Properties

Label 27.10.a.d.1.2
Level $27$
Weight $10$
Character 27.1
Self dual yes
Analytic conductor $13.906$
Analytic rank $0$
Dimension $4$
CM no
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 1.2
Root \(4.96988\) 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)
 
\(f(q)\) \(=\) \(q-12.7978 q^{2} -348.216 q^{4} -2019.77 q^{5} -665.864 q^{7} +11008.9 q^{8} +O(q^{10})\) \(q-12.7978 q^{2} -348.216 q^{4} -2019.77 q^{5} -665.864 q^{7} +11008.9 q^{8} +25848.6 q^{10} -85435.1 q^{11} +84838.5 q^{13} +8521.60 q^{14} +37397.1 q^{16} +139510. q^{17} +993145. q^{19} +703317. q^{20} +1.09338e6 q^{22} -758529. q^{23} +2.12635e6 q^{25} -1.08575e6 q^{26} +231865. q^{28} +4.04867e6 q^{29} -3.01466e6 q^{31} -6.11515e6 q^{32} -1.78543e6 q^{34} +1.34489e6 q^{35} +2.04252e6 q^{37} -1.27101e7 q^{38} -2.22354e7 q^{40} +7.85245e6 q^{41} -2.72306e7 q^{43} +2.97499e7 q^{44} +9.70751e6 q^{46} +3.29865e7 q^{47} -3.99102e7 q^{49} -2.72126e7 q^{50} -2.95421e7 q^{52} -1.02265e8 q^{53} +1.72559e8 q^{55} -7.33042e6 q^{56} -5.18141e7 q^{58} +1.15657e8 q^{59} +1.29214e8 q^{61} +3.85811e7 q^{62} +5.91132e7 q^{64} -1.71354e8 q^{65} -1.95575e7 q^{67} -4.85797e7 q^{68} -1.72117e7 q^{70} +2.18178e8 q^{71} +4.41427e8 q^{73} -2.61398e7 q^{74} -3.45829e8 q^{76} +5.68882e7 q^{77} -1.09562e7 q^{79} -7.55335e7 q^{80} -1.00494e8 q^{82} +2.86843e8 q^{83} -2.81779e8 q^{85} +3.48492e8 q^{86} -9.40544e8 q^{88} -7.64398e8 q^{89} -5.64909e7 q^{91} +2.64132e8 q^{92} -4.22154e8 q^{94} -2.00593e9 q^{95} +815669. q^{97} +5.10764e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2236 q^{4} + 11852 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40} + 11585024 q^{43} + 13428000 q^{46} - 73622328 q^{49} - 44411860 q^{52} + 314722080 q^{55} - 407685888 q^{58} + 219882572 q^{61} + 1519285360 q^{64} + 390561236 q^{67} - 650481120 q^{70} + 1197949508 q^{73} - 2121017572 q^{76} + 49814324 q^{79} - 1767126528 q^{82} + 370444320 q^{85} - 653198976 q^{88} - 47017172 q^{91} - 4146926688 q^{94} + 1935734516 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.7978 −0.565589 −0.282794 0.959181i \(-0.591261\pi\)
−0.282794 + 0.959181i \(0.591261\pi\)
\(3\) 0 0
\(4\) −348.216 −0.680110
\(5\) −2019.77 −1.44523 −0.722615 0.691251i \(-0.757060\pi\)
−0.722615 + 0.691251i \(0.757060\pi\)
\(6\) 0 0
\(7\) −665.864 −0.104820 −0.0524100 0.998626i \(-0.516690\pi\)
−0.0524100 + 0.998626i \(0.516690\pi\)
\(8\) 11008.9 0.950251
\(9\) 0 0
\(10\) 25848.6 0.817406
\(11\) −85435.1 −1.75942 −0.879709 0.475512i \(-0.842263\pi\)
−0.879709 + 0.475512i \(0.842263\pi\)
\(12\) 0 0
\(13\) 84838.5 0.823850 0.411925 0.911218i \(-0.364857\pi\)
0.411925 + 0.911218i \(0.364857\pi\)
\(14\) 8521.60 0.0592851
\(15\) 0 0
\(16\) 37397.1 0.142658
\(17\) 139510. 0.405122 0.202561 0.979270i \(-0.435074\pi\)
0.202561 + 0.979270i \(0.435074\pi\)
\(18\) 0 0
\(19\) 993145. 1.74832 0.874161 0.485636i \(-0.161412\pi\)
0.874161 + 0.485636i \(0.161412\pi\)
\(20\) 703317. 0.982915
\(21\) 0 0
\(22\) 1.09338e6 0.995107
\(23\) −758529. −0.565193 −0.282597 0.959239i \(-0.591196\pi\)
−0.282597 + 0.959239i \(0.591196\pi\)
\(24\) 0 0
\(25\) 2.12635e6 1.08869
\(26\) −1.08575e6 −0.465960
\(27\) 0 0
\(28\) 231865. 0.0712891
\(29\) 4.04867e6 1.06297 0.531486 0.847067i \(-0.321634\pi\)
0.531486 + 0.847067i \(0.321634\pi\)
\(30\) 0 0
\(31\) −3.01466e6 −0.586288 −0.293144 0.956068i \(-0.594702\pi\)
−0.293144 + 0.956068i \(0.594702\pi\)
\(32\) −6.11515e6 −1.03094
\(33\) 0 0
\(34\) −1.78543e6 −0.229133
\(35\) 1.34489e6 0.151489
\(36\) 0 0
\(37\) 2.04252e6 0.179168 0.0895838 0.995979i \(-0.471446\pi\)
0.0895838 + 0.995979i \(0.471446\pi\)
\(38\) −1.27101e7 −0.988831
\(39\) 0 0
\(40\) −2.22354e7 −1.37333
\(41\) 7.85245e6 0.433988 0.216994 0.976173i \(-0.430375\pi\)
0.216994 + 0.976173i \(0.430375\pi\)
\(42\) 0 0
\(43\) −2.72306e7 −1.21464 −0.607322 0.794456i \(-0.707756\pi\)
−0.607322 + 0.794456i \(0.707756\pi\)
\(44\) 2.97499e7 1.19660
\(45\) 0 0
\(46\) 9.70751e6 0.319667
\(47\) 3.29865e7 0.986042 0.493021 0.870017i \(-0.335893\pi\)
0.493021 + 0.870017i \(0.335893\pi\)
\(48\) 0 0
\(49\) −3.99102e7 −0.989013
\(50\) −2.72126e7 −0.615751
\(51\) 0 0
\(52\) −2.95421e7 −0.560308
\(53\) −1.02265e8 −1.78026 −0.890131 0.455704i \(-0.849388\pi\)
−0.890131 + 0.455704i \(0.849388\pi\)
\(54\) 0 0
\(55\) 1.72559e8 2.54276
\(56\) −7.33042e6 −0.0996054
\(57\) 0 0
\(58\) −5.18141e7 −0.601204
\(59\) 1.15657e8 1.24262 0.621308 0.783566i \(-0.286601\pi\)
0.621308 + 0.783566i \(0.286601\pi\)
\(60\) 0 0
\(61\) 1.29214e8 1.19488 0.597439 0.801914i \(-0.296185\pi\)
0.597439 + 0.801914i \(0.296185\pi\)
\(62\) 3.85811e7 0.331598
\(63\) 0 0
\(64\) 5.91132e7 0.440428
\(65\) −1.71354e8 −1.19065
\(66\) 0 0
\(67\) −1.95575e7 −0.118570 −0.0592852 0.998241i \(-0.518882\pi\)
−0.0592852 + 0.998241i \(0.518882\pi\)
\(68\) −4.85797e7 −0.275528
\(69\) 0 0
\(70\) −1.72117e7 −0.0856806
\(71\) 2.18178e8 1.01894 0.509471 0.860488i \(-0.329841\pi\)
0.509471 + 0.860488i \(0.329841\pi\)
\(72\) 0 0
\(73\) 4.41427e8 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(74\) −2.61398e7 −0.101335
\(75\) 0 0
\(76\) −3.45829e8 −1.18905
\(77\) 5.68882e7 0.184422
\(78\) 0 0
\(79\) −1.09562e7 −0.0316475 −0.0158237 0.999875i \(-0.505037\pi\)
−0.0158237 + 0.999875i \(0.505037\pi\)
\(80\) −7.55335e7 −0.206174
\(81\) 0 0
\(82\) −1.00494e8 −0.245459
\(83\) 2.86843e8 0.663427 0.331714 0.943380i \(-0.392373\pi\)
0.331714 + 0.943380i \(0.392373\pi\)
\(84\) 0 0
\(85\) −2.81779e8 −0.585495
\(86\) 3.48492e8 0.686988
\(87\) 0 0
\(88\) −9.40544e8 −1.67189
\(89\) −7.64398e8 −1.29141 −0.645706 0.763586i \(-0.723437\pi\)
−0.645706 + 0.763586i \(0.723437\pi\)
\(90\) 0 0
\(91\) −5.64909e7 −0.0863560
\(92\) 2.64132e8 0.384393
\(93\) 0 0
\(94\) −4.22154e8 −0.557694
\(95\) −2.00593e9 −2.52673
\(96\) 0 0
\(97\) 815669. 0.000935494 0 0.000467747 1.00000i \(-0.499851\pi\)
0.000467747 1.00000i \(0.499851\pi\)
\(98\) 5.10764e8 0.559374
\(99\) 0 0
\(100\) −7.40429e8 −0.740429
\(101\) 8.80693e8 0.842129 0.421064 0.907031i \(-0.361657\pi\)
0.421064 + 0.907031i \(0.361657\pi\)
\(102\) 0 0
\(103\) 1.47976e9 1.29546 0.647730 0.761870i \(-0.275718\pi\)
0.647730 + 0.761870i \(0.275718\pi\)
\(104\) 9.33977e8 0.782864
\(105\) 0 0
\(106\) 1.30876e9 1.00690
\(107\) 1.64812e9 1.21552 0.607759 0.794122i \(-0.292069\pi\)
0.607759 + 0.794122i \(0.292069\pi\)
\(108\) 0 0
\(109\) −3.21670e8 −0.218268 −0.109134 0.994027i \(-0.534808\pi\)
−0.109134 + 0.994027i \(0.534808\pi\)
\(110\) −2.20838e9 −1.43816
\(111\) 0 0
\(112\) −2.49014e7 −0.0149535
\(113\) 1.91951e9 1.10748 0.553741 0.832689i \(-0.313200\pi\)
0.553741 + 0.832689i \(0.313200\pi\)
\(114\) 0 0
\(115\) 1.53206e9 0.816835
\(116\) −1.40981e9 −0.722937
\(117\) 0 0
\(118\) −1.48015e9 −0.702810
\(119\) −9.28949e7 −0.0424650
\(120\) 0 0
\(121\) 4.94120e9 2.09555
\(122\) −1.65365e9 −0.675810
\(123\) 0 0
\(124\) 1.04975e9 0.398740
\(125\) −3.49874e8 −0.128179
\(126\) 0 0
\(127\) −2.81305e9 −0.959535 −0.479767 0.877396i \(-0.659279\pi\)
−0.479767 + 0.877396i \(0.659279\pi\)
\(128\) 2.37444e9 0.781836
\(129\) 0 0
\(130\) 2.19296e9 0.673419
\(131\) 7.34372e8 0.217869 0.108934 0.994049i \(-0.465256\pi\)
0.108934 + 0.994049i \(0.465256\pi\)
\(132\) 0 0
\(133\) −6.61300e8 −0.183259
\(134\) 2.50293e8 0.0670621
\(135\) 0 0
\(136\) 1.53585e9 0.384968
\(137\) −1.00181e9 −0.242964 −0.121482 0.992594i \(-0.538765\pi\)
−0.121482 + 0.992594i \(0.538765\pi\)
\(138\) 0 0
\(139\) −2.12304e9 −0.482382 −0.241191 0.970478i \(-0.577538\pi\)
−0.241191 + 0.970478i \(0.577538\pi\)
\(140\) −4.68313e8 −0.103029
\(141\) 0 0
\(142\) −2.79221e9 −0.576302
\(143\) −7.24818e9 −1.44950
\(144\) 0 0
\(145\) −8.17739e9 −1.53624
\(146\) −5.64930e9 −1.02898
\(147\) 0 0
\(148\) −7.11240e8 −0.121854
\(149\) 6.02059e9 1.00069 0.500346 0.865825i \(-0.333206\pi\)
0.500346 + 0.865825i \(0.333206\pi\)
\(150\) 0 0
\(151\) −8.86951e9 −1.38836 −0.694182 0.719800i \(-0.744234\pi\)
−0.694182 + 0.719800i \(0.744234\pi\)
\(152\) 1.09334e10 1.66135
\(153\) 0 0
\(154\) −7.28044e8 −0.104307
\(155\) 6.08893e9 0.847321
\(156\) 0 0
\(157\) −2.51721e9 −0.330652 −0.165326 0.986239i \(-0.552868\pi\)
−0.165326 + 0.986239i \(0.552868\pi\)
\(158\) 1.40216e8 0.0178994
\(159\) 0 0
\(160\) 1.23512e10 1.48994
\(161\) 5.05078e8 0.0592436
\(162\) 0 0
\(163\) −6.17153e8 −0.0684776 −0.0342388 0.999414i \(-0.510901\pi\)
−0.0342388 + 0.999414i \(0.510901\pi\)
\(164\) −2.73435e9 −0.295160
\(165\) 0 0
\(166\) −3.67096e9 −0.375227
\(167\) −1.06520e10 −1.05976 −0.529882 0.848072i \(-0.677764\pi\)
−0.529882 + 0.848072i \(0.677764\pi\)
\(168\) 0 0
\(169\) −3.40693e9 −0.321272
\(170\) 3.60615e9 0.331149
\(171\) 0 0
\(172\) 9.48212e9 0.826090
\(173\) −4.55410e8 −0.0386540 −0.0193270 0.999813i \(-0.506152\pi\)
−0.0193270 + 0.999813i \(0.506152\pi\)
\(174\) 0 0
\(175\) −1.41586e9 −0.114117
\(176\) −3.19502e9 −0.250996
\(177\) 0 0
\(178\) 9.78262e9 0.730408
\(179\) −9.54934e9 −0.695240 −0.347620 0.937636i \(-0.613010\pi\)
−0.347620 + 0.937636i \(0.613010\pi\)
\(180\) 0 0
\(181\) 2.10546e10 1.45812 0.729061 0.684449i \(-0.239957\pi\)
0.729061 + 0.684449i \(0.239957\pi\)
\(182\) 7.22960e8 0.0488420
\(183\) 0 0
\(184\) −8.35056e9 −0.537075
\(185\) −4.12543e9 −0.258938
\(186\) 0 0
\(187\) −1.19191e10 −0.712779
\(188\) −1.14864e10 −0.670617
\(189\) 0 0
\(190\) 2.56715e10 1.42909
\(191\) 2.71245e10 1.47473 0.737364 0.675496i \(-0.236070\pi\)
0.737364 + 0.675496i \(0.236070\pi\)
\(192\) 0 0
\(193\) 5.06145e9 0.262583 0.131292 0.991344i \(-0.458088\pi\)
0.131292 + 0.991344i \(0.458088\pi\)
\(194\) −1.04388e7 −0.000529105 0
\(195\) 0 0
\(196\) 1.38974e10 0.672637
\(197\) 1.23458e10 0.584010 0.292005 0.956417i \(-0.405678\pi\)
0.292005 + 0.956417i \(0.405678\pi\)
\(198\) 0 0
\(199\) 1.20900e10 0.546497 0.273249 0.961943i \(-0.411902\pi\)
0.273249 + 0.961943i \(0.411902\pi\)
\(200\) 2.34087e10 1.03453
\(201\) 0 0
\(202\) −1.12709e10 −0.476299
\(203\) −2.69587e9 −0.111421
\(204\) 0 0
\(205\) −1.58602e10 −0.627213
\(206\) −1.89377e10 −0.732698
\(207\) 0 0
\(208\) 3.17271e9 0.117529
\(209\) −8.48494e10 −3.07603
\(210\) 0 0
\(211\) −1.72325e10 −0.598516 −0.299258 0.954172i \(-0.596739\pi\)
−0.299258 + 0.954172i \(0.596739\pi\)
\(212\) 3.56102e10 1.21077
\(213\) 0 0
\(214\) −2.10923e10 −0.687483
\(215\) 5.49995e10 1.75544
\(216\) 0 0
\(217\) 2.00736e9 0.0614548
\(218\) 4.11667e9 0.123450
\(219\) 0 0
\(220\) −6.00879e10 −1.72936
\(221\) 1.18358e10 0.333760
\(222\) 0 0
\(223\) −3.12781e10 −0.846972 −0.423486 0.905903i \(-0.639194\pi\)
−0.423486 + 0.905903i \(0.639194\pi\)
\(224\) 4.07186e9 0.108063
\(225\) 0 0
\(226\) −2.45655e10 −0.626379
\(227\) 9.86598e9 0.246617 0.123309 0.992368i \(-0.460649\pi\)
0.123309 + 0.992368i \(0.460649\pi\)
\(228\) 0 0
\(229\) 4.08107e10 0.980651 0.490325 0.871539i \(-0.336878\pi\)
0.490325 + 0.871539i \(0.336878\pi\)
\(230\) −1.96070e10 −0.461992
\(231\) 0 0
\(232\) 4.45713e10 1.01009
\(233\) −3.65168e10 −0.811693 −0.405846 0.913941i \(-0.633023\pi\)
−0.405846 + 0.913941i \(0.633023\pi\)
\(234\) 0 0
\(235\) −6.66251e10 −1.42506
\(236\) −4.02736e10 −0.845115
\(237\) 0 0
\(238\) 1.18885e9 0.0240177
\(239\) −5.78935e9 −0.114773 −0.0573865 0.998352i \(-0.518277\pi\)
−0.0573865 + 0.998352i \(0.518277\pi\)
\(240\) 0 0
\(241\) 2.33566e10 0.445998 0.222999 0.974819i \(-0.428415\pi\)
0.222999 + 0.974819i \(0.428415\pi\)
\(242\) −6.32366e10 −1.18522
\(243\) 0 0
\(244\) −4.49942e10 −0.812649
\(245\) 8.06095e10 1.42935
\(246\) 0 0
\(247\) 8.42570e10 1.44035
\(248\) −3.31881e10 −0.557121
\(249\) 0 0
\(250\) 4.47761e9 0.0724964
\(251\) −1.44384e10 −0.229608 −0.114804 0.993388i \(-0.536624\pi\)
−0.114804 + 0.993388i \(0.536624\pi\)
\(252\) 0 0
\(253\) 6.48050e10 0.994411
\(254\) 3.60009e10 0.542702
\(255\) 0 0
\(256\) −6.06535e10 −0.882625
\(257\) 9.77486e10 1.39769 0.698846 0.715272i \(-0.253697\pi\)
0.698846 + 0.715272i \(0.253697\pi\)
\(258\) 0 0
\(259\) −1.36004e9 −0.0187804
\(260\) 5.96683e10 0.809774
\(261\) 0 0
\(262\) −9.39835e9 −0.123224
\(263\) −2.94535e10 −0.379609 −0.189804 0.981822i \(-0.560785\pi\)
−0.189804 + 0.981822i \(0.560785\pi\)
\(264\) 0 0
\(265\) 2.06551e11 2.57289
\(266\) 8.46319e9 0.103649
\(267\) 0 0
\(268\) 6.81023e9 0.0806409
\(269\) 6.86011e10 0.798814 0.399407 0.916774i \(-0.369216\pi\)
0.399407 + 0.916774i \(0.369216\pi\)
\(270\) 0 0
\(271\) −6.74978e10 −0.760200 −0.380100 0.924945i \(-0.624110\pi\)
−0.380100 + 0.924945i \(0.624110\pi\)
\(272\) 5.21728e9 0.0577941
\(273\) 0 0
\(274\) 1.28209e10 0.137417
\(275\) −1.81665e11 −1.91546
\(276\) 0 0
\(277\) 6.24779e10 0.637628 0.318814 0.947817i \(-0.396715\pi\)
0.318814 + 0.947817i \(0.396715\pi\)
\(278\) 2.71702e10 0.272830
\(279\) 0 0
\(280\) 1.48058e10 0.143953
\(281\) −8.68104e10 −0.830603 −0.415301 0.909684i \(-0.636324\pi\)
−0.415301 + 0.909684i \(0.636324\pi\)
\(282\) 0 0
\(283\) 1.37271e11 1.27216 0.636078 0.771625i \(-0.280556\pi\)
0.636078 + 0.771625i \(0.280556\pi\)
\(284\) −7.59733e10 −0.692992
\(285\) 0 0
\(286\) 9.27609e10 0.819818
\(287\) −5.22867e9 −0.0454907
\(288\) 0 0
\(289\) −9.91248e10 −0.835876
\(290\) 1.04653e11 0.868879
\(291\) 0 0
\(292\) −1.53712e11 −1.23733
\(293\) 1.06733e11 0.846050 0.423025 0.906118i \(-0.360968\pi\)
0.423025 + 0.906118i \(0.360968\pi\)
\(294\) 0 0
\(295\) −2.33600e11 −1.79587
\(296\) 2.24859e10 0.170254
\(297\) 0 0
\(298\) −7.70503e10 −0.565980
\(299\) −6.43525e10 −0.465634
\(300\) 0 0
\(301\) 1.81319e10 0.127319
\(302\) 1.13510e11 0.785243
\(303\) 0 0
\(304\) 3.71407e10 0.249413
\(305\) −2.60982e11 −1.72688
\(306\) 0 0
\(307\) −1.13729e11 −0.730716 −0.365358 0.930867i \(-0.619054\pi\)
−0.365358 + 0.930867i \(0.619054\pi\)
\(308\) −1.98094e10 −0.125427
\(309\) 0 0
\(310\) −7.79249e10 −0.479235
\(311\) 1.66986e9 0.0101218 0.00506091 0.999987i \(-0.498389\pi\)
0.00506091 + 0.999987i \(0.498389\pi\)
\(312\) 0 0
\(313\) 5.22551e10 0.307736 0.153868 0.988091i \(-0.450827\pi\)
0.153868 + 0.988091i \(0.450827\pi\)
\(314\) 3.22148e10 0.187013
\(315\) 0 0
\(316\) 3.81513e9 0.0215237
\(317\) 1.76926e11 0.984070 0.492035 0.870575i \(-0.336253\pi\)
0.492035 + 0.870575i \(0.336253\pi\)
\(318\) 0 0
\(319\) −3.45898e11 −1.87021
\(320\) −1.19395e11 −0.636519
\(321\) 0 0
\(322\) −6.46389e9 −0.0335075
\(323\) 1.38554e11 0.708284
\(324\) 0 0
\(325\) 1.80396e11 0.896918
\(326\) 7.89820e9 0.0387301
\(327\) 0 0
\(328\) 8.64467e10 0.412398
\(329\) −2.19645e10 −0.103357
\(330\) 0 0
\(331\) −1.59523e11 −0.730461 −0.365231 0.930917i \(-0.619010\pi\)
−0.365231 + 0.930917i \(0.619010\pi\)
\(332\) −9.98834e10 −0.451203
\(333\) 0 0
\(334\) 1.36323e11 0.599390
\(335\) 3.95017e10 0.171362
\(336\) 0 0
\(337\) 1.07204e11 0.452769 0.226384 0.974038i \(-0.427309\pi\)
0.226384 + 0.974038i \(0.427309\pi\)
\(338\) 4.36012e10 0.181708
\(339\) 0 0
\(340\) 9.81199e10 0.398201
\(341\) 2.57558e11 1.03153
\(342\) 0 0
\(343\) 5.34448e10 0.208489
\(344\) −2.99778e11 −1.15422
\(345\) 0 0
\(346\) 5.82824e9 0.0218623
\(347\) 4.80474e11 1.77904 0.889522 0.456892i \(-0.151037\pi\)
0.889522 + 0.456892i \(0.151037\pi\)
\(348\) 0 0
\(349\) −2.05930e11 −0.743027 −0.371514 0.928428i \(-0.621161\pi\)
−0.371514 + 0.928428i \(0.621161\pi\)
\(350\) 1.81199e10 0.0645431
\(351\) 0 0
\(352\) 5.22448e11 1.81385
\(353\) −2.84250e11 −0.974348 −0.487174 0.873305i \(-0.661972\pi\)
−0.487174 + 0.873305i \(0.661972\pi\)
\(354\) 0 0
\(355\) −4.40671e11 −1.47261
\(356\) 2.66176e11 0.878301
\(357\) 0 0
\(358\) 1.22211e11 0.393220
\(359\) −3.45110e11 −1.09656 −0.548280 0.836295i \(-0.684717\pi\)
−0.548280 + 0.836295i \(0.684717\pi\)
\(360\) 0 0
\(361\) 6.63650e11 2.05663
\(362\) −2.69453e11 −0.824697
\(363\) 0 0
\(364\) 1.96711e10 0.0587315
\(365\) −8.91581e11 −2.62932
\(366\) 0 0
\(367\) 7.68807e10 0.221218 0.110609 0.993864i \(-0.464720\pi\)
0.110609 + 0.993864i \(0.464720\pi\)
\(368\) −2.83668e10 −0.0806296
\(369\) 0 0
\(370\) 5.27965e10 0.146453
\(371\) 6.80944e10 0.186607
\(372\) 0 0
\(373\) 3.76354e11 1.00671 0.503357 0.864078i \(-0.332098\pi\)
0.503357 + 0.864078i \(0.332098\pi\)
\(374\) 1.52538e11 0.403140
\(375\) 0 0
\(376\) 3.63144e11 0.936987
\(377\) 3.43483e11 0.875728
\(378\) 0 0
\(379\) −3.18048e11 −0.791801 −0.395900 0.918294i \(-0.629567\pi\)
−0.395900 + 0.918294i \(0.629567\pi\)
\(380\) 6.98496e11 1.71845
\(381\) 0 0
\(382\) −3.47135e11 −0.834089
\(383\) 1.21289e11 0.288023 0.144011 0.989576i \(-0.454000\pi\)
0.144011 + 0.989576i \(0.454000\pi\)
\(384\) 0 0
\(385\) −1.14901e11 −0.266533
\(386\) −6.47755e10 −0.148514
\(387\) 0 0
\(388\) −2.84029e8 −0.000636239 0
\(389\) 7.63532e11 1.69065 0.845325 0.534252i \(-0.179407\pi\)
0.845325 + 0.534252i \(0.179407\pi\)
\(390\) 0 0
\(391\) −1.05823e11 −0.228972
\(392\) −4.39367e11 −0.939810
\(393\) 0 0
\(394\) −1.57999e11 −0.330309
\(395\) 2.21291e10 0.0457379
\(396\) 0 0
\(397\) −5.37941e11 −1.08687 −0.543435 0.839451i \(-0.682876\pi\)
−0.543435 + 0.839451i \(0.682876\pi\)
\(398\) −1.54726e11 −0.309093
\(399\) 0 0
\(400\) 7.95192e10 0.155311
\(401\) −2.23876e11 −0.432373 −0.216187 0.976352i \(-0.569362\pi\)
−0.216187 + 0.976352i \(0.569362\pi\)
\(402\) 0 0
\(403\) −2.55760e11 −0.483013
\(404\) −3.06672e11 −0.572740
\(405\) 0 0
\(406\) 3.45012e10 0.0630183
\(407\) −1.74503e11 −0.315231
\(408\) 0 0
\(409\) −9.59201e11 −1.69494 −0.847471 0.530842i \(-0.821876\pi\)
−0.847471 + 0.530842i \(0.821876\pi\)
\(410\) 2.02975e11 0.354745
\(411\) 0 0
\(412\) −5.15277e11 −0.881055
\(413\) −7.70117e10 −0.130251
\(414\) 0 0
\(415\) −5.79358e11 −0.958805
\(416\) −5.18800e11 −0.849337
\(417\) 0 0
\(418\) 1.08589e12 1.73977
\(419\) 8.85619e11 1.40373 0.701865 0.712310i \(-0.252351\pi\)
0.701865 + 0.712310i \(0.252351\pi\)
\(420\) 0 0
\(421\) 1.06418e12 1.65100 0.825498 0.564405i \(-0.190894\pi\)
0.825498 + 0.564405i \(0.190894\pi\)
\(422\) 2.20538e11 0.338514
\(423\) 0 0
\(424\) −1.12582e12 −1.69170
\(425\) 2.96648e11 0.441053
\(426\) 0 0
\(427\) −8.60387e10 −0.125247
\(428\) −5.73901e11 −0.826685
\(429\) 0 0
\(430\) −7.03873e11 −0.992857
\(431\) −4.41717e11 −0.616589 −0.308295 0.951291i \(-0.599758\pi\)
−0.308295 + 0.951291i \(0.599758\pi\)
\(432\) 0 0
\(433\) 8.91127e11 1.21827 0.609136 0.793066i \(-0.291516\pi\)
0.609136 + 0.793066i \(0.291516\pi\)
\(434\) −2.56898e10 −0.0347581
\(435\) 0 0
\(436\) 1.12011e11 0.148446
\(437\) −7.53330e11 −0.988140
\(438\) 0 0
\(439\) −3.66122e11 −0.470474 −0.235237 0.971938i \(-0.575587\pi\)
−0.235237 + 0.971938i \(0.575587\pi\)
\(440\) 1.89968e12 2.41626
\(441\) 0 0
\(442\) −1.51473e11 −0.188771
\(443\) −6.97003e11 −0.859840 −0.429920 0.902867i \(-0.641458\pi\)
−0.429920 + 0.902867i \(0.641458\pi\)
\(444\) 0 0
\(445\) 1.54391e12 1.86639
\(446\) 4.00292e11 0.479038
\(447\) 0 0
\(448\) −3.93614e10 −0.0461657
\(449\) −8.70295e11 −1.01055 −0.505275 0.862958i \(-0.668609\pi\)
−0.505275 + 0.862958i \(0.668609\pi\)
\(450\) 0 0
\(451\) −6.70875e11 −0.763567
\(452\) −6.68403e11 −0.753208
\(453\) 0 0
\(454\) −1.26263e11 −0.139484
\(455\) 1.14099e11 0.124804
\(456\) 0 0
\(457\) 8.32855e11 0.893195 0.446598 0.894735i \(-0.352636\pi\)
0.446598 + 0.894735i \(0.352636\pi\)
\(458\) −5.22287e11 −0.554645
\(459\) 0 0
\(460\) −5.33486e11 −0.555537
\(461\) −1.63782e11 −0.168893 −0.0844467 0.996428i \(-0.526912\pi\)
−0.0844467 + 0.996428i \(0.526912\pi\)
\(462\) 0 0
\(463\) 7.51302e11 0.759801 0.379901 0.925027i \(-0.375958\pi\)
0.379901 + 0.925027i \(0.375958\pi\)
\(464\) 1.51408e11 0.151642
\(465\) 0 0
\(466\) 4.67336e11 0.459084
\(467\) −3.65892e11 −0.355981 −0.177991 0.984032i \(-0.556960\pi\)
−0.177991 + 0.984032i \(0.556960\pi\)
\(468\) 0 0
\(469\) 1.30226e10 0.0124286
\(470\) 8.52655e11 0.805996
\(471\) 0 0
\(472\) 1.27325e12 1.18080
\(473\) 2.32645e12 2.13707
\(474\) 0 0
\(475\) 2.11177e12 1.90338
\(476\) 3.23475e10 0.0288808
\(477\) 0 0
\(478\) 7.40910e10 0.0649142
\(479\) 5.30846e11 0.460743 0.230371 0.973103i \(-0.426006\pi\)
0.230371 + 0.973103i \(0.426006\pi\)
\(480\) 0 0
\(481\) 1.73285e11 0.147607
\(482\) −2.98913e11 −0.252251
\(483\) 0 0
\(484\) −1.72061e12 −1.42520
\(485\) −1.64746e9 −0.00135200
\(486\) 0 0
\(487\) 9.23142e11 0.743684 0.371842 0.928296i \(-0.378726\pi\)
0.371842 + 0.928296i \(0.378726\pi\)
\(488\) 1.42250e12 1.13543
\(489\) 0 0
\(490\) −1.03163e12 −0.808425
\(491\) −1.62628e12 −1.26278 −0.631391 0.775465i \(-0.717516\pi\)
−0.631391 + 0.775465i \(0.717516\pi\)
\(492\) 0 0
\(493\) 5.64831e11 0.430633
\(494\) −1.07830e12 −0.814648
\(495\) 0 0
\(496\) −1.12740e11 −0.0836390
\(497\) −1.45277e11 −0.106806
\(498\) 0 0
\(499\) −6.59222e11 −0.475970 −0.237985 0.971269i \(-0.576487\pi\)
−0.237985 + 0.971269i \(0.576487\pi\)
\(500\) 1.21832e11 0.0871756
\(501\) 0 0
\(502\) 1.84780e11 0.129864
\(503\) 6.69047e11 0.466016 0.233008 0.972475i \(-0.425143\pi\)
0.233008 + 0.972475i \(0.425143\pi\)
\(504\) 0 0
\(505\) −1.77880e12 −1.21707
\(506\) −8.29362e11 −0.562428
\(507\) 0 0
\(508\) 9.79549e11 0.652589
\(509\) −4.58197e11 −0.302568 −0.151284 0.988490i \(-0.548341\pi\)
−0.151284 + 0.988490i \(0.548341\pi\)
\(510\) 0 0
\(511\) −2.93930e11 −0.190700
\(512\) −4.39479e11 −0.282633
\(513\) 0 0
\(514\) −1.25097e12 −0.790518
\(515\) −2.98878e12 −1.87224
\(516\) 0 0
\(517\) −2.81820e12 −1.73486
\(518\) 1.74056e10 0.0106220
\(519\) 0 0
\(520\) −1.88642e12 −1.13142
\(521\) 3.34551e12 1.98926 0.994632 0.103472i \(-0.0329952\pi\)
0.994632 + 0.103472i \(0.0329952\pi\)
\(522\) 0 0
\(523\) 2.03945e12 1.19194 0.595970 0.803007i \(-0.296768\pi\)
0.595970 + 0.803007i \(0.296768\pi\)
\(524\) −2.55720e11 −0.148175
\(525\) 0 0
\(526\) 3.76940e11 0.214702
\(527\) −4.20577e11 −0.237518
\(528\) 0 0
\(529\) −1.22579e12 −0.680556
\(530\) −2.64340e12 −1.45520
\(531\) 0 0
\(532\) 2.30275e11 0.124636
\(533\) 6.66190e11 0.357541
\(534\) 0 0
\(535\) −3.32882e12 −1.75670
\(536\) −2.15306e11 −0.112672
\(537\) 0 0
\(538\) −8.77944e11 −0.451800
\(539\) 3.40973e12 1.74009
\(540\) 0 0
\(541\) 3.69312e12 1.85355 0.926777 0.375612i \(-0.122567\pi\)
0.926777 + 0.375612i \(0.122567\pi\)
\(542\) 8.63825e11 0.429961
\(543\) 0 0
\(544\) −8.53126e11 −0.417655
\(545\) 6.49699e11 0.315448
\(546\) 0 0
\(547\) 5.88944e11 0.281275 0.140637 0.990061i \(-0.455085\pi\)
0.140637 + 0.990061i \(0.455085\pi\)
\(548\) 3.48845e11 0.165242
\(549\) 0 0
\(550\) 2.32491e12 1.08336
\(551\) 4.02092e12 1.85842
\(552\) 0 0
\(553\) 7.29536e9 0.00331729
\(554\) −7.99580e11 −0.360635
\(555\) 0 0
\(556\) 7.39275e11 0.328072
\(557\) −3.85457e12 −1.69679 −0.848394 0.529366i \(-0.822430\pi\)
−0.848394 + 0.529366i \(0.822430\pi\)
\(558\) 0 0
\(559\) −2.31020e12 −1.00068
\(560\) 5.02951e10 0.0216112
\(561\) 0 0
\(562\) 1.11098e12 0.469779
\(563\) −3.03834e12 −1.27453 −0.637264 0.770646i \(-0.719934\pi\)
−0.637264 + 0.770646i \(0.719934\pi\)
\(564\) 0 0
\(565\) −3.87696e12 −1.60057
\(566\) −1.75677e12 −0.719517
\(567\) 0 0
\(568\) 2.40190e12 0.968250
\(569\) −5.83227e11 −0.233256 −0.116628 0.993176i \(-0.537208\pi\)
−0.116628 + 0.993176i \(0.537208\pi\)
\(570\) 0 0
\(571\) −2.09092e12 −0.823142 −0.411571 0.911378i \(-0.635020\pi\)
−0.411571 + 0.911378i \(0.635020\pi\)
\(572\) 2.52393e12 0.985816
\(573\) 0 0
\(574\) 6.69155e10 0.0257290
\(575\) −1.61290e12 −0.615321
\(576\) 0 0
\(577\) 3.05766e12 1.14841 0.574206 0.818711i \(-0.305311\pi\)
0.574206 + 0.818711i \(0.305311\pi\)
\(578\) 1.26858e12 0.472762
\(579\) 0 0
\(580\) 2.84750e12 1.04481
\(581\) −1.90999e11 −0.0695405
\(582\) 0 0
\(583\) 8.73699e12 3.13223
\(584\) 4.85962e12 1.72880
\(585\) 0 0
\(586\) −1.36595e12 −0.478516
\(587\) 1.81796e12 0.631995 0.315997 0.948760i \(-0.397661\pi\)
0.315997 + 0.948760i \(0.397661\pi\)
\(588\) 0 0
\(589\) −2.99400e12 −1.02502
\(590\) 2.98957e12 1.01572
\(591\) 0 0
\(592\) 7.63844e10 0.0255598
\(593\) −1.51787e12 −0.504067 −0.252034 0.967719i \(-0.581099\pi\)
−0.252034 + 0.967719i \(0.581099\pi\)
\(594\) 0 0
\(595\) 1.87626e11 0.0613716
\(596\) −2.09647e12 −0.680581
\(597\) 0 0
\(598\) 8.23571e11 0.263357
\(599\) 2.38179e12 0.755932 0.377966 0.925819i \(-0.376624\pi\)
0.377966 + 0.925819i \(0.376624\pi\)
\(600\) 0 0
\(601\) −1.56564e12 −0.489506 −0.244753 0.969586i \(-0.578707\pi\)
−0.244753 + 0.969586i \(0.578707\pi\)
\(602\) −2.32048e11 −0.0720102
\(603\) 0 0
\(604\) 3.08851e12 0.944239
\(605\) −9.98009e12 −3.02855
\(606\) 0 0
\(607\) 4.20030e12 1.25583 0.627915 0.778282i \(-0.283908\pi\)
0.627915 + 0.778282i \(0.283908\pi\)
\(608\) −6.07323e12 −1.80241
\(609\) 0 0
\(610\) 3.34000e12 0.976701
\(611\) 2.79852e12 0.812350
\(612\) 0 0
\(613\) −3.01770e12 −0.863186 −0.431593 0.902068i \(-0.642048\pi\)
−0.431593 + 0.902068i \(0.642048\pi\)
\(614\) 1.45548e12 0.413285
\(615\) 0 0
\(616\) 6.26275e11 0.175247
\(617\) 1.00060e12 0.277958 0.138979 0.990295i \(-0.455618\pi\)
0.138979 + 0.990295i \(0.455618\pi\)
\(618\) 0 0
\(619\) −4.40437e12 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) −2.12026e12 −0.576271
\(621\) 0 0
\(622\) −2.13706e10 −0.00572479
\(623\) 5.08985e11 0.135366
\(624\) 0 0
\(625\) −3.44636e12 −0.903443
\(626\) −6.68750e11 −0.174052
\(627\) 0 0
\(628\) 8.76534e11 0.224880
\(629\) 2.84953e11 0.0725848
\(630\) 0 0
\(631\) −4.71463e11 −0.118390 −0.0591950 0.998246i \(-0.518853\pi\)
−0.0591950 + 0.998246i \(0.518853\pi\)
\(632\) −1.20616e11 −0.0300730
\(633\) 0 0
\(634\) −2.26427e12 −0.556579
\(635\) 5.68172e12 1.38675
\(636\) 0 0
\(637\) −3.38592e12 −0.814798
\(638\) 4.42674e12 1.05777
\(639\) 0 0
\(640\) −4.79582e12 −1.12993
\(641\) 3.52736e12 0.825256 0.412628 0.910900i \(-0.364611\pi\)
0.412628 + 0.910900i \(0.364611\pi\)
\(642\) 0 0
\(643\) 4.60256e12 1.06182 0.530909 0.847429i \(-0.321851\pi\)
0.530909 + 0.847429i \(0.321851\pi\)
\(644\) −1.75876e11 −0.0402922
\(645\) 0 0
\(646\) −1.77319e12 −0.400598
\(647\) −5.52821e12 −1.24027 −0.620134 0.784496i \(-0.712922\pi\)
−0.620134 + 0.784496i \(0.712922\pi\)
\(648\) 0 0
\(649\) −9.88115e12 −2.18628
\(650\) −2.30868e12 −0.507286
\(651\) 0 0
\(652\) 2.14902e11 0.0465723
\(653\) 6.02691e12 1.29714 0.648568 0.761157i \(-0.275368\pi\)
0.648568 + 0.761157i \(0.275368\pi\)
\(654\) 0 0
\(655\) −1.48326e12 −0.314871
\(656\) 2.93659e11 0.0619121
\(657\) 0 0
\(658\) 2.81098e11 0.0584575
\(659\) 2.97106e11 0.0613658 0.0306829 0.999529i \(-0.490232\pi\)
0.0306829 + 0.999529i \(0.490232\pi\)
\(660\) 0 0
\(661\) 5.47716e12 1.11596 0.557981 0.829854i \(-0.311576\pi\)
0.557981 + 0.829854i \(0.311576\pi\)
\(662\) 2.04154e12 0.413141
\(663\) 0 0
\(664\) 3.15782e12 0.630422
\(665\) 1.33567e12 0.264852
\(666\) 0 0
\(667\) −3.07103e12 −0.600784
\(668\) 3.70921e12 0.720755
\(669\) 0 0
\(670\) −5.05535e11 −0.0969202
\(671\) −1.10394e13 −2.10229
\(672\) 0 0
\(673\) −4.06690e12 −0.764180 −0.382090 0.924125i \(-0.624795\pi\)
−0.382090 + 0.924125i \(0.624795\pi\)
\(674\) −1.37198e12 −0.256081
\(675\) 0 0
\(676\) 1.18635e12 0.218500
\(677\) 4.84588e12 0.886591 0.443295 0.896376i \(-0.353809\pi\)
0.443295 + 0.896376i \(0.353809\pi\)
\(678\) 0 0
\(679\) −5.43125e8 −9.80586e−5 0
\(680\) −3.10207e12 −0.556367
\(681\) 0 0
\(682\) −3.29618e12 −0.583419
\(683\) −5.80468e12 −1.02067 −0.510334 0.859976i \(-0.670478\pi\)
−0.510334 + 0.859976i \(0.670478\pi\)
\(684\) 0 0
\(685\) 2.02342e12 0.351138
\(686\) −6.83977e11 −0.117919
\(687\) 0 0
\(688\) −1.01834e12 −0.173279
\(689\) −8.67598e12 −1.46667
\(690\) 0 0
\(691\) 9.46733e12 1.57971 0.789853 0.613296i \(-0.210157\pi\)
0.789853 + 0.613296i \(0.210157\pi\)
\(692\) 1.58581e11 0.0262890
\(693\) 0 0
\(694\) −6.14901e12 −1.00621
\(695\) 4.28805e12 0.697153
\(696\) 0 0
\(697\) 1.09550e12 0.175818
\(698\) 2.63545e12 0.420248
\(699\) 0 0
\(700\) 4.93025e11 0.0776118
\(701\) 7.16819e12 1.12119 0.560594 0.828091i \(-0.310573\pi\)
0.560594 + 0.828091i \(0.310573\pi\)
\(702\) 0 0
\(703\) 2.02852e12 0.313243
\(704\) −5.05034e12 −0.774896
\(705\) 0 0
\(706\) 3.63778e12 0.551080
\(707\) −5.86422e11 −0.0882720
\(708\) 0 0
\(709\) 3.05585e12 0.454175 0.227088 0.973874i \(-0.427080\pi\)
0.227088 + 0.973874i \(0.427080\pi\)
\(710\) 5.63962e12 0.832889
\(711\) 0 0
\(712\) −8.41517e12 −1.22716
\(713\) 2.28671e12 0.331366
\(714\) 0 0
\(715\) 1.46397e13 2.09486
\(716\) 3.32523e12 0.472839
\(717\) 0 0
\(718\) 4.41665e12 0.620202
\(719\) −5.43142e12 −0.757938 −0.378969 0.925409i \(-0.623721\pi\)
−0.378969 + 0.925409i \(0.623721\pi\)
\(720\) 0 0
\(721\) −9.85320e11 −0.135790
\(722\) −8.49327e12 −1.16321
\(723\) 0 0
\(724\) −7.33155e12 −0.991682
\(725\) 8.60889e12 1.15725
\(726\) 0 0
\(727\) 7.25700e12 0.963502 0.481751 0.876308i \(-0.340001\pi\)
0.481751 + 0.876308i \(0.340001\pi\)
\(728\) −6.21902e11 −0.0820599
\(729\) 0 0
\(730\) 1.14103e13 1.48711
\(731\) −3.79895e12 −0.492079
\(732\) 0 0
\(733\) 8.75487e11 0.112016 0.0560082 0.998430i \(-0.482163\pi\)
0.0560082 + 0.998430i \(0.482163\pi\)
\(734\) −9.83905e11 −0.125118
\(735\) 0 0
\(736\) 4.63852e12 0.582679
\(737\) 1.67090e12 0.208615
\(738\) 0 0
\(739\) 3.05185e12 0.376411 0.188206 0.982130i \(-0.439733\pi\)
0.188206 + 0.982130i \(0.439733\pi\)
\(740\) 1.43654e12 0.176106
\(741\) 0 0
\(742\) −8.71459e11 −0.105543
\(743\) −8.89001e11 −0.107017 −0.0535085 0.998567i \(-0.517040\pi\)
−0.0535085 + 0.998567i \(0.517040\pi\)
\(744\) 0 0
\(745\) −1.21602e13 −1.44623
\(746\) −4.81650e12 −0.569386
\(747\) 0 0
\(748\) 4.15041e12 0.484768
\(749\) −1.09742e12 −0.127411
\(750\) 0 0
\(751\) 6.97996e12 0.800707 0.400353 0.916361i \(-0.368887\pi\)
0.400353 + 0.916361i \(0.368887\pi\)
\(752\) 1.23360e12 0.140667
\(753\) 0 0
\(754\) −4.39583e12 −0.495302
\(755\) 1.79144e13 2.00651
\(756\) 0 0
\(757\) 1.54121e13 1.70581 0.852905 0.522066i \(-0.174838\pi\)
0.852905 + 0.522066i \(0.174838\pi\)
\(758\) 4.07031e12 0.447833
\(759\) 0 0
\(760\) −2.20830e13 −2.40103
\(761\) −8.97482e12 −0.970052 −0.485026 0.874500i \(-0.661190\pi\)
−0.485026 + 0.874500i \(0.661190\pi\)
\(762\) 0 0
\(763\) 2.14188e11 0.0228789
\(764\) −9.44520e12 −1.00298
\(765\) 0 0
\(766\) −1.55223e12 −0.162902
\(767\) 9.81215e12 1.02373
\(768\) 0 0
\(769\) 2.29561e12 0.236718 0.118359 0.992971i \(-0.462237\pi\)
0.118359 + 0.992971i \(0.462237\pi\)
\(770\) 1.47048e12 0.150748
\(771\) 0 0
\(772\) −1.76248e12 −0.178585
\(773\) −4.85355e12 −0.488936 −0.244468 0.969657i \(-0.578613\pi\)
−0.244468 + 0.969657i \(0.578613\pi\)
\(774\) 0 0
\(775\) −6.41023e12 −0.638287
\(776\) 8.97960e9 0.000888954 0
\(777\) 0 0
\(778\) −9.77154e12 −0.956213
\(779\) 7.79863e12 0.758752
\(780\) 0 0
\(781\) −1.86401e13 −1.79274
\(782\) 1.35430e12 0.129504
\(783\) 0 0
\(784\) −1.49253e12 −0.141091
\(785\) 5.08419e12 0.477869
\(786\) 0 0
\(787\) −1.25927e13 −1.17013 −0.585063 0.810987i \(-0.698930\pi\)
−0.585063 + 0.810987i \(0.698930\pi\)
\(788\) −4.29899e12 −0.397191
\(789\) 0 0
\(790\) −2.83203e11 −0.0258688
\(791\) −1.27813e12 −0.116086
\(792\) 0 0
\(793\) 1.09623e13 0.984400
\(794\) 6.88447e12 0.614721
\(795\) 0 0
\(796\) −4.20994e12 −0.371678
\(797\) 5.33654e12 0.468487 0.234243 0.972178i \(-0.424739\pi\)
0.234243 + 0.972178i \(0.424739\pi\)
\(798\) 0 0
\(799\) 4.60195e12 0.399468
\(800\) −1.30029e13 −1.12237
\(801\) 0 0
\(802\) 2.86513e12 0.244545
\(803\) −3.77133e13 −3.20092
\(804\) 0 0
\(805\) −1.02014e12 −0.0856207
\(806\) 3.27316e12 0.273187
\(807\) 0 0
\(808\) 9.69545e12 0.800234
\(809\) −2.28889e13 −1.87870 −0.939348 0.342964i \(-0.888569\pi\)
−0.939348 + 0.342964i \(0.888569\pi\)
\(810\) 0 0
\(811\) −9.01947e11 −0.0732128 −0.0366064 0.999330i \(-0.511655\pi\)
−0.0366064 + 0.999330i \(0.511655\pi\)
\(812\) 9.38744e11 0.0757783
\(813\) 0 0
\(814\) 2.23326e12 0.178291
\(815\) 1.24651e12 0.0989659
\(816\) 0 0
\(817\) −2.70439e13 −2.12359
\(818\) 1.22757e13 0.958640
\(819\) 0 0
\(820\) 5.52276e12 0.426574
\(821\) −7.51091e12 −0.576963 −0.288482 0.957485i \(-0.593150\pi\)
−0.288482 + 0.957485i \(0.593150\pi\)
\(822\) 0 0
\(823\) −1.53129e13 −1.16348 −0.581738 0.813377i \(-0.697627\pi\)
−0.581738 + 0.813377i \(0.697627\pi\)
\(824\) 1.62905e13 1.23101
\(825\) 0 0
\(826\) 9.85582e11 0.0736686
\(827\) 1.24239e13 0.923602 0.461801 0.886984i \(-0.347203\pi\)
0.461801 + 0.886984i \(0.347203\pi\)
\(828\) 0 0
\(829\) 7.14034e12 0.525078 0.262539 0.964921i \(-0.415440\pi\)
0.262539 + 0.964921i \(0.415440\pi\)
\(830\) 7.41451e12 0.542289
\(831\) 0 0
\(832\) 5.01508e12 0.362846
\(833\) −5.56789e12 −0.400671
\(834\) 0 0
\(835\) 2.15147e13 1.53160
\(836\) 2.95459e13 2.09204
\(837\) 0 0
\(838\) −1.13340e13 −0.793934
\(839\) 1.84306e13 1.28414 0.642068 0.766648i \(-0.278077\pi\)
0.642068 + 0.766648i \(0.278077\pi\)
\(840\) 0 0
\(841\) 1.88459e12 0.129907
\(842\) −1.36192e13 −0.933785
\(843\) 0 0
\(844\) 6.00062e12 0.407057
\(845\) 6.88121e12 0.464312
\(846\) 0 0
\(847\) −3.29017e12 −0.219656
\(848\) −3.82440e12 −0.253970
\(849\) 0 0
\(850\) −3.79644e12 −0.249455
\(851\) −1.54931e12 −0.101264
\(852\) 0 0
\(853\) 5.76438e12 0.372805 0.186403 0.982473i \(-0.440317\pi\)
0.186403 + 0.982473i \(0.440317\pi\)
\(854\) 1.10111e12 0.0708385
\(855\) 0 0
\(856\) 1.81439e13 1.15505
\(857\) 4.75199e12 0.300927 0.150464 0.988616i \(-0.451923\pi\)
0.150464 + 0.988616i \(0.451923\pi\)
\(858\) 0 0
\(859\) 4.45715e12 0.279311 0.139656 0.990200i \(-0.455400\pi\)
0.139656 + 0.990200i \(0.455400\pi\)
\(860\) −1.91517e13 −1.19389
\(861\) 0 0
\(862\) 5.65301e12 0.348736
\(863\) −1.65466e13 −1.01545 −0.507727 0.861518i \(-0.669514\pi\)
−0.507727 + 0.861518i \(0.669514\pi\)
\(864\) 0 0
\(865\) 9.19823e11 0.0558640
\(866\) −1.14045e13 −0.689041
\(867\) 0 0
\(868\) −6.98994e11 −0.0417960
\(869\) 9.36045e11 0.0556811
\(870\) 0 0
\(871\) −1.65923e12 −0.0976842
\(872\) −3.54122e12 −0.207410
\(873\) 0 0
\(874\) 9.64097e12 0.558881
\(875\) 2.32968e11 0.0134357
\(876\) 0 0
\(877\) 3.05493e13 1.74382 0.871912 0.489663i \(-0.162880\pi\)
0.871912 + 0.489663i \(0.162880\pi\)
\(878\) 4.68556e12 0.266095
\(879\) 0 0
\(880\) 6.45321e12 0.362747
\(881\) 6.84641e12 0.382887 0.191444 0.981504i \(-0.438683\pi\)
0.191444 + 0.981504i \(0.438683\pi\)
\(882\) 0 0
\(883\) −2.29262e13 −1.26914 −0.634570 0.772865i \(-0.718823\pi\)
−0.634570 + 0.772865i \(0.718823\pi\)
\(884\) −4.12143e12 −0.226993
\(885\) 0 0
\(886\) 8.92011e12 0.486316
\(887\) 2.46814e13 1.33879 0.669397 0.742905i \(-0.266553\pi\)
0.669397 + 0.742905i \(0.266553\pi\)
\(888\) 0 0
\(889\) 1.87311e12 0.100578
\(890\) −1.97587e13 −1.05561
\(891\) 0 0
\(892\) 1.08916e13 0.576034
\(893\) 3.27604e13 1.72392
\(894\) 0 0
\(895\) 1.92875e13 1.00478
\(896\) −1.58105e12 −0.0819521
\(897\) 0 0
\(898\) 1.11379e13 0.571556
\(899\) −1.22054e13 −0.623207
\(900\) 0 0
\(901\) −1.42670e13 −0.721224
\(902\) 8.58573e12 0.431865
\(903\) 0 0
\(904\) 2.11316e13 1.05238
\(905\) −4.25255e13 −2.10732
\(906\) 0 0
\(907\) −2.45774e13 −1.20588 −0.602938 0.797788i \(-0.706003\pi\)
−0.602938 + 0.797788i \(0.706003\pi\)
\(908\) −3.43549e12 −0.167727
\(909\) 0 0
\(910\) −1.46021e12 −0.0705879
\(911\) 5.71188e12 0.274756 0.137378 0.990519i \(-0.456133\pi\)
0.137378 + 0.990519i \(0.456133\pi\)
\(912\) 0 0
\(913\) −2.45065e13 −1.16725
\(914\) −1.06587e13 −0.505181
\(915\) 0 0
\(916\) −1.42109e13 −0.666950
\(917\) −4.88992e11 −0.0228370
\(918\) 0 0
\(919\) −1.90767e13 −0.882231 −0.441116 0.897450i \(-0.645417\pi\)
−0.441116 + 0.897450i \(0.645417\pi\)
\(920\) 1.68662e13 0.776198
\(921\) 0 0
\(922\) 2.09605e12 0.0955242
\(923\) 1.85099e13 0.839455
\(924\) 0 0
\(925\) 4.34312e12 0.195058
\(926\) −9.61502e12 −0.429735
\(927\) 0 0
\(928\) −2.47582e13 −1.09586
\(929\) −2.72807e13 −1.20167 −0.600835 0.799373i \(-0.705165\pi\)
−0.600835 + 0.799373i \(0.705165\pi\)
\(930\) 0 0
\(931\) −3.96367e13 −1.72911
\(932\) 1.27158e13 0.552040
\(933\) 0 0
\(934\) 4.68261e12 0.201339
\(935\) 2.40738e13 1.03013
\(936\) 0 0
\(937\) 1.38541e13 0.587152 0.293576 0.955936i \(-0.405155\pi\)
0.293576 + 0.955936i \(0.405155\pi\)
\(938\) −1.66661e11 −0.00702946
\(939\) 0 0
\(940\) 2.31999e13 0.969195
\(941\) 1.33886e12 0.0556648 0.0278324 0.999613i \(-0.491140\pi\)
0.0278324 + 0.999613i \(0.491140\pi\)
\(942\) 0 0
\(943\) −5.95632e12 −0.245287
\(944\) 4.32522e12 0.177270
\(945\) 0 0
\(946\) −2.97734e13 −1.20870
\(947\) −4.26511e13 −1.72328 −0.861639 0.507521i \(-0.830562\pi\)
−0.861639 + 0.507521i \(0.830562\pi\)
\(948\) 0 0
\(949\) 3.74500e13 1.49883
\(950\) −2.70261e13 −1.07653
\(951\) 0 0
\(952\) −1.02267e12 −0.0403524
\(953\) 2.62256e13 1.02993 0.514964 0.857212i \(-0.327805\pi\)
0.514964 + 0.857212i \(0.327805\pi\)
\(954\) 0 0
\(955\) −5.47853e13 −2.13132
\(956\) 2.01595e12 0.0780581
\(957\) 0 0
\(958\) −6.79366e12 −0.260591
\(959\) 6.67067e11 0.0254675
\(960\) 0 0
\(961\) −1.73514e13 −0.656266
\(962\) −2.21766e12 −0.0834849
\(963\) 0 0
\(964\) −8.13314e12 −0.303328
\(965\) −1.02230e13 −0.379493
\(966\) 0 0
\(967\) −1.79584e11 −0.00660463 −0.00330232 0.999995i \(-0.501051\pi\)
−0.00330232 + 0.999995i \(0.501051\pi\)
\(968\) 5.43971e13 1.99130
\(969\) 0 0
\(970\) 2.10839e10 0.000764679 0
\(971\) 2.95293e13 1.06602 0.533012 0.846108i \(-0.321060\pi\)
0.533012 + 0.846108i \(0.321060\pi\)
\(972\) 0 0
\(973\) 1.41365e12 0.0505633
\(974\) −1.18142e13 −0.420619
\(975\) 0 0
\(976\) 4.83221e12 0.170460
\(977\) 4.93042e13 1.73124 0.865622 0.500699i \(-0.166924\pi\)
0.865622 + 0.500699i \(0.166924\pi\)
\(978\) 0 0
\(979\) 6.53064e13 2.27213
\(980\) −2.80695e13 −0.972115
\(981\) 0 0
\(982\) 2.08128e13 0.714215
\(983\) −8.89206e12 −0.303747 −0.151873 0.988400i \(-0.548531\pi\)
−0.151873 + 0.988400i \(0.548531\pi\)
\(984\) 0 0
\(985\) −2.49356e13 −0.844029
\(986\) −7.22860e12 −0.243561
\(987\) 0 0
\(988\) −2.93396e13 −0.979599
\(989\) 2.06552e13 0.686508
\(990\) 0 0
\(991\) −1.67054e13 −0.550207 −0.275103 0.961415i \(-0.588712\pi\)
−0.275103 + 0.961415i \(0.588712\pi\)
\(992\) 1.84351e13 0.604426
\(993\) 0 0
\(994\) 1.85923e12 0.0604080
\(995\) −2.44191e13 −0.789814
\(996\) 0 0
\(997\) 2.62911e12 0.0842714 0.0421357 0.999112i \(-0.486584\pi\)
0.0421357 + 0.999112i \(0.486584\pi\)
\(998\) 8.43660e12 0.269203
\(999\) 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.1.2 4
3.2 odd 2 inner 27.10.a.d.1.3 yes 4
9.2 odd 6 81.10.c.j.28.2 8
9.4 even 3 81.10.c.j.55.3 8
9.5 odd 6 81.10.c.j.55.2 8
9.7 even 3 81.10.c.j.28.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.10.a.d.1.2 4 1.1 even 1 trivial
27.10.a.d.1.3 yes 4 3.2 odd 2 inner
81.10.c.j.28.2 8 9.2 odd 6
81.10.c.j.28.3 8 9.7 even 3
81.10.c.j.55.2 8 9.5 odd 6
81.10.c.j.55.3 8 9.4 even 3