Properties

Label 9.22.a.a.1.1
Level $9$
Weight $22$
Character 9.1
Self dual yes
Analytic conductor $25.153$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,22,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1728.00 q^{2} +888832. q^{4} +4.15128e7 q^{5} +5.38430e8 q^{7} +2.08798e9 q^{8} +O(q^{10})\) \(q-1728.00 q^{2} +888832. q^{4} +4.15128e7 q^{5} +5.38430e8 q^{7} +2.08798e9 q^{8} -7.17341e10 q^{10} +6.41130e10 q^{11} -1.30980e11 q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} +1.34921e13 q^{19} +3.68979e13 q^{20} -1.10787e14 q^{22} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.26334e14 q^{26} +4.78574e14 q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} +1.42422e16 q^{34} +2.23517e16 q^{35} +3.44400e15 q^{37} -2.33144e16 q^{38} +8.66777e16 q^{40} +2.18424e16 q^{41} -7.17928e16 q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -2.68639e17 q^{49} -2.15391e18 q^{50} -1.16419e17 q^{52} +2.17229e18 q^{53} +2.66151e18 q^{55} +1.12423e18 q^{56} -3.49844e18 q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} +2.70285e18 q^{64} -5.43735e18 q^{65} +9.24391e18 q^{67} -7.32578e18 q^{68} -3.86238e19 q^{70} +2.03874e19 q^{71} +1.66178e19 q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} +6.79403e19 q^{79} -2.27160e20 q^{80} -3.77437e19 q^{82} -3.95037e19 q^{83} -3.42149e20 q^{85} +1.24058e20 q^{86} +1.33867e20 q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} +4.89965e20 q^{94} +5.60095e20 q^{95} +5.71815e19 q^{97} +4.64209e20 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1728.00 −1.19324 −0.596621 0.802523i \(-0.703491\pi\)
−0.596621 + 0.802523i \(0.703491\pi\)
\(3\) 0 0
\(4\) 888832. 0.423828
\(5\) 4.15128e7 1.90106 0.950532 0.310627i \(-0.100539\pi\)
0.950532 + 0.310627i \(0.100539\pi\)
\(6\) 0 0
\(7\) 5.38430e8 0.720443 0.360222 0.932867i \(-0.382701\pi\)
0.360222 + 0.932867i \(0.382701\pi\)
\(8\) 2.08798e9 0.687513
\(9\) 0 0
\(10\) −7.17341e10 −2.26843
\(11\) 6.41130e10 0.745286 0.372643 0.927975i \(-0.378452\pi\)
0.372643 + 0.927975i \(0.378452\pi\)
\(12\) 0 0
\(13\) −1.30980e11 −0.263512 −0.131756 0.991282i \(-0.542062\pi\)
−0.131756 + 0.991282i \(0.542062\pi\)
\(14\) −9.30407e11 −0.859663
\(15\) 0 0
\(16\) −5.47204e12 −1.24420
\(17\) −8.24203e12 −0.991563 −0.495782 0.868447i \(-0.665118\pi\)
−0.495782 + 0.868447i \(0.665118\pi\)
\(18\) 0 0
\(19\) 1.34921e13 0.504855 0.252428 0.967616i \(-0.418771\pi\)
0.252428 + 0.967616i \(0.418771\pi\)
\(20\) 3.68979e13 0.805724
\(21\) 0 0
\(22\) −1.10787e14 −0.889308
\(23\) 2.33185e14 1.17370 0.586851 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 0 0
\(25\) 1.24647e15 2.61404
\(26\) 2.26334e14 0.314434
\(27\) 0 0
\(28\) 4.78574e14 0.305344
\(29\) 2.02456e15 0.893618 0.446809 0.894629i \(-0.352560\pi\)
0.446809 + 0.894629i \(0.352560\pi\)
\(30\) 0 0
\(31\) −6.86919e15 −1.50525 −0.752624 0.658451i \(-0.771212\pi\)
−0.752624 + 0.658451i \(0.771212\pi\)
\(32\) 5.07688e15 0.797117
\(33\) 0 0
\(34\) 1.42422e16 1.18318
\(35\) 2.23517e16 1.36961
\(36\) 0 0
\(37\) 3.44400e15 0.117746 0.0588728 0.998265i \(-0.481249\pi\)
0.0588728 + 0.998265i \(0.481249\pi\)
\(38\) −2.33144e16 −0.602415
\(39\) 0 0
\(40\) 8.66777e16 1.30701
\(41\) 2.18424e16 0.254138 0.127069 0.991894i \(-0.459443\pi\)
0.127069 + 0.991894i \(0.459443\pi\)
\(42\) 0 0
\(43\) −7.17928e16 −0.506597 −0.253298 0.967388i \(-0.581515\pi\)
−0.253298 + 0.967388i \(0.581515\pi\)
\(44\) 5.69857e16 0.315873
\(45\) 0 0
\(46\) −4.02943e17 −1.40051
\(47\) −2.83545e17 −0.786310 −0.393155 0.919472i \(-0.628617\pi\)
−0.393155 + 0.919472i \(0.628617\pi\)
\(48\) 0 0
\(49\) −2.68639e17 −0.480962
\(50\) −2.15391e18 −3.11919
\(51\) 0 0
\(52\) −1.16419e17 −0.111684
\(53\) 2.17229e18 1.70616 0.853081 0.521779i \(-0.174731\pi\)
0.853081 + 0.521779i \(0.174731\pi\)
\(54\) 0 0
\(55\) 2.66151e18 1.41684
\(56\) 1.12423e18 0.495314
\(57\) 0 0
\(58\) −3.49844e18 −1.06630
\(59\) −1.53483e18 −0.390944 −0.195472 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 1.18700e19 1.79613
\(63\) 0 0
\(64\) 2.70285e18 0.293044
\(65\) −5.43735e18 −0.500953
\(66\) 0 0
\(67\) 9.24391e18 0.619541 0.309771 0.950811i \(-0.399748\pi\)
0.309771 + 0.950811i \(0.399748\pi\)
\(68\) −7.32578e18 −0.420252
\(69\) 0 0
\(70\) −3.86238e19 −1.63427
\(71\) 2.03874e19 0.743273 0.371636 0.928378i \(-0.378797\pi\)
0.371636 + 0.928378i \(0.378797\pi\)
\(72\) 0 0
\(73\) 1.66178e19 0.452566 0.226283 0.974062i \(-0.427343\pi\)
0.226283 + 0.974062i \(0.427343\pi\)
\(74\) −5.95123e18 −0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19 0.536936
\(78\) 0 0
\(79\) 6.79403e19 0.807315 0.403658 0.914910i \(-0.367739\pi\)
0.403658 + 0.914910i \(0.367739\pi\)
\(80\) −2.27160e20 −2.36530
\(81\) 0 0
\(82\) −3.77437e19 −0.303249
\(83\) −3.95037e19 −0.279459 −0.139730 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) 0 0
\(85\) −3.42149e20 −1.88503
\(86\) 1.24058e20 0.604493
\(87\) 0 0
\(88\) 1.33867e20 0.512394
\(89\) −4.16117e19 −0.141456 −0.0707278 0.997496i \(-0.522532\pi\)
−0.0707278 + 0.997496i \(0.522532\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20 0.497448
\(93\) 0 0
\(94\) 4.89965e20 0.938259
\(95\) 5.60095e20 0.959762
\(96\) 0 0
\(97\) 5.71815e19 0.0787322 0.0393661 0.999225i \(-0.487466\pi\)
0.0393661 + 0.999225i \(0.487466\pi\)
\(98\) 4.64209e20 0.573904
\(99\) 0 0
\(100\) 1.10791e21 1.10791
\(101\) −4.32417e20 −0.389518 −0.194759 0.980851i \(-0.562393\pi\)
−0.194759 + 0.980851i \(0.562393\pi\)
\(102\) 0 0
\(103\) 1.84123e21 1.34995 0.674974 0.737841i \(-0.264155\pi\)
0.674974 + 0.737841i \(0.264155\pi\)
\(104\) −2.73483e20 −0.181168
\(105\) 0 0
\(106\) −3.75371e21 −2.03587
\(107\) 2.43805e21 1.19815 0.599077 0.800691i \(-0.295534\pi\)
0.599077 + 0.800691i \(0.295534\pi\)
\(108\) 0 0
\(109\) −4.13676e21 −1.67372 −0.836859 0.547418i \(-0.815611\pi\)
−0.836859 + 0.547418i \(0.815611\pi\)
\(110\) −4.59909e21 −1.69063
\(111\) 0 0
\(112\) −2.94631e21 −0.896374
\(113\) −3.47910e21 −0.964146 −0.482073 0.876131i \(-0.660116\pi\)
−0.482073 + 0.876131i \(0.660116\pi\)
\(114\) 0 0
\(115\) 9.68015e21 2.23128
\(116\) 1.79950e21 0.378741
\(117\) 0 0
\(118\) 2.65219e21 0.466491
\(119\) −4.43775e21 −0.714365
\(120\) 0 0
\(121\) −3.28977e21 −0.444548
\(122\) −7.45043e21 −0.923428
\(123\) 0 0
\(124\) −6.10556e21 −0.637966
\(125\) 3.19497e22 3.06840
\(126\) 0 0
\(127\) 1.37141e21 0.111488 0.0557438 0.998445i \(-0.482247\pi\)
0.0557438 + 0.998445i \(0.482247\pi\)
\(128\) −1.53175e22 −1.14679
\(129\) 0 0
\(130\) 9.39574e21 0.597758
\(131\) 2.45276e22 1.43981 0.719907 0.694071i \(-0.244185\pi\)
0.719907 + 0.694071i \(0.244185\pi\)
\(132\) 0 0
\(133\) 7.26455e21 0.363719
\(134\) −1.59735e22 −0.739263
\(135\) 0 0
\(136\) −1.72092e22 −0.681713
\(137\) −1.02835e22 −0.377204 −0.188602 0.982054i \(-0.560396\pi\)
−0.188602 + 0.982054i \(0.560396\pi\)
\(138\) 0 0
\(139\) 8.70692e21 0.274289 0.137145 0.990551i \(-0.456207\pi\)
0.137145 + 0.990551i \(0.456207\pi\)
\(140\) 1.98669e22 0.580478
\(141\) 0 0
\(142\) −3.52294e22 −0.886905
\(143\) −8.39753e21 −0.196392
\(144\) 0 0
\(145\) 8.40452e22 1.69883
\(146\) −2.87155e22 −0.540022
\(147\) 0 0
\(148\) 3.06114e21 0.0499039
\(149\) 9.03997e22 1.37313 0.686564 0.727069i \(-0.259118\pi\)
0.686564 + 0.727069i \(0.259118\pi\)
\(150\) 0 0
\(151\) −4.75206e22 −0.627514 −0.313757 0.949503i \(-0.601588\pi\)
−0.313757 + 0.949503i \(0.601588\pi\)
\(152\) 2.81712e22 0.347094
\(153\) 0 0
\(154\) −5.96512e22 −0.640695
\(155\) −2.85159e23 −2.86157
\(156\) 0 0
\(157\) −1.50901e23 −1.32356 −0.661781 0.749697i \(-0.730199\pi\)
−0.661781 + 0.749697i \(0.730199\pi\)
\(158\) −1.17401e23 −0.963323
\(159\) 0 0
\(160\) 2.10755e23 1.51537
\(161\) 1.25554e23 0.845586
\(162\) 0 0
\(163\) −4.83503e22 −0.286042 −0.143021 0.989720i \(-0.545682\pi\)
−0.143021 + 0.989720i \(0.545682\pi\)
\(164\) 1.94142e22 0.107711
\(165\) 0 0
\(166\) 6.82624e22 0.333462
\(167\) −4.78731e20 −0.00219568 −0.00109784 0.999999i \(-0.500349\pi\)
−0.00109784 + 0.999999i \(0.500349\pi\)
\(168\) 0 0
\(169\) −2.29909e23 −0.930562
\(170\) 5.91234e23 2.24929
\(171\) 0 0
\(172\) −6.38118e22 −0.214710
\(173\) 1.61804e23 0.512277 0.256139 0.966640i \(-0.417550\pi\)
0.256139 + 0.966640i \(0.417550\pi\)
\(174\) 0 0
\(175\) 6.71138e23 1.88327
\(176\) −3.50829e23 −0.927284
\(177\) 0 0
\(178\) 7.19050e22 0.168791
\(179\) 8.76377e22 0.193970 0.0969849 0.995286i \(-0.469080\pi\)
0.0969849 + 0.995286i \(0.469080\pi\)
\(180\) 0 0
\(181\) 9.36624e22 0.184476 0.0922381 0.995737i \(-0.470598\pi\)
0.0922381 + 0.995737i \(0.470598\pi\)
\(182\) 1.21865e23 0.226532
\(183\) 0 0
\(184\) 4.86885e23 0.806936
\(185\) 1.42970e23 0.223842
\(186\) 0 0
\(187\) −5.28422e23 −0.738999
\(188\) −2.52024e23 −0.333260
\(189\) 0 0
\(190\) −9.67843e23 −1.14523
\(191\) −1.20858e24 −1.35340 −0.676699 0.736260i \(-0.736590\pi\)
−0.676699 + 0.736260i \(0.736590\pi\)
\(192\) 0 0
\(193\) −1.78822e24 −1.79502 −0.897509 0.440997i \(-0.854625\pi\)
−0.897509 + 0.440997i \(0.854625\pi\)
\(194\) −9.88096e22 −0.0939466
\(195\) 0 0
\(196\) −2.38775e23 −0.203845
\(197\) −1.90963e24 −1.54545 −0.772723 0.634743i \(-0.781106\pi\)
−0.772723 + 0.634743i \(0.781106\pi\)
\(198\) 0 0
\(199\) 1.44254e24 1.04995 0.524977 0.851116i \(-0.324074\pi\)
0.524977 + 0.851116i \(0.324074\pi\)
\(200\) 2.60261e24 1.79719
\(201\) 0 0
\(202\) 7.47216e23 0.464790
\(203\) 1.09008e24 0.643801
\(204\) 0 0
\(205\) 9.06739e23 0.483133
\(206\) −3.18165e24 −1.61082
\(207\) 0 0
\(208\) 7.16728e23 0.327861
\(209\) 8.65020e23 0.376262
\(210\) 0 0
\(211\) 3.98848e24 1.56979 0.784895 0.619629i \(-0.212717\pi\)
0.784895 + 0.619629i \(0.212717\pi\)
\(212\) 1.93080e24 0.723119
\(213\) 0 0
\(214\) −4.21295e24 −1.42969
\(215\) −2.98032e24 −0.963072
\(216\) 0 0
\(217\) −3.69858e24 −1.08445
\(218\) 7.14833e24 1.99715
\(219\) 0 0
\(220\) 2.36564e24 0.600495
\(221\) 1.07954e24 0.261289
\(222\) 0 0
\(223\) −4.62963e24 −1.01940 −0.509700 0.860352i \(-0.670244\pi\)
−0.509700 + 0.860352i \(0.670244\pi\)
\(224\) 2.73354e24 0.574278
\(225\) 0 0
\(226\) 6.01188e24 1.15046
\(227\) 3.43010e24 0.626664 0.313332 0.949644i \(-0.398555\pi\)
0.313332 + 0.949644i \(0.398555\pi\)
\(228\) 0 0
\(229\) 8.11792e23 0.135261 0.0676304 0.997710i \(-0.478456\pi\)
0.0676304 + 0.997710i \(0.478456\pi\)
\(230\) −1.67273e25 −2.66246
\(231\) 0 0
\(232\) 4.22724e24 0.614374
\(233\) −8.22188e23 −0.114218 −0.0571089 0.998368i \(-0.518188\pi\)
−0.0571089 + 0.998368i \(0.518188\pi\)
\(234\) 0 0
\(235\) −1.17707e25 −1.49483
\(236\) −1.36421e24 −0.165693
\(237\) 0 0
\(238\) 7.66844e24 0.852411
\(239\) 8.85525e24 0.941940 0.470970 0.882149i \(-0.343904\pi\)
0.470970 + 0.882149i \(0.343904\pi\)
\(240\) 0 0
\(241\) 7.46934e24 0.727953 0.363977 0.931408i \(-0.381419\pi\)
0.363977 + 0.931408i \(0.381419\pi\)
\(242\) 5.68472e24 0.530454
\(243\) 0 0
\(244\) 3.83228e24 0.327992
\(245\) −1.11520e25 −0.914339
\(246\) 0 0
\(247\) −1.76720e24 −0.133035
\(248\) −1.43427e25 −1.03488
\(249\) 0 0
\(250\) −5.52091e25 −3.66134
\(251\) −9.46474e23 −0.0601914 −0.0300957 0.999547i \(-0.509581\pi\)
−0.0300957 + 0.999547i \(0.509581\pi\)
\(252\) 0 0
\(253\) 1.49502e25 0.874744
\(254\) −2.36979e24 −0.133032
\(255\) 0 0
\(256\) 2.08004e25 1.07535
\(257\) 1.91825e25 0.951936 0.475968 0.879463i \(-0.342098\pi\)
0.475968 + 0.879463i \(0.342098\pi\)
\(258\) 0 0
\(259\) 1.85435e24 0.0848290
\(260\) −4.83289e24 −0.212318
\(261\) 0 0
\(262\) −4.23837e25 −1.71805
\(263\) −8.88429e23 −0.0346009 −0.0173004 0.999850i \(-0.505507\pi\)
−0.0173004 + 0.999850i \(0.505507\pi\)
\(264\) 0 0
\(265\) 9.01776e25 3.24352
\(266\) −1.25531e25 −0.434006
\(267\) 0 0
\(268\) 8.21628e24 0.262579
\(269\) 2.13847e25 0.657211 0.328605 0.944467i \(-0.393421\pi\)
0.328605 + 0.944467i \(0.393421\pi\)
\(270\) 0 0
\(271\) −1.56435e25 −0.444791 −0.222395 0.974957i \(-0.571388\pi\)
−0.222395 + 0.974957i \(0.571388\pi\)
\(272\) 4.51007e25 1.23370
\(273\) 0 0
\(274\) 1.77699e25 0.450095
\(275\) 7.99152e25 1.94821
\(276\) 0 0
\(277\) −8.04973e25 −1.81863 −0.909313 0.416112i \(-0.863392\pi\)
−0.909313 + 0.416112i \(0.863392\pi\)
\(278\) −1.50456e25 −0.327294
\(279\) 0 0
\(280\) 4.66699e25 0.941623
\(281\) −8.33171e25 −1.61926 −0.809632 0.586938i \(-0.800333\pi\)
−0.809632 + 0.586938i \(0.800333\pi\)
\(282\) 0 0
\(283\) −4.46130e24 −0.0804829 −0.0402415 0.999190i \(-0.512813\pi\)
−0.0402415 + 0.999190i \(0.512813\pi\)
\(284\) 1.81209e25 0.315020
\(285\) 0 0
\(286\) 1.45109e25 0.234343
\(287\) 1.17606e25 0.183092
\(288\) 0 0
\(289\) −1.16088e24 −0.0168020
\(290\) −1.45230e26 −2.02711
\(291\) 0 0
\(292\) 1.47704e25 0.191810
\(293\) −9.67128e25 −1.21164 −0.605820 0.795602i \(-0.707155\pi\)
−0.605820 + 0.795602i \(0.707155\pi\)
\(294\) 0 0
\(295\) −6.37151e25 −0.743209
\(296\) 7.19099e24 0.0809516
\(297\) 0 0
\(298\) −1.56211e26 −1.63848
\(299\) −3.05426e25 −0.309284
\(300\) 0 0
\(301\) −3.86554e25 −0.364974
\(302\) 8.21155e25 0.748777
\(303\) 0 0
\(304\) −7.38293e25 −0.628140
\(305\) 1.78986e26 1.47120
\(306\) 0 0
\(307\) 1.68163e26 1.29056 0.645278 0.763948i \(-0.276742\pi\)
0.645278 + 0.763948i \(0.276742\pi\)
\(308\) 3.06828e25 0.227569
\(309\) 0 0
\(310\) 4.92755e26 3.41455
\(311\) −2.30370e26 −1.54327 −0.771636 0.636065i \(-0.780561\pi\)
−0.771636 + 0.636065i \(0.780561\pi\)
\(312\) 0 0
\(313\) −2.79658e26 −1.75151 −0.875753 0.482759i \(-0.839635\pi\)
−0.875753 + 0.482759i \(0.839635\pi\)
\(314\) 2.60756e26 1.57933
\(315\) 0 0
\(316\) 6.03875e25 0.342163
\(317\) −2.98501e25 −0.163615 −0.0818075 0.996648i \(-0.526069\pi\)
−0.0818075 + 0.996648i \(0.526069\pi\)
\(318\) 0 0
\(319\) 1.29801e26 0.666002
\(320\) 1.12203e26 0.557095
\(321\) 0 0
\(322\) −2.16957e26 −1.00899
\(323\) −1.11202e26 −0.500596
\(324\) 0 0
\(325\) −1.63263e26 −0.688831
\(326\) 8.35494e25 0.341317
\(327\) 0 0
\(328\) 4.56064e25 0.174723
\(329\) −1.52669e26 −0.566492
\(330\) 0 0
\(331\) 2.55594e26 0.889933 0.444967 0.895547i \(-0.353216\pi\)
0.444967 + 0.895547i \(0.353216\pi\)
\(332\) −3.51122e25 −0.118443
\(333\) 0 0
\(334\) 8.27248e23 0.00261998
\(335\) 3.83740e26 1.17779
\(336\) 0 0
\(337\) −4.91931e25 −0.141837 −0.0709187 0.997482i \(-0.522593\pi\)
−0.0709187 + 0.997482i \(0.522593\pi\)
\(338\) 3.97282e26 1.11039
\(339\) 0 0
\(340\) −3.04113e26 −0.798927
\(341\) −4.40405e26 −1.12184
\(342\) 0 0
\(343\) −4.45381e26 −1.06695
\(344\) −1.49902e26 −0.348292
\(345\) 0 0
\(346\) −2.79597e26 −0.611271
\(347\) −2.98136e26 −0.632345 −0.316173 0.948702i \(-0.602398\pi\)
−0.316173 + 0.948702i \(0.602398\pi\)
\(348\) 0 0
\(349\) 7.72834e26 1.54319 0.771595 0.636115i \(-0.219459\pi\)
0.771595 + 0.636115i \(0.219459\pi\)
\(350\) −1.15973e27 −2.24720
\(351\) 0 0
\(352\) 3.25494e26 0.594081
\(353\) 7.30755e26 1.29461 0.647303 0.762233i \(-0.275897\pi\)
0.647303 + 0.762233i \(0.275897\pi\)
\(354\) 0 0
\(355\) 8.46336e26 1.41301
\(356\) −3.69858e25 −0.0599529
\(357\) 0 0
\(358\) −1.51438e26 −0.231453
\(359\) 1.58936e25 0.0235901 0.0117951 0.999930i \(-0.496245\pi\)
0.0117951 + 0.999930i \(0.496245\pi\)
\(360\) 0 0
\(361\) −5.32173e26 −0.745121
\(362\) −1.61849e26 −0.220125
\(363\) 0 0
\(364\) −6.26836e25 −0.0804618
\(365\) 6.89849e26 0.860358
\(366\) 0 0
\(367\) −1.40734e27 −1.65732 −0.828660 0.559752i \(-0.810896\pi\)
−0.828660 + 0.559752i \(0.810896\pi\)
\(368\) −1.27600e27 −1.46032
\(369\) 0 0
\(370\) −2.47052e26 −0.267098
\(371\) 1.16962e27 1.22919
\(372\) 0 0
\(373\) −9.30077e26 −0.923797 −0.461898 0.886933i \(-0.652831\pi\)
−0.461898 + 0.886933i \(0.652831\pi\)
\(374\) 9.13112e26 0.881805
\(375\) 0 0
\(376\) −5.92035e26 −0.540599
\(377\) −2.65177e26 −0.235479
\(378\) 0 0
\(379\) 2.18541e27 1.83578 0.917892 0.396830i \(-0.129890\pi\)
0.917892 + 0.396830i \(0.129890\pi\)
\(380\) 4.97830e26 0.406774
\(381\) 0 0
\(382\) 2.08843e27 1.61493
\(383\) −2.10347e27 −1.58252 −0.791258 0.611482i \(-0.790574\pi\)
−0.791258 + 0.611482i \(0.790574\pi\)
\(384\) 0 0
\(385\) 1.43304e27 1.02075
\(386\) 3.09004e27 2.14189
\(387\) 0 0
\(388\) 5.08247e25 0.0333689
\(389\) 2.97815e26 0.190316 0.0951582 0.995462i \(-0.469664\pi\)
0.0951582 + 0.995462i \(0.469664\pi\)
\(390\) 0 0
\(391\) −1.92192e27 −1.16380
\(392\) −5.60912e26 −0.330667
\(393\) 0 0
\(394\) 3.29984e27 1.84409
\(395\) 2.82039e27 1.53476
\(396\) 0 0
\(397\) 6.36504e26 0.328474 0.164237 0.986421i \(-0.447484\pi\)
0.164237 + 0.986421i \(0.447484\pi\)
\(398\) −2.49271e27 −1.25285
\(399\) 0 0
\(400\) −6.82075e27 −3.25239
\(401\) −2.43888e27 −1.13286 −0.566428 0.824111i \(-0.691675\pi\)
−0.566428 + 0.824111i \(0.691675\pi\)
\(402\) 0 0
\(403\) 8.99728e26 0.396651
\(404\) −3.84346e26 −0.165089
\(405\) 0 0
\(406\) −1.88367e27 −0.768211
\(407\) 2.20805e26 0.0877542
\(408\) 0 0
\(409\) 5.48032e26 0.206876 0.103438 0.994636i \(-0.467016\pi\)
0.103438 + 0.994636i \(0.467016\pi\)
\(410\) −1.56684e27 −0.576495
\(411\) 0 0
\(412\) 1.63654e27 0.572146
\(413\) −8.26399e26 −0.281653
\(414\) 0 0
\(415\) −1.63991e27 −0.531269
\(416\) −6.64970e26 −0.210050
\(417\) 0 0
\(418\) −1.49475e27 −0.448971
\(419\) 6.08246e27 1.78169 0.890844 0.454309i \(-0.150114\pi\)
0.890844 + 0.454309i \(0.150114\pi\)
\(420\) 0 0
\(421\) −4.05990e27 −1.13124 −0.565618 0.824667i \(-0.691362\pi\)
−0.565618 + 0.824667i \(0.691362\pi\)
\(422\) −6.89209e27 −1.87314
\(423\) 0 0
\(424\) 4.53568e27 1.17301
\(425\) −1.02735e28 −2.59199
\(426\) 0 0
\(427\) 2.32149e27 0.557537
\(428\) 2.16702e27 0.507811
\(429\) 0 0
\(430\) 5.14999e27 1.14918
\(431\) 7.87214e27 1.71428 0.857140 0.515084i \(-0.172239\pi\)
0.857140 + 0.515084i \(0.172239\pi\)
\(432\) 0 0
\(433\) −1.73785e27 −0.360486 −0.180243 0.983622i \(-0.557688\pi\)
−0.180243 + 0.983622i \(0.557688\pi\)
\(434\) 6.39115e27 1.29401
\(435\) 0 0
\(436\) −3.67689e27 −0.709369
\(437\) 3.14615e27 0.592550
\(438\) 0 0
\(439\) 8.37416e27 1.50336 0.751681 0.659526i \(-0.229243\pi\)
0.751681 + 0.659526i \(0.229243\pi\)
\(440\) 5.55717e27 0.974094
\(441\) 0 0
\(442\) −1.86545e27 −0.311781
\(443\) 3.30286e25 0.00539077 0.00269539 0.999996i \(-0.499142\pi\)
0.00269539 + 0.999996i \(0.499142\pi\)
\(444\) 0 0
\(445\) −1.72742e27 −0.268916
\(446\) 7.99999e27 1.21639
\(447\) 0 0
\(448\) 1.45530e27 0.211121
\(449\) −5.21713e27 −0.739341 −0.369670 0.929163i \(-0.620529\pi\)
−0.369670 + 0.929163i \(0.620529\pi\)
\(450\) 0 0
\(451\) 1.40038e27 0.189406
\(452\) −3.09233e27 −0.408632
\(453\) 0 0
\(454\) −5.92721e27 −0.747763
\(455\) −2.92763e27 −0.360908
\(456\) 0 0
\(457\) 2.15211e26 0.0253363 0.0126682 0.999920i \(-0.495967\pi\)
0.0126682 + 0.999920i \(0.495967\pi\)
\(458\) −1.40278e27 −0.161399
\(459\) 0 0
\(460\) 8.60403e27 0.945681
\(461\) −1.68699e28 −1.81239 −0.906197 0.422855i \(-0.861028\pi\)
−0.906197 + 0.422855i \(0.861028\pi\)
\(462\) 0 0
\(463\) −1.90352e28 −1.95415 −0.977074 0.212898i \(-0.931710\pi\)
−0.977074 + 0.212898i \(0.931710\pi\)
\(464\) −1.10785e28 −1.11184
\(465\) 0 0
\(466\) 1.42074e27 0.136290
\(467\) 1.21027e28 1.13515 0.567576 0.823321i \(-0.307881\pi\)
0.567576 + 0.823321i \(0.307881\pi\)
\(468\) 0 0
\(469\) 4.97720e27 0.446344
\(470\) 2.03398e28 1.78369
\(471\) 0 0
\(472\) −3.20469e27 −0.268779
\(473\) −4.60286e27 −0.377559
\(474\) 0 0
\(475\) 1.68175e28 1.31971
\(476\) −3.94442e27 −0.302768
\(477\) 0 0
\(478\) −1.53019e28 −1.12396
\(479\) −6.95253e27 −0.499597 −0.249798 0.968298i \(-0.580364\pi\)
−0.249798 + 0.968298i \(0.580364\pi\)
\(480\) 0 0
\(481\) −4.51095e26 −0.0310274
\(482\) −1.29070e28 −0.868625
\(483\) 0 0
\(484\) −2.92405e27 −0.188412
\(485\) 2.37376e27 0.149675
\(486\) 0 0
\(487\) −1.06412e28 −0.642596 −0.321298 0.946978i \(-0.604119\pi\)
−0.321298 + 0.946978i \(0.604119\pi\)
\(488\) 9.00250e27 0.532053
\(489\) 0 0
\(490\) 1.92706e28 1.09103
\(491\) −1.68064e28 −0.931361 −0.465681 0.884953i \(-0.654190\pi\)
−0.465681 + 0.884953i \(0.654190\pi\)
\(492\) 0 0
\(493\) −1.66865e28 −0.886079
\(494\) 3.05372e27 0.158743
\(495\) 0 0
\(496\) 3.75885e28 1.87283
\(497\) 1.09772e28 0.535486
\(498\) 0 0
\(499\) −5.12285e27 −0.239583 −0.119792 0.992799i \(-0.538223\pi\)
−0.119792 + 0.992799i \(0.538223\pi\)
\(500\) 2.83979e28 1.30047
\(501\) 0 0
\(502\) 1.63551e27 0.0718229
\(503\) −1.99606e28 −0.858442 −0.429221 0.903200i \(-0.641212\pi\)
−0.429221 + 0.903200i \(0.641212\pi\)
\(504\) 0 0
\(505\) −1.79508e28 −0.740499
\(506\) −2.58339e28 −1.04378
\(507\) 0 0
\(508\) 1.21895e27 0.0472516
\(509\) 2.57966e27 0.0979550 0.0489775 0.998800i \(-0.484404\pi\)
0.0489775 + 0.998800i \(0.484404\pi\)
\(510\) 0 0
\(511\) 8.94749e27 0.326048
\(512\) −3.81989e27 −0.136369
\(513\) 0 0
\(514\) −3.31474e28 −1.13589
\(515\) 7.64346e28 2.56634
\(516\) 0 0
\(517\) −1.81789e28 −0.586026
\(518\) −3.20432e27 −0.101222
\(519\) 0 0
\(520\) −1.13531e28 −0.344412
\(521\) 2.61230e28 0.776652 0.388326 0.921522i \(-0.373053\pi\)
0.388326 + 0.921522i \(0.373053\pi\)
\(522\) 0 0
\(523\) 6.70750e28 1.91555 0.957774 0.287523i \(-0.0928317\pi\)
0.957774 + 0.287523i \(0.0928317\pi\)
\(524\) 2.18009e28 0.610233
\(525\) 0 0
\(526\) 1.53520e27 0.0412873
\(527\) 5.66161e28 1.49255
\(528\) 0 0
\(529\) 1.49036e28 0.377577
\(530\) −1.55827e29 −3.87031
\(531\) 0 0
\(532\) 6.45696e27 0.154155
\(533\) −2.86092e27 −0.0669684
\(534\) 0 0
\(535\) 1.01210e29 2.27777
\(536\) 1.93011e28 0.425943
\(537\) 0 0
\(538\) −3.69528e28 −0.784212
\(539\) −1.72233e28 −0.358454
\(540\) 0 0
\(541\) −2.15196e28 −0.430787 −0.215394 0.976527i \(-0.569103\pi\)
−0.215394 + 0.976527i \(0.569103\pi\)
\(542\) 2.70319e28 0.530743
\(543\) 0 0
\(544\) −4.18438e28 −0.790392
\(545\) −1.71728e29 −3.18185
\(546\) 0 0
\(547\) −7.46789e28 −1.33147 −0.665734 0.746189i \(-0.731882\pi\)
−0.665734 + 0.746189i \(0.731882\pi\)
\(548\) −9.14032e27 −0.159870
\(549\) 0 0
\(550\) −1.38093e29 −2.32469
\(551\) 2.73156e28 0.451148
\(552\) 0 0
\(553\) 3.65811e28 0.581625
\(554\) 1.39099e29 2.17006
\(555\) 0 0
\(556\) 7.73899e27 0.116252
\(557\) 7.95166e28 1.17214 0.586068 0.810262i \(-0.300675\pi\)
0.586068 + 0.810262i \(0.300675\pi\)
\(558\) 0 0
\(559\) 9.40343e27 0.133494
\(560\) −1.22309e29 −1.70406
\(561\) 0 0
\(562\) 1.43972e29 1.93217
\(563\) −5.46305e28 −0.719609 −0.359805 0.933028i \(-0.617157\pi\)
−0.359805 + 0.933028i \(0.617157\pi\)
\(564\) 0 0
\(565\) −1.44427e29 −1.83290
\(566\) 7.70912e27 0.0960356
\(567\) 0 0
\(568\) 4.25683e28 0.511010
\(569\) −9.43478e28 −1.11187 −0.555933 0.831227i \(-0.687639\pi\)
−0.555933 + 0.831227i \(0.687639\pi\)
\(570\) 0 0
\(571\) 8.05027e28 0.914390 0.457195 0.889367i \(-0.348854\pi\)
0.457195 + 0.889367i \(0.348854\pi\)
\(572\) −7.46400e27 −0.0832364
\(573\) 0 0
\(574\) −2.03223e28 −0.218473
\(575\) 2.90659e29 3.06811
\(576\) 0 0
\(577\) −1.67132e28 −0.170104 −0.0850519 0.996377i \(-0.527106\pi\)
−0.0850519 + 0.996377i \(0.527106\pi\)
\(578\) 2.00600e27 0.0200488
\(579\) 0 0
\(580\) 7.47020e28 0.720010
\(581\) −2.12700e28 −0.201334
\(582\) 0 0
\(583\) 1.39272e29 1.27158
\(584\) 3.46975e28 0.311145
\(585\) 0 0
\(586\) 1.67120e29 1.44578
\(587\) −3.15730e28 −0.268297 −0.134149 0.990961i \(-0.542830\pi\)
−0.134149 + 0.990961i \(0.542830\pi\)
\(588\) 0 0
\(589\) −9.26799e28 −0.759932
\(590\) 1.10100e29 0.886829
\(591\) 0 0
\(592\) −1.88457e28 −0.146499
\(593\) 5.48493e27 0.0418887 0.0209443 0.999781i \(-0.493333\pi\)
0.0209443 + 0.999781i \(0.493333\pi\)
\(594\) 0 0
\(595\) −1.84223e29 −1.35805
\(596\) 8.03502e28 0.581971
\(597\) 0 0
\(598\) 5.27776e28 0.369051
\(599\) −1.25621e29 −0.863136 −0.431568 0.902080i \(-0.642040\pi\)
−0.431568 + 0.902080i \(0.642040\pi\)
\(600\) 0 0
\(601\) 3.99325e28 0.264938 0.132469 0.991187i \(-0.457709\pi\)
0.132469 + 0.991187i \(0.457709\pi\)
\(602\) 6.67965e28 0.435503
\(603\) 0 0
\(604\) −4.22378e28 −0.265958
\(605\) −1.36567e29 −0.845115
\(606\) 0 0
\(607\) −2.46990e29 −1.47638 −0.738189 0.674594i \(-0.764319\pi\)
−0.738189 + 0.674594i \(0.764319\pi\)
\(608\) 6.84978e28 0.402429
\(609\) 0 0
\(610\) −3.09288e29 −1.75549
\(611\) 3.71387e28 0.207202
\(612\) 0 0
\(613\) 2.63911e28 0.142273 0.0711364 0.997467i \(-0.477337\pi\)
0.0711364 + 0.997467i \(0.477337\pi\)
\(614\) −2.90585e29 −1.53995
\(615\) 0 0
\(616\) 7.20777e28 0.369151
\(617\) −3.09820e29 −1.55997 −0.779984 0.625800i \(-0.784773\pi\)
−0.779984 + 0.625800i \(0.784773\pi\)
\(618\) 0 0
\(619\) −2.50758e29 −1.22040 −0.610202 0.792246i \(-0.708912\pi\)
−0.610202 + 0.792246i \(0.708912\pi\)
\(620\) −2.53459e29 −1.21281
\(621\) 0 0
\(622\) 3.98080e29 1.84150
\(623\) −2.24050e28 −0.101911
\(624\) 0 0
\(625\) 7.31956e29 3.21918
\(626\) 4.83249e29 2.08997
\(627\) 0 0
\(628\) −1.34125e29 −0.560963
\(629\) −2.83855e28 −0.116752
\(630\) 0 0
\(631\) −4.32770e28 −0.172167 −0.0860833 0.996288i \(-0.527435\pi\)
−0.0860833 + 0.996288i \(0.527435\pi\)
\(632\) 1.41858e29 0.555039
\(633\) 0 0
\(634\) 5.15809e28 0.195232
\(635\) 5.69309e28 0.211945
\(636\) 0 0
\(637\) 3.51864e28 0.126739
\(638\) −2.24296e29 −0.794702
\(639\) 0 0
\(640\) −6.35872e29 −2.18012
\(641\) 8.73381e28 0.294574 0.147287 0.989094i \(-0.452946\pi\)
0.147287 + 0.989094i \(0.452946\pi\)
\(642\) 0 0
\(643\) −4.72013e29 −1.54077 −0.770386 0.637578i \(-0.779937\pi\)
−0.770386 + 0.637578i \(0.779937\pi\)
\(644\) 1.11596e29 0.358383
\(645\) 0 0
\(646\) 1.92158e29 0.597332
\(647\) −1.26799e28 −0.0387812 −0.0193906 0.999812i \(-0.506173\pi\)
−0.0193906 + 0.999812i \(0.506173\pi\)
\(648\) 0 0
\(649\) −9.84027e28 −0.291365
\(650\) 2.82119e29 0.821943
\(651\) 0 0
\(652\) −4.29753e28 −0.121233
\(653\) 2.76226e29 0.766790 0.383395 0.923584i \(-0.374755\pi\)
0.383395 + 0.923584i \(0.374755\pi\)
\(654\) 0 0
\(655\) 1.01821e30 2.73718
\(656\) −1.19523e29 −0.316198
\(657\) 0 0
\(658\) 2.63812e29 0.675962
\(659\) −6.46511e28 −0.163034 −0.0815172 0.996672i \(-0.525977\pi\)
−0.0815172 + 0.996672i \(0.525977\pi\)
\(660\) 0 0
\(661\) −2.27730e29 −0.556295 −0.278147 0.960538i \(-0.589720\pi\)
−0.278147 + 0.960538i \(0.589720\pi\)
\(662\) −4.41667e29 −1.06191
\(663\) 0 0
\(664\) −8.24829e28 −0.192132
\(665\) 3.01572e29 0.691454
\(666\) 0 0
\(667\) 4.72097e29 1.04884
\(668\) −4.25512e26 −0.000930591 0
\(669\) 0 0
\(670\) −6.63103e29 −1.40539
\(671\) 2.76429e29 0.576763
\(672\) 0 0
\(673\) −3.79243e29 −0.766936 −0.383468 0.923554i \(-0.625270\pi\)
−0.383468 + 0.923554i \(0.625270\pi\)
\(674\) 8.50057e28 0.169246
\(675\) 0 0
\(676\) −2.04350e29 −0.394398
\(677\) 3.39717e29 0.645559 0.322780 0.946474i \(-0.395383\pi\)
0.322780 + 0.946474i \(0.395383\pi\)
\(678\) 0 0
\(679\) 3.07882e28 0.0567220
\(680\) −7.14400e29 −1.29598
\(681\) 0 0
\(682\) 7.61020e29 1.33863
\(683\) −4.54742e29 −0.787677 −0.393838 0.919180i \(-0.628853\pi\)
−0.393838 + 0.919180i \(0.628853\pi\)
\(684\) 0 0
\(685\) −4.26897e29 −0.717088
\(686\) 7.69619e29 1.27313
\(687\) 0 0
\(688\) 3.92853e29 0.630306
\(689\) −2.84526e29 −0.449594
\(690\) 0 0
\(691\) −3.71838e29 −0.569947 −0.284974 0.958535i \(-0.591985\pi\)
−0.284974 + 0.958535i \(0.591985\pi\)
\(692\) 1.43817e29 0.217118
\(693\) 0 0
\(694\) 5.15178e29 0.754542
\(695\) 3.61448e29 0.521442
\(696\) 0 0
\(697\) −1.80026e29 −0.251994
\(698\) −1.33546e30 −1.84140
\(699\) 0 0
\(700\) 5.96529e29 0.798183
\(701\) 9.37969e29 1.23637 0.618186 0.786032i \(-0.287868\pi\)
0.618186 + 0.786032i \(0.287868\pi\)
\(702\) 0 0
\(703\) 4.64668e28 0.0594445
\(704\) 1.73288e29 0.218401
\(705\) 0 0
\(706\) −1.26275e30 −1.54478
\(707\) −2.32826e29 −0.280626
\(708\) 0 0
\(709\) −7.54578e29 −0.882914 −0.441457 0.897282i \(-0.645538\pi\)
−0.441457 + 0.897282i \(0.645538\pi\)
\(710\) −1.46247e30 −1.68606
\(711\) 0 0
\(712\) −8.68842e28 −0.0972525
\(713\) −1.60179e30 −1.76671
\(714\) 0 0
\(715\) −3.48605e29 −0.373353
\(716\) 7.78952e28 0.0822098
\(717\) 0 0
\(718\) −2.74641e28 −0.0281488
\(719\) −1.22754e30 −1.23989 −0.619944 0.784646i \(-0.712845\pi\)
−0.619944 + 0.784646i \(0.712845\pi\)
\(720\) 0 0
\(721\) 9.91374e29 0.972561
\(722\) 9.19594e29 0.889111
\(723\) 0 0
\(724\) 8.32502e28 0.0781862
\(725\) 2.52356e30 2.33596
\(726\) 0 0
\(727\) 9.04407e29 0.813303 0.406652 0.913583i \(-0.366696\pi\)
0.406652 + 0.913583i \(0.366696\pi\)
\(728\) −1.47252e29 −0.130521
\(729\) 0 0
\(730\) −1.19206e30 −1.02662
\(731\) 5.91719e29 0.502323
\(732\) 0 0
\(733\) −1.38874e30 −1.14559 −0.572795 0.819699i \(-0.694141\pi\)
−0.572795 + 0.819699i \(0.694141\pi\)
\(734\) 2.43189e30 1.97759
\(735\) 0 0
\(736\) 1.18385e30 0.935578
\(737\) 5.92655e29 0.461736
\(738\) 0 0
\(739\) 2.11506e30 1.60161 0.800804 0.598927i \(-0.204406\pi\)
0.800804 + 0.598927i \(0.204406\pi\)
\(740\) 1.27076e29 0.0948705
\(741\) 0 0
\(742\) −2.02111e30 −1.46673
\(743\) −2.26805e30 −1.62282 −0.811411 0.584476i \(-0.801300\pi\)
−0.811411 + 0.584476i \(0.801300\pi\)
\(744\) 0 0
\(745\) 3.75274e30 2.61040
\(746\) 1.60717e30 1.10231
\(747\) 0 0
\(748\) −4.69678e29 −0.313208
\(749\) 1.31272e30 0.863202
\(750\) 0 0
\(751\) −6.98349e29 −0.446533 −0.223266 0.974757i \(-0.571672\pi\)
−0.223266 + 0.974757i \(0.571672\pi\)
\(752\) 1.55157e30 0.978326
\(753\) 0 0
\(754\) 4.58226e29 0.280984
\(755\) −1.97271e30 −1.19294
\(756\) 0 0
\(757\) −3.79778e29 −0.223369 −0.111685 0.993744i \(-0.535625\pi\)
−0.111685 + 0.993744i \(0.535625\pi\)
\(758\) −3.77639e30 −2.19054
\(759\) 0 0
\(760\) 1.16946e30 0.659849
\(761\) 7.50371e29 0.417577 0.208789 0.977961i \(-0.433048\pi\)
0.208789 + 0.977961i \(0.433048\pi\)
\(762\) 0 0
\(763\) −2.22736e30 −1.20582
\(764\) −1.07423e30 −0.573608
\(765\) 0 0
\(766\) 3.63479e30 1.88833
\(767\) 2.01032e29 0.103018
\(768\) 0 0
\(769\) 2.16884e29 0.108144 0.0540719 0.998537i \(-0.482780\pi\)
0.0540719 + 0.998537i \(0.482780\pi\)
\(770\) −2.47629e30 −1.21800
\(771\) 0 0
\(772\) −1.58943e30 −0.760779
\(773\) 3.08128e30 1.45495 0.727473 0.686136i \(-0.240695\pi\)
0.727473 + 0.686136i \(0.240695\pi\)
\(774\) 0 0
\(775\) −8.56227e30 −3.93478
\(776\) 1.19394e29 0.0541294
\(777\) 0 0
\(778\) −5.14625e29 −0.227094
\(779\) 2.94700e29 0.128303
\(780\) 0 0
\(781\) 1.30710e30 0.553951
\(782\) 3.32107e30 1.38870
\(783\) 0 0
\(784\) 1.47000e30 0.598412
\(785\) −6.26430e30 −2.51618
\(786\) 0 0
\(787\) 1.46460e30 0.572775 0.286387 0.958114i \(-0.407546\pi\)
0.286387 + 0.958114i \(0.407546\pi\)
\(788\) −1.69734e30 −0.655004
\(789\) 0 0
\(790\) −4.87363e30 −1.83134
\(791\) −1.87325e30 −0.694612
\(792\) 0 0
\(793\) −5.64732e29 −0.203927
\(794\) −1.09988e30 −0.391949
\(795\) 0 0
\(796\) 1.28217e30 0.445000
\(797\) −1.27731e30 −0.437505 −0.218752 0.975780i \(-0.570199\pi\)
−0.218752 + 0.975780i \(0.570199\pi\)
\(798\) 0 0
\(799\) 2.33698e30 0.779677
\(800\) 6.32819e30 2.08370
\(801\) 0 0
\(802\) 4.21439e30 1.35177
\(803\) 1.06541e30 0.337292
\(804\) 0 0
\(805\) 5.21208e30 1.60751
\(806\) −1.55473e30 −0.473301
\(807\) 0 0
\(808\) −9.02876e29 −0.267799
\(809\) 4.24975e30 1.24424 0.622120 0.782922i \(-0.286272\pi\)
0.622120 + 0.782922i \(0.286272\pi\)
\(810\) 0 0
\(811\) 2.05863e30 0.587298 0.293649 0.955913i \(-0.405130\pi\)
0.293649 + 0.955913i \(0.405130\pi\)
\(812\) 9.68902e29 0.272861
\(813\) 0 0
\(814\) −3.81551e29 −0.104712
\(815\) −2.00716e30 −0.543784
\(816\) 0 0
\(817\) −9.68636e29 −0.255758
\(818\) −9.46999e29 −0.246854
\(819\) 0 0
\(820\) 8.05938e29 0.204765
\(821\) 1.80717e29 0.0453309 0.0226655 0.999743i \(-0.492785\pi\)
0.0226655 + 0.999743i \(0.492785\pi\)
\(822\) 0 0
\(823\) −6.23532e30 −1.52462 −0.762308 0.647214i \(-0.775934\pi\)
−0.762308 + 0.647214i \(0.775934\pi\)
\(824\) 3.84445e30 0.928107
\(825\) 0 0
\(826\) 1.42802e30 0.336080
\(827\) −3.37179e30 −0.783524 −0.391762 0.920067i \(-0.628134\pi\)
−0.391762 + 0.920067i \(0.628134\pi\)
\(828\) 0 0
\(829\) −2.70711e29 −0.0613315 −0.0306658 0.999530i \(-0.509763\pi\)
−0.0306658 + 0.999530i \(0.509763\pi\)
\(830\) 2.83376e30 0.633933
\(831\) 0 0
\(832\) −3.54020e29 −0.0772205
\(833\) 2.21413e30 0.476904
\(834\) 0 0
\(835\) −1.98735e28 −0.00417413
\(836\) 7.68857e29 0.159470
\(837\) 0 0
\(838\) −1.05105e31 −2.12599
\(839\) 7.45457e30 1.48909 0.744547 0.667570i \(-0.232666\pi\)
0.744547 + 0.667570i \(0.232666\pi\)
\(840\) 0 0
\(841\) −1.03399e30 −0.201446
\(842\) 7.01551e30 1.34984
\(843\) 0 0
\(844\) 3.54509e30 0.665321
\(845\) −9.54415e30 −1.76906
\(846\) 0 0
\(847\) −1.77131e30 −0.320272
\(848\) −1.18868e31 −2.12280
\(849\) 0 0
\(850\) 1.77526e31 3.09287
\(851\) 8.03088e29 0.138198
\(852\) 0 0
\(853\) −6.10653e30 −1.02525 −0.512625 0.858613i \(-0.671327\pi\)
−0.512625 + 0.858613i \(0.671327\pi\)
\(854\) −4.01153e30 −0.665277
\(855\) 0 0
\(856\) 5.09059e30 0.823746
\(857\) 4.08307e30 0.652662 0.326331 0.945256i \(-0.394188\pi\)
0.326331 + 0.945256i \(0.394188\pi\)
\(858\) 0 0
\(859\) 5.27189e30 0.822316 0.411158 0.911564i \(-0.365125\pi\)
0.411158 + 0.911564i \(0.365125\pi\)
\(860\) −2.64900e30 −0.408177
\(861\) 0 0
\(862\) −1.36031e31 −2.04555
\(863\) −1.05537e31 −1.56780 −0.783902 0.620884i \(-0.786774\pi\)
−0.783902 + 0.620884i \(0.786774\pi\)
\(864\) 0 0
\(865\) 6.71693e30 0.973872
\(866\) 3.00300e30 0.430147
\(867\) 0 0
\(868\) −3.28742e30 −0.459619
\(869\) 4.35586e30 0.601681
\(870\) 0 0
\(871\) −1.21077e30 −0.163256
\(872\) −8.63747e30 −1.15070
\(873\) 0 0
\(874\) −5.43655e30 −0.707056
\(875\) 1.72027e31 2.21061
\(876\) 0 0
\(877\) 7.16069e30 0.898378 0.449189 0.893437i \(-0.351713\pi\)
0.449189 + 0.893437i \(0.351713\pi\)
\(878\) −1.44706e31 −1.79388
\(879\) 0 0
\(880\) −1.45639e31 −1.76283
\(881\) 1.05943e31 1.26714 0.633568 0.773687i \(-0.281590\pi\)
0.633568 + 0.773687i \(0.281590\pi\)
\(882\) 0 0
\(883\) 6.60744e30 0.771695 0.385848 0.922562i \(-0.373909\pi\)
0.385848 + 0.922562i \(0.373909\pi\)
\(884\) 9.59531e29 0.110742
\(885\) 0 0
\(886\) −5.70734e28 −0.00643250
\(887\) 2.74467e30 0.305698 0.152849 0.988250i \(-0.451155\pi\)
0.152849 + 0.988250i \(0.451155\pi\)
\(888\) 0 0
\(889\) 7.38406e29 0.0803205
\(890\) 2.98497e30 0.320882
\(891\) 0 0
\(892\) −4.11496e30 −0.432051
\(893\) −3.82561e30 −0.396973
\(894\) 0 0
\(895\) 3.63808e30 0.368749
\(896\) −8.24741e30 −0.826196
\(897\) 0 0
\(898\) 9.01519e30 0.882213
\(899\) −1.39071e31 −1.34512
\(900\) 0 0
\(901\) −1.79040e31 −1.69177
\(902\) −2.41986e30 −0.226007
\(903\) 0 0
\(904\) −7.26427e30 −0.662863
\(905\) 3.88819e30 0.350701
\(906\) 0 0
\(907\) 5.45568e30 0.480809 0.240404 0.970673i \(-0.422720\pi\)
0.240404 + 0.970673i \(0.422720\pi\)
\(908\) 3.04878e30 0.265598
\(909\) 0 0
\(910\) 5.05894e30 0.430651
\(911\) 8.75739e30 0.736940 0.368470 0.929640i \(-0.379882\pi\)
0.368470 + 0.929640i \(0.379882\pi\)
\(912\) 0 0
\(913\) −2.53270e30 −0.208277
\(914\) −3.71884e29 −0.0302324
\(915\) 0 0
\(916\) 7.21547e29 0.0573274
\(917\) 1.32064e31 1.03730
\(918\) 0 0
\(919\) 1.24295e31 0.954206 0.477103 0.878847i \(-0.341687\pi\)
0.477103 + 0.878847i \(0.341687\pi\)
\(920\) 2.02119e31 1.53404
\(921\) 0 0
\(922\) 2.91512e31 2.16263
\(923\) −2.67034e30 −0.195861
\(924\) 0 0
\(925\) 4.29285e30 0.307792
\(926\) 3.28929e31 2.33177
\(927\) 0 0
\(928\) 1.02785e31 0.712319
\(929\) −2.24310e31 −1.53703 −0.768517 0.639829i \(-0.779005\pi\)
−0.768517 + 0.639829i \(0.779005\pi\)
\(930\) 0 0
\(931\) −3.62451e30 −0.242816
\(932\) −7.30787e29 −0.0484087
\(933\) 0 0
\(934\) −2.09134e31 −1.35451
\(935\) −2.19362e31 −1.40488
\(936\) 0 0
\(937\) −1.10052e31 −0.689176 −0.344588 0.938754i \(-0.611981\pi\)
−0.344588 + 0.938754i \(0.611981\pi\)
\(938\) −8.60060e30 −0.532597
\(939\) 0 0
\(940\) −1.04622e31 −0.633549
\(941\) 2.38036e31 1.42545 0.712725 0.701444i \(-0.247461\pi\)
0.712725 + 0.701444i \(0.247461\pi\)
\(942\) 0 0
\(943\) 5.09332e30 0.298283
\(944\) 8.39866e30 0.486412
\(945\) 0 0
\(946\) 7.95373e30 0.450520
\(947\) −8.09762e30 −0.453610 −0.226805 0.973940i \(-0.572828\pi\)
−0.226805 + 0.973940i \(0.572828\pi\)
\(948\) 0 0
\(949\) −2.17660e30 −0.119257
\(950\) −2.90607e31 −1.57474
\(951\) 0 0
\(952\) −9.26593e30 −0.491135
\(953\) 3.42232e30 0.179410 0.0897048 0.995968i \(-0.471408\pi\)
0.0897048 + 0.995968i \(0.471408\pi\)
\(954\) 0 0
\(955\) −5.01715e31 −2.57290
\(956\) 7.87083e30 0.399220
\(957\) 0 0
\(958\) 1.20140e31 0.596140
\(959\) −5.53695e30 −0.271754
\(960\) 0 0
\(961\) 2.63603e31 1.26577
\(962\) 7.79493e29 0.0370232
\(963\) 0 0
\(964\) 6.63899e30 0.308527
\(965\) −7.42339e31 −3.41244
\(966\) 0 0
\(967\) 1.33121e31 0.598780 0.299390 0.954131i \(-0.403217\pi\)
0.299390 + 0.954131i \(0.403217\pi\)
\(968\) −6.86896e30 −0.305633
\(969\) 0 0
\(970\) −4.10186e30 −0.178598
\(971\) −9.71774e30 −0.418565 −0.209283 0.977855i \(-0.567113\pi\)
−0.209283 + 0.977855i \(0.567113\pi\)
\(972\) 0 0
\(973\) 4.68806e30 0.197610
\(974\) 1.83881e31 0.766773
\(975\) 0 0
\(976\) −2.35932e31 −0.962861
\(977\) 1.40637e31 0.567816 0.283908 0.958851i \(-0.408369\pi\)
0.283908 + 0.958851i \(0.408369\pi\)
\(978\) 0 0
\(979\) −2.66785e30 −0.105425
\(980\) −9.91222e30 −0.387523
\(981\) 0 0
\(982\) 2.90414e31 1.11134
\(983\) 4.87291e31 1.84492 0.922458 0.386098i \(-0.126177\pi\)
0.922458 + 0.386098i \(0.126177\pi\)
\(984\) 0 0
\(985\) −7.92741e31 −2.93799
\(986\) 2.88343e31 1.05731
\(987\) 0 0
\(988\) −1.57074e30 −0.0563841
\(989\) −1.67410e31 −0.594594
\(990\) 0 0
\(991\) 1.17847e30 0.0409773 0.0204887 0.999790i \(-0.493478\pi\)
0.0204887 + 0.999790i \(0.493478\pi\)
\(992\) −3.48741e31 −1.19986
\(993\) 0 0
\(994\) −1.89685e31 −0.638965
\(995\) 5.98838e31 1.99603
\(996\) 0 0
\(997\) 5.05438e31 1.64956 0.824781 0.565453i \(-0.191299\pi\)
0.824781 + 0.565453i \(0.191299\pi\)
\(998\) 8.85229e30 0.285881
\(999\) 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 9.22.a.a.1.1 1
3.2 odd 2 3.22.a.b.1.1 1
12.11 even 2 48.22.a.d.1.1 1
15.2 even 4 75.22.b.b.49.2 2
15.8 even 4 75.22.b.b.49.1 2
15.14 odd 2 75.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 3.2 odd 2
9.22.a.a.1.1 1 1.1 even 1 trivial
48.22.a.d.1.1 1 12.11 even 2
75.22.a.a.1.1 1 15.14 odd 2
75.22.b.b.49.1 2 15.8 even 4
75.22.b.b.49.2 2 15.2 even 4