Properties

Label 140.15.d.a.41.18
Level $140$
Weight $15$
Character 140.41
Analytic conductor $174.061$
Analytic rank $0$
Dimension $36$
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] = [140,15,Mod(41,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.41");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 140.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(174.060555413\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 41.18
Character \(\chi\) \(=\) 140.41
Dual form 140.15.d.a.41.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-303.599i q^{3} +34938.6i q^{5} +(-415571. + 711002. i) q^{7} +4.69080e6 q^{9} +O(q^{10})\) \(q-303.599i q^{3} +34938.6i q^{5} +(-415571. + 711002. i) q^{7} +4.69080e6 q^{9} +1.67561e7 q^{11} +1.13953e8i q^{13} +1.06073e7 q^{15} +7.03769e8i q^{17} +5.27331e8i q^{19} +(2.15859e8 + 1.26167e8i) q^{21} -3.75456e9 q^{23} -1.22070e9 q^{25} -2.87622e9i q^{27} +7.44902e9 q^{29} +4.18796e10i q^{31} -5.08712e9i q^{33} +(-2.48414e10 - 1.45194e10i) q^{35} +1.57759e11 q^{37} +3.45961e10 q^{39} -1.56891e10i q^{41} +3.48374e11 q^{43} +1.63890e11i q^{45} +5.96594e11i q^{47} +(-3.32825e11 - 5.90943e11i) q^{49} +2.13663e11 q^{51} -1.35416e12 q^{53} +5.85433e11i q^{55} +1.60097e11 q^{57} -2.06767e11i q^{59} -5.42825e12i q^{61} +(-1.94936e12 + 3.33517e12i) q^{63} -3.98137e12 q^{65} -3.85879e12 q^{67} +1.13988e12i q^{69} +1.27085e13 q^{71} -5.67859e12i q^{73} +3.70604e11i q^{75} +(-6.96333e12 + 1.19136e13i) q^{77} +2.24296e13 q^{79} +2.15627e13 q^{81} +8.87401e12i q^{83} -2.45887e13 q^{85} -2.26151e12i q^{87} -5.93459e13i q^{89} +(-8.10211e13 - 4.73557e13i) q^{91} +1.27146e13 q^{93} -1.84242e13 q^{95} +2.97418e13i q^{97} +7.85993e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 1364266 q^{7} - 54790830 q^{9} - 26192606 q^{11} + 44843750 q^{15} + 1512952694 q^{21} - 8670648636 q^{23} - 43945312500 q^{25} - 43956395706 q^{29} + 44839531250 q^{35} - 169523027308 q^{37} + 805671747486 q^{39} + 554691319560 q^{43} + 1095688125176 q^{49} + 1032170625826 q^{51} - 4262050556480 q^{53} - 3162001614828 q^{57} - 15828953775898 q^{63} - 3014492656250 q^{65} - 23495876471600 q^{67} + 22887953193352 q^{71} + 56411959501488 q^{77} + 8995204220854 q^{79} + 132868621377344 q^{81} - 2034215156250 q^{85} - 53912825209186 q^{91} + 101093199187348 q^{93} + 3862990000000 q^{95} - 416078903388420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 303.599i 0.138820i −0.997588 0.0694098i \(-0.977888\pi\)
0.997588 0.0694098i \(-0.0221116\pi\)
\(4\) 0 0
\(5\) 34938.6i 0.447214i
\(6\) 0 0
\(7\) −415571. + 711002.i −0.504613 + 0.863345i
\(8\) 0 0
\(9\) 4.69080e6 0.980729
\(10\) 0 0
\(11\) 1.67561e7 0.859851 0.429925 0.902864i \(-0.358540\pi\)
0.429925 + 0.902864i \(0.358540\pi\)
\(12\) 0 0
\(13\) 1.13953e8i 1.81603i 0.418934 + 0.908017i \(0.362404\pi\)
−0.418934 + 0.908017i \(0.637596\pi\)
\(14\) 0 0
\(15\) 1.06073e7 0.0620821
\(16\) 0 0
\(17\) 7.03769e8i 1.71509i 0.514406 + 0.857547i \(0.328013\pi\)
−0.514406 + 0.857547i \(0.671987\pi\)
\(18\) 0 0
\(19\) 5.27331e8i 0.589940i 0.955506 + 0.294970i \(0.0953097\pi\)
−0.955506 + 0.294970i \(0.904690\pi\)
\(20\) 0 0
\(21\) 2.15859e8 + 1.26167e8i 0.119849 + 0.0700503i
\(22\) 0 0
\(23\) −3.75456e9 −1.10272 −0.551358 0.834269i \(-0.685890\pi\)
−0.551358 + 0.834269i \(0.685890\pi\)
\(24\) 0 0
\(25\) −1.22070e9 −0.200000
\(26\) 0 0
\(27\) 2.87622e9i 0.274964i
\(28\) 0 0
\(29\) 7.44902e9 0.431830 0.215915 0.976412i \(-0.430727\pi\)
0.215915 + 0.976412i \(0.430727\pi\)
\(30\) 0 0
\(31\) 4.18796e10i 1.52220i 0.648636 + 0.761099i \(0.275340\pi\)
−0.648636 + 0.761099i \(0.724660\pi\)
\(32\) 0 0
\(33\) 5.08712e9i 0.119364i
\(34\) 0 0
\(35\) −2.48414e10 1.45194e10i −0.386100 0.225670i
\(36\) 0 0
\(37\) 1.57759e11 1.66182 0.830908 0.556410i \(-0.187821\pi\)
0.830908 + 0.556410i \(0.187821\pi\)
\(38\) 0 0
\(39\) 3.45961e10 0.252101
\(40\) 0 0
\(41\) 1.56891e10i 0.0805586i −0.999188 0.0402793i \(-0.987175\pi\)
0.999188 0.0402793i \(-0.0128248\pi\)
\(42\) 0 0
\(43\) 3.48374e11 1.28164 0.640821 0.767690i \(-0.278594\pi\)
0.640821 + 0.767690i \(0.278594\pi\)
\(44\) 0 0
\(45\) 1.63890e11i 0.438595i
\(46\) 0 0
\(47\) 5.96594e11i 1.17759i 0.808282 + 0.588795i \(0.200398\pi\)
−0.808282 + 0.588795i \(0.799602\pi\)
\(48\) 0 0
\(49\) −3.32825e11 5.90943e11i −0.490731 0.871311i
\(50\) 0 0
\(51\) 2.13663e11 0.238089
\(52\) 0 0
\(53\) −1.35416e12 −1.15276 −0.576378 0.817183i \(-0.695534\pi\)
−0.576378 + 0.817183i \(0.695534\pi\)
\(54\) 0 0
\(55\) 5.85433e11i 0.384537i
\(56\) 0 0
\(57\) 1.60097e11 0.0818953
\(58\) 0 0
\(59\) 2.06767e11i 0.0830841i −0.999137 0.0415421i \(-0.986773\pi\)
0.999137 0.0415421i \(-0.0132271\pi\)
\(60\) 0 0
\(61\) 5.42825e12i 1.72723i −0.504148 0.863617i \(-0.668193\pi\)
0.504148 0.863617i \(-0.331807\pi\)
\(62\) 0 0
\(63\) −1.94936e12 + 3.33517e12i −0.494889 + 0.846708i
\(64\) 0 0
\(65\) −3.98137e12 −0.812155
\(66\) 0 0
\(67\) −3.85879e12 −0.636689 −0.318345 0.947975i \(-0.603127\pi\)
−0.318345 + 0.947975i \(0.603127\pi\)
\(68\) 0 0
\(69\) 1.13988e12i 0.153079i
\(70\) 0 0
\(71\) 1.27085e13 1.39729 0.698646 0.715468i \(-0.253786\pi\)
0.698646 + 0.715468i \(0.253786\pi\)
\(72\) 0 0
\(73\) 5.67859e12i 0.514021i −0.966409 0.257011i \(-0.917263\pi\)
0.966409 0.257011i \(-0.0827375\pi\)
\(74\) 0 0
\(75\) 3.70604e11i 0.0277639i
\(76\) 0 0
\(77\) −6.96333e12 + 1.19136e13i −0.433892 + 0.742348i
\(78\) 0 0
\(79\) 2.24296e13 1.16797 0.583985 0.811765i \(-0.301493\pi\)
0.583985 + 0.811765i \(0.301493\pi\)
\(80\) 0 0
\(81\) 2.15627e13 0.942559
\(82\) 0 0
\(83\) 8.87401e12i 0.327019i 0.986542 + 0.163510i \(0.0522815\pi\)
−0.986542 + 0.163510i \(0.947719\pi\)
\(84\) 0 0
\(85\) −2.45887e13 −0.767013
\(86\) 0 0
\(87\) 2.26151e12i 0.0599465i
\(88\) 0 0
\(89\) 5.93459e13i 1.34172i −0.741586 0.670858i \(-0.765926\pi\)
0.741586 0.670858i \(-0.234074\pi\)
\(90\) 0 0
\(91\) −8.10211e13 4.73557e13i −1.56786 0.916395i
\(92\) 0 0
\(93\) 1.27146e13 0.211311
\(94\) 0 0
\(95\) −1.84242e13 −0.263829
\(96\) 0 0
\(97\) 2.97418e13i 0.368099i 0.982917 + 0.184050i \(0.0589207\pi\)
−0.982917 + 0.184050i \(0.941079\pi\)
\(98\) 0 0
\(99\) 7.85993e13 0.843281
\(100\) 0 0
\(101\) 1.69586e13i 0.158176i 0.996868 + 0.0790881i \(0.0252008\pi\)
−0.996868 + 0.0790881i \(0.974799\pi\)
\(102\) 0 0
\(103\) 1.51463e14i 1.23153i 0.787930 + 0.615765i \(0.211153\pi\)
−0.787930 + 0.615765i \(0.788847\pi\)
\(104\) 0 0
\(105\) −4.40809e12 + 7.54181e12i −0.0313274 + 0.0535983i
\(106\) 0 0
\(107\) −2.73318e14 −1.70209 −0.851043 0.525096i \(-0.824029\pi\)
−0.851043 + 0.525096i \(0.824029\pi\)
\(108\) 0 0
\(109\) 1.26975e14 0.694594 0.347297 0.937755i \(-0.387100\pi\)
0.347297 + 0.937755i \(0.387100\pi\)
\(110\) 0 0
\(111\) 4.78955e13i 0.230693i
\(112\) 0 0
\(113\) 3.12851e14 1.32981 0.664903 0.746930i \(-0.268473\pi\)
0.664903 + 0.746930i \(0.268473\pi\)
\(114\) 0 0
\(115\) 1.31179e14i 0.493150i
\(116\) 0 0
\(117\) 5.34532e14i 1.78104i
\(118\) 0 0
\(119\) −5.00381e14 2.92466e14i −1.48072 0.865459i
\(120\) 0 0
\(121\) −9.89843e13 −0.260656
\(122\) 0 0
\(123\) −4.76320e12 −0.0111831
\(124\) 0 0
\(125\) 4.26496e13i 0.0894427i
\(126\) 0 0
\(127\) 2.00611e14 0.376469 0.188235 0.982124i \(-0.439723\pi\)
0.188235 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) 1.05766e14i 0.177917i
\(130\) 0 0
\(131\) 7.24410e13i 0.109417i −0.998502 0.0547085i \(-0.982577\pi\)
0.998502 0.0547085i \(-0.0174230\pi\)
\(132\) 0 0
\(133\) −3.74933e14 2.19143e14i −0.509322 0.297692i
\(134\) 0 0
\(135\) 1.00491e14 0.122968
\(136\) 0 0
\(137\) 1.46428e15 1.61652 0.808261 0.588825i \(-0.200409\pi\)
0.808261 + 0.588825i \(0.200409\pi\)
\(138\) 0 0
\(139\) 1.84937e14i 0.184467i 0.995737 + 0.0922337i \(0.0294007\pi\)
−0.995737 + 0.0922337i \(0.970599\pi\)
\(140\) 0 0
\(141\) 1.81125e14 0.163473
\(142\) 0 0
\(143\) 1.90941e15i 1.56152i
\(144\) 0 0
\(145\) 2.60258e14i 0.193120i
\(146\) 0 0
\(147\) −1.79410e14 + 1.01045e14i −0.120955 + 0.0681231i
\(148\) 0 0
\(149\) −1.76431e15 −1.08211 −0.541055 0.840987i \(-0.681975\pi\)
−0.541055 + 0.840987i \(0.681975\pi\)
\(150\) 0 0
\(151\) −3.45390e14 −0.192962 −0.0964808 0.995335i \(-0.530759\pi\)
−0.0964808 + 0.995335i \(0.530759\pi\)
\(152\) 0 0
\(153\) 3.30124e15i 1.68204i
\(154\) 0 0
\(155\) −1.46321e15 −0.680748
\(156\) 0 0
\(157\) 4.63544e15i 1.97148i −0.168264 0.985742i \(-0.553816\pi\)
0.168264 0.985742i \(-0.446184\pi\)
\(158\) 0 0
\(159\) 4.11120e14i 0.160025i
\(160\) 0 0
\(161\) 1.56028e15 2.66950e15i 0.556446 0.952025i
\(162\) 0 0
\(163\) −5.99814e15 −1.96202 −0.981010 0.193957i \(-0.937868\pi\)
−0.981010 + 0.193957i \(0.937868\pi\)
\(164\) 0 0
\(165\) 1.77737e14 0.0533813
\(166\) 0 0
\(167\) 3.50351e15i 0.967138i −0.875306 0.483569i \(-0.839340\pi\)
0.875306 0.483569i \(-0.160660\pi\)
\(168\) 0 0
\(169\) −9.04800e15 −2.29798
\(170\) 0 0
\(171\) 2.47360e15i 0.578572i
\(172\) 0 0
\(173\) 5.27794e15i 1.13800i 0.822338 + 0.568999i \(0.192669\pi\)
−0.822338 + 0.568999i \(0.807331\pi\)
\(174\) 0 0
\(175\) 5.07289e14 8.67922e14i 0.100923 0.172669i
\(176\) 0 0
\(177\) −6.27743e13 −0.0115337
\(178\) 0 0
\(179\) 6.27090e15 1.06502 0.532511 0.846423i \(-0.321249\pi\)
0.532511 + 0.846423i \(0.321249\pi\)
\(180\) 0 0
\(181\) 7.20214e14i 0.113165i 0.998398 + 0.0565825i \(0.0180204\pi\)
−0.998398 + 0.0565825i \(0.981980\pi\)
\(182\) 0 0
\(183\) −1.64801e15 −0.239774
\(184\) 0 0
\(185\) 5.51188e15i 0.743187i
\(186\) 0 0
\(187\) 1.17924e16i 1.47472i
\(188\) 0 0
\(189\) 2.04500e15 + 1.19527e15i 0.237389 + 0.138751i
\(190\) 0 0
\(191\) 9.75823e15 1.05229 0.526147 0.850393i \(-0.323636\pi\)
0.526147 + 0.850393i \(0.323636\pi\)
\(192\) 0 0
\(193\) −1.24929e16 −1.25246 −0.626230 0.779639i \(-0.715403\pi\)
−0.626230 + 0.779639i \(0.715403\pi\)
\(194\) 0 0
\(195\) 1.20874e15i 0.112743i
\(196\) 0 0
\(197\) −3.75841e15 −0.326393 −0.163196 0.986594i \(-0.552180\pi\)
−0.163196 + 0.986594i \(0.552180\pi\)
\(198\) 0 0
\(199\) 3.08791e15i 0.249858i −0.992166 0.124929i \(-0.960130\pi\)
0.992166 0.124929i \(-0.0398703\pi\)
\(200\) 0 0
\(201\) 1.17152e15i 0.0883850i
\(202\) 0 0
\(203\) −3.09559e15 + 5.29627e15i −0.217907 + 0.372819i
\(204\) 0 0
\(205\) 5.48156e14 0.0360269
\(206\) 0 0
\(207\) −1.76119e16 −1.08147
\(208\) 0 0
\(209\) 8.83599e15i 0.507261i
\(210\) 0 0
\(211\) −6.71504e15 −0.360638 −0.180319 0.983608i \(-0.557713\pi\)
−0.180319 + 0.983608i \(0.557713\pi\)
\(212\) 0 0
\(213\) 3.85829e15i 0.193972i
\(214\) 0 0
\(215\) 1.21717e16i 0.573168i
\(216\) 0 0
\(217\) −2.97765e16 1.74040e16i −1.31418 0.768122i
\(218\) 0 0
\(219\) −1.72401e15 −0.0713562
\(220\) 0 0
\(221\) −8.01969e16 −3.11467
\(222\) 0 0
\(223\) 1.03931e16i 0.378977i −0.981883 0.189489i \(-0.939317\pi\)
0.981883 0.189489i \(-0.0606830\pi\)
\(224\) 0 0
\(225\) −5.72607e15 −0.196146
\(226\) 0 0
\(227\) 5.42849e16i 1.74783i 0.486082 + 0.873913i \(0.338425\pi\)
−0.486082 + 0.873913i \(0.661575\pi\)
\(228\) 0 0
\(229\) 2.13192e16i 0.645539i 0.946478 + 0.322770i \(0.104614\pi\)
−0.946478 + 0.322770i \(0.895386\pi\)
\(230\) 0 0
\(231\) 3.61695e15 + 2.11406e15i 0.103053 + 0.0602328i
\(232\) 0 0
\(233\) 2.50693e16 0.672435 0.336218 0.941784i \(-0.390852\pi\)
0.336218 + 0.941784i \(0.390852\pi\)
\(234\) 0 0
\(235\) −2.08442e16 −0.526634
\(236\) 0 0
\(237\) 6.80959e15i 0.162137i
\(238\) 0 0
\(239\) −3.81043e15 −0.0855439 −0.0427719 0.999085i \(-0.513619\pi\)
−0.0427719 + 0.999085i \(0.513619\pi\)
\(240\) 0 0
\(241\) 2.77294e16i 0.587248i 0.955921 + 0.293624i \(0.0948614\pi\)
−0.955921 + 0.293624i \(0.905139\pi\)
\(242\) 0 0
\(243\) 2.03033e16i 0.405810i
\(244\) 0 0
\(245\) 2.06467e16 1.16284e16i 0.389662 0.219461i
\(246\) 0 0
\(247\) −6.00912e16 −1.07135
\(248\) 0 0
\(249\) 2.69414e15 0.0453967
\(250\) 0 0
\(251\) 3.53313e16i 0.562916i −0.959573 0.281458i \(-0.909182\pi\)
0.959573 0.281458i \(-0.0908181\pi\)
\(252\) 0 0
\(253\) −6.29116e16 −0.948172
\(254\) 0 0
\(255\) 7.46509e15i 0.106477i
\(256\) 0 0
\(257\) 6.07557e16i 0.820455i 0.911983 + 0.410228i \(0.134551\pi\)
−0.911983 + 0.410228i \(0.865449\pi\)
\(258\) 0 0
\(259\) −6.55602e16 + 1.12167e17i −0.838575 + 1.43472i
\(260\) 0 0
\(261\) 3.49418e16 0.423508
\(262\) 0 0
\(263\) −8.71447e16 −1.00127 −0.500635 0.865659i \(-0.666900\pi\)
−0.500635 + 0.865659i \(0.666900\pi\)
\(264\) 0 0
\(265\) 4.73122e16i 0.515528i
\(266\) 0 0
\(267\) −1.80173e16 −0.186257
\(268\) 0 0
\(269\) 2.05481e15i 0.0201607i −0.999949 0.0100804i \(-0.996791\pi\)
0.999949 0.0100804i \(-0.00320874\pi\)
\(270\) 0 0
\(271\) 6.63939e16i 0.618504i 0.950980 + 0.309252i \(0.100079\pi\)
−0.950980 + 0.309252i \(0.899921\pi\)
\(272\) 0 0
\(273\) −1.43771e16 + 2.45979e16i −0.127214 + 0.217650i
\(274\) 0 0
\(275\) −2.04542e16 −0.171970
\(276\) 0 0
\(277\) 1.72127e17 1.37560 0.687798 0.725903i \(-0.258578\pi\)
0.687798 + 0.725903i \(0.258578\pi\)
\(278\) 0 0
\(279\) 1.96449e17i 1.49286i
\(280\) 0 0
\(281\) −1.56765e17 −1.13320 −0.566599 0.823994i \(-0.691741\pi\)
−0.566599 + 0.823994i \(0.691741\pi\)
\(282\) 0 0
\(283\) 1.04416e17i 0.718224i 0.933294 + 0.359112i \(0.116920\pi\)
−0.933294 + 0.359112i \(0.883080\pi\)
\(284\) 0 0
\(285\) 5.59356e15i 0.0366247i
\(286\) 0 0
\(287\) 1.11550e16 + 6.51995e15i 0.0695499 + 0.0406510i
\(288\) 0 0
\(289\) −3.26913e17 −1.94155
\(290\) 0 0
\(291\) 9.02957e15 0.0510994
\(292\) 0 0
\(293\) 1.22909e17i 0.662996i 0.943456 + 0.331498i \(0.107554\pi\)
−0.943456 + 0.331498i \(0.892446\pi\)
\(294\) 0 0
\(295\) 7.22416e15 0.0371563
\(296\) 0 0
\(297\) 4.81942e16i 0.236428i
\(298\) 0 0
\(299\) 4.27845e17i 2.00257i
\(300\) 0 0
\(301\) −1.44774e17 + 2.47695e17i −0.646734 + 1.10650i
\(302\) 0 0
\(303\) 5.14862e15 0.0219580
\(304\) 0 0
\(305\) 1.89655e17 0.772443
\(306\) 0 0
\(307\) 2.65358e16i 0.103244i 0.998667 + 0.0516219i \(0.0164391\pi\)
−0.998667 + 0.0516219i \(0.983561\pi\)
\(308\) 0 0
\(309\) 4.59839e16 0.170961
\(310\) 0 0
\(311\) 5.75983e16i 0.204685i −0.994749 0.102343i \(-0.967366\pi\)
0.994749 0.102343i \(-0.0326338\pi\)
\(312\) 0 0
\(313\) 4.04739e17i 1.37520i −0.726092 0.687598i \(-0.758665\pi\)
0.726092 0.687598i \(-0.241335\pi\)
\(314\) 0 0
\(315\) −1.16526e17 6.81078e16i −0.378659 0.221321i
\(316\) 0 0
\(317\) 1.23747e17 0.384699 0.192349 0.981327i \(-0.438389\pi\)
0.192349 + 0.981327i \(0.438389\pi\)
\(318\) 0 0
\(319\) 1.24816e17 0.371309
\(320\) 0 0
\(321\) 8.29789e16i 0.236283i
\(322\) 0 0
\(323\) −3.71119e17 −1.01180
\(324\) 0 0
\(325\) 1.39103e17i 0.363207i
\(326\) 0 0
\(327\) 3.85493e16i 0.0964234i
\(328\) 0 0
\(329\) −4.24180e17 2.47927e17i −1.01667 0.594228i
\(330\) 0 0
\(331\) −3.51196e16 −0.0806778 −0.0403389 0.999186i \(-0.512844\pi\)
−0.0403389 + 0.999186i \(0.512844\pi\)
\(332\) 0 0
\(333\) 7.40017e17 1.62979
\(334\) 0 0
\(335\) 1.34821e17i 0.284736i
\(336\) 0 0
\(337\) −2.67045e17 −0.540973 −0.270486 0.962724i \(-0.587185\pi\)
−0.270486 + 0.962724i \(0.587185\pi\)
\(338\) 0 0
\(339\) 9.49811e16i 0.184603i
\(340\) 0 0
\(341\) 7.01738e17i 1.30886i
\(342\) 0 0
\(343\) 5.58474e17 + 8.93980e15i 0.999872 + 0.0160055i
\(344\) 0 0
\(345\) −3.98257e16 −0.0684589
\(346\) 0 0
\(347\) −7.24503e17 −1.19601 −0.598004 0.801493i \(-0.704039\pi\)
−0.598004 + 0.801493i \(0.704039\pi\)
\(348\) 0 0
\(349\) 2.68419e17i 0.425633i 0.977092 + 0.212817i \(0.0682637\pi\)
−0.977092 + 0.212817i \(0.931736\pi\)
\(350\) 0 0
\(351\) 3.27755e17 0.499344
\(352\) 0 0
\(353\) 2.88463e16i 0.0422346i −0.999777 0.0211173i \(-0.993278\pi\)
0.999777 0.0211173i \(-0.00672234\pi\)
\(354\) 0 0
\(355\) 4.44018e17i 0.624888i
\(356\) 0 0
\(357\) −8.87923e16 + 1.51915e17i −0.120143 + 0.205553i
\(358\) 0 0
\(359\) 1.09399e18 1.42348 0.711739 0.702444i \(-0.247908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(360\) 0 0
\(361\) 5.20929e17 0.651970
\(362\) 0 0
\(363\) 3.00515e16i 0.0361843i
\(364\) 0 0
\(365\) 1.98402e17 0.229877
\(366\) 0 0
\(367\) 8.66474e17i 0.966258i −0.875549 0.483129i \(-0.839500\pi\)
0.875549 0.483129i \(-0.160500\pi\)
\(368\) 0 0
\(369\) 7.35946e16i 0.0790062i
\(370\) 0 0
\(371\) 5.62747e17 9.62807e17i 0.581696 0.995226i
\(372\) 0 0
\(373\) 1.29444e18 1.28860 0.644302 0.764771i \(-0.277148\pi\)
0.644302 + 0.764771i \(0.277148\pi\)
\(374\) 0 0
\(375\) −1.29484e16 −0.0124164
\(376\) 0 0
\(377\) 8.48841e17i 0.784218i
\(378\) 0 0
\(379\) 3.18349e17 0.283419 0.141710 0.989908i \(-0.454740\pi\)
0.141710 + 0.989908i \(0.454740\pi\)
\(380\) 0 0
\(381\) 6.09053e16i 0.0522613i
\(382\) 0 0
\(383\) 4.48862e16i 0.0371297i 0.999828 + 0.0185648i \(0.00590971\pi\)
−0.999828 + 0.0185648i \(0.994090\pi\)
\(384\) 0 0
\(385\) −4.16244e17 2.43289e17i −0.331988 0.194043i
\(386\) 0 0
\(387\) 1.63415e18 1.25694
\(388\) 0 0
\(389\) 1.44079e18 1.06894 0.534471 0.845187i \(-0.320511\pi\)
0.534471 + 0.845187i \(0.320511\pi\)
\(390\) 0 0
\(391\) 2.64234e18i 1.89126i
\(392\) 0 0
\(393\) −2.19930e16 −0.0151892
\(394\) 0 0
\(395\) 7.83657e17i 0.522332i
\(396\) 0 0
\(397\) 2.32098e18i 1.49327i −0.665236 0.746633i \(-0.731669\pi\)
0.665236 0.746633i \(-0.268331\pi\)
\(398\) 0 0
\(399\) −6.65316e16 + 1.13829e17i −0.0413255 + 0.0707040i
\(400\) 0 0
\(401\) 7.46571e17 0.447776 0.223888 0.974615i \(-0.428125\pi\)
0.223888 + 0.974615i \(0.428125\pi\)
\(402\) 0 0
\(403\) −4.77233e18 −2.76436
\(404\) 0 0
\(405\) 7.53370e17i 0.421525i
\(406\) 0 0
\(407\) 2.64343e18 1.42891
\(408\) 0 0
\(409\) 2.71052e18i 1.41576i −0.706334 0.707879i \(-0.749652\pi\)
0.706334 0.707879i \(-0.250348\pi\)
\(410\) 0 0
\(411\) 4.44555e17i 0.224405i
\(412\) 0 0
\(413\) 1.47012e17 + 8.59265e16i 0.0717303 + 0.0419254i
\(414\) 0 0
\(415\) −3.10045e17 −0.146248
\(416\) 0 0
\(417\) 5.61466e16 0.0256077
\(418\) 0 0
\(419\) 1.72227e18i 0.759632i 0.925062 + 0.379816i \(0.124013\pi\)
−0.925062 + 0.379816i \(0.875987\pi\)
\(420\) 0 0
\(421\) 1.49543e18 0.637957 0.318978 0.947762i \(-0.396660\pi\)
0.318978 + 0.947762i \(0.396660\pi\)
\(422\) 0 0
\(423\) 2.79850e18i 1.15490i
\(424\) 0 0
\(425\) 8.59093e17i 0.343019i
\(426\) 0 0
\(427\) 3.85950e18 + 2.25582e18i 1.49120 + 0.871586i
\(428\) 0 0
\(429\) 5.79694e17 0.216769
\(430\) 0 0
\(431\) 1.34090e18 0.485349 0.242675 0.970108i \(-0.421975\pi\)
0.242675 + 0.970108i \(0.421975\pi\)
\(432\) 0 0
\(433\) 3.11557e18i 1.09175i −0.837867 0.545874i \(-0.816198\pi\)
0.837867 0.545874i \(-0.183802\pi\)
\(434\) 0 0
\(435\) 7.90140e16 0.0268089
\(436\) 0 0
\(437\) 1.97989e18i 0.650537i
\(438\) 0 0
\(439\) 4.73894e18i 1.50810i 0.656818 + 0.754049i \(0.271902\pi\)
−0.656818 + 0.754049i \(0.728098\pi\)
\(440\) 0 0
\(441\) −1.56121e18 2.77200e18i −0.481274 0.854520i
\(442\) 0 0
\(443\) 1.25427e18 0.374597 0.187298 0.982303i \(-0.440027\pi\)
0.187298 + 0.982303i \(0.440027\pi\)
\(444\) 0 0
\(445\) 2.07346e18 0.600034
\(446\) 0 0
\(447\) 5.35643e17i 0.150218i
\(448\) 0 0
\(449\) −2.61692e18 −0.711322 −0.355661 0.934615i \(-0.615744\pi\)
−0.355661 + 0.934615i \(0.615744\pi\)
\(450\) 0 0
\(451\) 2.62888e17i 0.0692684i
\(452\) 0 0
\(453\) 1.04860e17i 0.0267869i
\(454\) 0 0
\(455\) 1.65454e18 2.83076e18i 0.409824 0.701170i
\(456\) 0 0
\(457\) 1.08592e18 0.260846 0.130423 0.991458i \(-0.458366\pi\)
0.130423 + 0.991458i \(0.458366\pi\)
\(458\) 0 0
\(459\) 2.02420e18 0.471589
\(460\) 0 0
\(461\) 8.75216e18i 1.97792i −0.148185 0.988960i \(-0.547343\pi\)
0.148185 0.988960i \(-0.452657\pi\)
\(462\) 0 0
\(463\) 1.63944e18 0.359442 0.179721 0.983718i \(-0.442480\pi\)
0.179721 + 0.983718i \(0.442480\pi\)
\(464\) 0 0
\(465\) 4.44230e17i 0.0945012i
\(466\) 0 0
\(467\) 8.89533e18i 1.83630i −0.396232 0.918151i \(-0.629682\pi\)
0.396232 0.918151i \(-0.370318\pi\)
\(468\) 0 0
\(469\) 1.60360e18 2.74361e18i 0.321282 0.549683i
\(470\) 0 0
\(471\) −1.40731e18 −0.273681
\(472\) 0 0
\(473\) 5.83738e18 1.10202
\(474\) 0 0
\(475\) 6.43715e17i 0.117988i
\(476\) 0 0
\(477\) −6.35207e18 −1.13054
\(478\) 0 0
\(479\) 2.35669e18i 0.407337i 0.979040 + 0.203669i \(0.0652865\pi\)
−0.979040 + 0.203669i \(0.934713\pi\)
\(480\) 0 0
\(481\) 1.79772e19i 3.01791i
\(482\) 0 0
\(483\) −8.10456e17 4.73700e17i −0.132160 0.0772456i
\(484\) 0 0
\(485\) −1.03914e18 −0.164619
\(486\) 0 0
\(487\) 9.09589e18 1.40005 0.700023 0.714120i \(-0.253173\pi\)
0.700023 + 0.714120i \(0.253173\pi\)
\(488\) 0 0
\(489\) 1.82103e18i 0.272367i
\(490\) 0 0
\(491\) −8.07479e18 −1.17371 −0.586855 0.809692i \(-0.699634\pi\)
−0.586855 + 0.809692i \(0.699634\pi\)
\(492\) 0 0
\(493\) 5.24239e18i 0.740629i
\(494\) 0 0
\(495\) 2.74615e18i 0.377127i
\(496\) 0 0
\(497\) −5.28130e18 + 9.03580e18i −0.705092 + 1.20635i
\(498\) 0 0
\(499\) −1.10936e19 −1.44002 −0.720008 0.693966i \(-0.755862\pi\)
−0.720008 + 0.693966i \(0.755862\pi\)
\(500\) 0 0
\(501\) −1.06366e18 −0.134258
\(502\) 0 0
\(503\) 8.61754e18i 1.05781i −0.848681 0.528905i \(-0.822603\pi\)
0.848681 0.528905i \(-0.177397\pi\)
\(504\) 0 0
\(505\) −5.92510e17 −0.0707386
\(506\) 0 0
\(507\) 2.74696e18i 0.319004i
\(508\) 0 0
\(509\) 9.44910e18i 1.06749i −0.845644 0.533747i \(-0.820783\pi\)
0.845644 0.533747i \(-0.179217\pi\)
\(510\) 0 0
\(511\) 4.03749e18 + 2.35986e18i 0.443778 + 0.259382i
\(512\) 0 0
\(513\) 1.51672e18 0.162212
\(514\) 0 0
\(515\) −5.29189e18 −0.550757
\(516\) 0 0
\(517\) 9.99657e18i 1.01255i
\(518\) 0 0
\(519\) 1.60238e18 0.157977
\(520\) 0 0
\(521\) 1.04200e19i 1.00000i 0.866024 + 0.500002i \(0.166668\pi\)
−0.866024 + 0.500002i \(0.833332\pi\)
\(522\) 0 0
\(523\) 7.90486e18i 0.738555i −0.929319 0.369277i \(-0.879605\pi\)
0.929319 0.369277i \(-0.120395\pi\)
\(524\) 0 0
\(525\) −2.63500e17 1.54012e17i −0.0239699 0.0140101i
\(526\) 0 0
\(527\) −2.94736e19 −2.61071
\(528\) 0 0
\(529\) 2.50387e18 0.215984
\(530\) 0 0
\(531\) 9.69904e17i 0.0814830i
\(532\) 0 0
\(533\) 1.78783e18 0.146297
\(534\) 0 0
\(535\) 9.54933e18i 0.761196i
\(536\) 0 0
\(537\) 1.90384e18i 0.147846i
\(538\) 0 0
\(539\) −5.57683e18 9.90189e18i −0.421955 0.749198i
\(540\) 0 0
\(541\) 7.66413e18 0.565044 0.282522 0.959261i \(-0.408829\pi\)
0.282522 + 0.959261i \(0.408829\pi\)
\(542\) 0 0
\(543\) 2.18656e17 0.0157095
\(544\) 0 0
\(545\) 4.43631e18i 0.310632i
\(546\) 0 0
\(547\) 1.75508e19 1.19780 0.598900 0.800824i \(-0.295605\pi\)
0.598900 + 0.800824i \(0.295605\pi\)
\(548\) 0 0
\(549\) 2.54628e19i 1.69395i
\(550\) 0 0
\(551\) 3.92810e18i 0.254754i
\(552\) 0 0
\(553\) −9.32108e18 + 1.59475e19i −0.589373 + 1.00836i
\(554\) 0 0
\(555\) 1.67340e18 0.103169
\(556\) 0 0
\(557\) −1.24906e19 −0.750924 −0.375462 0.926838i \(-0.622516\pi\)
−0.375462 + 0.926838i \(0.622516\pi\)
\(558\) 0 0
\(559\) 3.96984e19i 2.32750i
\(560\) 0 0
\(561\) 3.58016e18 0.204721
\(562\) 0 0
\(563\) 2.19445e19i 1.22396i 0.790874 + 0.611979i \(0.209626\pi\)
−0.790874 + 0.611979i \(0.790374\pi\)
\(564\) 0 0
\(565\) 1.09306e19i 0.594707i
\(566\) 0 0
\(567\) −8.96084e18 + 1.53311e19i −0.475628 + 0.813754i
\(568\) 0 0
\(569\) 1.92000e19 0.994294 0.497147 0.867666i \(-0.334381\pi\)
0.497147 + 0.867666i \(0.334381\pi\)
\(570\) 0 0
\(571\) −1.75307e19 −0.885824 −0.442912 0.896565i \(-0.646055\pi\)
−0.442912 + 0.896565i \(0.646055\pi\)
\(572\) 0 0
\(573\) 2.96259e18i 0.146079i
\(574\) 0 0
\(575\) 4.58320e18 0.220543
\(576\) 0 0
\(577\) 2.86957e18i 0.134768i −0.997727 0.0673840i \(-0.978535\pi\)
0.997727 0.0673840i \(-0.0214652\pi\)
\(578\) 0 0
\(579\) 3.79284e18i 0.173866i
\(580\) 0 0
\(581\) −6.30944e18 3.68778e18i −0.282331 0.165018i
\(582\) 0 0
\(583\) −2.26903e19 −0.991198
\(584\) 0 0
\(585\) −1.86758e19 −0.796504
\(586\) 0 0
\(587\) 1.52698e19i 0.635868i 0.948113 + 0.317934i \(0.102989\pi\)
−0.948113 + 0.317934i \(0.897011\pi\)
\(588\) 0 0
\(589\) −2.20844e19 −0.898006
\(590\) 0 0
\(591\) 1.14105e18i 0.0453098i
\(592\) 0 0
\(593\) 2.76713e19i 1.07312i −0.843863 0.536559i \(-0.819724\pi\)
0.843863 0.536559i \(-0.180276\pi\)
\(594\) 0 0
\(595\) 1.02183e19 1.74826e19i 0.387045 0.662197i
\(596\) 0 0
\(597\) −9.37487e17 −0.0346852
\(598\) 0 0
\(599\) −4.04578e19 −1.46223 −0.731113 0.682256i \(-0.760999\pi\)
−0.731113 + 0.682256i \(0.760999\pi\)
\(600\) 0 0
\(601\) 2.10600e19i 0.743595i −0.928314 0.371797i \(-0.878742\pi\)
0.928314 0.371797i \(-0.121258\pi\)
\(602\) 0 0
\(603\) −1.81008e19 −0.624420
\(604\) 0 0
\(605\) 3.45837e18i 0.116569i
\(606\) 0 0
\(607\) 1.49208e19i 0.491442i −0.969341 0.245721i \(-0.920975\pi\)
0.969341 0.245721i \(-0.0790247\pi\)
\(608\) 0 0
\(609\) 1.60794e18 + 9.39818e17i 0.0517546 + 0.0302498i
\(610\) 0 0
\(611\) −6.79840e19 −2.13854
\(612\) 0 0
\(613\) −2.61547e19 −0.804129 −0.402065 0.915611i \(-0.631707\pi\)
−0.402065 + 0.915611i \(0.631707\pi\)
\(614\) 0 0
\(615\) 1.66419e17i 0.00500125i
\(616\) 0 0
\(617\) 3.65280e18 0.107307 0.0536537 0.998560i \(-0.482913\pi\)
0.0536537 + 0.998560i \(0.482913\pi\)
\(618\) 0 0
\(619\) 1.85715e18i 0.0533352i 0.999644 + 0.0266676i \(0.00848956\pi\)
−0.999644 + 0.0266676i \(0.991510\pi\)
\(620\) 0 0
\(621\) 1.07989e19i 0.303208i
\(622\) 0 0
\(623\) 4.21951e19 + 2.46624e19i 1.15836 + 0.677048i
\(624\) 0 0
\(625\) 1.49012e18 0.0400000
\(626\) 0 0
\(627\) 2.68260e18 0.0704178
\(628\) 0 0
\(629\) 1.11026e20i 2.85017i
\(630\) 0 0
\(631\) −2.86578e19 −0.719511 −0.359755 0.933047i \(-0.617140\pi\)
−0.359755 + 0.933047i \(0.617140\pi\)
\(632\) 0 0
\(633\) 2.03868e18i 0.0500636i
\(634\) 0 0
\(635\) 7.00907e18i 0.168362i
\(636\) 0 0
\(637\) 6.73400e19 3.79265e19i 1.58233 0.891183i
\(638\) 0 0
\(639\) 5.96132e19 1.37036
\(640\) 0 0
\(641\) 4.72028e19 1.06160 0.530801 0.847496i \(-0.321891\pi\)
0.530801 + 0.847496i \(0.321891\pi\)
\(642\) 0 0
\(643\) 3.74689e19i 0.824507i −0.911069 0.412253i \(-0.864742\pi\)
0.911069 0.412253i \(-0.135258\pi\)
\(644\) 0 0
\(645\) 3.69531e18 0.0795670
\(646\) 0 0
\(647\) 5.28437e19i 1.11343i 0.830704 + 0.556715i \(0.187938\pi\)
−0.830704 + 0.556715i \(0.812062\pi\)
\(648\) 0 0
\(649\) 3.46461e18i 0.0714399i
\(650\) 0 0
\(651\) −5.28382e18 + 9.04011e18i −0.106630 + 0.182434i
\(652\) 0 0
\(653\) 4.17691e19 0.825017 0.412509 0.910954i \(-0.364653\pi\)
0.412509 + 0.910954i \(0.364653\pi\)
\(654\) 0 0
\(655\) 2.53098e18 0.0489328
\(656\) 0 0
\(657\) 2.66371e19i 0.504115i
\(658\) 0 0
\(659\) −1.20234e19 −0.222757 −0.111378 0.993778i \(-0.535527\pi\)
−0.111378 + 0.993778i \(0.535527\pi\)
\(660\) 0 0
\(661\) 3.50077e19i 0.634970i −0.948263 0.317485i \(-0.897162\pi\)
0.948263 0.317485i \(-0.102838\pi\)
\(662\) 0 0
\(663\) 2.43477e19i 0.432377i
\(664\) 0 0
\(665\) 7.65656e18 1.30996e19i 0.133132 0.227776i
\(666\) 0 0
\(667\) −2.79678e19 −0.476186
\(668\) 0 0
\(669\) −3.15535e18 −0.0526095
\(670\) 0 0
\(671\) 9.09561e19i 1.48516i
\(672\) 0 0
\(673\) −8.60462e19 −1.37602 −0.688012 0.725699i \(-0.741516\pi\)
−0.688012 + 0.725699i \(0.741516\pi\)
\(674\) 0 0
\(675\) 3.51101e18i 0.0549928i
\(676\) 0 0
\(677\) 8.68945e19i 1.33313i 0.745448 + 0.666564i \(0.232236\pi\)
−0.745448 + 0.666564i \(0.767764\pi\)
\(678\) 0 0
\(679\) −2.11465e19 1.23598e19i −0.317797 0.185748i
\(680\) 0 0
\(681\) 1.64808e19 0.242633
\(682\) 0 0
\(683\) 4.93433e19 0.711677 0.355839 0.934547i \(-0.384195\pi\)
0.355839 + 0.934547i \(0.384195\pi\)
\(684\) 0 0
\(685\) 5.11600e19i 0.722931i
\(686\) 0 0
\(687\) 6.47248e18 0.0896136
\(688\) 0 0
\(689\) 1.54311e20i 2.09344i
\(690\) 0 0
\(691\) 1.34242e20i 1.78461i −0.451433 0.892305i \(-0.649087\pi\)
0.451433 0.892305i \(-0.350913\pi\)
\(692\) 0 0
\(693\) −3.26636e19 + 5.58842e19i −0.425531 + 0.728043i
\(694\) 0 0
\(695\) −6.46143e18 −0.0824963
\(696\) 0 0
\(697\) 1.10415e19 0.138166
\(698\) 0 0
\(699\) 7.61100e18i 0.0933473i
\(700\) 0 0
\(701\) 9.76404e19 1.17383 0.586913 0.809650i \(-0.300343\pi\)
0.586913 + 0.809650i \(0.300343\pi\)
\(702\) 0 0
\(703\) 8.31914e19i 0.980373i
\(704\) 0 0
\(705\) 6.32826e18i 0.0731072i
\(706\) 0 0
\(707\) −1.20576e19 7.04751e18i −0.136561 0.0798178i
\(708\) 0 0
\(709\) 6.73303e19 0.747630 0.373815 0.927503i \(-0.378050\pi\)
0.373815 + 0.927503i \(0.378050\pi\)
\(710\) 0 0
\(711\) 1.05213e20 1.14546
\(712\) 0 0
\(713\) 1.57240e20i 1.67855i
\(714\) 0 0
\(715\) −6.67120e19 −0.698332
\(716\) 0 0
\(717\) 1.15684e18i 0.0118752i
\(718\) 0 0
\(719\) 6.34330e19i 0.638576i 0.947658 + 0.319288i \(0.103444\pi\)
−0.947658 + 0.319288i \(0.896556\pi\)
\(720\) 0 0
\(721\) −1.07690e20 6.29435e19i −1.06324 0.621447i
\(722\) 0 0
\(723\) 8.41862e18 0.0815216
\(724\) 0 0
\(725\) −9.09304e18 −0.0863660
\(726\) 0 0
\(727\) 1.16939e20i 1.08948i −0.838606 0.544739i \(-0.816629\pi\)
0.838606 0.544739i \(-0.183371\pi\)
\(728\) 0 0
\(729\) 9.69698e19 0.886224
\(730\) 0 0
\(731\) 2.45175e20i 2.19814i
\(732\) 0 0
\(733\) 4.13181e19i 0.363423i 0.983352 + 0.181712i \(0.0581638\pi\)
−0.983352 + 0.181712i \(0.941836\pi\)
\(734\) 0 0
\(735\) −3.53037e18 6.26832e18i −0.0304656 0.0540928i
\(736\) 0 0
\(737\) −6.46581e19 −0.547458
\(738\) 0 0
\(739\) 1.68737e20 1.40184 0.700921 0.713239i \(-0.252773\pi\)
0.700921 + 0.713239i \(0.252773\pi\)
\(740\) 0 0
\(741\) 1.82436e19i 0.148725i
\(742\) 0 0
\(743\) 1.44810e20 1.15845 0.579223 0.815169i \(-0.303356\pi\)
0.579223 + 0.815169i \(0.303356\pi\)
\(744\) 0 0
\(745\) 6.16425e19i 0.483934i
\(746\) 0 0
\(747\) 4.16262e19i 0.320717i
\(748\) 0 0
\(749\) 1.13583e20 1.94330e20i 0.858895 1.46949i
\(750\) 0 0
\(751\) 5.26508e19 0.390774 0.195387 0.980726i \(-0.437404\pi\)
0.195387 + 0.980726i \(0.437404\pi\)
\(752\) 0 0
\(753\) −1.07265e19 −0.0781439
\(754\) 0 0
\(755\) 1.20674e19i 0.0862951i
\(756\) 0 0
\(757\) 1.67968e20 1.17911 0.589557 0.807727i \(-0.299303\pi\)
0.589557 + 0.807727i \(0.299303\pi\)
\(758\) 0 0
\(759\) 1.90999e19i 0.131625i
\(760\) 0 0
\(761\) 2.33435e20i 1.57933i −0.613539 0.789664i \(-0.710255\pi\)
0.613539 0.789664i \(-0.289745\pi\)
\(762\) 0 0
\(763\) −5.27669e19 + 9.02792e19i −0.350502 + 0.599675i
\(764\) 0 0
\(765\) −1.15341e20 −0.752232
\(766\) 0 0
\(767\) 2.35618e19 0.150884
\(768\) 0 0
\(769\) 1.19790e19i 0.0753248i 0.999291 + 0.0376624i \(0.0119911\pi\)
−0.999291 + 0.0376624i \(0.988009\pi\)
\(770\) 0 0
\(771\) 1.84453e19 0.113895
\(772\) 0 0
\(773\) 3.79396e19i 0.230058i −0.993362 0.115029i \(-0.963304\pi\)
0.993362 0.115029i \(-0.0366960\pi\)
\(774\) 0 0
\(775\) 5.11226e19i 0.304440i
\(776\) 0 0
\(777\) 3.40538e19 + 1.99040e19i 0.199168 + 0.116411i
\(778\) 0 0
\(779\) 8.27337e18 0.0475248
\(780\) 0 0
\(781\) 2.12945e20 1.20146
\(782\) 0 0
\(783\) 2.14250e19i 0.118738i
\(784\) 0 0
\(785\) 1.61956e20 0.881674
\(786\) 0 0
\(787\) 3.98290e19i 0.212998i 0.994313 + 0.106499i \(0.0339642\pi\)
−0.994313 + 0.106499i \(0.966036\pi\)
\(788\) 0 0
\(789\) 2.64570e19i 0.138996i
\(790\) 0 0
\(791\) −1.30012e20 + 2.22438e20i −0.671038 + 1.14808i
\(792\) 0 0
\(793\) 6.18568e20 3.13671
\(794\) 0 0
\(795\) −1.43639e19 −0.0715654
\(796\) 0 0
\(797\) 1.42042e20i 0.695356i 0.937614 + 0.347678i \(0.113030\pi\)
−0.937614 + 0.347678i \(0.886970\pi\)
\(798\) 0 0
\(799\) −4.19865e20 −2.01968
\(800\) 0 0
\(801\) 2.78380e20i 1.31586i
\(802\) 0 0
\(803\) 9.51509e19i 0.441981i
\(804\) 0 0
\(805\) 9.32684e19 + 5.45141e19i 0.425759 + 0.248850i
\(806\) 0 0
\(807\) −6.23839e17 −0.00279871
\(808\) 0 0
\(809\) 1.55543e20 0.685819 0.342910 0.939368i \(-0.388588\pi\)
0.342910 + 0.939368i \(0.388588\pi\)
\(810\) 0 0
\(811\) 4.05657e19i 0.175797i −0.996129 0.0878987i \(-0.971985\pi\)
0.996129 0.0878987i \(-0.0280152\pi\)
\(812\) 0 0
\(813\) 2.01571e19 0.0858606
\(814\) 0 0
\(815\) 2.09566e20i 0.877442i
\(816\) 0 0
\(817\) 1.83708e20i 0.756092i
\(818\) 0 0
\(819\) −3.80053e20 2.22136e20i −1.53765 0.898735i
\(820\) 0 0
\(821\) −3.62216e20 −1.44067 −0.720337 0.693625i \(-0.756013\pi\)
−0.720337 + 0.693625i \(0.756013\pi\)
\(822\) 0 0
\(823\) −5.04171e20 −1.97142 −0.985710 0.168450i \(-0.946124\pi\)
−0.985710 + 0.168450i \(0.946124\pi\)
\(824\) 0 0
\(825\) 6.20986e18i 0.0238728i
\(826\) 0 0
\(827\) 3.49173e20 1.31978 0.659890 0.751362i \(-0.270603\pi\)
0.659890 + 0.751362i \(0.270603\pi\)
\(828\) 0 0
\(829\) 2.72366e20i 1.01221i 0.862472 + 0.506105i \(0.168915\pi\)
−0.862472 + 0.506105i \(0.831085\pi\)
\(830\) 0 0
\(831\) 5.22575e19i 0.190960i
\(832\) 0 0
\(833\) 4.15888e20 2.34232e20i 1.49438 0.841649i
\(834\) 0 0
\(835\) 1.22408e20 0.432517
\(836\) 0 0
\(837\) 1.20455e20 0.418550
\(838\) 0 0
\(839\) 1.74521e20i 0.596368i 0.954508 + 0.298184i \(0.0963809\pi\)
−0.954508 + 0.298184i \(0.903619\pi\)
\(840\) 0 0
\(841\) −2.42070e20 −0.813523
\(842\) 0 0
\(843\) 4.75938e19i 0.157310i
\(844\) 0 0
\(845\) 3.16124e20i 1.02769i
\(846\) 0 0
\(847\) 4.11350e19 7.03780e19i 0.131531 0.225037i
\(848\) 0 0
\(849\) 3.17004e19 0.0997037
\(850\) 0 0
\(851\) −5.92317e20 −1.83251
\(852\) 0 0
\(853\) 4.51180e20i 1.37311i −0.727076 0.686557i \(-0.759121\pi\)
0.727076 0.686557i \(-0.240879\pi\)
\(854\) 0 0
\(855\) −8.64241e19 −0.258745
\(856\) 0 0
\(857\) 3.31340e20i 0.975905i 0.872870 + 0.487952i \(0.162256\pi\)
−0.872870 + 0.487952i \(0.837744\pi\)
\(858\) 0 0
\(859\) 4.95753e20i 1.43652i 0.695773 + 0.718262i \(0.255062\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(860\) 0 0
\(861\) 1.97945e18 3.38665e18i 0.00564316 0.00965490i
\(862\) 0 0
\(863\) −1.50051e19 −0.0420886 −0.0210443 0.999779i \(-0.506699\pi\)
−0.0210443 + 0.999779i \(0.506699\pi\)
\(864\) 0 0
\(865\) −1.84404e20 −0.508928
\(866\) 0 0
\(867\) 9.92505e19i 0.269525i
\(868\) 0 0
\(869\) 3.75831e20 1.00428
\(870\) 0 0
\(871\) 4.39722e20i 1.15625i
\(872\) 0 0
\(873\) 1.39513e20i 0.361006i
\(874\) 0 0
\(875\) 3.03240e19 + 1.77239e19i 0.0772200 + 0.0451340i
\(876\) 0 0
\(877\) −9.44892e19 −0.236802 −0.118401 0.992966i \(-0.537777\pi\)
−0.118401 + 0.992966i \(0.537777\pi\)
\(878\) 0 0
\(879\) 3.73150e19 0.0920369
\(880\) 0 0
\(881\) 2.92647e20i 0.710416i 0.934787 + 0.355208i \(0.115590\pi\)
−0.934787 + 0.355208i \(0.884410\pi\)
\(882\) 0 0
\(883\) −8.14106e20 −1.94516 −0.972582 0.232561i \(-0.925289\pi\)
−0.972582 + 0.232561i \(0.925289\pi\)
\(884\) 0 0
\(885\) 2.19324e18i 0.00515803i
\(886\) 0 0
\(887\) 1.73996e19i 0.0402785i 0.999797 + 0.0201392i \(0.00641095\pi\)
−0.999797 + 0.0201392i \(0.993589\pi\)
\(888\) 0 0
\(889\) −8.33682e19 + 1.42635e20i −0.189971 + 0.325023i
\(890\) 0 0
\(891\) 3.61306e20 0.810460
\(892\) 0 0
\(893\) −3.14603e20 −0.694708
\(894\) 0 0
\(895\) 2.19096e20i 0.476292i
\(896\) 0 0
\(897\) −1.29893e20 −0.277996
\(898\) 0 0
\(899\) 3.11962e20i 0.657331i
\(900\) 0 0
\(901\) 9.53013e20i 1.97708i
\(902\) 0 0
\(903\) 7.51998e19 + 4.39532e19i 0.153604 + 0.0897794i
\(904\) 0 0
\(905\) −2.51633e19 −0.0506089
\(906\) 0 0
\(907\) 6.34150e20 1.25586 0.627930 0.778270i \(-0.283902\pi\)
0.627930 + 0.778270i \(0.283902\pi\)
\(908\) 0 0
\(909\) 7.95495e19i 0.155128i
\(910\) 0 0
\(911\) −7.49270e20 −1.43883 −0.719415 0.694581i \(-0.755590\pi\)
−0.719415 + 0.694581i \(0.755590\pi\)
\(912\) 0 0
\(913\) 1.48694e20i 0.281188i
\(914\) 0 0
\(915\) 5.75791e19i 0.107230i
\(916\) 0 0
\(917\) 5.15057e19 + 3.01044e19i 0.0944647 + 0.0552133i
\(918\) 0 0
\(919\) 6.04762e20 1.09238 0.546192 0.837660i \(-0.316077\pi\)
0.546192 + 0.837660i \(0.316077\pi\)
\(920\) 0 0
\(921\) 8.05624e18 0.0143323
\(922\) 0 0
\(923\) 1.44818e21i 2.53753i
\(924\) 0 0
\(925\) −1.92577e20 −0.332363
\(926\) 0 0
\(927\) 7.10481e20i 1.20780i
\(928\) 0 0
\(929\) 7.82789e19i 0.131079i 0.997850 + 0.0655397i \(0.0208769\pi\)
−0.997850 + 0.0655397i \(0.979123\pi\)
\(930\) 0 0
\(931\) 3.11623e20 1.75509e20i 0.514022 0.289502i
\(932\) 0 0
\(933\) −1.74868e19 −0.0284143
\(934\) 0 0
\(935\) −4.12010e20 −0.659517
\(936\) 0 0
\(937\) 9.26304e20i 1.46075i 0.683045 + 0.730376i \(0.260655\pi\)
−0.683045 + 0.730376i \(0.739345\pi\)
\(938\) 0 0
\(939\) −1.22878e20 −0.190904
\(940\) 0 0
\(941\) 3.61063e20i 0.552657i −0.961063 0.276329i \(-0.910882\pi\)
0.961063 0.276329i \(-0.0891178\pi\)
\(942\) 0 0
\(943\) 5.89058e19i 0.0888334i
\(944\) 0 0
\(945\) −4.17612e19 + 7.14494e19i −0.0620512 + 0.106164i
\(946\) 0 0
\(947\) −1.18296e21 −1.73189 −0.865944 0.500141i \(-0.833282\pi\)
−0.865944 + 0.500141i \(0.833282\pi\)
\(948\) 0 0
\(949\) 6.47095e20 0.933479
\(950\) 0 0
\(951\) 3.75695e19i 0.0534037i
\(952\) 0 0
\(953\) 4.82570e20 0.675943 0.337971 0.941156i \(-0.390259\pi\)
0.337971 + 0.941156i \(0.390259\pi\)
\(954\) 0 0
\(955\) 3.40938e20i 0.470601i
\(956\) 0 0
\(957\) 3.78940e19i 0.0515451i
\(958\) 0 0
\(959\) −6.08514e20 + 1.04111e21i −0.815719 + 1.39562i
\(960\) 0 0
\(961\) −9.96961e20 −1.31709
\(962\) 0 0
\(963\) −1.28208e21 −1.66929
\(964\) 0 0
\(965\) 4.36485e20i 0.560117i
\(966\) 0 0
\(967\) −7.01512e20 −0.887257 −0.443629 0.896211i \(-0.646309\pi\)
−0.443629 + 0.896211i \(0.646309\pi\)
\(968\) 0 0
\(969\) 1.12671e20i 0.140458i
\(970\) 0 0
\(971\) 6.31484e20i 0.775939i −0.921672 0.387970i \(-0.873177\pi\)
0.921672 0.387970i \(-0.126823\pi\)
\(972\) 0 0
\(973\) −1.31490e20 7.68543e19i −0.159259 0.0930847i
\(974\) 0 0
\(975\) −4.22316e19 −0.0504202
\(976\) 0 0
\(977\) −2.01865e20 −0.237574 −0.118787 0.992920i \(-0.537901\pi\)
−0.118787 + 0.992920i \(0.537901\pi\)
\(978\) 0 0
\(979\) 9.94404e20i 1.15368i
\(980\) 0 0
\(981\) 5.95612e20 0.681209
\(982\) 0 0
\(983\) 4.07405e20i 0.459358i −0.973266 0.229679i \(-0.926232\pi\)
0.973266 0.229679i \(-0.0737676\pi\)
\(984\) 0 0
\(985\) 1.31314e20i 0.145967i
\(986\) 0 0
\(987\) −7.52704e19 + 1.28780e20i −0.0824905 + 0.141133i
\(988\) 0 0
\(989\) −1.30799e21 −1.41329
\(990\) 0 0
\(991\) 3.37255e20 0.359288 0.179644 0.983732i \(-0.442505\pi\)
0.179644 + 0.983732i \(0.442505\pi\)
\(992\) 0 0
\(993\) 1.06623e19i 0.0111997i
\(994\) 0 0
\(995\) 1.07887e20 0.111740
\(996\) 0 0
\(997\) 3.09145e20i 0.315715i 0.987462 + 0.157858i \(0.0504587\pi\)
−0.987462 + 0.157858i \(0.949541\pi\)
\(998\) 0 0
\(999\) 4.53751e20i 0.456940i
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 140.15.d.a.41.18 36
7.6 odd 2 inner 140.15.d.a.41.19 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.15.d.a.41.18 36 1.1 even 1 trivial
140.15.d.a.41.19 yes 36 7.6 odd 2 inner