Properties

Label 135.6.a.a.1.1
Level $135$
Weight $6$
Character 135.1
Self dual yes
Analytic conductor $21.652$
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] = [135,6,Mod(1,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, 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("135.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(21.6518156748\)
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 \(\chi\) \(=\) 135.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000 q^{2} +68.0000 q^{4} +25.0000 q^{5} +6.00000 q^{7} -360.000 q^{8} +O(q^{10})\) \(q-10.0000 q^{2} +68.0000 q^{4} +25.0000 q^{5} +6.00000 q^{7} -360.000 q^{8} -250.000 q^{10} -685.000 q^{11} +685.000 q^{13} -60.0000 q^{14} +1424.00 q^{16} -1045.00 q^{17} +1108.00 q^{19} +1700.00 q^{20} +6850.00 q^{22} +3855.00 q^{23} +625.000 q^{25} -6850.00 q^{26} +408.000 q^{28} +1315.00 q^{29} -9909.00 q^{31} -2720.00 q^{32} +10450.0 q^{34} +150.000 q^{35} -6826.00 q^{37} -11080.0 q^{38} -9000.00 q^{40} -4520.00 q^{41} +9097.00 q^{43} -46580.0 q^{44} -38550.0 q^{46} -2095.00 q^{47} -16771.0 q^{49} -6250.00 q^{50} +46580.0 q^{52} -10060.0 q^{53} -17125.0 q^{55} -2160.00 q^{56} -13150.0 q^{58} -24820.0 q^{59} -46286.0 q^{61} +99090.0 q^{62} -18368.0 q^{64} +17125.0 q^{65} +13860.0 q^{67} -71060.0 q^{68} -1500.00 q^{70} +75580.0 q^{71} -32738.0 q^{73} +68260.0 q^{74} +75344.0 q^{76} -4110.00 q^{77} +74877.0 q^{79} +35600.0 q^{80} +45200.0 q^{82} -93930.0 q^{83} -26125.0 q^{85} -90970.0 q^{86} +246600. q^{88} -123540. q^{89} +4110.00 q^{91} +262140. q^{92} +20950.0 q^{94} +27700.0 q^{95} -85966.0 q^{97} +167710. q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 68.0000 2.12500
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 6.00000 0.0462814 0.0231407 0.999732i \(-0.492633\pi\)
0.0231407 + 0.999732i \(0.492633\pi\)
\(8\) −360.000 −1.98874
\(9\) 0 0
\(10\) −250.000 −0.790569
\(11\) −685.000 −1.70690 −0.853452 0.521172i \(-0.825495\pi\)
−0.853452 + 0.521172i \(0.825495\pi\)
\(12\) 0 0
\(13\) 685.000 1.12417 0.562085 0.827079i \(-0.309999\pi\)
0.562085 + 0.827079i \(0.309999\pi\)
\(14\) −60.0000 −0.0818147
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) −1045.00 −0.876989 −0.438494 0.898734i \(-0.644488\pi\)
−0.438494 + 0.898734i \(0.644488\pi\)
\(18\) 0 0
\(19\) 1108.00 0.704135 0.352067 0.935975i \(-0.385479\pi\)
0.352067 + 0.935975i \(0.385479\pi\)
\(20\) 1700.00 0.950329
\(21\) 0 0
\(22\) 6850.00 3.01741
\(23\) 3855.00 1.51951 0.759757 0.650207i \(-0.225318\pi\)
0.759757 + 0.650207i \(0.225318\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −6850.00 −1.98727
\(27\) 0 0
\(28\) 408.000 0.0983479
\(29\) 1315.00 0.290356 0.145178 0.989406i \(-0.453625\pi\)
0.145178 + 0.989406i \(0.453625\pi\)
\(30\) 0 0
\(31\) −9909.00 −1.85193 −0.925967 0.377604i \(-0.876748\pi\)
−0.925967 + 0.377604i \(0.876748\pi\)
\(32\) −2720.00 −0.469563
\(33\) 0 0
\(34\) 10450.0 1.55031
\(35\) 150.000 0.0206977
\(36\) 0 0
\(37\) −6826.00 −0.819713 −0.409857 0.912150i \(-0.634421\pi\)
−0.409857 + 0.912150i \(0.634421\pi\)
\(38\) −11080.0 −1.24475
\(39\) 0 0
\(40\) −9000.00 −0.889391
\(41\) −4520.00 −0.419932 −0.209966 0.977709i \(-0.567335\pi\)
−0.209966 + 0.977709i \(0.567335\pi\)
\(42\) 0 0
\(43\) 9097.00 0.750286 0.375143 0.926967i \(-0.377594\pi\)
0.375143 + 0.926967i \(0.377594\pi\)
\(44\) −46580.0 −3.62717
\(45\) 0 0
\(46\) −38550.0 −2.68615
\(47\) −2095.00 −0.138337 −0.0691687 0.997605i \(-0.522035\pi\)
−0.0691687 + 0.997605i \(0.522035\pi\)
\(48\) 0 0
\(49\) −16771.0 −0.997858
\(50\) −6250.00 −0.353553
\(51\) 0 0
\(52\) 46580.0 2.38886
\(53\) −10060.0 −0.491936 −0.245968 0.969278i \(-0.579106\pi\)
−0.245968 + 0.969278i \(0.579106\pi\)
\(54\) 0 0
\(55\) −17125.0 −0.763350
\(56\) −2160.00 −0.0920415
\(57\) 0 0
\(58\) −13150.0 −0.513282
\(59\) −24820.0 −0.928265 −0.464132 0.885766i \(-0.653634\pi\)
−0.464132 + 0.885766i \(0.653634\pi\)
\(60\) 0 0
\(61\) −46286.0 −1.59267 −0.796334 0.604858i \(-0.793230\pi\)
−0.796334 + 0.604858i \(0.793230\pi\)
\(62\) 99090.0 3.27379
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 17125.0 0.502744
\(66\) 0 0
\(67\) 13860.0 0.377204 0.188602 0.982054i \(-0.439604\pi\)
0.188602 + 0.982054i \(0.439604\pi\)
\(68\) −71060.0 −1.86360
\(69\) 0 0
\(70\) −1500.00 −0.0365886
\(71\) 75580.0 1.77935 0.889674 0.456596i \(-0.150931\pi\)
0.889674 + 0.456596i \(0.150931\pi\)
\(72\) 0 0
\(73\) −32738.0 −0.719027 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(74\) 68260.0 1.44906
\(75\) 0 0
\(76\) 75344.0 1.49629
\(77\) −4110.00 −0.0789978
\(78\) 0 0
\(79\) 74877.0 1.34984 0.674918 0.737893i \(-0.264179\pi\)
0.674918 + 0.737893i \(0.264179\pi\)
\(80\) 35600.0 0.621906
\(81\) 0 0
\(82\) 45200.0 0.742342
\(83\) −93930.0 −1.49661 −0.748306 0.663354i \(-0.769132\pi\)
−0.748306 + 0.663354i \(0.769132\pi\)
\(84\) 0 0
\(85\) −26125.0 −0.392201
\(86\) −90970.0 −1.32633
\(87\) 0 0
\(88\) 246600. 3.39458
\(89\) −123540. −1.65323 −0.826614 0.562770i \(-0.809736\pi\)
−0.826614 + 0.562770i \(0.809736\pi\)
\(90\) 0 0
\(91\) 4110.00 0.0520281
\(92\) 262140. 3.22897
\(93\) 0 0
\(94\) 20950.0 0.244548
\(95\) 27700.0 0.314899
\(96\) 0 0
\(97\) −85966.0 −0.927678 −0.463839 0.885919i \(-0.653528\pi\)
−0.463839 + 0.885919i \(0.653528\pi\)
\(98\) 167710. 1.76398
\(99\) 0 0
\(100\) 42500.0 0.425000
\(101\) −62415.0 −0.608815 −0.304408 0.952542i \(-0.598458\pi\)
−0.304408 + 0.952542i \(0.598458\pi\)
\(102\) 0 0
\(103\) −169522. −1.57446 −0.787232 0.616656i \(-0.788487\pi\)
−0.787232 + 0.616656i \(0.788487\pi\)
\(104\) −246600. −2.23568
\(105\) 0 0
\(106\) 100600. 0.869628
\(107\) −2850.00 −0.0240650 −0.0120325 0.999928i \(-0.503830\pi\)
−0.0120325 + 0.999928i \(0.503830\pi\)
\(108\) 0 0
\(109\) −210628. −1.69805 −0.849024 0.528355i \(-0.822809\pi\)
−0.849024 + 0.528355i \(0.822809\pi\)
\(110\) 171250. 1.34943
\(111\) 0 0
\(112\) 8544.00 0.0643600
\(113\) 81335.0 0.599213 0.299607 0.954063i \(-0.403145\pi\)
0.299607 + 0.954063i \(0.403145\pi\)
\(114\) 0 0
\(115\) 96375.0 0.679547
\(116\) 89420.0 0.617006
\(117\) 0 0
\(118\) 248200. 1.64096
\(119\) −6270.00 −0.0405882
\(120\) 0 0
\(121\) 308174. 1.91352
\(122\) 462860. 2.81546
\(123\) 0 0
\(124\) −673812. −3.93536
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −237410. −1.30614 −0.653070 0.757298i \(-0.726519\pi\)
−0.653070 + 0.757298i \(0.726519\pi\)
\(128\) 270720. 1.46048
\(129\) 0 0
\(130\) −171250. −0.888735
\(131\) −158115. −0.804998 −0.402499 0.915420i \(-0.631858\pi\)
−0.402499 + 0.915420i \(0.631858\pi\)
\(132\) 0 0
\(133\) 6648.00 0.0325883
\(134\) −138600. −0.666809
\(135\) 0 0
\(136\) 376200. 1.74410
\(137\) 195270. 0.888862 0.444431 0.895813i \(-0.353406\pi\)
0.444431 + 0.895813i \(0.353406\pi\)
\(138\) 0 0
\(139\) 13238.0 0.0581146 0.0290573 0.999578i \(-0.490749\pi\)
0.0290573 + 0.999578i \(0.490749\pi\)
\(140\) 10200.0 0.0439825
\(141\) 0 0
\(142\) −755800. −3.14547
\(143\) −469225. −1.91885
\(144\) 0 0
\(145\) 32875.0 0.129851
\(146\) 327380. 1.27107
\(147\) 0 0
\(148\) −464168. −1.74189
\(149\) −242405. −0.894491 −0.447245 0.894411i \(-0.647595\pi\)
−0.447245 + 0.894411i \(0.647595\pi\)
\(150\) 0 0
\(151\) −446459. −1.59345 −0.796726 0.604340i \(-0.793437\pi\)
−0.796726 + 0.604340i \(0.793437\pi\)
\(152\) −398880. −1.40034
\(153\) 0 0
\(154\) 41100.0 0.139650
\(155\) −247725. −0.828210
\(156\) 0 0
\(157\) 332411. 1.07628 0.538141 0.842855i \(-0.319127\pi\)
0.538141 + 0.842855i \(0.319127\pi\)
\(158\) −748770. −2.38619
\(159\) 0 0
\(160\) −68000.0 −0.209995
\(161\) 23130.0 0.0703252
\(162\) 0 0
\(163\) 442925. 1.30575 0.652877 0.757464i \(-0.273562\pi\)
0.652877 + 0.757464i \(0.273562\pi\)
\(164\) −307360. −0.892355
\(165\) 0 0
\(166\) 939300. 2.64566
\(167\) 7140.00 0.0198110 0.00990551 0.999951i \(-0.496847\pi\)
0.00990551 + 0.999951i \(0.496847\pi\)
\(168\) 0 0
\(169\) 97932.0 0.263759
\(170\) 261250. 0.693320
\(171\) 0 0
\(172\) 618596. 1.59436
\(173\) −292650. −0.743418 −0.371709 0.928349i \(-0.621228\pi\)
−0.371709 + 0.928349i \(0.621228\pi\)
\(174\) 0 0
\(175\) 3750.00 0.00925627
\(176\) −975440. −2.37366
\(177\) 0 0
\(178\) 1.23540e6 2.92252
\(179\) 609980. 1.42293 0.711464 0.702722i \(-0.248032\pi\)
0.711464 + 0.702722i \(0.248032\pi\)
\(180\) 0 0
\(181\) −79852.0 −0.181171 −0.0905856 0.995889i \(-0.528874\pi\)
−0.0905856 + 0.995889i \(0.528874\pi\)
\(182\) −41100.0 −0.0919736
\(183\) 0 0
\(184\) −1.38780e6 −3.02192
\(185\) −170650. −0.366587
\(186\) 0 0
\(187\) 715825. 1.49693
\(188\) −142460. −0.293967
\(189\) 0 0
\(190\) −277000. −0.556667
\(191\) 150910. 0.299319 0.149660 0.988738i \(-0.452182\pi\)
0.149660 + 0.988738i \(0.452182\pi\)
\(192\) 0 0
\(193\) 170702. 0.329872 0.164936 0.986304i \(-0.447258\pi\)
0.164936 + 0.986304i \(0.447258\pi\)
\(194\) 859660. 1.63992
\(195\) 0 0
\(196\) −1.14043e6 −2.12045
\(197\) −333360. −0.611995 −0.305998 0.952032i \(-0.598990\pi\)
−0.305998 + 0.952032i \(0.598990\pi\)
\(198\) 0 0
\(199\) 57511.0 0.102948 0.0514740 0.998674i \(-0.483608\pi\)
0.0514740 + 0.998674i \(0.483608\pi\)
\(200\) −225000. −0.397748
\(201\) 0 0
\(202\) 624150. 1.07624
\(203\) 7890.00 0.0134381
\(204\) 0 0
\(205\) −113000. −0.187799
\(206\) 1.69522e6 2.78329
\(207\) 0 0
\(208\) 975440. 1.56330
\(209\) −758980. −1.20189
\(210\) 0 0
\(211\) 428102. 0.661974 0.330987 0.943635i \(-0.392618\pi\)
0.330987 + 0.943635i \(0.392618\pi\)
\(212\) −684080. −1.04536
\(213\) 0 0
\(214\) 28500.0 0.0425413
\(215\) 227425. 0.335538
\(216\) 0 0
\(217\) −59454.0 −0.0857100
\(218\) 2.10628e6 3.00175
\(219\) 0 0
\(220\) −1.16450e6 −1.62212
\(221\) −715825. −0.985885
\(222\) 0 0
\(223\) 699368. 0.941767 0.470884 0.882195i \(-0.343935\pi\)
0.470884 + 0.882195i \(0.343935\pi\)
\(224\) −16320.0 −0.0217320
\(225\) 0 0
\(226\) −813350. −1.05927
\(227\) −471200. −0.606933 −0.303466 0.952842i \(-0.598144\pi\)
−0.303466 + 0.952842i \(0.598144\pi\)
\(228\) 0 0
\(229\) 1.44071e6 1.81546 0.907731 0.419552i \(-0.137813\pi\)
0.907731 + 0.419552i \(0.137813\pi\)
\(230\) −963750. −1.20128
\(231\) 0 0
\(232\) −473400. −0.577442
\(233\) 294090. 0.354887 0.177444 0.984131i \(-0.443217\pi\)
0.177444 + 0.984131i \(0.443217\pi\)
\(234\) 0 0
\(235\) −52375.0 −0.0618663
\(236\) −1.68776e6 −1.97256
\(237\) 0 0
\(238\) 62700.0 0.0717505
\(239\) 833330. 0.943675 0.471837 0.881686i \(-0.343591\pi\)
0.471837 + 0.881686i \(0.343591\pi\)
\(240\) 0 0
\(241\) 436477. 0.484082 0.242041 0.970266i \(-0.422183\pi\)
0.242041 + 0.970266i \(0.422183\pi\)
\(242\) −3.08174e6 −3.38265
\(243\) 0 0
\(244\) −3.14745e6 −3.38442
\(245\) −419275. −0.446256
\(246\) 0 0
\(247\) 758980. 0.791567
\(248\) 3.56724e6 3.68301
\(249\) 0 0
\(250\) −156250. −0.158114
\(251\) 676695. 0.677967 0.338984 0.940792i \(-0.389917\pi\)
0.338984 + 0.940792i \(0.389917\pi\)
\(252\) 0 0
\(253\) −2.64068e6 −2.59366
\(254\) 2.37410e6 2.30895
\(255\) 0 0
\(256\) −2.11942e6 −2.02124
\(257\) 669735. 0.632514 0.316257 0.948674i \(-0.397574\pi\)
0.316257 + 0.948674i \(0.397574\pi\)
\(258\) 0 0
\(259\) −40956.0 −0.0379374
\(260\) 1.16450e6 1.06833
\(261\) 0 0
\(262\) 1.58115e6 1.42305
\(263\) −1.92650e6 −1.71743 −0.858716 0.512451i \(-0.828737\pi\)
−0.858716 + 0.512451i \(0.828737\pi\)
\(264\) 0 0
\(265\) −251500. −0.220000
\(266\) −66480.0 −0.0576085
\(267\) 0 0
\(268\) 942480. 0.801558
\(269\) −1.06518e6 −0.897512 −0.448756 0.893654i \(-0.648133\pi\)
−0.448756 + 0.893654i \(0.648133\pi\)
\(270\) 0 0
\(271\) −1.39806e6 −1.15638 −0.578191 0.815901i \(-0.696241\pi\)
−0.578191 + 0.815901i \(0.696241\pi\)
\(272\) −1.48808e6 −1.21956
\(273\) 0 0
\(274\) −1.95270e6 −1.57130
\(275\) −428125. −0.341381
\(276\) 0 0
\(277\) −91866.0 −0.0719375 −0.0359688 0.999353i \(-0.511452\pi\)
−0.0359688 + 0.999353i \(0.511452\pi\)
\(278\) −132380. −0.102733
\(279\) 0 0
\(280\) −54000.0 −0.0411622
\(281\) 1.81323e6 1.36989 0.684947 0.728593i \(-0.259825\pi\)
0.684947 + 0.728593i \(0.259825\pi\)
\(282\) 0 0
\(283\) 1.24205e6 0.921876 0.460938 0.887432i \(-0.347513\pi\)
0.460938 + 0.887432i \(0.347513\pi\)
\(284\) 5.13944e6 3.78112
\(285\) 0 0
\(286\) 4.69225e6 3.39208
\(287\) −27120.0 −0.0194350
\(288\) 0 0
\(289\) −327832. −0.230891
\(290\) −328750. −0.229547
\(291\) 0 0
\(292\) −2.22618e6 −1.52793
\(293\) −453690. −0.308738 −0.154369 0.988013i \(-0.549334\pi\)
−0.154369 + 0.988013i \(0.549334\pi\)
\(294\) 0 0
\(295\) −620500. −0.415133
\(296\) 2.45736e6 1.63019
\(297\) 0 0
\(298\) 2.42405e6 1.58125
\(299\) 2.64068e6 1.70819
\(300\) 0 0
\(301\) 54582.0 0.0347243
\(302\) 4.46459e6 2.81685
\(303\) 0 0
\(304\) 1.57779e6 0.979187
\(305\) −1.15715e6 −0.712262
\(306\) 0 0
\(307\) −2.37090e6 −1.43571 −0.717856 0.696192i \(-0.754876\pi\)
−0.717856 + 0.696192i \(0.754876\pi\)
\(308\) −279480. −0.167870
\(309\) 0 0
\(310\) 2.47725e6 1.46408
\(311\) 981600. 0.575484 0.287742 0.957708i \(-0.407095\pi\)
0.287742 + 0.957708i \(0.407095\pi\)
\(312\) 0 0
\(313\) 1.16432e6 0.671756 0.335878 0.941906i \(-0.390967\pi\)
0.335878 + 0.941906i \(0.390967\pi\)
\(314\) −3.32411e6 −1.90262
\(315\) 0 0
\(316\) 5.09164e6 2.86840
\(317\) −855800. −0.478326 −0.239163 0.970979i \(-0.576873\pi\)
−0.239163 + 0.970979i \(0.576873\pi\)
\(318\) 0 0
\(319\) −900775. −0.495609
\(320\) −459200. −0.250684
\(321\) 0 0
\(322\) −231300. −0.124319
\(323\) −1.15786e6 −0.617518
\(324\) 0 0
\(325\) 428125. 0.224834
\(326\) −4.42925e6 −2.30827
\(327\) 0 0
\(328\) 1.62720e6 0.835134
\(329\) −12570.0 −0.00640244
\(330\) 0 0
\(331\) 65994.0 0.0331081 0.0165541 0.999863i \(-0.494730\pi\)
0.0165541 + 0.999863i \(0.494730\pi\)
\(332\) −6.38724e6 −3.18030
\(333\) 0 0
\(334\) −71400.0 −0.0350213
\(335\) 346500. 0.168691
\(336\) 0 0
\(337\) −820040. −0.393333 −0.196666 0.980470i \(-0.563012\pi\)
−0.196666 + 0.980470i \(0.563012\pi\)
\(338\) −979320. −0.466265
\(339\) 0 0
\(340\) −1.77650e6 −0.833428
\(341\) 6.78766e6 3.16107
\(342\) 0 0
\(343\) −201468. −0.0924636
\(344\) −3.27492e6 −1.49212
\(345\) 0 0
\(346\) 2.92650e6 1.31419
\(347\) 851960. 0.379835 0.189918 0.981800i \(-0.439178\pi\)
0.189918 + 0.981800i \(0.439178\pi\)
\(348\) 0 0
\(349\) −1.11612e6 −0.490510 −0.245255 0.969459i \(-0.578872\pi\)
−0.245255 + 0.969459i \(0.578872\pi\)
\(350\) −37500.0 −0.0163629
\(351\) 0 0
\(352\) 1.86320e6 0.801499
\(353\) −4.18808e6 −1.78887 −0.894433 0.447202i \(-0.852420\pi\)
−0.894433 + 0.447202i \(0.852420\pi\)
\(354\) 0 0
\(355\) 1.88950e6 0.795749
\(356\) −8.40072e6 −3.51311
\(357\) 0 0
\(358\) −6.09980e6 −2.51541
\(359\) 1.37034e6 0.561167 0.280584 0.959830i \(-0.409472\pi\)
0.280584 + 0.959830i \(0.409472\pi\)
\(360\) 0 0
\(361\) −1.24843e6 −0.504194
\(362\) 798520. 0.320269
\(363\) 0 0
\(364\) 279480. 0.110560
\(365\) −818450. −0.321558
\(366\) 0 0
\(367\) −1.42773e6 −0.553326 −0.276663 0.960967i \(-0.589229\pi\)
−0.276663 + 0.960967i \(0.589229\pi\)
\(368\) 5.48952e6 2.11307
\(369\) 0 0
\(370\) 1.70650e6 0.648040
\(371\) −60360.0 −0.0227675
\(372\) 0 0
\(373\) 821623. 0.305774 0.152887 0.988244i \(-0.451143\pi\)
0.152887 + 0.988244i \(0.451143\pi\)
\(374\) −7.15825e6 −2.64623
\(375\) 0 0
\(376\) 754200. 0.275117
\(377\) 900775. 0.326410
\(378\) 0 0
\(379\) −609536. −0.217972 −0.108986 0.994043i \(-0.534760\pi\)
−0.108986 + 0.994043i \(0.534760\pi\)
\(380\) 1.88360e6 0.669160
\(381\) 0 0
\(382\) −1.50910e6 −0.529127
\(383\) 1.08650e6 0.378469 0.189235 0.981932i \(-0.439399\pi\)
0.189235 + 0.981932i \(0.439399\pi\)
\(384\) 0 0
\(385\) −102750. −0.0353289
\(386\) −1.70702e6 −0.583137
\(387\) 0 0
\(388\) −5.84569e6 −1.97132
\(389\) −2.26594e6 −0.759233 −0.379617 0.925144i \(-0.623944\pi\)
−0.379617 + 0.925144i \(0.623944\pi\)
\(390\) 0 0
\(391\) −4.02847e6 −1.33260
\(392\) 6.03756e6 1.98448
\(393\) 0 0
\(394\) 3.33360e6 1.08186
\(395\) 1.87192e6 0.603665
\(396\) 0 0
\(397\) 3.85253e6 1.22679 0.613394 0.789777i \(-0.289804\pi\)
0.613394 + 0.789777i \(0.289804\pi\)
\(398\) −575110. −0.181988
\(399\) 0 0
\(400\) 890000. 0.278125
\(401\) 293700. 0.0912101 0.0456051 0.998960i \(-0.485478\pi\)
0.0456051 + 0.998960i \(0.485478\pi\)
\(402\) 0 0
\(403\) −6.78766e6 −2.08189
\(404\) −4.24422e6 −1.29373
\(405\) 0 0
\(406\) −78900.0 −0.0237554
\(407\) 4.67581e6 1.39917
\(408\) 0 0
\(409\) −1.54596e6 −0.456973 −0.228486 0.973547i \(-0.573378\pi\)
−0.228486 + 0.973547i \(0.573378\pi\)
\(410\) 1.13000e6 0.331985
\(411\) 0 0
\(412\) −1.15275e7 −3.34574
\(413\) −148920. −0.0429613
\(414\) 0 0
\(415\) −2.34825e6 −0.669305
\(416\) −1.86320e6 −0.527869
\(417\) 0 0
\(418\) 7.58980e6 2.12466
\(419\) −274355. −0.0763445 −0.0381723 0.999271i \(-0.512154\pi\)
−0.0381723 + 0.999271i \(0.512154\pi\)
\(420\) 0 0
\(421\) 122836. 0.0337769 0.0168885 0.999857i \(-0.494624\pi\)
0.0168885 + 0.999857i \(0.494624\pi\)
\(422\) −4.28102e6 −1.17022
\(423\) 0 0
\(424\) 3.62160e6 0.978331
\(425\) −653125. −0.175398
\(426\) 0 0
\(427\) −277716. −0.0737108
\(428\) −193800. −0.0511381
\(429\) 0 0
\(430\) −2.27425e6 −0.593153
\(431\) 3.85980e6 1.00086 0.500428 0.865778i \(-0.333176\pi\)
0.500428 + 0.865778i \(0.333176\pi\)
\(432\) 0 0
\(433\) 2.18754e6 0.560707 0.280353 0.959897i \(-0.409548\pi\)
0.280353 + 0.959897i \(0.409548\pi\)
\(434\) 594540. 0.151515
\(435\) 0 0
\(436\) −1.43227e7 −3.60835
\(437\) 4.27134e6 1.06994
\(438\) 0 0
\(439\) 1.26527e6 0.313345 0.156672 0.987651i \(-0.449923\pi\)
0.156672 + 0.987651i \(0.449923\pi\)
\(440\) 6.16500e6 1.51810
\(441\) 0 0
\(442\) 7.15825e6 1.74281
\(443\) −672180. −0.162733 −0.0813666 0.996684i \(-0.525928\pi\)
−0.0813666 + 0.996684i \(0.525928\pi\)
\(444\) 0 0
\(445\) −3.08850e6 −0.739346
\(446\) −6.99368e6 −1.66483
\(447\) 0 0
\(448\) −110208. −0.0259429
\(449\) −1.26013e6 −0.294985 −0.147492 0.989063i \(-0.547120\pi\)
−0.147492 + 0.989063i \(0.547120\pi\)
\(450\) 0 0
\(451\) 3.09620e6 0.716783
\(452\) 5.53078e6 1.27333
\(453\) 0 0
\(454\) 4.71200e6 1.07292
\(455\) 102750. 0.0232677
\(456\) 0 0
\(457\) 6.81838e6 1.52718 0.763591 0.645700i \(-0.223435\pi\)
0.763591 + 0.645700i \(0.223435\pi\)
\(458\) −1.44071e7 −3.20931
\(459\) 0 0
\(460\) 6.55350e6 1.44404
\(461\) −6.55773e6 −1.43715 −0.718574 0.695451i \(-0.755205\pi\)
−0.718574 + 0.695451i \(0.755205\pi\)
\(462\) 0 0
\(463\) 3.15148e6 0.683223 0.341611 0.939841i \(-0.389027\pi\)
0.341611 + 0.939841i \(0.389027\pi\)
\(464\) 1.87256e6 0.403776
\(465\) 0 0
\(466\) −2.94090e6 −0.627358
\(467\) 7.80490e6 1.65606 0.828028 0.560686i \(-0.189463\pi\)
0.828028 + 0.560686i \(0.189463\pi\)
\(468\) 0 0
\(469\) 83160.0 0.0174575
\(470\) 523750. 0.109365
\(471\) 0 0
\(472\) 8.93520e6 1.84607
\(473\) −6.23145e6 −1.28067
\(474\) 0 0
\(475\) 692500. 0.140827
\(476\) −426360. −0.0862500
\(477\) 0 0
\(478\) −8.33330e6 −1.66820
\(479\) 729660. 0.145305 0.0726527 0.997357i \(-0.476854\pi\)
0.0726527 + 0.997357i \(0.476854\pi\)
\(480\) 0 0
\(481\) −4.67581e6 −0.921498
\(482\) −4.36477e6 −0.855744
\(483\) 0 0
\(484\) 2.09558e7 4.06623
\(485\) −2.14915e6 −0.414870
\(486\) 0 0
\(487\) 9.35510e6 1.78742 0.893709 0.448647i \(-0.148094\pi\)
0.893709 + 0.448647i \(0.148094\pi\)
\(488\) 1.66630e7 3.16740
\(489\) 0 0
\(490\) 4.19275e6 0.788876
\(491\) 9.39394e6 1.75851 0.879253 0.476354i \(-0.158042\pi\)
0.879253 + 0.476354i \(0.158042\pi\)
\(492\) 0 0
\(493\) −1.37418e6 −0.254639
\(494\) −7.58980e6 −1.39931
\(495\) 0 0
\(496\) −1.41104e7 −2.57535
\(497\) 453480. 0.0823507
\(498\) 0 0
\(499\) 6.54473e6 1.17663 0.588316 0.808631i \(-0.299791\pi\)
0.588316 + 0.808631i \(0.299791\pi\)
\(500\) 1.06250e6 0.190066
\(501\) 0 0
\(502\) −6.76695e6 −1.19849
\(503\) −6.68478e6 −1.17806 −0.589030 0.808111i \(-0.700490\pi\)
−0.589030 + 0.808111i \(0.700490\pi\)
\(504\) 0 0
\(505\) −1.56037e6 −0.272270
\(506\) 2.64068e7 4.58499
\(507\) 0 0
\(508\) −1.61439e7 −2.77555
\(509\) 3.61198e6 0.617946 0.308973 0.951071i \(-0.400015\pi\)
0.308973 + 0.951071i \(0.400015\pi\)
\(510\) 0 0
\(511\) −196428. −0.0332775
\(512\) 1.25312e7 2.11260
\(513\) 0 0
\(514\) −6.69735e6 −1.11814
\(515\) −4.23805e6 −0.704122
\(516\) 0 0
\(517\) 1.43508e6 0.236128
\(518\) 409560. 0.0670646
\(519\) 0 0
\(520\) −6.16500e6 −0.999827
\(521\) 572740. 0.0924407 0.0462203 0.998931i \(-0.485282\pi\)
0.0462203 + 0.998931i \(0.485282\pi\)
\(522\) 0 0
\(523\) 5.40826e6 0.864577 0.432288 0.901735i \(-0.357706\pi\)
0.432288 + 0.901735i \(0.357706\pi\)
\(524\) −1.07518e7 −1.71062
\(525\) 0 0
\(526\) 1.92650e7 3.03602
\(527\) 1.03549e7 1.62413
\(528\) 0 0
\(529\) 8.42468e6 1.30892
\(530\) 2.51500e6 0.388909
\(531\) 0 0
\(532\) 452064. 0.0692502
\(533\) −3.09620e6 −0.472075
\(534\) 0 0
\(535\) −71250.0 −0.0107622
\(536\) −4.98960e6 −0.750160
\(537\) 0 0
\(538\) 1.06518e7 1.58659
\(539\) 1.14881e7 1.70325
\(540\) 0 0
\(541\) −7.26253e6 −1.06683 −0.533414 0.845854i \(-0.679091\pi\)
−0.533414 + 0.845854i \(0.679091\pi\)
\(542\) 1.39806e7 2.04421
\(543\) 0 0
\(544\) 2.84240e6 0.411802
\(545\) −5.26570e6 −0.759390
\(546\) 0 0
\(547\) 30901.0 0.00441575 0.00220787 0.999998i \(-0.499297\pi\)
0.00220787 + 0.999998i \(0.499297\pi\)
\(548\) 1.32784e7 1.88883
\(549\) 0 0
\(550\) 4.28125e6 0.603481
\(551\) 1.45702e6 0.204450
\(552\) 0 0
\(553\) 449262. 0.0624722
\(554\) 918660. 0.127169
\(555\) 0 0
\(556\) 900184. 0.123494
\(557\) −6.29634e6 −0.859904 −0.429952 0.902852i \(-0.641470\pi\)
−0.429952 + 0.902852i \(0.641470\pi\)
\(558\) 0 0
\(559\) 6.23144e6 0.843450
\(560\) 213600. 0.0287827
\(561\) 0 0
\(562\) −1.81323e7 −2.42165
\(563\) −1.23597e6 −0.164338 −0.0821688 0.996618i \(-0.526185\pi\)
−0.0821688 + 0.996618i \(0.526185\pi\)
\(564\) 0 0
\(565\) 2.03338e6 0.267976
\(566\) −1.24205e7 −1.62966
\(567\) 0 0
\(568\) −2.72088e7 −3.53866
\(569\) −6.30000e6 −0.815755 −0.407878 0.913037i \(-0.633731\pi\)
−0.407878 + 0.913037i \(0.633731\pi\)
\(570\) 0 0
\(571\) −2.12369e6 −0.272584 −0.136292 0.990669i \(-0.543519\pi\)
−0.136292 + 0.990669i \(0.543519\pi\)
\(572\) −3.19073e7 −4.07756
\(573\) 0 0
\(574\) 271200. 0.0343566
\(575\) 2.40938e6 0.303903
\(576\) 0 0
\(577\) −7.58210e6 −0.948090 −0.474045 0.880501i \(-0.657207\pi\)
−0.474045 + 0.880501i \(0.657207\pi\)
\(578\) 3.27832e6 0.408161
\(579\) 0 0
\(580\) 2.23550e6 0.275934
\(581\) −563580. −0.0692652
\(582\) 0 0
\(583\) 6.89110e6 0.839686
\(584\) 1.17857e7 1.42996
\(585\) 0 0
\(586\) 4.53690e6 0.545777
\(587\) −1.24694e7 −1.49366 −0.746828 0.665017i \(-0.768424\pi\)
−0.746828 + 0.665017i \(0.768424\pi\)
\(588\) 0 0
\(589\) −1.09792e7 −1.30401
\(590\) 6.20500e6 0.733858
\(591\) 0 0
\(592\) −9.72022e6 −1.13991
\(593\) 2.94156e6 0.343511 0.171755 0.985140i \(-0.445056\pi\)
0.171755 + 0.985140i \(0.445056\pi\)
\(594\) 0 0
\(595\) −156750. −0.0181516
\(596\) −1.64835e7 −1.90079
\(597\) 0 0
\(598\) −2.64067e7 −3.01969
\(599\) −1.58779e7 −1.80812 −0.904058 0.427410i \(-0.859426\pi\)
−0.904058 + 0.427410i \(0.859426\pi\)
\(600\) 0 0
\(601\) −4.59290e6 −0.518681 −0.259341 0.965786i \(-0.583505\pi\)
−0.259341 + 0.965786i \(0.583505\pi\)
\(602\) −545820. −0.0613844
\(603\) 0 0
\(604\) −3.03592e7 −3.38609
\(605\) 7.70435e6 0.855751
\(606\) 0 0
\(607\) 5.59659e6 0.616527 0.308263 0.951301i \(-0.400252\pi\)
0.308263 + 0.951301i \(0.400252\pi\)
\(608\) −3.01376e6 −0.330636
\(609\) 0 0
\(610\) 1.15715e7 1.25911
\(611\) −1.43508e6 −0.155515
\(612\) 0 0
\(613\) 1.01497e7 1.09095 0.545473 0.838128i \(-0.316350\pi\)
0.545473 + 0.838128i \(0.316350\pi\)
\(614\) 2.37090e7 2.53800
\(615\) 0 0
\(616\) 1.47960e6 0.157106
\(617\) −1.53162e7 −1.61971 −0.809854 0.586631i \(-0.800454\pi\)
−0.809854 + 0.586631i \(0.800454\pi\)
\(618\) 0 0
\(619\) −5.20008e6 −0.545486 −0.272743 0.962087i \(-0.587931\pi\)
−0.272743 + 0.962087i \(0.587931\pi\)
\(620\) −1.68453e7 −1.75995
\(621\) 0 0
\(622\) −9.81600e6 −1.01732
\(623\) −741240. −0.0765136
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) −1.16432e7 −1.18751
\(627\) 0 0
\(628\) 2.26039e7 2.28710
\(629\) 7.13317e6 0.718879
\(630\) 0 0
\(631\) 7.34675e6 0.734551 0.367276 0.930112i \(-0.380291\pi\)
0.367276 + 0.930112i \(0.380291\pi\)
\(632\) −2.69557e7 −2.68447
\(633\) 0 0
\(634\) 8.55800e6 0.845569
\(635\) −5.93525e6 −0.584124
\(636\) 0 0
\(637\) −1.14881e7 −1.12176
\(638\) 9.00775e6 0.876122
\(639\) 0 0
\(640\) 6.76800e6 0.653146
\(641\) 1.17297e6 0.112757 0.0563783 0.998409i \(-0.482045\pi\)
0.0563783 + 0.998409i \(0.482045\pi\)
\(642\) 0 0
\(643\) −1.92135e7 −1.83264 −0.916322 0.400443i \(-0.868856\pi\)
−0.916322 + 0.400443i \(0.868856\pi\)
\(644\) 1.57284e6 0.149441
\(645\) 0 0
\(646\) 1.15786e7 1.09163
\(647\) 8.64470e6 0.811875 0.405938 0.913901i \(-0.366945\pi\)
0.405938 + 0.913901i \(0.366945\pi\)
\(648\) 0 0
\(649\) 1.70017e7 1.58446
\(650\) −4.28125e6 −0.397454
\(651\) 0 0
\(652\) 3.01189e7 2.77473
\(653\) −4.50206e6 −0.413170 −0.206585 0.978429i \(-0.566235\pi\)
−0.206585 + 0.978429i \(0.566235\pi\)
\(654\) 0 0
\(655\) −3.95288e6 −0.360006
\(656\) −6.43648e6 −0.583968
\(657\) 0 0
\(658\) 125700. 0.0113180
\(659\) 1.51573e7 1.35959 0.679795 0.733403i \(-0.262069\pi\)
0.679795 + 0.733403i \(0.262069\pi\)
\(660\) 0 0
\(661\) 1.03752e7 0.923618 0.461809 0.886979i \(-0.347201\pi\)
0.461809 + 0.886979i \(0.347201\pi\)
\(662\) −659940. −0.0585274
\(663\) 0 0
\(664\) 3.38148e7 2.97637
\(665\) 166200. 0.0145739
\(666\) 0 0
\(667\) 5.06932e6 0.441200
\(668\) 485520. 0.0420984
\(669\) 0 0
\(670\) −3.46500e6 −0.298206
\(671\) 3.17059e7 2.71853
\(672\) 0 0
\(673\) 1.49638e7 1.27352 0.636758 0.771064i \(-0.280275\pi\)
0.636758 + 0.771064i \(0.280275\pi\)
\(674\) 8.20040e6 0.695321
\(675\) 0 0
\(676\) 6.65938e6 0.560489
\(677\) −8.83758e6 −0.741074 −0.370537 0.928818i \(-0.620826\pi\)
−0.370537 + 0.928818i \(0.620826\pi\)
\(678\) 0 0
\(679\) −515796. −0.0429342
\(680\) 9.40500e6 0.779985
\(681\) 0 0
\(682\) −6.78766e7 −5.58804
\(683\) 1.09012e7 0.894172 0.447086 0.894491i \(-0.352462\pi\)
0.447086 + 0.894491i \(0.352462\pi\)
\(684\) 0 0
\(685\) 4.88175e6 0.397511
\(686\) 2.01468e6 0.163454
\(687\) 0 0
\(688\) 1.29541e7 1.04337
\(689\) −6.89110e6 −0.553020
\(690\) 0 0
\(691\) −9.02310e6 −0.718887 −0.359444 0.933167i \(-0.617033\pi\)
−0.359444 + 0.933167i \(0.617033\pi\)
\(692\) −1.99002e7 −1.57976
\(693\) 0 0
\(694\) −8.51960e6 −0.671461
\(695\) 330950. 0.0259896
\(696\) 0 0
\(697\) 4.72340e6 0.368275
\(698\) 1.11612e7 0.867108
\(699\) 0 0
\(700\) 255000. 0.0196696
\(701\) −2.15578e7 −1.65695 −0.828474 0.560028i \(-0.810790\pi\)
−0.828474 + 0.560028i \(0.810790\pi\)
\(702\) 0 0
\(703\) −7.56321e6 −0.577189
\(704\) 1.25821e7 0.956799
\(705\) 0 0
\(706\) 4.18808e7 3.16230
\(707\) −374490. −0.0281768
\(708\) 0 0
\(709\) −1.69473e7 −1.26615 −0.633076 0.774090i \(-0.718208\pi\)
−0.633076 + 0.774090i \(0.718208\pi\)
\(710\) −1.88950e7 −1.40670
\(711\) 0 0
\(712\) 4.44744e7 3.28784
\(713\) −3.81992e7 −2.81404
\(714\) 0 0
\(715\) −1.17306e7 −0.858136
\(716\) 4.14786e7 3.02372
\(717\) 0 0
\(718\) −1.37034e7 −0.992013
\(719\) −2.05676e7 −1.48375 −0.741874 0.670539i \(-0.766063\pi\)
−0.741874 + 0.670539i \(0.766063\pi\)
\(720\) 0 0
\(721\) −1.01713e6 −0.0728684
\(722\) 1.24844e7 0.891298
\(723\) 0 0
\(724\) −5.42994e6 −0.384989
\(725\) 821875. 0.0580712
\(726\) 0 0
\(727\) 1.66714e7 1.16986 0.584931 0.811083i \(-0.301121\pi\)
0.584931 + 0.811083i \(0.301121\pi\)
\(728\) −1.47960e6 −0.103470
\(729\) 0 0
\(730\) 8.18450e6 0.568440
\(731\) −9.50636e6 −0.657993
\(732\) 0 0
\(733\) −1.32270e7 −0.909284 −0.454642 0.890674i \(-0.650233\pi\)
−0.454642 + 0.890674i \(0.650233\pi\)
\(734\) 1.42773e7 0.978151
\(735\) 0 0
\(736\) −1.04856e7 −0.713508
\(737\) −9.49410e6 −0.643851
\(738\) 0 0
\(739\) −9.00924e6 −0.606844 −0.303422 0.952856i \(-0.598129\pi\)
−0.303422 + 0.952856i \(0.598129\pi\)
\(740\) −1.16042e7 −0.778997
\(741\) 0 0
\(742\) 603600. 0.0402476
\(743\) 2.44168e7 1.62262 0.811309 0.584618i \(-0.198756\pi\)
0.811309 + 0.584618i \(0.198756\pi\)
\(744\) 0 0
\(745\) −6.06012e6 −0.400028
\(746\) −8.21623e6 −0.540537
\(747\) 0 0
\(748\) 4.86761e7 3.18099
\(749\) −17100.0 −0.00111376
\(750\) 0 0
\(751\) −2.35907e7 −1.52630 −0.763151 0.646220i \(-0.776349\pi\)
−0.763151 + 0.646220i \(0.776349\pi\)
\(752\) −2.98328e6 −0.192375
\(753\) 0 0
\(754\) −9.00775e6 −0.577016
\(755\) −1.11615e7 −0.712614
\(756\) 0 0
\(757\) 2.76229e7 1.75198 0.875991 0.482328i \(-0.160209\pi\)
0.875991 + 0.482328i \(0.160209\pi\)
\(758\) 6.09536e6 0.385324
\(759\) 0 0
\(760\) −9.97200e6 −0.626251
\(761\) −1.20776e7 −0.755992 −0.377996 0.925807i \(-0.623387\pi\)
−0.377996 + 0.925807i \(0.623387\pi\)
\(762\) 0 0
\(763\) −1.26377e6 −0.0785880
\(764\) 1.02619e7 0.636053
\(765\) 0 0
\(766\) −1.08650e7 −0.669046
\(767\) −1.70017e7 −1.04353
\(768\) 0 0
\(769\) 3.49831e6 0.213325 0.106663 0.994295i \(-0.465983\pi\)
0.106663 + 0.994295i \(0.465983\pi\)
\(770\) 1.02750e6 0.0624532
\(771\) 0 0
\(772\) 1.16077e7 0.700978
\(773\) −2.99115e6 −0.180048 −0.0900242 0.995940i \(-0.528694\pi\)
−0.0900242 + 0.995940i \(0.528694\pi\)
\(774\) 0 0
\(775\) −6.19312e6 −0.370387
\(776\) 3.09478e7 1.84491
\(777\) 0 0
\(778\) 2.26594e7 1.34215
\(779\) −5.00816e6 −0.295689
\(780\) 0 0
\(781\) −5.17723e7 −3.03718
\(782\) 4.02848e7 2.35572
\(783\) 0 0
\(784\) −2.38819e7 −1.38765
\(785\) 8.31028e6 0.481328
\(786\) 0 0
\(787\) −1.38920e7 −0.799520 −0.399760 0.916620i \(-0.630906\pi\)
−0.399760 + 0.916620i \(0.630906\pi\)
\(788\) −2.26685e7 −1.30049
\(789\) 0 0
\(790\) −1.87192e7 −1.06714
\(791\) 488010. 0.0277324
\(792\) 0 0
\(793\) −3.17059e7 −1.79043
\(794\) −3.85253e7 −2.16868
\(795\) 0 0
\(796\) 3.91075e6 0.218765
\(797\) −8.13974e6 −0.453905 −0.226952 0.973906i \(-0.572876\pi\)
−0.226952 + 0.973906i \(0.572876\pi\)
\(798\) 0 0
\(799\) 2.18928e6 0.121320
\(800\) −1.70000e6 −0.0939126
\(801\) 0 0
\(802\) −2.93700e6 −0.161238
\(803\) 2.24255e7 1.22731
\(804\) 0 0
\(805\) 578250. 0.0314504
\(806\) 6.78766e7 3.68030
\(807\) 0 0
\(808\) 2.24694e7 1.21077
\(809\) 2.68094e7 1.44017 0.720087 0.693883i \(-0.244102\pi\)
0.720087 + 0.693883i \(0.244102\pi\)
\(810\) 0 0
\(811\) −1.97282e7 −1.05326 −0.526629 0.850095i \(-0.676544\pi\)
−0.526629 + 0.850095i \(0.676544\pi\)
\(812\) 536520. 0.0285559
\(813\) 0 0
\(814\) −4.67581e7 −2.47341
\(815\) 1.10731e7 0.583951
\(816\) 0 0
\(817\) 1.00795e7 0.528303
\(818\) 1.54596e7 0.807821
\(819\) 0 0
\(820\) −7.68400e6 −0.399073
\(821\) −2.16090e7 −1.11886 −0.559432 0.828876i \(-0.688981\pi\)
−0.559432 + 0.828876i \(0.688981\pi\)
\(822\) 0 0
\(823\) 6.99026e6 0.359744 0.179872 0.983690i \(-0.442432\pi\)
0.179872 + 0.983690i \(0.442432\pi\)
\(824\) 6.10279e7 3.13120
\(825\) 0 0
\(826\) 1.48920e6 0.0759457
\(827\) 2.68441e7 1.36485 0.682425 0.730955i \(-0.260925\pi\)
0.682425 + 0.730955i \(0.260925\pi\)
\(828\) 0 0
\(829\) 2.11268e7 1.06770 0.533848 0.845580i \(-0.320745\pi\)
0.533848 + 0.845580i \(0.320745\pi\)
\(830\) 2.34825e7 1.18318
\(831\) 0 0
\(832\) −1.25821e7 −0.630150
\(833\) 1.75257e7 0.875110
\(834\) 0 0
\(835\) 178500. 0.00885976
\(836\) −5.16106e7 −2.55402
\(837\) 0 0
\(838\) 2.74355e6 0.134959
\(839\) 2.28583e7 1.12108 0.560542 0.828126i \(-0.310593\pi\)
0.560542 + 0.828126i \(0.310593\pi\)
\(840\) 0 0
\(841\) −1.87819e7 −0.915693
\(842\) −1.22836e6 −0.0597098
\(843\) 0 0
\(844\) 2.91109e7 1.40670
\(845\) 2.44830e6 0.117957
\(846\) 0 0
\(847\) 1.84904e6 0.0885602
\(848\) −1.43254e7 −0.684098
\(849\) 0 0
\(850\) 6.53125e6 0.310062
\(851\) −2.63142e7 −1.24557
\(852\) 0 0
\(853\) 5.75945e6 0.271024 0.135512 0.990776i \(-0.456732\pi\)
0.135512 + 0.990776i \(0.456732\pi\)
\(854\) 2.77716e6 0.130304
\(855\) 0 0
\(856\) 1.02600e6 0.0478589
\(857\) 1.43880e7 0.669188 0.334594 0.942362i \(-0.391401\pi\)
0.334594 + 0.942362i \(0.391401\pi\)
\(858\) 0 0
\(859\) 3.28619e7 1.51953 0.759766 0.650197i \(-0.225314\pi\)
0.759766 + 0.650197i \(0.225314\pi\)
\(860\) 1.54649e7 0.713019
\(861\) 0 0
\(862\) −3.85980e7 −1.76928
\(863\) 4.34132e6 0.198425 0.0992123 0.995066i \(-0.468368\pi\)
0.0992123 + 0.995066i \(0.468368\pi\)
\(864\) 0 0
\(865\) −7.31625e6 −0.332467
\(866\) −2.18754e7 −0.991199
\(867\) 0 0
\(868\) −4.04287e6 −0.182134
\(869\) −5.12907e7 −2.30404
\(870\) 0 0
\(871\) 9.49410e6 0.424042
\(872\) 7.58261e7 3.37697
\(873\) 0 0
\(874\) −4.27134e7 −1.89141
\(875\) 93750.0 0.00413953
\(876\) 0 0
\(877\) 2.70727e7 1.18859 0.594296 0.804246i \(-0.297431\pi\)
0.594296 + 0.804246i \(0.297431\pi\)
\(878\) −1.26527e7 −0.553921
\(879\) 0 0
\(880\) −2.43860e7 −1.06153
\(881\) −4.98787e6 −0.216509 −0.108254 0.994123i \(-0.534526\pi\)
−0.108254 + 0.994123i \(0.534526\pi\)
\(882\) 0 0
\(883\) 1.69054e7 0.729667 0.364833 0.931073i \(-0.381126\pi\)
0.364833 + 0.931073i \(0.381126\pi\)
\(884\) −4.86761e7 −2.09501
\(885\) 0 0
\(886\) 6.72180e6 0.287674
\(887\) −3.06531e7 −1.30817 −0.654087 0.756419i \(-0.726947\pi\)
−0.654087 + 0.756419i \(0.726947\pi\)
\(888\) 0 0
\(889\) −1.42446e6 −0.0604499
\(890\) 3.08850e7 1.30699
\(891\) 0 0
\(892\) 4.75570e7 2.00126
\(893\) −2.32126e6 −0.0974081
\(894\) 0 0
\(895\) 1.52495e7 0.636353
\(896\) 1.62432e6 0.0675930
\(897\) 0 0
\(898\) 1.26013e7 0.521464
\(899\) −1.30303e7 −0.537720
\(900\) 0 0
\(901\) 1.05127e7 0.431422
\(902\) −3.09620e7 −1.26711
\(903\) 0 0
\(904\) −2.92806e7 −1.19168
\(905\) −1.99630e6 −0.0810223
\(906\) 0 0
\(907\) 1.73344e7 0.699667 0.349833 0.936812i \(-0.386238\pi\)
0.349833 + 0.936812i \(0.386238\pi\)
\(908\) −3.20416e7 −1.28973
\(909\) 0 0
\(910\) −1.02750e6 −0.0411319
\(911\) 9.85196e6 0.393302 0.196651 0.980474i \(-0.436993\pi\)
0.196651 + 0.980474i \(0.436993\pi\)
\(912\) 0 0
\(913\) 6.43420e7 2.55457
\(914\) −6.81838e7 −2.69970
\(915\) 0 0
\(916\) 9.79681e7 3.85786
\(917\) −948690. −0.0372564
\(918\) 0 0
\(919\) −3.67691e7 −1.43613 −0.718066 0.695975i \(-0.754973\pi\)
−0.718066 + 0.695975i \(0.754973\pi\)
\(920\) −3.46950e7 −1.35144
\(921\) 0 0
\(922\) 6.55773e7 2.54054
\(923\) 5.17723e7 2.00029
\(924\) 0 0
\(925\) −4.26625e6 −0.163943
\(926\) −3.15148e7 −1.20778
\(927\) 0 0
\(928\) −3.57680e6 −0.136340
\(929\) −1.27151e7 −0.483371 −0.241685 0.970355i \(-0.577700\pi\)
−0.241685 + 0.970355i \(0.577700\pi\)
\(930\) 0 0
\(931\) −1.85823e7 −0.702626
\(932\) 1.99981e7 0.754136
\(933\) 0 0
\(934\) −7.80490e7 −2.92752
\(935\) 1.78956e7 0.669450
\(936\) 0 0
\(937\) −5.26250e7 −1.95814 −0.979068 0.203533i \(-0.934758\pi\)
−0.979068 + 0.203533i \(0.934758\pi\)
\(938\) −831600. −0.0308608
\(939\) 0 0
\(940\) −3.56150e6 −0.131466
\(941\) −1.26678e6 −0.0466368 −0.0233184 0.999728i \(-0.507423\pi\)
−0.0233184 + 0.999728i \(0.507423\pi\)
\(942\) 0 0
\(943\) −1.74246e7 −0.638092
\(944\) −3.53437e7 −1.29087
\(945\) 0 0
\(946\) 6.23144e7 2.26392
\(947\) 2.45637e7 0.890058 0.445029 0.895516i \(-0.353193\pi\)
0.445029 + 0.895516i \(0.353193\pi\)
\(948\) 0 0
\(949\) −2.24255e7 −0.808309
\(950\) −6.92500e6 −0.248949
\(951\) 0 0
\(952\) 2.25720e6 0.0807194
\(953\) −2.58428e7 −0.921737 −0.460868 0.887469i \(-0.652462\pi\)
−0.460868 + 0.887469i \(0.652462\pi\)
\(954\) 0 0
\(955\) 3.77275e6 0.133860
\(956\) 5.66664e7 2.00531
\(957\) 0 0
\(958\) −7.29660e6 −0.256866
\(959\) 1.17162e6 0.0411377
\(960\) 0 0
\(961\) 6.95591e7 2.42966
\(962\) 4.67581e7 1.62899
\(963\) 0 0
\(964\) 2.96804e7 1.02867
\(965\) 4.26755e6 0.147523
\(966\) 0 0
\(967\) −1.34437e7 −0.462331 −0.231165 0.972914i \(-0.574254\pi\)
−0.231165 + 0.972914i \(0.574254\pi\)
\(968\) −1.10943e8 −3.80549
\(969\) 0 0
\(970\) 2.14915e7 0.733394
\(971\) −4.08522e7 −1.39049 −0.695244 0.718774i \(-0.744704\pi\)
−0.695244 + 0.718774i \(0.744704\pi\)
\(972\) 0 0
\(973\) 79428.0 0.00268962
\(974\) −9.35510e7 −3.15974
\(975\) 0 0
\(976\) −6.59113e7 −2.21480
\(977\) 2.66025e7 0.891634 0.445817 0.895124i \(-0.352913\pi\)
0.445817 + 0.895124i \(0.352913\pi\)
\(978\) 0 0
\(979\) 8.46249e7 2.82190
\(980\) −2.85107e7 −0.948293
\(981\) 0 0
\(982\) −9.39394e7 −3.10863
\(983\) 5.59346e6 0.184627 0.0923137 0.995730i \(-0.470574\pi\)
0.0923137 + 0.995730i \(0.470574\pi\)
\(984\) 0 0
\(985\) −8.33400e6 −0.273693
\(986\) 1.37417e7 0.450142
\(987\) 0 0
\(988\) 5.16106e7 1.68208
\(989\) 3.50689e7 1.14007
\(990\) 0 0
\(991\) −1.67296e7 −0.541130 −0.270565 0.962702i \(-0.587211\pi\)
−0.270565 + 0.962702i \(0.587211\pi\)
\(992\) 2.69525e7 0.869600
\(993\) 0 0
\(994\) −4.53480e6 −0.145577
\(995\) 1.43778e6 0.0460398
\(996\) 0 0
\(997\) 4.48356e7 1.42851 0.714257 0.699883i \(-0.246765\pi\)
0.714257 + 0.699883i \(0.246765\pi\)
\(998\) −6.54473e7 −2.08001
\(999\) 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 135.6.a.a.1.1 1
3.2 odd 2 135.6.a.b.1.1 yes 1
5.4 even 2 675.6.a.e.1.1 1
15.14 odd 2 675.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.6.a.a.1.1 1 1.1 even 1 trivial
135.6.a.b.1.1 yes 1 3.2 odd 2
675.6.a.a.1.1 1 15.14 odd 2
675.6.a.e.1.1 1 5.4 even 2