Properties

Label 11.6.a.a.1.1
Level $11$
Weight $6$
Character 11.1
Self dual yes
Analytic conductor $1.764$
Analytic rank $1$
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] = [11,6,Mod(1,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 11.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: \(1.76422201794\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 11.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -15.0000 q^{3} -16.0000 q^{4} -19.0000 q^{5} +60.0000 q^{6} +10.0000 q^{7} +192.000 q^{8} -18.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -15.0000 q^{3} -16.0000 q^{4} -19.0000 q^{5} +60.0000 q^{6} +10.0000 q^{7} +192.000 q^{8} -18.0000 q^{9} +76.0000 q^{10} -121.000 q^{11} +240.000 q^{12} -1148.00 q^{13} -40.0000 q^{14} +285.000 q^{15} -256.000 q^{16} +686.000 q^{17} +72.0000 q^{18} -384.000 q^{19} +304.000 q^{20} -150.000 q^{21} +484.000 q^{22} +3709.00 q^{23} -2880.00 q^{24} -2764.00 q^{25} +4592.00 q^{26} +3915.00 q^{27} -160.000 q^{28} -5424.00 q^{29} -1140.00 q^{30} -6443.00 q^{31} -5120.00 q^{32} +1815.00 q^{33} -2744.00 q^{34} -190.000 q^{35} +288.000 q^{36} +12063.0 q^{37} +1536.00 q^{38} +17220.0 q^{39} -3648.00 q^{40} -1528.00 q^{41} +600.000 q^{42} -4026.00 q^{43} +1936.00 q^{44} +342.000 q^{45} -14836.0 q^{46} +7168.00 q^{47} +3840.00 q^{48} -16707.0 q^{49} +11056.0 q^{50} -10290.0 q^{51} +18368.0 q^{52} -29862.0 q^{53} -15660.0 q^{54} +2299.00 q^{55} +1920.00 q^{56} +5760.00 q^{57} +21696.0 q^{58} -6461.00 q^{59} -4560.00 q^{60} -16980.0 q^{61} +25772.0 q^{62} -180.000 q^{63} +28672.0 q^{64} +21812.0 q^{65} -7260.00 q^{66} +29999.0 q^{67} -10976.0 q^{68} -55635.0 q^{69} +760.000 q^{70} +31023.0 q^{71} -3456.00 q^{72} +1924.00 q^{73} -48252.0 q^{74} +41460.0 q^{75} +6144.00 q^{76} -1210.00 q^{77} -68880.0 q^{78} +65138.0 q^{79} +4864.00 q^{80} -54351.0 q^{81} +6112.00 q^{82} -102714. q^{83} +2400.00 q^{84} -13034.0 q^{85} +16104.0 q^{86} +81360.0 q^{87} -23232.0 q^{88} +17415.0 q^{89} -1368.00 q^{90} -11480.0 q^{91} -59344.0 q^{92} +96645.0 q^{93} -28672.0 q^{94} +7296.00 q^{95} +76800.0 q^{96} +66905.0 q^{97} +66828.0 q^{98} +2178.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −15.0000 −0.962250 −0.481125 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) −16.0000 −0.500000
\(5\) −19.0000 −0.339882 −0.169941 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(6\) 60.0000 0.680414
\(7\) 10.0000 0.0771356 0.0385678 0.999256i \(-0.487720\pi\)
0.0385678 + 0.999256i \(0.487720\pi\)
\(8\) 192.000 1.06066
\(9\) −18.0000 −0.0740741
\(10\) 76.0000 0.240333
\(11\) −121.000 −0.301511
\(12\) 240.000 0.481125
\(13\) −1148.00 −1.88401 −0.942006 0.335597i \(-0.891062\pi\)
−0.942006 + 0.335597i \(0.891062\pi\)
\(14\) −40.0000 −0.0545431
\(15\) 285.000 0.327052
\(16\) −256.000 −0.250000
\(17\) 686.000 0.575707 0.287854 0.957674i \(-0.407058\pi\)
0.287854 + 0.957674i \(0.407058\pi\)
\(18\) 72.0000 0.0523783
\(19\) −384.000 −0.244032 −0.122016 0.992528i \(-0.538936\pi\)
−0.122016 + 0.992528i \(0.538936\pi\)
\(20\) 304.000 0.169941
\(21\) −150.000 −0.0742238
\(22\) 484.000 0.213201
\(23\) 3709.00 1.46197 0.730983 0.682396i \(-0.239062\pi\)
0.730983 + 0.682396i \(0.239062\pi\)
\(24\) −2880.00 −1.02062
\(25\) −2764.00 −0.884480
\(26\) 4592.00 1.33220
\(27\) 3915.00 1.03353
\(28\) −160.000 −0.0385678
\(29\) −5424.00 −1.19764 −0.598818 0.800885i \(-0.704363\pi\)
−0.598818 + 0.800885i \(0.704363\pi\)
\(30\) −1140.00 −0.231261
\(31\) −6443.00 −1.20416 −0.602080 0.798436i \(-0.705661\pi\)
−0.602080 + 0.798436i \(0.705661\pi\)
\(32\) −5120.00 −0.883883
\(33\) 1815.00 0.290129
\(34\) −2744.00 −0.407087
\(35\) −190.000 −0.0262170
\(36\) 288.000 0.0370370
\(37\) 12063.0 1.44861 0.724304 0.689481i \(-0.242161\pi\)
0.724304 + 0.689481i \(0.242161\pi\)
\(38\) 1536.00 0.172557
\(39\) 17220.0 1.81289
\(40\) −3648.00 −0.360500
\(41\) −1528.00 −0.141959 −0.0709796 0.997478i \(-0.522613\pi\)
−0.0709796 + 0.997478i \(0.522613\pi\)
\(42\) 600.000 0.0524841
\(43\) −4026.00 −0.332049 −0.166025 0.986122i \(-0.553093\pi\)
−0.166025 + 0.986122i \(0.553093\pi\)
\(44\) 1936.00 0.150756
\(45\) 342.000 0.0251765
\(46\) −14836.0 −1.03377
\(47\) 7168.00 0.473318 0.236659 0.971593i \(-0.423948\pi\)
0.236659 + 0.971593i \(0.423948\pi\)
\(48\) 3840.00 0.240563
\(49\) −16707.0 −0.994050
\(50\) 11056.0 0.625422
\(51\) −10290.0 −0.553975
\(52\) 18368.0 0.942006
\(53\) −29862.0 −1.46026 −0.730128 0.683310i \(-0.760540\pi\)
−0.730128 + 0.683310i \(0.760540\pi\)
\(54\) −15660.0 −0.730815
\(55\) 2299.00 0.102478
\(56\) 1920.00 0.0818147
\(57\) 5760.00 0.234820
\(58\) 21696.0 0.846856
\(59\) −6461.00 −0.241640 −0.120820 0.992674i \(-0.538552\pi\)
−0.120820 + 0.992674i \(0.538552\pi\)
\(60\) −4560.00 −0.163526
\(61\) −16980.0 −0.584269 −0.292135 0.956377i \(-0.594366\pi\)
−0.292135 + 0.956377i \(0.594366\pi\)
\(62\) 25772.0 0.851469
\(63\) −180.000 −0.00571375
\(64\) 28672.0 0.875000
\(65\) 21812.0 0.640342
\(66\) −7260.00 −0.205152
\(67\) 29999.0 0.816432 0.408216 0.912885i \(-0.366151\pi\)
0.408216 + 0.912885i \(0.366151\pi\)
\(68\) −10976.0 −0.287854
\(69\) −55635.0 −1.40678
\(70\) 760.000 0.0185382
\(71\) 31023.0 0.730362 0.365181 0.930937i \(-0.381007\pi\)
0.365181 + 0.930937i \(0.381007\pi\)
\(72\) −3456.00 −0.0785674
\(73\) 1924.00 0.0422569 0.0211285 0.999777i \(-0.493274\pi\)
0.0211285 + 0.999777i \(0.493274\pi\)
\(74\) −48252.0 −1.02432
\(75\) 41460.0 0.851091
\(76\) 6144.00 0.122016
\(77\) −1210.00 −0.0232573
\(78\) −68880.0 −1.28191
\(79\) 65138.0 1.17427 0.587133 0.809490i \(-0.300256\pi\)
0.587133 + 0.809490i \(0.300256\pi\)
\(80\) 4864.00 0.0849706
\(81\) −54351.0 −0.920439
\(82\) 6112.00 0.100380
\(83\) −102714. −1.63657 −0.818285 0.574813i \(-0.805075\pi\)
−0.818285 + 0.574813i \(0.805075\pi\)
\(84\) 2400.00 0.0371119
\(85\) −13034.0 −0.195673
\(86\) 16104.0 0.234794
\(87\) 81360.0 1.15243
\(88\) −23232.0 −0.319801
\(89\) 17415.0 0.233050 0.116525 0.993188i \(-0.462825\pi\)
0.116525 + 0.993188i \(0.462825\pi\)
\(90\) −1368.00 −0.0178025
\(91\) −11480.0 −0.145324
\(92\) −59344.0 −0.730983
\(93\) 96645.0 1.15870
\(94\) −28672.0 −0.334687
\(95\) 7296.00 0.0829422
\(96\) 76800.0 0.850517
\(97\) 66905.0 0.721987 0.360993 0.932568i \(-0.382438\pi\)
0.360993 + 0.932568i \(0.382438\pi\)
\(98\) 66828.0 0.702900
\(99\) 2178.00 0.0223342
\(100\) 44224.0 0.442240
\(101\) 96730.0 0.943534 0.471767 0.881723i \(-0.343616\pi\)
0.471767 + 0.881723i \(0.343616\pi\)
\(102\) 41160.0 0.391719
\(103\) −95704.0 −0.888868 −0.444434 0.895812i \(-0.646595\pi\)
−0.444434 + 0.895812i \(0.646595\pi\)
\(104\) −220416. −1.99830
\(105\) 2850.00 0.0252273
\(106\) 119448. 1.03256
\(107\) −32658.0 −0.275759 −0.137880 0.990449i \(-0.544029\pi\)
−0.137880 + 0.990449i \(0.544029\pi\)
\(108\) −62640.0 −0.516764
\(109\) −185438. −1.49497 −0.747485 0.664279i \(-0.768739\pi\)
−0.747485 + 0.664279i \(0.768739\pi\)
\(110\) −9196.00 −0.0724632
\(111\) −180945. −1.39392
\(112\) −2560.00 −0.0192839
\(113\) 72849.0 0.536695 0.268347 0.963322i \(-0.413522\pi\)
0.268347 + 0.963322i \(0.413522\pi\)
\(114\) −23040.0 −0.166043
\(115\) −70471.0 −0.496896
\(116\) 86784.0 0.598818
\(117\) 20664.0 0.139556
\(118\) 25844.0 0.170866
\(119\) 6860.00 0.0444075
\(120\) 54720.0 0.346891
\(121\) 14641.0 0.0909091
\(122\) 67920.0 0.413141
\(123\) 22920.0 0.136600
\(124\) 103088. 0.602080
\(125\) 111891. 0.640501
\(126\) 720.000 0.00404023
\(127\) −78184.0 −0.430139 −0.215069 0.976599i \(-0.568998\pi\)
−0.215069 + 0.976599i \(0.568998\pi\)
\(128\) 49152.0 0.265165
\(129\) 60390.0 0.319515
\(130\) −87248.0 −0.452790
\(131\) −462.000 −0.00235214 −0.00117607 0.999999i \(-0.500374\pi\)
−0.00117607 + 0.999999i \(0.500374\pi\)
\(132\) −29040.0 −0.145065
\(133\) −3840.00 −0.0188236
\(134\) −119996. −0.577304
\(135\) −74385.0 −0.351278
\(136\) 131712. 0.610630
\(137\) 296233. 1.34844 0.674221 0.738530i \(-0.264480\pi\)
0.674221 + 0.738530i \(0.264480\pi\)
\(138\) 222540. 0.994742
\(139\) −399818. −1.75519 −0.877597 0.479398i \(-0.840855\pi\)
−0.877597 + 0.479398i \(0.840855\pi\)
\(140\) 3040.00 0.0131085
\(141\) −107520. −0.455451
\(142\) −124092. −0.516444
\(143\) 138908. 0.568051
\(144\) 4608.00 0.0185185
\(145\) 103056. 0.407055
\(146\) −7696.00 −0.0298802
\(147\) 250605. 0.956525
\(148\) −193008. −0.724304
\(149\) 72670.0 0.268157 0.134079 0.990971i \(-0.457193\pi\)
0.134079 + 0.990971i \(0.457193\pi\)
\(150\) −165840. −0.601812
\(151\) −303082. −1.08173 −0.540864 0.841110i \(-0.681902\pi\)
−0.540864 + 0.841110i \(0.681902\pi\)
\(152\) −73728.0 −0.258835
\(153\) −12348.0 −0.0426450
\(154\) 4840.00 0.0164454
\(155\) 122417. 0.409272
\(156\) −275520. −0.906445
\(157\) −532987. −1.72571 −0.862854 0.505453i \(-0.831326\pi\)
−0.862854 + 0.505453i \(0.831326\pi\)
\(158\) −260552. −0.830332
\(159\) 447930. 1.40513
\(160\) 97280.0 0.300416
\(161\) 37090.0 0.112770
\(162\) 217404. 0.650849
\(163\) 282076. 0.831567 0.415783 0.909464i \(-0.363507\pi\)
0.415783 + 0.909464i \(0.363507\pi\)
\(164\) 24448.0 0.0709796
\(165\) −34485.0 −0.0986099
\(166\) 410856. 1.15723
\(167\) −573588. −1.59151 −0.795754 0.605620i \(-0.792925\pi\)
−0.795754 + 0.605620i \(0.792925\pi\)
\(168\) −28800.0 −0.0787262
\(169\) 946611. 2.54950
\(170\) 52136.0 0.138362
\(171\) 6912.00 0.0180765
\(172\) 64416.0 0.166025
\(173\) −386286. −0.981282 −0.490641 0.871362i \(-0.663237\pi\)
−0.490641 + 0.871362i \(0.663237\pi\)
\(174\) −325440. −0.814888
\(175\) −27640.0 −0.0682249
\(176\) 30976.0 0.0753778
\(177\) 96915.0 0.232519
\(178\) −69660.0 −0.164791
\(179\) 545079. 1.27153 0.635765 0.771882i \(-0.280685\pi\)
0.635765 + 0.771882i \(0.280685\pi\)
\(180\) −5472.00 −0.0125882
\(181\) −279485. −0.634106 −0.317053 0.948408i \(-0.602693\pi\)
−0.317053 + 0.948408i \(0.602693\pi\)
\(182\) 45920.0 0.102760
\(183\) 254700. 0.562213
\(184\) 712128. 1.55065
\(185\) −229197. −0.492356
\(186\) −386580. −0.819327
\(187\) −83006.0 −0.173582
\(188\) −114688. −0.236659
\(189\) 39150.0 0.0797218
\(190\) −29184.0 −0.0586490
\(191\) −444437. −0.881509 −0.440755 0.897628i \(-0.645289\pi\)
−0.440755 + 0.897628i \(0.645289\pi\)
\(192\) −430080. −0.841969
\(193\) −18476.0 −0.0357038 −0.0178519 0.999841i \(-0.505683\pi\)
−0.0178519 + 0.999841i \(0.505683\pi\)
\(194\) −267620. −0.510522
\(195\) −327180. −0.616170
\(196\) 267312. 0.497025
\(197\) 270182. 0.496010 0.248005 0.968759i \(-0.420225\pi\)
0.248005 + 0.968759i \(0.420225\pi\)
\(198\) −8712.00 −0.0157926
\(199\) 43320.0 0.0775453 0.0387727 0.999248i \(-0.487655\pi\)
0.0387727 + 0.999248i \(0.487655\pi\)
\(200\) −530688. −0.938133
\(201\) −449985. −0.785612
\(202\) −386920. −0.667180
\(203\) −54240.0 −0.0923803
\(204\) 164640. 0.276987
\(205\) 29032.0 0.0482494
\(206\) 382816. 0.628524
\(207\) −66762.0 −0.108294
\(208\) 293888. 0.471003
\(209\) 46464.0 0.0735785
\(210\) −11400.0 −0.0178384
\(211\) 1.02968e6 1.59220 0.796100 0.605165i \(-0.206893\pi\)
0.796100 + 0.605165i \(0.206893\pi\)
\(212\) 477792. 0.730128
\(213\) −465345. −0.702791
\(214\) 130632. 0.194991
\(215\) 76494.0 0.112858
\(216\) 751680. 1.09622
\(217\) −64430.0 −0.0928835
\(218\) 741752. 1.05710
\(219\) −28860.0 −0.0406617
\(220\) −36784.0 −0.0512392
\(221\) −787528. −1.08464
\(222\) 723780. 0.985653
\(223\) 461281. 0.621160 0.310580 0.950547i \(-0.399477\pi\)
0.310580 + 0.950547i \(0.399477\pi\)
\(224\) −51200.0 −0.0681789
\(225\) 49752.0 0.0655170
\(226\) −291396. −0.379501
\(227\) −855570. −1.10202 −0.551012 0.834497i \(-0.685758\pi\)
−0.551012 + 0.834497i \(0.685758\pi\)
\(228\) −92160.0 −0.117410
\(229\) −665805. −0.838993 −0.419497 0.907757i \(-0.637793\pi\)
−0.419497 + 0.907757i \(0.637793\pi\)
\(230\) 281884. 0.351359
\(231\) 18150.0 0.0223793
\(232\) −1.04141e6 −1.27028
\(233\) 1.20934e6 1.45934 0.729671 0.683798i \(-0.239673\pi\)
0.729671 + 0.683798i \(0.239673\pi\)
\(234\) −82656.0 −0.0986813
\(235\) −136192. −0.160873
\(236\) 103376. 0.120820
\(237\) −977070. −1.12994
\(238\) −27440.0 −0.0314009
\(239\) −571482. −0.647154 −0.323577 0.946202i \(-0.604886\pi\)
−0.323577 + 0.946202i \(0.604886\pi\)
\(240\) −72960.0 −0.0817630
\(241\) −267080. −0.296209 −0.148105 0.988972i \(-0.547317\pi\)
−0.148105 + 0.988972i \(0.547317\pi\)
\(242\) −58564.0 −0.0642824
\(243\) −136080. −0.147835
\(244\) 271680. 0.292135
\(245\) 317433. 0.337860
\(246\) −91680.0 −0.0965910
\(247\) 440832. 0.459760
\(248\) −1.23706e6 −1.27720
\(249\) 1.54071e6 1.57479
\(250\) −447564. −0.452903
\(251\) 1.38737e6 1.38998 0.694988 0.719022i \(-0.255410\pi\)
0.694988 + 0.719022i \(0.255410\pi\)
\(252\) 2880.00 0.00285687
\(253\) −448789. −0.440799
\(254\) 312736. 0.304154
\(255\) 195510. 0.188286
\(256\) −1.11411e6 −1.06250
\(257\) −885922. −0.836686 −0.418343 0.908289i \(-0.637389\pi\)
−0.418343 + 0.908289i \(0.637389\pi\)
\(258\) −241560. −0.225931
\(259\) 120630. 0.111739
\(260\) −348992. −0.320171
\(261\) 97632.0 0.0887137
\(262\) 1848.00 0.00166322
\(263\) 1.44687e6 1.28986 0.644928 0.764243i \(-0.276887\pi\)
0.644928 + 0.764243i \(0.276887\pi\)
\(264\) 348480. 0.307729
\(265\) 567378. 0.496315
\(266\) 15360.0 0.0133103
\(267\) −261225. −0.224252
\(268\) −479984. −0.408216
\(269\) −353878. −0.298176 −0.149088 0.988824i \(-0.547634\pi\)
−0.149088 + 0.988824i \(0.547634\pi\)
\(270\) 297540. 0.248391
\(271\) 525260. 0.434461 0.217231 0.976120i \(-0.430298\pi\)
0.217231 + 0.976120i \(0.430298\pi\)
\(272\) −175616. −0.143927
\(273\) 172200. 0.139838
\(274\) −1.18493e6 −0.953492
\(275\) 334444. 0.266681
\(276\) 890160. 0.703389
\(277\) −595610. −0.466404 −0.233202 0.972428i \(-0.574920\pi\)
−0.233202 + 0.972428i \(0.574920\pi\)
\(278\) 1.59927e6 1.24111
\(279\) 115974. 0.0891970
\(280\) −36480.0 −0.0278074
\(281\) 732318. 0.553266 0.276633 0.960976i \(-0.410781\pi\)
0.276633 + 0.960976i \(0.410781\pi\)
\(282\) 430080. 0.322052
\(283\) 2.23380e6 1.65798 0.828989 0.559264i \(-0.188916\pi\)
0.828989 + 0.559264i \(0.188916\pi\)
\(284\) −496368. −0.365181
\(285\) −109440. −0.0798112
\(286\) −555632. −0.401673
\(287\) −15280.0 −0.0109501
\(288\) 92160.0 0.0654729
\(289\) −949261. −0.668561
\(290\) −412224. −0.287831
\(291\) −1.00358e6 −0.694732
\(292\) −30784.0 −0.0211285
\(293\) −1.53108e6 −1.04191 −0.520953 0.853585i \(-0.674423\pi\)
−0.520953 + 0.853585i \(0.674423\pi\)
\(294\) −1.00242e6 −0.676365
\(295\) 122759. 0.0821293
\(296\) 2.31610e6 1.53648
\(297\) −473715. −0.311620
\(298\) −290680. −0.189616
\(299\) −4.25793e6 −2.75436
\(300\) −663360. −0.425546
\(301\) −40260.0 −0.0256128
\(302\) 1.21233e6 0.764897
\(303\) −1.45095e6 −0.907916
\(304\) 98304.0 0.0610081
\(305\) 322620. 0.198583
\(306\) 49392.0 0.0301546
\(307\) −1.14268e6 −0.691956 −0.345978 0.938243i \(-0.612453\pi\)
−0.345978 + 0.938243i \(0.612453\pi\)
\(308\) 19360.0 0.0116286
\(309\) 1.43556e6 0.855313
\(310\) −489668. −0.289399
\(311\) 586956. 0.344116 0.172058 0.985087i \(-0.444958\pi\)
0.172058 + 0.985087i \(0.444958\pi\)
\(312\) 3.30624e6 1.92286
\(313\) −233857. −0.134924 −0.0674621 0.997722i \(-0.521490\pi\)
−0.0674621 + 0.997722i \(0.521490\pi\)
\(314\) 2.13195e6 1.22026
\(315\) 3420.00 0.00194200
\(316\) −1.04221e6 −0.587133
\(317\) −935503. −0.522874 −0.261437 0.965221i \(-0.584196\pi\)
−0.261437 + 0.965221i \(0.584196\pi\)
\(318\) −1.79172e6 −0.993579
\(319\) 656304. 0.361101
\(320\) −544768. −0.297397
\(321\) 489870. 0.265349
\(322\) −148360. −0.0797402
\(323\) −263424. −0.140491
\(324\) 869616. 0.460219
\(325\) 3.17307e6 1.66637
\(326\) −1.12830e6 −0.588007
\(327\) 2.78157e6 1.43854
\(328\) −293376. −0.150571
\(329\) 71680.0 0.0365097
\(330\) 137940. 0.0697277
\(331\) −1.05823e6 −0.530897 −0.265449 0.964125i \(-0.585520\pi\)
−0.265449 + 0.964125i \(0.585520\pi\)
\(332\) 1.64342e6 0.818285
\(333\) −217134. −0.107304
\(334\) 2.29435e6 1.12537
\(335\) −569981. −0.277491
\(336\) 38400.0 0.0185559
\(337\) 506186. 0.242793 0.121396 0.992604i \(-0.461263\pi\)
0.121396 + 0.992604i \(0.461263\pi\)
\(338\) −3.78644e6 −1.80277
\(339\) −1.09274e6 −0.516435
\(340\) 208544. 0.0978364
\(341\) 779603. 0.363068
\(342\) −27648.0 −0.0127820
\(343\) −335140. −0.153812
\(344\) −772992. −0.352192
\(345\) 1.05706e6 0.478139
\(346\) 1.54514e6 0.693871
\(347\) 467636. 0.208490 0.104245 0.994552i \(-0.466757\pi\)
0.104245 + 0.994552i \(0.466757\pi\)
\(348\) −1.30176e6 −0.576213
\(349\) 304470. 0.133808 0.0669038 0.997759i \(-0.478688\pi\)
0.0669038 + 0.997759i \(0.478688\pi\)
\(350\) 110560. 0.0482423
\(351\) −4.49442e6 −1.94718
\(352\) 619520. 0.266501
\(353\) 2.51868e6 1.07581 0.537906 0.843005i \(-0.319215\pi\)
0.537906 + 0.843005i \(0.319215\pi\)
\(354\) −387660. −0.164416
\(355\) −589437. −0.248237
\(356\) −278640. −0.116525
\(357\) −102900. −0.0427312
\(358\) −2.18032e6 −0.899108
\(359\) −3.01841e6 −1.23607 −0.618034 0.786151i \(-0.712071\pi\)
−0.618034 + 0.786151i \(0.712071\pi\)
\(360\) 65664.0 0.0267037
\(361\) −2.32864e6 −0.940448
\(362\) 1.11794e6 0.448381
\(363\) −219615. −0.0874773
\(364\) 183680. 0.0726622
\(365\) −36556.0 −0.0143624
\(366\) −1.01880e6 −0.397545
\(367\) 994429. 0.385397 0.192699 0.981258i \(-0.438276\pi\)
0.192699 + 0.981258i \(0.438276\pi\)
\(368\) −949504. −0.365491
\(369\) 27504.0 0.0105155
\(370\) 916788. 0.348149
\(371\) −298620. −0.112638
\(372\) −1.54632e6 −0.579351
\(373\) 1.72896e6 0.643446 0.321723 0.946834i \(-0.395738\pi\)
0.321723 + 0.946834i \(0.395738\pi\)
\(374\) 332024. 0.122741
\(375\) −1.67836e6 −0.616323
\(376\) 1.37626e6 0.502030
\(377\) 6.22675e6 2.25636
\(378\) −156600. −0.0563718
\(379\) 454765. 0.162626 0.0813128 0.996689i \(-0.474089\pi\)
0.0813128 + 0.996689i \(0.474089\pi\)
\(380\) −116736. −0.0414711
\(381\) 1.17276e6 0.413901
\(382\) 1.77775e6 0.623321
\(383\) 2.27557e6 0.792673 0.396336 0.918105i \(-0.370281\pi\)
0.396336 + 0.918105i \(0.370281\pi\)
\(384\) −737280. −0.255155
\(385\) 22990.0 0.00790473
\(386\) 73904.0 0.0252464
\(387\) 72468.0 0.0245962
\(388\) −1.07048e6 −0.360993
\(389\) 389781. 0.130601 0.0653005 0.997866i \(-0.479199\pi\)
0.0653005 + 0.997866i \(0.479199\pi\)
\(390\) 1.30872e6 0.435698
\(391\) 2.54437e6 0.841665
\(392\) −3.20774e6 −1.05435
\(393\) 6930.00 0.00226335
\(394\) −1.08073e6 −0.350732
\(395\) −1.23762e6 −0.399112
\(396\) −34848.0 −0.0111671
\(397\) −1.61933e6 −0.515655 −0.257827 0.966191i \(-0.583007\pi\)
−0.257827 + 0.966191i \(0.583007\pi\)
\(398\) −173280. −0.0548328
\(399\) 57600.0 0.0181130
\(400\) 707584. 0.221120
\(401\) −5.54368e6 −1.72162 −0.860810 0.508927i \(-0.830042\pi\)
−0.860810 + 0.508927i \(0.830042\pi\)
\(402\) 1.79994e6 0.555511
\(403\) 7.39656e6 2.26865
\(404\) −1.54768e6 −0.471767
\(405\) 1.03267e6 0.312841
\(406\) 216960. 0.0653228
\(407\) −1.45962e6 −0.436772
\(408\) −1.97568e6 −0.587579
\(409\) −2.70493e6 −0.799553 −0.399776 0.916613i \(-0.630912\pi\)
−0.399776 + 0.916613i \(0.630912\pi\)
\(410\) −116128. −0.0341175
\(411\) −4.44350e6 −1.29754
\(412\) 1.53126e6 0.444434
\(413\) −64610.0 −0.0186391
\(414\) 267048. 0.0765753
\(415\) 1.95157e6 0.556241
\(416\) 5.87776e6 1.66525
\(417\) 5.99727e6 1.68894
\(418\) −185856. −0.0520279
\(419\) 3.37337e6 0.938705 0.469353 0.883011i \(-0.344487\pi\)
0.469353 + 0.883011i \(0.344487\pi\)
\(420\) −45600.0 −0.0126137
\(421\) −4.52551e6 −1.24441 −0.622204 0.782855i \(-0.713762\pi\)
−0.622204 + 0.782855i \(0.713762\pi\)
\(422\) −4.11874e6 −1.12586
\(423\) −129024. −0.0350606
\(424\) −5.73350e6 −1.54884
\(425\) −1.89610e6 −0.509202
\(426\) 1.86138e6 0.496948
\(427\) −169800. −0.0450680
\(428\) 522528. 0.137880
\(429\) −2.08362e6 −0.546607
\(430\) −305976. −0.0798024
\(431\) −684534. −0.177501 −0.0887507 0.996054i \(-0.528287\pi\)
−0.0887507 + 0.996054i \(0.528287\pi\)
\(432\) −1.00224e6 −0.258382
\(433\) −4.22591e6 −1.08318 −0.541589 0.840643i \(-0.682177\pi\)
−0.541589 + 0.840643i \(0.682177\pi\)
\(434\) 257720. 0.0656786
\(435\) −1.54584e6 −0.391689
\(436\) 2.96701e6 0.747485
\(437\) −1.42426e6 −0.356767
\(438\) 115440. 0.0287522
\(439\) −2.09185e6 −0.518047 −0.259023 0.965871i \(-0.583401\pi\)
−0.259023 + 0.965871i \(0.583401\pi\)
\(440\) 441408. 0.108695
\(441\) 300726. 0.0736333
\(442\) 3.15011e6 0.766956
\(443\) 1.56284e6 0.378361 0.189180 0.981942i \(-0.439417\pi\)
0.189180 + 0.981942i \(0.439417\pi\)
\(444\) 2.89512e6 0.696962
\(445\) −330885. −0.0792095
\(446\) −1.84512e6 −0.439226
\(447\) −1.09005e6 −0.258034
\(448\) 286720. 0.0674937
\(449\) −3.00449e6 −0.703324 −0.351662 0.936127i \(-0.614383\pi\)
−0.351662 + 0.936127i \(0.614383\pi\)
\(450\) −199008. −0.0463275
\(451\) 184888. 0.0428023
\(452\) −1.16558e6 −0.268347
\(453\) 4.54623e6 1.04089
\(454\) 3.42228e6 0.779248
\(455\) 218120. 0.0493932
\(456\) 1.10592e6 0.249064
\(457\) −2.44552e6 −0.547747 −0.273874 0.961766i \(-0.588305\pi\)
−0.273874 + 0.961766i \(0.588305\pi\)
\(458\) 2.66322e6 0.593258
\(459\) 2.68569e6 0.595010
\(460\) 1.12754e6 0.248448
\(461\) 7.79104e6 1.70743 0.853715 0.520741i \(-0.174344\pi\)
0.853715 + 0.520741i \(0.174344\pi\)
\(462\) −72600.0 −0.0158246
\(463\) −1.05196e6 −0.228059 −0.114029 0.993477i \(-0.536376\pi\)
−0.114029 + 0.993477i \(0.536376\pi\)
\(464\) 1.38854e6 0.299409
\(465\) −1.83626e6 −0.393823
\(466\) −4.83734e6 −1.03191
\(467\) 3.97003e6 0.842369 0.421184 0.906975i \(-0.361615\pi\)
0.421184 + 0.906975i \(0.361615\pi\)
\(468\) −330624. −0.0697782
\(469\) 299990. 0.0629759
\(470\) 544768. 0.113754
\(471\) 7.99480e6 1.66056
\(472\) −1.24051e6 −0.256298
\(473\) 487146. 0.100117
\(474\) 3.90828e6 0.798987
\(475\) 1.06138e6 0.215842
\(476\) −109760. −0.0222038
\(477\) 537516. 0.108167
\(478\) 2.28593e6 0.457607
\(479\) −8.53908e6 −1.70048 −0.850241 0.526393i \(-0.823544\pi\)
−0.850241 + 0.526393i \(0.823544\pi\)
\(480\) −1.45920e6 −0.289076
\(481\) −1.38483e7 −2.72919
\(482\) 1.06832e6 0.209452
\(483\) −556350. −0.108513
\(484\) −234256. −0.0454545
\(485\) −1.27120e6 −0.245391
\(486\) 544320. 0.104535
\(487\) −1.86487e6 −0.356308 −0.178154 0.984003i \(-0.557013\pi\)
−0.178154 + 0.984003i \(0.557013\pi\)
\(488\) −3.26016e6 −0.619711
\(489\) −4.23114e6 −0.800175
\(490\) −1.26973e6 −0.238903
\(491\) 5.15727e6 0.965420 0.482710 0.875780i \(-0.339653\pi\)
0.482710 + 0.875780i \(0.339653\pi\)
\(492\) −366720. −0.0683002
\(493\) −3.72086e6 −0.689488
\(494\) −1.76333e6 −0.325099
\(495\) −41382.0 −0.00759099
\(496\) 1.64941e6 0.301040
\(497\) 310230. 0.0563369
\(498\) −6.16284e6 −1.11354
\(499\) 4.53340e6 0.815029 0.407514 0.913199i \(-0.366396\pi\)
0.407514 + 0.913199i \(0.366396\pi\)
\(500\) −1.79026e6 −0.320251
\(501\) 8.60382e6 1.53143
\(502\) −5.54947e6 −0.982861
\(503\) 1.71163e6 0.301641 0.150821 0.988561i \(-0.451808\pi\)
0.150821 + 0.988561i \(0.451808\pi\)
\(504\) −34560.0 −0.00606035
\(505\) −1.83787e6 −0.320691
\(506\) 1.79516e6 0.311692
\(507\) −1.41992e7 −2.45326
\(508\) 1.25094e6 0.215069
\(509\) 9.73822e6 1.66604 0.833019 0.553244i \(-0.186610\pi\)
0.833019 + 0.553244i \(0.186610\pi\)
\(510\) −782040. −0.133138
\(511\) 19240.0 0.00325951
\(512\) 2.88358e6 0.486136
\(513\) −1.50336e6 −0.252214
\(514\) 3.54369e6 0.591627
\(515\) 1.81838e6 0.302110
\(516\) −966240. −0.159757
\(517\) −867328. −0.142711
\(518\) −482520. −0.0790116
\(519\) 5.79429e6 0.944239
\(520\) 4.18790e6 0.679185
\(521\) 4.30279e6 0.694474 0.347237 0.937777i \(-0.387120\pi\)
0.347237 + 0.937777i \(0.387120\pi\)
\(522\) −390528. −0.0627301
\(523\) 2.62280e6 0.419287 0.209643 0.977778i \(-0.432770\pi\)
0.209643 + 0.977778i \(0.432770\pi\)
\(524\) 7392.00 0.00117607
\(525\) 414600. 0.0656494
\(526\) −5.78750e6 −0.912066
\(527\) −4.41990e6 −0.693243
\(528\) −464640. −0.0725324
\(529\) 7.32034e6 1.13734
\(530\) −2.26951e6 −0.350948
\(531\) 116298. 0.0178993
\(532\) 61440.0 0.00941179
\(533\) 1.75414e6 0.267453
\(534\) 1.04490e6 0.158570
\(535\) 620502. 0.0937257
\(536\) 5.75981e6 0.865956
\(537\) −8.17618e6 −1.22353
\(538\) 1.41551e6 0.210842
\(539\) 2.02155e6 0.299717
\(540\) 1.19016e6 0.175639
\(541\) −2.49634e6 −0.366700 −0.183350 0.983048i \(-0.558694\pi\)
−0.183350 + 0.983048i \(0.558694\pi\)
\(542\) −2.10104e6 −0.307211
\(543\) 4.19228e6 0.610169
\(544\) −3.51232e6 −0.508858
\(545\) 3.52332e6 0.508114
\(546\) −688800. −0.0988807
\(547\) 1.14323e7 1.63368 0.816838 0.576868i \(-0.195725\pi\)
0.816838 + 0.576868i \(0.195725\pi\)
\(548\) −4.73973e6 −0.674221
\(549\) 305640. 0.0432792
\(550\) −1.33778e6 −0.188572
\(551\) 2.08282e6 0.292262
\(552\) −1.06819e7 −1.49211
\(553\) 651380. 0.0905778
\(554\) 2.38244e6 0.329798
\(555\) 3.43796e6 0.473770
\(556\) 6.39709e6 0.877597
\(557\) −9.81529e6 −1.34049 −0.670247 0.742138i \(-0.733812\pi\)
−0.670247 + 0.742138i \(0.733812\pi\)
\(558\) −463896. −0.0630718
\(559\) 4.62185e6 0.625585
\(560\) 48640.0 0.00655426
\(561\) 1.24509e6 0.167030
\(562\) −2.92927e6 −0.391218
\(563\) −8.19192e6 −1.08922 −0.544609 0.838690i \(-0.683322\pi\)
−0.544609 + 0.838690i \(0.683322\pi\)
\(564\) 1.72032e6 0.227725
\(565\) −1.38413e6 −0.182413
\(566\) −8.93522e6 −1.17237
\(567\) −543510. −0.0709986
\(568\) 5.95642e6 0.774665
\(569\) −7.54286e6 −0.976687 −0.488344 0.872651i \(-0.662399\pi\)
−0.488344 + 0.872651i \(0.662399\pi\)
\(570\) 437760. 0.0564351
\(571\) −8.69400e6 −1.11591 −0.557956 0.829871i \(-0.688414\pi\)
−0.557956 + 0.829871i \(0.688414\pi\)
\(572\) −2.22253e6 −0.284025
\(573\) 6.66656e6 0.848233
\(574\) 61120.0 0.00774290
\(575\) −1.02517e7 −1.29308
\(576\) −516096. −0.0648148
\(577\) 2.03379e6 0.254312 0.127156 0.991883i \(-0.459415\pi\)
0.127156 + 0.991883i \(0.459415\pi\)
\(578\) 3.79704e6 0.472744
\(579\) 277140. 0.0343560
\(580\) −1.64890e6 −0.203528
\(581\) −1.02714e6 −0.126238
\(582\) 4.01430e6 0.491250
\(583\) 3.61330e6 0.440284
\(584\) 369408. 0.0448202
\(585\) −392616. −0.0474328
\(586\) 6.12432e6 0.736739
\(587\) 3.51780e6 0.421381 0.210691 0.977553i \(-0.432429\pi\)
0.210691 + 0.977553i \(0.432429\pi\)
\(588\) −4.00968e6 −0.478263
\(589\) 2.47411e6 0.293854
\(590\) −491036. −0.0580742
\(591\) −4.05273e6 −0.477286
\(592\) −3.08813e6 −0.362152
\(593\) −8.34535e6 −0.974558 −0.487279 0.873246i \(-0.662011\pi\)
−0.487279 + 0.873246i \(0.662011\pi\)
\(594\) 1.89486e6 0.220349
\(595\) −130340. −0.0150933
\(596\) −1.16272e6 −0.134079
\(597\) −649800. −0.0746180
\(598\) 1.70317e7 1.94763
\(599\) 6.15022e6 0.700364 0.350182 0.936682i \(-0.386120\pi\)
0.350182 + 0.936682i \(0.386120\pi\)
\(600\) 7.96032e6 0.902719
\(601\) −6.86232e6 −0.774970 −0.387485 0.921876i \(-0.626656\pi\)
−0.387485 + 0.921876i \(0.626656\pi\)
\(602\) 161040. 0.0181110
\(603\) −539982. −0.0604764
\(604\) 4.84931e6 0.540864
\(605\) −278179. −0.0308984
\(606\) 5.80380e6 0.641994
\(607\) −9.45536e6 −1.04161 −0.520807 0.853675i \(-0.674369\pi\)
−0.520807 + 0.853675i \(0.674369\pi\)
\(608\) 1.96608e6 0.215696
\(609\) 813600. 0.0888930
\(610\) −1.29048e6 −0.140419
\(611\) −8.22886e6 −0.891737
\(612\) 197568. 0.0213225
\(613\) −4.63658e6 −0.498363 −0.249182 0.968457i \(-0.580162\pi\)
−0.249182 + 0.968457i \(0.580162\pi\)
\(614\) 4.57072e6 0.489287
\(615\) −435480. −0.0464280
\(616\) −232320. −0.0246680
\(617\) 6.05704e6 0.640542 0.320271 0.947326i \(-0.396226\pi\)
0.320271 + 0.947326i \(0.396226\pi\)
\(618\) −5.74224e6 −0.604798
\(619\) −5.63994e6 −0.591626 −0.295813 0.955246i \(-0.595591\pi\)
−0.295813 + 0.955246i \(0.595591\pi\)
\(620\) −1.95867e6 −0.204636
\(621\) 1.45207e7 1.51098
\(622\) −2.34782e6 −0.243327
\(623\) 174150. 0.0179764
\(624\) −4.40832e6 −0.453223
\(625\) 6.51157e6 0.666785
\(626\) 935428. 0.0954057
\(627\) −696960. −0.0708009
\(628\) 8.52779e6 0.862854
\(629\) 8.27522e6 0.833975
\(630\) −13680.0 −0.00137320
\(631\) 1.12616e6 0.112597 0.0562987 0.998414i \(-0.482070\pi\)
0.0562987 + 0.998414i \(0.482070\pi\)
\(632\) 1.25065e7 1.24550
\(633\) −1.54453e7 −1.53210
\(634\) 3.74201e6 0.369728
\(635\) 1.48550e6 0.146197
\(636\) −7.16688e6 −0.702566
\(637\) 1.91796e7 1.87280
\(638\) −2.62522e6 −0.255337
\(639\) −558414. −0.0541009
\(640\) −933888. −0.0901249
\(641\) −1.42020e7 −1.36522 −0.682611 0.730782i \(-0.739156\pi\)
−0.682611 + 0.730782i \(0.739156\pi\)
\(642\) −1.95948e6 −0.187630
\(643\) 1.60794e6 0.153371 0.0766853 0.997055i \(-0.475566\pi\)
0.0766853 + 0.997055i \(0.475566\pi\)
\(644\) −593440. −0.0563848
\(645\) −1.14741e6 −0.108597
\(646\) 1.05370e6 0.0993423
\(647\) 3.10236e6 0.291361 0.145680 0.989332i \(-0.453463\pi\)
0.145680 + 0.989332i \(0.453463\pi\)
\(648\) −1.04354e7 −0.976273
\(649\) 781781. 0.0728573
\(650\) −1.26923e7 −1.17830
\(651\) 966450. 0.0893772
\(652\) −4.51322e6 −0.415783
\(653\) 6.88852e6 0.632183 0.316091 0.948729i \(-0.397629\pi\)
0.316091 + 0.948729i \(0.397629\pi\)
\(654\) −1.11263e7 −1.01720
\(655\) 8778.00 0.000799452 0
\(656\) 391168. 0.0354898
\(657\) −34632.0 −0.00313014
\(658\) −286720. −0.0258163
\(659\) −1.24134e7 −1.11347 −0.556735 0.830690i \(-0.687946\pi\)
−0.556735 + 0.830690i \(0.687946\pi\)
\(660\) 551760. 0.0493049
\(661\) −8.10994e6 −0.721961 −0.360980 0.932573i \(-0.617558\pi\)
−0.360980 + 0.932573i \(0.617558\pi\)
\(662\) 4.23292e6 0.375401
\(663\) 1.18129e7 1.04369
\(664\) −1.97211e7 −1.73584
\(665\) 72960.0 0.00639780
\(666\) 868536. 0.0758756
\(667\) −2.01176e7 −1.75090
\(668\) 9.17741e6 0.795754
\(669\) −6.91922e6 −0.597711
\(670\) 2.27992e6 0.196216
\(671\) 2.05458e6 0.176164
\(672\) 768000. 0.0656052
\(673\) 1.78063e7 1.51543 0.757717 0.652584i \(-0.226315\pi\)
0.757717 + 0.652584i \(0.226315\pi\)
\(674\) −2.02474e6 −0.171680
\(675\) −1.08211e7 −0.914135
\(676\) −1.51458e7 −1.27475
\(677\) 1.55179e7 1.30125 0.650626 0.759398i \(-0.274507\pi\)
0.650626 + 0.759398i \(0.274507\pi\)
\(678\) 4.37094e6 0.365175
\(679\) 669050. 0.0556909
\(680\) −2.50253e6 −0.207542
\(681\) 1.28336e7 1.06042
\(682\) −3.11841e6 −0.256728
\(683\) −2.18106e6 −0.178902 −0.0894510 0.995991i \(-0.528511\pi\)
−0.0894510 + 0.995991i \(0.528511\pi\)
\(684\) −110592. −0.00903823
\(685\) −5.62843e6 −0.458311
\(686\) 1.34056e6 0.108762
\(687\) 9.98708e6 0.807321
\(688\) 1.03066e6 0.0830123
\(689\) 3.42816e7 2.75114
\(690\) −4.22826e6 −0.338095
\(691\) 2.29892e7 1.83159 0.915795 0.401647i \(-0.131562\pi\)
0.915795 + 0.401647i \(0.131562\pi\)
\(692\) 6.18058e6 0.490641
\(693\) 21780.0 0.00172276
\(694\) −1.87054e6 −0.147424
\(695\) 7.59654e6 0.596560
\(696\) 1.56211e7 1.22233
\(697\) −1.04821e6 −0.0817270
\(698\) −1.21788e6 −0.0946163
\(699\) −1.81400e7 −1.40425
\(700\) 442240. 0.0341125
\(701\) 2.34092e6 0.179925 0.0899626 0.995945i \(-0.471325\pi\)
0.0899626 + 0.995945i \(0.471325\pi\)
\(702\) 1.79777e7 1.37686
\(703\) −4.63219e6 −0.353507
\(704\) −3.46931e6 −0.263822
\(705\) 2.04288e6 0.154800
\(706\) −1.00747e7 −0.760715
\(707\) 967300. 0.0727801
\(708\) −1.55064e6 −0.116259
\(709\) −1.92694e7 −1.43964 −0.719820 0.694161i \(-0.755775\pi\)
−0.719820 + 0.694161i \(0.755775\pi\)
\(710\) 2.35775e6 0.175530
\(711\) −1.17248e6 −0.0869827
\(712\) 3.34368e6 0.247186
\(713\) −2.38971e7 −1.76044
\(714\) 411600. 0.0302155
\(715\) −2.63925e6 −0.193070
\(716\) −8.72126e6 −0.635765
\(717\) 8.57223e6 0.622724
\(718\) 1.20736e7 0.874032
\(719\) −2.14665e7 −1.54860 −0.774300 0.632819i \(-0.781898\pi\)
−0.774300 + 0.632819i \(0.781898\pi\)
\(720\) −87552.0 −0.00629412
\(721\) −957040. −0.0685633
\(722\) 9.31457e6 0.664997
\(723\) 4.00620e6 0.285028
\(724\) 4.47176e6 0.317053
\(725\) 1.49919e7 1.05928
\(726\) 878460. 0.0618558
\(727\) −1.67705e7 −1.17682 −0.588411 0.808562i \(-0.700246\pi\)
−0.588411 + 0.808562i \(0.700246\pi\)
\(728\) −2.20416e6 −0.154140
\(729\) 1.52485e7 1.06269
\(730\) 146224. 0.0101557
\(731\) −2.76184e6 −0.191163
\(732\) −4.07520e6 −0.281107
\(733\) −1.75373e7 −1.20560 −0.602798 0.797894i \(-0.705948\pi\)
−0.602798 + 0.797894i \(0.705948\pi\)
\(734\) −3.97772e6 −0.272517
\(735\) −4.76150e6 −0.325106
\(736\) −1.89901e7 −1.29221
\(737\) −3.62988e6 −0.246163
\(738\) −110016. −0.00743558
\(739\) 1.47387e7 0.992766 0.496383 0.868104i \(-0.334661\pi\)
0.496383 + 0.868104i \(0.334661\pi\)
\(740\) 3.66715e6 0.246178
\(741\) −6.61248e6 −0.442404
\(742\) 1.19448e6 0.0796469
\(743\) −4.80946e6 −0.319613 −0.159806 0.987148i \(-0.551087\pi\)
−0.159806 + 0.987148i \(0.551087\pi\)
\(744\) 1.85558e7 1.22899
\(745\) −1.38073e6 −0.0911419
\(746\) −6.91583e6 −0.454985
\(747\) 1.84885e6 0.121227
\(748\) 1.32810e6 0.0867912
\(749\) −326580. −0.0212709
\(750\) 6.71346e6 0.435806
\(751\) 8.29317e6 0.536563 0.268282 0.963341i \(-0.413544\pi\)
0.268282 + 0.963341i \(0.413544\pi\)
\(752\) −1.83501e6 −0.118330
\(753\) −2.08105e7 −1.33750
\(754\) −2.49070e7 −1.59549
\(755\) 5.75856e6 0.367660
\(756\) −626400. −0.0398609
\(757\) −352294. −0.0223442 −0.0111721 0.999938i \(-0.503556\pi\)
−0.0111721 + 0.999938i \(0.503556\pi\)
\(758\) −1.81906e6 −0.114994
\(759\) 6.73184e6 0.424159
\(760\) 1.40083e6 0.0879735
\(761\) 1.68985e7 1.05776 0.528878 0.848698i \(-0.322613\pi\)
0.528878 + 0.848698i \(0.322613\pi\)
\(762\) −4.69104e6 −0.292672
\(763\) −1.85438e6 −0.115315
\(764\) 7.11099e6 0.440755
\(765\) 234612. 0.0144943
\(766\) −9.10229e6 −0.560504
\(767\) 7.41723e6 0.455253
\(768\) 1.67117e7 1.02239
\(769\) −36652.0 −0.00223502 −0.00111751 0.999999i \(-0.500356\pi\)
−0.00111751 + 0.999999i \(0.500356\pi\)
\(770\) −91960.0 −0.00558949
\(771\) 1.32888e7 0.805102
\(772\) 295616. 0.0178519
\(773\) −3.17462e7 −1.91093 −0.955463 0.295113i \(-0.904643\pi\)
−0.955463 + 0.295113i \(0.904643\pi\)
\(774\) −289872. −0.0173922
\(775\) 1.78085e7 1.06505
\(776\) 1.28458e7 0.765783
\(777\) −1.80945e6 −0.107521
\(778\) −1.55912e6 −0.0923489
\(779\) 586752. 0.0346426
\(780\) 5.23488e6 0.308085
\(781\) −3.75378e6 −0.220212
\(782\) −1.01775e7 −0.595147
\(783\) −2.12350e7 −1.23779
\(784\) 4.27699e6 0.248513
\(785\) 1.01268e7 0.586538
\(786\) −27720.0 −0.00160043
\(787\) 2.01985e7 1.16247 0.581236 0.813735i \(-0.302569\pi\)
0.581236 + 0.813735i \(0.302569\pi\)
\(788\) −4.32291e6 −0.248005
\(789\) −2.17031e7 −1.24116
\(790\) 4.95049e6 0.282215
\(791\) 728490. 0.0413983
\(792\) 418176. 0.0236890
\(793\) 1.94930e7 1.10077
\(794\) 6.47732e6 0.364623
\(795\) −8.51067e6 −0.477580
\(796\) −693120. −0.0387727
\(797\) 1.55660e7 0.868023 0.434011 0.900907i \(-0.357098\pi\)
0.434011 + 0.900907i \(0.357098\pi\)
\(798\) −230400. −0.0128078
\(799\) 4.91725e6 0.272493
\(800\) 1.41517e7 0.781777
\(801\) −313470. −0.0172629
\(802\) 2.21747e7 1.21737
\(803\) −232804. −0.0127409
\(804\) 7.19976e6 0.392806
\(805\) −704710. −0.0383284
\(806\) −2.95863e7 −1.60418
\(807\) 5.30817e6 0.286920
\(808\) 1.85722e7 1.00077
\(809\) −2.91667e7 −1.56681 −0.783404 0.621513i \(-0.786518\pi\)
−0.783404 + 0.621513i \(0.786518\pi\)
\(810\) −4.13068e6 −0.221212
\(811\) −1.65215e7 −0.882057 −0.441029 0.897493i \(-0.645386\pi\)
−0.441029 + 0.897493i \(0.645386\pi\)
\(812\) 867840. 0.0461902
\(813\) −7.87890e6 −0.418061
\(814\) 5.83849e6 0.308844
\(815\) −5.35944e6 −0.282635
\(816\) 2.63424e6 0.138494
\(817\) 1.54598e6 0.0810307
\(818\) 1.08197e7 0.565369
\(819\) 206640. 0.0107648
\(820\) −464512. −0.0241247
\(821\) 5.56614e6 0.288202 0.144101 0.989563i \(-0.453971\pi\)
0.144101 + 0.989563i \(0.453971\pi\)
\(822\) 1.77740e7 0.917498
\(823\) 1.18801e7 0.611391 0.305696 0.952129i \(-0.401111\pi\)
0.305696 + 0.952129i \(0.401111\pi\)
\(824\) −1.83752e7 −0.942786
\(825\) −5.01666e6 −0.256614
\(826\) 258440. 0.0131798
\(827\) −1.32856e7 −0.675489 −0.337745 0.941238i \(-0.609664\pi\)
−0.337745 + 0.941238i \(0.609664\pi\)
\(828\) 1.06819e6 0.0541469
\(829\) −653987. −0.0330509 −0.0165254 0.999863i \(-0.505260\pi\)
−0.0165254 + 0.999863i \(0.505260\pi\)
\(830\) −7.80626e6 −0.393322
\(831\) 8.93415e6 0.448798
\(832\) −3.29155e7 −1.64851
\(833\) −1.14610e7 −0.572282
\(834\) −2.39891e7 −1.19426
\(835\) 1.08982e7 0.540926
\(836\) −743424. −0.0367892
\(837\) −2.52243e7 −1.24453
\(838\) −1.34935e7 −0.663765
\(839\) 2.47747e7 1.21508 0.607538 0.794290i \(-0.292157\pi\)
0.607538 + 0.794290i \(0.292157\pi\)
\(840\) 547200. 0.0267576
\(841\) 8.90863e6 0.434331
\(842\) 1.81021e7 0.879929
\(843\) −1.09848e7 −0.532380
\(844\) −1.64749e7 −0.796100
\(845\) −1.79856e7 −0.866530
\(846\) 516096. 0.0247916
\(847\) 146410. 0.00701233
\(848\) 7.64467e6 0.365064
\(849\) −3.35071e7 −1.59539
\(850\) 7.58442e6 0.360060
\(851\) 4.47417e7 2.11782
\(852\) 7.44552e6 0.351395
\(853\) 2.71291e7 1.27662 0.638311 0.769779i \(-0.279633\pi\)
0.638311 + 0.769779i \(0.279633\pi\)
\(854\) 679200. 0.0318679
\(855\) −131328. −0.00614387
\(856\) −6.27034e6 −0.292487
\(857\) −2.84232e7 −1.32197 −0.660984 0.750400i \(-0.729861\pi\)
−0.660984 + 0.750400i \(0.729861\pi\)
\(858\) 8.33448e6 0.386510
\(859\) 2.65922e7 1.22962 0.614810 0.788675i \(-0.289233\pi\)
0.614810 + 0.788675i \(0.289233\pi\)
\(860\) −1.22390e6 −0.0564289
\(861\) 229200. 0.0105368
\(862\) 2.73814e6 0.125512
\(863\) −2.22500e7 −1.01696 −0.508479 0.861074i \(-0.669792\pi\)
−0.508479 + 0.861074i \(0.669792\pi\)
\(864\) −2.00448e7 −0.913519
\(865\) 7.33943e6 0.333520
\(866\) 1.69036e7 0.765923
\(867\) 1.42389e7 0.643323
\(868\) 1.03088e6 0.0464418
\(869\) −7.88170e6 −0.354055
\(870\) 6.18336e6 0.276966
\(871\) −3.44389e7 −1.53817
\(872\) −3.56041e7 −1.58566
\(873\) −1.20429e6 −0.0534805
\(874\) 5.69702e6 0.252272
\(875\) 1.11891e6 0.0494055
\(876\) 461760. 0.0203309
\(877\) 2.83428e7 1.24435 0.622176 0.782877i \(-0.286249\pi\)
0.622176 + 0.782877i \(0.286249\pi\)
\(878\) 8.36739e6 0.366314
\(879\) 2.29662e7 1.00258
\(880\) −588544. −0.0256196
\(881\) 3.66445e7 1.59063 0.795315 0.606196i \(-0.207305\pi\)
0.795315 + 0.606196i \(0.207305\pi\)
\(882\) −1.20290e6 −0.0520666
\(883\) 1.68772e7 0.728447 0.364223 0.931312i \(-0.381334\pi\)
0.364223 + 0.931312i \(0.381334\pi\)
\(884\) 1.26004e7 0.542320
\(885\) −1.84138e6 −0.0790290
\(886\) −6.25137e6 −0.267541
\(887\) 2.73941e6 0.116909 0.0584544 0.998290i \(-0.481383\pi\)
0.0584544 + 0.998290i \(0.481383\pi\)
\(888\) −3.47414e7 −1.47848
\(889\) −781840. −0.0331790
\(890\) 1.32354e6 0.0560095
\(891\) 6.57647e6 0.277523
\(892\) −7.38050e6 −0.310580
\(893\) −2.75251e6 −0.115505
\(894\) 4.36020e6 0.182458
\(895\) −1.03565e7 −0.432171
\(896\) 491520. 0.0204537
\(897\) 6.38690e7 2.65038
\(898\) 1.20180e7 0.497325
\(899\) 3.49468e7 1.44214
\(900\) −796032. −0.0327585
\(901\) −2.04853e7 −0.840681
\(902\) −739552. −0.0302658
\(903\) 603900. 0.0246460
\(904\) 1.39870e7 0.569251
\(905\) 5.31022e6 0.215522
\(906\) −1.81849e7 −0.736022
\(907\) −3.13286e7 −1.26451 −0.632255 0.774760i \(-0.717871\pi\)
−0.632255 + 0.774760i \(0.717871\pi\)
\(908\) 1.36891e7 0.551012
\(909\) −1.74114e6 −0.0698914
\(910\) −872480. −0.0349263
\(911\) −2.49762e7 −0.997081 −0.498541 0.866866i \(-0.666131\pi\)
−0.498541 + 0.866866i \(0.666131\pi\)
\(912\) −1.47456e6 −0.0587050
\(913\) 1.24284e7 0.493444
\(914\) 9.78206e6 0.387316
\(915\) −4.83930e6 −0.191086
\(916\) 1.06529e7 0.419497
\(917\) −4620.00 −0.000181434 0
\(918\) −1.07428e7 −0.420736
\(919\) −1.10613e7 −0.432032 −0.216016 0.976390i \(-0.569306\pi\)
−0.216016 + 0.976390i \(0.569306\pi\)
\(920\) −1.35304e7 −0.527038
\(921\) 1.71402e7 0.665835
\(922\) −3.11641e7 −1.20734
\(923\) −3.56144e7 −1.37601
\(924\) −290400. −0.0111897
\(925\) −3.33421e7 −1.28127
\(926\) 4.20784e6 0.161262
\(927\) 1.72267e6 0.0658420
\(928\) 2.77709e7 1.05857
\(929\) −2.01739e7 −0.766919 −0.383460 0.923558i \(-0.625267\pi\)
−0.383460 + 0.923558i \(0.625267\pi\)
\(930\) 7.34502e6 0.278475
\(931\) 6.41549e6 0.242580
\(932\) −1.93494e7 −0.729671
\(933\) −8.80434e6 −0.331126
\(934\) −1.58801e7 −0.595645
\(935\) 1.57711e6 0.0589976
\(936\) 3.96749e6 0.148022
\(937\) 9.10734e6 0.338877 0.169439 0.985541i \(-0.445805\pi\)
0.169439 + 0.985541i \(0.445805\pi\)
\(938\) −1.19996e6 −0.0445307
\(939\) 3.50785e6 0.129831
\(940\) 2.17907e6 0.0804363
\(941\) −3.67709e7 −1.35372 −0.676861 0.736110i \(-0.736660\pi\)
−0.676861 + 0.736110i \(0.736660\pi\)
\(942\) −3.19792e7 −1.17420
\(943\) −5.66735e6 −0.207540
\(944\) 1.65402e6 0.0604101
\(945\) −743850. −0.0270960
\(946\) −1.94858e6 −0.0707932
\(947\) −4.95743e7 −1.79631 −0.898156 0.439677i \(-0.855093\pi\)
−0.898156 + 0.439677i \(0.855093\pi\)
\(948\) 1.56331e7 0.564969
\(949\) −2.20875e6 −0.0796125
\(950\) −4.24550e6 −0.152623
\(951\) 1.40325e7 0.503136
\(952\) 1.31712e6 0.0471013
\(953\) 3.53787e7 1.26186 0.630928 0.775841i \(-0.282674\pi\)
0.630928 + 0.775841i \(0.282674\pi\)
\(954\) −2.15006e6 −0.0764857
\(955\) 8.44430e6 0.299609
\(956\) 9.14371e6 0.323577
\(957\) −9.84456e6 −0.347469
\(958\) 3.41563e7 1.20242
\(959\) 2.96233e6 0.104013
\(960\) 8.17152e6 0.286170
\(961\) 1.28831e7 0.449999
\(962\) 5.53933e7 1.92983
\(963\) 587844. 0.0204266
\(964\) 4.27328e6 0.148105
\(965\) 351044. 0.0121351
\(966\) 2.22540e6 0.0767300
\(967\) 2.78059e7 0.956248 0.478124 0.878292i \(-0.341317\pi\)
0.478124 + 0.878292i \(0.341317\pi\)
\(968\) 2.81107e6 0.0964237
\(969\) 3.95136e6 0.135188
\(970\) 5.08478e6 0.173517
\(971\) 1.56835e7 0.533821 0.266910 0.963721i \(-0.413997\pi\)
0.266910 + 0.963721i \(0.413997\pi\)
\(972\) 2.17728e6 0.0739177
\(973\) −3.99818e6 −0.135388
\(974\) 7.45947e6 0.251948
\(975\) −4.75961e7 −1.60347
\(976\) 4.34688e6 0.146067
\(977\) 2.01140e7 0.674157 0.337079 0.941476i \(-0.390561\pi\)
0.337079 + 0.941476i \(0.390561\pi\)
\(978\) 1.69246e7 0.565810
\(979\) −2.10722e6 −0.0702671
\(980\) −5.07893e6 −0.168930
\(981\) 3.33788e6 0.110739
\(982\) −2.06291e7 −0.682655
\(983\) −2.09269e7 −0.690750 −0.345375 0.938465i \(-0.612248\pi\)
−0.345375 + 0.938465i \(0.612248\pi\)
\(984\) 4.40064e6 0.144887
\(985\) −5.13346e6 −0.168585
\(986\) 1.48835e7 0.487541
\(987\) −1.07520e6 −0.0351315
\(988\) −7.05331e6 −0.229880
\(989\) −1.49324e7 −0.485445
\(990\) 165528. 0.00536764
\(991\) −3.18663e7 −1.03074 −0.515368 0.856969i \(-0.672345\pi\)
−0.515368 + 0.856969i \(0.672345\pi\)
\(992\) 3.29882e7 1.06434
\(993\) 1.58735e7 0.510856
\(994\) −1.24092e6 −0.0398362
\(995\) −823080. −0.0263563
\(996\) −2.46514e7 −0.787395
\(997\) 1.38913e6 0.0442595 0.0221297 0.999755i \(-0.492955\pi\)
0.0221297 + 0.999755i \(0.492955\pi\)
\(998\) −1.81336e7 −0.576313
\(999\) 4.72266e7 1.49718
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 11.6.a.a.1.1 1
3.2 odd 2 99.6.a.c.1.1 1
4.3 odd 2 176.6.a.c.1.1 1
5.2 odd 4 275.6.b.a.199.1 2
5.3 odd 4 275.6.b.a.199.2 2
5.4 even 2 275.6.a.a.1.1 1
7.6 odd 2 539.6.a.c.1.1 1
8.3 odd 2 704.6.a.c.1.1 1
8.5 even 2 704.6.a.h.1.1 1
11.10 odd 2 121.6.a.b.1.1 1
33.32 even 2 1089.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.a.1.1 1 1.1 even 1 trivial
99.6.a.c.1.1 1 3.2 odd 2
121.6.a.b.1.1 1 11.10 odd 2
176.6.a.c.1.1 1 4.3 odd 2
275.6.a.a.1.1 1 5.4 even 2
275.6.b.a.199.1 2 5.2 odd 4
275.6.b.a.199.2 2 5.3 odd 4
539.6.a.c.1.1 1 7.6 odd 2
704.6.a.c.1.1 1 8.3 odd 2
704.6.a.h.1.1 1 8.5 even 2
1089.6.a.c.1.1 1 33.32 even 2