Properties

Label 75.22.a.a.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
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 \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1728.00 q^{2} +59049.0 q^{3} +888832. q^{4} -1.02037e8 q^{6} -5.38430e8 q^{7} +2.08798e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-1728.00 q^{2} +59049.0 q^{3} +888832. q^{4} -1.02037e8 q^{6} -5.38430e8 q^{7} +2.08798e9 q^{8} +3.48678e9 q^{9} -6.41130e10 q^{11} +5.24846e10 q^{12} +1.30980e11 q^{13} +9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} -6.02516e12 q^{18} +1.34921e13 q^{19} -3.17937e13 q^{21} +1.10787e14 q^{22} +2.33185e14 q^{23} +1.23293e14 q^{24} -2.26334e14 q^{26} +2.05891e14 q^{27} -4.78574e14 q^{28} -2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} -3.78581e15 q^{33} +1.42422e16 q^{34} +3.09917e15 q^{36} -3.44400e15 q^{37} -2.33144e16 q^{38} +7.73424e15 q^{39} -2.18424e16 q^{41} +5.49396e16 q^{42} +7.17928e16 q^{43} -5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -3.23118e17 q^{48} -2.68639e17 q^{49} -4.86684e17 q^{51} +1.16419e17 q^{52} +2.17229e18 q^{53} -3.55780e17 q^{54} -1.12423e18 q^{56} +7.96695e17 q^{57} +3.49844e18 q^{58} +1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} -1.87739e18 q^{63} +2.70285e18 q^{64} +6.54188e18 q^{66} -9.24391e18 q^{67} -7.32578e18 q^{68} +1.37693e19 q^{69} -2.03874e19 q^{71} +7.28033e18 q^{72} -1.66178e19 q^{73} +5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} -1.33648e19 q^{78} +6.79403e19 q^{79} +1.21577e19 q^{81} +3.77437e19 q^{82} -3.95037e19 q^{83} -2.82593e19 q^{84} -1.24058e20 q^{86} -1.19548e20 q^{87} -1.33867e20 q^{88} +4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} -4.05619e20 q^{93} +4.89965e20 q^{94} +2.99785e20 q^{96} -5.71815e19 q^{97} +4.64209e20 q^{98} -2.23548e20 q^{99} +O(q^{100})\)

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 \(0.703491\pi\)
\(3\) 59049.0 0.577350
\(4\) 888832. 0.423828
\(5\) 0 0
\(6\) −1.02037e8 −0.688919
\(7\) −5.38430e8 −0.720443 −0.360222 0.932867i \(-0.617299\pi\)
−0.360222 + 0.932867i \(0.617299\pi\)
\(8\) 2.08798e9 0.687513
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −6.41130e10 −0.745286 −0.372643 0.927975i \(-0.621548\pi\)
−0.372643 + 0.927975i \(0.621548\pi\)
\(12\) 5.24846e10 0.244697
\(13\) 1.30980e11 0.263512 0.131756 0.991282i \(-0.457938\pi\)
0.131756 + 0.991282i \(0.457938\pi\)
\(14\) 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 \(0.665118\pi\)
\(18\) −6.02516e12 −0.397748
\(19\) 1.34921e13 0.504855 0.252428 0.967616i \(-0.418771\pi\)
0.252428 + 0.967616i \(0.418771\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 1.10787e14 0.889308
\(23\) 2.33185e14 1.17370 0.586851 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 1.23293e14 0.396936
\(25\) 0 0
\(26\) −2.26334e14 −0.314434
\(27\) 2.05891e14 0.192450
\(28\) −4.78574e14 −0.305344
\(29\) −2.02456e15 −0.893618 −0.446809 0.894629i \(-0.647440\pi\)
−0.446809 + 0.894629i \(0.647440\pi\)
\(30\) 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
\(33\) −3.78581e15 −0.430291
\(34\) 1.42422e16 1.18318
\(35\) 0 0
\(36\) 3.09917e15 0.141276
\(37\) −3.44400e15 −0.117746 −0.0588728 0.998265i \(-0.518751\pi\)
−0.0588728 + 0.998265i \(0.518751\pi\)
\(38\) −2.33144e16 −0.602415
\(39\) 7.73424e15 0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 5.49396e16 0.496327
\(43\) 7.17928e16 0.506597 0.253298 0.967388i \(-0.418485\pi\)
0.253298 + 0.967388i \(0.418485\pi\)
\(44\) −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 \(0.628617\pi\)
\(48\) −3.23118e17 −0.718338
\(49\) −2.68639e17 −0.480962
\(50\) 0 0
\(51\) −4.86684e17 −0.572479
\(52\) 1.16419e17 0.111684
\(53\) 2.17229e18 1.70616 0.853081 0.521779i \(-0.174731\pi\)
0.853081 + 0.521779i \(0.174731\pi\)
\(54\) −3.55780e17 −0.229640
\(55\) 0 0
\(56\) −1.12423e18 −0.495314
\(57\) 7.96695e17 0.291478
\(58\) 3.49844e18 1.06630
\(59\) 1.53483e18 0.390944 0.195472 0.980709i \(-0.437376\pi\)
0.195472 + 0.980709i \(0.437376\pi\)
\(60\) 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
\(63\) −1.87739e18 −0.240148
\(64\) 2.70285e18 0.293044
\(65\) 0 0
\(66\) 6.54188e18 0.513442
\(67\) −9.24391e18 −0.619541 −0.309771 0.950811i \(-0.600252\pi\)
−0.309771 + 0.950811i \(0.600252\pi\)
\(68\) −7.32578e18 −0.420252
\(69\) 1.37693e19 0.677637
\(70\) 0 0
\(71\) −2.03874e19 −0.743273 −0.371636 0.928378i \(-0.621203\pi\)
−0.371636 + 0.928378i \(0.621203\pi\)
\(72\) 7.28033e18 0.229171
\(73\) −1.66178e19 −0.452566 −0.226283 0.974062i \(-0.572657\pi\)
−0.226283 + 0.974062i \(0.572657\pi\)
\(74\) 5.95123e18 0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19 0.536936
\(78\) −1.33648e19 −0.181538
\(79\) 6.79403e19 0.807315 0.403658 0.914910i \(-0.367739\pi\)
0.403658 + 0.914910i \(0.367739\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 3.77437e19 0.303249
\(83\) −3.95037e19 −0.279459 −0.139730 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) −2.82593e19 −0.176290
\(85\) 0 0
\(86\) −1.24058e20 −0.604493
\(87\) −1.19548e20 −0.515931
\(88\) −1.33867e20 −0.512394
\(89\) 4.16117e19 0.141456 0.0707278 0.997496i \(-0.477468\pi\)
0.0707278 + 0.997496i \(0.477468\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20 0.497448
\(93\) −4.05619e20 −0.869055
\(94\) 4.89965e20 0.938259
\(95\) 0 0
\(96\) 2.99785e20 0.460216
\(97\) −5.71815e19 −0.0787322 −0.0393661 0.999225i \(-0.512534\pi\)
−0.0393661 + 0.999225i \(0.512534\pi\)
\(98\) 4.64209e20 0.573904
\(99\) −2.23548e20 −0.248429
\(100\) 0 0
\(101\) 4.32417e20 0.389518 0.194759 0.980851i \(-0.437607\pi\)
0.194759 + 0.980851i \(0.437607\pi\)
\(102\) 8.40989e20 0.683107
\(103\) −1.84123e21 −1.34995 −0.674974 0.737841i \(-0.735845\pi\)
−0.674974 + 0.737841i \(0.735845\pi\)
\(104\) 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 \(0.295534\pi\)
\(108\) 1.83003e20 0.0815658
\(109\) −4.13676e21 −1.67372 −0.836859 0.547418i \(-0.815611\pi\)
−0.836859 + 0.547418i \(0.815611\pi\)
\(110\) 0 0
\(111\) −2.03365e20 −0.0679805
\(112\) 2.94631e21 0.896374
\(113\) −3.47910e21 −0.964146 −0.482073 0.876131i \(-0.660116\pi\)
−0.482073 + 0.876131i \(0.660116\pi\)
\(114\) −1.37669e21 −0.347804
\(115\) 0 0
\(116\) −1.79950e21 −0.378741
\(117\) 4.56699e20 0.0878373
\(118\) −2.65219e21 −0.466491
\(119\) 4.43775e21 0.714365
\(120\) 0 0
\(121\) −3.28977e21 −0.444548
\(122\) −7.45043e21 −0.923428
\(123\) −1.28977e21 −0.146727
\(124\) −6.10556e21 −0.637966
\(125\) 0 0
\(126\) 3.24413e21 0.286554
\(127\) −1.37141e21 −0.111488 −0.0557438 0.998445i \(-0.517753\pi\)
−0.0557438 + 0.998445i \(0.517753\pi\)
\(128\) −1.53175e22 −1.14679
\(129\) 4.23929e21 0.292484
\(130\) 0 0
\(131\) −2.45276e22 −1.43981 −0.719907 0.694071i \(-0.755815\pi\)
−0.719907 + 0.694071i \(0.755815\pi\)
\(132\) −3.36495e21 −0.182370
\(133\) −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 \(0.560396\pi\)
\(138\) −2.37934e22 −0.808586
\(139\) 8.70692e21 0.274289 0.137145 0.990551i \(-0.456207\pi\)
0.137145 + 0.990551i \(0.456207\pi\)
\(140\) 0 0
\(141\) −1.67430e22 −0.453977
\(142\) 3.52294e22 0.886905
\(143\) −8.39753e21 −0.196392
\(144\) −1.90798e22 −0.414733
\(145\) 0 0
\(146\) 2.87155e22 0.540022
\(147\) −1.58629e22 −0.277683
\(148\) −3.06114e21 −0.0499039
\(149\) −9.03997e22 −1.37313 −0.686564 0.727069i \(-0.740882\pi\)
−0.686564 + 0.727069i \(0.740882\pi\)
\(150\) 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
\(153\) −2.87382e22 −0.330521
\(154\) −5.96512e22 −0.640695
\(155\) 0 0
\(156\) 6.87444e21 0.0644806
\(157\) 1.50901e23 1.32356 0.661781 0.749697i \(-0.269801\pi\)
0.661781 + 0.749697i \(0.269801\pi\)
\(158\) −1.17401e23 −0.963323
\(159\) 1.28271e23 0.985053
\(160\) 0 0
\(161\) −1.25554e23 −0.845586
\(162\) −2.10084e22 −0.132583
\(163\) 4.83503e22 0.286042 0.143021 0.989720i \(-0.454318\pi\)
0.143021 + 0.989720i \(0.454318\pi\)
\(164\) −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 \(0.500349\pi\)
\(168\) −6.63846e22 −0.285970
\(169\) −2.29909e23 −0.930562
\(170\) 0 0
\(171\) 4.70440e22 0.168285
\(172\) 6.38118e22 0.214710
\(173\) 1.61804e23 0.512277 0.256139 0.966640i \(-0.417550\pi\)
0.256139 + 0.966640i \(0.417550\pi\)
\(174\) 2.06580e23 0.615631
\(175\) 0 0
\(176\) 3.50829e23 0.927284
\(177\) 9.06303e22 0.225712
\(178\) −7.19050e22 −0.168791
\(179\) −8.76377e22 −0.193970 −0.0969849 0.995286i \(-0.530920\pi\)
−0.0969849 + 0.995286i \(0.530920\pi\)
\(180\) 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
\(183\) 2.54595e23 0.446800
\(184\) 4.86885e23 0.806936
\(185\) 0 0
\(186\) 7.00910e23 1.03699
\(187\) 5.28422e23 0.738999
\(188\) −2.52024e23 −0.333260
\(189\) −1.10858e23 −0.138649
\(190\) 0 0
\(191\) 1.20858e24 1.35340 0.676699 0.736260i \(-0.263410\pi\)
0.676699 + 0.736260i \(0.263410\pi\)
\(192\) 1.59601e23 0.169189
\(193\) 1.78822e24 1.79502 0.897509 0.440997i \(-0.145375\pi\)
0.897509 + 0.440997i \(0.145375\pi\)
\(194\) 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 \(0.781106\pi\)
\(198\) 3.86292e23 0.296436
\(199\) 1.44254e24 1.04995 0.524977 0.851116i \(-0.324074\pi\)
0.524977 + 0.851116i \(0.324074\pi\)
\(200\) 0 0
\(201\) −5.45844e23 −0.357692
\(202\) −7.47216e23 −0.464790
\(203\) 1.09008e24 0.643801
\(204\) −4.32580e23 −0.242633
\(205\) 0 0
\(206\) 3.18165e24 1.61082
\(207\) 8.13065e23 0.391234
\(208\) −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
\(213\) −1.20385e24 −0.429129
\(214\) −4.21295e24 −1.42969
\(215\) 0 0
\(216\) 4.29896e23 0.132312
\(217\) 3.69858e24 1.08445
\(218\) 7.14833e24 1.99715
\(219\) −9.81262e23 −0.261289
\(220\) 0 0
\(221\) −1.07954e24 −0.261289
\(222\) 3.51414e23 0.0811172
\(223\) 4.62963e24 1.01940 0.509700 0.860352i \(-0.329756\pi\)
0.509700 + 0.860352i \(0.329756\pi\)
\(224\) −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 \(0.398555\pi\)
\(228\) 7.08128e23 0.123537
\(229\) 8.11792e23 0.135261 0.0676304 0.997710i \(-0.478456\pi\)
0.0676304 + 0.997710i \(0.478456\pi\)
\(230\) 0 0
\(231\) 2.03839e24 0.310000
\(232\) −4.22724e24 −0.614374
\(233\) −8.22188e23 −0.114218 −0.0571089 0.998368i \(-0.518188\pi\)
−0.0571089 + 0.998368i \(0.518188\pi\)
\(234\) −7.89177e23 −0.104811
\(235\) 0 0
\(236\) 1.36421e24 0.165693
\(237\) 4.01181e24 0.466104
\(238\) −7.66844e24 −0.852411
\(239\) −8.85525e24 −0.941940 −0.470970 0.882149i \(-0.656096\pi\)
−0.470970 + 0.882149i \(0.656096\pi\)
\(240\) 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
\(243\) 7.17898e23 0.0641500
\(244\) 3.83228e24 0.327992
\(245\) 0 0
\(246\) 2.22873e24 0.175081
\(247\) 1.76720e24 0.133035
\(248\) −1.43427e25 −1.03488
\(249\) −2.33266e24 −0.161346
\(250\) 0 0
\(251\) 9.46474e23 0.0601914 0.0300957 0.999547i \(-0.490419\pi\)
0.0300957 + 0.999547i \(0.490419\pi\)
\(252\) −1.66868e24 −0.101781
\(253\) −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 \(0.342098\pi\)
\(258\) −7.32550e24 −0.349004
\(259\) 1.85435e24 0.0848290
\(260\) 0 0
\(261\) −7.05921e24 −0.297873
\(262\) 4.23837e25 1.71805
\(263\) −8.88429e23 −0.0346009 −0.0173004 0.999850i \(-0.505507\pi\)
−0.0173004 + 0.999850i \(0.505507\pi\)
\(264\) −7.90469e24 −0.295831
\(265\) 0 0
\(266\) 1.25531e25 0.434006
\(267\) 2.45713e24 0.0816694
\(268\) −8.21628e24 −0.262579
\(269\) −2.13847e25 −0.657211 −0.328605 0.944467i \(-0.606579\pi\)
−0.328605 + 0.944467i \(0.606579\pi\)
\(270\) 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
\(273\) −4.16435e24 −0.109607
\(274\) 1.77699e25 0.450095
\(275\) 0 0
\(276\) 1.22386e25 0.287202
\(277\) 8.04973e25 1.81863 0.909313 0.416112i \(-0.136608\pi\)
0.909313 + 0.416112i \(0.136608\pi\)
\(278\) −1.50456e25 −0.327294
\(279\) −2.39514e25 −0.501749
\(280\) 0 0
\(281\) 8.33171e25 1.61926 0.809632 0.586938i \(-0.199667\pi\)
0.809632 + 0.586938i \(0.199667\pi\)
\(282\) 2.89320e25 0.541704
\(283\) 4.46130e24 0.0804829 0.0402415 0.999190i \(-0.487187\pi\)
0.0402415 + 0.999190i \(0.487187\pi\)
\(284\) −1.81209e25 −0.315020
\(285\) 0 0
\(286\) 1.45109e25 0.234343
\(287\) 1.17606e25 0.183092
\(288\) 1.77020e25 0.265706
\(289\) −1.16088e24 −0.0168020
\(290\) 0 0
\(291\) −3.37651e24 −0.0454560
\(292\) −1.47704e25 −0.191810
\(293\) −9.67128e25 −1.21164 −0.605820 0.795602i \(-0.707155\pi\)
−0.605820 + 0.795602i \(0.707155\pi\)
\(294\) 2.74111e25 0.331344
\(295\) 0 0
\(296\) −7.19099e24 −0.0809516
\(297\) −1.32003e25 −0.143430
\(298\) 1.56211e26 1.63848
\(299\) 3.05426e25 0.309284
\(300\) 0 0
\(301\) −3.86554e25 −0.364974
\(302\) 8.21155e25 0.748777
\(303\) 2.55338e25 0.224889
\(304\) −7.38293e25 −0.628140
\(305\) 0 0
\(306\) 4.96596e25 0.394392
\(307\) −1.68163e26 −1.29056 −0.645278 0.763948i \(-0.723258\pi\)
−0.645278 + 0.763948i \(0.723258\pi\)
\(308\) 3.06828e25 0.227569
\(309\) −1.08723e26 −0.779393
\(310\) 0 0
\(311\) 2.30370e26 1.54327 0.771636 0.636065i \(-0.219439\pi\)
0.771636 + 0.636065i \(0.219439\pi\)
\(312\) 1.61489e25 0.104597
\(313\) 2.79658e26 1.75151 0.875753 0.482759i \(-0.160365\pi\)
0.875753 + 0.482759i \(0.160365\pi\)
\(314\) −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 \(0.526069\pi\)
\(318\) −2.21653e26 −1.17541
\(319\) 1.29801e26 0.666002
\(320\) 0 0
\(321\) 1.43964e26 0.691755
\(322\) 2.16957e26 1.00899
\(323\) −1.11202e26 −0.500596
\(324\) 1.08061e25 0.0470920
\(325\) 0 0
\(326\) −8.35494e25 −0.341317
\(327\) −2.44272e26 −0.966322
\(328\) −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
\(333\) −1.20085e25 −0.0392485
\(334\) 8.27248e23 0.00261998
\(335\) 0 0
\(336\) 1.73977e26 0.517522
\(337\) 4.91931e25 0.141837 0.0709187 0.997482i \(-0.477407\pi\)
0.0709187 + 0.997482i \(0.477407\pi\)
\(338\) 3.97282e26 1.11039
\(339\) −2.05437e26 −0.556650
\(340\) 0 0
\(341\) 4.40405e26 1.12184
\(342\) −8.12921e25 −0.200805
\(343\) 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 \(0.602398\pi\)
\(348\) −1.06258e26 −0.218666
\(349\) 7.72834e26 1.54319 0.771595 0.636115i \(-0.219459\pi\)
0.771595 + 0.636115i \(0.219459\pi\)
\(350\) 0 0
\(351\) 2.69676e25 0.0507129
\(352\) −3.25494e26 −0.594081
\(353\) 7.30755e26 1.29461 0.647303 0.762233i \(-0.275897\pi\)
0.647303 + 0.762233i \(0.275897\pi\)
\(354\) −1.56609e26 −0.269329
\(355\) 0 0
\(356\) 3.69858e25 0.0599529
\(357\) 2.62045e26 0.412439
\(358\) 1.51438e26 0.231453
\(359\) −1.58936e25 −0.0235901 −0.0117951 0.999930i \(-0.503755\pi\)
−0.0117951 + 0.999930i \(0.503755\pi\)
\(360\) 0 0
\(361\) −5.32173e26 −0.745121
\(362\) −1.61849e26 −0.220125
\(363\) −1.94258e26 −0.256660
\(364\) −6.26836e25 −0.0804618
\(365\) 0 0
\(366\) −4.39940e26 −0.533141
\(367\) 1.40734e27 1.65732 0.828660 0.559752i \(-0.189104\pi\)
0.828660 + 0.559752i \(0.189104\pi\)
\(368\) −1.27600e27 −1.46032
\(369\) −7.61598e25 −0.0847127
\(370\) 0 0
\(371\) −1.16962e27 −1.22919
\(372\) −3.60527e26 −0.368330
\(373\) 9.30077e26 0.923797 0.461898 0.886933i \(-0.347169\pi\)
0.461898 + 0.886933i \(0.347169\pi\)
\(374\) −9.13112e26 −0.881805
\(375\) 0 0
\(376\) −5.92035e26 −0.540599
\(377\) −2.65177e26 −0.235479
\(378\) 1.91562e26 0.165442
\(379\) 2.18541e27 1.83578 0.917892 0.396830i \(-0.129890\pi\)
0.917892 + 0.396830i \(0.129890\pi\)
\(380\) 0 0
\(381\) −8.09801e25 −0.0643674
\(382\) −2.08843e27 −1.61493
\(383\) −2.10347e27 −1.58252 −0.791258 0.611482i \(-0.790574\pi\)
−0.791258 + 0.611482i \(0.790574\pi\)
\(384\) −9.04484e26 −0.662099
\(385\) 0 0
\(386\) −3.09004e27 −2.14189
\(387\) 2.50326e26 0.168866
\(388\) −5.08247e25 −0.0333689
\(389\) −2.97815e26 −0.190316 −0.0951582 0.995462i \(-0.530336\pi\)
−0.0951582 + 0.995462i \(0.530336\pi\)
\(390\) 0 0
\(391\) −1.92192e27 −1.16380
\(392\) −5.60912e26 −0.330667
\(393\) −1.44833e27 −0.831277
\(394\) 3.29984e27 1.84409
\(395\) 0 0
\(396\) −1.98697e26 −0.105291
\(397\) −6.36504e26 −0.328474 −0.164237 0.986421i \(-0.552516\pi\)
−0.164237 + 0.986421i \(0.552516\pi\)
\(398\) −2.49271e27 −1.25285
\(399\) −4.28964e26 −0.209993
\(400\) 0 0
\(401\) 2.43888e27 1.13286 0.566428 0.824111i \(-0.308325\pi\)
0.566428 + 0.824111i \(0.308325\pi\)
\(402\) 9.43218e26 0.426814
\(403\) −8.99728e26 −0.396651
\(404\) 3.84346e26 0.165089
\(405\) 0 0
\(406\) −1.88367e27 −0.768211
\(407\) 2.20805e26 0.0877542
\(408\) −1.01618e27 −0.393587
\(409\) 5.48032e26 0.206876 0.103438 0.994636i \(-0.467016\pi\)
0.103438 + 0.994636i \(0.467016\pi\)
\(410\) 0 0
\(411\) −6.07232e26 −0.217779
\(412\) −1.63654e27 −0.572146
\(413\) −8.26399e26 −0.281653
\(414\) −1.40498e27 −0.466837
\(415\) 0 0
\(416\) 6.64970e26 0.210050
\(417\) 5.14135e26 0.158361
\(418\) 1.49475e27 0.448971
\(419\) −6.08246e27 −1.78169 −0.890844 0.454309i \(-0.849886\pi\)
−0.890844 + 0.454309i \(0.849886\pi\)
\(420\) 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
\(423\) −9.88659e26 −0.262103
\(424\) 4.53568e27 1.17301
\(425\) 0 0
\(426\) 2.08026e27 0.512055
\(427\) −2.32149e27 −0.557537
\(428\) 2.16702e27 0.507811
\(429\) −4.95866e26 −0.113387
\(430\) 0 0
\(431\) −7.87214e27 −1.71428 −0.857140 0.515084i \(-0.827761\pi\)
−0.857140 + 0.515084i \(0.827761\pi\)
\(432\) −1.12664e27 −0.239446
\(433\) 1.73785e27 0.360486 0.180243 0.983622i \(-0.442312\pi\)
0.180243 + 0.983622i \(0.442312\pi\)
\(434\) −6.39115e27 −1.29401
\(435\) 0 0
\(436\) −3.67689e27 −0.709369
\(437\) 3.14615e27 0.592550
\(438\) 1.69562e27 0.311782
\(439\) 8.37416e27 1.50336 0.751681 0.659526i \(-0.229243\pi\)
0.751681 + 0.659526i \(0.229243\pi\)
\(440\) 0 0
\(441\) −9.36687e26 −0.160321
\(442\) 1.86545e27 0.311781
\(443\) 3.30286e25 0.00539077 0.00269539 0.999996i \(-0.499142\pi\)
0.00269539 + 0.999996i \(0.499142\pi\)
\(444\) −1.80757e26 −0.0288120
\(445\) 0 0
\(446\) −7.99999e27 −1.21639
\(447\) −5.33801e27 −0.792776
\(448\) −1.45530e27 −0.211121
\(449\) 5.21713e27 0.739341 0.369670 0.929163i \(-0.379471\pi\)
0.369670 + 0.929163i \(0.379471\pi\)
\(450\) 0 0
\(451\) 1.40038e27 0.189406
\(452\) −3.09233e27 −0.408632
\(453\) −2.80604e27 −0.362296
\(454\) −5.92721e27 −0.747763
\(455\) 0 0
\(456\) 1.66348e27 0.200395
\(457\) −2.15211e26 −0.0253363 −0.0126682 0.999920i \(-0.504033\pi\)
−0.0126682 + 0.999920i \(0.504033\pi\)
\(458\) −1.40278e27 −0.161399
\(459\) −1.69696e27 −0.190826
\(460\) 0 0
\(461\) 1.68699e28 1.81239 0.906197 0.422855i \(-0.138972\pi\)
0.906197 + 0.422855i \(0.138972\pi\)
\(462\) −3.52234e27 −0.369906
\(463\) 1.90352e28 1.95415 0.977074 0.212898i \(-0.0682901\pi\)
0.977074 + 0.212898i \(0.0682901\pi\)
\(464\) 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 \(0.307881\pi\)
\(468\) 4.05929e26 0.0372279
\(469\) 4.97720e27 0.446344
\(470\) 0 0
\(471\) 8.91053e27 0.764159
\(472\) 3.20469e27 0.268779
\(473\) −4.60286e27 −0.377559
\(474\) −6.93240e27 −0.556175
\(475\) 0 0
\(476\) 3.94442e27 0.302768
\(477\) 7.57429e27 0.568721
\(478\) 1.53019e28 1.12396
\(479\) 6.95253e27 0.499597 0.249798 0.968298i \(-0.419636\pi\)
0.249798 + 0.968298i \(0.419636\pi\)
\(480\) 0 0
\(481\) −4.51095e26 −0.0310274
\(482\) −1.29070e28 −0.868625
\(483\) −7.41382e27 −0.488199
\(484\) −2.92405e27 −0.188412
\(485\) 0 0
\(486\) −1.24053e27 −0.0765466
\(487\) 1.06412e28 0.642596 0.321298 0.946978i \(-0.395881\pi\)
0.321298 + 0.946978i \(0.395881\pi\)
\(488\) 9.00250e27 0.532053
\(489\) 2.85504e27 0.165146
\(490\) 0 0
\(491\) 1.68064e28 0.931361 0.465681 0.884953i \(-0.345810\pi\)
0.465681 + 0.884953i \(0.345810\pi\)
\(492\) −1.14639e27 −0.0621869
\(493\) 1.66865e28 0.886079
\(494\) −3.05372e27 −0.158743
\(495\) 0 0
\(496\) 3.75885e28 1.87283
\(497\) 1.09772e28 0.535486
\(498\) 4.03083e27 0.192525
\(499\) −5.12285e27 −0.239583 −0.119792 0.992799i \(-0.538223\pi\)
−0.119792 + 0.992799i \(0.538223\pi\)
\(500\) 0 0
\(501\) −2.82686e25 −0.00126768
\(502\) −1.63551e27 −0.0718229
\(503\) −1.99606e28 −0.858442 −0.429221 0.903200i \(-0.641212\pi\)
−0.429221 + 0.903200i \(0.641212\pi\)
\(504\) −3.91994e27 −0.165105
\(505\) 0 0
\(506\) 2.58339e28 1.04378
\(507\) −1.35759e28 −0.537260
\(508\) −1.21895e27 −0.0472516
\(509\) −2.57966e27 −0.0979550 −0.0489775 0.998800i \(-0.515596\pi\)
−0.0489775 + 0.998800i \(0.515596\pi\)
\(510\) 0 0
\(511\) 8.94749e27 0.326048
\(512\) −3.81989e27 −0.136369
\(513\) 2.77790e27 0.0971594
\(514\) −3.31474e28 −1.13589
\(515\) 0 0
\(516\) 3.76802e27 0.123963
\(517\) 1.81789e28 0.586026
\(518\) −3.20432e27 −0.101222
\(519\) 9.55436e27 0.295763
\(520\) 0 0
\(521\) −2.61230e28 −0.776652 −0.388326 0.921522i \(-0.626947\pi\)
−0.388326 + 0.921522i \(0.626947\pi\)
\(522\) 1.21983e28 0.355435
\(523\) −6.70750e28 −1.91555 −0.957774 0.287523i \(-0.907168\pi\)
−0.957774 + 0.287523i \(0.907168\pi\)
\(524\) −2.18009e28 −0.610233
\(525\) 0 0
\(526\) 1.53520e27 0.0412873
\(527\) 5.66161e28 1.49255
\(528\) 2.07161e28 0.535367
\(529\) 1.49036e28 0.377577
\(530\) 0 0
\(531\) 5.35163e27 0.130315
\(532\) −6.45696e27 −0.154155
\(533\) −2.86092e27 −0.0669684
\(534\) −4.24592e27 −0.0974514
\(535\) 0 0
\(536\) −1.93011e28 −0.425943
\(537\) −5.17492e27 −0.111988
\(538\) 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
\(543\) 5.53067e27 0.106507
\(544\) −4.18438e28 −0.790392
\(545\) 0 0
\(546\) 7.19599e27 0.130788
\(547\) 7.46789e28 1.33147 0.665734 0.746189i \(-0.268118\pi\)
0.665734 + 0.746189i \(0.268118\pi\)
\(548\) −9.14032e27 −0.159870
\(549\) 1.50336e28 0.257960
\(550\) 0 0
\(551\) −2.73156e28 −0.451148
\(552\) 2.87500e28 0.465884
\(553\) −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 \(0.300675\pi\)
\(558\) 4.13880e28 0.598709
\(559\) 9.40343e27 0.133494
\(560\) 0 0
\(561\) 3.12028e28 0.426661
\(562\) −1.43972e29 −1.93217
\(563\) −5.46305e28 −0.719609 −0.359805 0.933028i \(-0.617157\pi\)
−0.359805 + 0.933028i \(0.617157\pi\)
\(564\) −1.48817e28 −0.192408
\(565\) 0 0
\(566\) −7.70912e27 −0.0960356
\(567\) −6.54605e27 −0.0800492
\(568\) −4.25683e28 −0.511010
\(569\) 9.43478e28 1.11187 0.555933 0.831227i \(-0.312361\pi\)
0.555933 + 0.831227i \(0.312361\pi\)
\(570\) 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
\(573\) 7.13655e28 0.781384
\(574\) −2.03223e28 −0.218473
\(575\) 0 0
\(576\) 9.42426e27 0.0976812
\(577\) 1.67132e28 0.170104 0.0850519 0.996377i \(-0.472894\pi\)
0.0850519 + 0.996377i \(0.472894\pi\)
\(578\) 2.00600e27 0.0200488
\(579\) 1.05592e29 1.03635
\(580\) 0 0
\(581\) 2.12700e28 0.201334
\(582\) 5.83461e27 0.0542401
\(583\) −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 \(0.542830\pi\)
\(588\) −1.40994e28 −0.117690
\(589\) −9.26799e28 −0.759932
\(590\) 0 0
\(591\) −1.12762e29 −0.892264
\(592\) 1.88457e28 0.146499
\(593\) 5.48493e27 0.0418887 0.0209443 0.999781i \(-0.493333\pi\)
0.0209443 + 0.999781i \(0.493333\pi\)
\(594\) 2.28101e28 0.171147
\(595\) 0 0
\(596\) −8.03502e28 −0.581971
\(597\) 8.51805e28 0.606191
\(598\) −5.27776e28 −0.369051
\(599\) 1.25621e29 0.863136 0.431568 0.902080i \(-0.357960\pi\)
0.431568 + 0.902080i \(0.357960\pi\)
\(600\) 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
\(603\) −3.22315e28 −0.206514
\(604\) −4.22378e28 −0.265958
\(605\) 0 0
\(606\) −4.41223e28 −0.268347
\(607\) 2.46990e29 1.47638 0.738189 0.674594i \(-0.235681\pi\)
0.738189 + 0.674594i \(0.235681\pi\)
\(608\) 6.84978e28 0.402429
\(609\) 6.43684e28 0.371699
\(610\) 0 0
\(611\) −3.71387e28 −0.207202
\(612\) −2.55434e28 −0.140084
\(613\) −2.63911e28 −0.142273 −0.0711364 0.997467i \(-0.522663\pi\)
−0.0711364 + 0.997467i \(0.522663\pi\)
\(614\) 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 \(0.784773\pi\)
\(618\) 1.87873e29 0.930005
\(619\) −2.50758e29 −1.22040 −0.610202 0.792246i \(-0.708912\pi\)
−0.610202 + 0.792246i \(0.708912\pi\)
\(620\) 0 0
\(621\) 4.80107e28 0.225879
\(622\) −3.98080e29 −1.84150
\(623\) −2.24050e28 −0.101911
\(624\) −4.23221e28 −0.189291
\(625\) 0 0
\(626\) −4.83249e29 −2.08997
\(627\) −5.10785e28 −0.217235
\(628\) 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
\(633\) 2.35516e29 0.906318
\(634\) 5.15809e28 0.195232
\(635\) 0 0
\(636\) 1.14012e29 0.417493
\(637\) −3.51864e28 −0.126739
\(638\) −2.24296e29 −0.794702
\(639\) −7.10863e28 −0.247758
\(640\) 0 0
\(641\) −8.73381e28 −0.294574 −0.147287 0.989094i \(-0.547054\pi\)
−0.147287 + 0.989094i \(0.547054\pi\)
\(642\) −2.48770e29 −0.825431
\(643\) 4.72013e29 1.54077 0.770386 0.637578i \(-0.220063\pi\)
0.770386 + 0.637578i \(0.220063\pi\)
\(644\) −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 \(0.506173\pi\)
\(648\) 2.53849e28 0.0763903
\(649\) −9.84027e28 −0.291365
\(650\) 0 0
\(651\) 2.18397e29 0.626105
\(652\) 4.29753e28 0.121233
\(653\) 2.76226e29 0.766790 0.383395 0.923584i \(-0.374755\pi\)
0.383395 + 0.923584i \(0.374755\pi\)
\(654\) 4.22102e29 1.15306
\(655\) 0 0
\(656\) 1.19523e29 0.316198
\(657\) −5.79425e28 −0.150855
\(658\) −2.63812e29 −0.675962
\(659\) 6.46511e28 0.163034 0.0815172 0.996672i \(-0.474023\pi\)
0.0815172 + 0.996672i \(0.474023\pi\)
\(660\) 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
\(663\) −6.37459e28 −0.150855
\(664\) −8.24829e28 −0.192132
\(665\) 0 0
\(666\) 2.07507e28 0.0468330
\(667\) −4.72097e29 −1.04884
\(668\) −4.25512e26 −0.000930591 0
\(669\) 2.73375e29 0.588551
\(670\) 0 0
\(671\) −2.76429e29 −0.576763
\(672\) −1.61413e29 −0.331559
\(673\) 3.79243e29 0.766936 0.383468 0.923554i \(-0.374730\pi\)
0.383468 + 0.923554i \(0.374730\pi\)
\(674\) −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 \(0.395383\pi\)
\(678\) 3.54995e29 0.664218
\(679\) 3.07882e28 0.0567220
\(680\) 0 0
\(681\) 2.02544e29 0.361805
\(682\) −7.61020e29 −1.33863
\(683\) −4.54742e29 −0.787677 −0.393838 0.919180i \(-0.628853\pi\)
−0.393838 + 0.919180i \(0.628853\pi\)
\(684\) 4.18143e28 0.0713239
\(685\) 0 0
\(686\) −7.69619e29 −1.27313
\(687\) 4.79355e28 0.0780929
\(688\) −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
\(693\) 1.20365e29 0.178979
\(694\) 5.15178e29 0.754542
\(695\) 0 0
\(696\) −2.49614e29 −0.354709
\(697\) 1.80026e29 0.251994
\(698\) −1.33546e30 −1.84140
\(699\) −4.85494e28 −0.0659437
\(700\) 0 0
\(701\) −9.37969e29 −1.23637 −0.618186 0.786032i \(-0.712132\pi\)
−0.618186 + 0.786032i \(0.712132\pi\)
\(702\) −4.66001e28 −0.0605128
\(703\) −4.64668e28 −0.0594445
\(704\) −1.73288e29 −0.218401
\(705\) 0 0
\(706\) −1.26275e30 −1.54478
\(707\) −2.32826e29 −0.280626
\(708\) 8.05551e28 0.0956629
\(709\) −7.54578e29 −0.882914 −0.441457 0.897282i \(-0.645538\pi\)
−0.441457 + 0.897282i \(0.645538\pi\)
\(710\) 0 0
\(711\) 2.36893e29 0.269105
\(712\) 8.68842e28 0.0972525
\(713\) −1.60179e30 −1.76671
\(714\) −4.52814e29 −0.492140
\(715\) 0 0
\(716\) −7.78952e28 −0.0822098
\(717\) −5.22894e29 −0.543829
\(718\) 2.74641e28 0.0281488
\(719\) 1.22754e30 1.23989 0.619944 0.784646i \(-0.287155\pi\)
0.619944 + 0.784646i \(0.287155\pi\)
\(720\) 0 0
\(721\) 9.91374e29 0.972561
\(722\) 9.19594e29 0.889111
\(723\) 4.41057e29 0.420284
\(724\) 8.32502e28 0.0781862
\(725\) 0 0
\(726\) 3.35677e29 0.306258
\(727\) −9.04407e29 −0.813303 −0.406652 0.913583i \(-0.633304\pi\)
−0.406652 + 0.913583i \(0.633304\pi\)
\(728\) −1.47252e29 −0.130521
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −5.91719e29 −0.502323
\(732\) 2.26292e29 0.189367
\(733\) 1.38874e30 1.14559 0.572795 0.819699i \(-0.305859\pi\)
0.572795 + 0.819699i \(0.305859\pi\)
\(734\) −2.43189e30 −1.97759
\(735\) 0 0
\(736\) 1.18385e30 0.935578
\(737\) 5.92655e29 0.461736
\(738\) 1.31604e29 0.101083
\(739\) 2.11506e30 1.60161 0.800804 0.598927i \(-0.204406\pi\)
0.800804 + 0.598927i \(0.204406\pi\)
\(740\) 0 0
\(741\) 1.04351e29 0.0768080
\(742\) 2.02111e30 1.46673
\(743\) −2.26805e30 −1.62282 −0.811411 0.584476i \(-0.801300\pi\)
−0.811411 + 0.584476i \(0.801300\pi\)
\(744\) −8.46923e29 −0.597487
\(745\) 0 0
\(746\) −1.60717e30 −1.10231
\(747\) −1.37741e29 −0.0931530
\(748\) 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
\(753\) 5.58883e28 0.0347515
\(754\) 4.58226e29 0.280984
\(755\) 0 0
\(756\) −9.85341e28 −0.0587635
\(757\) 3.79778e29 0.223369 0.111685 0.993744i \(-0.464375\pi\)
0.111685 + 0.993744i \(0.464375\pi\)
\(758\) −3.77639e30 −2.19054
\(759\) −8.82794e29 −0.505034
\(760\) 0 0
\(761\) −7.50371e29 −0.417577 −0.208789 0.977961i \(-0.566952\pi\)
−0.208789 + 0.977961i \(0.566952\pi\)
\(762\) 1.39934e29 0.0768060
\(763\) 2.22736e30 1.20582
\(764\) 1.07423e30 0.573608
\(765\) 0 0
\(766\) 3.63479e30 1.88833
\(767\) 2.01032e29 0.103018
\(768\) 1.22824e30 0.620856
\(769\) 2.16884e29 0.108144 0.0540719 0.998537i \(-0.482780\pi\)
0.0540719 + 0.998537i \(0.482780\pi\)
\(770\) 0 0
\(771\) 1.13271e30 0.549600
\(772\) 1.58943e30 0.760779
\(773\) 3.08128e30 1.45495 0.727473 0.686136i \(-0.240695\pi\)
0.727473 + 0.686136i \(0.240695\pi\)
\(774\) −4.32563e29 −0.201498
\(775\) 0 0
\(776\) −1.19394e29 −0.0541294
\(777\) 1.09498e29 0.0489761
\(778\) 5.14625e29 0.227094
\(779\) −2.94700e29 −0.128303
\(780\) 0 0
\(781\) 1.30710e30 0.553951
\(782\) 3.32107e30 1.38870
\(783\) −4.16839e29 −0.171977
\(784\) 1.47000e30 0.598412
\(785\) 0 0
\(786\) 2.50271e30 0.991915
\(787\) −1.46460e30 −0.572775 −0.286387 0.958114i \(-0.592454\pi\)
−0.286387 + 0.958114i \(0.592454\pi\)
\(788\) −1.69734e30 −0.655004
\(789\) −5.24608e28 −0.0199768
\(790\) 0 0
\(791\) 1.87325e30 0.694612
\(792\) −4.66764e29 −0.170798
\(793\) 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 \(0.570199\pi\)
\(798\) 7.41250e29 0.250573
\(799\) 2.33698e30 0.779677
\(800\) 0 0
\(801\) 1.45091e29 0.0471519
\(802\) −4.21439e30 −1.35177
\(803\) 1.06541e30 0.337292
\(804\) −4.85163e29 −0.151600
\(805\) 0 0
\(806\) 1.55473e30 0.473301
\(807\) −1.26275e30 −0.379441
\(808\) 9.02876e29 0.267799
\(809\) −4.24975e30 −1.24424 −0.622120 0.782922i \(-0.713728\pi\)
−0.622120 + 0.782922i \(0.713728\pi\)
\(810\) 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
\(813\) −9.23732e29 −0.256800
\(814\) −3.81551e29 −0.104712
\(815\) 0 0
\(816\) 2.66315e30 0.712278
\(817\) 9.68636e29 0.255758
\(818\) −9.46999e29 −0.246854
\(819\) −2.45901e29 −0.0632818
\(820\) 0 0
\(821\) −1.80717e29 −0.0453309 −0.0226655 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(822\) 1.04930e30 0.259863
\(823\) 6.23532e30 1.52462 0.762308 0.647214i \(-0.224066\pi\)
0.762308 + 0.647214i \(0.224066\pi\)
\(824\) −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 \(0.628134\pi\)
\(828\) 7.22678e29 0.165816
\(829\) −2.70711e29 −0.0613315 −0.0306658 0.999530i \(-0.509763\pi\)
−0.0306658 + 0.999530i \(0.509763\pi\)
\(830\) 0 0
\(831\) 4.75328e30 1.04998
\(832\) 3.54020e29 0.0772205
\(833\) 2.21413e30 0.476904
\(834\) −8.88425e29 −0.188963
\(835\) 0 0
\(836\) −7.68857e29 −0.159470
\(837\) −1.41431e30 −0.289685
\(838\) 1.05105e31 2.12599
\(839\) −7.45457e30 −1.48909 −0.744547 0.667570i \(-0.767334\pi\)
−0.744547 + 0.667570i \(0.767334\pi\)
\(840\) 0 0
\(841\) −1.03399e30 −0.201446
\(842\) 7.01551e30 1.34984
\(843\) 4.91979e30 0.934882
\(844\) 3.54509e30 0.665321
\(845\) 0 0
\(846\) 1.70840e30 0.312753
\(847\) 1.77131e30 0.320272
\(848\) −1.18868e31 −2.12280
\(849\) 2.63435e29 0.0464668
\(850\) 0 0
\(851\) −8.03088e29 −0.138198
\(852\) −1.07002e30 −0.181877
\(853\) 6.10653e30 1.02525 0.512625 0.858613i \(-0.328673\pi\)
0.512625 + 0.858613i \(0.328673\pi\)
\(854\) 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 \(0.394188\pi\)
\(858\) 8.56856e29 0.135298
\(859\) 5.27189e30 0.822316 0.411158 0.911564i \(-0.365125\pi\)
0.411158 + 0.911564i \(0.365125\pi\)
\(860\) 0 0
\(861\) 6.94452e29 0.105708
\(862\) 1.36031e31 2.04555
\(863\) −1.05537e31 −1.56780 −0.783902 0.620884i \(-0.786774\pi\)
−0.783902 + 0.620884i \(0.786774\pi\)
\(864\) 1.04528e30 0.153405
\(865\) 0 0
\(866\) −3.00300e30 −0.430147
\(867\) −6.85488e28 −0.00970062
\(868\) 3.28742e30 0.459619
\(869\) −4.35586e30 −0.601681
\(870\) 0 0
\(871\) −1.21077e30 −0.163256
\(872\) −8.63747e30 −1.15070
\(873\) −1.99379e29 −0.0262441
\(874\) −5.43655e30 −0.707056
\(875\) 0 0
\(876\) −8.72177e29 −0.110742
\(877\) −7.16069e30 −0.898378 −0.449189 0.893437i \(-0.648287\pi\)
−0.449189 + 0.893437i \(0.648287\pi\)
\(878\) −1.44706e31 −1.79388
\(879\) −5.71079e30 −0.699541
\(880\) 0 0
\(881\) −1.05943e31 −1.26714 −0.633568 0.773687i \(-0.718410\pi\)
−0.633568 + 0.773687i \(0.718410\pi\)
\(882\) 1.61860e30 0.191301
\(883\) −6.60744e30 −0.771695 −0.385848 0.922562i \(-0.626091\pi\)
−0.385848 + 0.922562i \(0.626091\pi\)
\(884\) −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 \(0.451155\pi\)
\(888\) −4.24621e29 −0.0467374
\(889\) 7.38406e29 0.0803205
\(890\) 0 0
\(891\) −7.79465e29 −0.0828096
\(892\) 4.11496e30 0.432051
\(893\) −3.82561e30 −0.396973
\(894\) 9.22409e30 0.945974
\(895\) 0 0
\(896\) 8.24741e30 0.826196
\(897\) 1.80351e30 0.178565
\(898\) −9.01519e30 −0.882213
\(899\) 1.39071e31 1.34512
\(900\) 0 0
\(901\) −1.79040e31 −1.69177
\(902\) −2.41986e30 −0.226007
\(903\) −2.28256e30 −0.210718
\(904\) −7.26427e30 −0.662863
\(905\) 0 0
\(906\) 4.84884e30 0.432307
\(907\) −5.45568e30 −0.480809 −0.240404 0.970673i \(-0.577280\pi\)
−0.240404 + 0.970673i \(0.577280\pi\)
\(908\) 3.04878e30 0.265598
\(909\) 1.50774e30 0.129839
\(910\) 0 0
\(911\) −8.75739e30 −0.736940 −0.368470 0.929640i \(-0.620118\pi\)
−0.368470 + 0.929640i \(0.620118\pi\)
\(912\) −4.35955e30 −0.362657
\(913\) 2.53270e30 0.208277
\(914\) 3.71884e29 0.0302324
\(915\) 0 0
\(916\) 7.21547e29 0.0573274
\(917\) 1.32064e31 1.03730
\(918\) 2.93235e30 0.227702
\(919\) 1.24295e31 0.954206 0.477103 0.878847i \(-0.341687\pi\)
0.477103 + 0.878847i \(0.341687\pi\)
\(920\) 0 0
\(921\) −9.92984e30 −0.745102
\(922\) −2.91512e31 −2.16263
\(923\) −2.67034e30 −0.195861
\(924\) 1.81179e30 0.131387
\(925\) 0 0
\(926\) −3.28929e31 −2.33177
\(927\) −6.41997e30 −0.449983
\(928\) −1.02785e31 −0.712319
\(929\) 2.24310e31 1.53703 0.768517 0.639829i \(-0.220995\pi\)
0.768517 + 0.639829i \(0.220995\pi\)
\(930\) 0 0
\(931\) −3.62451e30 −0.242816
\(932\) −7.30787e29 −0.0484087
\(933\) 1.36031e31 0.891008
\(934\) −2.09134e31 −1.35451
\(935\) 0 0
\(936\) 9.53578e29 0.0603893
\(937\) 1.10052e31 0.689176 0.344588 0.938754i \(-0.388019\pi\)
0.344588 + 0.938754i \(0.388019\pi\)
\(938\) −8.60060e30 −0.532597
\(939\) 1.65135e31 1.01123
\(940\) 0 0
\(941\) −2.38036e31 −1.42545 −0.712725 0.701444i \(-0.752539\pi\)
−0.712725 + 0.701444i \(0.752539\pi\)
\(942\) −1.53974e31 −0.911828
\(943\) −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 \(0.572828\pi\)
\(948\) 3.56582e30 0.197548
\(949\) −2.17660e30 −0.119257
\(950\) 0 0
\(951\) −1.76262e30 −0.0944631
\(952\) 9.26593e30 0.491135
\(953\) 3.42232e30 0.179410 0.0897048 0.995968i \(-0.471408\pi\)
0.0897048 + 0.995968i \(0.471408\pi\)
\(954\) −1.30884e31 −0.678622
\(955\) 0 0
\(956\) −7.87083e30 −0.399220
\(957\) 7.66461e30 0.384516
\(958\) −1.20140e31 −0.596140
\(959\) 5.53695e30 0.271754
\(960\) 0 0
\(961\) 2.63603e31 1.26577
\(962\) 7.79493e29 0.0370232
\(963\) 8.50095e30 0.399385
\(964\) 6.63899e30 0.308527
\(965\) 0 0
\(966\) 1.28111e31 0.582540
\(967\) −1.33121e31 −0.598780 −0.299390 0.954131i \(-0.596783\pi\)
−0.299390 + 0.954131i \(0.596783\pi\)
\(968\) −6.86896e30 −0.305633
\(969\) −6.56638e30 −0.289019
\(970\) 0 0
\(971\) 9.71774e30 0.418565 0.209283 0.977855i \(-0.432887\pi\)
0.209283 + 0.977855i \(0.432887\pi\)
\(972\) 6.38091e29 0.0271886
\(973\) −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 \(0.408369\pi\)
\(978\) −4.93351e30 −0.197060
\(979\) −2.66785e30 −0.105425
\(980\) 0 0
\(981\) −1.44240e31 −0.557906
\(982\) −2.90414e31 −1.11134
\(983\) 4.87291e31 1.84492 0.922458 0.386098i \(-0.126177\pi\)
0.922458 + 0.386098i \(0.126177\pi\)
\(984\) −2.69301e30 −0.100877
\(985\) 0 0
\(986\) −2.88343e31 −1.05731
\(987\) 9.01495e30 0.327064
\(988\) 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
\(993\) 1.50926e31 0.513803
\(994\) −1.89685e31 −0.638965
\(995\) 0 0
\(996\) −2.07334e30 −0.0683829
\(997\) −5.05438e31 −1.64956 −0.824781 0.565453i \(-0.808701\pi\)
−0.824781 + 0.565453i \(0.808701\pi\)
\(998\) 8.85229e30 0.285881
\(999\) −7.09089e29 −0.0226602
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 75.22.a.a.1.1 1
5.2 odd 4 75.22.b.b.49.1 2
5.3 odd 4 75.22.b.b.49.2 2
5.4 even 2 3.22.a.b.1.1 1
15.14 odd 2 9.22.a.a.1.1 1
20.19 odd 2 48.22.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 5.4 even 2
9.22.a.a.1.1 1 15.14 odd 2
48.22.a.d.1.1 1 20.19 odd 2
75.22.a.a.1.1 1 1.1 even 1 trivial
75.22.b.b.49.1 2 5.2 odd 4
75.22.b.b.49.2 2 5.3 odd 4