Properties

Label 338.4.a.e.1.1
Level $338$
Weight $4$
Character 338.1
Self dual yes
Analytic conductor $19.943$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.0000 q^{5} +6.00000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -18.0000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.0000 q^{5} +6.00000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -18.0000 q^{9} -22.0000 q^{10} +38.0000 q^{11} +12.0000 q^{12} -38.0000 q^{14} -33.0000 q^{15} +16.0000 q^{16} -51.0000 q^{17} -36.0000 q^{18} -90.0000 q^{19} -44.0000 q^{20} -57.0000 q^{21} +76.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} -4.00000 q^{25} -135.000 q^{27} -76.0000 q^{28} -190.000 q^{29} -66.0000 q^{30} -292.000 q^{31} +32.0000 q^{32} +114.000 q^{33} -102.000 q^{34} +209.000 q^{35} -72.0000 q^{36} +441.000 q^{37} -180.000 q^{38} -88.0000 q^{40} -312.000 q^{41} -114.000 q^{42} +373.000 q^{43} +152.000 q^{44} +198.000 q^{45} -104.000 q^{46} +41.0000 q^{47} +48.0000 q^{48} +18.0000 q^{49} -8.00000 q^{50} -153.000 q^{51} +468.000 q^{53} -270.000 q^{54} -418.000 q^{55} -152.000 q^{56} -270.000 q^{57} -380.000 q^{58} -530.000 q^{59} -132.000 q^{60} +592.000 q^{61} -584.000 q^{62} +342.000 q^{63} +64.0000 q^{64} +228.000 q^{66} +206.000 q^{67} -204.000 q^{68} -156.000 q^{69} +418.000 q^{70} +863.000 q^{71} -144.000 q^{72} +322.000 q^{73} +882.000 q^{74} -12.0000 q^{75} -360.000 q^{76} -722.000 q^{77} -460.000 q^{79} -176.000 q^{80} +81.0000 q^{81} -624.000 q^{82} -528.000 q^{83} -228.000 q^{84} +561.000 q^{85} +746.000 q^{86} -570.000 q^{87} +304.000 q^{88} -870.000 q^{89} +396.000 q^{90} -208.000 q^{92} -876.000 q^{93} +82.0000 q^{94} +990.000 q^{95} +96.0000 q^{96} +346.000 q^{97} +36.0000 q^{98} -684.000 q^{99} +O(q^{100})\)

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 \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 4.00000 0.500000
\(5\) −11.0000 −0.983870 −0.491935 0.870632i \(-0.663710\pi\)
−0.491935 + 0.870632i \(0.663710\pi\)
\(6\) 6.00000 0.408248
\(7\) −19.0000 −1.02590 −0.512952 0.858417i \(-0.671448\pi\)
−0.512952 + 0.858417i \(0.671448\pi\)
\(8\) 8.00000 0.353553
\(9\) −18.0000 −0.666667
\(10\) −22.0000 −0.695701
\(11\) 38.0000 1.04158 0.520792 0.853683i \(-0.325637\pi\)
0.520792 + 0.853683i \(0.325637\pi\)
\(12\) 12.0000 0.288675
\(13\) 0 0
\(14\) −38.0000 −0.725423
\(15\) −33.0000 −0.568038
\(16\) 16.0000 0.250000
\(17\) −51.0000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −36.0000 −0.471405
\(19\) −90.0000 −1.08671 −0.543353 0.839504i \(-0.682845\pi\)
−0.543353 + 0.839504i \(0.682845\pi\)
\(20\) −44.0000 −0.491935
\(21\) −57.0000 −0.592306
\(22\) 76.0000 0.736512
\(23\) −52.0000 −0.471424 −0.235712 0.971823i \(-0.575742\pi\)
−0.235712 + 0.971823i \(0.575742\pi\)
\(24\) 24.0000 0.204124
\(25\) −4.00000 −0.0320000
\(26\) 0 0
\(27\) −135.000 −0.962250
\(28\) −76.0000 −0.512952
\(29\) −190.000 −1.21662 −0.608312 0.793698i \(-0.708153\pi\)
−0.608312 + 0.793698i \(0.708153\pi\)
\(30\) −66.0000 −0.401663
\(31\) −292.000 −1.69177 −0.845883 0.533368i \(-0.820926\pi\)
−0.845883 + 0.533368i \(0.820926\pi\)
\(32\) 32.0000 0.176777
\(33\) 114.000 0.601359
\(34\) −102.000 −0.514496
\(35\) 209.000 1.00936
\(36\) −72.0000 −0.333333
\(37\) 441.000 1.95946 0.979729 0.200327i \(-0.0642004\pi\)
0.979729 + 0.200327i \(0.0642004\pi\)
\(38\) −180.000 −0.768417
\(39\) 0 0
\(40\) −88.0000 −0.347851
\(41\) −312.000 −1.18844 −0.594222 0.804301i \(-0.702540\pi\)
−0.594222 + 0.804301i \(0.702540\pi\)
\(42\) −114.000 −0.418823
\(43\) 373.000 1.32284 0.661418 0.750017i \(-0.269955\pi\)
0.661418 + 0.750017i \(0.269955\pi\)
\(44\) 152.000 0.520792
\(45\) 198.000 0.655913
\(46\) −104.000 −0.333347
\(47\) 41.0000 0.127244 0.0636220 0.997974i \(-0.479735\pi\)
0.0636220 + 0.997974i \(0.479735\pi\)
\(48\) 48.0000 0.144338
\(49\) 18.0000 0.0524781
\(50\) −8.00000 −0.0226274
\(51\) −153.000 −0.420084
\(52\) 0 0
\(53\) 468.000 1.21292 0.606460 0.795114i \(-0.292589\pi\)
0.606460 + 0.795114i \(0.292589\pi\)
\(54\) −270.000 −0.680414
\(55\) −418.000 −1.02478
\(56\) −152.000 −0.362712
\(57\) −270.000 −0.627410
\(58\) −380.000 −0.860284
\(59\) −530.000 −1.16949 −0.584747 0.811216i \(-0.698806\pi\)
−0.584747 + 0.811216i \(0.698806\pi\)
\(60\) −132.000 −0.284019
\(61\) 592.000 1.24259 0.621294 0.783578i \(-0.286607\pi\)
0.621294 + 0.783578i \(0.286607\pi\)
\(62\) −584.000 −1.19626
\(63\) 342.000 0.683936
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 228.000 0.425225
\(67\) 206.000 0.375625 0.187813 0.982205i \(-0.439860\pi\)
0.187813 + 0.982205i \(0.439860\pi\)
\(68\) −204.000 −0.363803
\(69\) −156.000 −0.272177
\(70\) 418.000 0.713722
\(71\) 863.000 1.44252 0.721262 0.692662i \(-0.243562\pi\)
0.721262 + 0.692662i \(0.243562\pi\)
\(72\) −144.000 −0.235702
\(73\) 322.000 0.516264 0.258132 0.966110i \(-0.416893\pi\)
0.258132 + 0.966110i \(0.416893\pi\)
\(74\) 882.000 1.38555
\(75\) −12.0000 −0.0184752
\(76\) −360.000 −0.543353
\(77\) −722.000 −1.06857
\(78\) 0 0
\(79\) −460.000 −0.655114 −0.327557 0.944831i \(-0.606225\pi\)
−0.327557 + 0.944831i \(0.606225\pi\)
\(80\) −176.000 −0.245967
\(81\) 81.0000 0.111111
\(82\) −624.000 −0.840357
\(83\) −528.000 −0.698259 −0.349130 0.937074i \(-0.613523\pi\)
−0.349130 + 0.937074i \(0.613523\pi\)
\(84\) −228.000 −0.296153
\(85\) 561.000 0.715871
\(86\) 746.000 0.935387
\(87\) −570.000 −0.702419
\(88\) 304.000 0.368256
\(89\) −870.000 −1.03618 −0.518089 0.855327i \(-0.673356\pi\)
−0.518089 + 0.855327i \(0.673356\pi\)
\(90\) 396.000 0.463801
\(91\) 0 0
\(92\) −208.000 −0.235712
\(93\) −876.000 −0.976742
\(94\) 82.0000 0.0899750
\(95\) 990.000 1.06918
\(96\) 96.0000 0.102062
\(97\) 346.000 0.362175 0.181088 0.983467i \(-0.442038\pi\)
0.181088 + 0.983467i \(0.442038\pi\)
\(98\) 36.0000 0.0371076
\(99\) −684.000 −0.694390
\(100\) −16.0000 −0.0160000
\(101\) 1492.00 1.46990 0.734948 0.678123i \(-0.237206\pi\)
0.734948 + 0.678123i \(0.237206\pi\)
\(102\) −306.000 −0.297044
\(103\) −152.000 −0.145408 −0.0727039 0.997354i \(-0.523163\pi\)
−0.0727039 + 0.997354i \(0.523163\pi\)
\(104\) 0 0
\(105\) 627.000 0.582752
\(106\) 936.000 0.857664
\(107\) 764.000 0.690268 0.345134 0.938553i \(-0.387833\pi\)
0.345134 + 0.938553i \(0.387833\pi\)
\(108\) −540.000 −0.481125
\(109\) −1135.00 −0.997370 −0.498685 0.866783i \(-0.666183\pi\)
−0.498685 + 0.866783i \(0.666183\pi\)
\(110\) −836.000 −0.724632
\(111\) 1323.00 1.13129
\(112\) −304.000 −0.256476
\(113\) −1822.00 −1.51681 −0.758404 0.651785i \(-0.774021\pi\)
−0.758404 + 0.651785i \(0.774021\pi\)
\(114\) −540.000 −0.443646
\(115\) 572.000 0.463820
\(116\) −760.000 −0.608312
\(117\) 0 0
\(118\) −1060.00 −0.826957
\(119\) 969.000 0.746454
\(120\) −264.000 −0.200832
\(121\) 113.000 0.0848986
\(122\) 1184.00 0.878642
\(123\) −936.000 −0.686149
\(124\) −1168.00 −0.845883
\(125\) 1419.00 1.01535
\(126\) 684.000 0.483616
\(127\) −1256.00 −0.877575 −0.438787 0.898591i \(-0.644592\pi\)
−0.438787 + 0.898591i \(0.644592\pi\)
\(128\) 128.000 0.0883883
\(129\) 1119.00 0.763740
\(130\) 0 0
\(131\) 1097.00 0.731644 0.365822 0.930685i \(-0.380788\pi\)
0.365822 + 0.930685i \(0.380788\pi\)
\(132\) 456.000 0.300680
\(133\) 1710.00 1.11486
\(134\) 412.000 0.265607
\(135\) 1485.00 0.946729
\(136\) −408.000 −0.257248
\(137\) 156.000 0.0972845 0.0486423 0.998816i \(-0.484511\pi\)
0.0486423 + 0.998816i \(0.484511\pi\)
\(138\) −312.000 −0.192458
\(139\) −2015.00 −1.22957 −0.614784 0.788695i \(-0.710757\pi\)
−0.614784 + 0.788695i \(0.710757\pi\)
\(140\) 836.000 0.504678
\(141\) 123.000 0.0734643
\(142\) 1726.00 1.02002
\(143\) 0 0
\(144\) −288.000 −0.166667
\(145\) 2090.00 1.19700
\(146\) 644.000 0.365054
\(147\) 54.0000 0.0302983
\(148\) 1764.00 0.979729
\(149\) 1050.00 0.577311 0.288656 0.957433i \(-0.406792\pi\)
0.288656 + 0.957433i \(0.406792\pi\)
\(150\) −24.0000 −0.0130639
\(151\) −1917.00 −1.03313 −0.516567 0.856247i \(-0.672790\pi\)
−0.516567 + 0.856247i \(0.672790\pi\)
\(152\) −720.000 −0.384209
\(153\) 918.000 0.485071
\(154\) −1444.00 −0.755590
\(155\) 3212.00 1.66448
\(156\) 0 0
\(157\) −1546.00 −0.785887 −0.392943 0.919563i \(-0.628543\pi\)
−0.392943 + 0.919563i \(0.628543\pi\)
\(158\) −920.000 −0.463236
\(159\) 1404.00 0.700280
\(160\) −352.000 −0.173925
\(161\) 988.000 0.483635
\(162\) 162.000 0.0785674
\(163\) −668.000 −0.320993 −0.160496 0.987036i \(-0.551309\pi\)
−0.160496 + 0.987036i \(0.551309\pi\)
\(164\) −1248.00 −0.594222
\(165\) −1254.00 −0.591659
\(166\) −1056.00 −0.493744
\(167\) 936.000 0.433712 0.216856 0.976204i \(-0.430420\pi\)
0.216856 + 0.976204i \(0.430420\pi\)
\(168\) −456.000 −0.209412
\(169\) 0 0
\(170\) 1122.00 0.506197
\(171\) 1620.00 0.724471
\(172\) 1492.00 0.661418
\(173\) 1508.00 0.662723 0.331362 0.943504i \(-0.392492\pi\)
0.331362 + 0.943504i \(0.392492\pi\)
\(174\) −1140.00 −0.496685
\(175\) 76.0000 0.0328289
\(176\) 608.000 0.260396
\(177\) −1590.00 −0.675207
\(178\) −1740.00 −0.732688
\(179\) 1785.00 0.745347 0.372674 0.927962i \(-0.378441\pi\)
0.372674 + 0.927962i \(0.378441\pi\)
\(180\) 792.000 0.327957
\(181\) −1008.00 −0.413945 −0.206973 0.978347i \(-0.566361\pi\)
−0.206973 + 0.978347i \(0.566361\pi\)
\(182\) 0 0
\(183\) 1776.00 0.717408
\(184\) −416.000 −0.166674
\(185\) −4851.00 −1.92785
\(186\) −1752.00 −0.690661
\(187\) −1938.00 −0.757864
\(188\) 164.000 0.0636220
\(189\) 2565.00 0.987176
\(190\) 1980.00 0.756023
\(191\) −2138.00 −0.809949 −0.404974 0.914328i \(-0.632720\pi\)
−0.404974 + 0.914328i \(0.632720\pi\)
\(192\) 192.000 0.0721688
\(193\) −4688.00 −1.74844 −0.874222 0.485527i \(-0.838628\pi\)
−0.874222 + 0.485527i \(0.838628\pi\)
\(194\) 692.000 0.256096
\(195\) 0 0
\(196\) 72.0000 0.0262391
\(197\) 891.000 0.322239 0.161120 0.986935i \(-0.448490\pi\)
0.161120 + 0.986935i \(0.448490\pi\)
\(198\) −1368.00 −0.491008
\(199\) −1630.00 −0.580641 −0.290321 0.956929i \(-0.593762\pi\)
−0.290321 + 0.956929i \(0.593762\pi\)
\(200\) −32.0000 −0.0113137
\(201\) 618.000 0.216867
\(202\) 2984.00 1.03937
\(203\) 3610.00 1.24814
\(204\) −612.000 −0.210042
\(205\) 3432.00 1.16927
\(206\) −304.000 −0.102819
\(207\) 936.000 0.314283
\(208\) 0 0
\(209\) −3420.00 −1.13190
\(210\) 1254.00 0.412068
\(211\) 5057.00 1.64994 0.824972 0.565173i \(-0.191191\pi\)
0.824972 + 0.565173i \(0.191191\pi\)
\(212\) 1872.00 0.606460
\(213\) 2589.00 0.832842
\(214\) 1528.00 0.488093
\(215\) −4103.00 −1.30150
\(216\) −1080.00 −0.340207
\(217\) 5548.00 1.73559
\(218\) −2270.00 −0.705247
\(219\) 966.000 0.298065
\(220\) −1672.00 −0.512392
\(221\) 0 0
\(222\) 2646.00 0.799945
\(223\) −2913.00 −0.874748 −0.437374 0.899280i \(-0.644091\pi\)
−0.437374 + 0.899280i \(0.644091\pi\)
\(224\) −608.000 −0.181356
\(225\) 72.0000 0.0213333
\(226\) −3644.00 −1.07255
\(227\) −3744.00 −1.09470 −0.547352 0.836902i \(-0.684364\pi\)
−0.547352 + 0.836902i \(0.684364\pi\)
\(228\) −1080.00 −0.313705
\(229\) 1755.00 0.506435 0.253218 0.967409i \(-0.418511\pi\)
0.253218 + 0.967409i \(0.418511\pi\)
\(230\) 1144.00 0.327970
\(231\) −2166.00 −0.616937
\(232\) −1520.00 −0.430142
\(233\) −2027.00 −0.569928 −0.284964 0.958538i \(-0.591982\pi\)
−0.284964 + 0.958538i \(0.591982\pi\)
\(234\) 0 0
\(235\) −451.000 −0.125191
\(236\) −2120.00 −0.584747
\(237\) −1380.00 −0.378231
\(238\) 1938.00 0.527823
\(239\) 4605.00 1.24633 0.623165 0.782091i \(-0.285847\pi\)
0.623165 + 0.782091i \(0.285847\pi\)
\(240\) −528.000 −0.142009
\(241\) 798.000 0.213293 0.106647 0.994297i \(-0.465989\pi\)
0.106647 + 0.994297i \(0.465989\pi\)
\(242\) 226.000 0.0600324
\(243\) 3888.00 1.02640
\(244\) 2368.00 0.621294
\(245\) −198.000 −0.0516317
\(246\) −1872.00 −0.485180
\(247\) 0 0
\(248\) −2336.00 −0.598130
\(249\) −1584.00 −0.403140
\(250\) 2838.00 0.717964
\(251\) −2088.00 −0.525073 −0.262537 0.964922i \(-0.584559\pi\)
−0.262537 + 0.964922i \(0.584559\pi\)
\(252\) 1368.00 0.341968
\(253\) −1976.00 −0.491028
\(254\) −2512.00 −0.620539
\(255\) 1683.00 0.413308
\(256\) 256.000 0.0625000
\(257\) −6111.00 −1.48324 −0.741622 0.670818i \(-0.765943\pi\)
−0.741622 + 0.670818i \(0.765943\pi\)
\(258\) 2238.00 0.540046
\(259\) −8379.00 −2.01022
\(260\) 0 0
\(261\) 3420.00 0.811083
\(262\) 2194.00 0.517350
\(263\) −2532.00 −0.593649 −0.296825 0.954932i \(-0.595928\pi\)
−0.296825 + 0.954932i \(0.595928\pi\)
\(264\) 912.000 0.212613
\(265\) −5148.00 −1.19336
\(266\) 3420.00 0.788322
\(267\) −2610.00 −0.598237
\(268\) 824.000 0.187813
\(269\) 2400.00 0.543980 0.271990 0.962300i \(-0.412318\pi\)
0.271990 + 0.962300i \(0.412318\pi\)
\(270\) 2970.00 0.669439
\(271\) 4793.00 1.07437 0.537185 0.843465i \(-0.319488\pi\)
0.537185 + 0.843465i \(0.319488\pi\)
\(272\) −816.000 −0.181902
\(273\) 0 0
\(274\) 312.000 0.0687905
\(275\) −152.000 −0.0333307
\(276\) −624.000 −0.136088
\(277\) −5676.00 −1.23118 −0.615592 0.788065i \(-0.711083\pi\)
−0.615592 + 0.788065i \(0.711083\pi\)
\(278\) −4030.00 −0.869436
\(279\) 5256.00 1.12784
\(280\) 1672.00 0.356861
\(281\) −5542.00 −1.17654 −0.588270 0.808664i \(-0.700191\pi\)
−0.588270 + 0.808664i \(0.700191\pi\)
\(282\) 246.000 0.0519471
\(283\) −6332.00 −1.33003 −0.665015 0.746830i \(-0.731575\pi\)
−0.665015 + 0.746830i \(0.731575\pi\)
\(284\) 3452.00 0.721262
\(285\) 2970.00 0.617290
\(286\) 0 0
\(287\) 5928.00 1.21923
\(288\) −576.000 −0.117851
\(289\) −2312.00 −0.470588
\(290\) 4180.00 0.846407
\(291\) 1038.00 0.209102
\(292\) 1288.00 0.258132
\(293\) 3077.00 0.613516 0.306758 0.951788i \(-0.400756\pi\)
0.306758 + 0.951788i \(0.400756\pi\)
\(294\) 108.000 0.0214241
\(295\) 5830.00 1.15063
\(296\) 3528.00 0.692773
\(297\) −5130.00 −1.00227
\(298\) 2100.00 0.408221
\(299\) 0 0
\(300\) −48.0000 −0.00923760
\(301\) −7087.00 −1.35710
\(302\) −3834.00 −0.730536
\(303\) 4476.00 0.848645
\(304\) −1440.00 −0.271677
\(305\) −6512.00 −1.22254
\(306\) 1836.00 0.342997
\(307\) 3286.00 0.610886 0.305443 0.952210i \(-0.401195\pi\)
0.305443 + 0.952210i \(0.401195\pi\)
\(308\) −2888.00 −0.534283
\(309\) −456.000 −0.0839512
\(310\) 6424.00 1.17696
\(311\) 3462.00 0.631228 0.315614 0.948888i \(-0.397789\pi\)
0.315614 + 0.948888i \(0.397789\pi\)
\(312\) 0 0
\(313\) −8737.00 −1.57778 −0.788889 0.614536i \(-0.789343\pi\)
−0.788889 + 0.614536i \(0.789343\pi\)
\(314\) −3092.00 −0.555706
\(315\) −3762.00 −0.672904
\(316\) −1840.00 −0.327557
\(317\) −6054.00 −1.07264 −0.536319 0.844015i \(-0.680186\pi\)
−0.536319 + 0.844015i \(0.680186\pi\)
\(318\) 2808.00 0.495172
\(319\) −7220.00 −1.26722
\(320\) −704.000 −0.122984
\(321\) 2292.00 0.398526
\(322\) 1976.00 0.341982
\(323\) 4590.00 0.790695
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) −1336.00 −0.226976
\(327\) −3405.00 −0.575832
\(328\) −2496.00 −0.420178
\(329\) −779.000 −0.130540
\(330\) −2508.00 −0.418366
\(331\) −932.000 −0.154765 −0.0773827 0.997001i \(-0.524656\pi\)
−0.0773827 + 0.997001i \(0.524656\pi\)
\(332\) −2112.00 −0.349130
\(333\) −7938.00 −1.30631
\(334\) 1872.00 0.306680
\(335\) −2266.00 −0.369567
\(336\) −912.000 −0.148076
\(337\) −1921.00 −0.310515 −0.155257 0.987874i \(-0.549621\pi\)
−0.155257 + 0.987874i \(0.549621\pi\)
\(338\) 0 0
\(339\) −5466.00 −0.875730
\(340\) 2244.00 0.357935
\(341\) −11096.0 −1.76212
\(342\) 3240.00 0.512278
\(343\) 6175.00 0.972066
\(344\) 2984.00 0.467693
\(345\) 1716.00 0.267786
\(346\) 3016.00 0.468616
\(347\) 2289.00 0.354121 0.177060 0.984200i \(-0.443341\pi\)
0.177060 + 0.984200i \(0.443341\pi\)
\(348\) −2280.00 −0.351209
\(349\) −1195.00 −0.183286 −0.0916431 0.995792i \(-0.529212\pi\)
−0.0916431 + 0.995792i \(0.529212\pi\)
\(350\) 152.000 0.0232135
\(351\) 0 0
\(352\) 1216.00 0.184128
\(353\) −7588.00 −1.14410 −0.572052 0.820218i \(-0.693852\pi\)
−0.572052 + 0.820218i \(0.693852\pi\)
\(354\) −3180.00 −0.477444
\(355\) −9493.00 −1.41926
\(356\) −3480.00 −0.518089
\(357\) 2907.00 0.430966
\(358\) 3570.00 0.527040
\(359\) −6240.00 −0.917367 −0.458683 0.888600i \(-0.651679\pi\)
−0.458683 + 0.888600i \(0.651679\pi\)
\(360\) 1584.00 0.231900
\(361\) 1241.00 0.180930
\(362\) −2016.00 −0.292703
\(363\) 339.000 0.0490162
\(364\) 0 0
\(365\) −3542.00 −0.507936
\(366\) 3552.00 0.507284
\(367\) 9074.00 1.29062 0.645312 0.763919i \(-0.276727\pi\)
0.645312 + 0.763919i \(0.276727\pi\)
\(368\) −832.000 −0.117856
\(369\) 5616.00 0.792296
\(370\) −9702.00 −1.36320
\(371\) −8892.00 −1.24434
\(372\) −3504.00 −0.488371
\(373\) −7732.00 −1.07332 −0.536659 0.843799i \(-0.680314\pi\)
−0.536659 + 0.843799i \(0.680314\pi\)
\(374\) −3876.00 −0.535891
\(375\) 4257.00 0.586215
\(376\) 328.000 0.0449875
\(377\) 0 0
\(378\) 5130.00 0.698039
\(379\) 9320.00 1.26316 0.631578 0.775312i \(-0.282407\pi\)
0.631578 + 0.775312i \(0.282407\pi\)
\(380\) 3960.00 0.534589
\(381\) −3768.00 −0.506668
\(382\) −4276.00 −0.572720
\(383\) 927.000 0.123675 0.0618375 0.998086i \(-0.480304\pi\)
0.0618375 + 0.998086i \(0.480304\pi\)
\(384\) 384.000 0.0510310
\(385\) 7942.00 1.05133
\(386\) −9376.00 −1.23634
\(387\) −6714.00 −0.881891
\(388\) 1384.00 0.181088
\(389\) −1710.00 −0.222880 −0.111440 0.993771i \(-0.535546\pi\)
−0.111440 + 0.993771i \(0.535546\pi\)
\(390\) 0 0
\(391\) 2652.00 0.343011
\(392\) 144.000 0.0185538
\(393\) 3291.00 0.422415
\(394\) 1782.00 0.227858
\(395\) 5060.00 0.644547
\(396\) −2736.00 −0.347195
\(397\) 4606.00 0.582288 0.291144 0.956679i \(-0.405964\pi\)
0.291144 + 0.956679i \(0.405964\pi\)
\(398\) −3260.00 −0.410575
\(399\) 5130.00 0.643662
\(400\) −64.0000 −0.00800000
\(401\) 7248.00 0.902613 0.451307 0.892369i \(-0.350958\pi\)
0.451307 + 0.892369i \(0.350958\pi\)
\(402\) 1236.00 0.153348
\(403\) 0 0
\(404\) 5968.00 0.734948
\(405\) −891.000 −0.109319
\(406\) 7220.00 0.882568
\(407\) 16758.0 2.04094
\(408\) −1224.00 −0.148522
\(409\) −10540.0 −1.27425 −0.637126 0.770759i \(-0.719877\pi\)
−0.637126 + 0.770759i \(0.719877\pi\)
\(410\) 6864.00 0.826802
\(411\) 468.000 0.0561672
\(412\) −608.000 −0.0727039
\(413\) 10070.0 1.19979
\(414\) 1872.00 0.222231
\(415\) 5808.00 0.686996
\(416\) 0 0
\(417\) −6045.00 −0.709892
\(418\) −6840.00 −0.800372
\(419\) 10635.0 1.23999 0.619993 0.784608i \(-0.287136\pi\)
0.619993 + 0.784608i \(0.287136\pi\)
\(420\) 2508.00 0.291376
\(421\) −12487.0 −1.44556 −0.722778 0.691080i \(-0.757135\pi\)
−0.722778 + 0.691080i \(0.757135\pi\)
\(422\) 10114.0 1.16669
\(423\) −738.000 −0.0848293
\(424\) 3744.00 0.428832
\(425\) 204.000 0.0232834
\(426\) 5178.00 0.588908
\(427\) −11248.0 −1.27477
\(428\) 3056.00 0.345134
\(429\) 0 0
\(430\) −8206.00 −0.920299
\(431\) 14613.0 1.63314 0.816570 0.577246i \(-0.195873\pi\)
0.816570 + 0.577246i \(0.195873\pi\)
\(432\) −2160.00 −0.240563
\(433\) −6977.00 −0.774349 −0.387175 0.922006i \(-0.626549\pi\)
−0.387175 + 0.922006i \(0.626549\pi\)
\(434\) 11096.0 1.22725
\(435\) 6270.00 0.691088
\(436\) −4540.00 −0.498685
\(437\) 4680.00 0.512299
\(438\) 1932.00 0.210764
\(439\) 1430.00 0.155467 0.0777337 0.996974i \(-0.475232\pi\)
0.0777337 + 0.996974i \(0.475232\pi\)
\(440\) −3344.00 −0.362316
\(441\) −324.000 −0.0349854
\(442\) 0 0
\(443\) 12423.0 1.33236 0.666179 0.745792i \(-0.267929\pi\)
0.666179 + 0.745792i \(0.267929\pi\)
\(444\) 5292.00 0.565647
\(445\) 9570.00 1.01946
\(446\) −5826.00 −0.618541
\(447\) 3150.00 0.333311
\(448\) −1216.00 −0.128238
\(449\) 9890.00 1.03951 0.519753 0.854317i \(-0.326024\pi\)
0.519753 + 0.854317i \(0.326024\pi\)
\(450\) 144.000 0.0150849
\(451\) −11856.0 −1.23787
\(452\) −7288.00 −0.758404
\(453\) −5751.00 −0.596480
\(454\) −7488.00 −0.774073
\(455\) 0 0
\(456\) −2160.00 −0.221823
\(457\) 9926.00 1.01601 0.508007 0.861353i \(-0.330382\pi\)
0.508007 + 0.861353i \(0.330382\pi\)
\(458\) 3510.00 0.358104
\(459\) 6885.00 0.700140
\(460\) 2288.00 0.231910
\(461\) 2793.00 0.282176 0.141088 0.989997i \(-0.454940\pi\)
0.141088 + 0.989997i \(0.454940\pi\)
\(462\) −4332.00 −0.436240
\(463\) 6872.00 0.689782 0.344891 0.938643i \(-0.387916\pi\)
0.344891 + 0.938643i \(0.387916\pi\)
\(464\) −3040.00 −0.304156
\(465\) 9636.00 0.960987
\(466\) −4054.00 −0.403000
\(467\) −3676.00 −0.364251 −0.182125 0.983275i \(-0.558298\pi\)
−0.182125 + 0.983275i \(0.558298\pi\)
\(468\) 0 0
\(469\) −3914.00 −0.385355
\(470\) −902.000 −0.0885237
\(471\) −4638.00 −0.453732
\(472\) −4240.00 −0.413478
\(473\) 14174.0 1.37785
\(474\) −2760.00 −0.267449
\(475\) 360.000 0.0347746
\(476\) 3876.00 0.373227
\(477\) −8424.00 −0.808613
\(478\) 9210.00 0.881288
\(479\) −13575.0 −1.29490 −0.647451 0.762108i \(-0.724165\pi\)
−0.647451 + 0.762108i \(0.724165\pi\)
\(480\) −1056.00 −0.100416
\(481\) 0 0
\(482\) 1596.00 0.150821
\(483\) 2964.00 0.279227
\(484\) 452.000 0.0424493
\(485\) −3806.00 −0.356333
\(486\) 7776.00 0.725775
\(487\) −11864.0 −1.10392 −0.551960 0.833871i \(-0.686120\pi\)
−0.551960 + 0.833871i \(0.686120\pi\)
\(488\) 4736.00 0.439321
\(489\) −2004.00 −0.185325
\(490\) −396.000 −0.0365091
\(491\) 4837.00 0.444584 0.222292 0.974980i \(-0.428646\pi\)
0.222292 + 0.974980i \(0.428646\pi\)
\(492\) −3744.00 −0.343074
\(493\) 9690.00 0.885224
\(494\) 0 0
\(495\) 7524.00 0.683189
\(496\) −4672.00 −0.422942
\(497\) −16397.0 −1.47989
\(498\) −3168.00 −0.285063
\(499\) −9160.00 −0.821759 −0.410880 0.911690i \(-0.634778\pi\)
−0.410880 + 0.911690i \(0.634778\pi\)
\(500\) 5676.00 0.507677
\(501\) 2808.00 0.250404
\(502\) −4176.00 −0.371283
\(503\) −842.000 −0.0746380 −0.0373190 0.999303i \(-0.511882\pi\)
−0.0373190 + 0.999303i \(0.511882\pi\)
\(504\) 2736.00 0.241808
\(505\) −16412.0 −1.44619
\(506\) −3952.00 −0.347209
\(507\) 0 0
\(508\) −5024.00 −0.438787
\(509\) −250.000 −0.0217702 −0.0108851 0.999941i \(-0.503465\pi\)
−0.0108851 + 0.999941i \(0.503465\pi\)
\(510\) 3366.00 0.292253
\(511\) −6118.00 −0.529637
\(512\) 512.000 0.0441942
\(513\) 12150.0 1.04568
\(514\) −12222.0 −1.04881
\(515\) 1672.00 0.143062
\(516\) 4476.00 0.381870
\(517\) 1558.00 0.132535
\(518\) −16758.0 −1.42144
\(519\) 4524.00 0.382623
\(520\) 0 0
\(521\) 18087.0 1.52093 0.760466 0.649377i \(-0.224970\pi\)
0.760466 + 0.649377i \(0.224970\pi\)
\(522\) 6840.00 0.573522
\(523\) 20028.0 1.67450 0.837250 0.546821i \(-0.184162\pi\)
0.837250 + 0.546821i \(0.184162\pi\)
\(524\) 4388.00 0.365822
\(525\) 228.000 0.0189538
\(526\) −5064.00 −0.419774
\(527\) 14892.0 1.23094
\(528\) 1824.00 0.150340
\(529\) −9463.00 −0.777760
\(530\) −10296.0 −0.843830
\(531\) 9540.00 0.779662
\(532\) 6840.00 0.557428
\(533\) 0 0
\(534\) −5220.00 −0.423018
\(535\) −8404.00 −0.679134
\(536\) 1648.00 0.132804
\(537\) 5355.00 0.430326
\(538\) 4800.00 0.384652
\(539\) 684.000 0.0546604
\(540\) 5940.00 0.473365
\(541\) 6763.00 0.537457 0.268728 0.963216i \(-0.413397\pi\)
0.268728 + 0.963216i \(0.413397\pi\)
\(542\) 9586.00 0.759694
\(543\) −3024.00 −0.238991
\(544\) −1632.00 −0.128624
\(545\) 12485.0 0.981282
\(546\) 0 0
\(547\) 2539.00 0.198464 0.0992320 0.995064i \(-0.468361\pi\)
0.0992320 + 0.995064i \(0.468361\pi\)
\(548\) 624.000 0.0486423
\(549\) −10656.0 −0.828392
\(550\) −304.000 −0.0235684
\(551\) 17100.0 1.32211
\(552\) −1248.00 −0.0962290
\(553\) 8740.00 0.672084
\(554\) −11352.0 −0.870578
\(555\) −14553.0 −1.11305
\(556\) −8060.00 −0.614784
\(557\) 7611.00 0.578974 0.289487 0.957182i \(-0.406515\pi\)
0.289487 + 0.957182i \(0.406515\pi\)
\(558\) 10512.0 0.797506
\(559\) 0 0
\(560\) 3344.00 0.252339
\(561\) −5814.00 −0.437553
\(562\) −11084.0 −0.831940
\(563\) 3653.00 0.273456 0.136728 0.990609i \(-0.456341\pi\)
0.136728 + 0.990609i \(0.456341\pi\)
\(564\) 492.000 0.0367322
\(565\) 20042.0 1.49234
\(566\) −12664.0 −0.940473
\(567\) −1539.00 −0.113989
\(568\) 6904.00 0.510010
\(569\) 23095.0 1.70157 0.850785 0.525515i \(-0.176127\pi\)
0.850785 + 0.525515i \(0.176127\pi\)
\(570\) 5940.00 0.436490
\(571\) −21273.0 −1.55910 −0.779551 0.626339i \(-0.784553\pi\)
−0.779551 + 0.626339i \(0.784553\pi\)
\(572\) 0 0
\(573\) −6414.00 −0.467624
\(574\) 11856.0 0.862125
\(575\) 208.000 0.0150856
\(576\) −1152.00 −0.0833333
\(577\) −15114.0 −1.09047 −0.545237 0.838282i \(-0.683561\pi\)
−0.545237 + 0.838282i \(0.683561\pi\)
\(578\) −4624.00 −0.332756
\(579\) −14064.0 −1.00946
\(580\) 8360.00 0.598500
\(581\) 10032.0 0.716347
\(582\) 2076.00 0.147857
\(583\) 17784.0 1.26336
\(584\) 2576.00 0.182527
\(585\) 0 0
\(586\) 6154.00 0.433821
\(587\) 22156.0 1.55788 0.778940 0.627098i \(-0.215757\pi\)
0.778940 + 0.627098i \(0.215757\pi\)
\(588\) 216.000 0.0151491
\(589\) 26280.0 1.83845
\(590\) 11660.0 0.813618
\(591\) 2673.00 0.186045
\(592\) 7056.00 0.489865
\(593\) −12338.0 −0.854403 −0.427201 0.904156i \(-0.640500\pi\)
−0.427201 + 0.904156i \(0.640500\pi\)
\(594\) −10260.0 −0.708709
\(595\) −10659.0 −0.734414
\(596\) 4200.00 0.288656
\(597\) −4890.00 −0.335233
\(598\) 0 0
\(599\) −2750.00 −0.187583 −0.0937913 0.995592i \(-0.529899\pi\)
−0.0937913 + 0.995592i \(0.529899\pi\)
\(600\) −96.0000 −0.00653197
\(601\) 23317.0 1.58256 0.791281 0.611452i \(-0.209414\pi\)
0.791281 + 0.611452i \(0.209414\pi\)
\(602\) −14174.0 −0.959616
\(603\) −3708.00 −0.250417
\(604\) −7668.00 −0.516567
\(605\) −1243.00 −0.0835292
\(606\) 8952.00 0.600083
\(607\) −19686.0 −1.31636 −0.658180 0.752861i \(-0.728673\pi\)
−0.658180 + 0.752861i \(0.728673\pi\)
\(608\) −2880.00 −0.192104
\(609\) 10830.0 0.720614
\(610\) −13024.0 −0.864469
\(611\) 0 0
\(612\) 3672.00 0.242536
\(613\) 1822.00 0.120049 0.0600244 0.998197i \(-0.480882\pi\)
0.0600244 + 0.998197i \(0.480882\pi\)
\(614\) 6572.00 0.431961
\(615\) 10296.0 0.675081
\(616\) −5776.00 −0.377795
\(617\) −6304.00 −0.411328 −0.205664 0.978623i \(-0.565935\pi\)
−0.205664 + 0.978623i \(0.565935\pi\)
\(618\) −912.000 −0.0593625
\(619\) −18340.0 −1.19087 −0.595434 0.803404i \(-0.703020\pi\)
−0.595434 + 0.803404i \(0.703020\pi\)
\(620\) 12848.0 0.832239
\(621\) 7020.00 0.453628
\(622\) 6924.00 0.446346
\(623\) 16530.0 1.06302
\(624\) 0 0
\(625\) −15109.0 −0.966976
\(626\) −17474.0 −1.11566
\(627\) −10260.0 −0.653501
\(628\) −6184.00 −0.392943
\(629\) −22491.0 −1.42572
\(630\) −7524.00 −0.475815
\(631\) −10057.0 −0.634489 −0.317245 0.948344i \(-0.602758\pi\)
−0.317245 + 0.948344i \(0.602758\pi\)
\(632\) −3680.00 −0.231618
\(633\) 15171.0 0.952596
\(634\) −12108.0 −0.758470
\(635\) 13816.0 0.863419
\(636\) 5616.00 0.350140
\(637\) 0 0
\(638\) −14440.0 −0.896058
\(639\) −15534.0 −0.961683
\(640\) −1408.00 −0.0869626
\(641\) −25058.0 −1.54404 −0.772021 0.635596i \(-0.780754\pi\)
−0.772021 + 0.635596i \(0.780754\pi\)
\(642\) 4584.00 0.281801
\(643\) −3698.00 −0.226804 −0.113402 0.993549i \(-0.536175\pi\)
−0.113402 + 0.993549i \(0.536175\pi\)
\(644\) 3952.00 0.241818
\(645\) −12309.0 −0.751421
\(646\) 9180.00 0.559106
\(647\) −11786.0 −0.716160 −0.358080 0.933691i \(-0.616568\pi\)
−0.358080 + 0.933691i \(0.616568\pi\)
\(648\) 648.000 0.0392837
\(649\) −20140.0 −1.21813
\(650\) 0 0
\(651\) 16644.0 1.00204
\(652\) −2672.00 −0.160496
\(653\) −9672.00 −0.579624 −0.289812 0.957084i \(-0.593593\pi\)
−0.289812 + 0.957084i \(0.593593\pi\)
\(654\) −6810.00 −0.407174
\(655\) −12067.0 −0.719842
\(656\) −4992.00 −0.297111
\(657\) −5796.00 −0.344176
\(658\) −1558.00 −0.0923057
\(659\) 17460.0 1.03209 0.516043 0.856563i \(-0.327404\pi\)
0.516043 + 0.856563i \(0.327404\pi\)
\(660\) −5016.00 −0.295830
\(661\) −8702.00 −0.512055 −0.256028 0.966669i \(-0.582414\pi\)
−0.256028 + 0.966669i \(0.582414\pi\)
\(662\) −1864.00 −0.109436
\(663\) 0 0
\(664\) −4224.00 −0.246872
\(665\) −18810.0 −1.09687
\(666\) −15876.0 −0.923697
\(667\) 9880.00 0.573546
\(668\) 3744.00 0.216856
\(669\) −8739.00 −0.505036
\(670\) −4532.00 −0.261323
\(671\) 22496.0 1.29426
\(672\) −1824.00 −0.104706
\(673\) −22667.0 −1.29829 −0.649145 0.760665i \(-0.724873\pi\)
−0.649145 + 0.760665i \(0.724873\pi\)
\(674\) −3842.00 −0.219567
\(675\) 540.000 0.0307920
\(676\) 0 0
\(677\) −18516.0 −1.05115 −0.525574 0.850748i \(-0.676150\pi\)
−0.525574 + 0.850748i \(0.676150\pi\)
\(678\) −10932.0 −0.619234
\(679\) −6574.00 −0.371557
\(680\) 4488.00 0.253098
\(681\) −11232.0 −0.632028
\(682\) −22192.0 −1.24601
\(683\) 4772.00 0.267343 0.133672 0.991026i \(-0.457323\pi\)
0.133672 + 0.991026i \(0.457323\pi\)
\(684\) 6480.00 0.362235
\(685\) −1716.00 −0.0957153
\(686\) 12350.0 0.687355
\(687\) 5265.00 0.292391
\(688\) 5968.00 0.330709
\(689\) 0 0
\(690\) 3432.00 0.189354
\(691\) −19672.0 −1.08301 −0.541504 0.840698i \(-0.682145\pi\)
−0.541504 + 0.840698i \(0.682145\pi\)
\(692\) 6032.00 0.331362
\(693\) 12996.0 0.712377
\(694\) 4578.00 0.250401
\(695\) 22165.0 1.20974
\(696\) −4560.00 −0.248342
\(697\) 15912.0 0.864720
\(698\) −2390.00 −0.129603
\(699\) −6081.00 −0.329048
\(700\) 304.000 0.0164145
\(701\) −9828.00 −0.529527 −0.264764 0.964313i \(-0.585294\pi\)
−0.264764 + 0.964313i \(0.585294\pi\)
\(702\) 0 0
\(703\) −39690.0 −2.12936
\(704\) 2432.00 0.130198
\(705\) −1353.00 −0.0722793
\(706\) −15176.0 −0.809003
\(707\) −28348.0 −1.50797
\(708\) −6360.00 −0.337604
\(709\) 15730.0 0.833219 0.416610 0.909085i \(-0.363218\pi\)
0.416610 + 0.909085i \(0.363218\pi\)
\(710\) −18986.0 −1.00357
\(711\) 8280.00 0.436743
\(712\) −6960.00 −0.366344
\(713\) 15184.0 0.797539
\(714\) 5814.00 0.304739
\(715\) 0 0
\(716\) 7140.00 0.372674
\(717\) 13815.0 0.719569
\(718\) −12480.0 −0.648676
\(719\) −17890.0 −0.927934 −0.463967 0.885853i \(-0.653574\pi\)
−0.463967 + 0.885853i \(0.653574\pi\)
\(720\) 3168.00 0.163978
\(721\) 2888.00 0.149174
\(722\) 2482.00 0.127937
\(723\) 2394.00 0.123145
\(724\) −4032.00 −0.206973
\(725\) 760.000 0.0389320
\(726\) 678.000 0.0346597
\(727\) −6386.00 −0.325782 −0.162891 0.986644i \(-0.552082\pi\)
−0.162891 + 0.986644i \(0.552082\pi\)
\(728\) 0 0
\(729\) 9477.00 0.481481
\(730\) −7084.00 −0.359165
\(731\) −19023.0 −0.962505
\(732\) 7104.00 0.358704
\(733\) −2273.00 −0.114536 −0.0572682 0.998359i \(-0.518239\pi\)
−0.0572682 + 0.998359i \(0.518239\pi\)
\(734\) 18148.0 0.912609
\(735\) −594.000 −0.0298096
\(736\) −1664.00 −0.0833368
\(737\) 7828.00 0.391246
\(738\) 11232.0 0.560238
\(739\) 4980.00 0.247892 0.123946 0.992289i \(-0.460445\pi\)
0.123946 + 0.992289i \(0.460445\pi\)
\(740\) −19404.0 −0.963926
\(741\) 0 0
\(742\) −17784.0 −0.879880
\(743\) −7483.00 −0.369481 −0.184741 0.982787i \(-0.559145\pi\)
−0.184741 + 0.982787i \(0.559145\pi\)
\(744\) −7008.00 −0.345330
\(745\) −11550.0 −0.567999
\(746\) −15464.0 −0.758951
\(747\) 9504.00 0.465506
\(748\) −7752.00 −0.378932
\(749\) −14516.0 −0.708148
\(750\) 8514.00 0.414516
\(751\) 31632.0 1.53697 0.768487 0.639865i \(-0.221010\pi\)
0.768487 + 0.639865i \(0.221010\pi\)
\(752\) 656.000 0.0318110
\(753\) −6264.00 −0.303151
\(754\) 0 0
\(755\) 21087.0 1.01647
\(756\) 10260.0 0.493588
\(757\) −16116.0 −0.773773 −0.386886 0.922127i \(-0.626449\pi\)
−0.386886 + 0.922127i \(0.626449\pi\)
\(758\) 18640.0 0.893186
\(759\) −5928.00 −0.283495
\(760\) 7920.00 0.378011
\(761\) −14622.0 −0.696514 −0.348257 0.937399i \(-0.613226\pi\)
−0.348257 + 0.937399i \(0.613226\pi\)
\(762\) −7536.00 −0.358268
\(763\) 21565.0 1.02321
\(764\) −8552.00 −0.404974
\(765\) −10098.0 −0.477247
\(766\) 1854.00 0.0874514
\(767\) 0 0
\(768\) 768.000 0.0360844
\(769\) −9840.00 −0.461430 −0.230715 0.973021i \(-0.574106\pi\)
−0.230715 + 0.973021i \(0.574106\pi\)
\(770\) 15884.0 0.743402
\(771\) −18333.0 −0.856351
\(772\) −18752.0 −0.874222
\(773\) −4823.00 −0.224413 −0.112207 0.993685i \(-0.535792\pi\)
−0.112207 + 0.993685i \(0.535792\pi\)
\(774\) −13428.0 −0.623591
\(775\) 1168.00 0.0541365
\(776\) 2768.00 0.128048
\(777\) −25137.0 −1.16060
\(778\) −3420.00 −0.157600
\(779\) 28080.0 1.29149
\(780\) 0 0
\(781\) 32794.0 1.50251
\(782\) 5304.00 0.242546
\(783\) 25650.0 1.17070
\(784\) 288.000 0.0131195
\(785\) 17006.0 0.773210
\(786\) 6582.00 0.298692
\(787\) −28424.0 −1.28743 −0.643714 0.765266i \(-0.722607\pi\)
−0.643714 + 0.765266i \(0.722607\pi\)
\(788\) 3564.00 0.161120
\(789\) −7596.00 −0.342744
\(790\) 10120.0 0.455764
\(791\) 34618.0 1.55610
\(792\) −5472.00 −0.245504
\(793\) 0 0
\(794\) 9212.00 0.411740
\(795\) −15444.0 −0.688984
\(796\) −6520.00 −0.290321
\(797\) 33294.0 1.47972 0.739858 0.672763i \(-0.234893\pi\)
0.739858 + 0.672763i \(0.234893\pi\)
\(798\) 10260.0 0.455138
\(799\) −2091.00 −0.0925836
\(800\) −128.000 −0.00565685
\(801\) 15660.0 0.690785
\(802\) 14496.0 0.638244
\(803\) 12236.0 0.537732
\(804\) 2472.00 0.108434
\(805\) −10868.0 −0.475834
\(806\) 0 0
\(807\) 7200.00 0.314067
\(808\) 11936.0 0.519687
\(809\) −18585.0 −0.807681 −0.403840 0.914829i \(-0.632325\pi\)
−0.403840 + 0.914829i \(0.632325\pi\)
\(810\) −1782.00 −0.0773001
\(811\) 19348.0 0.837731 0.418866 0.908048i \(-0.362428\pi\)
0.418866 + 0.908048i \(0.362428\pi\)
\(812\) 14440.0 0.624070
\(813\) 14379.0 0.620287
\(814\) 33516.0 1.44316
\(815\) 7348.00 0.315815
\(816\) −2448.00 −0.105021
\(817\) −33570.0 −1.43753
\(818\) −21080.0 −0.901033
\(819\) 0 0
\(820\) 13728.0 0.584637
\(821\) 15853.0 0.673902 0.336951 0.941522i \(-0.390604\pi\)
0.336951 + 0.941522i \(0.390604\pi\)
\(822\) 936.000 0.0397162
\(823\) 23898.0 1.01219 0.506095 0.862478i \(-0.331089\pi\)
0.506095 + 0.862478i \(0.331089\pi\)
\(824\) −1216.00 −0.0514094
\(825\) −456.000 −0.0192435
\(826\) 20140.0 0.848378
\(827\) −35634.0 −1.49833 −0.749163 0.662386i \(-0.769544\pi\)
−0.749163 + 0.662386i \(0.769544\pi\)
\(828\) 3744.00 0.157141
\(829\) −29390.0 −1.23131 −0.615656 0.788015i \(-0.711109\pi\)
−0.615656 + 0.788015i \(0.711109\pi\)
\(830\) 11616.0 0.485780
\(831\) −17028.0 −0.710824
\(832\) 0 0
\(833\) −918.000 −0.0381835
\(834\) −12090.0 −0.501969
\(835\) −10296.0 −0.426716
\(836\) −13680.0 −0.565948
\(837\) 39420.0 1.62790
\(838\) 21270.0 0.876802
\(839\) 27040.0 1.11266 0.556332 0.830960i \(-0.312208\pi\)
0.556332 + 0.830960i \(0.312208\pi\)
\(840\) 5016.00 0.206034
\(841\) 11711.0 0.480175
\(842\) −24974.0 −1.02216
\(843\) −16626.0 −0.679276
\(844\) 20228.0 0.824972
\(845\) 0 0
\(846\) −1476.00 −0.0599834
\(847\) −2147.00 −0.0870977
\(848\) 7488.00 0.303230
\(849\) −18996.0 −0.767893
\(850\) 408.000 0.0164639
\(851\) −22932.0 −0.923735
\(852\) 10356.0 0.416421
\(853\) 15467.0 0.620844 0.310422 0.950599i \(-0.399530\pi\)
0.310422 + 0.950599i \(0.399530\pi\)
\(854\) −22496.0 −0.901402
\(855\) −17820.0 −0.712785
\(856\) 6112.00 0.244047
\(857\) 15694.0 0.625551 0.312775 0.949827i \(-0.398741\pi\)
0.312775 + 0.949827i \(0.398741\pi\)
\(858\) 0 0
\(859\) 2180.00 0.0865898 0.0432949 0.999062i \(-0.486214\pi\)
0.0432949 + 0.999062i \(0.486214\pi\)
\(860\) −16412.0 −0.650749
\(861\) 17784.0 0.703922
\(862\) 29226.0 1.15480
\(863\) 26537.0 1.04673 0.523366 0.852108i \(-0.324676\pi\)
0.523366 + 0.852108i \(0.324676\pi\)
\(864\) −4320.00 −0.170103
\(865\) −16588.0 −0.652033
\(866\) −13954.0 −0.547548
\(867\) −6936.00 −0.271694
\(868\) 22192.0 0.867794
\(869\) −17480.0 −0.682357
\(870\) 12540.0 0.488673
\(871\) 0 0
\(872\) −9080.00 −0.352623
\(873\) −6228.00 −0.241450
\(874\) 9360.00 0.362250
\(875\) −26961.0 −1.04166
\(876\) 3864.00 0.149032
\(877\) −23829.0 −0.917501 −0.458750 0.888565i \(-0.651703\pi\)
−0.458750 + 0.888565i \(0.651703\pi\)
\(878\) 2860.00 0.109932
\(879\) 9231.00 0.354214
\(880\) −6688.00 −0.256196
\(881\) 24117.0 0.922273 0.461136 0.887329i \(-0.347442\pi\)
0.461136 + 0.887329i \(0.347442\pi\)
\(882\) −648.000 −0.0247384
\(883\) −14537.0 −0.554031 −0.277015 0.960866i \(-0.589345\pi\)
−0.277015 + 0.960866i \(0.589345\pi\)
\(884\) 0 0
\(885\) 17490.0 0.664316
\(886\) 24846.0 0.942119
\(887\) 12064.0 0.456674 0.228337 0.973582i \(-0.426671\pi\)
0.228337 + 0.973582i \(0.426671\pi\)
\(888\) 10584.0 0.399973
\(889\) 23864.0 0.900307
\(890\) 19140.0 0.720870
\(891\) 3078.00 0.115732
\(892\) −11652.0 −0.437374
\(893\) −3690.00 −0.138277
\(894\) 6300.00 0.235686
\(895\) −19635.0 −0.733325
\(896\) −2432.00 −0.0906779
\(897\) 0 0
\(898\) 19780.0 0.735041
\(899\) 55480.0 2.05824
\(900\) 288.000 0.0106667
\(901\) −23868.0 −0.882529
\(902\) −23712.0 −0.875303
\(903\) −21261.0 −0.783524
\(904\) −14576.0 −0.536273
\(905\) 11088.0 0.407268
\(906\) −11502.0 −0.421775
\(907\) 38409.0 1.40612 0.703059 0.711131i \(-0.251817\pi\)
0.703059 + 0.711131i \(0.251817\pi\)
\(908\) −14976.0 −0.547352
\(909\) −26856.0 −0.979931
\(910\) 0 0
\(911\) −49578.0 −1.80307 −0.901533 0.432711i \(-0.857557\pi\)
−0.901533 + 0.432711i \(0.857557\pi\)
\(912\) −4320.00 −0.156853
\(913\) −20064.0 −0.727296
\(914\) 19852.0 0.718431
\(915\) −19536.0 −0.705836
\(916\) 7020.00 0.253218
\(917\) −20843.0 −0.750596
\(918\) 13770.0 0.495074
\(919\) 8280.00 0.297206 0.148603 0.988897i \(-0.452522\pi\)
0.148603 + 0.988897i \(0.452522\pi\)
\(920\) 4576.00 0.163985
\(921\) 9858.00 0.352695
\(922\) 5586.00 0.199528
\(923\) 0 0
\(924\) −8664.00 −0.308468
\(925\) −1764.00 −0.0627027
\(926\) 13744.0 0.487749
\(927\) 2736.00 0.0969385
\(928\) −6080.00 −0.215071
\(929\) −3180.00 −0.112306 −0.0561531 0.998422i \(-0.517883\pi\)
−0.0561531 + 0.998422i \(0.517883\pi\)
\(930\) 19272.0 0.679520
\(931\) −1620.00 −0.0570283
\(932\) −8108.00 −0.284964
\(933\) 10386.0 0.364440
\(934\) −7352.00 −0.257564
\(935\) 21318.0 0.745640
\(936\) 0 0
\(937\) 14214.0 0.495572 0.247786 0.968815i \(-0.420297\pi\)
0.247786 + 0.968815i \(0.420297\pi\)
\(938\) −7828.00 −0.272487
\(939\) −26211.0 −0.910930
\(940\) −1804.00 −0.0625957
\(941\) −8917.00 −0.308912 −0.154456 0.988000i \(-0.549362\pi\)
−0.154456 + 0.988000i \(0.549362\pi\)
\(942\) −9276.00 −0.320837
\(943\) 16224.0 0.560261
\(944\) −8480.00 −0.292373
\(945\) −28215.0 −0.971253
\(946\) 28348.0 0.974284
\(947\) −34074.0 −1.16923 −0.584613 0.811313i \(-0.698753\pi\)
−0.584613 + 0.811313i \(0.698753\pi\)
\(948\) −5520.00 −0.189115
\(949\) 0 0
\(950\) 720.000 0.0245894
\(951\) −18162.0 −0.619288
\(952\) 7752.00 0.263912
\(953\) 24383.0 0.828796 0.414398 0.910096i \(-0.363992\pi\)
0.414398 + 0.910096i \(0.363992\pi\)
\(954\) −16848.0 −0.571776
\(955\) 23518.0 0.796884
\(956\) 18420.0 0.623165
\(957\) −21660.0 −0.731628
\(958\) −27150.0 −0.915633
\(959\) −2964.00 −0.0998045
\(960\) −2112.00 −0.0710047
\(961\) 55473.0 1.86207
\(962\) 0 0
\(963\) −13752.0 −0.460179
\(964\) 3192.00 0.106647
\(965\) 51568.0 1.72024
\(966\) 5928.00 0.197443
\(967\) 16171.0 0.537771 0.268885 0.963172i \(-0.413345\pi\)
0.268885 + 0.963172i \(0.413345\pi\)
\(968\) 904.000 0.0300162
\(969\) 13770.0 0.456508
\(970\) −7612.00 −0.251966
\(971\) −18513.0 −0.611854 −0.305927 0.952055i \(-0.598966\pi\)
−0.305927 + 0.952055i \(0.598966\pi\)
\(972\) 15552.0 0.513200
\(973\) 38285.0 1.26142
\(974\) −23728.0 −0.780589
\(975\) 0 0
\(976\) 9472.00 0.310647
\(977\) 39966.0 1.30873 0.654363 0.756180i \(-0.272937\pi\)
0.654363 + 0.756180i \(0.272937\pi\)
\(978\) −4008.00 −0.131045
\(979\) −33060.0 −1.07927
\(980\) −792.000 −0.0258158
\(981\) 20430.0 0.664913
\(982\) 9674.00 0.314368
\(983\) 44317.0 1.43794 0.718969 0.695042i \(-0.244614\pi\)
0.718969 + 0.695042i \(0.244614\pi\)
\(984\) −7488.00 −0.242590
\(985\) −9801.00 −0.317041
\(986\) 19380.0 0.625948
\(987\) −2337.00 −0.0753673
\(988\) 0 0
\(989\) −19396.0 −0.623617
\(990\) 15048.0 0.483088
\(991\) 52422.0 1.68036 0.840181 0.542305i \(-0.182448\pi\)
0.840181 + 0.542305i \(0.182448\pi\)
\(992\) −9344.00 −0.299065
\(993\) −2796.00 −0.0893539
\(994\) −32794.0 −1.04644
\(995\) 17930.0 0.571276
\(996\) −6336.00 −0.201570
\(997\) −2026.00 −0.0643571 −0.0321786 0.999482i \(-0.510245\pi\)
−0.0321786 + 0.999482i \(0.510245\pi\)
\(998\) −18320.0 −0.581072
\(999\) −59535.0 −1.88549
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.e.1.1 1
13.2 odd 12 338.4.e.b.147.2 4
13.3 even 3 338.4.c.b.191.1 2
13.4 even 6 338.4.c.f.315.1 2
13.5 odd 4 338.4.b.c.337.1 2
13.6 odd 12 338.4.e.b.23.1 4
13.7 odd 12 338.4.e.b.23.2 4
13.8 odd 4 338.4.b.c.337.2 2
13.9 even 3 338.4.c.b.315.1 2
13.10 even 6 338.4.c.f.191.1 2
13.11 odd 12 338.4.e.b.147.1 4
13.12 even 2 26.4.a.a.1.1 1
39.38 odd 2 234.4.a.g.1.1 1
52.51 odd 2 208.4.a.c.1.1 1
65.12 odd 4 650.4.b.b.599.1 2
65.38 odd 4 650.4.b.b.599.2 2
65.64 even 2 650.4.a.f.1.1 1
91.90 odd 2 1274.4.a.b.1.1 1
104.51 odd 2 832.4.a.m.1.1 1
104.77 even 2 832.4.a.e.1.1 1
156.155 even 2 1872.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.a.1.1 1 13.12 even 2
208.4.a.c.1.1 1 52.51 odd 2
234.4.a.g.1.1 1 39.38 odd 2
338.4.a.e.1.1 1 1.1 even 1 trivial
338.4.b.c.337.1 2 13.5 odd 4
338.4.b.c.337.2 2 13.8 odd 4
338.4.c.b.191.1 2 13.3 even 3
338.4.c.b.315.1 2 13.9 even 3
338.4.c.f.191.1 2 13.10 even 6
338.4.c.f.315.1 2 13.4 even 6
338.4.e.b.23.1 4 13.6 odd 12
338.4.e.b.23.2 4 13.7 odd 12
338.4.e.b.147.1 4 13.11 odd 12
338.4.e.b.147.2 4 13.2 odd 12
650.4.a.f.1.1 1 65.64 even 2
650.4.b.b.599.1 2 65.12 odd 4
650.4.b.b.599.2 2 65.38 odd 4
832.4.a.e.1.1 1 104.77 even 2
832.4.a.m.1.1 1 104.51 odd 2
1274.4.a.b.1.1 1 91.90 odd 2
1872.4.a.c.1.1 1 156.155 even 2