Properties

Label 42.6.a.e.1.1
Level $42$
Weight $6$
Character 42.1
Self dual yes
Analytic conductor $6.736$
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] = [42,6,Mod(1,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.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: \(6.73612043215\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 42.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +76.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +76.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +304.000 q^{10} +650.000 q^{11} -144.000 q^{12} +762.000 q^{13} -196.000 q^{14} -684.000 q^{15} +256.000 q^{16} -556.000 q^{17} +324.000 q^{18} -2452.00 q^{19} +1216.00 q^{20} +441.000 q^{21} +2600.00 q^{22} -2950.00 q^{23} -576.000 q^{24} +2651.00 q^{25} +3048.00 q^{26} -729.000 q^{27} -784.000 q^{28} -674.000 q^{29} -2736.00 q^{30} -3024.00 q^{31} +1024.00 q^{32} -5850.00 q^{33} -2224.00 q^{34} -3724.00 q^{35} +1296.00 q^{36} +7730.00 q^{37} -9808.00 q^{38} -6858.00 q^{39} +4864.00 q^{40} -17016.0 q^{41} +1764.00 q^{42} +21836.0 q^{43} +10400.0 q^{44} +6156.00 q^{45} -11800.0 q^{46} -23940.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +10604.0 q^{50} +5004.00 q^{51} +12192.0 q^{52} +15594.0 q^{53} -2916.00 q^{54} +49400.0 q^{55} -3136.00 q^{56} +22068.0 q^{57} -2696.00 q^{58} +5608.00 q^{59} -10944.0 q^{60} +150.000 q^{61} -12096.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} +57912.0 q^{65} -23400.0 q^{66} -43784.0 q^{67} -8896.00 q^{68} +26550.0 q^{69} -14896.0 q^{70} -39178.0 q^{71} +5184.00 q^{72} -23570.0 q^{73} +30920.0 q^{74} -23859.0 q^{75} -39232.0 q^{76} -31850.0 q^{77} -27432.0 q^{78} -17892.0 q^{79} +19456.0 q^{80} +6561.00 q^{81} -68064.0 q^{82} +38972.0 q^{83} +7056.00 q^{84} -42256.0 q^{85} +87344.0 q^{86} +6066.00 q^{87} +41600.0 q^{88} +6024.00 q^{89} +24624.0 q^{90} -37338.0 q^{91} -47200.0 q^{92} +27216.0 q^{93} -95760.0 q^{94} -186352. q^{95} -9216.00 q^{96} +108430. q^{97} +9604.00 q^{98} +52650.0 q^{99} +O(q^{100})\)

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
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) 76.0000 1.35953 0.679765 0.733430i \(-0.262082\pi\)
0.679765 + 0.733430i \(0.262082\pi\)
\(6\) −36.0000 −0.408248
\(7\) −49.0000 −0.377964
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 304.000 0.961332
\(11\) 650.000 1.61969 0.809845 0.586645i \(-0.199551\pi\)
0.809845 + 0.586645i \(0.199551\pi\)
\(12\) −144.000 −0.288675
\(13\) 762.000 1.25054 0.625269 0.780410i \(-0.284989\pi\)
0.625269 + 0.780410i \(0.284989\pi\)
\(14\) −196.000 −0.267261
\(15\) −684.000 −0.784925
\(16\) 256.000 0.250000
\(17\) −556.000 −0.466608 −0.233304 0.972404i \(-0.574954\pi\)
−0.233304 + 0.972404i \(0.574954\pi\)
\(18\) 324.000 0.235702
\(19\) −2452.00 −1.55825 −0.779124 0.626870i \(-0.784336\pi\)
−0.779124 + 0.626870i \(0.784336\pi\)
\(20\) 1216.00 0.679765
\(21\) 441.000 0.218218
\(22\) 2600.00 1.14529
\(23\) −2950.00 −1.16279 −0.581397 0.813620i \(-0.697493\pi\)
−0.581397 + 0.813620i \(0.697493\pi\)
\(24\) −576.000 −0.204124
\(25\) 2651.00 0.848320
\(26\) 3048.00 0.884263
\(27\) −729.000 −0.192450
\(28\) −784.000 −0.188982
\(29\) −674.000 −0.148821 −0.0744106 0.997228i \(-0.523708\pi\)
−0.0744106 + 0.997228i \(0.523708\pi\)
\(30\) −2736.00 −0.555026
\(31\) −3024.00 −0.565168 −0.282584 0.959243i \(-0.591192\pi\)
−0.282584 + 0.959243i \(0.591192\pi\)
\(32\) 1024.00 0.176777
\(33\) −5850.00 −0.935128
\(34\) −2224.00 −0.329942
\(35\) −3724.00 −0.513854
\(36\) 1296.00 0.166667
\(37\) 7730.00 0.928272 0.464136 0.885764i \(-0.346365\pi\)
0.464136 + 0.885764i \(0.346365\pi\)
\(38\) −9808.00 −1.10185
\(39\) −6858.00 −0.721998
\(40\) 4864.00 0.480666
\(41\) −17016.0 −1.58088 −0.790438 0.612542i \(-0.790147\pi\)
−0.790438 + 0.612542i \(0.790147\pi\)
\(42\) 1764.00 0.154303
\(43\) 21836.0 1.80095 0.900476 0.434907i \(-0.143219\pi\)
0.900476 + 0.434907i \(0.143219\pi\)
\(44\) 10400.0 0.809845
\(45\) 6156.00 0.453176
\(46\) −11800.0 −0.822219
\(47\) −23940.0 −1.58081 −0.790405 0.612585i \(-0.790130\pi\)
−0.790405 + 0.612585i \(0.790130\pi\)
\(48\) −2304.00 −0.144338
\(49\) 2401.00 0.142857
\(50\) 10604.0 0.599853
\(51\) 5004.00 0.269396
\(52\) 12192.0 0.625269
\(53\) 15594.0 0.762549 0.381275 0.924462i \(-0.375485\pi\)
0.381275 + 0.924462i \(0.375485\pi\)
\(54\) −2916.00 −0.136083
\(55\) 49400.0 2.20201
\(56\) −3136.00 −0.133631
\(57\) 22068.0 0.899655
\(58\) −2696.00 −0.105233
\(59\) 5608.00 0.209738 0.104869 0.994486i \(-0.466558\pi\)
0.104869 + 0.994486i \(0.466558\pi\)
\(60\) −10944.0 −0.392462
\(61\) 150.000 0.00516139 0.00258069 0.999997i \(-0.499179\pi\)
0.00258069 + 0.999997i \(0.499179\pi\)
\(62\) −12096.0 −0.399634
\(63\) −3969.00 −0.125988
\(64\) 4096.00 0.125000
\(65\) 57912.0 1.70014
\(66\) −23400.0 −0.661235
\(67\) −43784.0 −1.19159 −0.595797 0.803135i \(-0.703164\pi\)
−0.595797 + 0.803135i \(0.703164\pi\)
\(68\) −8896.00 −0.233304
\(69\) 26550.0 0.671339
\(70\) −14896.0 −0.363349
\(71\) −39178.0 −0.922351 −0.461176 0.887309i \(-0.652572\pi\)
−0.461176 + 0.887309i \(0.652572\pi\)
\(72\) 5184.00 0.117851
\(73\) −23570.0 −0.517669 −0.258835 0.965922i \(-0.583338\pi\)
−0.258835 + 0.965922i \(0.583338\pi\)
\(74\) 30920.0 0.656387
\(75\) −23859.0 −0.489778
\(76\) −39232.0 −0.779124
\(77\) −31850.0 −0.612185
\(78\) −27432.0 −0.510530
\(79\) −17892.0 −0.322546 −0.161273 0.986910i \(-0.551560\pi\)
−0.161273 + 0.986910i \(0.551560\pi\)
\(80\) 19456.0 0.339882
\(81\) 6561.00 0.111111
\(82\) −68064.0 −1.11785
\(83\) 38972.0 0.620951 0.310476 0.950581i \(-0.399512\pi\)
0.310476 + 0.950581i \(0.399512\pi\)
\(84\) 7056.00 0.109109
\(85\) −42256.0 −0.634368
\(86\) 87344.0 1.27346
\(87\) 6066.00 0.0859220
\(88\) 41600.0 0.572647
\(89\) 6024.00 0.0806139 0.0403070 0.999187i \(-0.487166\pi\)
0.0403070 + 0.999187i \(0.487166\pi\)
\(90\) 24624.0 0.320444
\(91\) −37338.0 −0.472659
\(92\) −47200.0 −0.581397
\(93\) 27216.0 0.326300
\(94\) −95760.0 −1.11780
\(95\) −186352. −2.11848
\(96\) −9216.00 −0.102062
\(97\) 108430. 1.17009 0.585046 0.811000i \(-0.301076\pi\)
0.585046 + 0.811000i \(0.301076\pi\)
\(98\) 9604.00 0.101015
\(99\) 52650.0 0.539896
\(100\) 42416.0 0.424160
\(101\) −70424.0 −0.686938 −0.343469 0.939164i \(-0.611602\pi\)
−0.343469 + 0.939164i \(0.611602\pi\)
\(102\) 20016.0 0.190492
\(103\) −31552.0 −0.293045 −0.146522 0.989207i \(-0.546808\pi\)
−0.146522 + 0.989207i \(0.546808\pi\)
\(104\) 48768.0 0.442132
\(105\) 33516.0 0.296674
\(106\) 62376.0 0.539204
\(107\) 108282. 0.914317 0.457159 0.889385i \(-0.348867\pi\)
0.457159 + 0.889385i \(0.348867\pi\)
\(108\) −11664.0 −0.0962250
\(109\) −72146.0 −0.581629 −0.290814 0.956779i \(-0.593926\pi\)
−0.290814 + 0.956779i \(0.593926\pi\)
\(110\) 197600. 1.55706
\(111\) −69570.0 −0.535938
\(112\) −12544.0 −0.0944911
\(113\) 220906. 1.62746 0.813732 0.581240i \(-0.197432\pi\)
0.813732 + 0.581240i \(0.197432\pi\)
\(114\) 88272.0 0.636152
\(115\) −224200. −1.58085
\(116\) −10784.0 −0.0744106
\(117\) 61722.0 0.416846
\(118\) 22432.0 0.148307
\(119\) 27244.0 0.176361
\(120\) −43776.0 −0.277513
\(121\) 261449. 1.62339
\(122\) 600.000 0.00364965
\(123\) 153144. 0.912719
\(124\) −48384.0 −0.282584
\(125\) −36024.0 −0.206213
\(126\) −15876.0 −0.0890871
\(127\) −239652. −1.31847 −0.659237 0.751935i \(-0.729121\pi\)
−0.659237 + 0.751935i \(0.729121\pi\)
\(128\) 16384.0 0.0883883
\(129\) −196524. −1.03978
\(130\) 231648. 1.20218
\(131\) −274172. −1.39587 −0.697935 0.716161i \(-0.745897\pi\)
−0.697935 + 0.716161i \(0.745897\pi\)
\(132\) −93600.0 −0.467564
\(133\) 120148. 0.588962
\(134\) −175136. −0.842584
\(135\) −55404.0 −0.261642
\(136\) −35584.0 −0.164971
\(137\) −391154. −1.78052 −0.890259 0.455455i \(-0.849477\pi\)
−0.890259 + 0.455455i \(0.849477\pi\)
\(138\) 106200. 0.474708
\(139\) 339364. 1.48980 0.744901 0.667175i \(-0.232497\pi\)
0.744901 + 0.667175i \(0.232497\pi\)
\(140\) −59584.0 −0.256927
\(141\) 215460. 0.912681
\(142\) −156712. −0.652201
\(143\) 495300. 2.02548
\(144\) 20736.0 0.0833333
\(145\) −51224.0 −0.202327
\(146\) −94280.0 −0.366047
\(147\) −21609.0 −0.0824786
\(148\) 123680. 0.464136
\(149\) −29334.0 −0.108244 −0.0541222 0.998534i \(-0.517236\pi\)
−0.0541222 + 0.998534i \(0.517236\pi\)
\(150\) −95436.0 −0.346325
\(151\) 71608.0 0.255575 0.127788 0.991802i \(-0.459212\pi\)
0.127788 + 0.991802i \(0.459212\pi\)
\(152\) −156928. −0.550924
\(153\) −45036.0 −0.155536
\(154\) −127400. −0.432880
\(155\) −229824. −0.768362
\(156\) −109728. −0.360999
\(157\) 296318. 0.959420 0.479710 0.877427i \(-0.340742\pi\)
0.479710 + 0.877427i \(0.340742\pi\)
\(158\) −71568.0 −0.228074
\(159\) −140346. −0.440258
\(160\) 77824.0 0.240333
\(161\) 144550. 0.439494
\(162\) 26244.0 0.0785674
\(163\) −480400. −1.41623 −0.708115 0.706097i \(-0.750454\pi\)
−0.708115 + 0.706097i \(0.750454\pi\)
\(164\) −272256. −0.790438
\(165\) −444600. −1.27133
\(166\) 155888. 0.439079
\(167\) 160180. 0.444444 0.222222 0.974996i \(-0.428669\pi\)
0.222222 + 0.974996i \(0.428669\pi\)
\(168\) 28224.0 0.0771517
\(169\) 209351. 0.563843
\(170\) −169024. −0.448566
\(171\) −198612. −0.519416
\(172\) 349376. 0.900476
\(173\) −8984.00 −0.0228220 −0.0114110 0.999935i \(-0.503632\pi\)
−0.0114110 + 0.999935i \(0.503632\pi\)
\(174\) 24264.0 0.0607560
\(175\) −129899. −0.320635
\(176\) 166400. 0.404922
\(177\) −50472.0 −0.121093
\(178\) 24096.0 0.0570026
\(179\) 182886. 0.426627 0.213313 0.976984i \(-0.431575\pi\)
0.213313 + 0.976984i \(0.431575\pi\)
\(180\) 98496.0 0.226588
\(181\) 138330. 0.313848 0.156924 0.987611i \(-0.449842\pi\)
0.156924 + 0.987611i \(0.449842\pi\)
\(182\) −149352. −0.334220
\(183\) −1350.00 −0.00297993
\(184\) −188800. −0.411109
\(185\) 587480. 1.26201
\(186\) 108864. 0.230729
\(187\) −361400. −0.755760
\(188\) −383040. −0.790405
\(189\) 35721.0 0.0727393
\(190\) −745408. −1.49799
\(191\) 327222. 0.649021 0.324511 0.945882i \(-0.394800\pi\)
0.324511 + 0.945882i \(0.394800\pi\)
\(192\) −36864.0 −0.0721688
\(193\) 786902. 1.52064 0.760322 0.649547i \(-0.225041\pi\)
0.760322 + 0.649547i \(0.225041\pi\)
\(194\) 433720. 0.827380
\(195\) −521208. −0.981577
\(196\) 38416.0 0.0714286
\(197\) 423098. 0.776740 0.388370 0.921504i \(-0.373038\pi\)
0.388370 + 0.921504i \(0.373038\pi\)
\(198\) 210600. 0.381764
\(199\) 1.02392e6 1.83288 0.916439 0.400175i \(-0.131051\pi\)
0.916439 + 0.400175i \(0.131051\pi\)
\(200\) 169664. 0.299926
\(201\) 394056. 0.687967
\(202\) −281696. −0.485738
\(203\) 33026.0 0.0562491
\(204\) 80064.0 0.134698
\(205\) −1.29322e6 −2.14925
\(206\) −126208. −0.207214
\(207\) −238950. −0.387598
\(208\) 195072. 0.312634
\(209\) −1.59380e6 −2.52388
\(210\) 134064. 0.209780
\(211\) 461516. 0.713642 0.356821 0.934173i \(-0.383861\pi\)
0.356821 + 0.934173i \(0.383861\pi\)
\(212\) 249504. 0.381275
\(213\) 352602. 0.532520
\(214\) 433128. 0.646520
\(215\) 1.65954e6 2.44845
\(216\) −46656.0 −0.0680414
\(217\) 148176. 0.213613
\(218\) −288584. −0.411274
\(219\) 212130. 0.298877
\(220\) 790400. 1.10101
\(221\) −423672. −0.583511
\(222\) −278280. −0.378965
\(223\) 995048. 1.33993 0.669965 0.742393i \(-0.266309\pi\)
0.669965 + 0.742393i \(0.266309\pi\)
\(224\) −50176.0 −0.0668153
\(225\) 214731. 0.282773
\(226\) 883624. 1.15079
\(227\) −95568.0 −0.123097 −0.0615486 0.998104i \(-0.519604\pi\)
−0.0615486 + 0.998104i \(0.519604\pi\)
\(228\) 353088. 0.449827
\(229\) −1.04409e6 −1.31567 −0.657836 0.753161i \(-0.728528\pi\)
−0.657836 + 0.753161i \(0.728528\pi\)
\(230\) −896800. −1.11783
\(231\) 286650. 0.353445
\(232\) −43136.0 −0.0526163
\(233\) −1.16941e6 −1.41116 −0.705581 0.708629i \(-0.749314\pi\)
−0.705581 + 0.708629i \(0.749314\pi\)
\(234\) 246888. 0.294754
\(235\) −1.81944e6 −2.14916
\(236\) 89728.0 0.104869
\(237\) 161028. 0.186222
\(238\) 108976. 0.124706
\(239\) −27342.0 −0.0309625 −0.0154812 0.999880i \(-0.504928\pi\)
−0.0154812 + 0.999880i \(0.504928\pi\)
\(240\) −175104. −0.196231
\(241\) −907714. −1.00671 −0.503357 0.864078i \(-0.667902\pi\)
−0.503357 + 0.864078i \(0.667902\pi\)
\(242\) 1.04580e6 1.14791
\(243\) −59049.0 −0.0641500
\(244\) 2400.00 0.00258069
\(245\) 182476. 0.194218
\(246\) 612576. 0.645390
\(247\) −1.86842e6 −1.94865
\(248\) −193536. −0.199817
\(249\) −350748. −0.358506
\(250\) −144096. −0.145815
\(251\) 44088.0 0.0441709 0.0220854 0.999756i \(-0.492969\pi\)
0.0220854 + 0.999756i \(0.492969\pi\)
\(252\) −63504.0 −0.0629941
\(253\) −1.91750e6 −1.88336
\(254\) −958608. −0.932302
\(255\) 380304. 0.366252
\(256\) 65536.0 0.0625000
\(257\) −829200. −0.783117 −0.391558 0.920153i \(-0.628064\pi\)
−0.391558 + 0.920153i \(0.628064\pi\)
\(258\) −786096. −0.735235
\(259\) −378770. −0.350854
\(260\) 926592. 0.850071
\(261\) −54594.0 −0.0496071
\(262\) −1.09669e6 −0.987029
\(263\) 1.31947e6 1.17627 0.588137 0.808761i \(-0.299861\pi\)
0.588137 + 0.808761i \(0.299861\pi\)
\(264\) −374400. −0.330618
\(265\) 1.18514e6 1.03671
\(266\) 480592. 0.416459
\(267\) −54216.0 −0.0465425
\(268\) −700544. −0.595797
\(269\) −783788. −0.660416 −0.330208 0.943908i \(-0.607119\pi\)
−0.330208 + 0.943908i \(0.607119\pi\)
\(270\) −221616. −0.185009
\(271\) 955080. 0.789981 0.394990 0.918685i \(-0.370748\pi\)
0.394990 + 0.918685i \(0.370748\pi\)
\(272\) −142336. −0.116652
\(273\) 336042. 0.272890
\(274\) −1.56462e6 −1.25902
\(275\) 1.72315e6 1.37401
\(276\) 424800. 0.335669
\(277\) 1.91273e6 1.49780 0.748901 0.662682i \(-0.230582\pi\)
0.748901 + 0.662682i \(0.230582\pi\)
\(278\) 1.35746e6 1.05345
\(279\) −244944. −0.188389
\(280\) −238336. −0.181675
\(281\) −1.02620e6 −0.775295 −0.387648 0.921808i \(-0.626712\pi\)
−0.387648 + 0.921808i \(0.626712\pi\)
\(282\) 861840. 0.645363
\(283\) 1.74668e6 1.29642 0.648211 0.761461i \(-0.275518\pi\)
0.648211 + 0.761461i \(0.275518\pi\)
\(284\) −626848. −0.461176
\(285\) 1.67717e6 1.22311
\(286\) 1.98120e6 1.43223
\(287\) 833784. 0.597515
\(288\) 82944.0 0.0589256
\(289\) −1.11072e6 −0.782277
\(290\) −204896. −0.143067
\(291\) −975870. −0.675553
\(292\) −377120. −0.258835
\(293\) 2.23212e6 1.51897 0.759484 0.650526i \(-0.225452\pi\)
0.759484 + 0.650526i \(0.225452\pi\)
\(294\) −86436.0 −0.0583212
\(295\) 426208. 0.285146
\(296\) 494720. 0.328194
\(297\) −473850. −0.311709
\(298\) −117336. −0.0765404
\(299\) −2.24790e6 −1.45412
\(300\) −381744. −0.244889
\(301\) −1.06996e6 −0.680696
\(302\) 286432. 0.180719
\(303\) 633816. 0.396604
\(304\) −627712. −0.389562
\(305\) 11400.0 0.00701706
\(306\) −180144. −0.109981
\(307\) 1.85324e6 1.12224 0.561119 0.827735i \(-0.310371\pi\)
0.561119 + 0.827735i \(0.310371\pi\)
\(308\) −509600. −0.306092
\(309\) 283968. 0.169189
\(310\) −919296. −0.543314
\(311\) −450956. −0.264383 −0.132191 0.991224i \(-0.542201\pi\)
−0.132191 + 0.991224i \(0.542201\pi\)
\(312\) −438912. −0.255265
\(313\) 1.60263e6 0.924642 0.462321 0.886713i \(-0.347017\pi\)
0.462321 + 0.886713i \(0.347017\pi\)
\(314\) 1.18527e6 0.678413
\(315\) −301644. −0.171285
\(316\) −286272. −0.161273
\(317\) −20862.0 −0.0116602 −0.00583012 0.999983i \(-0.501856\pi\)
−0.00583012 + 0.999983i \(0.501856\pi\)
\(318\) −561384. −0.311309
\(319\) −438100. −0.241044
\(320\) 311296. 0.169941
\(321\) −974538. −0.527881
\(322\) 578200. 0.310770
\(323\) 1.36331e6 0.727091
\(324\) 104976. 0.0555556
\(325\) 2.02006e6 1.06086
\(326\) −1.92160e6 −1.00143
\(327\) 649314. 0.335804
\(328\) −1.08902e6 −0.558924
\(329\) 1.17306e6 0.597490
\(330\) −1.77840e6 −0.898969
\(331\) 2.07621e6 1.04160 0.520801 0.853678i \(-0.325633\pi\)
0.520801 + 0.853678i \(0.325633\pi\)
\(332\) 623552. 0.310476
\(333\) 626130. 0.309424
\(334\) 640720. 0.314269
\(335\) −3.32758e6 −1.62001
\(336\) 112896. 0.0545545
\(337\) 1.20508e6 0.578019 0.289009 0.957326i \(-0.406674\pi\)
0.289009 + 0.957326i \(0.406674\pi\)
\(338\) 837404. 0.398697
\(339\) −1.98815e6 −0.939617
\(340\) −676096. −0.317184
\(341\) −1.96560e6 −0.915396
\(342\) −794448. −0.367282
\(343\) −117649. −0.0539949
\(344\) 1.39750e6 0.636732
\(345\) 2.01780e6 0.912705
\(346\) −35936.0 −0.0161376
\(347\) −876642. −0.390840 −0.195420 0.980720i \(-0.562607\pi\)
−0.195420 + 0.980720i \(0.562607\pi\)
\(348\) 97056.0 0.0429610
\(349\) −1.29593e6 −0.569532 −0.284766 0.958597i \(-0.591916\pi\)
−0.284766 + 0.958597i \(0.591916\pi\)
\(350\) −519596. −0.226723
\(351\) −555498. −0.240666
\(352\) 665600. 0.286323
\(353\) −3.99040e6 −1.70443 −0.852215 0.523192i \(-0.824741\pi\)
−0.852215 + 0.523192i \(0.824741\pi\)
\(354\) −201888. −0.0856253
\(355\) −2.97753e6 −1.25396
\(356\) 96384.0 0.0403070
\(357\) −245196. −0.101822
\(358\) 731544. 0.301671
\(359\) 4.06452e6 1.66446 0.832229 0.554432i \(-0.187064\pi\)
0.832229 + 0.554432i \(0.187064\pi\)
\(360\) 393984. 0.160222
\(361\) 3.53620e6 1.42814
\(362\) 553320. 0.221924
\(363\) −2.35304e6 −0.937266
\(364\) −597408. −0.236329
\(365\) −1.79132e6 −0.703787
\(366\) −5400.00 −0.00210713
\(367\) −1.67243e6 −0.648162 −0.324081 0.946029i \(-0.605055\pi\)
−0.324081 + 0.946029i \(0.605055\pi\)
\(368\) −755200. −0.290698
\(369\) −1.37830e6 −0.526959
\(370\) 2.34992e6 0.892378
\(371\) −764106. −0.288216
\(372\) 435456. 0.163150
\(373\) 3.16769e6 1.17888 0.589441 0.807812i \(-0.299348\pi\)
0.589441 + 0.807812i \(0.299348\pi\)
\(374\) −1.44560e6 −0.534403
\(375\) 324216. 0.119057
\(376\) −1.53216e6 −0.558901
\(377\) −513588. −0.186106
\(378\) 142884. 0.0514344
\(379\) −4.20388e6 −1.50332 −0.751662 0.659548i \(-0.770748\pi\)
−0.751662 + 0.659548i \(0.770748\pi\)
\(380\) −2.98163e6 −1.05924
\(381\) 2.15687e6 0.761222
\(382\) 1.30889e6 0.458927
\(383\) −342616. −0.119347 −0.0596734 0.998218i \(-0.519006\pi\)
−0.0596734 + 0.998218i \(0.519006\pi\)
\(384\) −147456. −0.0510310
\(385\) −2.42060e6 −0.832283
\(386\) 3.14761e6 1.07526
\(387\) 1.76872e6 0.600317
\(388\) 1.73488e6 0.585046
\(389\) −3.83959e6 −1.28650 −0.643252 0.765654i \(-0.722415\pi\)
−0.643252 + 0.765654i \(0.722415\pi\)
\(390\) −2.08483e6 −0.694080
\(391\) 1.64020e6 0.542569
\(392\) 153664. 0.0505076
\(393\) 2.46755e6 0.805906
\(394\) 1.69239e6 0.549238
\(395\) −1.35979e6 −0.438510
\(396\) 842400. 0.269948
\(397\) 3.43894e6 1.09509 0.547543 0.836777i \(-0.315563\pi\)
0.547543 + 0.836777i \(0.315563\pi\)
\(398\) 4.09568e6 1.29604
\(399\) −1.08133e6 −0.340038
\(400\) 678656. 0.212080
\(401\) −3.89421e6 −1.20937 −0.604684 0.796466i \(-0.706701\pi\)
−0.604684 + 0.796466i \(0.706701\pi\)
\(402\) 1.57622e6 0.486466
\(403\) −2.30429e6 −0.706764
\(404\) −1.12678e6 −0.343469
\(405\) 498636. 0.151059
\(406\) 132104. 0.0397741
\(407\) 5.02450e6 1.50351
\(408\) 320256. 0.0952460
\(409\) −1.64679e6 −0.486778 −0.243389 0.969929i \(-0.578259\pi\)
−0.243389 + 0.969929i \(0.578259\pi\)
\(410\) −5.17286e6 −1.51975
\(411\) 3.52039e6 1.02798
\(412\) −504832. −0.146522
\(413\) −274792. −0.0792737
\(414\) −955800. −0.274073
\(415\) 2.96187e6 0.844201
\(416\) 780288. 0.221066
\(417\) −3.05428e6 −0.860138
\(418\) −6.37520e6 −1.78465
\(419\) −1.67659e6 −0.466544 −0.233272 0.972412i \(-0.574943\pi\)
−0.233272 + 0.972412i \(0.574943\pi\)
\(420\) 536256. 0.148337
\(421\) −566742. −0.155840 −0.0779202 0.996960i \(-0.524828\pi\)
−0.0779202 + 0.996960i \(0.524828\pi\)
\(422\) 1.84606e6 0.504621
\(423\) −1.93914e6 −0.526936
\(424\) 998016. 0.269602
\(425\) −1.47396e6 −0.395833
\(426\) 1.41041e6 0.376548
\(427\) −7350.00 −0.00195082
\(428\) 1.73251e6 0.457159
\(429\) −4.45770e6 −1.16941
\(430\) 6.63814e6 1.73131
\(431\) 6.68468e6 1.73335 0.866677 0.498870i \(-0.166251\pi\)
0.866677 + 0.498870i \(0.166251\pi\)
\(432\) −186624. −0.0481125
\(433\) 6.91337e6 1.77203 0.886013 0.463661i \(-0.153464\pi\)
0.886013 + 0.463661i \(0.153464\pi\)
\(434\) 592704. 0.151047
\(435\) 461016. 0.116813
\(436\) −1.15434e6 −0.290814
\(437\) 7.23340e6 1.81192
\(438\) 848520. 0.211338
\(439\) −4.56281e6 −1.12998 −0.564990 0.825098i \(-0.691120\pi\)
−0.564990 + 0.825098i \(0.691120\pi\)
\(440\) 3.16160e6 0.778530
\(441\) 194481. 0.0476190
\(442\) −1.69469e6 −0.412605
\(443\) 4.59760e6 1.11307 0.556534 0.830825i \(-0.312131\pi\)
0.556534 + 0.830825i \(0.312131\pi\)
\(444\) −1.11312e6 −0.267969
\(445\) 457824. 0.109597
\(446\) 3.98019e6 0.947473
\(447\) 264006. 0.0624950
\(448\) −200704. −0.0472456
\(449\) 1.70658e6 0.399494 0.199747 0.979848i \(-0.435988\pi\)
0.199747 + 0.979848i \(0.435988\pi\)
\(450\) 858924. 0.199951
\(451\) −1.10604e7 −2.56053
\(452\) 3.53450e6 0.813732
\(453\) −644472. −0.147557
\(454\) −382272. −0.0870428
\(455\) −2.83769e6 −0.642593
\(456\) 1.41235e6 0.318076
\(457\) −6.93916e6 −1.55423 −0.777117 0.629356i \(-0.783319\pi\)
−0.777117 + 0.629356i \(0.783319\pi\)
\(458\) −4.17634e6 −0.930320
\(459\) 405324. 0.0897988
\(460\) −3.58720e6 −0.790426
\(461\) −2.61805e6 −0.573753 −0.286877 0.957968i \(-0.592617\pi\)
−0.286877 + 0.957968i \(0.592617\pi\)
\(462\) 1.14660e6 0.249923
\(463\) 7.13602e6 1.54705 0.773524 0.633767i \(-0.218492\pi\)
0.773524 + 0.633767i \(0.218492\pi\)
\(464\) −172544. −0.0372053
\(465\) 2.06842e6 0.443614
\(466\) −4.67764e6 −0.997843
\(467\) −2.17398e6 −0.461278 −0.230639 0.973039i \(-0.574082\pi\)
−0.230639 + 0.973039i \(0.574082\pi\)
\(468\) 987552. 0.208423
\(469\) 2.14542e6 0.450380
\(470\) −7.27776e6 −1.51968
\(471\) −2.66686e6 −0.553922
\(472\) 358912. 0.0741537
\(473\) 1.41934e7 2.91698
\(474\) 644112. 0.131679
\(475\) −6.50025e6 −1.32189
\(476\) 435904. 0.0881807
\(477\) 1.26311e6 0.254183
\(478\) −109368. −0.0218938
\(479\) −4.63294e6 −0.922609 −0.461305 0.887242i \(-0.652619\pi\)
−0.461305 + 0.887242i \(0.652619\pi\)
\(480\) −700416. −0.138756
\(481\) 5.89026e6 1.16084
\(482\) −3.63086e6 −0.711855
\(483\) −1.30095e6 −0.253742
\(484\) 4.18318e6 0.811696
\(485\) 8.24068e6 1.59077
\(486\) −236196. −0.0453609
\(487\) −4.56645e6 −0.872481 −0.436241 0.899830i \(-0.643690\pi\)
−0.436241 + 0.899830i \(0.643690\pi\)
\(488\) 9600.00 0.00182483
\(489\) 4.32360e6 0.817661
\(490\) 729904. 0.137333
\(491\) −5.31429e6 −0.994813 −0.497407 0.867518i \(-0.665714\pi\)
−0.497407 + 0.867518i \(0.665714\pi\)
\(492\) 2.45030e6 0.456360
\(493\) 374744. 0.0694412
\(494\) −7.47370e6 −1.37790
\(495\) 4.00140e6 0.734005
\(496\) −774144. −0.141292
\(497\) 1.91972e6 0.348616
\(498\) −1.40299e6 −0.253502
\(499\) −2.46314e6 −0.442831 −0.221415 0.975180i \(-0.571068\pi\)
−0.221415 + 0.975180i \(0.571068\pi\)
\(500\) −576384. −0.103107
\(501\) −1.44162e6 −0.256600
\(502\) 176352. 0.0312335
\(503\) 2.79924e6 0.493310 0.246655 0.969103i \(-0.420669\pi\)
0.246655 + 0.969103i \(0.420669\pi\)
\(504\) −254016. −0.0445435
\(505\) −5.35222e6 −0.933912
\(506\) −7.67000e6 −1.33174
\(507\) −1.88416e6 −0.325535
\(508\) −3.83443e6 −0.659237
\(509\) 1.99914e6 0.342018 0.171009 0.985269i \(-0.445297\pi\)
0.171009 + 0.985269i \(0.445297\pi\)
\(510\) 1.52122e6 0.258980
\(511\) 1.15493e6 0.195661
\(512\) 262144. 0.0441942
\(513\) 1.78751e6 0.299885
\(514\) −3.31680e6 −0.553747
\(515\) −2.39795e6 −0.398403
\(516\) −3.14438e6 −0.519890
\(517\) −1.55610e7 −2.56042
\(518\) −1.51508e6 −0.248091
\(519\) 80856.0 0.0131763
\(520\) 3.70637e6 0.601091
\(521\) 3.52160e6 0.568390 0.284195 0.958767i \(-0.408274\pi\)
0.284195 + 0.958767i \(0.408274\pi\)
\(522\) −218376. −0.0350775
\(523\) 2.60685e6 0.416737 0.208369 0.978050i \(-0.433185\pi\)
0.208369 + 0.978050i \(0.433185\pi\)
\(524\) −4.38675e6 −0.697935
\(525\) 1.16909e6 0.185119
\(526\) 5.27786e6 0.831752
\(527\) 1.68134e6 0.263712
\(528\) −1.49760e6 −0.233782
\(529\) 2.26616e6 0.352088
\(530\) 4.74058e6 0.733063
\(531\) 454248. 0.0699128
\(532\) 1.92237e6 0.294481
\(533\) −1.29662e7 −1.97694
\(534\) −216864. −0.0329105
\(535\) 8.22943e6 1.24304
\(536\) −2.80218e6 −0.421292
\(537\) −1.64597e6 −0.246313
\(538\) −3.13515e6 −0.466985
\(539\) 1.56065e6 0.231384
\(540\) −886464. −0.130821
\(541\) −1.37441e6 −0.201894 −0.100947 0.994892i \(-0.532187\pi\)
−0.100947 + 0.994892i \(0.532187\pi\)
\(542\) 3.82032e6 0.558601
\(543\) −1.24497e6 −0.181200
\(544\) −569344. −0.0824855
\(545\) −5.48310e6 −0.790742
\(546\) 1.34417e6 0.192962
\(547\) −8.78398e6 −1.25523 −0.627614 0.778524i \(-0.715968\pi\)
−0.627614 + 0.778524i \(0.715968\pi\)
\(548\) −6.25846e6 −0.890259
\(549\) 12150.0 0.00172046
\(550\) 6.89260e6 0.971575
\(551\) 1.65265e6 0.231900
\(552\) 1.69920e6 0.237354
\(553\) 876708. 0.121911
\(554\) 7.65092e6 1.05911
\(555\) −5.28732e6 −0.728623
\(556\) 5.42982e6 0.744901
\(557\) 6.29262e6 0.859396 0.429698 0.902973i \(-0.358620\pi\)
0.429698 + 0.902973i \(0.358620\pi\)
\(558\) −979776. −0.133211
\(559\) 1.66390e7 2.25216
\(560\) −953344. −0.128463
\(561\) 3.25260e6 0.436338
\(562\) −4.10481e6 −0.548216
\(563\) 4.86582e6 0.646971 0.323485 0.946233i \(-0.395145\pi\)
0.323485 + 0.946233i \(0.395145\pi\)
\(564\) 3.44736e6 0.456340
\(565\) 1.67889e7 2.21259
\(566\) 6.98670e6 0.916709
\(567\) −321489. −0.0419961
\(568\) −2.50739e6 −0.326100
\(569\) −4.46383e6 −0.577998 −0.288999 0.957329i \(-0.593322\pi\)
−0.288999 + 0.957329i \(0.593322\pi\)
\(570\) 6.70867e6 0.864867
\(571\) 8.17054e6 1.04872 0.524361 0.851496i \(-0.324304\pi\)
0.524361 + 0.851496i \(0.324304\pi\)
\(572\) 7.92480e6 1.01274
\(573\) −2.94500e6 −0.374713
\(574\) 3.33514e6 0.422507
\(575\) −7.82045e6 −0.986421
\(576\) 331776. 0.0416667
\(577\) −5.50343e6 −0.688167 −0.344084 0.938939i \(-0.611810\pi\)
−0.344084 + 0.938939i \(0.611810\pi\)
\(578\) −4.44288e6 −0.553153
\(579\) −7.08212e6 −0.877944
\(580\) −819584. −0.101163
\(581\) −1.90963e6 −0.234697
\(582\) −3.90348e6 −0.477688
\(583\) 1.01361e7 1.23509
\(584\) −1.50848e6 −0.183024
\(585\) 4.69087e6 0.566714
\(586\) 8.92848e6 1.07407
\(587\) 8.14251e6 0.975356 0.487678 0.873024i \(-0.337844\pi\)
0.487678 + 0.873024i \(0.337844\pi\)
\(588\) −345744. −0.0412393
\(589\) 7.41485e6 0.880672
\(590\) 1.70483e6 0.201628
\(591\) −3.80788e6 −0.448451
\(592\) 1.97888e6 0.232068
\(593\) −2.73136e6 −0.318964 −0.159482 0.987201i \(-0.550982\pi\)
−0.159482 + 0.987201i \(0.550982\pi\)
\(594\) −1.89540e6 −0.220412
\(595\) 2.07054e6 0.239768
\(596\) −469344. −0.0541222
\(597\) −9.21528e6 −1.05821
\(598\) −8.99160e6 −1.02822
\(599\) 1.23733e6 0.140902 0.0704510 0.997515i \(-0.477556\pi\)
0.0704510 + 0.997515i \(0.477556\pi\)
\(600\) −1.52698e6 −0.173163
\(601\) −1.59756e7 −1.80414 −0.902071 0.431587i \(-0.857954\pi\)
−0.902071 + 0.431587i \(0.857954\pi\)
\(602\) −4.27986e6 −0.481324
\(603\) −3.54650e6 −0.397198
\(604\) 1.14573e6 0.127788
\(605\) 1.98701e7 2.20705
\(606\) 2.53526e6 0.280441
\(607\) −1.88275e6 −0.207406 −0.103703 0.994608i \(-0.533069\pi\)
−0.103703 + 0.994608i \(0.533069\pi\)
\(608\) −2.51085e6 −0.275462
\(609\) −297234. −0.0324755
\(610\) 45600.0 0.00496181
\(611\) −1.82423e7 −1.97686
\(612\) −720576. −0.0777681
\(613\) −9.82804e6 −1.05637 −0.528185 0.849130i \(-0.677127\pi\)
−0.528185 + 0.849130i \(0.677127\pi\)
\(614\) 7.41294e6 0.793542
\(615\) 1.16389e7 1.24087
\(616\) −2.03840e6 −0.216440
\(617\) −8.21262e6 −0.868498 −0.434249 0.900793i \(-0.642986\pi\)
−0.434249 + 0.900793i \(0.642986\pi\)
\(618\) 1.13587e6 0.119635
\(619\) 6.98465e6 0.732686 0.366343 0.930480i \(-0.380610\pi\)
0.366343 + 0.930480i \(0.380610\pi\)
\(620\) −3.67718e6 −0.384181
\(621\) 2.15055e6 0.223780
\(622\) −1.80382e6 −0.186947
\(623\) −295176. −0.0304692
\(624\) −1.75565e6 −0.180499
\(625\) −1.10222e7 −1.12867
\(626\) 6.41054e6 0.653820
\(627\) 1.43442e7 1.45716
\(628\) 4.74109e6 0.479710
\(629\) −4.29788e6 −0.433139
\(630\) −1.20658e6 −0.121116
\(631\) 1.26789e7 1.26767 0.633837 0.773467i \(-0.281479\pi\)
0.633837 + 0.773467i \(0.281479\pi\)
\(632\) −1.14509e6 −0.114037
\(633\) −4.15364e6 −0.412022
\(634\) −83448.0 −0.00824504
\(635\) −1.82136e7 −1.79250
\(636\) −2.24554e6 −0.220129
\(637\) 1.82956e6 0.178648
\(638\) −1.75240e6 −0.170444
\(639\) −3.17342e6 −0.307450
\(640\) 1.24518e6 0.120167
\(641\) −1.40324e7 −1.34892 −0.674460 0.738311i \(-0.735624\pi\)
−0.674460 + 0.738311i \(0.735624\pi\)
\(642\) −3.89815e6 −0.373268
\(643\) 1.30368e6 0.124349 0.0621745 0.998065i \(-0.480196\pi\)
0.0621745 + 0.998065i \(0.480196\pi\)
\(644\) 2.31280e6 0.219747
\(645\) −1.49358e7 −1.41361
\(646\) 5.45325e6 0.514131
\(647\) 1.57110e6 0.147551 0.0737757 0.997275i \(-0.476495\pi\)
0.0737757 + 0.997275i \(0.476495\pi\)
\(648\) 419904. 0.0392837
\(649\) 3.64520e6 0.339711
\(650\) 8.08025e6 0.750138
\(651\) −1.33358e6 −0.123330
\(652\) −7.68640e6 −0.708115
\(653\) −8.34115e6 −0.765496 −0.382748 0.923853i \(-0.625022\pi\)
−0.382748 + 0.923853i \(0.625022\pi\)
\(654\) 2.59726e6 0.237449
\(655\) −2.08371e7 −1.89773
\(656\) −4.35610e6 −0.395219
\(657\) −1.90917e6 −0.172556
\(658\) 4.69224e6 0.422489
\(659\) 6.18334e6 0.554638 0.277319 0.960778i \(-0.410554\pi\)
0.277319 + 0.960778i \(0.410554\pi\)
\(660\) −7.11360e6 −0.635667
\(661\) 928966. 0.0826982 0.0413491 0.999145i \(-0.486834\pi\)
0.0413491 + 0.999145i \(0.486834\pi\)
\(662\) 8.30485e6 0.736524
\(663\) 3.81305e6 0.336890
\(664\) 2.49421e6 0.219539
\(665\) 9.13125e6 0.800711
\(666\) 2.50452e6 0.218796
\(667\) 1.98830e6 0.173048
\(668\) 2.56288e6 0.222222
\(669\) −8.95543e6 −0.773609
\(670\) −1.33103e7 −1.14552
\(671\) 97500.0 0.00835985
\(672\) 451584. 0.0385758
\(673\) 1.79131e7 1.52452 0.762259 0.647272i \(-0.224090\pi\)
0.762259 + 0.647272i \(0.224090\pi\)
\(674\) 4.82033e6 0.408721
\(675\) −1.93258e6 −0.163259
\(676\) 3.34962e6 0.281922
\(677\) −4.96397e6 −0.416253 −0.208126 0.978102i \(-0.566737\pi\)
−0.208126 + 0.978102i \(0.566737\pi\)
\(678\) −7.95262e6 −0.664409
\(679\) −5.31307e6 −0.442253
\(680\) −2.70438e6 −0.224283
\(681\) 860112. 0.0710701
\(682\) −7.86240e6 −0.647283
\(683\) 89526.0 0.00734340 0.00367170 0.999993i \(-0.498831\pi\)
0.00367170 + 0.999993i \(0.498831\pi\)
\(684\) −3.17779e6 −0.259708
\(685\) −2.97277e7 −2.42067
\(686\) −470596. −0.0381802
\(687\) 9.39677e6 0.759603
\(688\) 5.59002e6 0.450238
\(689\) 1.18826e7 0.953596
\(690\) 8.07120e6 0.645380
\(691\) −142396. −0.0113450 −0.00567248 0.999984i \(-0.501806\pi\)
−0.00567248 + 0.999984i \(0.501806\pi\)
\(692\) −143744. −0.0114110
\(693\) −2.57985e6 −0.204062
\(694\) −3.50657e6 −0.276365
\(695\) 2.57917e7 2.02543
\(696\) 388224. 0.0303780
\(697\) 9.46090e6 0.737650
\(698\) −5.18372e6 −0.402720
\(699\) 1.05247e7 0.814735
\(700\) −2.07838e6 −0.160317
\(701\) 1.03935e7 0.798852 0.399426 0.916765i \(-0.369209\pi\)
0.399426 + 0.916765i \(0.369209\pi\)
\(702\) −2.22199e6 −0.170177
\(703\) −1.89540e7 −1.44648
\(704\) 2.66240e6 0.202461
\(705\) 1.63750e7 1.24082
\(706\) −1.59616e7 −1.20521
\(707\) 3.45078e6 0.259638
\(708\) −807552. −0.0605463
\(709\) 4.65503e6 0.347782 0.173891 0.984765i \(-0.444366\pi\)
0.173891 + 0.984765i \(0.444366\pi\)
\(710\) −1.19101e7 −0.886686
\(711\) −1.44925e6 −0.107515
\(712\) 385536. 0.0285013
\(713\) 8.92080e6 0.657173
\(714\) −980784. −0.0719992
\(715\) 3.76428e7 2.75370
\(716\) 2.92618e6 0.213313
\(717\) 246078. 0.0178762
\(718\) 1.62581e7 1.17695
\(719\) −6.72134e6 −0.484880 −0.242440 0.970166i \(-0.577948\pi\)
−0.242440 + 0.970166i \(0.577948\pi\)
\(720\) 1.57594e6 0.113294
\(721\) 1.54605e6 0.110760
\(722\) 1.41448e7 1.00984
\(723\) 8.16943e6 0.581227
\(724\) 2.21328e6 0.156924
\(725\) −1.78677e6 −0.126248
\(726\) −9.41216e6 −0.662747
\(727\) −1.24076e7 −0.870670 −0.435335 0.900269i \(-0.643370\pi\)
−0.435335 + 0.900269i \(0.643370\pi\)
\(728\) −2.38963e6 −0.167110
\(729\) 531441. 0.0370370
\(730\) −7.16528e6 −0.497652
\(731\) −1.21408e7 −0.840339
\(732\) −21600.0 −0.00148996
\(733\) 1.35958e7 0.934641 0.467321 0.884088i \(-0.345219\pi\)
0.467321 + 0.884088i \(0.345219\pi\)
\(734\) −6.68973e6 −0.458319
\(735\) −1.64228e6 −0.112132
\(736\) −3.02080e6 −0.205555
\(737\) −2.84596e7 −1.93001
\(738\) −5.51318e6 −0.372616
\(739\) 2.56819e6 0.172988 0.0864941 0.996252i \(-0.472434\pi\)
0.0864941 + 0.996252i \(0.472434\pi\)
\(740\) 9.39968e6 0.631006
\(741\) 1.68158e7 1.12505
\(742\) −3.05642e6 −0.203800
\(743\) −2.02133e7 −1.34327 −0.671637 0.740880i \(-0.734409\pi\)
−0.671637 + 0.740880i \(0.734409\pi\)
\(744\) 1.74182e6 0.115364
\(745\) −2.22938e6 −0.147161
\(746\) 1.26707e7 0.833595
\(747\) 3.15673e6 0.206984
\(748\) −5.78240e6 −0.377880
\(749\) −5.30582e6 −0.345579
\(750\) 1.29686e6 0.0841863
\(751\) 7.04813e6 0.456010 0.228005 0.973660i \(-0.426780\pi\)
0.228005 + 0.973660i \(0.426780\pi\)
\(752\) −6.12864e6 −0.395202
\(753\) −396792. −0.0255021
\(754\) −2.05435e6 −0.131597
\(755\) 5.44221e6 0.347462
\(756\) 571536. 0.0363696
\(757\) −2.04120e7 −1.29463 −0.647315 0.762223i \(-0.724108\pi\)
−0.647315 + 0.762223i \(0.724108\pi\)
\(758\) −1.68155e7 −1.06301
\(759\) 1.72575e7 1.08736
\(760\) −1.19265e7 −0.748997
\(761\) −5.07974e6 −0.317965 −0.158983 0.987281i \(-0.550821\pi\)
−0.158983 + 0.987281i \(0.550821\pi\)
\(762\) 8.62747e6 0.538265
\(763\) 3.53515e6 0.219835
\(764\) 5.23555e6 0.324511
\(765\) −3.42274e6 −0.211456
\(766\) −1.37046e6 −0.0843909
\(767\) 4.27330e6 0.262286
\(768\) −589824. −0.0360844
\(769\) 2.33898e7 1.42630 0.713149 0.701012i \(-0.247268\pi\)
0.713149 + 0.701012i \(0.247268\pi\)
\(770\) −9.68240e6 −0.588513
\(771\) 7.46280e6 0.452133
\(772\) 1.25904e7 0.760322
\(773\) −1.11253e6 −0.0669672 −0.0334836 0.999439i \(-0.510660\pi\)
−0.0334836 + 0.999439i \(0.510660\pi\)
\(774\) 7.07486e6 0.424488
\(775\) −8.01662e6 −0.479443
\(776\) 6.93952e6 0.413690
\(777\) 3.40893e6 0.202566
\(778\) −1.53584e7 −0.909696
\(779\) 4.17232e7 2.46340
\(780\) −8.33933e6 −0.490789
\(781\) −2.54657e7 −1.49392
\(782\) 6.56080e6 0.383654
\(783\) 491346. 0.0286407
\(784\) 614656. 0.0357143
\(785\) 2.25202e7 1.30436
\(786\) 9.87019e6 0.569861
\(787\) 2.00812e6 0.115572 0.0577859 0.998329i \(-0.481596\pi\)
0.0577859 + 0.998329i \(0.481596\pi\)
\(788\) 6.76957e6 0.388370
\(789\) −1.18752e7 −0.679123
\(790\) −5.43917e6 −0.310074
\(791\) −1.08244e7 −0.615124
\(792\) 3.36960e6 0.190882
\(793\) 114300. 0.00645451
\(794\) 1.37558e7 0.774343
\(795\) −1.06663e7 −0.598544
\(796\) 1.63827e7 0.916439
\(797\) 3.00897e7 1.67792 0.838961 0.544191i \(-0.183163\pi\)
0.838961 + 0.544191i \(0.183163\pi\)
\(798\) −4.32533e6 −0.240443
\(799\) 1.33106e7 0.737619
\(800\) 2.71462e6 0.149963
\(801\) 487944. 0.0268713
\(802\) −1.55768e7 −0.855152
\(803\) −1.53205e7 −0.838463
\(804\) 6.30490e6 0.343984
\(805\) 1.09858e7 0.597506
\(806\) −9.21715e6 −0.499757
\(807\) 7.05409e6 0.381292
\(808\) −4.50714e6 −0.242869
\(809\) −1.88207e6 −0.101103 −0.0505515 0.998721i \(-0.516098\pi\)
−0.0505515 + 0.998721i \(0.516098\pi\)
\(810\) 1.99454e6 0.106815
\(811\) 4.88220e6 0.260654 0.130327 0.991471i \(-0.458397\pi\)
0.130327 + 0.991471i \(0.458397\pi\)
\(812\) 528416. 0.0281246
\(813\) −8.59572e6 −0.456096
\(814\) 2.00980e7 1.06314
\(815\) −3.65104e7 −1.92541
\(816\) 1.28102e6 0.0673491
\(817\) −5.35419e7 −2.80633
\(818\) −6.58718e6 −0.344204
\(819\) −3.02438e6 −0.157553
\(820\) −2.06915e7 −1.07462
\(821\) 8.37096e6 0.433429 0.216714 0.976235i \(-0.430466\pi\)
0.216714 + 0.976235i \(0.430466\pi\)
\(822\) 1.40815e7 0.726893
\(823\) −2.02090e7 −1.04003 −0.520015 0.854157i \(-0.674074\pi\)
−0.520015 + 0.854157i \(0.674074\pi\)
\(824\) −2.01933e6 −0.103607
\(825\) −1.55084e7 −0.793288
\(826\) −1.09917e6 −0.0560549
\(827\) −1.31059e7 −0.666352 −0.333176 0.942865i \(-0.608120\pi\)
−0.333176 + 0.942865i \(0.608120\pi\)
\(828\) −3.82320e6 −0.193799
\(829\) 3.18667e7 1.61046 0.805232 0.592960i \(-0.202041\pi\)
0.805232 + 0.592960i \(0.202041\pi\)
\(830\) 1.18475e7 0.596941
\(831\) −1.72146e7 −0.864756
\(832\) 3.12115e6 0.156317
\(833\) −1.33496e6 −0.0666583
\(834\) −1.22171e7 −0.608209
\(835\) 1.21737e7 0.604235
\(836\) −2.55008e7 −1.26194
\(837\) 2.20450e6 0.108767
\(838\) −6.70637e6 −0.329896
\(839\) 9.94742e6 0.487872 0.243936 0.969791i \(-0.421561\pi\)
0.243936 + 0.969791i \(0.421561\pi\)
\(840\) 2.14502e6 0.104890
\(841\) −2.00569e7 −0.977852
\(842\) −2.26697e6 −0.110196
\(843\) 9.23582e6 0.447617
\(844\) 7.38426e6 0.356821
\(845\) 1.59107e7 0.766561
\(846\) −7.75656e6 −0.372600
\(847\) −1.28110e7 −0.613585
\(848\) 3.99206e6 0.190637
\(849\) −1.57201e7 −0.748489
\(850\) −5.89582e6 −0.279896
\(851\) −2.28035e7 −1.07939
\(852\) 5.64163e6 0.266260
\(853\) −6.52611e6 −0.307102 −0.153551 0.988141i \(-0.549071\pi\)
−0.153551 + 0.988141i \(0.549071\pi\)
\(854\) −29400.0 −0.00137944
\(855\) −1.50945e7 −0.706161
\(856\) 6.93005e6 0.323260
\(857\) −8.76238e6 −0.407540 −0.203770 0.979019i \(-0.565319\pi\)
−0.203770 + 0.979019i \(0.565319\pi\)
\(858\) −1.78308e7 −0.826899
\(859\) 6.47942e6 0.299608 0.149804 0.988716i \(-0.452136\pi\)
0.149804 + 0.988716i \(0.452136\pi\)
\(860\) 2.65526e7 1.22422
\(861\) −7.50406e6 −0.344975
\(862\) 2.67387e7 1.22567
\(863\) −1.83417e7 −0.838323 −0.419162 0.907912i \(-0.637676\pi\)
−0.419162 + 0.907912i \(0.637676\pi\)
\(864\) −746496. −0.0340207
\(865\) −682784. −0.0310272
\(866\) 2.76535e7 1.25301
\(867\) 9.99649e6 0.451648
\(868\) 2.37082e6 0.106807
\(869\) −1.16298e7 −0.522424
\(870\) 1.84406e6 0.0825996
\(871\) −3.33634e7 −1.49013
\(872\) −4.61734e6 −0.205637
\(873\) 8.78283e6 0.390031
\(874\) 2.89336e7 1.28122
\(875\) 1.76518e6 0.0779413
\(876\) 3.39408e6 0.149438
\(877\) 2.69065e7 1.18129 0.590647 0.806930i \(-0.298873\pi\)
0.590647 + 0.806930i \(0.298873\pi\)
\(878\) −1.82512e7 −0.799017
\(879\) −2.00891e7 −0.876976
\(880\) 1.26464e7 0.550504
\(881\) −1.52174e7 −0.660542 −0.330271 0.943886i \(-0.607140\pi\)
−0.330271 + 0.943886i \(0.607140\pi\)
\(882\) 777924. 0.0336718
\(883\) −2.61520e7 −1.12877 −0.564383 0.825513i \(-0.690886\pi\)
−0.564383 + 0.825513i \(0.690886\pi\)
\(884\) −6.77875e6 −0.291756
\(885\) −3.83587e6 −0.164629
\(886\) 1.83904e7 0.787058
\(887\) −1.08021e7 −0.460997 −0.230499 0.973073i \(-0.574036\pi\)
−0.230499 + 0.973073i \(0.574036\pi\)
\(888\) −4.45248e6 −0.189483
\(889\) 1.17429e7 0.498337
\(890\) 1.83130e6 0.0774968
\(891\) 4.26465e6 0.179965
\(892\) 1.59208e7 0.669965
\(893\) 5.87009e7 2.46329
\(894\) 1.05602e6 0.0441906
\(895\) 1.38993e7 0.580011
\(896\) −802816. −0.0334077
\(897\) 2.02311e7 0.839534
\(898\) 6.82631e6 0.282485
\(899\) 2.03818e6 0.0841090
\(900\) 3.43570e6 0.141387
\(901\) −8.67026e6 −0.355812
\(902\) −4.42416e7 −1.81057
\(903\) 9.62968e6 0.393000
\(904\) 1.41380e7 0.575395
\(905\) 1.05131e7 0.426686
\(906\) −2.57789e6 −0.104338
\(907\) −9.84167e6 −0.397238 −0.198619 0.980077i \(-0.563646\pi\)
−0.198619 + 0.980077i \(0.563646\pi\)
\(908\) −1.52909e6 −0.0615486
\(909\) −5.70434e6 −0.228979
\(910\) −1.13508e7 −0.454382
\(911\) 2.72509e7 1.08789 0.543945 0.839121i \(-0.316930\pi\)
0.543945 + 0.839121i \(0.316930\pi\)
\(912\) 5.64941e6 0.224914
\(913\) 2.53318e7 1.00575
\(914\) −2.77566e7 −1.09901
\(915\) −102600. −0.00405130
\(916\) −1.67054e7 −0.657836
\(917\) 1.34344e7 0.527589
\(918\) 1.62130e6 0.0634974
\(919\) 2.86432e7 1.11875 0.559374 0.828916i \(-0.311042\pi\)
0.559374 + 0.828916i \(0.311042\pi\)
\(920\) −1.43488e7 −0.558915
\(921\) −1.66791e7 −0.647924
\(922\) −1.04722e7 −0.405705
\(923\) −2.98536e7 −1.15343
\(924\) 4.58640e6 0.176723
\(925\) 2.04922e7 0.787472
\(926\) 2.85441e7 1.09393
\(927\) −2.55571e6 −0.0976816
\(928\) −690176. −0.0263081
\(929\) 6.78492e6 0.257932 0.128966 0.991649i \(-0.458834\pi\)
0.128966 + 0.991649i \(0.458834\pi\)
\(930\) 8.27366e6 0.313683
\(931\) −5.88725e6 −0.222607
\(932\) −1.87106e7 −0.705581
\(933\) 4.05860e6 0.152641
\(934\) −8.69590e6 −0.326173
\(935\) −2.74664e7 −1.02748
\(936\) 3.95021e6 0.147377
\(937\) 3.00308e7 1.11742 0.558712 0.829362i \(-0.311296\pi\)
0.558712 + 0.829362i \(0.311296\pi\)
\(938\) 8.58166e6 0.318467
\(939\) −1.44237e7 −0.533842
\(940\) −2.91110e7 −1.07458
\(941\) 2.30725e7 0.849415 0.424707 0.905331i \(-0.360377\pi\)
0.424707 + 0.905331i \(0.360377\pi\)
\(942\) −1.06674e7 −0.391682
\(943\) 5.01972e7 1.83823
\(944\) 1.43565e6 0.0524346
\(945\) 2.71480e6 0.0988912
\(946\) 5.67736e7 2.06262
\(947\) 2.71433e7 0.983531 0.491765 0.870728i \(-0.336352\pi\)
0.491765 + 0.870728i \(0.336352\pi\)
\(948\) 2.57645e6 0.0931109
\(949\) −1.79603e7 −0.647365
\(950\) −2.60010e7 −0.934719
\(951\) 187758. 0.00673205
\(952\) 1.74362e6 0.0623532
\(953\) −1.61552e7 −0.576209 −0.288104 0.957599i \(-0.593025\pi\)
−0.288104 + 0.957599i \(0.593025\pi\)
\(954\) 5.05246e6 0.179735
\(955\) 2.48689e7 0.882364
\(956\) −437472. −0.0154812
\(957\) 3.94290e6 0.139167
\(958\) −1.85318e7 −0.652383
\(959\) 1.91665e7 0.672973
\(960\) −2.80166e6 −0.0981156
\(961\) −1.94846e7 −0.680585
\(962\) 2.35610e7 0.820837
\(963\) 8.77084e6 0.304772
\(964\) −1.45234e7 −0.503357
\(965\) 5.98046e7 2.06736
\(966\) −5.20380e6 −0.179423
\(967\) −3.80323e7 −1.30793 −0.653967 0.756523i \(-0.726897\pi\)
−0.653967 + 0.756523i \(0.726897\pi\)
\(968\) 1.67327e7 0.573956
\(969\) −1.22698e7 −0.419786
\(970\) 3.29627e7 1.12485
\(971\) −2.23104e7 −0.759379 −0.379689 0.925114i \(-0.623969\pi\)
−0.379689 + 0.925114i \(0.623969\pi\)
\(972\) −944784. −0.0320750
\(973\) −1.66288e7 −0.563093
\(974\) −1.82658e7 −0.616937
\(975\) −1.81806e7 −0.612485
\(976\) 38400.0 0.00129035
\(977\) 3.06930e7 1.02873 0.514367 0.857570i \(-0.328027\pi\)
0.514367 + 0.857570i \(0.328027\pi\)
\(978\) 1.72944e7 0.578174
\(979\) 3.91560e6 0.130569
\(980\) 2.91962e6 0.0971092
\(981\) −5.84383e6 −0.193876
\(982\) −2.12572e7 −0.703439
\(983\) −1.52706e7 −0.504048 −0.252024 0.967721i \(-0.581096\pi\)
−0.252024 + 0.967721i \(0.581096\pi\)
\(984\) 9.80122e6 0.322695
\(985\) 3.21554e7 1.05600
\(986\) 1.49898e6 0.0491024
\(987\) −1.05575e7 −0.344961
\(988\) −2.98948e7 −0.974323
\(989\) −6.44162e7 −2.09413
\(990\) 1.60056e7 0.519020
\(991\) 3.16279e7 1.02303 0.511513 0.859276i \(-0.329085\pi\)
0.511513 + 0.859276i \(0.329085\pi\)
\(992\) −3.09658e6 −0.0999085
\(993\) −1.86859e7 −0.601369
\(994\) 7.67889e6 0.246509
\(995\) 7.78179e7 2.49185
\(996\) −5.61197e6 −0.179253
\(997\) −3.55842e7 −1.13376 −0.566878 0.823802i \(-0.691849\pi\)
−0.566878 + 0.823802i \(0.691849\pi\)
\(998\) −9.85256e6 −0.313129
\(999\) −5.63517e6 −0.178646
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 42.6.a.e.1.1 1
3.2 odd 2 126.6.a.a.1.1 1
4.3 odd 2 336.6.a.q.1.1 1
5.2 odd 4 1050.6.g.h.799.2 2
5.3 odd 4 1050.6.g.h.799.1 2
5.4 even 2 1050.6.a.f.1.1 1
7.2 even 3 294.6.e.d.67.1 2
7.3 odd 6 294.6.e.c.79.1 2
7.4 even 3 294.6.e.d.79.1 2
7.5 odd 6 294.6.e.c.67.1 2
7.6 odd 2 294.6.a.k.1.1 1
12.11 even 2 1008.6.a.d.1.1 1
21.20 even 2 882.6.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.e.1.1 1 1.1 even 1 trivial
126.6.a.a.1.1 1 3.2 odd 2
294.6.a.k.1.1 1 7.6 odd 2
294.6.e.c.67.1 2 7.5 odd 6
294.6.e.c.79.1 2 7.3 odd 6
294.6.e.d.67.1 2 7.2 even 3
294.6.e.d.79.1 2 7.4 even 3
336.6.a.q.1.1 1 4.3 odd 2
882.6.a.j.1.1 1 21.20 even 2
1008.6.a.d.1.1 1 12.11 even 2
1050.6.a.f.1.1 1 5.4 even 2
1050.6.g.h.799.1 2 5.3 odd 4
1050.6.g.h.799.2 2 5.2 odd 4