Properties

Label 27.10.a.d.1.1
Level $27$
Weight $10$
Character 27.1
Self dual yes
Analytic conductor $13.906$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,10,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.203942560.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 83x^{2} + 1440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7.63546\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-44.4771 q^{2} +1466.22 q^{4} +1050.62 q^{5} +6591.86 q^{7} -42440.8 q^{8} -46728.6 q^{10} -14465.9 q^{11} +5003.49 q^{13} -293187. q^{14} +1.13694e6 q^{16} +444500. q^{17} -487431. q^{19} +1.54044e6 q^{20} +643402. q^{22} +67304.5 q^{23} -849319. q^{25} -222541. q^{26} +9.66510e6 q^{28} +3.41813e6 q^{29} +3.45923e6 q^{31} -2.88383e7 q^{32} -1.97701e7 q^{34} +6.92555e6 q^{35} -3.94510e6 q^{37} +2.16796e7 q^{38} -4.45892e7 q^{40} +1.76061e7 q^{41} +3.30231e7 q^{43} -2.12102e7 q^{44} -2.99351e6 q^{46} +3.71271e7 q^{47} +3.09907e6 q^{49} +3.77753e7 q^{50} +7.33620e6 q^{52} +7.86818e7 q^{53} -1.51982e7 q^{55} -2.79764e8 q^{56} -1.52029e8 q^{58} -5.13054e7 q^{59} -1.92723e7 q^{61} -1.53857e8 q^{62} +7.00529e8 q^{64} +5.25678e6 q^{65} +2.14838e8 q^{67} +6.51733e8 q^{68} -3.08029e8 q^{70} -3.25802e8 q^{71} +1.57548e8 q^{73} +1.75467e8 q^{74} -7.14680e8 q^{76} -9.53573e7 q^{77} +3.58634e7 q^{79} +1.19450e9 q^{80} -7.83069e8 q^{82} -8.07984e6 q^{83} +4.67001e8 q^{85} -1.46877e9 q^{86} +6.13945e8 q^{88} -9.43654e8 q^{89} +3.29823e7 q^{91} +9.86830e7 q^{92} -1.65131e9 q^{94} -5.12106e8 q^{95} +9.67052e8 q^{97} -1.37838e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40}+ \cdots + 1935734516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −44.4771 −1.96563 −0.982815 0.184592i \(-0.940904\pi\)
−0.982815 + 0.184592i \(0.940904\pi\)
\(3\) 0 0
\(4\) 1466.22 2.86370
\(5\) 1050.62 0.751764 0.375882 0.926668i \(-0.377340\pi\)
0.375882 + 0.926668i \(0.377340\pi\)
\(6\) 0 0
\(7\) 6591.86 1.03769 0.518844 0.854869i \(-0.326362\pi\)
0.518844 + 0.854869i \(0.326362\pi\)
\(8\) −42440.8 −3.66335
\(9\) 0 0
\(10\) −46728.6 −1.47769
\(11\) −14465.9 −0.297906 −0.148953 0.988844i \(-0.547590\pi\)
−0.148953 + 0.988844i \(0.547590\pi\)
\(12\) 0 0
\(13\) 5003.49 0.0485879 0.0242940 0.999705i \(-0.492266\pi\)
0.0242940 + 0.999705i \(0.492266\pi\)
\(14\) −293187. −2.03971
\(15\) 0 0
\(16\) 1.13694e6 4.33709
\(17\) 444500. 1.29078 0.645389 0.763854i \(-0.276695\pi\)
0.645389 + 0.763854i \(0.276695\pi\)
\(18\) 0 0
\(19\) −487431. −0.858069 −0.429035 0.903288i \(-0.641146\pi\)
−0.429035 + 0.903288i \(0.641146\pi\)
\(20\) 1.54044e6 2.15283
\(21\) 0 0
\(22\) 643402. 0.585572
\(23\) 67304.5 0.0501498 0.0250749 0.999686i \(-0.492018\pi\)
0.0250749 + 0.999686i \(0.492018\pi\)
\(24\) 0 0
\(25\) −849319. −0.434852
\(26\) −222541. −0.0955059
\(27\) 0 0
\(28\) 9.66510e6 2.97163
\(29\) 3.41813e6 0.897425 0.448712 0.893676i \(-0.351883\pi\)
0.448712 + 0.893676i \(0.351883\pi\)
\(30\) 0 0
\(31\) 3.45923e6 0.672747 0.336374 0.941729i \(-0.390800\pi\)
0.336374 + 0.941729i \(0.390800\pi\)
\(32\) −2.88383e7 −4.86177
\(33\) 0 0
\(34\) −1.97701e7 −2.53719
\(35\) 6.92555e6 0.780097
\(36\) 0 0
\(37\) −3.94510e6 −0.346059 −0.173030 0.984917i \(-0.555356\pi\)
−0.173030 + 0.984917i \(0.555356\pi\)
\(38\) 2.16796e7 1.68665
\(39\) 0 0
\(40\) −4.45892e7 −2.75397
\(41\) 1.76061e7 0.973052 0.486526 0.873666i \(-0.338264\pi\)
0.486526 + 0.873666i \(0.338264\pi\)
\(42\) 0 0
\(43\) 3.30231e7 1.47302 0.736512 0.676425i \(-0.236472\pi\)
0.736512 + 0.676425i \(0.236472\pi\)
\(44\) −2.12102e7 −0.853113
\(45\) 0 0
\(46\) −2.99351e6 −0.0985759
\(47\) 3.71271e7 1.10982 0.554908 0.831912i \(-0.312753\pi\)
0.554908 + 0.831912i \(0.312753\pi\)
\(48\) 0 0
\(49\) 3.09907e6 0.0767978
\(50\) 3.77753e7 0.854757
\(51\) 0 0
\(52\) 7.33620e6 0.139141
\(53\) 7.86818e7 1.36972 0.684861 0.728673i \(-0.259863\pi\)
0.684861 + 0.728673i \(0.259863\pi\)
\(54\) 0 0
\(55\) −1.51982e7 −0.223955
\(56\) −2.79764e8 −3.80142
\(57\) 0 0
\(58\) −1.52029e8 −1.76401
\(59\) −5.13054e7 −0.551225 −0.275613 0.961269i \(-0.588881\pi\)
−0.275613 + 0.961269i \(0.588881\pi\)
\(60\) 0 0
\(61\) −1.92723e7 −0.178217 −0.0891085 0.996022i \(-0.528402\pi\)
−0.0891085 + 0.996022i \(0.528402\pi\)
\(62\) −1.53857e8 −1.32237
\(63\) 0 0
\(64\) 7.00529e8 5.21935
\(65\) 5.25678e6 0.0365266
\(66\) 0 0
\(67\) 2.14838e8 1.30249 0.651245 0.758867i \(-0.274247\pi\)
0.651245 + 0.758867i \(0.274247\pi\)
\(68\) 6.51733e8 3.69640
\(69\) 0 0
\(70\) −3.08029e8 −1.53338
\(71\) −3.25802e8 −1.52157 −0.760784 0.649006i \(-0.775185\pi\)
−0.760784 + 0.649006i \(0.775185\pi\)
\(72\) 0 0
\(73\) 1.57548e8 0.649322 0.324661 0.945830i \(-0.394750\pi\)
0.324661 + 0.945830i \(0.394750\pi\)
\(74\) 1.75467e8 0.680225
\(75\) 0 0
\(76\) −7.14680e8 −2.45726
\(77\) −9.53573e7 −0.309133
\(78\) 0 0
\(79\) 3.58634e7 0.103593 0.0517964 0.998658i \(-0.483505\pi\)
0.0517964 + 0.998658i \(0.483505\pi\)
\(80\) 1.19450e9 3.26047
\(81\) 0 0
\(82\) −7.83069e8 −1.91266
\(83\) −8.07984e6 −0.0186875 −0.00934375 0.999956i \(-0.502974\pi\)
−0.00934375 + 0.999956i \(0.502974\pi\)
\(84\) 0 0
\(85\) 4.67001e8 0.970359
\(86\) −1.46877e9 −2.89542
\(87\) 0 0
\(88\) 6.13945e8 1.09133
\(89\) −9.43654e8 −1.59425 −0.797127 0.603812i \(-0.793648\pi\)
−0.797127 + 0.603812i \(0.793648\pi\)
\(90\) 0 0
\(91\) 3.29823e7 0.0504191
\(92\) 9.86830e7 0.143614
\(93\) 0 0
\(94\) −1.65131e9 −2.18149
\(95\) −5.12106e8 −0.645065
\(96\) 0 0
\(97\) 9.67052e8 1.10912 0.554558 0.832145i \(-0.312887\pi\)
0.554558 + 0.832145i \(0.312887\pi\)
\(98\) −1.37838e8 −0.150956
\(99\) 0 0
\(100\) −1.24529e9 −1.24529
\(101\) 1.68398e9 1.61024 0.805121 0.593110i \(-0.202100\pi\)
0.805121 + 0.593110i \(0.202100\pi\)
\(102\) 0 0
\(103\) 1.00611e8 0.0880805 0.0440403 0.999030i \(-0.485977\pi\)
0.0440403 + 0.999030i \(0.485977\pi\)
\(104\) −2.12352e8 −0.177995
\(105\) 0 0
\(106\) −3.49954e9 −2.69237
\(107\) −5.50494e8 −0.406000 −0.203000 0.979179i \(-0.565069\pi\)
−0.203000 + 0.979179i \(0.565069\pi\)
\(108\) 0 0
\(109\) 1.31023e9 0.889055 0.444527 0.895765i \(-0.353372\pi\)
0.444527 + 0.895765i \(0.353372\pi\)
\(110\) 6.75972e8 0.440212
\(111\) 0 0
\(112\) 7.49457e9 4.50055
\(113\) 7.75402e7 0.0447377 0.0223689 0.999750i \(-0.492879\pi\)
0.0223689 + 0.999750i \(0.492879\pi\)
\(114\) 0 0
\(115\) 7.07116e7 0.0377008
\(116\) 5.01172e9 2.56996
\(117\) 0 0
\(118\) 2.28192e9 1.08351
\(119\) 2.93008e9 1.33943
\(120\) 0 0
\(121\) −2.14869e9 −0.911252
\(122\) 8.57176e8 0.350309
\(123\) 0 0
\(124\) 5.07198e9 1.92655
\(125\) −2.94431e9 −1.07867
\(126\) 0 0
\(127\) −1.89287e9 −0.645661 −0.322830 0.946457i \(-0.604634\pi\)
−0.322830 + 0.946457i \(0.604634\pi\)
\(128\) −1.63923e10 −5.39755
\(129\) 0 0
\(130\) −2.33806e8 −0.0717978
\(131\) 4.62187e9 1.37119 0.685593 0.727985i \(-0.259543\pi\)
0.685593 + 0.727985i \(0.259543\pi\)
\(132\) 0 0
\(133\) −3.21308e9 −0.890409
\(134\) −9.55538e9 −2.56022
\(135\) 0 0
\(136\) −1.88649e10 −4.72857
\(137\) 6.82694e9 1.65571 0.827854 0.560944i \(-0.189562\pi\)
0.827854 + 0.560944i \(0.189562\pi\)
\(138\) 0 0
\(139\) 4.87668e9 1.10805 0.554023 0.832501i \(-0.313092\pi\)
0.554023 + 0.832501i \(0.313092\pi\)
\(140\) 1.01544e10 2.23397
\(141\) 0 0
\(142\) 1.44907e10 2.99084
\(143\) −7.23801e7 −0.0144746
\(144\) 0 0
\(145\) 3.59117e9 0.674651
\(146\) −7.00728e9 −1.27633
\(147\) 0 0
\(148\) −5.78437e9 −0.991011
\(149\) −7.64937e9 −1.27142 −0.635708 0.771930i \(-0.719292\pi\)
−0.635708 + 0.771930i \(0.719292\pi\)
\(150\) 0 0
\(151\) −4.19589e9 −0.656791 −0.328396 0.944540i \(-0.606508\pi\)
−0.328396 + 0.944540i \(0.606508\pi\)
\(152\) 2.06870e10 3.14341
\(153\) 0 0
\(154\) 4.24122e9 0.607642
\(155\) 3.63434e9 0.505747
\(156\) 0 0
\(157\) −7.60479e8 −0.0998938 −0.0499469 0.998752i \(-0.515905\pi\)
−0.0499469 + 0.998752i \(0.515905\pi\)
\(158\) −1.59510e9 −0.203625
\(159\) 0 0
\(160\) −3.02981e10 −3.65490
\(161\) 4.43662e8 0.0520398
\(162\) 0 0
\(163\) −7.55113e9 −0.837853 −0.418926 0.908020i \(-0.637593\pi\)
−0.418926 + 0.908020i \(0.637593\pi\)
\(164\) 2.58143e10 2.78653
\(165\) 0 0
\(166\) 3.59368e8 0.0367327
\(167\) −8.05764e9 −0.801648 −0.400824 0.916155i \(-0.631276\pi\)
−0.400824 + 0.916155i \(0.631276\pi\)
\(168\) 0 0
\(169\) −1.05795e10 −0.997639
\(170\) −2.07709e10 −1.90737
\(171\) 0 0
\(172\) 4.84190e10 4.21830
\(173\) −4.12318e9 −0.349965 −0.174982 0.984572i \(-0.555987\pi\)
−0.174982 + 0.984572i \(0.555987\pi\)
\(174\) 0 0
\(175\) −5.59860e9 −0.451240
\(176\) −1.64469e10 −1.29204
\(177\) 0 0
\(178\) 4.19710e10 3.13371
\(179\) −2.46170e10 −1.79224 −0.896120 0.443811i \(-0.853626\pi\)
−0.896120 + 0.443811i \(0.853626\pi\)
\(180\) 0 0
\(181\) −4.17661e9 −0.289248 −0.144624 0.989487i \(-0.546197\pi\)
−0.144624 + 0.989487i \(0.546197\pi\)
\(182\) −1.46696e9 −0.0991054
\(183\) 0 0
\(184\) −2.85646e9 −0.183716
\(185\) −4.14481e9 −0.260155
\(186\) 0 0
\(187\) −6.43009e9 −0.384530
\(188\) 5.44364e10 3.17818
\(189\) 0 0
\(190\) 2.27770e10 1.26796
\(191\) −1.46430e10 −0.796122 −0.398061 0.917359i \(-0.630317\pi\)
−0.398061 + 0.917359i \(0.630317\pi\)
\(192\) 0 0
\(193\) 1.58666e10 0.823144 0.411572 0.911377i \(-0.364980\pi\)
0.411572 + 0.911377i \(0.364980\pi\)
\(194\) −4.30117e10 −2.18011
\(195\) 0 0
\(196\) 4.54390e9 0.219926
\(197\) 4.01582e9 0.189966 0.0949832 0.995479i \(-0.469720\pi\)
0.0949832 + 0.995479i \(0.469720\pi\)
\(198\) 0 0
\(199\) −3.90309e10 −1.76429 −0.882144 0.470980i \(-0.843901\pi\)
−0.882144 + 0.470980i \(0.843901\pi\)
\(200\) 3.60458e10 1.59301
\(201\) 0 0
\(202\) −7.48987e10 −3.16514
\(203\) 2.25319e10 0.931248
\(204\) 0 0
\(205\) 1.84973e10 0.731505
\(206\) −4.47491e9 −0.173134
\(207\) 0 0
\(208\) 5.68869e9 0.210730
\(209\) 7.05114e9 0.255624
\(210\) 0 0
\(211\) 2.70099e10 0.938108 0.469054 0.883170i \(-0.344595\pi\)
0.469054 + 0.883170i \(0.344595\pi\)
\(212\) 1.15364e11 3.92248
\(213\) 0 0
\(214\) 2.44844e10 0.798046
\(215\) 3.46948e10 1.10737
\(216\) 0 0
\(217\) 2.28028e10 0.698102
\(218\) −5.82753e10 −1.74755
\(219\) 0 0
\(220\) −2.22838e10 −0.641340
\(221\) 2.22405e9 0.0627162
\(222\) 0 0
\(223\) −5.73036e10 −1.55171 −0.775854 0.630912i \(-0.782681\pi\)
−0.775854 + 0.630912i \(0.782681\pi\)
\(224\) −1.90098e11 −5.04500
\(225\) 0 0
\(226\) −3.44877e9 −0.0879379
\(227\) −3.50057e10 −0.875029 −0.437514 0.899211i \(-0.644141\pi\)
−0.437514 + 0.899211i \(0.644141\pi\)
\(228\) 0 0
\(229\) 6.47530e10 1.55597 0.777984 0.628284i \(-0.216243\pi\)
0.777984 + 0.628284i \(0.216243\pi\)
\(230\) −3.14505e9 −0.0741058
\(231\) 0 0
\(232\) −1.45068e11 −3.28758
\(233\) 5.91061e10 1.31380 0.656902 0.753976i \(-0.271866\pi\)
0.656902 + 0.753976i \(0.271866\pi\)
\(234\) 0 0
\(235\) 3.90066e10 0.834319
\(236\) −7.52248e10 −1.57855
\(237\) 0 0
\(238\) −1.30322e11 −2.63281
\(239\) 4.04702e10 0.802315 0.401158 0.916009i \(-0.368608\pi\)
0.401158 + 0.916009i \(0.368608\pi\)
\(240\) 0 0
\(241\) 3.37089e9 0.0643677 0.0321839 0.999482i \(-0.489754\pi\)
0.0321839 + 0.999482i \(0.489754\pi\)
\(242\) 9.55674e10 1.79119
\(243\) 0 0
\(244\) −2.82573e10 −0.510361
\(245\) 3.25595e9 0.0577338
\(246\) 0 0
\(247\) −2.43886e9 −0.0416918
\(248\) −1.46813e11 −2.46451
\(249\) 0 0
\(250\) 1.30954e11 2.12026
\(251\) 4.69295e10 0.746301 0.373151 0.927771i \(-0.378277\pi\)
0.373151 + 0.927771i \(0.378277\pi\)
\(252\) 0 0
\(253\) −9.73621e8 −0.0149399
\(254\) 8.41895e10 1.26913
\(255\) 0 0
\(256\) 3.70414e11 5.39023
\(257\) 1.74946e10 0.250152 0.125076 0.992147i \(-0.460083\pi\)
0.125076 + 0.992147i \(0.460083\pi\)
\(258\) 0 0
\(259\) −2.60056e10 −0.359102
\(260\) 7.70757e9 0.104601
\(261\) 0 0
\(262\) −2.05567e11 −2.69525
\(263\) −1.00629e11 −1.29695 −0.648476 0.761235i \(-0.724593\pi\)
−0.648476 + 0.761235i \(0.724593\pi\)
\(264\) 0 0
\(265\) 8.26648e10 1.02971
\(266\) 1.42909e11 1.75021
\(267\) 0 0
\(268\) 3.14999e11 3.72995
\(269\) −2.18445e10 −0.254365 −0.127182 0.991879i \(-0.540593\pi\)
−0.127182 + 0.991879i \(0.540593\pi\)
\(270\) 0 0
\(271\) 5.19462e10 0.585049 0.292524 0.956258i \(-0.405505\pi\)
0.292524 + 0.956258i \(0.405505\pi\)
\(272\) 5.05371e11 5.59822
\(273\) 0 0
\(274\) −3.03643e11 −3.25451
\(275\) 1.22862e10 0.129545
\(276\) 0 0
\(277\) −8.37704e9 −0.0854932 −0.0427466 0.999086i \(-0.513611\pi\)
−0.0427466 + 0.999086i \(0.513611\pi\)
\(278\) −2.16901e11 −2.17801
\(279\) 0 0
\(280\) −2.93926e11 −2.85777
\(281\) −8.16325e10 −0.781061 −0.390530 0.920590i \(-0.627708\pi\)
−0.390530 + 0.920590i \(0.627708\pi\)
\(282\) 0 0
\(283\) −6.11475e10 −0.566683 −0.283341 0.959019i \(-0.591443\pi\)
−0.283341 + 0.959019i \(0.591443\pi\)
\(284\) −4.77696e11 −4.35732
\(285\) 0 0
\(286\) 3.21926e9 0.0284517
\(287\) 1.16057e11 1.00972
\(288\) 0 0
\(289\) 7.89921e10 0.666106
\(290\) −1.59725e11 −1.32612
\(291\) 0 0
\(292\) 2.30999e11 1.85946
\(293\) −1.73459e11 −1.37497 −0.687486 0.726198i \(-0.741286\pi\)
−0.687486 + 0.726198i \(0.741286\pi\)
\(294\) 0 0
\(295\) −5.39026e10 −0.414391
\(296\) 1.67433e11 1.26774
\(297\) 0 0
\(298\) 3.40222e11 2.49913
\(299\) 3.36758e8 0.00243667
\(300\) 0 0
\(301\) 2.17684e11 1.52854
\(302\) 1.86621e11 1.29101
\(303\) 0 0
\(304\) −5.54182e11 −3.72153
\(305\) −2.02479e10 −0.133977
\(306\) 0 0
\(307\) −1.89397e11 −1.21689 −0.608443 0.793598i \(-0.708205\pi\)
−0.608443 + 0.793598i \(0.708205\pi\)
\(308\) −1.39814e11 −0.885266
\(309\) 0 0
\(310\) −1.61645e11 −0.994112
\(311\) −1.47676e11 −0.895137 −0.447569 0.894250i \(-0.647710\pi\)
−0.447569 + 0.894250i \(0.647710\pi\)
\(312\) 0 0
\(313\) 4.64248e9 0.0273402 0.0136701 0.999907i \(-0.495649\pi\)
0.0136701 + 0.999907i \(0.495649\pi\)
\(314\) 3.38239e10 0.196354
\(315\) 0 0
\(316\) 5.25835e10 0.296659
\(317\) −1.76364e11 −0.980943 −0.490471 0.871457i \(-0.663175\pi\)
−0.490471 + 0.871457i \(0.663175\pi\)
\(318\) 0 0
\(319\) −4.94464e10 −0.267348
\(320\) 7.35991e11 3.92372
\(321\) 0 0
\(322\) −1.97328e10 −0.102291
\(323\) −2.16663e11 −1.10758
\(324\) 0 0
\(325\) −4.24956e9 −0.0211285
\(326\) 3.35853e11 1.64691
\(327\) 0 0
\(328\) −7.47217e11 −3.56463
\(329\) 2.44737e11 1.15164
\(330\) 0 0
\(331\) −3.09857e11 −1.41885 −0.709423 0.704783i \(-0.751044\pi\)
−0.709423 + 0.704783i \(0.751044\pi\)
\(332\) −1.18468e10 −0.0535155
\(333\) 0 0
\(334\) 3.58381e11 1.57574
\(335\) 2.25714e11 0.979165
\(336\) 0 0
\(337\) 3.13658e11 1.32471 0.662356 0.749189i \(-0.269557\pi\)
0.662356 + 0.749189i \(0.269557\pi\)
\(338\) 4.70544e11 1.96099
\(339\) 0 0
\(340\) 6.84724e11 2.77882
\(341\) −5.00409e10 −0.200415
\(342\) 0 0
\(343\) −2.45577e11 −0.957996
\(344\) −1.40153e12 −5.39620
\(345\) 0 0
\(346\) 1.83387e11 0.687902
\(347\) 8.10841e10 0.300229 0.150115 0.988669i \(-0.452036\pi\)
0.150115 + 0.988669i \(0.452036\pi\)
\(348\) 0 0
\(349\) 2.68777e11 0.969791 0.484896 0.874572i \(-0.338858\pi\)
0.484896 + 0.874572i \(0.338858\pi\)
\(350\) 2.49010e11 0.886972
\(351\) 0 0
\(352\) 4.17172e11 1.44835
\(353\) 3.92960e11 1.34698 0.673491 0.739195i \(-0.264794\pi\)
0.673491 + 0.739195i \(0.264794\pi\)
\(354\) 0 0
\(355\) −3.42295e11 −1.14386
\(356\) −1.38360e12 −4.56547
\(357\) 0 0
\(358\) 1.09489e12 3.52288
\(359\) −2.48781e11 −0.790484 −0.395242 0.918577i \(-0.629339\pi\)
−0.395242 + 0.918577i \(0.629339\pi\)
\(360\) 0 0
\(361\) −8.50984e10 −0.263718
\(362\) 1.85764e11 0.568555
\(363\) 0 0
\(364\) 4.83592e10 0.144385
\(365\) 1.65523e11 0.488136
\(366\) 0 0
\(367\) −3.25248e11 −0.935875 −0.467937 0.883762i \(-0.655003\pi\)
−0.467937 + 0.883762i \(0.655003\pi\)
\(368\) 7.65214e10 0.217504
\(369\) 0 0
\(370\) 1.84349e11 0.511368
\(371\) 5.18660e11 1.42135
\(372\) 0 0
\(373\) −3.55232e11 −0.950217 −0.475108 0.879927i \(-0.657591\pi\)
−0.475108 + 0.879927i \(0.657591\pi\)
\(374\) 2.85992e11 0.755844
\(375\) 0 0
\(376\) −1.57571e12 −4.06565
\(377\) 1.71026e10 0.0436040
\(378\) 0 0
\(379\) −4.17590e11 −1.03962 −0.519809 0.854283i \(-0.673997\pi\)
−0.519809 + 0.854283i \(0.673997\pi\)
\(380\) −7.50858e11 −1.84727
\(381\) 0 0
\(382\) 6.51279e11 1.56488
\(383\) 8.39723e9 0.0199408 0.00997038 0.999950i \(-0.496826\pi\)
0.00997038 + 0.999950i \(0.496826\pi\)
\(384\) 0 0
\(385\) −1.00184e11 −0.232395
\(386\) −7.05701e11 −1.61800
\(387\) 0 0
\(388\) 1.41791e12 3.17618
\(389\) 4.47803e11 0.991549 0.495774 0.868451i \(-0.334884\pi\)
0.495774 + 0.868451i \(0.334884\pi\)
\(390\) 0 0
\(391\) 2.99168e10 0.0647322
\(392\) −1.31527e11 −0.281337
\(393\) 0 0
\(394\) −1.78612e11 −0.373404
\(395\) 3.76788e10 0.0778773
\(396\) 0 0
\(397\) 8.93449e11 1.80515 0.902573 0.430537i \(-0.141676\pi\)
0.902573 + 0.430537i \(0.141676\pi\)
\(398\) 1.73598e12 3.46794
\(399\) 0 0
\(400\) −9.65628e11 −1.88599
\(401\) −2.21707e10 −0.0428183 −0.0214092 0.999771i \(-0.506815\pi\)
−0.0214092 + 0.999771i \(0.506815\pi\)
\(402\) 0 0
\(403\) 1.73082e10 0.0326874
\(404\) 2.46908e12 4.61126
\(405\) 0 0
\(406\) −1.00215e12 −1.83049
\(407\) 5.70695e10 0.103093
\(408\) 0 0
\(409\) 4.79724e11 0.847688 0.423844 0.905735i \(-0.360680\pi\)
0.423844 + 0.905735i \(0.360680\pi\)
\(410\) −8.22709e11 −1.43787
\(411\) 0 0
\(412\) 1.47518e11 0.252236
\(413\) −3.38198e11 −0.572000
\(414\) 0 0
\(415\) −8.48885e9 −0.0140486
\(416\) −1.44292e11 −0.236223
\(417\) 0 0
\(418\) −3.13614e11 −0.502462
\(419\) −8.35323e11 −1.32401 −0.662005 0.749500i \(-0.730294\pi\)
−0.662005 + 0.749500i \(0.730294\pi\)
\(420\) 0 0
\(421\) 2.06281e11 0.320029 0.160014 0.987115i \(-0.448846\pi\)
0.160014 + 0.987115i \(0.448846\pi\)
\(422\) −1.20133e12 −1.84397
\(423\) 0 0
\(424\) −3.33932e12 −5.01778
\(425\) −3.77522e11 −0.561297
\(426\) 0 0
\(427\) −1.27040e11 −0.184934
\(428\) −8.07144e11 −1.16266
\(429\) 0 0
\(430\) −1.54312e12 −2.17667
\(431\) −3.46311e11 −0.483414 −0.241707 0.970349i \(-0.577707\pi\)
−0.241707 + 0.970349i \(0.577707\pi\)
\(432\) 0 0
\(433\) −7.41282e9 −0.0101342 −0.00506708 0.999987i \(-0.501613\pi\)
−0.00506708 + 0.999987i \(0.501613\pi\)
\(434\) −1.01420e12 −1.37221
\(435\) 0 0
\(436\) 1.92108e12 2.54599
\(437\) −3.28063e10 −0.0430320
\(438\) 0 0
\(439\) 9.59814e11 1.23338 0.616690 0.787206i \(-0.288473\pi\)
0.616690 + 0.787206i \(0.288473\pi\)
\(440\) 6.45024e11 0.820425
\(441\) 0 0
\(442\) −9.89194e10 −0.123277
\(443\) −2.88512e11 −0.355915 −0.177958 0.984038i \(-0.556949\pi\)
−0.177958 + 0.984038i \(0.556949\pi\)
\(444\) 0 0
\(445\) −9.91423e11 −1.19850
\(446\) 2.54870e12 3.05008
\(447\) 0 0
\(448\) 4.61780e12 5.41606
\(449\) −1.00029e12 −1.16150 −0.580749 0.814083i \(-0.697240\pi\)
−0.580749 + 0.814083i \(0.697240\pi\)
\(450\) 0 0
\(451\) −2.54688e11 −0.289878
\(452\) 1.13691e11 0.128116
\(453\) 0 0
\(454\) 1.55695e12 1.71998
\(455\) 3.46520e10 0.0379033
\(456\) 0 0
\(457\) 2.83855e11 0.304420 0.152210 0.988348i \(-0.451361\pi\)
0.152210 + 0.988348i \(0.451361\pi\)
\(458\) −2.88003e12 −3.05846
\(459\) 0 0
\(460\) 1.03678e11 0.107964
\(461\) 6.45819e8 0.000665973 0 0.000332987 1.00000i \(-0.499894\pi\)
0.000332987 1.00000i \(0.499894\pi\)
\(462\) 0 0
\(463\) −4.83071e11 −0.488536 −0.244268 0.969708i \(-0.578548\pi\)
−0.244268 + 0.969708i \(0.578548\pi\)
\(464\) 3.88622e12 3.89222
\(465\) 0 0
\(466\) −2.62887e12 −2.58245
\(467\) 7.10128e11 0.690893 0.345447 0.938438i \(-0.387727\pi\)
0.345447 + 0.938438i \(0.387727\pi\)
\(468\) 0 0
\(469\) 1.41618e12 1.35158
\(470\) −1.73490e12 −1.63996
\(471\) 0 0
\(472\) 2.17744e12 2.01933
\(473\) −4.77709e11 −0.438822
\(474\) 0 0
\(475\) 4.13985e11 0.373133
\(476\) 4.29613e12 3.83572
\(477\) 0 0
\(478\) −1.80000e12 −1.57706
\(479\) −1.54619e10 −0.0134200 −0.00671002 0.999977i \(-0.502136\pi\)
−0.00671002 + 0.999977i \(0.502136\pi\)
\(480\) 0 0
\(481\) −1.97393e10 −0.0168143
\(482\) −1.49928e11 −0.126523
\(483\) 0 0
\(484\) −3.15044e12 −2.60956
\(485\) 1.01601e12 0.833793
\(486\) 0 0
\(487\) −5.65405e11 −0.455490 −0.227745 0.973721i \(-0.573135\pi\)
−0.227745 + 0.973721i \(0.573135\pi\)
\(488\) 8.17932e11 0.652872
\(489\) 0 0
\(490\) −1.44815e11 −0.113483
\(491\) 3.72733e11 0.289421 0.144711 0.989474i \(-0.453775\pi\)
0.144711 + 0.989474i \(0.453775\pi\)
\(492\) 0 0
\(493\) 1.51936e12 1.15838
\(494\) 1.08473e11 0.0819506
\(495\) 0 0
\(496\) 3.93295e12 2.91777
\(497\) −2.14764e12 −1.57891
\(498\) 0 0
\(499\) −3.57615e11 −0.258204 −0.129102 0.991631i \(-0.541209\pi\)
−0.129102 + 0.991631i \(0.541209\pi\)
\(500\) −4.31699e12 −3.08899
\(501\) 0 0
\(502\) −2.08729e12 −1.46695
\(503\) 1.08145e12 0.753267 0.376633 0.926362i \(-0.377082\pi\)
0.376633 + 0.926362i \(0.377082\pi\)
\(504\) 0 0
\(505\) 1.76923e12 1.21052
\(506\) 4.33039e10 0.0293663
\(507\) 0 0
\(508\) −2.77536e12 −1.84898
\(509\) 6.97300e11 0.460458 0.230229 0.973137i \(-0.426053\pi\)
0.230229 + 0.973137i \(0.426053\pi\)
\(510\) 0 0
\(511\) 1.03853e12 0.673794
\(512\) −8.08206e12 −5.19765
\(513\) 0 0
\(514\) −7.78108e11 −0.491706
\(515\) 1.05705e11 0.0662157
\(516\) 0 0
\(517\) −5.37078e11 −0.330621
\(518\) 1.15665e12 0.705861
\(519\) 0 0
\(520\) −2.23102e11 −0.133810
\(521\) 1.96135e12 1.16623 0.583116 0.812389i \(-0.301833\pi\)
0.583116 + 0.812389i \(0.301833\pi\)
\(522\) 0 0
\(523\) −6.03647e10 −0.0352797 −0.0176399 0.999844i \(-0.505615\pi\)
−0.0176399 + 0.999844i \(0.505615\pi\)
\(524\) 6.77665e12 3.92667
\(525\) 0 0
\(526\) 4.47570e12 2.54933
\(527\) 1.53763e12 0.868367
\(528\) 0 0
\(529\) −1.79662e12 −0.997485
\(530\) −3.67669e12 −2.02402
\(531\) 0 0
\(532\) −4.71107e12 −2.54987
\(533\) 8.80920e10 0.0472785
\(534\) 0 0
\(535\) −5.78361e11 −0.305216
\(536\) −9.11790e12 −4.77148
\(537\) 0 0
\(538\) 9.71580e11 0.499987
\(539\) −4.48308e10 −0.0228785
\(540\) 0 0
\(541\) −1.58437e12 −0.795185 −0.397592 0.917562i \(-0.630154\pi\)
−0.397592 + 0.917562i \(0.630154\pi\)
\(542\) −2.31042e12 −1.14999
\(543\) 0 0
\(544\) −1.28186e13 −6.27546
\(545\) 1.37656e12 0.668359
\(546\) 0 0
\(547\) 1.47236e12 0.703189 0.351595 0.936152i \(-0.385640\pi\)
0.351595 + 0.936152i \(0.385640\pi\)
\(548\) 1.00098e13 4.74146
\(549\) 0 0
\(550\) −5.46454e11 −0.254637
\(551\) −1.66611e12 −0.770052
\(552\) 0 0
\(553\) 2.36407e11 0.107497
\(554\) 3.72587e11 0.168048
\(555\) 0 0
\(556\) 7.15027e12 3.17311
\(557\) 1.23459e12 0.543470 0.271735 0.962372i \(-0.412403\pi\)
0.271735 + 0.962372i \(0.412403\pi\)
\(558\) 0 0
\(559\) 1.65231e11 0.0715711
\(560\) 7.87396e12 3.38335
\(561\) 0 0
\(562\) 3.63078e12 1.53528
\(563\) −3.22798e11 −0.135407 −0.0677037 0.997705i \(-0.521567\pi\)
−0.0677037 + 0.997705i \(0.521567\pi\)
\(564\) 0 0
\(565\) 8.14654e10 0.0336322
\(566\) 2.71967e12 1.11389
\(567\) 0 0
\(568\) 1.38273e13 5.57404
\(569\) 1.51796e12 0.607091 0.303546 0.952817i \(-0.401830\pi\)
0.303546 + 0.952817i \(0.401830\pi\)
\(570\) 0 0
\(571\) 6.04069e11 0.237807 0.118903 0.992906i \(-0.462062\pi\)
0.118903 + 0.992906i \(0.462062\pi\)
\(572\) −1.06125e11 −0.0414510
\(573\) 0 0
\(574\) −5.16188e12 −1.98475
\(575\) −5.71630e10 −0.0218077
\(576\) 0 0
\(577\) −4.16132e12 −1.56293 −0.781466 0.623948i \(-0.785528\pi\)
−0.781466 + 0.623948i \(0.785528\pi\)
\(578\) −3.51334e12 −1.30932
\(579\) 0 0
\(580\) 5.26542e12 1.93200
\(581\) −5.32612e10 −0.0193918
\(582\) 0 0
\(583\) −1.13820e12 −0.408048
\(584\) −6.68646e12 −2.37869
\(585\) 0 0
\(586\) 7.71498e12 2.70269
\(587\) −1.29792e12 −0.451207 −0.225604 0.974219i \(-0.572435\pi\)
−0.225604 + 0.974219i \(0.572435\pi\)
\(588\) 0 0
\(589\) −1.68614e12 −0.577264
\(590\) 2.39743e12 0.814540
\(591\) 0 0
\(592\) −4.48536e12 −1.50089
\(593\) −6.89642e11 −0.229022 −0.114511 0.993422i \(-0.536530\pi\)
−0.114511 + 0.993422i \(0.536530\pi\)
\(594\) 0 0
\(595\) 3.07841e12 1.00693
\(596\) −1.12156e13 −3.64096
\(597\) 0 0
\(598\) −1.49780e10 −0.00478960
\(599\) 6.81664e11 0.216346 0.108173 0.994132i \(-0.465500\pi\)
0.108173 + 0.994132i \(0.465500\pi\)
\(600\) 0 0
\(601\) −4.73397e12 −1.48010 −0.740049 0.672553i \(-0.765198\pi\)
−0.740049 + 0.672553i \(0.765198\pi\)
\(602\) −9.68195e12 −3.00454
\(603\) 0 0
\(604\) −6.15208e12 −1.88086
\(605\) −2.25745e12 −0.685046
\(606\) 0 0
\(607\) −3.38422e12 −1.01183 −0.505917 0.862582i \(-0.668846\pi\)
−0.505917 + 0.862582i \(0.668846\pi\)
\(608\) 1.40567e13 4.17173
\(609\) 0 0
\(610\) 9.00568e11 0.263349
\(611\) 1.85765e11 0.0539237
\(612\) 0 0
\(613\) −2.77900e12 −0.794906 −0.397453 0.917622i \(-0.630106\pi\)
−0.397453 + 0.917622i \(0.630106\pi\)
\(614\) 8.42382e12 2.39195
\(615\) 0 0
\(616\) 4.04704e12 1.13246
\(617\) −5.70613e12 −1.58511 −0.792554 0.609802i \(-0.791249\pi\)
−0.792554 + 0.609802i \(0.791249\pi\)
\(618\) 0 0
\(619\) 5.70022e12 1.56057 0.780285 0.625424i \(-0.215074\pi\)
0.780285 + 0.625424i \(0.215074\pi\)
\(620\) 5.32873e12 1.44831
\(621\) 0 0
\(622\) 6.56823e12 1.75951
\(623\) −6.22044e12 −1.65434
\(624\) 0 0
\(625\) −1.43453e12 −0.376053
\(626\) −2.06484e11 −0.0537406
\(627\) 0 0
\(628\) −1.11503e12 −0.286066
\(629\) −1.75360e12 −0.446685
\(630\) 0 0
\(631\) −7.58053e11 −0.190356 −0.0951782 0.995460i \(-0.530342\pi\)
−0.0951782 + 0.995460i \(0.530342\pi\)
\(632\) −1.52207e12 −0.379497
\(633\) 0 0
\(634\) 7.84417e12 1.92817
\(635\) −1.98869e12 −0.485384
\(636\) 0 0
\(637\) 1.55062e10 0.00373144
\(638\) 2.19924e12 0.525507
\(639\) 0 0
\(640\) −1.72222e13 −4.05768
\(641\) 3.07750e12 0.720007 0.360003 0.932951i \(-0.382776\pi\)
0.360003 + 0.932951i \(0.382776\pi\)
\(642\) 0 0
\(643\) 2.10425e12 0.485455 0.242727 0.970095i \(-0.421958\pi\)
0.242727 + 0.970095i \(0.421958\pi\)
\(644\) 6.50505e11 0.149027
\(645\) 0 0
\(646\) 9.63655e12 2.17709
\(647\) 1.41884e12 0.318319 0.159160 0.987253i \(-0.449122\pi\)
0.159160 + 0.987253i \(0.449122\pi\)
\(648\) 0 0
\(649\) 7.42180e11 0.164213
\(650\) 1.89008e11 0.0415309
\(651\) 0 0
\(652\) −1.10716e13 −2.39936
\(653\) −3.40288e11 −0.0732382 −0.0366191 0.999329i \(-0.511659\pi\)
−0.0366191 + 0.999329i \(0.511659\pi\)
\(654\) 0 0
\(655\) 4.85583e12 1.03081
\(656\) 2.00171e13 4.22022
\(657\) 0 0
\(658\) −1.08852e13 −2.26371
\(659\) 4.62024e12 0.954290 0.477145 0.878825i \(-0.341672\pi\)
0.477145 + 0.878825i \(0.341672\pi\)
\(660\) 0 0
\(661\) −4.63497e12 −0.944365 −0.472183 0.881501i \(-0.656534\pi\)
−0.472183 + 0.881501i \(0.656534\pi\)
\(662\) 1.37815e13 2.78893
\(663\) 0 0
\(664\) 3.42915e11 0.0684589
\(665\) −3.37573e12 −0.669377
\(666\) 0 0
\(667\) 2.30056e11 0.0450057
\(668\) −1.18142e13 −2.29568
\(669\) 0 0
\(670\) −1.00391e13 −1.92468
\(671\) 2.78791e11 0.0530918
\(672\) 0 0
\(673\) −1.00517e13 −1.88874 −0.944370 0.328886i \(-0.893327\pi\)
−0.944370 + 0.328886i \(0.893327\pi\)
\(674\) −1.39506e13 −2.60390
\(675\) 0 0
\(676\) −1.55118e13 −2.85694
\(677\) 8.83179e12 1.61584 0.807922 0.589289i \(-0.200592\pi\)
0.807922 + 0.589289i \(0.200592\pi\)
\(678\) 0 0
\(679\) 6.37467e12 1.15092
\(680\) −1.98199e13 −3.55477
\(681\) 0 0
\(682\) 2.22568e12 0.393942
\(683\) −1.10788e13 −1.94805 −0.974027 0.226430i \(-0.927294\pi\)
−0.974027 + 0.226430i \(0.927294\pi\)
\(684\) 0 0
\(685\) 7.17253e12 1.24470
\(686\) 1.09226e13 1.88307
\(687\) 0 0
\(688\) 3.75454e13 6.38864
\(689\) 3.93684e11 0.0665520
\(690\) 0 0
\(691\) −6.32931e11 −0.105610 −0.0528050 0.998605i \(-0.516816\pi\)
−0.0528050 + 0.998605i \(0.516816\pi\)
\(692\) −6.04547e12 −1.00220
\(693\) 0 0
\(694\) −3.60639e12 −0.590140
\(695\) 5.12355e12 0.832988
\(696\) 0 0
\(697\) 7.82591e12 1.25599
\(698\) −1.19545e13 −1.90625
\(699\) 0 0
\(700\) −8.20875e12 −1.29222
\(701\) 3.84382e12 0.601219 0.300609 0.953747i \(-0.402810\pi\)
0.300609 + 0.953747i \(0.402810\pi\)
\(702\) 0 0
\(703\) 1.92297e12 0.296943
\(704\) −1.01338e13 −1.55487
\(705\) 0 0
\(706\) −1.74777e13 −2.64767
\(707\) 1.11006e13 1.67093
\(708\) 0 0
\(709\) −6.15935e12 −0.915433 −0.457716 0.889098i \(-0.651333\pi\)
−0.457716 + 0.889098i \(0.651333\pi\)
\(710\) 1.52243e13 2.24840
\(711\) 0 0
\(712\) 4.00494e13 5.84031
\(713\) 2.32822e11 0.0337381
\(714\) 0 0
\(715\) −7.60441e10 −0.0108815
\(716\) −3.60938e13 −5.13245
\(717\) 0 0
\(718\) 1.10651e13 1.55380
\(719\) 6.51194e12 0.908721 0.454361 0.890818i \(-0.349868\pi\)
0.454361 + 0.890818i \(0.349868\pi\)
\(720\) 0 0
\(721\) 6.63217e11 0.0914001
\(722\) 3.78493e12 0.518371
\(723\) 0 0
\(724\) −6.12381e12 −0.828321
\(725\) −2.90309e12 −0.390247
\(726\) 0 0
\(727\) 2.05785e12 0.273218 0.136609 0.990625i \(-0.456380\pi\)
0.136609 + 0.990625i \(0.456380\pi\)
\(728\) −1.39980e12 −0.184703
\(729\) 0 0
\(730\) −7.36200e12 −0.959496
\(731\) 1.46788e13 1.90135
\(732\) 0 0
\(733\) 7.96245e12 1.01878 0.509388 0.860537i \(-0.329872\pi\)
0.509388 + 0.860537i \(0.329872\pi\)
\(734\) 1.44661e13 1.83958
\(735\) 0 0
\(736\) −1.94095e12 −0.243817
\(737\) −3.10783e12 −0.388019
\(738\) 0 0
\(739\) −4.20258e12 −0.518341 −0.259171 0.965832i \(-0.583449\pi\)
−0.259171 + 0.965832i \(0.583449\pi\)
\(740\) −6.07719e12 −0.745006
\(741\) 0 0
\(742\) −2.30685e13 −2.79384
\(743\) −5.46810e12 −0.658244 −0.329122 0.944287i \(-0.606753\pi\)
−0.329122 + 0.944287i \(0.606753\pi\)
\(744\) 0 0
\(745\) −8.03659e12 −0.955804
\(746\) 1.57997e13 1.86777
\(747\) 0 0
\(748\) −9.42791e12 −1.10118
\(749\) −3.62878e12 −0.421302
\(750\) 0 0
\(751\) 7.47523e12 0.857521 0.428760 0.903418i \(-0.358950\pi\)
0.428760 + 0.903418i \(0.358950\pi\)
\(752\) 4.22114e13 4.81338
\(753\) 0 0
\(754\) −7.60675e11 −0.0857093
\(755\) −4.40829e12 −0.493752
\(756\) 0 0
\(757\) 1.40395e12 0.155389 0.0776947 0.996977i \(-0.475244\pi\)
0.0776947 + 0.996977i \(0.475244\pi\)
\(758\) 1.85732e13 2.04350
\(759\) 0 0
\(760\) 2.17342e13 2.36310
\(761\) 7.95047e12 0.859334 0.429667 0.902987i \(-0.358631\pi\)
0.429667 + 0.902987i \(0.358631\pi\)
\(762\) 0 0
\(763\) 8.63686e12 0.922562
\(764\) −2.14698e13 −2.27986
\(765\) 0 0
\(766\) −3.73485e11 −0.0391961
\(767\) −2.56706e11 −0.0267829
\(768\) 0 0
\(769\) −6.95899e12 −0.717592 −0.358796 0.933416i \(-0.616813\pi\)
−0.358796 + 0.933416i \(0.616813\pi\)
\(770\) 4.45592e12 0.456803
\(771\) 0 0
\(772\) 2.32639e13 2.35724
\(773\) −1.22597e13 −1.23501 −0.617506 0.786566i \(-0.711857\pi\)
−0.617506 + 0.786566i \(0.711857\pi\)
\(774\) 0 0
\(775\) −2.93799e12 −0.292545
\(776\) −4.10424e13 −4.06308
\(777\) 0 0
\(778\) −1.99170e13 −1.94902
\(779\) −8.58177e12 −0.834945
\(780\) 0 0
\(781\) 4.71302e12 0.453283
\(782\) −1.33062e12 −0.127240
\(783\) 0 0
\(784\) 3.52346e12 0.333079
\(785\) −7.98975e11 −0.0750965
\(786\) 0 0
\(787\) 1.19814e13 1.11332 0.556662 0.830739i \(-0.312082\pi\)
0.556662 + 0.830739i \(0.312082\pi\)
\(788\) 5.88807e12 0.544007
\(789\) 0 0
\(790\) −1.67585e12 −0.153078
\(791\) 5.11135e11 0.0464238
\(792\) 0 0
\(793\) −9.64288e10 −0.00865919
\(794\) −3.97380e13 −3.54825
\(795\) 0 0
\(796\) −5.72277e13 −5.05240
\(797\) 2.90159e12 0.254726 0.127363 0.991856i \(-0.459349\pi\)
0.127363 + 0.991856i \(0.459349\pi\)
\(798\) 0 0
\(799\) 1.65030e13 1.43253
\(800\) 2.44929e13 2.11415
\(801\) 0 0
\(802\) 9.86089e11 0.0841650
\(803\) −2.27908e12 −0.193437
\(804\) 0 0
\(805\) 4.66121e11 0.0391217
\(806\) −7.69821e11 −0.0642513
\(807\) 0 0
\(808\) −7.14695e13 −5.89888
\(809\) 1.25435e13 1.02956 0.514779 0.857323i \(-0.327874\pi\)
0.514779 + 0.857323i \(0.327874\pi\)
\(810\) 0 0
\(811\) −1.31788e13 −1.06975 −0.534874 0.844932i \(-0.679641\pi\)
−0.534874 + 0.844932i \(0.679641\pi\)
\(812\) 3.30366e13 2.66682
\(813\) 0 0
\(814\) −2.53829e12 −0.202643
\(815\) −7.93338e12 −0.629867
\(816\) 0 0
\(817\) −1.60965e13 −1.26396
\(818\) −2.13367e13 −1.66624
\(819\) 0 0
\(820\) 2.71211e13 2.09481
\(821\) −1.39450e13 −1.07121 −0.535606 0.844468i \(-0.679917\pi\)
−0.535606 + 0.844468i \(0.679917\pi\)
\(822\) 0 0
\(823\) 1.81146e13 1.37635 0.688177 0.725543i \(-0.258411\pi\)
0.688177 + 0.725543i \(0.258411\pi\)
\(824\) −4.27003e12 −0.322670
\(825\) 0 0
\(826\) 1.50421e13 1.12434
\(827\) −9.15727e12 −0.680756 −0.340378 0.940289i \(-0.610555\pi\)
−0.340378 + 0.940289i \(0.610555\pi\)
\(828\) 0 0
\(829\) 1.41975e13 1.04404 0.522020 0.852933i \(-0.325179\pi\)
0.522020 + 0.852933i \(0.325179\pi\)
\(830\) 3.77560e11 0.0276143
\(831\) 0 0
\(832\) 3.50509e12 0.253597
\(833\) 1.37753e12 0.0991289
\(834\) 0 0
\(835\) −8.46553e12 −0.602650
\(836\) 1.03385e13 0.732030
\(837\) 0 0
\(838\) 3.71528e13 2.60251
\(839\) −2.26357e13 −1.57712 −0.788561 0.614956i \(-0.789174\pi\)
−0.788561 + 0.614956i \(0.789174\pi\)
\(840\) 0 0
\(841\) −2.82351e12 −0.194629
\(842\) −9.17478e12 −0.629059
\(843\) 0 0
\(844\) 3.96024e13 2.68646
\(845\) −1.11150e13 −0.749989
\(846\) 0 0
\(847\) −1.41638e13 −0.945596
\(848\) 8.94567e13 5.94062
\(849\) 0 0
\(850\) 1.67911e13 1.10330
\(851\) −2.65523e11 −0.0173548
\(852\) 0 0
\(853\) −2.75478e13 −1.78163 −0.890813 0.454370i \(-0.849864\pi\)
−0.890813 + 0.454370i \(0.849864\pi\)
\(854\) 5.65039e12 0.363511
\(855\) 0 0
\(856\) 2.33634e13 1.48732
\(857\) 1.19508e13 0.756801 0.378401 0.925642i \(-0.376474\pi\)
0.378401 + 0.925642i \(0.376474\pi\)
\(858\) 0 0
\(859\) −1.68633e13 −1.05675 −0.528377 0.849010i \(-0.677199\pi\)
−0.528377 + 0.849010i \(0.677199\pi\)
\(860\) 5.08700e13 3.17117
\(861\) 0 0
\(862\) 1.54029e13 0.950213
\(863\) 2.70257e13 1.65855 0.829275 0.558840i \(-0.188753\pi\)
0.829275 + 0.558840i \(0.188753\pi\)
\(864\) 0 0
\(865\) −4.33190e12 −0.263091
\(866\) 3.29701e11 0.0199200
\(867\) 0 0
\(868\) 3.34338e13 1.99916
\(869\) −5.18797e11 −0.0308609
\(870\) 0 0
\(871\) 1.07494e12 0.0632853
\(872\) −5.56072e13 −3.25692
\(873\) 0 0
\(874\) 1.45913e12 0.0845849
\(875\) −1.94085e13 −1.11932
\(876\) 0 0
\(877\) 1.47392e13 0.841350 0.420675 0.907211i \(-0.361793\pi\)
0.420675 + 0.907211i \(0.361793\pi\)
\(878\) −4.26898e13 −2.42437
\(879\) 0 0
\(880\) −1.72795e13 −0.971312
\(881\) 2.25153e13 1.25918 0.629589 0.776928i \(-0.283223\pi\)
0.629589 + 0.776928i \(0.283223\pi\)
\(882\) 0 0
\(883\) −2.11103e13 −1.16861 −0.584307 0.811533i \(-0.698634\pi\)
−0.584307 + 0.811533i \(0.698634\pi\)
\(884\) 3.26094e12 0.179601
\(885\) 0 0
\(886\) 1.28322e13 0.699598
\(887\) −4.72702e12 −0.256408 −0.128204 0.991748i \(-0.540921\pi\)
−0.128204 + 0.991748i \(0.540921\pi\)
\(888\) 0 0
\(889\) −1.24775e13 −0.669995
\(890\) 4.40956e13 2.35581
\(891\) 0 0
\(892\) −8.40195e13 −4.44363
\(893\) −1.80969e13 −0.952299
\(894\) 0 0
\(895\) −2.58631e13 −1.34734
\(896\) −1.08056e14 −5.60097
\(897\) 0 0
\(898\) 4.44902e13 2.28308
\(899\) 1.18241e13 0.603740
\(900\) 0 0
\(901\) 3.49740e13 1.76801
\(902\) 1.13278e13 0.569792
\(903\) 0 0
\(904\) −3.29087e12 −0.163890
\(905\) −4.38804e12 −0.217446
\(906\) 0 0
\(907\) 1.32865e13 0.651895 0.325948 0.945388i \(-0.394317\pi\)
0.325948 + 0.945388i \(0.394317\pi\)
\(908\) −5.13259e13 −2.50582
\(909\) 0 0
\(910\) −1.54122e12 −0.0745038
\(911\) 2.09108e13 1.00586 0.502930 0.864327i \(-0.332255\pi\)
0.502930 + 0.864327i \(0.332255\pi\)
\(912\) 0 0
\(913\) 1.16882e11 0.00556711
\(914\) −1.26251e13 −0.598378
\(915\) 0 0
\(916\) 9.49419e13 4.45583
\(917\) 3.04667e13 1.42286
\(918\) 0 0
\(919\) −1.12670e13 −0.521060 −0.260530 0.965466i \(-0.583897\pi\)
−0.260530 + 0.965466i \(0.583897\pi\)
\(920\) −3.00106e12 −0.138111
\(921\) 0 0
\(922\) −2.87242e10 −0.00130906
\(923\) −1.63015e12 −0.0739298
\(924\) 0 0
\(925\) 3.35065e12 0.150484
\(926\) 2.14856e13 0.960281
\(927\) 0 0
\(928\) −9.85731e13 −4.36307
\(929\) −2.48251e13 −1.09350 −0.546752 0.837294i \(-0.684136\pi\)
−0.546752 + 0.837294i \(0.684136\pi\)
\(930\) 0 0
\(931\) −1.51058e12 −0.0658978
\(932\) 8.66623e13 3.76235
\(933\) 0 0
\(934\) −3.15845e13 −1.35804
\(935\) −6.75560e12 −0.289076
\(936\) 0 0
\(937\) 2.05299e13 0.870081 0.435040 0.900411i \(-0.356734\pi\)
0.435040 + 0.900411i \(0.356734\pi\)
\(938\) −6.29878e13 −2.65671
\(939\) 0 0
\(940\) 5.71921e13 2.38924
\(941\) 2.89837e13 1.20504 0.602519 0.798104i \(-0.294164\pi\)
0.602519 + 0.798104i \(0.294164\pi\)
\(942\) 0 0
\(943\) 1.18497e12 0.0487983
\(944\) −5.83313e13 −2.39072
\(945\) 0 0
\(946\) 2.12471e13 0.862562
\(947\) 2.78208e13 1.12407 0.562036 0.827113i \(-0.310018\pi\)
0.562036 + 0.827113i \(0.310018\pi\)
\(948\) 0 0
\(949\) 7.88290e11 0.0315492
\(950\) −1.84129e13 −0.733441
\(951\) 0 0
\(952\) −1.24355e14 −4.90679
\(953\) −2.85882e13 −1.12271 −0.561357 0.827573i \(-0.689721\pi\)
−0.561357 + 0.827573i \(0.689721\pi\)
\(954\) 0 0
\(955\) −1.53842e13 −0.598496
\(956\) 5.93381e13 2.29759
\(957\) 0 0
\(958\) 6.87702e11 0.0263788
\(959\) 4.50023e13 1.71811
\(960\) 0 0
\(961\) −1.44733e13 −0.547411
\(962\) 8.77947e11 0.0330507
\(963\) 0 0
\(964\) 4.94246e12 0.184330
\(965\) 1.66698e13 0.618810
\(966\) 0 0
\(967\) −5.14935e13 −1.89380 −0.946898 0.321534i \(-0.895802\pi\)
−0.946898 + 0.321534i \(0.895802\pi\)
\(968\) 9.11919e13 3.33824
\(969\) 0 0
\(970\) −4.51890e13 −1.63893
\(971\) 3.75474e13 1.35548 0.677740 0.735301i \(-0.262959\pi\)
0.677740 + 0.735301i \(0.262959\pi\)
\(972\) 0 0
\(973\) 3.21464e13 1.14981
\(974\) 2.51476e13 0.895325
\(975\) 0 0
\(976\) −2.19115e13 −0.772944
\(977\) 3.25385e13 1.14254 0.571271 0.820762i \(-0.306451\pi\)
0.571271 + 0.820762i \(0.306451\pi\)
\(978\) 0 0
\(979\) 1.36508e13 0.474937
\(980\) 4.77392e12 0.165332
\(981\) 0 0
\(982\) −1.65781e13 −0.568895
\(983\) −5.63366e13 −1.92442 −0.962209 0.272310i \(-0.912212\pi\)
−0.962209 + 0.272310i \(0.912212\pi\)
\(984\) 0 0
\(985\) 4.21911e12 0.142810
\(986\) −6.75768e13 −2.27694
\(987\) 0 0
\(988\) −3.57589e12 −0.119393
\(989\) 2.22260e12 0.0738718
\(990\) 0 0
\(991\) −2.09674e13 −0.690578 −0.345289 0.938496i \(-0.612219\pi\)
−0.345289 + 0.938496i \(0.612219\pi\)
\(992\) −9.97583e13 −3.27074
\(993\) 0 0
\(994\) 9.55210e13 3.10356
\(995\) −4.10067e13 −1.32633
\(996\) 0 0
\(997\) 4.85780e13 1.55708 0.778541 0.627593i \(-0.215960\pi\)
0.778541 + 0.627593i \(0.215960\pi\)
\(998\) 1.59057e13 0.507533
\(999\) 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 27.10.a.d.1.1 4
3.2 odd 2 inner 27.10.a.d.1.4 yes 4
9.2 odd 6 81.10.c.j.28.1 8
9.4 even 3 81.10.c.j.55.4 8
9.5 odd 6 81.10.c.j.55.1 8
9.7 even 3 81.10.c.j.28.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.10.a.d.1.1 4 1.1 even 1 trivial
27.10.a.d.1.4 yes 4 3.2 odd 2 inner
81.10.c.j.28.1 8 9.2 odd 6
81.10.c.j.28.4 8 9.7 even 3
81.10.c.j.55.1 8 9.5 odd 6
81.10.c.j.55.4 8 9.4 even 3