Properties

Label 1694.4.a.c.1.1
Level $1694$
Weight $4$
Character 1694.1
Self dual yes
Analytic conductor $99.949$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1694,4,Mod(1,1694)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1694, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1694.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(99.9492355497\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1694.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} +4.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} +4.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} -23.0000 q^{9} -36.0000 q^{10} -8.00000 q^{12} -56.0000 q^{13} +14.0000 q^{14} -36.0000 q^{15} +16.0000 q^{16} -36.0000 q^{17} +46.0000 q^{18} +28.0000 q^{19} +72.0000 q^{20} +14.0000 q^{21} +180.000 q^{23} +16.0000 q^{24} +199.000 q^{25} +112.000 q^{26} +100.000 q^{27} -28.0000 q^{28} +54.0000 q^{29} +72.0000 q^{30} -334.000 q^{31} -32.0000 q^{32} +72.0000 q^{34} -126.000 q^{35} -92.0000 q^{36} +386.000 q^{37} -56.0000 q^{38} +112.000 q^{39} -144.000 q^{40} +444.000 q^{41} -28.0000 q^{42} +316.000 q^{43} -414.000 q^{45} -360.000 q^{46} -402.000 q^{47} -32.0000 q^{48} +49.0000 q^{49} -398.000 q^{50} +72.0000 q^{51} -224.000 q^{52} -486.000 q^{53} -200.000 q^{54} +56.0000 q^{56} -56.0000 q^{57} -108.000 q^{58} -282.000 q^{59} -144.000 q^{60} -380.000 q^{61} +668.000 q^{62} +161.000 q^{63} +64.0000 q^{64} -1008.00 q^{65} +176.000 q^{67} -144.000 q^{68} -360.000 q^{69} +252.000 q^{70} -324.000 q^{71} +184.000 q^{72} -800.000 q^{73} -772.000 q^{74} -398.000 q^{75} +112.000 q^{76} -224.000 q^{78} +1144.00 q^{79} +288.000 q^{80} +421.000 q^{81} -888.000 q^{82} -468.000 q^{83} +56.0000 q^{84} -648.000 q^{85} -632.000 q^{86} -108.000 q^{87} -870.000 q^{89} +828.000 q^{90} +392.000 q^{91} +720.000 q^{92} +668.000 q^{93} +804.000 q^{94} +504.000 q^{95} +64.0000 q^{96} -1330.00 q^{97} -98.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 4.00000 0.500000
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 4.00000 0.272166
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) −23.0000 −0.851852
\(10\) −36.0000 −1.13842
\(11\) 0 0
\(12\) −8.00000 −0.192450
\(13\) −56.0000 −1.19474 −0.597369 0.801966i \(-0.703787\pi\)
−0.597369 + 0.801966i \(0.703787\pi\)
\(14\) 14.0000 0.267261
\(15\) −36.0000 −0.619677
\(16\) 16.0000 0.250000
\(17\) −36.0000 −0.513605 −0.256802 0.966464i \(-0.582669\pi\)
−0.256802 + 0.966464i \(0.582669\pi\)
\(18\) 46.0000 0.602350
\(19\) 28.0000 0.338086 0.169043 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 72.0000 0.804984
\(21\) 14.0000 0.145479
\(22\) 0 0
\(23\) 180.000 1.63185 0.815926 0.578156i \(-0.196228\pi\)
0.815926 + 0.578156i \(0.196228\pi\)
\(24\) 16.0000 0.136083
\(25\) 199.000 1.59200
\(26\) 112.000 0.844808
\(27\) 100.000 0.712778
\(28\) −28.0000 −0.188982
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 72.0000 0.438178
\(31\) −334.000 −1.93510 −0.967551 0.252675i \(-0.918690\pi\)
−0.967551 + 0.252675i \(0.918690\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 72.0000 0.363173
\(35\) −126.000 −0.608511
\(36\) −92.0000 −0.425926
\(37\) 386.000 1.71508 0.857541 0.514416i \(-0.171991\pi\)
0.857541 + 0.514416i \(0.171991\pi\)
\(38\) −56.0000 −0.239063
\(39\) 112.000 0.459855
\(40\) −144.000 −0.569210
\(41\) 444.000 1.69125 0.845624 0.533779i \(-0.179229\pi\)
0.845624 + 0.533779i \(0.179229\pi\)
\(42\) −28.0000 −0.102869
\(43\) 316.000 1.12069 0.560344 0.828260i \(-0.310669\pi\)
0.560344 + 0.828260i \(0.310669\pi\)
\(44\) 0 0
\(45\) −414.000 −1.37146
\(46\) −360.000 −1.15389
\(47\) −402.000 −1.24761 −0.623806 0.781580i \(-0.714414\pi\)
−0.623806 + 0.781580i \(0.714414\pi\)
\(48\) −32.0000 −0.0962250
\(49\) 49.0000 0.142857
\(50\) −398.000 −1.12571
\(51\) 72.0000 0.197687
\(52\) −224.000 −0.597369
\(53\) −486.000 −1.25957 −0.629785 0.776769i \(-0.716857\pi\)
−0.629785 + 0.776769i \(0.716857\pi\)
\(54\) −200.000 −0.504010
\(55\) 0 0
\(56\) 56.0000 0.133631
\(57\) −56.0000 −0.130129
\(58\) −108.000 −0.244502
\(59\) −282.000 −0.622259 −0.311129 0.950368i \(-0.600707\pi\)
−0.311129 + 0.950368i \(0.600707\pi\)
\(60\) −144.000 −0.309839
\(61\) −380.000 −0.797607 −0.398803 0.917036i \(-0.630574\pi\)
−0.398803 + 0.917036i \(0.630574\pi\)
\(62\) 668.000 1.36832
\(63\) 161.000 0.321970
\(64\) 64.0000 0.125000
\(65\) −1008.00 −1.92349
\(66\) 0 0
\(67\) 176.000 0.320923 0.160461 0.987042i \(-0.448702\pi\)
0.160461 + 0.987042i \(0.448702\pi\)
\(68\) −144.000 −0.256802
\(69\) −360.000 −0.628100
\(70\) 252.000 0.430282
\(71\) −324.000 −0.541574 −0.270787 0.962639i \(-0.587284\pi\)
−0.270787 + 0.962639i \(0.587284\pi\)
\(72\) 184.000 0.301175
\(73\) −800.000 −1.28264 −0.641321 0.767272i \(-0.721613\pi\)
−0.641321 + 0.767272i \(0.721613\pi\)
\(74\) −772.000 −1.21275
\(75\) −398.000 −0.612761
\(76\) 112.000 0.169043
\(77\) 0 0
\(78\) −224.000 −0.325167
\(79\) 1144.00 1.62924 0.814621 0.579994i \(-0.196945\pi\)
0.814621 + 0.579994i \(0.196945\pi\)
\(80\) 288.000 0.402492
\(81\) 421.000 0.577503
\(82\) −888.000 −1.19589
\(83\) −468.000 −0.618912 −0.309456 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(84\) 56.0000 0.0727393
\(85\) −648.000 −0.826888
\(86\) −632.000 −0.792445
\(87\) −108.000 −0.133090
\(88\) 0 0
\(89\) −870.000 −1.03618 −0.518089 0.855327i \(-0.673356\pi\)
−0.518089 + 0.855327i \(0.673356\pi\)
\(90\) 828.000 0.969765
\(91\) 392.000 0.451569
\(92\) 720.000 0.815926
\(93\) 668.000 0.744821
\(94\) 804.000 0.882194
\(95\) 504.000 0.544309
\(96\) 64.0000 0.0680414
\(97\) −1330.00 −1.39218 −0.696088 0.717957i \(-0.745078\pi\)
−0.696088 + 0.717957i \(0.745078\pi\)
\(98\) −98.0000 −0.101015
\(99\) 0 0
\(100\) 796.000 0.796000
\(101\) 120.000 0.118222 0.0591111 0.998251i \(-0.481173\pi\)
0.0591111 + 0.998251i \(0.481173\pi\)
\(102\) −144.000 −0.139786
\(103\) −1210.00 −1.15752 −0.578761 0.815497i \(-0.696464\pi\)
−0.578761 + 0.815497i \(0.696464\pi\)
\(104\) 448.000 0.422404
\(105\) 252.000 0.234216
\(106\) 972.000 0.890651
\(107\) −1236.00 −1.11672 −0.558358 0.829600i \(-0.688568\pi\)
−0.558358 + 0.829600i \(0.688568\pi\)
\(108\) 400.000 0.356389
\(109\) 694.000 0.609845 0.304923 0.952377i \(-0.401369\pi\)
0.304923 + 0.952377i \(0.401369\pi\)
\(110\) 0 0
\(111\) −772.000 −0.660135
\(112\) −112.000 −0.0944911
\(113\) 978.000 0.814181 0.407091 0.913388i \(-0.366543\pi\)
0.407091 + 0.913388i \(0.366543\pi\)
\(114\) 112.000 0.0920154
\(115\) 3240.00 2.62723
\(116\) 216.000 0.172889
\(117\) 1288.00 1.01774
\(118\) 564.000 0.440003
\(119\) 252.000 0.194124
\(120\) 288.000 0.219089
\(121\) 0 0
\(122\) 760.000 0.563993
\(123\) −888.000 −0.650961
\(124\) −1336.00 −0.967551
\(125\) 1332.00 0.953102
\(126\) −322.000 −0.227667
\(127\) 1216.00 0.849626 0.424813 0.905281i \(-0.360340\pi\)
0.424813 + 0.905281i \(0.360340\pi\)
\(128\) −128.000 −0.0883883
\(129\) −632.000 −0.431353
\(130\) 2016.00 1.36011
\(131\) −1680.00 −1.12048 −0.560238 0.828332i \(-0.689290\pi\)
−0.560238 + 0.828332i \(0.689290\pi\)
\(132\) 0 0
\(133\) −196.000 −0.127785
\(134\) −352.000 −0.226927
\(135\) 1800.00 1.14755
\(136\) 288.000 0.181587
\(137\) 1062.00 0.662283 0.331142 0.943581i \(-0.392566\pi\)
0.331142 + 0.943581i \(0.392566\pi\)
\(138\) 720.000 0.444134
\(139\) 508.000 0.309986 0.154993 0.987916i \(-0.450465\pi\)
0.154993 + 0.987916i \(0.450465\pi\)
\(140\) −504.000 −0.304256
\(141\) 804.000 0.480206
\(142\) 648.000 0.382950
\(143\) 0 0
\(144\) −368.000 −0.212963
\(145\) 972.000 0.556691
\(146\) 1600.00 0.906965
\(147\) −98.0000 −0.0549857
\(148\) 1544.00 0.857541
\(149\) −2598.00 −1.42843 −0.714216 0.699925i \(-0.753217\pi\)
−0.714216 + 0.699925i \(0.753217\pi\)
\(150\) 796.000 0.433288
\(151\) −2648.00 −1.42709 −0.713547 0.700607i \(-0.752912\pi\)
−0.713547 + 0.700607i \(0.752912\pi\)
\(152\) −224.000 −0.119532
\(153\) 828.000 0.437515
\(154\) 0 0
\(155\) −6012.00 −3.11545
\(156\) 448.000 0.229928
\(157\) −790.000 −0.401585 −0.200793 0.979634i \(-0.564352\pi\)
−0.200793 + 0.979634i \(0.564352\pi\)
\(158\) −2288.00 −1.15205
\(159\) 972.000 0.484809
\(160\) −576.000 −0.284605
\(161\) −1260.00 −0.616782
\(162\) −842.000 −0.408357
\(163\) −160.000 −0.0768845 −0.0384422 0.999261i \(-0.512240\pi\)
−0.0384422 + 0.999261i \(0.512240\pi\)
\(164\) 1776.00 0.845624
\(165\) 0 0
\(166\) 936.000 0.437637
\(167\) −264.000 −0.122329 −0.0611645 0.998128i \(-0.519481\pi\)
−0.0611645 + 0.998128i \(0.519481\pi\)
\(168\) −112.000 −0.0514344
\(169\) 939.000 0.427401
\(170\) 1296.00 0.584698
\(171\) −644.000 −0.287999
\(172\) 1264.00 0.560344
\(173\) −1632.00 −0.717218 −0.358609 0.933488i \(-0.616749\pi\)
−0.358609 + 0.933488i \(0.616749\pi\)
\(174\) 216.000 0.0941087
\(175\) −1393.00 −0.601719
\(176\) 0 0
\(177\) 564.000 0.239508
\(178\) 1740.00 0.732688
\(179\) −708.000 −0.295634 −0.147817 0.989015i \(-0.547225\pi\)
−0.147817 + 0.989015i \(0.547225\pi\)
\(180\) −1656.00 −0.685728
\(181\) 902.000 0.370415 0.185208 0.982699i \(-0.440704\pi\)
0.185208 + 0.982699i \(0.440704\pi\)
\(182\) −784.000 −0.319307
\(183\) 760.000 0.306999
\(184\) −1440.00 −0.576947
\(185\) 6948.00 2.76123
\(186\) −1336.00 −0.526668
\(187\) 0 0
\(188\) −1608.00 −0.623806
\(189\) −700.000 −0.269405
\(190\) −1008.00 −0.384884
\(191\) 1824.00 0.690995 0.345497 0.938420i \(-0.387710\pi\)
0.345497 + 0.938420i \(0.387710\pi\)
\(192\) −128.000 −0.0481125
\(193\) −2090.00 −0.779490 −0.389745 0.920923i \(-0.627437\pi\)
−0.389745 + 0.920923i \(0.627437\pi\)
\(194\) 2660.00 0.984417
\(195\) 2016.00 0.740353
\(196\) 196.000 0.0714286
\(197\) 1602.00 0.579380 0.289690 0.957121i \(-0.406448\pi\)
0.289690 + 0.957121i \(0.406448\pi\)
\(198\) 0 0
\(199\) −3274.00 −1.16627 −0.583135 0.812375i \(-0.698174\pi\)
−0.583135 + 0.812375i \(0.698174\pi\)
\(200\) −1592.00 −0.562857
\(201\) −352.000 −0.123523
\(202\) −240.000 −0.0835957
\(203\) −378.000 −0.130692
\(204\) 288.000 0.0988433
\(205\) 7992.00 2.72286
\(206\) 2420.00 0.818492
\(207\) −4140.00 −1.39010
\(208\) −896.000 −0.298685
\(209\) 0 0
\(210\) −504.000 −0.165616
\(211\) 4948.00 1.61438 0.807190 0.590291i \(-0.200987\pi\)
0.807190 + 0.590291i \(0.200987\pi\)
\(212\) −1944.00 −0.629785
\(213\) 648.000 0.208452
\(214\) 2472.00 0.789638
\(215\) 5688.00 1.80427
\(216\) −800.000 −0.252005
\(217\) 2338.00 0.731400
\(218\) −1388.00 −0.431226
\(219\) 1600.00 0.493689
\(220\) 0 0
\(221\) 2016.00 0.613624
\(222\) 1544.00 0.466786
\(223\) 2342.00 0.703282 0.351641 0.936135i \(-0.385624\pi\)
0.351641 + 0.936135i \(0.385624\pi\)
\(224\) 224.000 0.0668153
\(225\) −4577.00 −1.35615
\(226\) −1956.00 −0.575713
\(227\) −2064.00 −0.603491 −0.301746 0.953388i \(-0.597569\pi\)
−0.301746 + 0.953388i \(0.597569\pi\)
\(228\) −224.000 −0.0650647
\(229\) −1666.00 −0.480753 −0.240376 0.970680i \(-0.577271\pi\)
−0.240376 + 0.970680i \(0.577271\pi\)
\(230\) −6480.00 −1.85773
\(231\) 0 0
\(232\) −432.000 −0.122251
\(233\) −4158.00 −1.16910 −0.584549 0.811359i \(-0.698728\pi\)
−0.584549 + 0.811359i \(0.698728\pi\)
\(234\) −2576.00 −0.719651
\(235\) −7236.00 −2.00862
\(236\) −1128.00 −0.311129
\(237\) −2288.00 −0.627095
\(238\) −504.000 −0.137267
\(239\) −72.0000 −0.0194866 −0.00974329 0.999953i \(-0.503101\pi\)
−0.00974329 + 0.999953i \(0.503101\pi\)
\(240\) −576.000 −0.154919
\(241\) −6860.00 −1.83357 −0.916787 0.399376i \(-0.869227\pi\)
−0.916787 + 0.399376i \(0.869227\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) −1520.00 −0.398803
\(245\) 882.000 0.229996
\(246\) 1776.00 0.460299
\(247\) −1568.00 −0.403925
\(248\) 2672.00 0.684162
\(249\) 936.000 0.238219
\(250\) −2664.00 −0.673945
\(251\) −150.000 −0.0377208 −0.0188604 0.999822i \(-0.506004\pi\)
−0.0188604 + 0.999822i \(0.506004\pi\)
\(252\) 644.000 0.160985
\(253\) 0 0
\(254\) −2432.00 −0.600777
\(255\) 1296.00 0.318269
\(256\) 256.000 0.0625000
\(257\) −2430.00 −0.589802 −0.294901 0.955528i \(-0.595287\pi\)
−0.294901 + 0.955528i \(0.595287\pi\)
\(258\) 1264.00 0.305012
\(259\) −2702.00 −0.648240
\(260\) −4032.00 −0.961746
\(261\) −1242.00 −0.294551
\(262\) 3360.00 0.792296
\(263\) −3048.00 −0.714630 −0.357315 0.933984i \(-0.616308\pi\)
−0.357315 + 0.933984i \(0.616308\pi\)
\(264\) 0 0
\(265\) −8748.00 −2.02787
\(266\) 392.000 0.0903574
\(267\) 1740.00 0.398825
\(268\) 704.000 0.160461
\(269\) −3834.00 −0.869008 −0.434504 0.900670i \(-0.643076\pi\)
−0.434504 + 0.900670i \(0.643076\pi\)
\(270\) −3600.00 −0.811441
\(271\) 3508.00 0.786331 0.393166 0.919468i \(-0.371380\pi\)
0.393166 + 0.919468i \(0.371380\pi\)
\(272\) −576.000 −0.128401
\(273\) −784.000 −0.173809
\(274\) −2124.00 −0.468305
\(275\) 0 0
\(276\) −1440.00 −0.314050
\(277\) −8294.00 −1.79905 −0.899527 0.436864i \(-0.856089\pi\)
−0.899527 + 0.436864i \(0.856089\pi\)
\(278\) −1016.00 −0.219193
\(279\) 7682.00 1.64842
\(280\) 1008.00 0.215141
\(281\) −8022.00 −1.70303 −0.851517 0.524327i \(-0.824317\pi\)
−0.851517 + 0.524327i \(0.824317\pi\)
\(282\) −1608.00 −0.339557
\(283\) −392.000 −0.0823392 −0.0411696 0.999152i \(-0.513108\pi\)
−0.0411696 + 0.999152i \(0.513108\pi\)
\(284\) −1296.00 −0.270787
\(285\) −1008.00 −0.209504
\(286\) 0 0
\(287\) −3108.00 −0.639231
\(288\) 736.000 0.150588
\(289\) −3617.00 −0.736210
\(290\) −1944.00 −0.393640
\(291\) 2660.00 0.535849
\(292\) −3200.00 −0.641321
\(293\) 2748.00 0.547918 0.273959 0.961741i \(-0.411667\pi\)
0.273959 + 0.961741i \(0.411667\pi\)
\(294\) 196.000 0.0388808
\(295\) −5076.00 −1.00182
\(296\) −3088.00 −0.606373
\(297\) 0 0
\(298\) 5196.00 1.01005
\(299\) −10080.0 −1.94964
\(300\) −1592.00 −0.306381
\(301\) −2212.00 −0.423580
\(302\) 5296.00 1.00911
\(303\) −240.000 −0.0455038
\(304\) 448.000 0.0845216
\(305\) −6840.00 −1.28412
\(306\) −1656.00 −0.309370
\(307\) 3064.00 0.569615 0.284807 0.958585i \(-0.408070\pi\)
0.284807 + 0.958585i \(0.408070\pi\)
\(308\) 0 0
\(309\) 2420.00 0.445531
\(310\) 12024.0 2.20296
\(311\) 4062.00 0.740627 0.370313 0.928907i \(-0.379250\pi\)
0.370313 + 0.928907i \(0.379250\pi\)
\(312\) −896.000 −0.162583
\(313\) −4870.00 −0.879453 −0.439726 0.898132i \(-0.644925\pi\)
−0.439726 + 0.898132i \(0.644925\pi\)
\(314\) 1580.00 0.283964
\(315\) 2898.00 0.518361
\(316\) 4576.00 0.814621
\(317\) 4806.00 0.851520 0.425760 0.904836i \(-0.360007\pi\)
0.425760 + 0.904836i \(0.360007\pi\)
\(318\) −1944.00 −0.342812
\(319\) 0 0
\(320\) 1152.00 0.201246
\(321\) 2472.00 0.429824
\(322\) 2520.00 0.436131
\(323\) −1008.00 −0.173643
\(324\) 1684.00 0.288752
\(325\) −11144.0 −1.90202
\(326\) 320.000 0.0543655
\(327\) −1388.00 −0.234730
\(328\) −3552.00 −0.597946
\(329\) 2814.00 0.471553
\(330\) 0 0
\(331\) 6620.00 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −1872.00 −0.309456
\(333\) −8878.00 −1.46100
\(334\) 528.000 0.0864996
\(335\) 3168.00 0.516676
\(336\) 224.000 0.0363696
\(337\) −1094.00 −0.176837 −0.0884184 0.996083i \(-0.528181\pi\)
−0.0884184 + 0.996083i \(0.528181\pi\)
\(338\) −1878.00 −0.302218
\(339\) −1956.00 −0.313379
\(340\) −2592.00 −0.413444
\(341\) 0 0
\(342\) 1288.00 0.203646
\(343\) −343.000 −0.0539949
\(344\) −2528.00 −0.396223
\(345\) −6480.00 −1.01122
\(346\) 3264.00 0.507150
\(347\) −3468.00 −0.536519 −0.268259 0.963347i \(-0.586448\pi\)
−0.268259 + 0.963347i \(0.586448\pi\)
\(348\) −432.000 −0.0665449
\(349\) 8188.00 1.25586 0.627928 0.778272i \(-0.283903\pi\)
0.627928 + 0.778272i \(0.283903\pi\)
\(350\) 2786.00 0.425480
\(351\) −5600.00 −0.851584
\(352\) 0 0
\(353\) −5070.00 −0.764444 −0.382222 0.924070i \(-0.624841\pi\)
−0.382222 + 0.924070i \(0.624841\pi\)
\(354\) −1128.00 −0.169357
\(355\) −5832.00 −0.871917
\(356\) −3480.00 −0.518089
\(357\) −504.000 −0.0747185
\(358\) 1416.00 0.209044
\(359\) −1656.00 −0.243455 −0.121727 0.992564i \(-0.538843\pi\)
−0.121727 + 0.992564i \(0.538843\pi\)
\(360\) 3312.00 0.484883
\(361\) −6075.00 −0.885698
\(362\) −1804.00 −0.261923
\(363\) 0 0
\(364\) 1568.00 0.225784
\(365\) −14400.0 −2.06501
\(366\) −1520.00 −0.217081
\(367\) 10166.0 1.44594 0.722971 0.690878i \(-0.242776\pi\)
0.722971 + 0.690878i \(0.242776\pi\)
\(368\) 2880.00 0.407963
\(369\) −10212.0 −1.44069
\(370\) −13896.0 −1.95248
\(371\) 3402.00 0.476073
\(372\) 2672.00 0.372411
\(373\) 2722.00 0.377855 0.188927 0.981991i \(-0.439499\pi\)
0.188927 + 0.981991i \(0.439499\pi\)
\(374\) 0 0
\(375\) −2664.00 −0.366849
\(376\) 3216.00 0.441097
\(377\) −3024.00 −0.413114
\(378\) 1400.00 0.190498
\(379\) −5872.00 −0.795843 −0.397921 0.917420i \(-0.630268\pi\)
−0.397921 + 0.917420i \(0.630268\pi\)
\(380\) 2016.00 0.272154
\(381\) −2432.00 −0.327021
\(382\) −3648.00 −0.488607
\(383\) 12330.0 1.64500 0.822498 0.568768i \(-0.192580\pi\)
0.822498 + 0.568768i \(0.192580\pi\)
\(384\) 256.000 0.0340207
\(385\) 0 0
\(386\) 4180.00 0.551182
\(387\) −7268.00 −0.954659
\(388\) −5320.00 −0.696088
\(389\) −14586.0 −1.90113 −0.950565 0.310526i \(-0.899495\pi\)
−0.950565 + 0.310526i \(0.899495\pi\)
\(390\) −4032.00 −0.523508
\(391\) −6480.00 −0.838127
\(392\) −392.000 −0.0505076
\(393\) 3360.00 0.431271
\(394\) −3204.00 −0.409683
\(395\) 20592.0 2.62303
\(396\) 0 0
\(397\) 1874.00 0.236910 0.118455 0.992959i \(-0.462206\pi\)
0.118455 + 0.992959i \(0.462206\pi\)
\(398\) 6548.00 0.824677
\(399\) 392.000 0.0491843
\(400\) 3184.00 0.398000
\(401\) 13338.0 1.66102 0.830509 0.557006i \(-0.188050\pi\)
0.830509 + 0.557006i \(0.188050\pi\)
\(402\) 704.000 0.0873441
\(403\) 18704.0 2.31194
\(404\) 480.000 0.0591111
\(405\) 7578.00 0.929763
\(406\) 756.000 0.0924129
\(407\) 0 0
\(408\) −576.000 −0.0698928
\(409\) 8200.00 0.991354 0.495677 0.868507i \(-0.334920\pi\)
0.495677 + 0.868507i \(0.334920\pi\)
\(410\) −15984.0 −1.92535
\(411\) −2124.00 −0.254913
\(412\) −4840.00 −0.578761
\(413\) 1974.00 0.235192
\(414\) 8280.00 0.982946
\(415\) −8424.00 −0.996429
\(416\) 1792.00 0.211202
\(417\) −1016.00 −0.119314
\(418\) 0 0
\(419\) −7362.00 −0.858370 −0.429185 0.903216i \(-0.641199\pi\)
−0.429185 + 0.903216i \(0.641199\pi\)
\(420\) 1008.00 0.117108
\(421\) −11710.0 −1.35561 −0.677803 0.735243i \(-0.737068\pi\)
−0.677803 + 0.735243i \(0.737068\pi\)
\(422\) −9896.00 −1.14154
\(423\) 9246.00 1.06278
\(424\) 3888.00 0.445325
\(425\) −7164.00 −0.817659
\(426\) −1296.00 −0.147398
\(427\) 2660.00 0.301467
\(428\) −4944.00 −0.558358
\(429\) 0 0
\(430\) −11376.0 −1.27581
\(431\) 936.000 0.104607 0.0523034 0.998631i \(-0.483344\pi\)
0.0523034 + 0.998631i \(0.483344\pi\)
\(432\) 1600.00 0.178195
\(433\) 9038.00 1.00309 0.501546 0.865131i \(-0.332765\pi\)
0.501546 + 0.865131i \(0.332765\pi\)
\(434\) −4676.00 −0.517178
\(435\) −1944.00 −0.214270
\(436\) 2776.00 0.304923
\(437\) 5040.00 0.551707
\(438\) −3200.00 −0.349091
\(439\) −1964.00 −0.213523 −0.106762 0.994285i \(-0.534048\pi\)
−0.106762 + 0.994285i \(0.534048\pi\)
\(440\) 0 0
\(441\) −1127.00 −0.121693
\(442\) −4032.00 −0.433897
\(443\) 10068.0 1.07979 0.539893 0.841734i \(-0.318465\pi\)
0.539893 + 0.841734i \(0.318465\pi\)
\(444\) −3088.00 −0.330068
\(445\) −15660.0 −1.66821
\(446\) −4684.00 −0.497296
\(447\) 5196.00 0.549804
\(448\) −448.000 −0.0472456
\(449\) 3270.00 0.343699 0.171849 0.985123i \(-0.445026\pi\)
0.171849 + 0.985123i \(0.445026\pi\)
\(450\) 9154.00 0.958942
\(451\) 0 0
\(452\) 3912.00 0.407091
\(453\) 5296.00 0.549289
\(454\) 4128.00 0.426733
\(455\) 7056.00 0.727012
\(456\) 448.000 0.0460077
\(457\) 15526.0 1.58922 0.794612 0.607117i \(-0.207674\pi\)
0.794612 + 0.607117i \(0.207674\pi\)
\(458\) 3332.00 0.339944
\(459\) −3600.00 −0.366086
\(460\) 12960.0 1.31362
\(461\) −10548.0 −1.06566 −0.532830 0.846222i \(-0.678872\pi\)
−0.532830 + 0.846222i \(0.678872\pi\)
\(462\) 0 0
\(463\) −3796.00 −0.381026 −0.190513 0.981685i \(-0.561015\pi\)
−0.190513 + 0.981685i \(0.561015\pi\)
\(464\) 864.000 0.0864444
\(465\) 12024.0 1.19914
\(466\) 8316.00 0.826677
\(467\) 7122.00 0.705711 0.352855 0.935678i \(-0.385211\pi\)
0.352855 + 0.935678i \(0.385211\pi\)
\(468\) 5152.00 0.508870
\(469\) −1232.00 −0.121297
\(470\) 14472.0 1.42031
\(471\) 1580.00 0.154570
\(472\) 2256.00 0.220002
\(473\) 0 0
\(474\) 4576.00 0.443423
\(475\) 5572.00 0.538233
\(476\) 1008.00 0.0970622
\(477\) 11178.0 1.07297
\(478\) 144.000 0.0137791
\(479\) −2292.00 −0.218631 −0.109315 0.994007i \(-0.534866\pi\)
−0.109315 + 0.994007i \(0.534866\pi\)
\(480\) 1152.00 0.109545
\(481\) −21616.0 −2.04907
\(482\) 13720.0 1.29653
\(483\) 2520.00 0.237400
\(484\) 0 0
\(485\) −23940.0 −2.24136
\(486\) 7084.00 0.661187
\(487\) 5132.00 0.477522 0.238761 0.971078i \(-0.423259\pi\)
0.238761 + 0.971078i \(0.423259\pi\)
\(488\) 3040.00 0.281997
\(489\) 320.000 0.0295928
\(490\) −1764.00 −0.162631
\(491\) −4188.00 −0.384932 −0.192466 0.981304i \(-0.561649\pi\)
−0.192466 + 0.981304i \(0.561649\pi\)
\(492\) −3552.00 −0.325481
\(493\) −1944.00 −0.177593
\(494\) 3136.00 0.285618
\(495\) 0 0
\(496\) −5344.00 −0.483776
\(497\) 2268.00 0.204696
\(498\) −1872.00 −0.168446
\(499\) 3848.00 0.345211 0.172605 0.984991i \(-0.444781\pi\)
0.172605 + 0.984991i \(0.444781\pi\)
\(500\) 5328.00 0.476551
\(501\) 528.000 0.0470844
\(502\) 300.000 0.0266726
\(503\) 1068.00 0.0946715 0.0473358 0.998879i \(-0.484927\pi\)
0.0473358 + 0.998879i \(0.484927\pi\)
\(504\) −1288.00 −0.113833
\(505\) 2160.00 0.190334
\(506\) 0 0
\(507\) −1878.00 −0.164507
\(508\) 4864.00 0.424813
\(509\) −6162.00 −0.536593 −0.268297 0.963336i \(-0.586461\pi\)
−0.268297 + 0.963336i \(0.586461\pi\)
\(510\) −2592.00 −0.225050
\(511\) 5600.00 0.484793
\(512\) −512.000 −0.0441942
\(513\) 2800.00 0.240981
\(514\) 4860.00 0.417053
\(515\) −21780.0 −1.86358
\(516\) −2528.00 −0.215676
\(517\) 0 0
\(518\) 5404.00 0.458375
\(519\) 3264.00 0.276057
\(520\) 8064.00 0.680057
\(521\) −20946.0 −1.76135 −0.880673 0.473725i \(-0.842909\pi\)
−0.880673 + 0.473725i \(0.842909\pi\)
\(522\) 2484.00 0.208279
\(523\) 4696.00 0.392623 0.196311 0.980542i \(-0.437104\pi\)
0.196311 + 0.980542i \(0.437104\pi\)
\(524\) −6720.00 −0.560238
\(525\) 2786.00 0.231602
\(526\) 6096.00 0.505320
\(527\) 12024.0 0.993878
\(528\) 0 0
\(529\) 20233.0 1.66294
\(530\) 17496.0 1.43392
\(531\) 6486.00 0.530072
\(532\) −784.000 −0.0638923
\(533\) −24864.0 −2.02060
\(534\) −3480.00 −0.282012
\(535\) −22248.0 −1.79788
\(536\) −1408.00 −0.113463
\(537\) 1416.00 0.113789
\(538\) 7668.00 0.614481
\(539\) 0 0
\(540\) 7200.00 0.573775
\(541\) −19358.0 −1.53838 −0.769192 0.639018i \(-0.779341\pi\)
−0.769192 + 0.639018i \(0.779341\pi\)
\(542\) −7016.00 −0.556020
\(543\) −1804.00 −0.142573
\(544\) 1152.00 0.0907934
\(545\) 12492.0 0.981832
\(546\) 1568.00 0.122901
\(547\) −18020.0 −1.40855 −0.704277 0.709925i \(-0.748729\pi\)
−0.704277 + 0.709925i \(0.748729\pi\)
\(548\) 4248.00 0.331142
\(549\) 8740.00 0.679443
\(550\) 0 0
\(551\) 1512.00 0.116903
\(552\) 2880.00 0.222067
\(553\) −8008.00 −0.615795
\(554\) 16588.0 1.27212
\(555\) −13896.0 −1.06280
\(556\) 2032.00 0.154993
\(557\) −14622.0 −1.11231 −0.556153 0.831080i \(-0.687723\pi\)
−0.556153 + 0.831080i \(0.687723\pi\)
\(558\) −15364.0 −1.16561
\(559\) −17696.0 −1.33893
\(560\) −2016.00 −0.152128
\(561\) 0 0
\(562\) 16044.0 1.20423
\(563\) 2244.00 0.167981 0.0839905 0.996467i \(-0.473233\pi\)
0.0839905 + 0.996467i \(0.473233\pi\)
\(564\) 3216.00 0.240103
\(565\) 17604.0 1.31081
\(566\) 784.000 0.0582226
\(567\) −2947.00 −0.218276
\(568\) 2592.00 0.191475
\(569\) 3258.00 0.240039 0.120020 0.992772i \(-0.461704\pi\)
0.120020 + 0.992772i \(0.461704\pi\)
\(570\) 2016.00 0.148142
\(571\) 6604.00 0.484008 0.242004 0.970275i \(-0.422195\pi\)
0.242004 + 0.970275i \(0.422195\pi\)
\(572\) 0 0
\(573\) −3648.00 −0.265964
\(574\) 6216.00 0.452005
\(575\) 35820.0 2.59791
\(576\) −1472.00 −0.106481
\(577\) −16594.0 −1.19726 −0.598628 0.801027i \(-0.704287\pi\)
−0.598628 + 0.801027i \(0.704287\pi\)
\(578\) 7234.00 0.520579
\(579\) 4180.00 0.300026
\(580\) 3888.00 0.278346
\(581\) 3276.00 0.233927
\(582\) −5320.00 −0.378902
\(583\) 0 0
\(584\) 6400.00 0.453483
\(585\) 23184.0 1.63853
\(586\) −5496.00 −0.387436
\(587\) −19062.0 −1.34033 −0.670164 0.742213i \(-0.733776\pi\)
−0.670164 + 0.742213i \(0.733776\pi\)
\(588\) −392.000 −0.0274929
\(589\) −9352.00 −0.654232
\(590\) 10152.0 0.708392
\(591\) −3204.00 −0.223003
\(592\) 6176.00 0.428770
\(593\) 4776.00 0.330737 0.165368 0.986232i \(-0.447119\pi\)
0.165368 + 0.986232i \(0.447119\pi\)
\(594\) 0 0
\(595\) 4536.00 0.312534
\(596\) −10392.0 −0.714216
\(597\) 6548.00 0.448897
\(598\) 20160.0 1.37860
\(599\) 7956.00 0.542693 0.271347 0.962482i \(-0.412531\pi\)
0.271347 + 0.962482i \(0.412531\pi\)
\(600\) 3184.00 0.216644
\(601\) −14348.0 −0.973822 −0.486911 0.873452i \(-0.661876\pi\)
−0.486911 + 0.873452i \(0.661876\pi\)
\(602\) 4424.00 0.299516
\(603\) −4048.00 −0.273379
\(604\) −10592.0 −0.713547
\(605\) 0 0
\(606\) 480.000 0.0321760
\(607\) −24488.0 −1.63746 −0.818729 0.574180i \(-0.805321\pi\)
−0.818729 + 0.574180i \(0.805321\pi\)
\(608\) −896.000 −0.0597658
\(609\) 756.000 0.0503032
\(610\) 13680.0 0.908011
\(611\) 22512.0 1.49057
\(612\) 3312.00 0.218758
\(613\) 19654.0 1.29497 0.647486 0.762078i \(-0.275821\pi\)
0.647486 + 0.762078i \(0.275821\pi\)
\(614\) −6128.00 −0.402778
\(615\) −15984.0 −1.04803
\(616\) 0 0
\(617\) 2694.00 0.175780 0.0878901 0.996130i \(-0.471988\pi\)
0.0878901 + 0.996130i \(0.471988\pi\)
\(618\) −4840.00 −0.315038
\(619\) 10178.0 0.660886 0.330443 0.943826i \(-0.392802\pi\)
0.330443 + 0.943826i \(0.392802\pi\)
\(620\) −24048.0 −1.55773
\(621\) 18000.0 1.16315
\(622\) −8124.00 −0.523702
\(623\) 6090.00 0.391638
\(624\) 1792.00 0.114964
\(625\) −899.000 −0.0575360
\(626\) 9740.00 0.621867
\(627\) 0 0
\(628\) −3160.00 −0.200793
\(629\) −13896.0 −0.880874
\(630\) −5796.00 −0.366537
\(631\) −7648.00 −0.482507 −0.241254 0.970462i \(-0.577559\pi\)
−0.241254 + 0.970462i \(0.577559\pi\)
\(632\) −9152.00 −0.576024
\(633\) −9896.00 −0.621375
\(634\) −9612.00 −0.602116
\(635\) 21888.0 1.36787
\(636\) 3888.00 0.242404
\(637\) −2744.00 −0.170677
\(638\) 0 0
\(639\) 7452.00 0.461340
\(640\) −2304.00 −0.142302
\(641\) 270.000 0.0166371 0.00831853 0.999965i \(-0.497352\pi\)
0.00831853 + 0.999965i \(0.497352\pi\)
\(642\) −4944.00 −0.303932
\(643\) 16250.0 0.996637 0.498318 0.866994i \(-0.333951\pi\)
0.498318 + 0.866994i \(0.333951\pi\)
\(644\) −5040.00 −0.308391
\(645\) −11376.0 −0.694464
\(646\) 2016.00 0.122784
\(647\) 10242.0 0.622341 0.311170 0.950354i \(-0.399279\pi\)
0.311170 + 0.950354i \(0.399279\pi\)
\(648\) −3368.00 −0.204178
\(649\) 0 0
\(650\) 22288.0 1.34493
\(651\) −4676.00 −0.281516
\(652\) −640.000 −0.0384422
\(653\) −17322.0 −1.03807 −0.519037 0.854752i \(-0.673709\pi\)
−0.519037 + 0.854752i \(0.673709\pi\)
\(654\) 2776.00 0.165979
\(655\) −30240.0 −1.80393
\(656\) 7104.00 0.422812
\(657\) 18400.0 1.09262
\(658\) −5628.00 −0.333438
\(659\) −11676.0 −0.690186 −0.345093 0.938569i \(-0.612153\pi\)
−0.345093 + 0.938569i \(0.612153\pi\)
\(660\) 0 0
\(661\) −20710.0 −1.21865 −0.609323 0.792922i \(-0.708559\pi\)
−0.609323 + 0.792922i \(0.708559\pi\)
\(662\) −13240.0 −0.777322
\(663\) −4032.00 −0.236184
\(664\) 3744.00 0.218818
\(665\) −3528.00 −0.205729
\(666\) 17756.0 1.03308
\(667\) 9720.00 0.564258
\(668\) −1056.00 −0.0611645
\(669\) −4684.00 −0.270693
\(670\) −6336.00 −0.365345
\(671\) 0 0
\(672\) −448.000 −0.0257172
\(673\) 10354.0 0.593042 0.296521 0.955026i \(-0.404173\pi\)
0.296521 + 0.955026i \(0.404173\pi\)
\(674\) 2188.00 0.125042
\(675\) 19900.0 1.13474
\(676\) 3756.00 0.213701
\(677\) 10920.0 0.619926 0.309963 0.950749i \(-0.399683\pi\)
0.309963 + 0.950749i \(0.399683\pi\)
\(678\) 3912.00 0.221592
\(679\) 9310.00 0.526193
\(680\) 5184.00 0.292349
\(681\) 4128.00 0.232284
\(682\) 0 0
\(683\) 27804.0 1.55767 0.778836 0.627227i \(-0.215810\pi\)
0.778836 + 0.627227i \(0.215810\pi\)
\(684\) −2576.00 −0.144000
\(685\) 19116.0 1.06626
\(686\) 686.000 0.0381802
\(687\) 3332.00 0.185042
\(688\) 5056.00 0.280172
\(689\) 27216.0 1.50486
\(690\) 12960.0 0.715042
\(691\) −25834.0 −1.42225 −0.711123 0.703068i \(-0.751813\pi\)
−0.711123 + 0.703068i \(0.751813\pi\)
\(692\) −6528.00 −0.358609
\(693\) 0 0
\(694\) 6936.00 0.379376
\(695\) 9144.00 0.499067
\(696\) 864.000 0.0470544
\(697\) −15984.0 −0.868633
\(698\) −16376.0 −0.888024
\(699\) 8316.00 0.449986
\(700\) −5572.00 −0.300860
\(701\) 10590.0 0.570583 0.285292 0.958441i \(-0.407910\pi\)
0.285292 + 0.958441i \(0.407910\pi\)
\(702\) 11200.0 0.602161
\(703\) 10808.0 0.579846
\(704\) 0 0
\(705\) 14472.0 0.773116
\(706\) 10140.0 0.540544
\(707\) −840.000 −0.0446838
\(708\) 2256.00 0.119754
\(709\) −6802.00 −0.360302 −0.180151 0.983639i \(-0.557659\pi\)
−0.180151 + 0.983639i \(0.557659\pi\)
\(710\) 11664.0 0.616538
\(711\) −26312.0 −1.38787
\(712\) 6960.00 0.366344
\(713\) −60120.0 −3.15780
\(714\) 1008.00 0.0528340
\(715\) 0 0
\(716\) −2832.00 −0.147817
\(717\) 144.000 0.00750039
\(718\) 3312.00 0.172149
\(719\) 23010.0 1.19350 0.596751 0.802426i \(-0.296458\pi\)
0.596751 + 0.802426i \(0.296458\pi\)
\(720\) −6624.00 −0.342864
\(721\) 8470.00 0.437502
\(722\) 12150.0 0.626283
\(723\) 13720.0 0.705743
\(724\) 3608.00 0.185208
\(725\) 10746.0 0.550478
\(726\) 0 0
\(727\) 4682.00 0.238853 0.119426 0.992843i \(-0.461894\pi\)
0.119426 + 0.992843i \(0.461894\pi\)
\(728\) −3136.00 −0.159654
\(729\) −4283.00 −0.217599
\(730\) 28800.0 1.46019
\(731\) −11376.0 −0.575590
\(732\) 3040.00 0.153499
\(733\) 17860.0 0.899965 0.449982 0.893037i \(-0.351430\pi\)
0.449982 + 0.893037i \(0.351430\pi\)
\(734\) −20332.0 −1.02244
\(735\) −1764.00 −0.0885253
\(736\) −5760.00 −0.288473
\(737\) 0 0
\(738\) 20424.0 1.01872
\(739\) −6860.00 −0.341474 −0.170737 0.985317i \(-0.554615\pi\)
−0.170737 + 0.985317i \(0.554615\pi\)
\(740\) 27792.0 1.38061
\(741\) 3136.00 0.155471
\(742\) −6804.00 −0.336634
\(743\) 22752.0 1.12341 0.561703 0.827339i \(-0.310147\pi\)
0.561703 + 0.827339i \(0.310147\pi\)
\(744\) −5344.00 −0.263334
\(745\) −46764.0 −2.29973
\(746\) −5444.00 −0.267184
\(747\) 10764.0 0.527221
\(748\) 0 0
\(749\) 8652.00 0.422079
\(750\) 5328.00 0.259401
\(751\) 7364.00 0.357811 0.178906 0.983866i \(-0.442744\pi\)
0.178906 + 0.983866i \(0.442744\pi\)
\(752\) −6432.00 −0.311903
\(753\) 300.000 0.0145187
\(754\) 6048.00 0.292116
\(755\) −47664.0 −2.29758
\(756\) −2800.00 −0.134702
\(757\) −34378.0 −1.65058 −0.825290 0.564709i \(-0.808989\pi\)
−0.825290 + 0.564709i \(0.808989\pi\)
\(758\) 11744.0 0.562746
\(759\) 0 0
\(760\) −4032.00 −0.192442
\(761\) −27456.0 −1.30786 −0.653929 0.756556i \(-0.726880\pi\)
−0.653929 + 0.756556i \(0.726880\pi\)
\(762\) 4864.00 0.231239
\(763\) −4858.00 −0.230500
\(764\) 7296.00 0.345497
\(765\) 14904.0 0.704386
\(766\) −24660.0 −1.16319
\(767\) 15792.0 0.743437
\(768\) −512.000 −0.0240563
\(769\) −7952.00 −0.372895 −0.186448 0.982465i \(-0.559697\pi\)
−0.186448 + 0.982465i \(0.559697\pi\)
\(770\) 0 0
\(771\) 4860.00 0.227015
\(772\) −8360.00 −0.389745
\(773\) −4986.00 −0.231997 −0.115999 0.993249i \(-0.537007\pi\)
−0.115999 + 0.993249i \(0.537007\pi\)
\(774\) 14536.0 0.675046
\(775\) −66466.0 −3.08068
\(776\) 10640.0 0.492208
\(777\) 5404.00 0.249508
\(778\) 29172.0 1.34430
\(779\) 12432.0 0.571788
\(780\) 8064.00 0.370176
\(781\) 0 0
\(782\) 12960.0 0.592645
\(783\) 5400.00 0.246463
\(784\) 784.000 0.0357143
\(785\) −14220.0 −0.646540
\(786\) −6720.00 −0.304955
\(787\) 42748.0 1.93622 0.968108 0.250534i \(-0.0806062\pi\)
0.968108 + 0.250534i \(0.0806062\pi\)
\(788\) 6408.00 0.289690
\(789\) 6096.00 0.275061
\(790\) −41184.0 −1.85476
\(791\) −6846.00 −0.307732
\(792\) 0 0
\(793\) 21280.0 0.952932
\(794\) −3748.00 −0.167521
\(795\) 17496.0 0.780527
\(796\) −13096.0 −0.583135
\(797\) 35610.0 1.58265 0.791324 0.611397i \(-0.209392\pi\)
0.791324 + 0.611397i \(0.209392\pi\)
\(798\) −784.000 −0.0347786
\(799\) 14472.0 0.640779
\(800\) −6368.00 −0.281428
\(801\) 20010.0 0.882670
\(802\) −26676.0 −1.17452
\(803\) 0 0
\(804\) −1408.00 −0.0617616
\(805\) −22680.0 −0.993000
\(806\) −37408.0 −1.63479
\(807\) 7668.00 0.334481
\(808\) −960.000 −0.0417979
\(809\) 17046.0 0.740798 0.370399 0.928873i \(-0.379221\pi\)
0.370399 + 0.928873i \(0.379221\pi\)
\(810\) −15156.0 −0.657441
\(811\) 2176.00 0.0942166 0.0471083 0.998890i \(-0.484999\pi\)
0.0471083 + 0.998890i \(0.484999\pi\)
\(812\) −1512.00 −0.0653458
\(813\) −7016.00 −0.302659
\(814\) 0 0
\(815\) −2880.00 −0.123782
\(816\) 1152.00 0.0494217
\(817\) 8848.00 0.378889
\(818\) −16400.0 −0.700993
\(819\) −9016.00 −0.384670
\(820\) 31968.0 1.36143
\(821\) −2094.00 −0.0890147 −0.0445074 0.999009i \(-0.514172\pi\)
−0.0445074 + 0.999009i \(0.514172\pi\)
\(822\) 4248.00 0.180251
\(823\) 7328.00 0.310374 0.155187 0.987885i \(-0.450402\pi\)
0.155187 + 0.987885i \(0.450402\pi\)
\(824\) 9680.00 0.409246
\(825\) 0 0
\(826\) −3948.00 −0.166306
\(827\) 12492.0 0.525259 0.262630 0.964897i \(-0.415410\pi\)
0.262630 + 0.964897i \(0.415410\pi\)
\(828\) −16560.0 −0.695048
\(829\) −37486.0 −1.57050 −0.785249 0.619180i \(-0.787465\pi\)
−0.785249 + 0.619180i \(0.787465\pi\)
\(830\) 16848.0 0.704581
\(831\) 16588.0 0.692456
\(832\) −3584.00 −0.149342
\(833\) −1764.00 −0.0733721
\(834\) 2032.00 0.0843674
\(835\) −4752.00 −0.196946
\(836\) 0 0
\(837\) −33400.0 −1.37930
\(838\) 14724.0 0.606960
\(839\) −17574.0 −0.723149 −0.361574 0.932343i \(-0.617761\pi\)
−0.361574 + 0.932343i \(0.617761\pi\)
\(840\) −2016.00 −0.0828079
\(841\) −21473.0 −0.880438
\(842\) 23420.0 0.958559
\(843\) 16044.0 0.655498
\(844\) 19792.0 0.807190
\(845\) 16902.0 0.688102
\(846\) −18492.0 −0.751499
\(847\) 0 0
\(848\) −7776.00 −0.314893
\(849\) 784.000 0.0316924
\(850\) 14328.0 0.578172
\(851\) 69480.0 2.79876
\(852\) 2592.00 0.104226
\(853\) −9440.00 −0.378921 −0.189460 0.981888i \(-0.560674\pi\)
−0.189460 + 0.981888i \(0.560674\pi\)
\(854\) −5320.00 −0.213169
\(855\) −11592.0 −0.463670
\(856\) 9888.00 0.394819
\(857\) 28440.0 1.13360 0.566798 0.823857i \(-0.308182\pi\)
0.566798 + 0.823857i \(0.308182\pi\)
\(858\) 0 0
\(859\) −24334.0 −0.966549 −0.483274 0.875469i \(-0.660553\pi\)
−0.483274 + 0.875469i \(0.660553\pi\)
\(860\) 22752.0 0.902136
\(861\) 6216.00 0.246040
\(862\) −1872.00 −0.0739682
\(863\) 39264.0 1.54874 0.774370 0.632733i \(-0.218067\pi\)
0.774370 + 0.632733i \(0.218067\pi\)
\(864\) −3200.00 −0.126003
\(865\) −29376.0 −1.15470
\(866\) −18076.0 −0.709293
\(867\) 7234.00 0.283367
\(868\) 9352.00 0.365700
\(869\) 0 0
\(870\) 3888.00 0.151512
\(871\) −9856.00 −0.383419
\(872\) −5552.00 −0.215613
\(873\) 30590.0 1.18593
\(874\) −10080.0 −0.390116
\(875\) −9324.00 −0.360239
\(876\) 6400.00 0.246845
\(877\) −32114.0 −1.23650 −0.618251 0.785981i \(-0.712159\pi\)
−0.618251 + 0.785981i \(0.712159\pi\)
\(878\) 3928.00 0.150984
\(879\) −5496.00 −0.210894
\(880\) 0 0
\(881\) 41454.0 1.58527 0.792634 0.609698i \(-0.208709\pi\)
0.792634 + 0.609698i \(0.208709\pi\)
\(882\) 2254.00 0.0860500
\(883\) 2876.00 0.109609 0.0548047 0.998497i \(-0.482546\pi\)
0.0548047 + 0.998497i \(0.482546\pi\)
\(884\) 8064.00 0.306812
\(885\) 10152.0 0.385600
\(886\) −20136.0 −0.763524
\(887\) −13932.0 −0.527385 −0.263693 0.964607i \(-0.584940\pi\)
−0.263693 + 0.964607i \(0.584940\pi\)
\(888\) 6176.00 0.233393
\(889\) −8512.00 −0.321129
\(890\) 31320.0 1.17961
\(891\) 0 0
\(892\) 9368.00 0.351641
\(893\) −11256.0 −0.421800
\(894\) −10392.0 −0.388770
\(895\) −12744.0 −0.475961
\(896\) 896.000 0.0334077
\(897\) 20160.0 0.750416
\(898\) −6540.00 −0.243032
\(899\) −18036.0 −0.669115
\(900\) −18308.0 −0.678074
\(901\) 17496.0 0.646921
\(902\) 0 0
\(903\) 4424.00 0.163036
\(904\) −7824.00 −0.287857
\(905\) 16236.0 0.596357
\(906\) −10592.0 −0.388406
\(907\) −19768.0 −0.723689 −0.361844 0.932239i \(-0.617853\pi\)
−0.361844 + 0.932239i \(0.617853\pi\)
\(908\) −8256.00 −0.301746
\(909\) −2760.00 −0.100708
\(910\) −14112.0 −0.514075
\(911\) 43836.0 1.59424 0.797119 0.603822i \(-0.206356\pi\)
0.797119 + 0.603822i \(0.206356\pi\)
\(912\) −896.000 −0.0325324
\(913\) 0 0
\(914\) −31052.0 −1.12375
\(915\) 13680.0 0.494259
\(916\) −6664.00 −0.240376
\(917\) 11760.0 0.423500
\(918\) 7200.00 0.258862
\(919\) −31544.0 −1.13225 −0.566127 0.824318i \(-0.691559\pi\)
−0.566127 + 0.824318i \(0.691559\pi\)
\(920\) −25920.0 −0.928866
\(921\) −6128.00 −0.219245
\(922\) 21096.0 0.753536
\(923\) 18144.0 0.647039
\(924\) 0 0
\(925\) 76814.0 2.73041
\(926\) 7592.00 0.269426
\(927\) 27830.0 0.986038
\(928\) −1728.00 −0.0611254
\(929\) 11118.0 0.392648 0.196324 0.980539i \(-0.437100\pi\)
0.196324 + 0.980539i \(0.437100\pi\)
\(930\) −24048.0 −0.847919
\(931\) 1372.00 0.0482980
\(932\) −16632.0 −0.584549
\(933\) −8124.00 −0.285067
\(934\) −14244.0 −0.499013
\(935\) 0 0
\(936\) −10304.0 −0.359826
\(937\) −10568.0 −0.368454 −0.184227 0.982884i \(-0.558978\pi\)
−0.184227 + 0.982884i \(0.558978\pi\)
\(938\) 2464.00 0.0857702
\(939\) 9740.00 0.338501
\(940\) −28944.0 −1.00431
\(941\) −14964.0 −0.518398 −0.259199 0.965824i \(-0.583459\pi\)
−0.259199 + 0.965824i \(0.583459\pi\)
\(942\) −3160.00 −0.109298
\(943\) 79920.0 2.75987
\(944\) −4512.00 −0.155565
\(945\) −12600.0 −0.433733
\(946\) 0 0
\(947\) 3324.00 0.114061 0.0570304 0.998372i \(-0.481837\pi\)
0.0570304 + 0.998372i \(0.481837\pi\)
\(948\) −9152.00 −0.313548
\(949\) 44800.0 1.53242
\(950\) −11144.0 −0.380589
\(951\) −9612.00 −0.327750
\(952\) −2016.00 −0.0686333
\(953\) −3906.00 −0.132768 −0.0663839 0.997794i \(-0.521146\pi\)
−0.0663839 + 0.997794i \(0.521146\pi\)
\(954\) −22356.0 −0.758703
\(955\) 32832.0 1.11248
\(956\) −288.000 −0.00974329
\(957\) 0 0
\(958\) 4584.00 0.154595
\(959\) −7434.00 −0.250319
\(960\) −2304.00 −0.0774597
\(961\) 81765.0 2.74462
\(962\) 43232.0 1.44891
\(963\) 28428.0 0.951277
\(964\) −27440.0 −0.916787
\(965\) −37620.0 −1.25495
\(966\) −5040.00 −0.167867
\(967\) 36448.0 1.21209 0.606044 0.795431i \(-0.292756\pi\)
0.606044 + 0.795431i \(0.292756\pi\)
\(968\) 0 0
\(969\) 2016.00 0.0668351
\(970\) 47880.0 1.58488
\(971\) −20526.0 −0.678384 −0.339192 0.940717i \(-0.610154\pi\)
−0.339192 + 0.940717i \(0.610154\pi\)
\(972\) −14168.0 −0.467530
\(973\) −3556.00 −0.117164
\(974\) −10264.0 −0.337659
\(975\) 22288.0 0.732089
\(976\) −6080.00 −0.199402
\(977\) 37434.0 1.22581 0.612907 0.790155i \(-0.290000\pi\)
0.612907 + 0.790155i \(0.290000\pi\)
\(978\) −640.000 −0.0209253
\(979\) 0 0
\(980\) 3528.00 0.114998
\(981\) −15962.0 −0.519498
\(982\) 8376.00 0.272188
\(983\) −52194.0 −1.69352 −0.846760 0.531975i \(-0.821450\pi\)
−0.846760 + 0.531975i \(0.821450\pi\)
\(984\) 7104.00 0.230150
\(985\) 28836.0 0.932783
\(986\) 3888.00 0.125577
\(987\) −5628.00 −0.181501
\(988\) −6272.00 −0.201962
\(989\) 56880.0 1.82880
\(990\) 0 0
\(991\) −15220.0 −0.487870 −0.243935 0.969792i \(-0.578438\pi\)
−0.243935 + 0.969792i \(0.578438\pi\)
\(992\) 10688.0 0.342081
\(993\) −13240.0 −0.423121
\(994\) −4536.00 −0.144742
\(995\) −58932.0 −1.87766
\(996\) 3744.00 0.119110
\(997\) −37664.0 −1.19642 −0.598210 0.801339i \(-0.704121\pi\)
−0.598210 + 0.801339i \(0.704121\pi\)
\(998\) −7696.00 −0.244101
\(999\) 38600.0 1.22247
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 1694.4.a.c.1.1 1
11.10 odd 2 154.4.a.d.1.1 1
33.32 even 2 1386.4.a.a.1.1 1
44.43 even 2 1232.4.a.f.1.1 1
77.76 even 2 1078.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.4.a.d.1.1 1 11.10 odd 2
1078.4.a.g.1.1 1 77.76 even 2
1232.4.a.f.1.1 1 44.43 even 2
1386.4.a.a.1.1 1 33.32 even 2
1694.4.a.c.1.1 1 1.1 even 1 trivial