Properties

Label 600.8.f.d.49.1
Level $600$
Weight $8$
Character 600.49
Analytic conductor $187.431$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,8,Mod(49,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 600.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(187.431015290\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 600.49
Dual form 600.8.f.d.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000i q^{3} +504.000i q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} +504.000i q^{7} -729.000 q^{9} +3812.00 q^{11} -9574.00i q^{13} +26098.0i q^{17} +38308.0 q^{19} +13608.0 q^{21} +71128.0i q^{23} +19683.0i q^{27} -74262.0 q^{29} -275680. q^{31} -102924. i q^{33} -266610. i q^{37} -258498. q^{39} +684762. q^{41} -245956. i q^{43} +478800. i q^{47} +569527. q^{49} +704646. q^{51} +569410. i q^{53} -1.03432e6i q^{57} +1.52532e6 q^{59} -2.64046e6 q^{61} -367416. i q^{63} +1.41624e6i q^{67} +1.92046e6 q^{69} -3.51130e6 q^{71} -4.73862e6i q^{73} +1.92125e6i q^{77} -4.66149e6 q^{79} +531441. q^{81} +5.72925e6i q^{83} +2.00507e6i q^{87} -1.19935e7 q^{89} +4.82530e6 q^{91} +7.44336e6i q^{93} +7.15075e6i q^{97} -2.77895e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1458 q^{9} + 7624 q^{11} + 76616 q^{19} + 27216 q^{21} - 148524 q^{29} - 551360 q^{31} - 516996 q^{39} + 1369524 q^{41} + 1139054 q^{49} + 1409292 q^{51} + 3050648 q^{59} - 5280916 q^{61} + 3840912 q^{69} - 7022608 q^{71} - 9322976 q^{79} + 1062882 q^{81} - 23987028 q^{89} + 9650592 q^{91} - 5557896 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 27.0000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 504.000i 0.555376i 0.960671 + 0.277688i \(0.0895682\pi\)
−0.960671 + 0.277688i \(0.910432\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) 3812.00 0.863532 0.431766 0.901986i \(-0.357891\pi\)
0.431766 + 0.901986i \(0.357891\pi\)
\(12\) 0 0
\(13\) − 9574.00i − 1.20863i −0.796747 0.604313i \(-0.793448\pi\)
0.796747 0.604313i \(-0.206552\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 26098.0i 1.28836i 0.764875 + 0.644178i \(0.222800\pi\)
−0.764875 + 0.644178i \(0.777200\pi\)
\(18\) 0 0
\(19\) 38308.0 1.28130 0.640652 0.767832i \(-0.278664\pi\)
0.640652 + 0.767832i \(0.278664\pi\)
\(20\) 0 0
\(21\) 13608.0 0.320647
\(22\) 0 0
\(23\) 71128.0i 1.21897i 0.792797 + 0.609485i \(0.208624\pi\)
−0.792797 + 0.609485i \(0.791376\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) −74262.0 −0.565423 −0.282712 0.959205i \(-0.591234\pi\)
−0.282712 + 0.959205i \(0.591234\pi\)
\(30\) 0 0
\(31\) −275680. −1.66203 −0.831016 0.556249i \(-0.812240\pi\)
−0.831016 + 0.556249i \(0.812240\pi\)
\(32\) 0 0
\(33\) − 102924.i − 0.498560i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 266610.i − 0.865307i −0.901560 0.432654i \(-0.857577\pi\)
0.901560 0.432654i \(-0.142423\pi\)
\(38\) 0 0
\(39\) −258498. −0.697800
\(40\) 0 0
\(41\) 684762. 1.55166 0.775829 0.630943i \(-0.217332\pi\)
0.775829 + 0.630943i \(0.217332\pi\)
\(42\) 0 0
\(43\) − 245956.i − 0.471756i −0.971783 0.235878i \(-0.924203\pi\)
0.971783 0.235878i \(-0.0757966\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 478800.i 0.672685i 0.941740 + 0.336342i \(0.109190\pi\)
−0.941740 + 0.336342i \(0.890810\pi\)
\(48\) 0 0
\(49\) 569527. 0.691557
\(50\) 0 0
\(51\) 704646. 0.743833
\(52\) 0 0
\(53\) 569410.i 0.525363i 0.964883 + 0.262682i \(0.0846069\pi\)
−0.964883 + 0.262682i \(0.915393\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.03432e6i − 0.739761i
\(58\) 0 0
\(59\) 1.52532e6 0.966897 0.483448 0.875373i \(-0.339384\pi\)
0.483448 + 0.875373i \(0.339384\pi\)
\(60\) 0 0
\(61\) −2.64046e6 −1.48945 −0.744723 0.667374i \(-0.767418\pi\)
−0.744723 + 0.667374i \(0.767418\pi\)
\(62\) 0 0
\(63\) − 367416.i − 0.185125i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.41624e6i 0.575273i 0.957740 + 0.287636i \(0.0928695\pi\)
−0.957740 + 0.287636i \(0.907131\pi\)
\(68\) 0 0
\(69\) 1.92046e6 0.703773
\(70\) 0 0
\(71\) −3.51130e6 −1.16430 −0.582149 0.813082i \(-0.697788\pi\)
−0.582149 + 0.813082i \(0.697788\pi\)
\(72\) 0 0
\(73\) − 4.73862e6i − 1.42568i −0.701327 0.712839i \(-0.747409\pi\)
0.701327 0.712839i \(-0.252591\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.92125e6i 0.479585i
\(78\) 0 0
\(79\) −4.66149e6 −1.06373 −0.531863 0.846830i \(-0.678508\pi\)
−0.531863 + 0.846830i \(0.678508\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.72925e6i 1.09983i 0.835221 + 0.549914i \(0.185339\pi\)
−0.835221 + 0.549914i \(0.814661\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00507e6i 0.326447i
\(88\) 0 0
\(89\) −1.19935e7 −1.80336 −0.901678 0.432408i \(-0.857664\pi\)
−0.901678 + 0.432408i \(0.857664\pi\)
\(90\) 0 0
\(91\) 4.82530e6 0.671242
\(92\) 0 0
\(93\) 7.44336e6i 0.959575i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.15075e6i 0.795519i 0.917490 + 0.397760i \(0.130212\pi\)
−0.917490 + 0.397760i \(0.869788\pi\)
\(98\) 0 0
\(99\) −2.77895e6 −0.287844
\(100\) 0 0
\(101\) −8.78373e6 −0.848309 −0.424155 0.905590i \(-0.639429\pi\)
−0.424155 + 0.905590i \(0.639429\pi\)
\(102\) 0 0
\(103\) 8.01610e6i 0.722825i 0.932406 + 0.361412i \(0.117705\pi\)
−0.932406 + 0.361412i \(0.882295\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 5.15123e6i − 0.406507i −0.979126 0.203253i \(-0.934849\pi\)
0.979126 0.203253i \(-0.0651515\pi\)
\(108\) 0 0
\(109\) 2.41280e7 1.78455 0.892274 0.451493i \(-0.149109\pi\)
0.892274 + 0.451493i \(0.149109\pi\)
\(110\) 0 0
\(111\) −7.19847e6 −0.499585
\(112\) 0 0
\(113\) − 2.04827e6i − 0.133541i −0.997768 0.0667703i \(-0.978731\pi\)
0.997768 0.0667703i \(-0.0212695\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.97945e6i 0.402875i
\(118\) 0 0
\(119\) −1.31534e7 −0.715523
\(120\) 0 0
\(121\) −4.95583e6 −0.254312
\(122\) 0 0
\(123\) − 1.84886e7i − 0.895850i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.36634e6i 0.0591895i 0.999562 + 0.0295947i \(0.00942167\pi\)
−0.999562 + 0.0295947i \(0.990578\pi\)
\(128\) 0 0
\(129\) −6.64081e6 −0.272369
\(130\) 0 0
\(131\) 3.84645e7 1.49489 0.747447 0.664321i \(-0.231279\pi\)
0.747447 + 0.664321i \(0.231279\pi\)
\(132\) 0 0
\(133\) 1.93072e7i 0.711605i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.62585e6i 0.253376i 0.991943 + 0.126688i \(0.0404348\pi\)
−0.991943 + 0.126688i \(0.959565\pi\)
\(138\) 0 0
\(139\) −5.32324e6 −0.168122 −0.0840609 0.996461i \(-0.526789\pi\)
−0.0840609 + 0.996461i \(0.526789\pi\)
\(140\) 0 0
\(141\) 1.29276e7 0.388375
\(142\) 0 0
\(143\) − 3.64961e7i − 1.04369i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.53772e7i − 0.399271i
\(148\) 0 0
\(149\) −7.61366e6 −0.188557 −0.0942783 0.995546i \(-0.530054\pi\)
−0.0942783 + 0.995546i \(0.530054\pi\)
\(150\) 0 0
\(151\) −2.50221e7 −0.591432 −0.295716 0.955276i \(-0.595558\pi\)
−0.295716 + 0.955276i \(0.595558\pi\)
\(152\) 0 0
\(153\) − 1.90254e7i − 0.429452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.93145e7i 0.810782i 0.914143 + 0.405391i \(0.132865\pi\)
−0.914143 + 0.405391i \(0.867135\pi\)
\(158\) 0 0
\(159\) 1.53741e7 0.303319
\(160\) 0 0
\(161\) −3.58485e7 −0.676987
\(162\) 0 0
\(163\) 6.28387e7i 1.13650i 0.822855 + 0.568252i \(0.192380\pi\)
−0.822855 + 0.568252i \(0.807620\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.04133e7i 0.173014i 0.996251 + 0.0865072i \(0.0275706\pi\)
−0.996251 + 0.0865072i \(0.972429\pi\)
\(168\) 0 0
\(169\) −2.89130e7 −0.460775
\(170\) 0 0
\(171\) −2.79265e7 −0.427101
\(172\) 0 0
\(173\) 8.03551e7i 1.17992i 0.807433 + 0.589959i \(0.200856\pi\)
−0.807433 + 0.589959i \(0.799144\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.11837e7i − 0.558238i
\(178\) 0 0
\(179\) 8.40084e7 1.09481 0.547403 0.836869i \(-0.315617\pi\)
0.547403 + 0.836869i \(0.315617\pi\)
\(180\) 0 0
\(181\) 1.15469e8 1.44741 0.723703 0.690112i \(-0.242439\pi\)
0.723703 + 0.690112i \(0.242439\pi\)
\(182\) 0 0
\(183\) 7.12924e7i 0.859932i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.94856e7i 1.11254i
\(188\) 0 0
\(189\) −9.92023e6 −0.106882
\(190\) 0 0
\(191\) 9.97154e7 1.03549 0.517744 0.855535i \(-0.326772\pi\)
0.517744 + 0.855535i \(0.326772\pi\)
\(192\) 0 0
\(193\) 1.86157e7i 0.186393i 0.995648 + 0.0931965i \(0.0297085\pi\)
−0.995648 + 0.0931965i \(0.970292\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.30384e7i 0.867022i 0.901148 + 0.433511i \(0.142726\pi\)
−0.901148 + 0.433511i \(0.857274\pi\)
\(198\) 0 0
\(199\) −7.39686e7 −0.665367 −0.332684 0.943038i \(-0.607954\pi\)
−0.332684 + 0.943038i \(0.607954\pi\)
\(200\) 0 0
\(201\) 3.82384e7 0.332134
\(202\) 0 0
\(203\) − 3.74280e7i − 0.314023i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 5.18523e7i − 0.406323i
\(208\) 0 0
\(209\) 1.46030e8 1.10645
\(210\) 0 0
\(211\) 1.85163e8 1.35695 0.678476 0.734623i \(-0.262641\pi\)
0.678476 + 0.734623i \(0.262641\pi\)
\(212\) 0 0
\(213\) 9.48052e7i 0.672208i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.38943e8i − 0.923053i
\(218\) 0 0
\(219\) −1.27943e8 −0.823116
\(220\) 0 0
\(221\) 2.49862e8 1.55714
\(222\) 0 0
\(223\) 1.20862e8i 0.729830i 0.931041 + 0.364915i \(0.118902\pi\)
−0.931041 + 0.364915i \(0.881098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.82315e8i − 1.60193i −0.598710 0.800966i \(-0.704320\pi\)
0.598710 0.800966i \(-0.295680\pi\)
\(228\) 0 0
\(229\) 8.91913e7 0.490793 0.245397 0.969423i \(-0.421082\pi\)
0.245397 + 0.969423i \(0.421082\pi\)
\(230\) 0 0
\(231\) 5.18737e7 0.276889
\(232\) 0 0
\(233\) 2.32240e8i 1.20279i 0.798951 + 0.601396i \(0.205389\pi\)
−0.798951 + 0.601396i \(0.794611\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.25860e8i 0.614142i
\(238\) 0 0
\(239\) −3.21986e8 −1.52561 −0.762807 0.646626i \(-0.776179\pi\)
−0.762807 + 0.646626i \(0.776179\pi\)
\(240\) 0 0
\(241\) −2.00366e8 −0.922072 −0.461036 0.887381i \(-0.652522\pi\)
−0.461036 + 0.887381i \(0.652522\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.66761e8i − 1.54862i
\(248\) 0 0
\(249\) 1.54690e8 0.634986
\(250\) 0 0
\(251\) −8.70560e7 −0.347489 −0.173744 0.984791i \(-0.555587\pi\)
−0.173744 + 0.984791i \(0.555587\pi\)
\(252\) 0 0
\(253\) 2.71140e8i 1.05262i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.22879e8i 1.92148i 0.277457 + 0.960738i \(0.410509\pi\)
−0.277457 + 0.960738i \(0.589491\pi\)
\(258\) 0 0
\(259\) 1.34371e8 0.480571
\(260\) 0 0
\(261\) 5.41370e7 0.188474
\(262\) 0 0
\(263\) 4.06215e8i 1.37693i 0.725270 + 0.688464i \(0.241715\pi\)
−0.725270 + 0.688464i \(0.758285\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.23825e8i 1.04117i
\(268\) 0 0
\(269\) −3.82347e8 −1.19764 −0.598818 0.800885i \(-0.704363\pi\)
−0.598818 + 0.800885i \(0.704363\pi\)
\(270\) 0 0
\(271\) 2.84165e8 0.867317 0.433658 0.901077i \(-0.357222\pi\)
0.433658 + 0.901077i \(0.357222\pi\)
\(272\) 0 0
\(273\) − 1.30283e8i − 0.387542i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.93752e8i 0.830427i 0.909724 + 0.415213i \(0.136293\pi\)
−0.909724 + 0.415213i \(0.863707\pi\)
\(278\) 0 0
\(279\) 2.00971e8 0.554011
\(280\) 0 0
\(281\) 4.15399e8 1.11685 0.558424 0.829556i \(-0.311406\pi\)
0.558424 + 0.829556i \(0.311406\pi\)
\(282\) 0 0
\(283\) − 5.06429e7i − 0.132821i −0.997792 0.0664104i \(-0.978845\pi\)
0.997792 0.0664104i \(-0.0211547\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.45120e8i 0.861754i
\(288\) 0 0
\(289\) −2.70767e8 −0.659862
\(290\) 0 0
\(291\) 1.93070e8 0.459293
\(292\) 0 0
\(293\) − 7.47714e7i − 0.173660i −0.996223 0.0868298i \(-0.972326\pi\)
0.996223 0.0868298i \(-0.0276736\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7.50316e7i 0.166187i
\(298\) 0 0
\(299\) 6.80979e8 1.47328
\(300\) 0 0
\(301\) 1.23962e8 0.262002
\(302\) 0 0
\(303\) 2.37161e8i 0.489772i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.52577e7i 0.168170i 0.996459 + 0.0840851i \(0.0267968\pi\)
−0.996459 + 0.0840851i \(0.973203\pi\)
\(308\) 0 0
\(309\) 2.16435e8 0.417323
\(310\) 0 0
\(311\) 9.39129e8 1.77037 0.885184 0.465240i \(-0.154032\pi\)
0.885184 + 0.465240i \(0.154032\pi\)
\(312\) 0 0
\(313\) 3.43040e8i 0.632323i 0.948705 + 0.316162i \(0.102394\pi\)
−0.948705 + 0.316162i \(0.897606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.03960e9i − 1.83298i −0.400054 0.916492i \(-0.631009\pi\)
0.400054 0.916492i \(-0.368991\pi\)
\(318\) 0 0
\(319\) −2.83087e8 −0.488261
\(320\) 0 0
\(321\) −1.39083e8 −0.234697
\(322\) 0 0
\(323\) 9.99762e8i 1.65077i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 6.51456e8i − 1.03031i
\(328\) 0 0
\(329\) −2.41315e8 −0.373593
\(330\) 0 0
\(331\) 1.10022e9 1.66756 0.833779 0.552098i \(-0.186173\pi\)
0.833779 + 0.552098i \(0.186173\pi\)
\(332\) 0 0
\(333\) 1.94359e8i 0.288436i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.28272e9i 1.82569i 0.408302 + 0.912847i \(0.366121\pi\)
−0.408302 + 0.912847i \(0.633879\pi\)
\(338\) 0 0
\(339\) −5.53034e7 −0.0770997
\(340\) 0 0
\(341\) −1.05089e9 −1.43522
\(342\) 0 0
\(343\) 7.02107e8i 0.939451i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.25822e9i 1.61660i 0.588770 + 0.808301i \(0.299612\pi\)
−0.588770 + 0.808301i \(0.700388\pi\)
\(348\) 0 0
\(349\) −1.35371e8 −0.170465 −0.0852327 0.996361i \(-0.527163\pi\)
−0.0852327 + 0.996361i \(0.527163\pi\)
\(350\) 0 0
\(351\) 1.88445e8 0.232600
\(352\) 0 0
\(353\) 8.49221e8i 1.02756i 0.857921 + 0.513782i \(0.171756\pi\)
−0.857921 + 0.513782i \(0.828244\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.55142e8i 0.413107i
\(358\) 0 0
\(359\) −2.20121e8 −0.251091 −0.125546 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(360\) 0 0
\(361\) 5.73631e8 0.641738
\(362\) 0 0
\(363\) 1.33807e8i 0.146827i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.66505e8i 0.492635i 0.969189 + 0.246317i \(0.0792205\pi\)
−0.969189 + 0.246317i \(0.920779\pi\)
\(368\) 0 0
\(369\) −4.99191e8 −0.517220
\(370\) 0 0
\(371\) −2.86983e8 −0.291774
\(372\) 0 0
\(373\) 2.98453e8i 0.297780i 0.988854 + 0.148890i \(0.0475700\pi\)
−0.988854 + 0.148890i \(0.952430\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.10984e8i 0.683385i
\(378\) 0 0
\(379\) −1.46218e9 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(380\) 0 0
\(381\) 3.68911e7 0.0341731
\(382\) 0 0
\(383\) − 1.58702e9i − 1.44340i −0.692206 0.721700i \(-0.743361\pi\)
0.692206 0.721700i \(-0.256639\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.79302e8i 0.157252i
\(388\) 0 0
\(389\) 3.14439e8 0.270840 0.135420 0.990788i \(-0.456762\pi\)
0.135420 + 0.990788i \(0.456762\pi\)
\(390\) 0 0
\(391\) −1.85630e9 −1.57047
\(392\) 0 0
\(393\) − 1.03854e9i − 0.863078i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8.52757e8i − 0.684004i −0.939699 0.342002i \(-0.888895\pi\)
0.939699 0.342002i \(-0.111105\pi\)
\(398\) 0 0
\(399\) 5.21295e8 0.410846
\(400\) 0 0
\(401\) 6.92522e8 0.536325 0.268163 0.963374i \(-0.413584\pi\)
0.268163 + 0.963374i \(0.413584\pi\)
\(402\) 0 0
\(403\) 2.63936e9i 2.00877i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.01632e9i − 0.747221i
\(408\) 0 0
\(409\) −6.17357e8 −0.446174 −0.223087 0.974799i \(-0.571613\pi\)
−0.223087 + 0.974799i \(0.571613\pi\)
\(410\) 0 0
\(411\) 2.05898e8 0.146287
\(412\) 0 0
\(413\) 7.68763e8i 0.536992i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.43727e8i 0.0970651i
\(418\) 0 0
\(419\) 1.65512e9 1.09921 0.549604 0.835425i \(-0.314779\pi\)
0.549604 + 0.835425i \(0.314779\pi\)
\(420\) 0 0
\(421\) 7.01472e7 0.0458166 0.0229083 0.999738i \(-0.492707\pi\)
0.0229083 + 0.999738i \(0.492707\pi\)
\(422\) 0 0
\(423\) − 3.49045e8i − 0.224228i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.33079e9i − 0.827203i
\(428\) 0 0
\(429\) −9.85394e8 −0.602573
\(430\) 0 0
\(431\) 1.81387e9 1.09128 0.545640 0.838020i \(-0.316287\pi\)
0.545640 + 0.838020i \(0.316287\pi\)
\(432\) 0 0
\(433\) 2.59970e9i 1.53892i 0.638695 + 0.769460i \(0.279475\pi\)
−0.638695 + 0.769460i \(0.720525\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.72477e9i 1.56187i
\(438\) 0 0
\(439\) 1.67431e9 0.944517 0.472258 0.881460i \(-0.343439\pi\)
0.472258 + 0.881460i \(0.343439\pi\)
\(440\) 0 0
\(441\) −4.15185e8 −0.230519
\(442\) 0 0
\(443\) 2.52711e8i 0.138105i 0.997613 + 0.0690527i \(0.0219977\pi\)
−0.997613 + 0.0690527i \(0.978002\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.05569e8i 0.108863i
\(448\) 0 0
\(449\) −7.55311e8 −0.393789 −0.196895 0.980425i \(-0.563086\pi\)
−0.196895 + 0.980425i \(0.563086\pi\)
\(450\) 0 0
\(451\) 2.61031e9 1.33991
\(452\) 0 0
\(453\) 6.75597e8i 0.341463i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.51584e8i − 0.0742928i −0.999310 0.0371464i \(-0.988173\pi\)
0.999310 0.0371464i \(-0.0118268\pi\)
\(458\) 0 0
\(459\) −5.13687e8 −0.247944
\(460\) 0 0
\(461\) −7.78405e8 −0.370043 −0.185022 0.982734i \(-0.559236\pi\)
−0.185022 + 0.982734i \(0.559236\pi\)
\(462\) 0 0
\(463\) − 2.41052e9i − 1.12870i −0.825536 0.564349i \(-0.809127\pi\)
0.825536 0.564349i \(-0.190873\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.76192e9i − 0.800527i −0.916400 0.400264i \(-0.868919\pi\)
0.916400 0.400264i \(-0.131081\pi\)
\(468\) 0 0
\(469\) −7.13783e8 −0.319493
\(470\) 0 0
\(471\) 1.06149e9 0.468105
\(472\) 0 0
\(473\) − 9.37584e8i − 0.407377i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4.15100e8i − 0.175121i
\(478\) 0 0
\(479\) −6.43811e8 −0.267661 −0.133830 0.991004i \(-0.542728\pi\)
−0.133830 + 0.991004i \(0.542728\pi\)
\(480\) 0 0
\(481\) −2.55252e9 −1.04583
\(482\) 0 0
\(483\) 9.67910e8i 0.390859i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.16421e9i 1.24141i 0.784045 + 0.620704i \(0.213153\pi\)
−0.784045 + 0.620704i \(0.786847\pi\)
\(488\) 0 0
\(489\) 1.69665e9 0.656160
\(490\) 0 0
\(491\) 3.62406e9 1.38169 0.690844 0.723004i \(-0.257239\pi\)
0.690844 + 0.723004i \(0.257239\pi\)
\(492\) 0 0
\(493\) − 1.93809e9i − 0.728467i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.76970e9i − 0.646624i
\(498\) 0 0
\(499\) 1.35483e9 0.488128 0.244064 0.969759i \(-0.421519\pi\)
0.244064 + 0.969759i \(0.421519\pi\)
\(500\) 0 0
\(501\) 2.81160e8 0.0998899
\(502\) 0 0
\(503\) 4.66389e9i 1.63403i 0.576616 + 0.817015i \(0.304373\pi\)
−0.576616 + 0.817015i \(0.695627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.80650e8i 0.266029i
\(508\) 0 0
\(509\) 1.34292e9 0.451376 0.225688 0.974200i \(-0.427537\pi\)
0.225688 + 0.974200i \(0.427537\pi\)
\(510\) 0 0
\(511\) 2.38826e9 0.791788
\(512\) 0 0
\(513\) 7.54016e8i 0.246587i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.82519e9i 0.580885i
\(518\) 0 0
\(519\) 2.16959e9 0.681226
\(520\) 0 0
\(521\) −1.45400e9 −0.450435 −0.225217 0.974309i \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(522\) 0 0
\(523\) − 4.90309e8i − 0.149870i −0.997188 0.0749349i \(-0.976125\pi\)
0.997188 0.0749349i \(-0.0238749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 7.19470e9i − 2.14129i
\(528\) 0 0
\(529\) −1.65437e9 −0.485889
\(530\) 0 0
\(531\) −1.11196e9 −0.322299
\(532\) 0 0
\(533\) − 6.55591e9i − 1.87537i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.26823e9i − 0.632086i
\(538\) 0 0
\(539\) 2.17104e9 0.597182
\(540\) 0 0
\(541\) −4.82889e9 −1.31116 −0.655582 0.755124i \(-0.727577\pi\)
−0.655582 + 0.755124i \(0.727577\pi\)
\(542\) 0 0
\(543\) − 3.11766e9i − 0.835660i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3.08793e9i − 0.806698i −0.915046 0.403349i \(-0.867846\pi\)
0.915046 0.403349i \(-0.132154\pi\)
\(548\) 0 0
\(549\) 1.92489e9 0.496482
\(550\) 0 0
\(551\) −2.84483e9 −0.724479
\(552\) 0 0
\(553\) − 2.34939e9i − 0.590768i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.09889e9i 1.00502i 0.864573 + 0.502508i \(0.167589\pi\)
−0.864573 + 0.502508i \(0.832411\pi\)
\(558\) 0 0
\(559\) −2.35478e9 −0.570177
\(560\) 0 0
\(561\) 2.68611e9 0.642324
\(562\) 0 0
\(563\) − 4.97105e9i − 1.17400i −0.809587 0.587001i \(-0.800309\pi\)
0.809587 0.587001i \(-0.199691\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.67846e8i 0.0617085i
\(568\) 0 0
\(569\) −3.71316e9 −0.844988 −0.422494 0.906366i \(-0.638845\pi\)
−0.422494 + 0.906366i \(0.638845\pi\)
\(570\) 0 0
\(571\) −2.36205e9 −0.530961 −0.265481 0.964116i \(-0.585531\pi\)
−0.265481 + 0.964116i \(0.585531\pi\)
\(572\) 0 0
\(573\) − 2.69232e9i − 0.597840i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.81146e8i − 0.0392566i −0.999807 0.0196283i \(-0.993752\pi\)
0.999807 0.0196283i \(-0.00624828\pi\)
\(578\) 0 0
\(579\) 5.02625e8 0.107614
\(580\) 0 0
\(581\) −2.88754e9 −0.610818
\(582\) 0 0
\(583\) 2.17059e9i 0.453668i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.31976e9i 0.473380i 0.971585 + 0.236690i \(0.0760626\pi\)
−0.971585 + 0.236690i \(0.923937\pi\)
\(588\) 0 0
\(589\) −1.05607e10 −2.12957
\(590\) 0 0
\(591\) 2.51204e9 0.500576
\(592\) 0 0
\(593\) − 2.27806e9i − 0.448615i −0.974518 0.224308i \(-0.927988\pi\)
0.974518 0.224308i \(-0.0720120\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.99715e9i 0.384150i
\(598\) 0 0
\(599\) 4.88253e9 0.928220 0.464110 0.885778i \(-0.346374\pi\)
0.464110 + 0.885778i \(0.346374\pi\)
\(600\) 0 0
\(601\) −6.74758e9 −1.26791 −0.633954 0.773371i \(-0.718569\pi\)
−0.633954 + 0.773371i \(0.718569\pi\)
\(602\) 0 0
\(603\) − 1.03244e9i − 0.191758i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 9.05928e9i − 1.64412i −0.569401 0.822060i \(-0.692825\pi\)
0.569401 0.822060i \(-0.307175\pi\)
\(608\) 0 0
\(609\) −1.01056e9 −0.181301
\(610\) 0 0
\(611\) 4.58403e9 0.813024
\(612\) 0 0
\(613\) 8.48777e9i 1.48827i 0.668029 + 0.744135i \(0.267138\pi\)
−0.668029 + 0.744135i \(0.732862\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.34407e9i 0.401766i 0.979615 + 0.200883i \(0.0643810\pi\)
−0.979615 + 0.200883i \(0.935619\pi\)
\(618\) 0 0
\(619\) 1.01541e9 0.172077 0.0860384 0.996292i \(-0.472579\pi\)
0.0860384 + 0.996292i \(0.472579\pi\)
\(620\) 0 0
\(621\) −1.40001e9 −0.234591
\(622\) 0 0
\(623\) − 6.04473e9i − 1.00154i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.94281e9i − 0.638807i
\(628\) 0 0
\(629\) 6.95799e9 1.11482
\(630\) 0 0
\(631\) 7.01911e9 1.11219 0.556095 0.831119i \(-0.312299\pi\)
0.556095 + 0.831119i \(0.312299\pi\)
\(632\) 0 0
\(633\) − 4.99939e9i − 0.783437i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 5.45265e9i − 0.835833i
\(638\) 0 0
\(639\) 2.55974e9 0.388099
\(640\) 0 0
\(641\) −4.52776e9 −0.679016 −0.339508 0.940603i \(-0.610261\pi\)
−0.339508 + 0.940603i \(0.610261\pi\)
\(642\) 0 0
\(643\) 8.63094e9i 1.28032i 0.768240 + 0.640162i \(0.221133\pi\)
−0.768240 + 0.640162i \(0.778867\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.57401e9i 0.373632i 0.982395 + 0.186816i \(0.0598169\pi\)
−0.982395 + 0.186816i \(0.940183\pi\)
\(648\) 0 0
\(649\) 5.81454e9 0.834946
\(650\) 0 0
\(651\) −3.75145e9 −0.532925
\(652\) 0 0
\(653\) 9.31827e9i 1.30960i 0.755802 + 0.654800i \(0.227247\pi\)
−0.755802 + 0.654800i \(0.772753\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.45445e9i 0.475226i
\(658\) 0 0
\(659\) −1.04422e10 −1.42133 −0.710663 0.703532i \(-0.751605\pi\)
−0.710663 + 0.703532i \(0.751605\pi\)
\(660\) 0 0
\(661\) −1.04761e10 −1.41090 −0.705449 0.708761i \(-0.749254\pi\)
−0.705449 + 0.708761i \(0.749254\pi\)
\(662\) 0 0
\(663\) − 6.74628e9i − 0.899015i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.28211e9i − 0.689234i
\(668\) 0 0
\(669\) 3.26327e9 0.421368
\(670\) 0 0
\(671\) −1.00654e10 −1.28618
\(672\) 0 0
\(673\) − 1.38891e10i − 1.75639i −0.478299 0.878197i \(-0.658746\pi\)
0.478299 0.878197i \(-0.341254\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 7.48893e8i − 0.0927598i −0.998924 0.0463799i \(-0.985232\pi\)
0.998924 0.0463799i \(-0.0147685\pi\)
\(678\) 0 0
\(679\) −3.60398e9 −0.441813
\(680\) 0 0
\(681\) −7.62251e9 −0.924875
\(682\) 0 0
\(683\) − 1.15581e10i − 1.38808i −0.719938 0.694038i \(-0.755830\pi\)
0.719938 0.694038i \(-0.244170\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 2.40817e9i − 0.283360i
\(688\) 0 0
\(689\) 5.45153e9 0.634967
\(690\) 0 0
\(691\) 3.34337e8 0.0385489 0.0192744 0.999814i \(-0.493864\pi\)
0.0192744 + 0.999814i \(0.493864\pi\)
\(692\) 0 0
\(693\) − 1.40059e9i − 0.159862i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.78709e10i 1.99909i
\(698\) 0 0
\(699\) 6.27047e9 0.694433
\(700\) 0 0
\(701\) −5.55383e9 −0.608948 −0.304474 0.952521i \(-0.598481\pi\)
−0.304474 + 0.952521i \(0.598481\pi\)
\(702\) 0 0
\(703\) − 1.02133e10i − 1.10872i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 4.42700e9i − 0.471131i
\(708\) 0 0
\(709\) −1.30817e10 −1.37849 −0.689243 0.724530i \(-0.742057\pi\)
−0.689243 + 0.724530i \(0.742057\pi\)
\(710\) 0 0
\(711\) 3.39822e9 0.354575
\(712\) 0 0
\(713\) − 1.96086e10i − 2.02597i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.69363e9i 0.880814i
\(718\) 0 0
\(719\) 1.10847e10 1.11217 0.556085 0.831125i \(-0.312303\pi\)
0.556085 + 0.831125i \(0.312303\pi\)
\(720\) 0 0
\(721\) −4.04012e9 −0.401440
\(722\) 0 0
\(723\) 5.40989e9i 0.532358i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.79416e9i − 0.752314i −0.926556 0.376157i \(-0.877245\pi\)
0.926556 0.376157i \(-0.122755\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) 6.41896e9 0.607790
\(732\) 0 0
\(733\) − 6.83552e9i − 0.641073i −0.947236 0.320537i \(-0.896137\pi\)
0.947236 0.320537i \(-0.103863\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.39869e9i 0.496767i
\(738\) 0 0
\(739\) −1.73862e10 −1.58471 −0.792356 0.610059i \(-0.791146\pi\)
−0.792356 + 0.610059i \(0.791146\pi\)
\(740\) 0 0
\(741\) −9.90254e9 −0.894094
\(742\) 0 0
\(743\) − 2.25537e9i − 0.201724i −0.994900 0.100862i \(-0.967840\pi\)
0.994900 0.100862i \(-0.0321600\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4.17662e9i − 0.366609i
\(748\) 0 0
\(749\) 2.59622e9 0.225764
\(750\) 0 0
\(751\) −2.05027e10 −1.76632 −0.883162 0.469068i \(-0.844590\pi\)
−0.883162 + 0.469068i \(0.844590\pi\)
\(752\) 0 0
\(753\) 2.35051e9i 0.200623i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.57872e8i − 0.0216057i −0.999942 0.0108029i \(-0.996561\pi\)
0.999942 0.0108029i \(-0.00343872\pi\)
\(758\) 0 0
\(759\) 7.32078e9 0.607731
\(760\) 0 0
\(761\) −1.34452e10 −1.10591 −0.552957 0.833210i \(-0.686501\pi\)
−0.552957 + 0.833210i \(0.686501\pi\)
\(762\) 0 0
\(763\) 1.21605e10i 0.991096i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.46035e10i − 1.16862i
\(768\) 0 0
\(769\) −8.28541e9 −0.657009 −0.328505 0.944502i \(-0.606545\pi\)
−0.328505 + 0.944502i \(0.606545\pi\)
\(770\) 0 0
\(771\) 1.41177e10 1.10936
\(772\) 0 0
\(773\) − 1.43430e10i − 1.11689i −0.829540 0.558447i \(-0.811397\pi\)
0.829540 0.558447i \(-0.188603\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3.62803e9i − 0.277458i
\(778\) 0 0
\(779\) 2.62319e10 1.98814
\(780\) 0 0
\(781\) −1.33851e10 −1.00541
\(782\) 0 0
\(783\) − 1.46170e9i − 0.108816i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 3.83137e9i − 0.280184i −0.990139 0.140092i \(-0.955260\pi\)
0.990139 0.140092i \(-0.0447398\pi\)
\(788\) 0 0
\(789\) 1.09678e10 0.794970
\(790\) 0 0
\(791\) 1.03233e9 0.0741653
\(792\) 0 0
\(793\) 2.52797e10i 1.80018i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.95859e9i − 0.137038i −0.997650 0.0685188i \(-0.978173\pi\)
0.997650 0.0685188i \(-0.0218273\pi\)
\(798\) 0 0
\(799\) −1.24957e10 −0.866658
\(800\) 0 0
\(801\) 8.74327e9 0.601119
\(802\) 0 0
\(803\) − 1.80636e10i − 1.23112i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.03234e10i 0.691455i
\(808\) 0 0
\(809\) −2.07415e9 −0.137727 −0.0688637 0.997626i \(-0.521937\pi\)
−0.0688637 + 0.997626i \(0.521937\pi\)
\(810\) 0 0
\(811\) −5.71508e9 −0.376227 −0.188113 0.982147i \(-0.560237\pi\)
−0.188113 + 0.982147i \(0.560237\pi\)
\(812\) 0 0
\(813\) − 7.67245e9i − 0.500746i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 9.42208e9i − 0.604463i
\(818\) 0 0
\(819\) −3.51764e9 −0.223747
\(820\) 0 0
\(821\) 2.82748e10 1.78319 0.891596 0.452832i \(-0.149586\pi\)
0.891596 + 0.452832i \(0.149586\pi\)
\(822\) 0 0
\(823\) 2.09283e9i 0.130868i 0.997857 + 0.0654342i \(0.0208432\pi\)
−0.997857 + 0.0654342i \(0.979157\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.71453e9i − 0.228368i −0.993460 0.114184i \(-0.963575\pi\)
0.993460 0.114184i \(-0.0364253\pi\)
\(828\) 0 0
\(829\) −3.37924e9 −0.206005 −0.103003 0.994681i \(-0.532845\pi\)
−0.103003 + 0.994681i \(0.532845\pi\)
\(830\) 0 0
\(831\) 7.93130e9 0.479447
\(832\) 0 0
\(833\) 1.48635e10i 0.890972i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 5.42621e9i − 0.319858i
\(838\) 0 0
\(839\) −1.64907e10 −0.963990 −0.481995 0.876174i \(-0.660088\pi\)
−0.481995 + 0.876174i \(0.660088\pi\)
\(840\) 0 0
\(841\) −1.17350e10 −0.680297
\(842\) 0 0
\(843\) − 1.12158e10i − 0.644812i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.49774e9i − 0.141239i
\(848\) 0 0
\(849\) −1.36736e9 −0.0766841
\(850\) 0 0
\(851\) 1.89634e10 1.05478
\(852\) 0 0
\(853\) 4.77028e9i 0.263162i 0.991305 + 0.131581i \(0.0420053\pi\)
−0.991305 + 0.131581i \(0.957995\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.61514e10i 0.876554i 0.898840 + 0.438277i \(0.144411\pi\)
−0.898840 + 0.438277i \(0.855589\pi\)
\(858\) 0 0
\(859\) 3.41593e8 0.0183879 0.00919397 0.999958i \(-0.497073\pi\)
0.00919397 + 0.999958i \(0.497073\pi\)
\(860\) 0 0
\(861\) 9.31824e9 0.497534
\(862\) 0 0
\(863\) − 6.07878e9i − 0.321943i −0.986959 0.160971i \(-0.948537\pi\)
0.986959 0.160971i \(-0.0514627\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.31071e9i 0.380972i
\(868\) 0 0
\(869\) −1.77696e10 −0.918562
\(870\) 0 0
\(871\) 1.35590e10 0.695289
\(872\) 0 0
\(873\) − 5.21290e9i − 0.265173i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.23852e10i 0.620020i 0.950733 + 0.310010i \(0.100332\pi\)
−0.950733 + 0.310010i \(0.899668\pi\)
\(878\) 0 0
\(879\) −2.01883e9 −0.100262
\(880\) 0 0
\(881\) −1.37801e10 −0.678949 −0.339475 0.940615i \(-0.610249\pi\)
−0.339475 + 0.940615i \(0.610249\pi\)
\(882\) 0 0
\(883\) − 1.89296e9i − 0.0925292i −0.998929 0.0462646i \(-0.985268\pi\)
0.998929 0.0462646i \(-0.0147317\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.23912e9i 0.203959i 0.994787 + 0.101979i \(0.0325176\pi\)
−0.994787 + 0.101979i \(0.967482\pi\)
\(888\) 0 0
\(889\) −6.88633e8 −0.0328724
\(890\) 0 0
\(891\) 2.02585e9 0.0959480
\(892\) 0 0
\(893\) 1.83419e10i 0.861913i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.83864e10i − 0.850598i
\(898\) 0 0
\(899\) 2.04725e10 0.939751
\(900\) 0 0
\(901\) −1.48605e10 −0.676855
\(902\) 0 0
\(903\) − 3.34697e9i − 0.151267i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 9.51367e9i − 0.423372i −0.977338 0.211686i \(-0.932105\pi\)
0.977338 0.211686i \(-0.0678955\pi\)
\(908\) 0 0
\(909\) 6.40334e9 0.282770
\(910\) 0 0
\(911\) −1.16235e10 −0.509359 −0.254680 0.967025i \(-0.581970\pi\)
−0.254680 + 0.967025i \(0.581970\pi\)
\(912\) 0 0
\(913\) 2.18399e10i 0.949736i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.93861e10i 0.830229i
\(918\) 0 0
\(919\) 9.22943e9 0.392257 0.196128 0.980578i \(-0.437163\pi\)
0.196128 + 0.980578i \(0.437163\pi\)
\(920\) 0 0
\(921\) 2.30196e9 0.0970931
\(922\) 0 0
\(923\) 3.36172e10i 1.40720i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 5.84374e9i − 0.240942i
\(928\) 0 0
\(929\) 4.09353e10 1.67511 0.837555 0.546353i \(-0.183984\pi\)
0.837555 + 0.546353i \(0.183984\pi\)
\(930\) 0 0
\(931\) 2.18174e10 0.886094
\(932\) 0 0
\(933\) − 2.53565e10i − 1.02212i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.75085e9i − 0.0695281i −0.999396 0.0347641i \(-0.988932\pi\)
0.999396 0.0347641i \(-0.0110680\pi\)
\(938\) 0 0
\(939\) 9.26207e9 0.365072
\(940\) 0 0
\(941\) 5.91102e9 0.231259 0.115630 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(942\) 0 0
\(943\) 4.87058e10i 1.89143i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.26089e10i − 0.865077i −0.901616 0.432538i \(-0.857618\pi\)
0.901616 0.432538i \(-0.142382\pi\)
\(948\) 0 0
\(949\) −4.53675e10 −1.72311
\(950\) 0 0
\(951\) −2.80692e10 −1.05827
\(952\) 0 0
\(953\) − 1.11773e10i − 0.418322i −0.977881 0.209161i \(-0.932927\pi\)
0.977881 0.209161i \(-0.0670733\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.64334e9i 0.281898i
\(958\) 0 0
\(959\) −3.84343e9 −0.140719
\(960\) 0 0
\(961\) 4.84868e10 1.76235
\(962\) 0 0
\(963\) 3.75525e9i 0.135502i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.55518e10i 0.553078i 0.961003 + 0.276539i \(0.0891875\pi\)
−0.961003 + 0.276539i \(0.910812\pi\)
\(968\) 0 0
\(969\) 2.69936e10 0.953075
\(970\) 0 0
\(971\) 8.34508e9 0.292525 0.146263 0.989246i \(-0.453276\pi\)
0.146263 + 0.989246i \(0.453276\pi\)
\(972\) 0 0
\(973\) − 2.68291e9i − 0.0933709i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 9.85180e9i − 0.337975i −0.985618 0.168988i \(-0.945950\pi\)
0.985618 0.168988i \(-0.0540498\pi\)
\(978\) 0 0
\(979\) −4.57193e10 −1.55726
\(980\) 0 0
\(981\) −1.75893e10 −0.594850
\(982\) 0 0
\(983\) − 3.70884e10i − 1.24538i −0.782470 0.622688i \(-0.786041\pi\)
0.782470 0.622688i \(-0.213959\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.51551e9i 0.215694i
\(988\) 0 0
\(989\) 1.74944e10 0.575057
\(990\) 0 0
\(991\) 6.43526e9 0.210043 0.105022 0.994470i \(-0.466509\pi\)
0.105022 + 0.994470i \(0.466509\pi\)
\(992\) 0 0
\(993\) − 2.97059e10i − 0.962765i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.23071e10i − 0.393299i −0.980474 0.196650i \(-0.936994\pi\)
0.980474 0.196650i \(-0.0630062\pi\)
\(998\) 0 0
\(999\) 5.24768e9 0.166528
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 600.8.f.d.49.1 2
5.2 odd 4 600.8.a.a.1.1 1
5.3 odd 4 24.8.a.c.1.1 1
5.4 even 2 inner 600.8.f.d.49.2 2
15.8 even 4 72.8.a.b.1.1 1
20.3 even 4 48.8.a.c.1.1 1
40.3 even 4 192.8.a.k.1.1 1
40.13 odd 4 192.8.a.c.1.1 1
60.23 odd 4 144.8.a.d.1.1 1
120.53 even 4 576.8.a.t.1.1 1
120.83 odd 4 576.8.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.8.a.c.1.1 1 5.3 odd 4
48.8.a.c.1.1 1 20.3 even 4
72.8.a.b.1.1 1 15.8 even 4
144.8.a.d.1.1 1 60.23 odd 4
192.8.a.c.1.1 1 40.13 odd 4
192.8.a.k.1.1 1 40.3 even 4
576.8.a.s.1.1 1 120.83 odd 4
576.8.a.t.1.1 1 120.53 even 4
600.8.a.a.1.1 1 5.2 odd 4
600.8.f.d.49.1 2 1.1 even 1 trivial
600.8.f.d.49.2 2 5.4 even 2 inner