0187124
\(890\) 8100.00 0.305070
\(891\) −4308.00 −0.161979
\(892\) 7088.00 0.266058
\(893\) 20400.0 0.764457
\(894\) −12000.0 −0.448926
\(895\) 15900.0 0.593831
\(896\) −512.000 −0.0190901
\(897\) 61248.0 2.27983
\(898\) 21540.0 0.800444
\(899\) −13680.0 −0.507512
\(900\) 3700.00 0.137037
\(901\) 14652.0 0.541763
\(902\) −10512.0 −0.388039
\(903\) 1024.00 0.0377371
\(904\) −8304.00 −0.305517
\(905\) −10490.0 −0.385303
\(906\) 7168.00 0.262849
\(907\) 18776.0 0.687372 0.343686 0.939085i \(-0.388324\pi\)
0.343686 + 0.939085i \(0.388324\pi\)
\(908\) −11136.0 −0.407006
\(909\) −9546.00 −0.348318
\(910\) 2320.00 0.0845135
\(911\) −20568.0 −0.748022 −0.374011 0.927424i \(-0.622018\pi\)
−0.374011 + 0.927424i \(0.622018\pi\)
\(912\) 12800.0 0.464748
\(913\) 864.000 0.0313190
\(914\) −13388.0 −0.484503
\(915\) −36080.0 −1.30357
\(916\) 1400.00 0.0504992
\(917\) −528.000 −0.0190143
\(918\) −10560.0 −0.379664
\(919\) −6280.00 −0.225417 −0.112708 0.993628i \(-0.535953\pi\)
−0.112708 + 0.993628i \(0.535953\pi\)
\(920\) 5280.00 0.189214
\(921\) 71552.0 2.55996
\(922\) −6036.00 −0.215602
\(923\) −25056.0 −0.893530
\(924\) 1536.00 0.0546869
\(925\) −850.000 −0.0302139
\(926\) 28984.0 1.02859
\(927\) −36556.0 −1.29521
\(928\) −2880.00 −0.101876
\(929\) −20430.0 −0.721514 −0.360757 0.932660i \(-0.617482\pi\)
−0.360757 + 0.932660i \(0.617482\pi\)
\(930\) −12160.0 −0.428755
\(931\) 32700.0 1.15113
\(932\) 7848.00 0.275826
\(933\) −11136.0 −0.390757
\(934\) 15552.0 0.544836
\(935\) 3960.00 0.138509
\(936\) −17168.0 −0.599523
\(937\) 8906.00 0.310508 0.155254 0.987875i \(-0.450380\pi\)
0.155254 + 0.987875i \(0.450380\pi\)
\(938\) 8192.00 0.285158
\(939\) 47024.0 1.63426
\(940\) −4080.00 −0.141569
\(941\) −17418.0 −0.603412 −0.301706 0.953401i \(-0.597556\pi\)
−0.301706 + 0.953401i \(0.597556\pi\)
\(942\) −35936.0 −1.24295
\(943\) −57816.0 −1.99655
\(944\) 6720.00 0.231692
\(945\) 1600.00 0.0550773
\(946\) 768.000 0.0263952
\(947\) −2544.00 −0.0872956 −0.0436478 0.999047i \(-0.513898\pi\)
−0.0436478 + 0.999047i \(0.513898\pi\)
\(948\) 5120.00 0.175411
\(949\) −20996.0 −0.718187
\(950\) −5000.00 −0.170759
\(951\) −82608.0 −2.81677
\(952\) −2112.00 −0.0719016
\(953\) 15402.0 0.523525 0.261763 0.965132i \(-0.415696\pi\)
0.261763 + 0.965132i \(0.415696\pi\)
\(954\) 16428.0 0.557522
\(955\) 21960.0 0.744093
\(956\) −17280.0 −0.584597
\(957\) 8640.00 0.291841
\(958\) −27360.0 −0.922716
\(959\) 5016.00 0.168900
\(960\) −2560.00 −0.0860663
\(961\) −6687.00 −0.224464
\(962\) 3944.00 0.132183
\(963\) −888.000 −0.0297148
\(964\) −1912.00 −0.0638811
\(965\) −10790.0 −0.359940
\(966\) 8448.00 0.281377
\(967\) −49444.0 −1.64427 −0.822136 0.569291i \(-0.807218\pi\)
−0.822136 + 0.569291i \(0.807218\pi\)
\(968\) −9496.00 −0.315303
\(969\) 52800.0 1.75044
\(970\) 11060.0 0.366098
\(971\) −25188.0 −0.832463 −0.416231 0.909259i \(-0.636649\pi\)
−0.416231 + 0.909259i \(0.636649\pi\)
\(972\) 20128.0 0.664204
\(973\) 11440.0 0.376927
\(974\) 15832.0 0.520832
\(975\) 11600.0 0.381023
\(976\) 14432.0 0.473317
\(977\) 2946.00 0.0964697 0.0482348 0.998836i \(-0.484640\pi\)
0.0482348 + 0.998836i \(0.484640\pi\)
\(978\) 9088.00 0.297139
\(979\) 9720.00 0.317316
\(980\) −6540.00 −0.213176
\(981\) 35150.0 1.14399
\(982\) 27864.0 0.905475
\(983\) 15012.0 0.487089 0.243544 0.969890i \(-0.421690\pi\)
0.243544 + 0.969890i \(0.421690\pi\)
\(984\) 28032.0 0.908158
\(985\) −5370.00 −0.173708
\(986\) −11880.0 −0.383708
\(987\) −6528.00 −0.210525
\(988\) 23200.0 0.747055
\(989\) 4224.00 0.135809
\(990\) 4440.00 0.142538
\(991\) −5128.00 −0.164376 −0.0821878 0.996617i \(-0.526191\pi\)
−0.0821878 + 0.996617i \(0.526191\pi\)
\(992\) 4864.00 0.155678
\(993\) 33824.0 1.08094
\(994\) −3456.00 −0.110279
\(995\) 14200.0 0.452432
\(996\) −2304.00 −0.0732982
\(997\) −49714.0 −1.57920 −0.789598 0.613625i \(-0.789711\pi\)
−0.789598 + 0.613625i \(0.789711\pi\)
\(998\) −16520.0 −0.523979
\(999\) 2720.00 0.0861431
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 10.4.a.a.1.1 1
3.2 odd 2 90.4.a.a.1.1 1
4.3 odd 2 80.4.a.f.1.1 1
5.2 odd 4 50.4.b.a.49.2 2
5.3 odd 4 50.4.b.a.49.1 2
5.4 even 2 50.4.a.c.1.1 1
7.2 even 3 490.4.e.i.361.1 2
7.3 odd 6 490.4.e.a.471.1 2
7.4 even 3 490.4.e.i.471.1 2
7.5 odd 6 490.4.e.a.361.1 2
7.6 odd 2 490.4.a.o.1.1 1
8.3 odd 2 320.4.a.b.1.1 1
8.5 even 2 320.4.a.m.1.1 1
9.2 odd 6 810.4.e.w.271.1 2
9.4 even 3 810.4.e.c.541.1 2
9.5 odd 6 810.4.e.w.541.1 2
9.7 even 3 810.4.e.c.271.1 2
11.10 odd 2 1210.4.a.b.1.1 1
12.11 even 2 720.4.a.j.1.1 1
13.12 even 2 1690.4.a.a.1.1 1
15.2 even 4 450.4.c.d.199.1 2
15.8 even 4 450.4.c.d.199.2 2
15.14 odd 2 450.4.a.q.1.1 1
16.3 odd 4 1280.4.d.g.641.2 2
16.5 even 4 1280.4.d.j.641.2 2
16.11 odd 4 1280.4.d.g.641.1 2
16.13 even 4 1280.4.d.j.641.1 2
20.3 even 4 400.4.c.c.49.2 2
20.7 even 4 400.4.c.c.49.1 2
20.19 odd 2 400.4.a.b.1.1 1
35.34 odd 2 2450.4.a.b.1.1 1
40.19 odd 2 1600.4.a.bx.1.1 1
40.29 even 2 1600.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 1.1 even 1 trivial
50.4.a.c.1.1 1 5.4 even 2
50.4.b.a.49.1 2 5.3 odd 4
50.4.b.a.49.2 2 5.2 odd 4
80.4.a.f.1.1 1 4.3 odd 2
90.4.a.a.1.1 1 3.2 odd 2
320.4.a.b.1.1 1 8.3 odd 2
320.4.a.m.1.1 1 8.5 even 2
400.4.a.b.1.1 1 20.19 odd 2
400.4.c.c.49.1 2 20.7 even 4
400.4.c.c.49.2 2 20.3 even 4
450.4.a.q.1.1 1 15.14 odd 2
450.4.c.d.199.1 2 15.2 even 4
450.4.c.d.199.2 2 15.8 even 4
490.4.a.o.1.1 1 7.6 odd 2
490.4.e.a.361.1 2 7.5 odd 6
490.4.e.a.471.1 2 7.3 odd 6
490.4.e.i.361.1 2 7.2 even 3
490.4.e.i.471.1 2 7.4 even 3
720.4.a.j.1.1 1 12.11 even 2
810.4.e.c.271.1 2 9.7 even 3
810.4.e.c.541.1 2 9.4 even 3
810.4.e.w.271.1 2 9.2 odd 6
810.4.e.w.541.1 2 9.5 odd 6
1210.4.a.b.1.1 1 11.10 odd 2
1280.4.d.g.641.1 2 16.11 odd 4
1280.4.d.g.641.2 2 16.3 odd 4
1280.4.d.j.641.1 2 16.13 even 4
1280.4.d.j.641.2 2 16.5 even 4
1600.4.a.d.1.1 1 40.29 even 2
1600.4.a.bx.1.1 1 40.19 odd 2
1690.4.a.a.1.1 1 13.12 even 2
2450.4.a.b.1.1 1 35.34 odd 2