Properties

Label 338.4.a.a.1.1
Level $338$
Weight $4$
Character 338.1
Self dual yes
Analytic conductor $19.943$
Analytic rank $0$
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] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} -18.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} -18.0000 q^{9} -4.00000 q^{10} +13.0000 q^{11} -12.0000 q^{12} +10.0000 q^{14} -6.00000 q^{15} +16.0000 q^{16} +27.0000 q^{17} +36.0000 q^{18} +75.0000 q^{19} +8.00000 q^{20} +15.0000 q^{21} -26.0000 q^{22} -187.000 q^{23} +24.0000 q^{24} -121.000 q^{25} +135.000 q^{27} -20.0000 q^{28} -13.0000 q^{29} +12.0000 q^{30} -104.000 q^{31} -32.0000 q^{32} -39.0000 q^{33} -54.0000 q^{34} -10.0000 q^{35} -72.0000 q^{36} +423.000 q^{37} -150.000 q^{38} -16.0000 q^{40} +195.000 q^{41} -30.0000 q^{42} +199.000 q^{43} +52.0000 q^{44} -36.0000 q^{45} +374.000 q^{46} +388.000 q^{47} -48.0000 q^{48} -318.000 q^{49} +242.000 q^{50} -81.0000 q^{51} +618.000 q^{53} -270.000 q^{54} +26.0000 q^{55} +40.0000 q^{56} -225.000 q^{57} +26.0000 q^{58} +491.000 q^{59} -24.0000 q^{60} +175.000 q^{61} +208.000 q^{62} +90.0000 q^{63} +64.0000 q^{64} +78.0000 q^{66} +817.000 q^{67} +108.000 q^{68} +561.000 q^{69} +20.0000 q^{70} +79.0000 q^{71} +144.000 q^{72} +230.000 q^{73} -846.000 q^{74} +363.000 q^{75} +300.000 q^{76} -65.0000 q^{77} +764.000 q^{79} +32.0000 q^{80} +81.0000 q^{81} -390.000 q^{82} -732.000 q^{83} +60.0000 q^{84} +54.0000 q^{85} -398.000 q^{86} +39.0000 q^{87} -104.000 q^{88} -1041.00 q^{89} +72.0000 q^{90} -748.000 q^{92} +312.000 q^{93} -776.000 q^{94} +150.000 q^{95} +96.0000 q^{96} -97.0000 q^{97} +636.000 q^{98} -234.000 q^{99} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 4.00000 0.500000
\(5\) 2.00000 0.178885 0.0894427 0.995992i \(-0.471491\pi\)
0.0894427 + 0.995992i \(0.471491\pi\)
\(6\) 6.00000 0.408248
\(7\) −5.00000 −0.269975 −0.134987 0.990847i \(-0.543099\pi\)
−0.134987 + 0.990847i \(0.543099\pi\)
\(8\) −8.00000 −0.353553
\(9\) −18.0000 −0.666667
\(10\) −4.00000 −0.126491
\(11\) 13.0000 0.356332 0.178166 0.984000i \(-0.442984\pi\)
0.178166 + 0.984000i \(0.442984\pi\)
\(12\) −12.0000 −0.288675
\(13\) 0 0
\(14\) 10.0000 0.190901
\(15\) −6.00000 −0.103280
\(16\) 16.0000 0.250000
\(17\) 27.0000 0.385204 0.192602 0.981277i \(-0.438307\pi\)
0.192602 + 0.981277i \(0.438307\pi\)
\(18\) 36.0000 0.471405
\(19\) 75.0000 0.905588 0.452794 0.891615i \(-0.350427\pi\)
0.452794 + 0.891615i \(0.350427\pi\)
\(20\) 8.00000 0.0894427
\(21\) 15.0000 0.155870
\(22\) −26.0000 −0.251964
\(23\) −187.000 −1.69531 −0.847656 0.530546i \(-0.821987\pi\)
−0.847656 + 0.530546i \(0.821987\pi\)
\(24\) 24.0000 0.204124
\(25\) −121.000 −0.968000
\(26\) 0 0
\(27\) 135.000 0.962250
\(28\) −20.0000 −0.134987
\(29\) −13.0000 −0.0832427 −0.0416214 0.999133i \(-0.513252\pi\)
−0.0416214 + 0.999133i \(0.513252\pi\)
\(30\) 12.0000 0.0730297
\(31\) −104.000 −0.602547 −0.301273 0.953538i \(-0.597412\pi\)
−0.301273 + 0.953538i \(0.597412\pi\)
\(32\) −32.0000 −0.176777
\(33\) −39.0000 −0.205728
\(34\) −54.0000 −0.272380
\(35\) −10.0000 −0.0482945
\(36\) −72.0000 −0.333333
\(37\) 423.000 1.87948 0.939740 0.341890i \(-0.111067\pi\)
0.939740 + 0.341890i \(0.111067\pi\)
\(38\) −150.000 −0.640348
\(39\) 0 0
\(40\) −16.0000 −0.0632456
\(41\) 195.000 0.742778 0.371389 0.928477i \(-0.378882\pi\)
0.371389 + 0.928477i \(0.378882\pi\)
\(42\) −30.0000 −0.110217
\(43\) 199.000 0.705749 0.352875 0.935671i \(-0.385204\pi\)
0.352875 + 0.935671i \(0.385204\pi\)
\(44\) 52.0000 0.178166
\(45\) −36.0000 −0.119257
\(46\) 374.000 1.19877
\(47\) 388.000 1.20416 0.602081 0.798435i \(-0.294338\pi\)
0.602081 + 0.798435i \(0.294338\pi\)
\(48\) −48.0000 −0.144338
\(49\) −318.000 −0.927114
\(50\) 242.000 0.684479
\(51\) −81.0000 −0.222397
\(52\) 0 0
\(53\) 618.000 1.60168 0.800838 0.598881i \(-0.204388\pi\)
0.800838 + 0.598881i \(0.204388\pi\)
\(54\) −270.000 −0.680414
\(55\) 26.0000 0.0637425
\(56\) 40.0000 0.0954504
\(57\) −225.000 −0.522842
\(58\) 26.0000 0.0588615
\(59\) 491.000 1.08344 0.541718 0.840560i \(-0.317774\pi\)
0.541718 + 0.840560i \(0.317774\pi\)
\(60\) −24.0000 −0.0516398
\(61\) 175.000 0.367319 0.183659 0.982990i \(-0.441206\pi\)
0.183659 + 0.982990i \(0.441206\pi\)
\(62\) 208.000 0.426065
\(63\) 90.0000 0.179983
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 78.0000 0.145472
\(67\) 817.000 1.48974 0.744869 0.667211i \(-0.232512\pi\)
0.744869 + 0.667211i \(0.232512\pi\)
\(68\) 108.000 0.192602
\(69\) 561.000 0.978789
\(70\) 20.0000 0.0341494
\(71\) 79.0000 0.132050 0.0660252 0.997818i \(-0.478968\pi\)
0.0660252 + 0.997818i \(0.478968\pi\)
\(72\) 144.000 0.235702
\(73\) 230.000 0.368760 0.184380 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(74\) −846.000 −1.32899
\(75\) 363.000 0.558875
\(76\) 300.000 0.452794
\(77\) −65.0000 −0.0962005
\(78\) 0 0
\(79\) 764.000 1.08806 0.544030 0.839066i \(-0.316898\pi\)
0.544030 + 0.839066i \(0.316898\pi\)
\(80\) 32.0000 0.0447214
\(81\) 81.0000 0.111111
\(82\) −390.000 −0.525223
\(83\) −732.000 −0.968041 −0.484021 0.875057i \(-0.660824\pi\)
−0.484021 + 0.875057i \(0.660824\pi\)
\(84\) 60.0000 0.0779350
\(85\) 54.0000 0.0689073
\(86\) −398.000 −0.499040
\(87\) 39.0000 0.0480602
\(88\) −104.000 −0.125982
\(89\) −1041.00 −1.23984 −0.619920 0.784665i \(-0.712835\pi\)
−0.619920 + 0.784665i \(0.712835\pi\)
\(90\) 72.0000 0.0843274
\(91\) 0 0
\(92\) −748.000 −0.847656
\(93\) 312.000 0.347881
\(94\) −776.000 −0.851471
\(95\) 150.000 0.161997
\(96\) 96.0000 0.102062
\(97\) −97.0000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 636.000 0.655568
\(99\) −234.000 −0.237554
\(100\) −484.000 −0.484000
\(101\) −809.000 −0.797015 −0.398507 0.917165i \(-0.630472\pi\)
−0.398507 + 0.917165i \(0.630472\pi\)
\(102\) 162.000 0.157259
\(103\) 1288.00 1.23214 0.616070 0.787691i \(-0.288724\pi\)
0.616070 + 0.787691i \(0.288724\pi\)
\(104\) 0 0
\(105\) 30.0000 0.0278829
\(106\) −1236.00 −1.13256
\(107\) 1277.00 1.15376 0.576880 0.816829i \(-0.304270\pi\)
0.576880 + 0.816829i \(0.304270\pi\)
\(108\) 540.000 0.481125
\(109\) 826.000 0.725839 0.362920 0.931820i \(-0.381780\pi\)
0.362920 + 0.931820i \(0.381780\pi\)
\(110\) −52.0000 −0.0450728
\(111\) −1269.00 −1.08512
\(112\) −80.0000 −0.0674937
\(113\) 947.000 0.788374 0.394187 0.919030i \(-0.371026\pi\)
0.394187 + 0.919030i \(0.371026\pi\)
\(114\) 450.000 0.369705
\(115\) −374.000 −0.303267
\(116\) −52.0000 −0.0416214
\(117\) 0 0
\(118\) −982.000 −0.766105
\(119\) −135.000 −0.103995
\(120\) 48.0000 0.0365148
\(121\) −1162.00 −0.873028
\(122\) −350.000 −0.259734
\(123\) −585.000 −0.428843
\(124\) −416.000 −0.301273
\(125\) −492.000 −0.352047
\(126\) −180.000 −0.127267
\(127\) 1177.00 0.822377 0.411188 0.911550i \(-0.365114\pi\)
0.411188 + 0.911550i \(0.365114\pi\)
\(128\) −128.000 −0.0883883
\(129\) −597.000 −0.407464
\(130\) 0 0
\(131\) −1420.00 −0.947069 −0.473534 0.880775i \(-0.657022\pi\)
−0.473534 + 0.880775i \(0.657022\pi\)
\(132\) −156.000 −0.102864
\(133\) −375.000 −0.244486
\(134\) −1634.00 −1.05340
\(135\) 270.000 0.172133
\(136\) −216.000 −0.136190
\(137\) −2409.00 −1.50230 −0.751149 0.660133i \(-0.770500\pi\)
−0.751149 + 0.660133i \(0.770500\pi\)
\(138\) −1122.00 −0.692109
\(139\) 2827.00 1.72506 0.862529 0.506008i \(-0.168879\pi\)
0.862529 + 0.506008i \(0.168879\pi\)
\(140\) −40.0000 −0.0241473
\(141\) −1164.00 −0.695223
\(142\) −158.000 −0.0933737
\(143\) 0 0
\(144\) −288.000 −0.166667
\(145\) −26.0000 −0.0148909
\(146\) −460.000 −0.260753
\(147\) 954.000 0.535269
\(148\) 1692.00 0.939740
\(149\) 855.000 0.470096 0.235048 0.971984i \(-0.424475\pi\)
0.235048 + 0.971984i \(0.424475\pi\)
\(150\) −726.000 −0.395184
\(151\) 2064.00 1.11236 0.556179 0.831063i \(-0.312267\pi\)
0.556179 + 0.831063i \(0.312267\pi\)
\(152\) −600.000 −0.320174
\(153\) −486.000 −0.256802
\(154\) 130.000 0.0680240
\(155\) −208.000 −0.107787
\(156\) 0 0
\(157\) −1894.00 −0.962788 −0.481394 0.876504i \(-0.659869\pi\)
−0.481394 + 0.876504i \(0.659869\pi\)
\(158\) −1528.00 −0.769374
\(159\) −1854.00 −0.924728
\(160\) −64.0000 −0.0316228
\(161\) 935.000 0.457691
\(162\) −162.000 −0.0785674
\(163\) −985.000 −0.473320 −0.236660 0.971593i \(-0.576053\pi\)
−0.236660 + 0.971593i \(0.576053\pi\)
\(164\) 780.000 0.371389
\(165\) −78.0000 −0.0368018
\(166\) 1464.00 0.684509
\(167\) −2355.00 −1.09123 −0.545615 0.838036i \(-0.683704\pi\)
−0.545615 + 0.838036i \(0.683704\pi\)
\(168\) −120.000 −0.0551083
\(169\) 0 0
\(170\) −108.000 −0.0487248
\(171\) −1350.00 −0.603726
\(172\) 796.000 0.352875
\(173\) −3889.00 −1.70911 −0.854553 0.519365i \(-0.826169\pi\)
−0.854553 + 0.519365i \(0.826169\pi\)
\(174\) −78.0000 −0.0339837
\(175\) 605.000 0.261335
\(176\) 208.000 0.0890829
\(177\) −1473.00 −0.625522
\(178\) 2082.00 0.876699
\(179\) 2229.00 0.930745 0.465372 0.885115i \(-0.345920\pi\)
0.465372 + 0.885115i \(0.345920\pi\)
\(180\) −144.000 −0.0596285
\(181\) −1038.00 −0.426265 −0.213132 0.977023i \(-0.568367\pi\)
−0.213132 + 0.977023i \(0.568367\pi\)
\(182\) 0 0
\(183\) −525.000 −0.212072
\(184\) 1496.00 0.599384
\(185\) 846.000 0.336212
\(186\) −624.000 −0.245989
\(187\) 351.000 0.137260
\(188\) 1552.00 0.602081
\(189\) −675.000 −0.259783
\(190\) −300.000 −0.114549
\(191\) −2141.00 −0.811085 −0.405543 0.914076i \(-0.632917\pi\)
−0.405543 + 0.914076i \(0.632917\pi\)
\(192\) −192.000 −0.0721688
\(193\) 2627.00 0.979770 0.489885 0.871787i \(-0.337039\pi\)
0.489885 + 0.871787i \(0.337039\pi\)
\(194\) 194.000 0.0717958
\(195\) 0 0
\(196\) −1272.00 −0.463557
\(197\) 1203.00 0.435077 0.217539 0.976052i \(-0.430197\pi\)
0.217539 + 0.976052i \(0.430197\pi\)
\(198\) 468.000 0.167976
\(199\) 743.000 0.264673 0.132336 0.991205i \(-0.457752\pi\)
0.132336 + 0.991205i \(0.457752\pi\)
\(200\) 968.000 0.342240
\(201\) −2451.00 −0.860101
\(202\) 1618.00 0.563575
\(203\) 65.0000 0.0224734
\(204\) −324.000 −0.111199
\(205\) 390.000 0.132872
\(206\) −2576.00 −0.871254
\(207\) 3366.00 1.13021
\(208\) 0 0
\(209\) 975.000 0.322690
\(210\) −60.0000 −0.0197162
\(211\) −355.000 −0.115826 −0.0579128 0.998322i \(-0.518445\pi\)
−0.0579128 + 0.998322i \(0.518445\pi\)
\(212\) 2472.00 0.800838
\(213\) −237.000 −0.0762393
\(214\) −2554.00 −0.815831
\(215\) 398.000 0.126248
\(216\) −1080.00 −0.340207
\(217\) 520.000 0.162672
\(218\) −1652.00 −0.513246
\(219\) −690.000 −0.212904
\(220\) 104.000 0.0318713
\(221\) 0 0
\(222\) 2538.00 0.767295
\(223\) −2283.00 −0.685565 −0.342782 0.939415i \(-0.611369\pi\)
−0.342782 + 0.939415i \(0.611369\pi\)
\(224\) 160.000 0.0477252
\(225\) 2178.00 0.645333
\(226\) −1894.00 −0.557465
\(227\) 2451.00 0.716646 0.358323 0.933598i \(-0.383349\pi\)
0.358323 + 0.933598i \(0.383349\pi\)
\(228\) −900.000 −0.261421
\(229\) −1878.00 −0.541929 −0.270964 0.962589i \(-0.587343\pi\)
−0.270964 + 0.962589i \(0.587343\pi\)
\(230\) 748.000 0.214442
\(231\) 195.000 0.0555414
\(232\) 104.000 0.0294308
\(233\) 1630.00 0.458304 0.229152 0.973391i \(-0.426405\pi\)
0.229152 + 0.973391i \(0.426405\pi\)
\(234\) 0 0
\(235\) 776.000 0.215407
\(236\) 1964.00 0.541718
\(237\) −2292.00 −0.628192
\(238\) 270.000 0.0735357
\(239\) −5544.00 −1.50047 −0.750233 0.661173i \(-0.770059\pi\)
−0.750233 + 0.661173i \(0.770059\pi\)
\(240\) −96.0000 −0.0258199
\(241\) 5523.00 1.47621 0.738107 0.674683i \(-0.235720\pi\)
0.738107 + 0.674683i \(0.235720\pi\)
\(242\) 2324.00 0.617324
\(243\) −3888.00 −1.02640
\(244\) 700.000 0.183659
\(245\) −636.000 −0.165847
\(246\) 1170.00 0.303238
\(247\) 0 0
\(248\) 832.000 0.213032
\(249\) 2196.00 0.558899
\(250\) 984.000 0.248934
\(251\) 2175.00 0.546951 0.273476 0.961879i \(-0.411827\pi\)
0.273476 + 0.961879i \(0.411827\pi\)
\(252\) 360.000 0.0899915
\(253\) −2431.00 −0.604094
\(254\) −2354.00 −0.581508
\(255\) −162.000 −0.0397837
\(256\) 256.000 0.0625000
\(257\) −5685.00 −1.37985 −0.689923 0.723883i \(-0.742356\pi\)
−0.689923 + 0.723883i \(0.742356\pi\)
\(258\) 1194.00 0.288121
\(259\) −2115.00 −0.507412
\(260\) 0 0
\(261\) 234.000 0.0554952
\(262\) 2840.00 0.669679
\(263\) 6117.00 1.43418 0.717092 0.696979i \(-0.245473\pi\)
0.717092 + 0.696979i \(0.245473\pi\)
\(264\) 312.000 0.0727359
\(265\) 1236.00 0.286517
\(266\) 750.000 0.172878
\(267\) 3123.00 0.715822
\(268\) 3268.00 0.744869
\(269\) −5109.00 −1.15800 −0.578999 0.815329i \(-0.696556\pi\)
−0.578999 + 0.815329i \(0.696556\pi\)
\(270\) −540.000 −0.121716
\(271\) 7549.00 1.69214 0.846068 0.533074i \(-0.178963\pi\)
0.846068 + 0.533074i \(0.178963\pi\)
\(272\) 432.000 0.0963009
\(273\) 0 0
\(274\) 4818.00 1.06228
\(275\) −1573.00 −0.344929
\(276\) 2244.00 0.489395
\(277\) −981.000 −0.212789 −0.106395 0.994324i \(-0.533931\pi\)
−0.106395 + 0.994324i \(0.533931\pi\)
\(278\) −5654.00 −1.21980
\(279\) 1872.00 0.401698
\(280\) 80.0000 0.0170747
\(281\) −2762.00 −0.586360 −0.293180 0.956057i \(-0.594713\pi\)
−0.293180 + 0.956057i \(0.594713\pi\)
\(282\) 2328.00 0.491597
\(283\) 3925.00 0.824442 0.412221 0.911084i \(-0.364753\pi\)
0.412221 + 0.911084i \(0.364753\pi\)
\(284\) 316.000 0.0660252
\(285\) −450.000 −0.0935288
\(286\) 0 0
\(287\) −975.000 −0.200531
\(288\) 576.000 0.117851
\(289\) −4184.00 −0.851618
\(290\) 52.0000 0.0105295
\(291\) 291.000 0.0586210
\(292\) 920.000 0.184380
\(293\) 7711.00 1.53748 0.768740 0.639562i \(-0.220884\pi\)
0.768740 + 0.639562i \(0.220884\pi\)
\(294\) −1908.00 −0.378493
\(295\) 982.000 0.193811
\(296\) −3384.00 −0.664497
\(297\) 1755.00 0.342880
\(298\) −1710.00 −0.332408
\(299\) 0 0
\(300\) 1452.00 0.279438
\(301\) −995.000 −0.190534
\(302\) −4128.00 −0.786555
\(303\) 2427.00 0.460157
\(304\) 1200.00 0.226397
\(305\) 350.000 0.0657080
\(306\) 972.000 0.181587
\(307\) 10388.0 1.93119 0.965594 0.260056i \(-0.0837409\pi\)
0.965594 + 0.260056i \(0.0837409\pi\)
\(308\) −260.000 −0.0481002
\(309\) −3864.00 −0.711376
\(310\) 416.000 0.0762168
\(311\) −7272.00 −1.32591 −0.662954 0.748660i \(-0.730697\pi\)
−0.662954 + 0.748660i \(0.730697\pi\)
\(312\) 0 0
\(313\) 7910.00 1.42843 0.714217 0.699925i \(-0.246783\pi\)
0.714217 + 0.699925i \(0.246783\pi\)
\(314\) 3788.00 0.680794
\(315\) 180.000 0.0321964
\(316\) 3056.00 0.544030
\(317\) 7398.00 1.31077 0.655383 0.755296i \(-0.272507\pi\)
0.655383 + 0.755296i \(0.272507\pi\)
\(318\) 3708.00 0.653881
\(319\) −169.000 −0.0296620
\(320\) 128.000 0.0223607
\(321\) −3831.00 −0.666123
\(322\) −1870.00 −0.323637
\(323\) 2025.00 0.348836
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) 1970.00 0.334688
\(327\) −2478.00 −0.419063
\(328\) −1560.00 −0.262612
\(329\) −1940.00 −0.325093
\(330\) 156.000 0.0260228
\(331\) −2377.00 −0.394718 −0.197359 0.980331i \(-0.563236\pi\)
−0.197359 + 0.980331i \(0.563236\pi\)
\(332\) −2928.00 −0.484021
\(333\) −7614.00 −1.25299
\(334\) 4710.00 0.771616
\(335\) 1634.00 0.266492
\(336\) 240.000 0.0389675
\(337\) −7618.00 −1.23139 −0.615696 0.787984i \(-0.711125\pi\)
−0.615696 + 0.787984i \(0.711125\pi\)
\(338\) 0 0
\(339\) −2841.00 −0.455168
\(340\) 216.000 0.0344537
\(341\) −1352.00 −0.214706
\(342\) 2700.00 0.426898
\(343\) 3305.00 0.520272
\(344\) −1592.00 −0.249520
\(345\) 1122.00 0.175091
\(346\) 7778.00 1.20852
\(347\) 375.000 0.0580146 0.0290073 0.999579i \(-0.490765\pi\)
0.0290073 + 0.999579i \(0.490765\pi\)
\(348\) 156.000 0.0240301
\(349\) 9727.00 1.49190 0.745952 0.666000i \(-0.231995\pi\)
0.745952 + 0.666000i \(0.231995\pi\)
\(350\) −1210.00 −0.184792
\(351\) 0 0
\(352\) −416.000 −0.0629911
\(353\) 2263.00 0.341211 0.170605 0.985339i \(-0.445428\pi\)
0.170605 + 0.985339i \(0.445428\pi\)
\(354\) 2946.00 0.442311
\(355\) 158.000 0.0236219
\(356\) −4164.00 −0.619920
\(357\) 405.000 0.0600417
\(358\) −4458.00 −0.658136
\(359\) −4488.00 −0.659798 −0.329899 0.944016i \(-0.607015\pi\)
−0.329899 + 0.944016i \(0.607015\pi\)
\(360\) 288.000 0.0421637
\(361\) −1234.00 −0.179910
\(362\) 2076.00 0.301415
\(363\) 3486.00 0.504043
\(364\) 0 0
\(365\) 460.000 0.0659658
\(366\) 1050.00 0.149957
\(367\) −1627.00 −0.231413 −0.115707 0.993283i \(-0.536913\pi\)
−0.115707 + 0.993283i \(0.536913\pi\)
\(368\) −2992.00 −0.423828
\(369\) −3510.00 −0.495185
\(370\) −1692.00 −0.237738
\(371\) −3090.00 −0.432412
\(372\) 1248.00 0.173940
\(373\) 2987.00 0.414641 0.207320 0.978273i \(-0.433526\pi\)
0.207320 + 0.978273i \(0.433526\pi\)
\(374\) −702.000 −0.0970576
\(375\) 1476.00 0.203254
\(376\) −3104.00 −0.425736
\(377\) 0 0
\(378\) 1350.00 0.183694
\(379\) −8867.00 −1.20176 −0.600880 0.799339i \(-0.705183\pi\)
−0.600880 + 0.799339i \(0.705183\pi\)
\(380\) 600.000 0.0809983
\(381\) −3531.00 −0.474800
\(382\) 4282.00 0.573524
\(383\) 11403.0 1.52132 0.760661 0.649150i \(-0.224875\pi\)
0.760661 + 0.649150i \(0.224875\pi\)
\(384\) 384.000 0.0510310
\(385\) −130.000 −0.0172089
\(386\) −5254.00 −0.692802
\(387\) −3582.00 −0.470499
\(388\) −388.000 −0.0507673
\(389\) 2622.00 0.341750 0.170875 0.985293i \(-0.445341\pi\)
0.170875 + 0.985293i \(0.445341\pi\)
\(390\) 0 0
\(391\) −5049.00 −0.653041
\(392\) 2544.00 0.327784
\(393\) 4260.00 0.546790
\(394\) −2406.00 −0.307646
\(395\) 1528.00 0.194638
\(396\) −936.000 −0.118777
\(397\) 659.000 0.0833105 0.0416552 0.999132i \(-0.486737\pi\)
0.0416552 + 0.999132i \(0.486737\pi\)
\(398\) −1486.00 −0.187152
\(399\) 1125.00 0.141154
\(400\) −1936.00 −0.242000
\(401\) −14685.0 −1.82876 −0.914381 0.404854i \(-0.867322\pi\)
−0.914381 + 0.404854i \(0.867322\pi\)
\(402\) 4902.00 0.608183
\(403\) 0 0
\(404\) −3236.00 −0.398507
\(405\) 162.000 0.0198762
\(406\) −130.000 −0.0158911
\(407\) 5499.00 0.669718
\(408\) 648.000 0.0786294
\(409\) −7829.00 −0.946502 −0.473251 0.880928i \(-0.656920\pi\)
−0.473251 + 0.880928i \(0.656920\pi\)
\(410\) −780.000 −0.0939548
\(411\) 7227.00 0.867352
\(412\) 5152.00 0.616070
\(413\) −2455.00 −0.292500
\(414\) −6732.00 −0.799178
\(415\) −1464.00 −0.173169
\(416\) 0 0
\(417\) −8481.00 −0.995962
\(418\) −1950.00 −0.228176
\(419\) −2919.00 −0.340340 −0.170170 0.985415i \(-0.554432\pi\)
−0.170170 + 0.985415i \(0.554432\pi\)
\(420\) 120.000 0.0139414
\(421\) −3110.00 −0.360029 −0.180014 0.983664i \(-0.557614\pi\)
−0.180014 + 0.983664i \(0.557614\pi\)
\(422\) 710.000 0.0819011
\(423\) −6984.00 −0.802775
\(424\) −4944.00 −0.566278
\(425\) −3267.00 −0.372877
\(426\) 474.000 0.0539093
\(427\) −875.000 −0.0991668
\(428\) 5108.00 0.576880
\(429\) 0 0
\(430\) −796.000 −0.0892710
\(431\) −9135.00 −1.02092 −0.510461 0.859901i \(-0.670525\pi\)
−0.510461 + 0.859901i \(0.670525\pi\)
\(432\) 2160.00 0.240563
\(433\) −11669.0 −1.29510 −0.647548 0.762025i \(-0.724205\pi\)
−0.647548 + 0.762025i \(0.724205\pi\)
\(434\) −1040.00 −0.115027
\(435\) 78.0000 0.00859727
\(436\) 3304.00 0.362920
\(437\) −14025.0 −1.53526
\(438\) 1380.00 0.150546
\(439\) 13529.0 1.47085 0.735426 0.677605i \(-0.236982\pi\)
0.735426 + 0.677605i \(0.236982\pi\)
\(440\) −208.000 −0.0225364
\(441\) 5724.00 0.618076
\(442\) 0 0
\(443\) −1932.00 −0.207206 −0.103603 0.994619i \(-0.533037\pi\)
−0.103603 + 0.994619i \(0.533037\pi\)
\(444\) −5076.00 −0.542559
\(445\) −2082.00 −0.221789
\(446\) 4566.00 0.484768
\(447\) −2565.00 −0.271410
\(448\) −320.000 −0.0337468
\(449\) −5357.00 −0.563057 −0.281528 0.959553i \(-0.590841\pi\)
−0.281528 + 0.959553i \(0.590841\pi\)
\(450\) −4356.00 −0.456320
\(451\) 2535.00 0.264675
\(452\) 3788.00 0.394187
\(453\) −6192.00 −0.642220
\(454\) −4902.00 −0.506745
\(455\) 0 0
\(456\) 1800.00 0.184852
\(457\) 19399.0 1.98566 0.992830 0.119532i \(-0.0381394\pi\)
0.992830 + 0.119532i \(0.0381394\pi\)
\(458\) 3756.00 0.383202
\(459\) 3645.00 0.370662
\(460\) −1496.00 −0.151633
\(461\) −15549.0 −1.57091 −0.785455 0.618919i \(-0.787571\pi\)
−0.785455 + 0.618919i \(0.787571\pi\)
\(462\) −390.000 −0.0392737
\(463\) 4072.00 0.408730 0.204365 0.978895i \(-0.434487\pi\)
0.204365 + 0.978895i \(0.434487\pi\)
\(464\) −208.000 −0.0208107
\(465\) 624.000 0.0622308
\(466\) −3260.00 −0.324070
\(467\) 15224.0 1.50853 0.754264 0.656571i \(-0.227994\pi\)
0.754264 + 0.656571i \(0.227994\pi\)
\(468\) 0 0
\(469\) −4085.00 −0.402191
\(470\) −1552.00 −0.152316
\(471\) 5682.00 0.555866
\(472\) −3928.00 −0.383053
\(473\) 2587.00 0.251481
\(474\) 4584.00 0.444199
\(475\) −9075.00 −0.876610
\(476\) −540.000 −0.0519976
\(477\) −11124.0 −1.06778
\(478\) 11088.0 1.06099
\(479\) −10335.0 −0.985842 −0.492921 0.870074i \(-0.664071\pi\)
−0.492921 + 0.870074i \(0.664071\pi\)
\(480\) 192.000 0.0182574
\(481\) 0 0
\(482\) −11046.0 −1.04384
\(483\) −2805.00 −0.264248
\(484\) −4648.00 −0.436514
\(485\) −194.000 −0.0181631
\(486\) 7776.00 0.725775
\(487\) 6455.00 0.600624 0.300312 0.953841i \(-0.402909\pi\)
0.300312 + 0.953841i \(0.402909\pi\)
\(488\) −1400.00 −0.129867
\(489\) 2955.00 0.273271
\(490\) 1272.00 0.117272
\(491\) 7777.00 0.714809 0.357404 0.933950i \(-0.383662\pi\)
0.357404 + 0.933950i \(0.383662\pi\)
\(492\) −2340.00 −0.214421
\(493\) −351.000 −0.0320654
\(494\) 0 0
\(495\) −468.000 −0.0424950
\(496\) −1664.00 −0.150637
\(497\) −395.000 −0.0356502
\(498\) −4392.00 −0.395201
\(499\) −3044.00 −0.273082 −0.136541 0.990634i \(-0.543599\pi\)
−0.136541 + 0.990634i \(0.543599\pi\)
\(500\) −1968.00 −0.176023
\(501\) 7065.00 0.630022
\(502\) −4350.00 −0.386753
\(503\) 11347.0 1.00584 0.502920 0.864333i \(-0.332259\pi\)
0.502920 + 0.864333i \(0.332259\pi\)
\(504\) −720.000 −0.0636336
\(505\) −1618.00 −0.142574
\(506\) 4862.00 0.427159
\(507\) 0 0
\(508\) 4708.00 0.411188
\(509\) 727.000 0.0633079 0.0316539 0.999499i \(-0.489923\pi\)
0.0316539 + 0.999499i \(0.489923\pi\)
\(510\) 324.000 0.0281313
\(511\) −1150.00 −0.0995558
\(512\) −512.000 −0.0441942
\(513\) 10125.0 0.871403
\(514\) 11370.0 0.975699
\(515\) 2576.00 0.220412
\(516\) −2388.00 −0.203732
\(517\) 5044.00 0.429081
\(518\) 4230.00 0.358794
\(519\) 11667.0 0.986752
\(520\) 0 0
\(521\) 9582.00 0.805749 0.402874 0.915255i \(-0.368011\pi\)
0.402874 + 0.915255i \(0.368011\pi\)
\(522\) −468.000 −0.0392410
\(523\) −10383.0 −0.868101 −0.434051 0.900889i \(-0.642916\pi\)
−0.434051 + 0.900889i \(0.642916\pi\)
\(524\) −5680.00 −0.473534
\(525\) −1815.00 −0.150882
\(526\) −12234.0 −1.01412
\(527\) −2808.00 −0.232103
\(528\) −624.000 −0.0514320
\(529\) 22802.0 1.87409
\(530\) −2472.00 −0.202598
\(531\) −8838.00 −0.722291
\(532\) −1500.00 −0.122243
\(533\) 0 0
\(534\) −6246.00 −0.506163
\(535\) 2554.00 0.206391
\(536\) −6536.00 −0.526702
\(537\) −6687.00 −0.537366
\(538\) 10218.0 0.818828
\(539\) −4134.00 −0.330360
\(540\) 1080.00 0.0860663
\(541\) 12230.0 0.971920 0.485960 0.873981i \(-0.338470\pi\)
0.485960 + 0.873981i \(0.338470\pi\)
\(542\) −15098.0 −1.19652
\(543\) 3114.00 0.246104
\(544\) −864.000 −0.0680950
\(545\) 1652.00 0.129842
\(546\) 0 0
\(547\) −14636.0 −1.14404 −0.572020 0.820239i \(-0.693840\pi\)
−0.572020 + 0.820239i \(0.693840\pi\)
\(548\) −9636.00 −0.751149
\(549\) −3150.00 −0.244879
\(550\) 3146.00 0.243902
\(551\) −975.000 −0.0753837
\(552\) −4488.00 −0.346054
\(553\) −3820.00 −0.293749
\(554\) 1962.00 0.150465
\(555\) −2538.00 −0.194112
\(556\) 11308.0 0.862529
\(557\) −765.000 −0.0581941 −0.0290970 0.999577i \(-0.509263\pi\)
−0.0290970 + 0.999577i \(0.509263\pi\)
\(558\) −3744.00 −0.284043
\(559\) 0 0
\(560\) −160.000 −0.0120736
\(561\) −1053.00 −0.0792472
\(562\) 5524.00 0.414619
\(563\) 5915.00 0.442784 0.221392 0.975185i \(-0.428940\pi\)
0.221392 + 0.975185i \(0.428940\pi\)
\(564\) −4656.00 −0.347612
\(565\) 1894.00 0.141029
\(566\) −7850.00 −0.582968
\(567\) −405.000 −0.0299972
\(568\) −632.000 −0.0466869
\(569\) −1217.00 −0.0896648 −0.0448324 0.998995i \(-0.514275\pi\)
−0.0448324 + 0.998995i \(0.514275\pi\)
\(570\) 900.000 0.0661348
\(571\) −23436.0 −1.71763 −0.858814 0.512287i \(-0.828798\pi\)
−0.858814 + 0.512287i \(0.828798\pi\)
\(572\) 0 0
\(573\) 6423.00 0.468280
\(574\) 1950.00 0.141797
\(575\) 22627.0 1.64106
\(576\) −1152.00 −0.0833333
\(577\) 7854.00 0.566666 0.283333 0.959022i \(-0.408560\pi\)
0.283333 + 0.959022i \(0.408560\pi\)
\(578\) 8368.00 0.602185
\(579\) −7881.00 −0.565670
\(580\) −104.000 −0.00744546
\(581\) 3660.00 0.261347
\(582\) −582.000 −0.0414513
\(583\) 8034.00 0.570728
\(584\) −1840.00 −0.130376
\(585\) 0 0
\(586\) −15422.0 −1.08716
\(587\) 17033.0 1.19766 0.598831 0.800876i \(-0.295632\pi\)
0.598831 + 0.800876i \(0.295632\pi\)
\(588\) 3816.00 0.267635
\(589\) −7800.00 −0.545659
\(590\) −1964.00 −0.137045
\(591\) −3609.00 −0.251192
\(592\) 6768.00 0.469870
\(593\) −14506.0 −1.00454 −0.502268 0.864712i \(-0.667501\pi\)
−0.502268 + 0.864712i \(0.667501\pi\)
\(594\) −3510.00 −0.242453
\(595\) −270.000 −0.0186032
\(596\) 3420.00 0.235048
\(597\) −2229.00 −0.152809
\(598\) 0 0
\(599\) 15388.0 1.04964 0.524822 0.851212i \(-0.324132\pi\)
0.524822 + 0.851212i \(0.324132\pi\)
\(600\) −2904.00 −0.197592
\(601\) −6077.00 −0.412456 −0.206228 0.978504i \(-0.566119\pi\)
−0.206228 + 0.978504i \(0.566119\pi\)
\(602\) 1990.00 0.134728
\(603\) −14706.0 −0.993159
\(604\) 8256.00 0.556179
\(605\) −2324.00 −0.156172
\(606\) −4854.00 −0.325380
\(607\) 10215.0 0.683054 0.341527 0.939872i \(-0.389056\pi\)
0.341527 + 0.939872i \(0.389056\pi\)
\(608\) −2400.00 −0.160087
\(609\) −195.000 −0.0129750
\(610\) −700.000 −0.0464626
\(611\) 0 0
\(612\) −1944.00 −0.128401
\(613\) −3457.00 −0.227776 −0.113888 0.993494i \(-0.536331\pi\)
−0.113888 + 0.993494i \(0.536331\pi\)
\(614\) −20776.0 −1.36556
\(615\) −1170.00 −0.0767137
\(616\) 520.000 0.0340120
\(617\) −7169.00 −0.467768 −0.233884 0.972264i \(-0.575144\pi\)
−0.233884 + 0.972264i \(0.575144\pi\)
\(618\) 7728.00 0.503019
\(619\) 20212.0 1.31242 0.656211 0.754578i \(-0.272158\pi\)
0.656211 + 0.754578i \(0.272158\pi\)
\(620\) −832.000 −0.0538934
\(621\) −25245.0 −1.63132
\(622\) 14544.0 0.937558
\(623\) 5205.00 0.334725
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) −15820.0 −1.01005
\(627\) −2925.00 −0.186305
\(628\) −7576.00 −0.481394
\(629\) 11421.0 0.723983
\(630\) −360.000 −0.0227663
\(631\) −8945.00 −0.564334 −0.282167 0.959365i \(-0.591053\pi\)
−0.282167 + 0.959365i \(0.591053\pi\)
\(632\) −6112.00 −0.384687
\(633\) 1065.00 0.0668720
\(634\) −14796.0 −0.926852
\(635\) 2354.00 0.147111
\(636\) −7416.00 −0.462364
\(637\) 0 0
\(638\) 338.000 0.0209742
\(639\) −1422.00 −0.0880336
\(640\) −256.000 −0.0158114
\(641\) 28243.0 1.74030 0.870149 0.492788i \(-0.164022\pi\)
0.870149 + 0.492788i \(0.164022\pi\)
\(642\) 7662.00 0.471020
\(643\) 5231.00 0.320825 0.160413 0.987050i \(-0.448718\pi\)
0.160413 + 0.987050i \(0.448718\pi\)
\(644\) 3740.00 0.228846
\(645\) −1194.00 −0.0728895
\(646\) −4050.00 −0.246664
\(647\) −4871.00 −0.295980 −0.147990 0.988989i \(-0.547280\pi\)
−0.147990 + 0.988989i \(0.547280\pi\)
\(648\) −648.000 −0.0392837
\(649\) 6383.00 0.386063
\(650\) 0 0
\(651\) −1560.00 −0.0939189
\(652\) −3940.00 −0.236660
\(653\) 12255.0 0.734418 0.367209 0.930138i \(-0.380313\pi\)
0.367209 + 0.930138i \(0.380313\pi\)
\(654\) 4956.00 0.296323
\(655\) −2840.00 −0.169417
\(656\) 3120.00 0.185694
\(657\) −4140.00 −0.245840
\(658\) 3880.00 0.229876
\(659\) −2145.00 −0.126794 −0.0633971 0.997988i \(-0.520193\pi\)
−0.0633971 + 0.997988i \(0.520193\pi\)
\(660\) −312.000 −0.0184009
\(661\) 2111.00 0.124218 0.0621092 0.998069i \(-0.480217\pi\)
0.0621092 + 0.998069i \(0.480217\pi\)
\(662\) 4754.00 0.279108
\(663\) 0 0
\(664\) 5856.00 0.342254
\(665\) −750.000 −0.0437350
\(666\) 15228.0 0.885996
\(667\) 2431.00 0.141122
\(668\) −9420.00 −0.545615
\(669\) 6849.00 0.395811
\(670\) −3268.00 −0.188439
\(671\) 2275.00 0.130887
\(672\) −480.000 −0.0275542
\(673\) −23273.0 −1.33300 −0.666499 0.745506i \(-0.732208\pi\)
−0.666499 + 0.745506i \(0.732208\pi\)
\(674\) 15236.0 0.870725
\(675\) −16335.0 −0.931458
\(676\) 0 0
\(677\) −5910.00 −0.335509 −0.167755 0.985829i \(-0.553652\pi\)
−0.167755 + 0.985829i \(0.553652\pi\)
\(678\) 5682.00 0.321852
\(679\) 485.000 0.0274118
\(680\) −432.000 −0.0243624
\(681\) −7353.00 −0.413756
\(682\) 2704.00 0.151820
\(683\) 16747.0 0.938223 0.469111 0.883139i \(-0.344574\pi\)
0.469111 + 0.883139i \(0.344574\pi\)
\(684\) −5400.00 −0.301863
\(685\) −4818.00 −0.268739
\(686\) −6610.00 −0.367888
\(687\) 5634.00 0.312883
\(688\) 3184.00 0.176437
\(689\) 0 0
\(690\) −2244.00 −0.123808
\(691\) 10309.0 0.567544 0.283772 0.958892i \(-0.408414\pi\)
0.283772 + 0.958892i \(0.408414\pi\)
\(692\) −15556.0 −0.854553
\(693\) 1170.00 0.0641337
\(694\) −750.000 −0.0410225
\(695\) 5654.00 0.308588
\(696\) −312.000 −0.0169919
\(697\) 5265.00 0.286121
\(698\) −19454.0 −1.05494
\(699\) −4890.00 −0.264602
\(700\) 2420.00 0.130668
\(701\) −24294.0 −1.30895 −0.654473 0.756085i \(-0.727110\pi\)
−0.654473 + 0.756085i \(0.727110\pi\)
\(702\) 0 0
\(703\) 31725.0 1.70204
\(704\) 832.000 0.0445414
\(705\) −2328.00 −0.124365
\(706\) −4526.00 −0.241272
\(707\) 4045.00 0.215174
\(708\) −5892.00 −0.312761
\(709\) 12659.0 0.670548 0.335274 0.942121i \(-0.391171\pi\)
0.335274 + 0.942121i \(0.391171\pi\)
\(710\) −316.000 −0.0167032
\(711\) −13752.0 −0.725373
\(712\) 8328.00 0.438350
\(713\) 19448.0 1.02151
\(714\) −810.000 −0.0424559
\(715\) 0 0
\(716\) 8916.00 0.465372
\(717\) 16632.0 0.866295
\(718\) 8976.00 0.466548
\(719\) 13091.0 0.679015 0.339508 0.940603i \(-0.389740\pi\)
0.339508 + 0.940603i \(0.389740\pi\)
\(720\) −576.000 −0.0298142
\(721\) −6440.00 −0.332647
\(722\) 2468.00 0.127215
\(723\) −16569.0 −0.852293
\(724\) −4152.00 −0.213132
\(725\) 1573.00 0.0805790
\(726\) −6972.00 −0.356412
\(727\) 10792.0 0.550555 0.275277 0.961365i \(-0.411230\pi\)
0.275277 + 0.961365i \(0.411230\pi\)
\(728\) 0 0
\(729\) 9477.00 0.481481
\(730\) −920.000 −0.0466448
\(731\) 5373.00 0.271857
\(732\) −2100.00 −0.106036
\(733\) −2698.00 −0.135952 −0.0679761 0.997687i \(-0.521654\pi\)
−0.0679761 + 0.997687i \(0.521654\pi\)
\(734\) 3254.00 0.163634
\(735\) 1908.00 0.0957519
\(736\) 5984.00 0.299692
\(737\) 10621.0 0.530841
\(738\) 7020.00 0.350149
\(739\) 2841.00 0.141418 0.0707090 0.997497i \(-0.477474\pi\)
0.0707090 + 0.997497i \(0.477474\pi\)
\(740\) 3384.00 0.168106
\(741\) 0 0
\(742\) 6180.00 0.305761
\(743\) −9191.00 −0.453816 −0.226908 0.973916i \(-0.572862\pi\)
−0.226908 + 0.973916i \(0.572862\pi\)
\(744\) −2496.00 −0.122994
\(745\) 1710.00 0.0840934
\(746\) −5974.00 −0.293195
\(747\) 13176.0 0.645361
\(748\) 1404.00 0.0686301
\(749\) −6385.00 −0.311486
\(750\) −2952.00 −0.143722
\(751\) −1659.00 −0.0806095 −0.0403048 0.999187i \(-0.512833\pi\)
−0.0403048 + 0.999187i \(0.512833\pi\)
\(752\) 6208.00 0.301041
\(753\) −6525.00 −0.315782
\(754\) 0 0
\(755\) 4128.00 0.198985
\(756\) −2700.00 −0.129892
\(757\) −13929.0 −0.668769 −0.334384 0.942437i \(-0.608528\pi\)
−0.334384 + 0.942437i \(0.608528\pi\)
\(758\) 17734.0 0.849773
\(759\) 7293.00 0.348774
\(760\) −1200.00 −0.0572744
\(761\) 4587.00 0.218500 0.109250 0.994014i \(-0.465155\pi\)
0.109250 + 0.994014i \(0.465155\pi\)
\(762\) 7062.00 0.335734
\(763\) −4130.00 −0.195958
\(764\) −8564.00 −0.405543
\(765\) −972.000 −0.0459382
\(766\) −22806.0 −1.07574
\(767\) 0 0
\(768\) −768.000 −0.0360844
\(769\) 14499.0 0.679905 0.339953 0.940443i \(-0.389589\pi\)
0.339953 + 0.940443i \(0.389589\pi\)
\(770\) 260.000 0.0121685
\(771\) 17055.0 0.796655
\(772\) 10508.0 0.489885
\(773\) 3059.00 0.142335 0.0711673 0.997464i \(-0.477328\pi\)
0.0711673 + 0.997464i \(0.477328\pi\)
\(774\) 7164.00 0.332693
\(775\) 12584.0 0.583265
\(776\) 776.000 0.0358979
\(777\) 6345.00 0.292954
\(778\) −5244.00 −0.241654
\(779\) 14625.0 0.672651
\(780\) 0 0
\(781\) 1027.00 0.0470537
\(782\) 10098.0 0.461769
\(783\) −1755.00 −0.0801004
\(784\) −5088.00 −0.231778
\(785\) −3788.00 −0.172229
\(786\) −8520.00 −0.386639
\(787\) 36407.0 1.64901 0.824504 0.565856i \(-0.191454\pi\)
0.824504 + 0.565856i \(0.191454\pi\)
\(788\) 4812.00 0.217539
\(789\) −18351.0 −0.828026
\(790\) −3056.00 −0.137630
\(791\) −4735.00 −0.212841
\(792\) 1872.00 0.0839882
\(793\) 0 0
\(794\) −1318.00 −0.0589094
\(795\) −3708.00 −0.165420
\(796\) 2972.00 0.132336
\(797\) −13137.0 −0.583860 −0.291930 0.956440i \(-0.594297\pi\)
−0.291930 + 0.956440i \(0.594297\pi\)
\(798\) −2250.00 −0.0998109
\(799\) 10476.0 0.463848
\(800\) 3872.00 0.171120
\(801\) 18738.0 0.826560
\(802\) 29370.0 1.29313
\(803\) 2990.00 0.131401
\(804\) −9804.00 −0.430050
\(805\) 1870.00 0.0818743
\(806\) 0 0
\(807\) 15327.0 0.668570
\(808\) 6472.00 0.281787
\(809\) 15411.0 0.669743 0.334871 0.942264i \(-0.391307\pi\)
0.334871 + 0.942264i \(0.391307\pi\)
\(810\) −324.000 −0.0140546
\(811\) −27664.0 −1.19780 −0.598899 0.800824i \(-0.704395\pi\)
−0.598899 + 0.800824i \(0.704395\pi\)
\(812\) 260.000 0.0112367
\(813\) −22647.0 −0.976956
\(814\) −10998.0 −0.473562
\(815\) −1970.00 −0.0846700
\(816\) −1296.00 −0.0555994
\(817\) 14925.0 0.639118
\(818\) 15658.0 0.669278
\(819\) 0 0
\(820\) 1560.00 0.0664361
\(821\) −21397.0 −0.909574 −0.454787 0.890600i \(-0.650285\pi\)
−0.454787 + 0.890600i \(0.650285\pi\)
\(822\) −14454.0 −0.613310
\(823\) −24249.0 −1.02706 −0.513528 0.858073i \(-0.671662\pi\)
−0.513528 + 0.858073i \(0.671662\pi\)
\(824\) −10304.0 −0.435627
\(825\) 4719.00 0.199145
\(826\) 4910.00 0.206829
\(827\) 14028.0 0.589844 0.294922 0.955521i \(-0.404706\pi\)
0.294922 + 0.955521i \(0.404706\pi\)
\(828\) 13464.0 0.565104
\(829\) 30451.0 1.27576 0.637881 0.770135i \(-0.279811\pi\)
0.637881 + 0.770135i \(0.279811\pi\)
\(830\) 2928.00 0.122449
\(831\) 2943.00 0.122854
\(832\) 0 0
\(833\) −8586.00 −0.357128
\(834\) 16962.0 0.704252
\(835\) −4710.00 −0.195205
\(836\) 3900.00 0.161345
\(837\) −14040.0 −0.579801
\(838\) 5838.00 0.240657
\(839\) 20591.0 0.847295 0.423647 0.905827i \(-0.360750\pi\)
0.423647 + 0.905827i \(0.360750\pi\)
\(840\) −240.000 −0.00985808
\(841\) −24220.0 −0.993071
\(842\) 6220.00 0.254579
\(843\) 8286.00 0.338535
\(844\) −1420.00 −0.0579128
\(845\) 0 0
\(846\) 13968.0 0.567647
\(847\) 5810.00 0.235695
\(848\) 9888.00 0.400419
\(849\) −11775.0 −0.475992
\(850\) 6534.00 0.263664
\(851\) −79101.0 −3.18631
\(852\) −948.000 −0.0381197
\(853\) −5798.00 −0.232731 −0.116366 0.993206i \(-0.537124\pi\)
−0.116366 + 0.993206i \(0.537124\pi\)
\(854\) 1750.00 0.0701215
\(855\) −2700.00 −0.107998
\(856\) −10216.0 −0.407916
\(857\) 5686.00 0.226640 0.113320 0.993559i \(-0.463852\pi\)
0.113320 + 0.993559i \(0.463852\pi\)
\(858\) 0 0
\(859\) −46708.0 −1.85525 −0.927623 0.373518i \(-0.878152\pi\)
−0.927623 + 0.373518i \(0.878152\pi\)
\(860\) 1592.00 0.0631241
\(861\) 2925.00 0.115777
\(862\) 18270.0 0.721901
\(863\) 25168.0 0.992733 0.496367 0.868113i \(-0.334667\pi\)
0.496367 + 0.868113i \(0.334667\pi\)
\(864\) −4320.00 −0.170103
\(865\) −7778.00 −0.305734
\(866\) 23338.0 0.915771
\(867\) 12552.0 0.491682
\(868\) 2080.00 0.0813362
\(869\) 9932.00 0.387710
\(870\) −156.000 −0.00607919
\(871\) 0 0
\(872\) −6608.00 −0.256623
\(873\) 1746.00 0.0676897
\(874\) 28050.0 1.08559
\(875\) 2460.00 0.0950436
\(876\) −2760.00 −0.106452
\(877\) 18663.0 0.718591 0.359296 0.933224i \(-0.383017\pi\)
0.359296 + 0.933224i \(0.383017\pi\)
\(878\) −27058.0 −1.04005
\(879\) −23133.0 −0.887664
\(880\) 416.000 0.0159356
\(881\) 4971.00 0.190099 0.0950495 0.995473i \(-0.469699\pi\)
0.0950495 + 0.995473i \(0.469699\pi\)
\(882\) −11448.0 −0.437046
\(883\) 6892.00 0.262666 0.131333 0.991338i \(-0.458074\pi\)
0.131333 + 0.991338i \(0.458074\pi\)
\(884\) 0 0
\(885\) −2946.00 −0.111897
\(886\) 3864.00 0.146516
\(887\) −24047.0 −0.910281 −0.455140 0.890420i \(-0.650411\pi\)
−0.455140 + 0.890420i \(0.650411\pi\)
\(888\) 10152.0 0.383647
\(889\) −5885.00 −0.222021
\(890\) 4164.00 0.156829
\(891\) 1053.00 0.0395924
\(892\) −9132.00 −0.342782
\(893\) 29100.0 1.09048
\(894\) 5130.00 0.191916
\(895\) 4458.00 0.166497
\(896\) 640.000 0.0238626
\(897\) 0 0
\(898\) 10714.0 0.398141
\(899\) 1352.00 0.0501576
\(900\) 8712.00 0.322667
\(901\) 16686.0 0.616971
\(902\) −5070.00 −0.187154
\(903\) 2985.00 0.110005
\(904\) −7576.00 −0.278732
\(905\) −2076.00 −0.0762526
\(906\) 12384.0 0.454118
\(907\) −12843.0 −0.470171 −0.235085 0.971975i \(-0.575537\pi\)
−0.235085 + 0.971975i \(0.575537\pi\)
\(908\) 9804.00 0.358323
\(909\) 14562.0 0.531343
\(910\) 0 0
\(911\) −144.000 −0.00523703 −0.00261851 0.999997i \(-0.500833\pi\)
−0.00261851 + 0.999997i \(0.500833\pi\)
\(912\) −3600.00 −0.130710
\(913\) −9516.00 −0.344944
\(914\) −38798.0 −1.40407
\(915\) −1050.00 −0.0379365
\(916\) −7512.00 −0.270964
\(917\) 7100.00 0.255684
\(918\) −7290.00 −0.262098
\(919\) −11061.0 −0.397028 −0.198514 0.980098i \(-0.563612\pi\)
−0.198514 + 0.980098i \(0.563612\pi\)
\(920\) 2992.00 0.107221
\(921\) −31164.0 −1.11497
\(922\) 31098.0 1.11080
\(923\) 0 0
\(924\) 780.000 0.0277707
\(925\) −51183.0 −1.81934
\(926\) −8144.00 −0.289016
\(927\) −23184.0 −0.821427
\(928\) 416.000 0.0147154
\(929\) 26307.0 0.929069 0.464534 0.885555i \(-0.346222\pi\)
0.464534 + 0.885555i \(0.346222\pi\)
\(930\) −1248.00 −0.0440038
\(931\) −23850.0 −0.839583
\(932\) 6520.00 0.229152
\(933\) 21816.0 0.765513
\(934\) −30448.0 −1.06669
\(935\) 702.000 0.0245539
\(936\) 0 0
\(937\) −46074.0 −1.60637 −0.803187 0.595727i \(-0.796864\pi\)
−0.803187 + 0.595727i \(0.796864\pi\)
\(938\) 8170.00 0.284392
\(939\) −23730.0 −0.824706
\(940\) 3104.00 0.107704
\(941\) 36118.0 1.25124 0.625618 0.780130i \(-0.284847\pi\)
0.625618 + 0.780130i \(0.284847\pi\)
\(942\) −11364.0 −0.393056
\(943\) −36465.0 −1.25924
\(944\) 7856.00 0.270859
\(945\) −1350.00 −0.0464714
\(946\) −5174.00 −0.177824
\(947\) −55515.0 −1.90496 −0.952479 0.304604i \(-0.901476\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(948\) −9168.00 −0.314096
\(949\) 0 0
\(950\) 18150.0 0.619857
\(951\) −22194.0 −0.756772
\(952\) 1080.00 0.0367679
\(953\) −5353.00 −0.181952 −0.0909762 0.995853i \(-0.528999\pi\)
−0.0909762 + 0.995853i \(0.528999\pi\)
\(954\) 22248.0 0.755037
\(955\) −4282.00 −0.145091
\(956\) −22176.0 −0.750233
\(957\) 507.000 0.0171254
\(958\) 20670.0 0.697095
\(959\) 12045.0 0.405582
\(960\) −384.000 −0.0129099
\(961\) −18975.0 −0.636937
\(962\) 0 0
\(963\) −22986.0 −0.769173
\(964\) 22092.0 0.738107
\(965\) 5254.00 0.175267
\(966\) 5610.00 0.186852
\(967\) −5488.00 −0.182505 −0.0912524 0.995828i \(-0.529087\pi\)
−0.0912524 + 0.995828i \(0.529087\pi\)
\(968\) 9296.00 0.308662
\(969\) −6075.00 −0.201401
\(970\) 388.000 0.0128432
\(971\) −37353.0 −1.23452 −0.617258 0.786761i \(-0.711756\pi\)
−0.617258 + 0.786761i \(0.711756\pi\)
\(972\) −15552.0 −0.513200
\(973\) −14135.0 −0.465722
\(974\) −12910.0 −0.424705
\(975\) 0 0
\(976\) 2800.00 0.0918297
\(977\) −12729.0 −0.416824 −0.208412 0.978041i \(-0.566829\pi\)
−0.208412 + 0.978041i \(0.566829\pi\)
\(978\) −5910.00 −0.193232
\(979\) −13533.0 −0.441794
\(980\) −2544.00 −0.0829236
\(981\) −14868.0 −0.483893
\(982\) −15554.0 −0.505446
\(983\) −56128.0 −1.82116 −0.910582 0.413327i \(-0.864367\pi\)
−0.910582 + 0.413327i \(0.864367\pi\)
\(984\) 4680.00 0.151619
\(985\) 2406.00 0.0778290
\(986\) 702.000 0.0226737
\(987\) 5820.00 0.187693
\(988\) 0 0
\(989\) −37213.0 −1.19647
\(990\) 936.000 0.0300485
\(991\) 47001.0 1.50660 0.753298 0.657680i \(-0.228462\pi\)
0.753298 + 0.657680i \(0.228462\pi\)
\(992\) 3328.00 0.106516
\(993\) 7131.00 0.227891
\(994\) 790.000 0.0252085
\(995\) 1486.00 0.0473461
\(996\) 8784.00 0.279449
\(997\) −24433.0 −0.776129 −0.388065 0.921632i \(-0.626856\pi\)
−0.388065 + 0.921632i \(0.626856\pi\)
\(998\) 6088.00 0.193098
\(999\) 57105.0 1.80853
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 338.4.a.a.1.1 1
13.2 odd 12 338.4.e.d.147.1 4
13.3 even 3 26.4.c.a.9.1 yes 2
13.4 even 6 338.4.c.d.315.1 2
13.5 odd 4 338.4.b.a.337.2 2
13.6 odd 12 338.4.e.d.23.2 4
13.7 odd 12 338.4.e.d.23.1 4
13.8 odd 4 338.4.b.a.337.1 2
13.9 even 3 26.4.c.a.3.1 2
13.10 even 6 338.4.c.d.191.1 2
13.11 odd 12 338.4.e.d.147.2 4
13.12 even 2 338.4.a.d.1.1 1
39.29 odd 6 234.4.h.b.217.1 2
39.35 odd 6 234.4.h.b.55.1 2
52.3 odd 6 208.4.i.a.113.1 2
52.35 odd 6 208.4.i.a.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.c.a.3.1 2 13.9 even 3
26.4.c.a.9.1 yes 2 13.3 even 3
208.4.i.a.81.1 2 52.35 odd 6
208.4.i.a.113.1 2 52.3 odd 6
234.4.h.b.55.1 2 39.35 odd 6
234.4.h.b.217.1 2 39.29 odd 6
338.4.a.a.1.1 1 1.1 even 1 trivial
338.4.a.d.1.1 1 13.12 even 2
338.4.b.a.337.1 2 13.8 odd 4
338.4.b.a.337.2 2 13.5 odd 4
338.4.c.d.191.1 2 13.10 even 6
338.4.c.d.315.1 2 13.4 even 6
338.4.e.d.23.1 4 13.7 odd 12
338.4.e.d.23.2 4 13.6 odd 12
338.4.e.d.147.1 4 13.2 odd 12
338.4.e.d.147.2 4 13.11 odd 12