Properties

Label 126.4.a.h.1.1
Level $126$
Weight $4$
Character 126.1
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,4,Mod(1,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, 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("126.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.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: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 126.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +28.0000 q^{10} +28.0000 q^{11} +18.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -74.0000 q^{17} +80.0000 q^{19} +56.0000 q^{20} +56.0000 q^{22} +112.000 q^{23} +71.0000 q^{25} +36.0000 q^{26} -28.0000 q^{28} -190.000 q^{29} +72.0000 q^{31} +32.0000 q^{32} -148.000 q^{34} -98.0000 q^{35} -346.000 q^{37} +160.000 q^{38} +112.000 q^{40} -162.000 q^{41} -412.000 q^{43} +112.000 q^{44} +224.000 q^{46} -24.0000 q^{47} +49.0000 q^{49} +142.000 q^{50} +72.0000 q^{52} -318.000 q^{53} +392.000 q^{55} -56.0000 q^{56} -380.000 q^{58} +200.000 q^{59} -198.000 q^{61} +144.000 q^{62} +64.0000 q^{64} +252.000 q^{65} -716.000 q^{67} -296.000 q^{68} -196.000 q^{70} -392.000 q^{71} +538.000 q^{73} -692.000 q^{74} +320.000 q^{76} -196.000 q^{77} +240.000 q^{79} +224.000 q^{80} -324.000 q^{82} +1072.00 q^{83} -1036.00 q^{85} -824.000 q^{86} +224.000 q^{88} -810.000 q^{89} -126.000 q^{91} +448.000 q^{92} -48.0000 q^{94} +1120.00 q^{95} +1354.00 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 14.0000 1.25220 0.626099 0.779744i \(-0.284651\pi\)
0.626099 + 0.779744i \(0.284651\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 28.0000 0.885438
\(11\) 28.0000 0.767483 0.383742 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 18.0000 0.384023 0.192012 0.981393i \(-0.438499\pi\)
0.192012 + 0.981393i \(0.438499\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −74.0000 −1.05574 −0.527872 0.849324i \(-0.677010\pi\)
−0.527872 + 0.849324i \(0.677010\pi\)
\(18\) 0 0
\(19\) 80.0000 0.965961 0.482980 0.875631i \(-0.339554\pi\)
0.482980 + 0.875631i \(0.339554\pi\)
\(20\) 56.0000 0.626099
\(21\) 0 0
\(22\) 56.0000 0.542693
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 0 0
\(25\) 71.0000 0.568000
\(26\) 36.0000 0.271545
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) −190.000 −1.21662 −0.608312 0.793698i \(-0.708153\pi\)
−0.608312 + 0.793698i \(0.708153\pi\)
\(30\) 0 0
\(31\) 72.0000 0.417148 0.208574 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −148.000 −0.746523
\(35\) −98.0000 −0.473286
\(36\) 0 0
\(37\) −346.000 −1.53735 −0.768676 0.639638i \(-0.779084\pi\)
−0.768676 + 0.639638i \(0.779084\pi\)
\(38\) 160.000 0.683038
\(39\) 0 0
\(40\) 112.000 0.442719
\(41\) −162.000 −0.617077 −0.308538 0.951212i \(-0.599840\pi\)
−0.308538 + 0.951212i \(0.599840\pi\)
\(42\) 0 0
\(43\) −412.000 −1.46115 −0.730575 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) 112.000 0.383742
\(45\) 0 0
\(46\) 224.000 0.717978
\(47\) −24.0000 −0.0744843 −0.0372421 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 142.000 0.401637
\(51\) 0 0
\(52\) 72.0000 0.192012
\(53\) −318.000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 392.000 0.961041
\(56\) −56.0000 −0.133631
\(57\) 0 0
\(58\) −380.000 −0.860284
\(59\) 200.000 0.441318 0.220659 0.975351i \(-0.429179\pi\)
0.220659 + 0.975351i \(0.429179\pi\)
\(60\) 0 0
\(61\) −198.000 −0.415595 −0.207798 0.978172i \(-0.566630\pi\)
−0.207798 + 0.978172i \(0.566630\pi\)
\(62\) 144.000 0.294968
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 252.000 0.480873
\(66\) 0 0
\(67\) −716.000 −1.30557 −0.652786 0.757542i \(-0.726400\pi\)
−0.652786 + 0.757542i \(0.726400\pi\)
\(68\) −296.000 −0.527872
\(69\) 0 0
\(70\) −196.000 −0.334664
\(71\) −392.000 −0.655237 −0.327619 0.944810i \(-0.606246\pi\)
−0.327619 + 0.944810i \(0.606246\pi\)
\(72\) 0 0
\(73\) 538.000 0.862577 0.431289 0.902214i \(-0.358059\pi\)
0.431289 + 0.902214i \(0.358059\pi\)
\(74\) −692.000 −1.08707
\(75\) 0 0
\(76\) 320.000 0.482980
\(77\) −196.000 −0.290081
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 224.000 0.313050
\(81\) 0 0
\(82\) −324.000 −0.436339
\(83\) 1072.00 1.41768 0.708839 0.705370i \(-0.249219\pi\)
0.708839 + 0.705370i \(0.249219\pi\)
\(84\) 0 0
\(85\) −1036.00 −1.32200
\(86\) −824.000 −1.03319
\(87\) 0 0
\(88\) 224.000 0.271346
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) −126.000 −0.145147
\(92\) 448.000 0.507687
\(93\) 0 0
\(94\) −48.0000 −0.0526683
\(95\) 1120.00 1.20957
\(96\) 0 0
\(97\) 1354.00 1.41730 0.708649 0.705561i \(-0.249305\pi\)
0.708649 + 0.705561i \(0.249305\pi\)
\(98\) 98.0000 0.101015
\(99\) 0 0
\(100\) 284.000 0.284000
\(101\) 1358.00 1.33788 0.668941 0.743316i \(-0.266748\pi\)
0.668941 + 0.743316i \(0.266748\pi\)
\(102\) 0 0
\(103\) −832.000 −0.795916 −0.397958 0.917404i \(-0.630281\pi\)
−0.397958 + 0.917404i \(0.630281\pi\)
\(104\) 144.000 0.135773
\(105\) 0 0
\(106\) −636.000 −0.582772
\(107\) −444.000 −0.401150 −0.200575 0.979678i \(-0.564281\pi\)
−0.200575 + 0.979678i \(0.564281\pi\)
\(108\) 0 0
\(109\) 1870.00 1.64324 0.821622 0.570033i \(-0.193070\pi\)
0.821622 + 0.570033i \(0.193070\pi\)
\(110\) 784.000 0.679559
\(111\) 0 0
\(112\) −112.000 −0.0944911
\(113\) −1378.00 −1.14718 −0.573590 0.819143i \(-0.694450\pi\)
−0.573590 + 0.819143i \(0.694450\pi\)
\(114\) 0 0
\(115\) 1568.00 1.27145
\(116\) −760.000 −0.608312
\(117\) 0 0
\(118\) 400.000 0.312059
\(119\) 518.000 0.399033
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) −396.000 −0.293870
\(123\) 0 0
\(124\) 288.000 0.208574
\(125\) −756.000 −0.540950
\(126\) 0 0
\(127\) 1944.00 1.35828 0.679142 0.734007i \(-0.262352\pi\)
0.679142 + 0.734007i \(0.262352\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 504.000 0.340029
\(131\) 848.000 0.565573 0.282787 0.959183i \(-0.408741\pi\)
0.282787 + 0.959183i \(0.408741\pi\)
\(132\) 0 0
\(133\) −560.000 −0.365099
\(134\) −1432.00 −0.923179
\(135\) 0 0
\(136\) −592.000 −0.373262
\(137\) 2966.00 1.84965 0.924827 0.380389i \(-0.124210\pi\)
0.924827 + 0.380389i \(0.124210\pi\)
\(138\) 0 0
\(139\) 2800.00 1.70858 0.854291 0.519795i \(-0.173992\pi\)
0.854291 + 0.519795i \(0.173992\pi\)
\(140\) −392.000 −0.236643
\(141\) 0 0
\(142\) −784.000 −0.463323
\(143\) 504.000 0.294731
\(144\) 0 0
\(145\) −2660.00 −1.52346
\(146\) 1076.00 0.609934
\(147\) 0 0
\(148\) −1384.00 −0.768676
\(149\) −510.000 −0.280408 −0.140204 0.990123i \(-0.544776\pi\)
−0.140204 + 0.990123i \(0.544776\pi\)
\(150\) 0 0
\(151\) 592.000 0.319048 0.159524 0.987194i \(-0.449004\pi\)
0.159524 + 0.987194i \(0.449004\pi\)
\(152\) 640.000 0.341519
\(153\) 0 0
\(154\) −392.000 −0.205119
\(155\) 1008.00 0.522352
\(156\) 0 0
\(157\) −2686.00 −1.36539 −0.682695 0.730704i \(-0.739192\pi\)
−0.682695 + 0.730704i \(0.739192\pi\)
\(158\) 480.000 0.241688
\(159\) 0 0
\(160\) 448.000 0.221359
\(161\) −784.000 −0.383776
\(162\) 0 0
\(163\) −1012.00 −0.486294 −0.243147 0.969989i \(-0.578180\pi\)
−0.243147 + 0.969989i \(0.578180\pi\)
\(164\) −648.000 −0.308538
\(165\) 0 0
\(166\) 2144.00 1.00245
\(167\) −544.000 −0.252072 −0.126036 0.992026i \(-0.540225\pi\)
−0.126036 + 0.992026i \(0.540225\pi\)
\(168\) 0 0
\(169\) −1873.00 −0.852526
\(170\) −2072.00 −0.934795
\(171\) 0 0
\(172\) −1648.00 −0.730575
\(173\) −1858.00 −0.816538 −0.408269 0.912862i \(-0.633868\pi\)
−0.408269 + 0.912862i \(0.633868\pi\)
\(174\) 0 0
\(175\) −497.000 −0.214684
\(176\) 448.000 0.191871
\(177\) 0 0
\(178\) −1620.00 −0.682158
\(179\) 300.000 0.125268 0.0626342 0.998037i \(-0.480050\pi\)
0.0626342 + 0.998037i \(0.480050\pi\)
\(180\) 0 0
\(181\) −2358.00 −0.968336 −0.484168 0.874975i \(-0.660878\pi\)
−0.484168 + 0.874975i \(0.660878\pi\)
\(182\) −252.000 −0.102635
\(183\) 0 0
\(184\) 896.000 0.358989
\(185\) −4844.00 −1.92507
\(186\) 0 0
\(187\) −2072.00 −0.810265
\(188\) −96.0000 −0.0372421
\(189\) 0 0
\(190\) 2240.00 0.855298
\(191\) −1392.00 −0.527338 −0.263669 0.964613i \(-0.584933\pi\)
−0.263669 + 0.964613i \(0.584933\pi\)
\(192\) 0 0
\(193\) 1778.00 0.663126 0.331563 0.943433i \(-0.392424\pi\)
0.331563 + 0.943433i \(0.392424\pi\)
\(194\) 2708.00 1.00218
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) −1214.00 −0.439055 −0.219528 0.975606i \(-0.570452\pi\)
−0.219528 + 0.975606i \(0.570452\pi\)
\(198\) 0 0
\(199\) 1040.00 0.370471 0.185235 0.982694i \(-0.440695\pi\)
0.185235 + 0.982694i \(0.440695\pi\)
\(200\) 568.000 0.200818
\(201\) 0 0
\(202\) 2716.00 0.946025
\(203\) 1330.00 0.459841
\(204\) 0 0
\(205\) −2268.00 −0.772702
\(206\) −1664.00 −0.562798
\(207\) 0 0
\(208\) 288.000 0.0960058
\(209\) 2240.00 0.741359
\(210\) 0 0
\(211\) −3868.00 −1.26201 −0.631005 0.775779i \(-0.717357\pi\)
−0.631005 + 0.775779i \(0.717357\pi\)
\(212\) −1272.00 −0.412082
\(213\) 0 0
\(214\) −888.000 −0.283656
\(215\) −5768.00 −1.82965
\(216\) 0 0
\(217\) −504.000 −0.157667
\(218\) 3740.00 1.16195
\(219\) 0 0
\(220\) 1568.00 0.480521
\(221\) −1332.00 −0.405430
\(222\) 0 0
\(223\) 3968.00 1.19156 0.595778 0.803149i \(-0.296844\pi\)
0.595778 + 0.803149i \(0.296844\pi\)
\(224\) −224.000 −0.0668153
\(225\) 0 0
\(226\) −2756.00 −0.811179
\(227\) 3936.00 1.15084 0.575422 0.817857i \(-0.304838\pi\)
0.575422 + 0.817857i \(0.304838\pi\)
\(228\) 0 0
\(229\) 4810.00 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(230\) 3136.00 0.899051
\(231\) 0 0
\(232\) −1520.00 −0.430142
\(233\) 2182.00 0.613509 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(234\) 0 0
\(235\) −336.000 −0.0932690
\(236\) 800.000 0.220659
\(237\) 0 0
\(238\) 1036.00 0.282159
\(239\) 3000.00 0.811941 0.405970 0.913886i \(-0.366934\pi\)
0.405970 + 0.913886i \(0.366934\pi\)
\(240\) 0 0
\(241\) 2042.00 0.545796 0.272898 0.962043i \(-0.412018\pi\)
0.272898 + 0.962043i \(0.412018\pi\)
\(242\) −1094.00 −0.290599
\(243\) 0 0
\(244\) −792.000 −0.207798
\(245\) 686.000 0.178885
\(246\) 0 0
\(247\) 1440.00 0.370951
\(248\) 576.000 0.147484
\(249\) 0 0
\(250\) −1512.00 −0.382509
\(251\) 528.000 0.132777 0.0663886 0.997794i \(-0.478852\pi\)
0.0663886 + 0.997794i \(0.478852\pi\)
\(252\) 0 0
\(253\) 3136.00 0.779283
\(254\) 3888.00 0.960452
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −5634.00 −1.36747 −0.683734 0.729731i \(-0.739645\pi\)
−0.683734 + 0.729731i \(0.739645\pi\)
\(258\) 0 0
\(259\) 2422.00 0.581065
\(260\) 1008.00 0.240437
\(261\) 0 0
\(262\) 1696.00 0.399921
\(263\) −168.000 −0.0393891 −0.0196945 0.999806i \(-0.506269\pi\)
−0.0196945 + 0.999806i \(0.506269\pi\)
\(264\) 0 0
\(265\) −4452.00 −1.03202
\(266\) −1120.00 −0.258164
\(267\) 0 0
\(268\) −2864.00 −0.652786
\(269\) 1310.00 0.296922 0.148461 0.988918i \(-0.452568\pi\)
0.148461 + 0.988918i \(0.452568\pi\)
\(270\) 0 0
\(271\) −2208.00 −0.494932 −0.247466 0.968897i \(-0.579598\pi\)
−0.247466 + 0.968897i \(0.579598\pi\)
\(272\) −1184.00 −0.263936
\(273\) 0 0
\(274\) 5932.00 1.30790
\(275\) 1988.00 0.435931
\(276\) 0 0
\(277\) 5294.00 1.14832 0.574162 0.818742i \(-0.305328\pi\)
0.574162 + 0.818742i \(0.305328\pi\)
\(278\) 5600.00 1.20815
\(279\) 0 0
\(280\) −784.000 −0.167332
\(281\) −3242.00 −0.688262 −0.344131 0.938922i \(-0.611826\pi\)
−0.344131 + 0.938922i \(0.611826\pi\)
\(282\) 0 0
\(283\) −1592.00 −0.334398 −0.167199 0.985923i \(-0.553472\pi\)
−0.167199 + 0.985923i \(0.553472\pi\)
\(284\) −1568.00 −0.327619
\(285\) 0 0
\(286\) 1008.00 0.208407
\(287\) 1134.00 0.233233
\(288\) 0 0
\(289\) 563.000 0.114594
\(290\) −5320.00 −1.07725
\(291\) 0 0
\(292\) 2152.00 0.431289
\(293\) 5022.00 1.00133 0.500663 0.865642i \(-0.333090\pi\)
0.500663 + 0.865642i \(0.333090\pi\)
\(294\) 0 0
\(295\) 2800.00 0.552618
\(296\) −2768.00 −0.543536
\(297\) 0 0
\(298\) −1020.00 −0.198279
\(299\) 2016.00 0.389927
\(300\) 0 0
\(301\) 2884.00 0.552262
\(302\) 1184.00 0.225601
\(303\) 0 0
\(304\) 1280.00 0.241490
\(305\) −2772.00 −0.520407
\(306\) 0 0
\(307\) −9536.00 −1.77280 −0.886398 0.462924i \(-0.846800\pi\)
−0.886398 + 0.462924i \(0.846800\pi\)
\(308\) −784.000 −0.145041
\(309\) 0 0
\(310\) 2016.00 0.369358
\(311\) 968.000 0.176496 0.0882480 0.996099i \(-0.471873\pi\)
0.0882480 + 0.996099i \(0.471873\pi\)
\(312\) 0 0
\(313\) 3058.00 0.552231 0.276116 0.961124i \(-0.410953\pi\)
0.276116 + 0.961124i \(0.410953\pi\)
\(314\) −5372.00 −0.965476
\(315\) 0 0
\(316\) 960.000 0.170899
\(317\) 4986.00 0.883412 0.441706 0.897160i \(-0.354373\pi\)
0.441706 + 0.897160i \(0.354373\pi\)
\(318\) 0 0
\(319\) −5320.00 −0.933739
\(320\) 896.000 0.156525
\(321\) 0 0
\(322\) −1568.00 −0.271370
\(323\) −5920.00 −1.01981
\(324\) 0 0
\(325\) 1278.00 0.218125
\(326\) −2024.00 −0.343862
\(327\) 0 0
\(328\) −1296.00 −0.218170
\(329\) 168.000 0.0281524
\(330\) 0 0
\(331\) 8612.00 1.43009 0.715043 0.699081i \(-0.246407\pi\)
0.715043 + 0.699081i \(0.246407\pi\)
\(332\) 4288.00 0.708839
\(333\) 0 0
\(334\) −1088.00 −0.178242
\(335\) −10024.0 −1.63483
\(336\) 0 0
\(337\) −10206.0 −1.64972 −0.824861 0.565336i \(-0.808747\pi\)
−0.824861 + 0.565336i \(0.808747\pi\)
\(338\) −3746.00 −0.602827
\(339\) 0 0
\(340\) −4144.00 −0.661000
\(341\) 2016.00 0.320154
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −3296.00 −0.516594
\(345\) 0 0
\(346\) −3716.00 −0.577380
\(347\) −2004.00 −0.310030 −0.155015 0.987912i \(-0.549543\pi\)
−0.155015 + 0.987912i \(0.549543\pi\)
\(348\) 0 0
\(349\) 1330.00 0.203992 0.101996 0.994785i \(-0.467477\pi\)
0.101996 + 0.994785i \(0.467477\pi\)
\(350\) −994.000 −0.151804
\(351\) 0 0
\(352\) 896.000 0.135673
\(353\) −978.000 −0.147461 −0.0737304 0.997278i \(-0.523490\pi\)
−0.0737304 + 0.997278i \(0.523490\pi\)
\(354\) 0 0
\(355\) −5488.00 −0.820487
\(356\) −3240.00 −0.482359
\(357\) 0 0
\(358\) 600.000 0.0885782
\(359\) 9680.00 1.42309 0.711547 0.702638i \(-0.247995\pi\)
0.711547 + 0.702638i \(0.247995\pi\)
\(360\) 0 0
\(361\) −459.000 −0.0669194
\(362\) −4716.00 −0.684717
\(363\) 0 0
\(364\) −504.000 −0.0725736
\(365\) 7532.00 1.08012
\(366\) 0 0
\(367\) −8656.00 −1.23117 −0.615585 0.788070i \(-0.711080\pi\)
−0.615585 + 0.788070i \(0.711080\pi\)
\(368\) 1792.00 0.253844
\(369\) 0 0
\(370\) −9688.00 −1.36123
\(371\) 2226.00 0.311504
\(372\) 0 0
\(373\) 5278.00 0.732666 0.366333 0.930484i \(-0.380613\pi\)
0.366333 + 0.930484i \(0.380613\pi\)
\(374\) −4144.00 −0.572944
\(375\) 0 0
\(376\) −192.000 −0.0263342
\(377\) −3420.00 −0.467212
\(378\) 0 0
\(379\) 6340.00 0.859272 0.429636 0.903002i \(-0.358642\pi\)
0.429636 + 0.903002i \(0.358642\pi\)
\(380\) 4480.00 0.604787
\(381\) 0 0
\(382\) −2784.00 −0.372884
\(383\) 6232.00 0.831437 0.415718 0.909493i \(-0.363530\pi\)
0.415718 + 0.909493i \(0.363530\pi\)
\(384\) 0 0
\(385\) −2744.00 −0.363239
\(386\) 3556.00 0.468901
\(387\) 0 0
\(388\) 5416.00 0.708649
\(389\) 14810.0 1.93033 0.965163 0.261649i \(-0.0842664\pi\)
0.965163 + 0.261649i \(0.0842664\pi\)
\(390\) 0 0
\(391\) −8288.00 −1.07197
\(392\) 392.000 0.0505076
\(393\) 0 0
\(394\) −2428.00 −0.310459
\(395\) 3360.00 0.428000
\(396\) 0 0
\(397\) 5154.00 0.651566 0.325783 0.945445i \(-0.394372\pi\)
0.325783 + 0.945445i \(0.394372\pi\)
\(398\) 2080.00 0.261962
\(399\) 0 0
\(400\) 1136.00 0.142000
\(401\) −3282.00 −0.408716 −0.204358 0.978896i \(-0.565511\pi\)
−0.204358 + 0.978896i \(0.565511\pi\)
\(402\) 0 0
\(403\) 1296.00 0.160194
\(404\) 5432.00 0.668941
\(405\) 0 0
\(406\) 2660.00 0.325157
\(407\) −9688.00 −1.17989
\(408\) 0 0
\(409\) 5810.00 0.702411 0.351205 0.936298i \(-0.385772\pi\)
0.351205 + 0.936298i \(0.385772\pi\)
\(410\) −4536.00 −0.546383
\(411\) 0 0
\(412\) −3328.00 −0.397958
\(413\) −1400.00 −0.166803
\(414\) 0 0
\(415\) 15008.0 1.77521
\(416\) 576.000 0.0678864
\(417\) 0 0
\(418\) 4480.00 0.524220
\(419\) −13560.0 −1.58102 −0.790512 0.612446i \(-0.790186\pi\)
−0.790512 + 0.612446i \(0.790186\pi\)
\(420\) 0 0
\(421\) −738.000 −0.0854345 −0.0427172 0.999087i \(-0.513601\pi\)
−0.0427172 + 0.999087i \(0.513601\pi\)
\(422\) −7736.00 −0.892376
\(423\) 0 0
\(424\) −2544.00 −0.291386
\(425\) −5254.00 −0.599662
\(426\) 0 0
\(427\) 1386.00 0.157080
\(428\) −1776.00 −0.200575
\(429\) 0 0
\(430\) −11536.0 −1.29376
\(431\) −1272.00 −0.142158 −0.0710790 0.997471i \(-0.522644\pi\)
−0.0710790 + 0.997471i \(0.522644\pi\)
\(432\) 0 0
\(433\) −5062.00 −0.561811 −0.280906 0.959735i \(-0.590635\pi\)
−0.280906 + 0.959735i \(0.590635\pi\)
\(434\) −1008.00 −0.111487
\(435\) 0 0
\(436\) 7480.00 0.821622
\(437\) 8960.00 0.980812
\(438\) 0 0
\(439\) 5640.00 0.613172 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(440\) 3136.00 0.339779
\(441\) 0 0
\(442\) −2664.00 −0.286682
\(443\) −13388.0 −1.43585 −0.717927 0.696119i \(-0.754909\pi\)
−0.717927 + 0.696119i \(0.754909\pi\)
\(444\) 0 0
\(445\) −11340.0 −1.20802
\(446\) 7936.00 0.842557
\(447\) 0 0
\(448\) −448.000 −0.0472456
\(449\) 3230.00 0.339495 0.169747 0.985488i \(-0.445705\pi\)
0.169747 + 0.985488i \(0.445705\pi\)
\(450\) 0 0
\(451\) −4536.00 −0.473596
\(452\) −5512.00 −0.573590
\(453\) 0 0
\(454\) 7872.00 0.813769
\(455\) −1764.00 −0.181753
\(456\) 0 0
\(457\) −10646.0 −1.08971 −0.544857 0.838529i \(-0.683416\pi\)
−0.544857 + 0.838529i \(0.683416\pi\)
\(458\) 9620.00 0.981470
\(459\) 0 0
\(460\) 6272.00 0.635725
\(461\) −7282.00 −0.735698 −0.367849 0.929886i \(-0.619906\pi\)
−0.367849 + 0.929886i \(0.619906\pi\)
\(462\) 0 0
\(463\) 12688.0 1.27357 0.636783 0.771043i \(-0.280265\pi\)
0.636783 + 0.771043i \(0.280265\pi\)
\(464\) −3040.00 −0.304156
\(465\) 0 0
\(466\) 4364.00 0.433816
\(467\) 2816.00 0.279034 0.139517 0.990220i \(-0.455445\pi\)
0.139517 + 0.990220i \(0.455445\pi\)
\(468\) 0 0
\(469\) 5012.00 0.493460
\(470\) −672.000 −0.0659512
\(471\) 0 0
\(472\) 1600.00 0.156030
\(473\) −11536.0 −1.12141
\(474\) 0 0
\(475\) 5680.00 0.548666
\(476\) 2072.00 0.199517
\(477\) 0 0
\(478\) 6000.00 0.574129
\(479\) 3160.00 0.301428 0.150714 0.988577i \(-0.451843\pi\)
0.150714 + 0.988577i \(0.451843\pi\)
\(480\) 0 0
\(481\) −6228.00 −0.590379
\(482\) 4084.00 0.385936
\(483\) 0 0
\(484\) −2188.00 −0.205485
\(485\) 18956.0 1.77474
\(486\) 0 0
\(487\) −14176.0 −1.31905 −0.659523 0.751684i \(-0.729242\pi\)
−0.659523 + 0.751684i \(0.729242\pi\)
\(488\) −1584.00 −0.146935
\(489\) 0 0
\(490\) 1372.00 0.126491
\(491\) 11268.0 1.03568 0.517839 0.855478i \(-0.326737\pi\)
0.517839 + 0.855478i \(0.326737\pi\)
\(492\) 0 0
\(493\) 14060.0 1.28444
\(494\) 2880.00 0.262302
\(495\) 0 0
\(496\) 1152.00 0.104287
\(497\) 2744.00 0.247656
\(498\) 0 0
\(499\) −4460.00 −0.400114 −0.200057 0.979784i \(-0.564113\pi\)
−0.200057 + 0.979784i \(0.564113\pi\)
\(500\) −3024.00 −0.270475
\(501\) 0 0
\(502\) 1056.00 0.0938876
\(503\) 1512.00 0.134029 0.0670147 0.997752i \(-0.478653\pi\)
0.0670147 + 0.997752i \(0.478653\pi\)
\(504\) 0 0
\(505\) 19012.0 1.67529
\(506\) 6272.00 0.551036
\(507\) 0 0
\(508\) 7776.00 0.679142
\(509\) 11790.0 1.02668 0.513342 0.858184i \(-0.328407\pi\)
0.513342 + 0.858184i \(0.328407\pi\)
\(510\) 0 0
\(511\) −3766.00 −0.326024
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −11268.0 −0.966946
\(515\) −11648.0 −0.996645
\(516\) 0 0
\(517\) −672.000 −0.0571654
\(518\) 4844.00 0.410875
\(519\) 0 0
\(520\) 2016.00 0.170014
\(521\) −1362.00 −0.114530 −0.0572652 0.998359i \(-0.518238\pi\)
−0.0572652 + 0.998359i \(0.518238\pi\)
\(522\) 0 0
\(523\) 6968.00 0.582580 0.291290 0.956635i \(-0.405916\pi\)
0.291290 + 0.956635i \(0.405916\pi\)
\(524\) 3392.00 0.282787
\(525\) 0 0
\(526\) −336.000 −0.0278523
\(527\) −5328.00 −0.440401
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) −8904.00 −0.729745
\(531\) 0 0
\(532\) −2240.00 −0.182549
\(533\) −2916.00 −0.236972
\(534\) 0 0
\(535\) −6216.00 −0.502320
\(536\) −5728.00 −0.461589
\(537\) 0 0
\(538\) 2620.00 0.209956
\(539\) 1372.00 0.109640
\(540\) 0 0
\(541\) 7062.00 0.561218 0.280609 0.959822i \(-0.409464\pi\)
0.280609 + 0.959822i \(0.409464\pi\)
\(542\) −4416.00 −0.349969
\(543\) 0 0
\(544\) −2368.00 −0.186631
\(545\) 26180.0 2.05767
\(546\) 0 0
\(547\) −8196.00 −0.640650 −0.320325 0.947308i \(-0.603792\pi\)
−0.320325 + 0.947308i \(0.603792\pi\)
\(548\) 11864.0 0.924827
\(549\) 0 0
\(550\) 3976.00 0.308249
\(551\) −15200.0 −1.17521
\(552\) 0 0
\(553\) −1680.00 −0.129188
\(554\) 10588.0 0.811987
\(555\) 0 0
\(556\) 11200.0 0.854291
\(557\) 7466.00 0.567944 0.283972 0.958833i \(-0.408348\pi\)
0.283972 + 0.958833i \(0.408348\pi\)
\(558\) 0 0
\(559\) −7416.00 −0.561115
\(560\) −1568.00 −0.118322
\(561\) 0 0
\(562\) −6484.00 −0.486674
\(563\) −24968.0 −1.86905 −0.934526 0.355896i \(-0.884176\pi\)
−0.934526 + 0.355896i \(0.884176\pi\)
\(564\) 0 0
\(565\) −19292.0 −1.43650
\(566\) −3184.00 −0.236455
\(567\) 0 0
\(568\) −3136.00 −0.231661
\(569\) −14250.0 −1.04990 −0.524948 0.851134i \(-0.675915\pi\)
−0.524948 + 0.851134i \(0.675915\pi\)
\(570\) 0 0
\(571\) 6372.00 0.467005 0.233503 0.972356i \(-0.424981\pi\)
0.233503 + 0.972356i \(0.424981\pi\)
\(572\) 2016.00 0.147366
\(573\) 0 0
\(574\) 2268.00 0.164921
\(575\) 7952.00 0.576733
\(576\) 0 0
\(577\) −8366.00 −0.603607 −0.301803 0.953370i \(-0.597589\pi\)
−0.301803 + 0.953370i \(0.597589\pi\)
\(578\) 1126.00 0.0810301
\(579\) 0 0
\(580\) −10640.0 −0.761728
\(581\) −7504.00 −0.535832
\(582\) 0 0
\(583\) −8904.00 −0.632532
\(584\) 4304.00 0.304967
\(585\) 0 0
\(586\) 10044.0 0.708044
\(587\) −20384.0 −1.43328 −0.716642 0.697441i \(-0.754322\pi\)
−0.716642 + 0.697441i \(0.754322\pi\)
\(588\) 0 0
\(589\) 5760.00 0.402948
\(590\) 5600.00 0.390760
\(591\) 0 0
\(592\) −5536.00 −0.384338
\(593\) −9378.00 −0.649424 −0.324712 0.945813i \(-0.605267\pi\)
−0.324712 + 0.945813i \(0.605267\pi\)
\(594\) 0 0
\(595\) 7252.00 0.499669
\(596\) −2040.00 −0.140204
\(597\) 0 0
\(598\) 4032.00 0.275720
\(599\) 9000.00 0.613907 0.306953 0.951725i \(-0.400690\pi\)
0.306953 + 0.951725i \(0.400690\pi\)
\(600\) 0 0
\(601\) 7562.00 0.513245 0.256623 0.966512i \(-0.417390\pi\)
0.256623 + 0.966512i \(0.417390\pi\)
\(602\) 5768.00 0.390509
\(603\) 0 0
\(604\) 2368.00 0.159524
\(605\) −7658.00 −0.514615
\(606\) 0 0
\(607\) −2976.00 −0.198999 −0.0994993 0.995038i \(-0.531724\pi\)
−0.0994993 + 0.995038i \(0.531724\pi\)
\(608\) 2560.00 0.170759
\(609\) 0 0
\(610\) −5544.00 −0.367984
\(611\) −432.000 −0.0286037
\(612\) 0 0
\(613\) 4278.00 0.281871 0.140935 0.990019i \(-0.454989\pi\)
0.140935 + 0.990019i \(0.454989\pi\)
\(614\) −19072.0 −1.25356
\(615\) 0 0
\(616\) −1568.00 −0.102559
\(617\) −18794.0 −1.22629 −0.613143 0.789972i \(-0.710095\pi\)
−0.613143 + 0.789972i \(0.710095\pi\)
\(618\) 0 0
\(619\) 18040.0 1.17139 0.585694 0.810532i \(-0.300822\pi\)
0.585694 + 0.810532i \(0.300822\pi\)
\(620\) 4032.00 0.261176
\(621\) 0 0
\(622\) 1936.00 0.124801
\(623\) 5670.00 0.364629
\(624\) 0 0
\(625\) −19459.0 −1.24538
\(626\) 6116.00 0.390486
\(627\) 0 0
\(628\) −10744.0 −0.682695
\(629\) 25604.0 1.62305
\(630\) 0 0
\(631\) −21688.0 −1.36828 −0.684141 0.729350i \(-0.739823\pi\)
−0.684141 + 0.729350i \(0.739823\pi\)
\(632\) 1920.00 0.120844
\(633\) 0 0
\(634\) 9972.00 0.624667
\(635\) 27216.0 1.70084
\(636\) 0 0
\(637\) 882.000 0.0548605
\(638\) −10640.0 −0.660253
\(639\) 0 0
\(640\) 1792.00 0.110680
\(641\) 10558.0 0.650571 0.325285 0.945616i \(-0.394540\pi\)
0.325285 + 0.945616i \(0.394540\pi\)
\(642\) 0 0
\(643\) −26152.0 −1.60394 −0.801971 0.597363i \(-0.796215\pi\)
−0.801971 + 0.597363i \(0.796215\pi\)
\(644\) −3136.00 −0.191888
\(645\) 0 0
\(646\) −11840.0 −0.721112
\(647\) −25584.0 −1.55458 −0.777288 0.629145i \(-0.783405\pi\)
−0.777288 + 0.629145i \(0.783405\pi\)
\(648\) 0 0
\(649\) 5600.00 0.338705
\(650\) 2556.00 0.154238
\(651\) 0 0
\(652\) −4048.00 −0.243147
\(653\) −15198.0 −0.910787 −0.455393 0.890290i \(-0.650501\pi\)
−0.455393 + 0.890290i \(0.650501\pi\)
\(654\) 0 0
\(655\) 11872.0 0.708210
\(656\) −2592.00 −0.154269
\(657\) 0 0
\(658\) 336.000 0.0199068
\(659\) 6100.00 0.360580 0.180290 0.983613i \(-0.442296\pi\)
0.180290 + 0.983613i \(0.442296\pi\)
\(660\) 0 0
\(661\) −2318.00 −0.136399 −0.0681995 0.997672i \(-0.521725\pi\)
−0.0681995 + 0.997672i \(0.521725\pi\)
\(662\) 17224.0 1.01122
\(663\) 0 0
\(664\) 8576.00 0.501225
\(665\) −7840.00 −0.457176
\(666\) 0 0
\(667\) −21280.0 −1.23533
\(668\) −2176.00 −0.126036
\(669\) 0 0
\(670\) −20048.0 −1.15600
\(671\) −5544.00 −0.318962
\(672\) 0 0
\(673\) −10222.0 −0.585482 −0.292741 0.956192i \(-0.594567\pi\)
−0.292741 + 0.956192i \(0.594567\pi\)
\(674\) −20412.0 −1.16653
\(675\) 0 0
\(676\) −7492.00 −0.426263
\(677\) −25434.0 −1.44388 −0.721941 0.691955i \(-0.756750\pi\)
−0.721941 + 0.691955i \(0.756750\pi\)
\(678\) 0 0
\(679\) −9478.00 −0.535688
\(680\) −8288.00 −0.467397
\(681\) 0 0
\(682\) 4032.00 0.226383
\(683\) 8532.00 0.477991 0.238996 0.971021i \(-0.423182\pi\)
0.238996 + 0.971021i \(0.423182\pi\)
\(684\) 0 0
\(685\) 41524.0 2.31613
\(686\) −686.000 −0.0381802
\(687\) 0 0
\(688\) −6592.00 −0.365287
\(689\) −5724.00 −0.316498
\(690\) 0 0
\(691\) 20672.0 1.13806 0.569030 0.822317i \(-0.307319\pi\)
0.569030 + 0.822317i \(0.307319\pi\)
\(692\) −7432.00 −0.408269
\(693\) 0 0
\(694\) −4008.00 −0.219224
\(695\) 39200.0 2.13948
\(696\) 0 0
\(697\) 11988.0 0.651475
\(698\) 2660.00 0.144244
\(699\) 0 0
\(700\) −1988.00 −0.107342
\(701\) 21458.0 1.15614 0.578072 0.815985i \(-0.303805\pi\)
0.578072 + 0.815985i \(0.303805\pi\)
\(702\) 0 0
\(703\) −27680.0 −1.48502
\(704\) 1792.00 0.0959354
\(705\) 0 0
\(706\) −1956.00 −0.104271
\(707\) −9506.00 −0.505672
\(708\) 0 0
\(709\) −9850.00 −0.521755 −0.260878 0.965372i \(-0.584012\pi\)
−0.260878 + 0.965372i \(0.584012\pi\)
\(710\) −10976.0 −0.580172
\(711\) 0 0
\(712\) −6480.00 −0.341079
\(713\) 8064.00 0.423561
\(714\) 0 0
\(715\) 7056.00 0.369062
\(716\) 1200.00 0.0626342
\(717\) 0 0
\(718\) 19360.0 1.00628
\(719\) 18840.0 0.977209 0.488605 0.872505i \(-0.337506\pi\)
0.488605 + 0.872505i \(0.337506\pi\)
\(720\) 0 0
\(721\) 5824.00 0.300828
\(722\) −918.000 −0.0473191
\(723\) 0 0
\(724\) −9432.00 −0.484168
\(725\) −13490.0 −0.691043
\(726\) 0 0
\(727\) 37504.0 1.91327 0.956634 0.291291i \(-0.0940849\pi\)
0.956634 + 0.291291i \(0.0940849\pi\)
\(728\) −1008.00 −0.0513173
\(729\) 0 0
\(730\) 15064.0 0.763758
\(731\) 30488.0 1.54260
\(732\) 0 0
\(733\) 13338.0 0.672101 0.336051 0.941844i \(-0.390909\pi\)
0.336051 + 0.941844i \(0.390909\pi\)
\(734\) −17312.0 −0.870569
\(735\) 0 0
\(736\) 3584.00 0.179495
\(737\) −20048.0 −1.00200
\(738\) 0 0
\(739\) 17100.0 0.851196 0.425598 0.904912i \(-0.360064\pi\)
0.425598 + 0.904912i \(0.360064\pi\)
\(740\) −19376.0 −0.962535
\(741\) 0 0
\(742\) 4452.00 0.220267
\(743\) 19632.0 0.969352 0.484676 0.874694i \(-0.338938\pi\)
0.484676 + 0.874694i \(0.338938\pi\)
\(744\) 0 0
\(745\) −7140.00 −0.351127
\(746\) 10556.0 0.518073
\(747\) 0 0
\(748\) −8288.00 −0.405133
\(749\) 3108.00 0.151621
\(750\) 0 0
\(751\) 33912.0 1.64776 0.823879 0.566766i \(-0.191805\pi\)
0.823879 + 0.566766i \(0.191805\pi\)
\(752\) −384.000 −0.0186211
\(753\) 0 0
\(754\) −6840.00 −0.330369
\(755\) 8288.00 0.399512
\(756\) 0 0
\(757\) −31386.0 −1.50693 −0.753463 0.657490i \(-0.771618\pi\)
−0.753463 + 0.657490i \(0.771618\pi\)
\(758\) 12680.0 0.607597
\(759\) 0 0
\(760\) 8960.00 0.427649
\(761\) 34558.0 1.64616 0.823079 0.567927i \(-0.192254\pi\)
0.823079 + 0.567927i \(0.192254\pi\)
\(762\) 0 0
\(763\) −13090.0 −0.621088
\(764\) −5568.00 −0.263669
\(765\) 0 0
\(766\) 12464.0 0.587915
\(767\) 3600.00 0.169476
\(768\) 0 0
\(769\) 39130.0 1.83493 0.917467 0.397812i \(-0.130231\pi\)
0.917467 + 0.397812i \(0.130231\pi\)
\(770\) −5488.00 −0.256849
\(771\) 0 0
\(772\) 7112.00 0.331563
\(773\) 25982.0 1.20894 0.604468 0.796629i \(-0.293386\pi\)
0.604468 + 0.796629i \(0.293386\pi\)
\(774\) 0 0
\(775\) 5112.00 0.236940
\(776\) 10832.0 0.501090
\(777\) 0 0
\(778\) 29620.0 1.36495
\(779\) −12960.0 −0.596072
\(780\) 0 0
\(781\) −10976.0 −0.502884
\(782\) −16576.0 −0.758001
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) −37604.0 −1.70974
\(786\) 0 0
\(787\) 35424.0 1.60448 0.802242 0.596999i \(-0.203640\pi\)
0.802242 + 0.596999i \(0.203640\pi\)
\(788\) −4856.00 −0.219528
\(789\) 0 0
\(790\) 6720.00 0.302642
\(791\) 9646.00 0.433593
\(792\) 0 0
\(793\) −3564.00 −0.159598
\(794\) 10308.0 0.460727
\(795\) 0 0
\(796\) 4160.00 0.185235
\(797\) 30606.0 1.36025 0.680126 0.733096i \(-0.261925\pi\)
0.680126 + 0.733096i \(0.261925\pi\)
\(798\) 0 0
\(799\) 1776.00 0.0786362
\(800\) 2272.00 0.100409
\(801\) 0 0
\(802\) −6564.00 −0.289006
\(803\) 15064.0 0.662014
\(804\) 0 0
\(805\) −10976.0 −0.480563
\(806\) 2592.00 0.113275
\(807\) 0 0
\(808\) 10864.0 0.473013
\(809\) −16810.0 −0.730542 −0.365271 0.930901i \(-0.619024\pi\)
−0.365271 + 0.930901i \(0.619024\pi\)
\(810\) 0 0
\(811\) −9368.00 −0.405616 −0.202808 0.979218i \(-0.565007\pi\)
−0.202808 + 0.979218i \(0.565007\pi\)
\(812\) 5320.00 0.229920
\(813\) 0 0
\(814\) −19376.0 −0.834310
\(815\) −14168.0 −0.608937
\(816\) 0 0
\(817\) −32960.0 −1.41141
\(818\) 11620.0 0.496679
\(819\) 0 0
\(820\) −9072.00 −0.386351
\(821\) −34382.0 −1.46156 −0.730780 0.682614i \(-0.760843\pi\)
−0.730780 + 0.682614i \(0.760843\pi\)
\(822\) 0 0
\(823\) −4472.00 −0.189410 −0.0947048 0.995505i \(-0.530191\pi\)
−0.0947048 + 0.995505i \(0.530191\pi\)
\(824\) −6656.00 −0.281399
\(825\) 0 0
\(826\) −2800.00 −0.117947
\(827\) 1716.00 0.0721538 0.0360769 0.999349i \(-0.488514\pi\)
0.0360769 + 0.999349i \(0.488514\pi\)
\(828\) 0 0
\(829\) −7910.00 −0.331394 −0.165697 0.986177i \(-0.552987\pi\)
−0.165697 + 0.986177i \(0.552987\pi\)
\(830\) 30016.0 1.25527
\(831\) 0 0
\(832\) 1152.00 0.0480029
\(833\) −3626.00 −0.150820
\(834\) 0 0
\(835\) −7616.00 −0.315644
\(836\) 8960.00 0.370680
\(837\) 0 0
\(838\) −27120.0 −1.11795
\(839\) 19360.0 0.796641 0.398320 0.917246i \(-0.369593\pi\)
0.398320 + 0.917246i \(0.369593\pi\)
\(840\) 0 0
\(841\) 11711.0 0.480175
\(842\) −1476.00 −0.0604113
\(843\) 0 0
\(844\) −15472.0 −0.631005
\(845\) −26222.0 −1.06753
\(846\) 0 0
\(847\) 3829.00 0.155332
\(848\) −5088.00 −0.206041
\(849\) 0 0
\(850\) −10508.0 −0.424025
\(851\) −38752.0 −1.56099
\(852\) 0 0
\(853\) 698.000 0.0280177 0.0140088 0.999902i \(-0.495541\pi\)
0.0140088 + 0.999902i \(0.495541\pi\)
\(854\) 2772.00 0.111072
\(855\) 0 0
\(856\) −3552.00 −0.141828
\(857\) 23406.0 0.932945 0.466472 0.884536i \(-0.345525\pi\)
0.466472 + 0.884536i \(0.345525\pi\)
\(858\) 0 0
\(859\) 7280.00 0.289162 0.144581 0.989493i \(-0.453817\pi\)
0.144581 + 0.989493i \(0.453817\pi\)
\(860\) −23072.0 −0.914824
\(861\) 0 0
\(862\) −2544.00 −0.100521
\(863\) −9808.00 −0.386869 −0.193435 0.981113i \(-0.561963\pi\)
−0.193435 + 0.981113i \(0.561963\pi\)
\(864\) 0 0
\(865\) −26012.0 −1.02247
\(866\) −10124.0 −0.397260
\(867\) 0 0
\(868\) −2016.00 −0.0788335
\(869\) 6720.00 0.262325
\(870\) 0 0
\(871\) −12888.0 −0.501370
\(872\) 14960.0 0.580974
\(873\) 0 0
\(874\) 17920.0 0.693539
\(875\) 5292.00 0.204460
\(876\) 0 0
\(877\) −8066.00 −0.310570 −0.155285 0.987870i \(-0.549630\pi\)
−0.155285 + 0.987870i \(0.549630\pi\)
\(878\) 11280.0 0.433578
\(879\) 0 0
\(880\) 6272.00 0.240260
\(881\) −25842.0 −0.988240 −0.494120 0.869394i \(-0.664510\pi\)
−0.494120 + 0.869394i \(0.664510\pi\)
\(882\) 0 0
\(883\) −5692.00 −0.216932 −0.108466 0.994100i \(-0.534594\pi\)
−0.108466 + 0.994100i \(0.534594\pi\)
\(884\) −5328.00 −0.202715
\(885\) 0 0
\(886\) −26776.0 −1.01530
\(887\) 13536.0 0.512395 0.256198 0.966624i \(-0.417530\pi\)
0.256198 + 0.966624i \(0.417530\pi\)
\(888\) 0 0
\(889\) −13608.0 −0.513383
\(890\) −22680.0 −0.854197
\(891\) 0 0
\(892\) 15872.0 0.595778
\(893\) −1920.00 −0.0719489
\(894\) 0 0
\(895\) 4200.00 0.156861
\(896\) −896.000 −0.0334077
\(897\) 0 0
\(898\) 6460.00 0.240059
\(899\) −13680.0 −0.507512
\(900\) 0 0
\(901\) 23532.0 0.870105
\(902\) −9072.00 −0.334883
\(903\) 0 0
\(904\) −11024.0 −0.405589
\(905\) −33012.0 −1.21255
\(906\) 0 0
\(907\) 17004.0 0.622501 0.311251 0.950328i \(-0.399252\pi\)
0.311251 + 0.950328i \(0.399252\pi\)
\(908\) 15744.0 0.575422
\(909\) 0 0
\(910\) −3528.00 −0.128519
\(911\) 14568.0 0.529813 0.264906 0.964274i \(-0.414659\pi\)
0.264906 + 0.964274i \(0.414659\pi\)
\(912\) 0 0
\(913\) 30016.0 1.08804
\(914\) −21292.0 −0.770544
\(915\) 0 0
\(916\) 19240.0 0.694004
\(917\) −5936.00 −0.213767
\(918\) 0 0
\(919\) −1400.00 −0.0502522 −0.0251261 0.999684i \(-0.507999\pi\)
−0.0251261 + 0.999684i \(0.507999\pi\)
\(920\) 12544.0 0.449525
\(921\) 0 0
\(922\) −14564.0 −0.520217
\(923\) −7056.00 −0.251626
\(924\) 0 0
\(925\) −24566.0 −0.873216
\(926\) 25376.0 0.900548
\(927\) 0 0
\(928\) −6080.00 −0.215071
\(929\) 13830.0 0.488426 0.244213 0.969722i \(-0.421470\pi\)
0.244213 + 0.969722i \(0.421470\pi\)
\(930\) 0 0
\(931\) 3920.00 0.137994
\(932\) 8728.00 0.306754
\(933\) 0 0
\(934\) 5632.00 0.197307
\(935\) −29008.0 −1.01461
\(936\) 0 0
\(937\) −24166.0 −0.842549 −0.421275 0.906933i \(-0.638417\pi\)
−0.421275 + 0.906933i \(0.638417\pi\)
\(938\) 10024.0 0.348929
\(939\) 0 0
\(940\) −1344.00 −0.0466345
\(941\) 10838.0 0.375461 0.187730 0.982221i \(-0.439887\pi\)
0.187730 + 0.982221i \(0.439887\pi\)
\(942\) 0 0
\(943\) −18144.0 −0.626564
\(944\) 3200.00 0.110330
\(945\) 0 0
\(946\) −23072.0 −0.792955
\(947\) 40916.0 1.40400 0.702002 0.712175i \(-0.252290\pi\)
0.702002 + 0.712175i \(0.252290\pi\)
\(948\) 0 0
\(949\) 9684.00 0.331250
\(950\) 11360.0 0.387965
\(951\) 0 0
\(952\) 4144.00 0.141080
\(953\) −56618.0 −1.92449 −0.962244 0.272189i \(-0.912253\pi\)
−0.962244 + 0.272189i \(0.912253\pi\)
\(954\) 0 0
\(955\) −19488.0 −0.660332
\(956\) 12000.0 0.405970
\(957\) 0 0
\(958\) 6320.00 0.213142
\(959\) −20762.0 −0.699103
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) −12456.0 −0.417461
\(963\) 0 0
\(964\) 8168.00 0.272898
\(965\) 24892.0 0.830365
\(966\) 0 0
\(967\) 17504.0 0.582100 0.291050 0.956708i \(-0.405995\pi\)
0.291050 + 0.956708i \(0.405995\pi\)
\(968\) −4376.00 −0.145300
\(969\) 0 0
\(970\) 37912.0 1.25493
\(971\) −23112.0 −0.763851 −0.381926 0.924193i \(-0.624739\pi\)
−0.381926 + 0.924193i \(0.624739\pi\)
\(972\) 0 0
\(973\) −19600.0 −0.645783
\(974\) −28352.0 −0.932707
\(975\) 0 0
\(976\) −3168.00 −0.103899
\(977\) −23874.0 −0.781778 −0.390889 0.920438i \(-0.627832\pi\)
−0.390889 + 0.920438i \(0.627832\pi\)
\(978\) 0 0
\(979\) −22680.0 −0.740404
\(980\) 2744.00 0.0894427
\(981\) 0 0
\(982\) 22536.0 0.732335
\(983\) 15312.0 0.496823 0.248411 0.968655i \(-0.420091\pi\)
0.248411 + 0.968655i \(0.420091\pi\)
\(984\) 0 0
\(985\) −16996.0 −0.549784
\(986\) 28120.0 0.908239
\(987\) 0 0
\(988\) 5760.00 0.185476
\(989\) −46144.0 −1.48361
\(990\) 0 0
\(991\) −16528.0 −0.529797 −0.264899 0.964276i \(-0.585339\pi\)
−0.264899 + 0.964276i \(0.585339\pi\)
\(992\) 2304.00 0.0737420
\(993\) 0 0
\(994\) 5488.00 0.175120
\(995\) 14560.0 0.463903
\(996\) 0 0
\(997\) −28606.0 −0.908687 −0.454344 0.890827i \(-0.650126\pi\)
−0.454344 + 0.890827i \(0.650126\pi\)
\(998\) −8920.00 −0.282924
\(999\) 0 0
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 126.4.a.h.1.1 1
3.2 odd 2 14.4.a.a.1.1 1
4.3 odd 2 1008.4.a.s.1.1 1
7.2 even 3 882.4.g.b.361.1 2
7.3 odd 6 882.4.g.k.667.1 2
7.4 even 3 882.4.g.b.667.1 2
7.5 odd 6 882.4.g.k.361.1 2
7.6 odd 2 882.4.a.i.1.1 1
12.11 even 2 112.4.a.a.1.1 1
15.2 even 4 350.4.c.b.99.1 2
15.8 even 4 350.4.c.b.99.2 2
15.14 odd 2 350.4.a.l.1.1 1
21.2 odd 6 98.4.c.d.67.1 2
21.5 even 6 98.4.c.f.67.1 2
21.11 odd 6 98.4.c.d.79.1 2
21.17 even 6 98.4.c.f.79.1 2
21.20 even 2 98.4.a.a.1.1 1
24.5 odd 2 448.4.a.b.1.1 1
24.11 even 2 448.4.a.o.1.1 1
33.32 even 2 1694.4.a.g.1.1 1
39.38 odd 2 2366.4.a.h.1.1 1
84.83 odd 2 784.4.a.s.1.1 1
105.104 even 2 2450.4.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.4.a.a.1.1 1 3.2 odd 2
98.4.a.a.1.1 1 21.20 even 2
98.4.c.d.67.1 2 21.2 odd 6
98.4.c.d.79.1 2 21.11 odd 6
98.4.c.f.67.1 2 21.5 even 6
98.4.c.f.79.1 2 21.17 even 6
112.4.a.a.1.1 1 12.11 even 2
126.4.a.h.1.1 1 1.1 even 1 trivial
350.4.a.l.1.1 1 15.14 odd 2
350.4.c.b.99.1 2 15.2 even 4
350.4.c.b.99.2 2 15.8 even 4
448.4.a.b.1.1 1 24.5 odd 2
448.4.a.o.1.1 1 24.11 even 2
784.4.a.s.1.1 1 84.83 odd 2
882.4.a.i.1.1 1 7.6 odd 2
882.4.g.b.361.1 2 7.2 even 3
882.4.g.b.667.1 2 7.4 even 3
882.4.g.k.361.1 2 7.5 odd 6
882.4.g.k.667.1 2 7.3 odd 6
1008.4.a.s.1.1 1 4.3 odd 2
1694.4.a.g.1.1 1 33.32 even 2
2366.4.a.h.1.1 1 39.38 odd 2
2450.4.a.bo.1.1 1 105.104 even 2