Properties

Label 48.13.g.a.31.1
Level $48$
Weight $13$
Character 48.31
Analytic conductor $43.872$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,13,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.g (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: \(43.8717032293\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.1
Root \(-32.6599 - 56.5686i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.13.g.a.31.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-420.888i q^{3} -21286.8 q^{5} +7111.90i q^{7} -177147. q^{9} +O(q^{10})\) \(q-420.888i q^{3} -21286.8 q^{5} +7111.90i q^{7} -177147. q^{9} -1.88540e6i q^{11} -8.56106e6 q^{13} +8.95935e6i q^{15} +6.77647e6 q^{17} +645743. i q^{19} +2.99331e6 q^{21} +2.15694e8i q^{23} +2.08985e8 q^{25} +7.45591e7i q^{27} -3.66716e8 q^{29} -1.03858e9i q^{31} -7.93544e8 q^{33} -1.51389e8i q^{35} +3.52823e9 q^{37} +3.60325e9i q^{39} +5.66160e9 q^{41} -4.86430e9i q^{43} +3.77089e9 q^{45} +1.89030e10i q^{47} +1.37907e10 q^{49} -2.85214e9i q^{51} +1.61412e9 q^{53} +4.01341e10i q^{55} +2.71786e8 q^{57} -1.28375e10i q^{59} +3.86920e10 q^{61} -1.25985e9i q^{63} +1.82237e11 q^{65} +1.56544e11i q^{67} +9.07829e10 q^{69} +4.16779e10i q^{71} +1.00107e11 q^{73} -8.79596e10i q^{75} +1.34088e10 q^{77} -1.99076e11i q^{79} +3.13811e10 q^{81} +2.30690e11i q^{83} -1.44249e11 q^{85} +1.54347e11i q^{87} -4.73312e11 q^{89} -6.08854e10i q^{91} -4.37126e11 q^{93} -1.37458e10i q^{95} +9.29103e11 q^{97} +3.33993e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 420.888i − 0.577350i
\(4\) 0 0
\(5\) −21286.8 −1.36235 −0.681176 0.732119i \(-0.738531\pi\)
−0.681176 + 0.732119i \(0.738531\pi\)
\(6\) 0 0
\(7\) 7111.90i 0.0604501i 0.999543 + 0.0302251i \(0.00962240\pi\)
−0.999543 + 0.0302251i \(0.990378\pi\)
\(8\) 0 0
\(9\) −177147. −0.333333
\(10\) 0 0
\(11\) − 1.88540e6i − 1.06426i −0.846663 0.532130i \(-0.821392\pi\)
0.846663 0.532130i \(-0.178608\pi\)
\(12\) 0 0
\(13\) −8.56106e6 −1.77365 −0.886824 0.462107i \(-0.847093\pi\)
−0.886824 + 0.462107i \(0.847093\pi\)
\(14\) 0 0
\(15\) 8.95935e6i 0.786555i
\(16\) 0 0
\(17\) 6.77647e6 0.280744 0.140372 0.990099i \(-0.455170\pi\)
0.140372 + 0.990099i \(0.455170\pi\)
\(18\) 0 0
\(19\) 645743.i 0.0137258i 0.999976 + 0.00686291i \(0.00218455\pi\)
−0.999976 + 0.00686291i \(0.997815\pi\)
\(20\) 0 0
\(21\) 2.99331e6 0.0349009
\(22\) 0 0
\(23\) 2.15694e8i 1.45704i 0.685027 + 0.728518i \(0.259790\pi\)
−0.685027 + 0.728518i \(0.740210\pi\)
\(24\) 0 0
\(25\) 2.08985e8 0.856005
\(26\) 0 0
\(27\) 7.45591e7i 0.192450i
\(28\) 0 0
\(29\) −3.66716e8 −0.616513 −0.308257 0.951303i \(-0.599746\pi\)
−0.308257 + 0.951303i \(0.599746\pi\)
\(30\) 0 0
\(31\) − 1.03858e9i − 1.17022i −0.810952 0.585112i \(-0.801050\pi\)
0.810952 0.585112i \(-0.198950\pi\)
\(32\) 0 0
\(33\) −7.93544e8 −0.614451
\(34\) 0 0
\(35\) − 1.51389e8i − 0.0823544i
\(36\) 0 0
\(37\) 3.52823e9 1.37514 0.687570 0.726118i \(-0.258678\pi\)
0.687570 + 0.726118i \(0.258678\pi\)
\(38\) 0 0
\(39\) 3.60325e9i 1.02402i
\(40\) 0 0
\(41\) 5.66160e9 1.19189 0.595945 0.803025i \(-0.296778\pi\)
0.595945 + 0.803025i \(0.296778\pi\)
\(42\) 0 0
\(43\) − 4.86430e9i − 0.769502i −0.923020 0.384751i \(-0.874287\pi\)
0.923020 0.384751i \(-0.125713\pi\)
\(44\) 0 0
\(45\) 3.77089e9 0.454118
\(46\) 0 0
\(47\) 1.89030e10i 1.75365i 0.480811 + 0.876824i \(0.340342\pi\)
−0.480811 + 0.876824i \(0.659658\pi\)
\(48\) 0 0
\(49\) 1.37907e10 0.996346
\(50\) 0 0
\(51\) − 2.85214e9i − 0.162087i
\(52\) 0 0
\(53\) 1.61412e9 0.0728251 0.0364126 0.999337i \(-0.488407\pi\)
0.0364126 + 0.999337i \(0.488407\pi\)
\(54\) 0 0
\(55\) 4.01341e10i 1.44990i
\(56\) 0 0
\(57\) 2.71786e8 0.00792460
\(58\) 0 0
\(59\) − 1.28375e10i − 0.304346i −0.988354 0.152173i \(-0.951373\pi\)
0.988354 0.152173i \(-0.0486271\pi\)
\(60\) 0 0
\(61\) 3.86920e10 0.751004 0.375502 0.926822i \(-0.377470\pi\)
0.375502 + 0.926822i \(0.377470\pi\)
\(62\) 0 0
\(63\) − 1.25985e9i − 0.0201500i
\(64\) 0 0
\(65\) 1.82237e11 2.41633
\(66\) 0 0
\(67\) 1.56544e11i 1.73056i 0.501289 + 0.865280i \(0.332859\pi\)
−0.501289 + 0.865280i \(0.667141\pi\)
\(68\) 0 0
\(69\) 9.07829e10 0.841220
\(70\) 0 0
\(71\) 4.16779e10i 0.325354i 0.986679 + 0.162677i \(0.0520128\pi\)
−0.986679 + 0.162677i \(0.947987\pi\)
\(72\) 0 0
\(73\) 1.00107e11 0.661497 0.330748 0.943719i \(-0.392699\pi\)
0.330748 + 0.943719i \(0.392699\pi\)
\(74\) 0 0
\(75\) − 8.79596e10i − 0.494214i
\(76\) 0 0
\(77\) 1.34088e10 0.0643347
\(78\) 0 0
\(79\) − 1.99076e11i − 0.818947i −0.912322 0.409473i \(-0.865712\pi\)
0.912322 0.409473i \(-0.134288\pi\)
\(80\) 0 0
\(81\) 3.13811e10 0.111111
\(82\) 0 0
\(83\) 2.30690e11i 0.705604i 0.935698 + 0.352802i \(0.114771\pi\)
−0.935698 + 0.352802i \(0.885229\pi\)
\(84\) 0 0
\(85\) −1.44249e11 −0.382472
\(86\) 0 0
\(87\) 1.54347e11i 0.355944i
\(88\) 0 0
\(89\) −4.73312e11 −0.952375 −0.476187 0.879344i \(-0.657982\pi\)
−0.476187 + 0.879344i \(0.657982\pi\)
\(90\) 0 0
\(91\) − 6.08854e10i − 0.107217i
\(92\) 0 0
\(93\) −4.37126e11 −0.675630
\(94\) 0 0
\(95\) − 1.37458e10i − 0.0186994i
\(96\) 0 0
\(97\) 9.29103e11 1.11541 0.557704 0.830040i \(-0.311682\pi\)
0.557704 + 0.830040i \(0.311682\pi\)
\(98\) 0 0
\(99\) 3.33993e11i 0.354754i
\(100\) 0 0
\(101\) −4.30494e11 −0.405545 −0.202773 0.979226i \(-0.564995\pi\)
−0.202773 + 0.979226i \(0.564995\pi\)
\(102\) 0 0
\(103\) − 1.24450e12i − 1.04225i −0.853482 0.521123i \(-0.825513\pi\)
0.853482 0.521123i \(-0.174487\pi\)
\(104\) 0 0
\(105\) −6.37180e10 −0.0475473
\(106\) 0 0
\(107\) 3.40478e11i 0.226875i 0.993545 + 0.113437i \(0.0361861\pi\)
−0.993545 + 0.113437i \(0.963814\pi\)
\(108\) 0 0
\(109\) −3.31888e12 −1.97894 −0.989470 0.144738i \(-0.953766\pi\)
−0.989470 + 0.144738i \(0.953766\pi\)
\(110\) 0 0
\(111\) − 1.48499e12i − 0.793937i
\(112\) 0 0
\(113\) −3.87063e12 −1.85913 −0.929567 0.368654i \(-0.879819\pi\)
−0.929567 + 0.368654i \(0.879819\pi\)
\(114\) 0 0
\(115\) − 4.59142e12i − 1.98500i
\(116\) 0 0
\(117\) 1.51657e12 0.591216
\(118\) 0 0
\(119\) 4.81935e10i 0.0169710i
\(120\) 0 0
\(121\) −4.16314e11 −0.132650
\(122\) 0 0
\(123\) − 2.38290e12i − 0.688138i
\(124\) 0 0
\(125\) 7.48339e11 0.196173
\(126\) 0 0
\(127\) 8.23613e12i 1.96291i 0.191688 + 0.981456i \(0.438604\pi\)
−0.191688 + 0.981456i \(0.561396\pi\)
\(128\) 0 0
\(129\) −2.04733e12 −0.444272
\(130\) 0 0
\(131\) − 1.61593e12i − 0.319738i −0.987138 0.159869i \(-0.948893\pi\)
0.987138 0.159869i \(-0.0511072\pi\)
\(132\) 0 0
\(133\) −4.59246e9 −0.000829727 0
\(134\) 0 0
\(135\) − 1.58712e12i − 0.262185i
\(136\) 0 0
\(137\) −4.29703e11 −0.0649897 −0.0324948 0.999472i \(-0.510345\pi\)
−0.0324948 + 0.999472i \(0.510345\pi\)
\(138\) 0 0
\(139\) 1.70332e12i 0.236160i 0.993004 + 0.118080i \(0.0376739\pi\)
−0.993004 + 0.118080i \(0.962326\pi\)
\(140\) 0 0
\(141\) 7.95603e12 1.01247
\(142\) 0 0
\(143\) 1.61410e13i 1.88762i
\(144\) 0 0
\(145\) 7.80621e12 0.839908
\(146\) 0 0
\(147\) − 5.80435e12i − 0.575241i
\(148\) 0 0
\(149\) −5.75418e12 −0.525854 −0.262927 0.964816i \(-0.584688\pi\)
−0.262927 + 0.964816i \(0.584688\pi\)
\(150\) 0 0
\(151\) − 1.73435e13i − 1.46311i −0.681785 0.731553i \(-0.738796\pi\)
0.681785 0.731553i \(-0.261204\pi\)
\(152\) 0 0
\(153\) −1.20043e12 −0.0935812
\(154\) 0 0
\(155\) 2.21080e13i 1.59426i
\(156\) 0 0
\(157\) 1.18262e13 0.789672 0.394836 0.918752i \(-0.370801\pi\)
0.394836 + 0.918752i \(0.370801\pi\)
\(158\) 0 0
\(159\) − 6.79365e11i − 0.0420456i
\(160\) 0 0
\(161\) −1.53399e12 −0.0880780
\(162\) 0 0
\(163\) − 1.72346e13i − 0.918917i −0.888199 0.459458i \(-0.848044\pi\)
0.888199 0.459458i \(-0.151956\pi\)
\(164\) 0 0
\(165\) 1.68920e13 0.837099
\(166\) 0 0
\(167\) 3.25246e13i 1.49939i 0.661785 + 0.749694i \(0.269799\pi\)
−0.661785 + 0.749694i \(0.730201\pi\)
\(168\) 0 0
\(169\) 4.99937e13 2.14583
\(170\) 0 0
\(171\) − 1.14391e11i − 0.00457527i
\(172\) 0 0
\(173\) 2.94963e13 1.10025 0.550125 0.835082i \(-0.314580\pi\)
0.550125 + 0.835082i \(0.314580\pi\)
\(174\) 0 0
\(175\) 1.48628e12i 0.0517456i
\(176\) 0 0
\(177\) −5.40314e12 −0.175714
\(178\) 0 0
\(179\) 2.17325e13i 0.660682i 0.943862 + 0.330341i \(0.107164\pi\)
−0.943862 + 0.330341i \(0.892836\pi\)
\(180\) 0 0
\(181\) −1.24786e12 −0.0354891 −0.0177446 0.999843i \(-0.505649\pi\)
−0.0177446 + 0.999843i \(0.505649\pi\)
\(182\) 0 0
\(183\) − 1.62850e13i − 0.433592i
\(184\) 0 0
\(185\) −7.51046e13 −1.87343
\(186\) 0 0
\(187\) − 1.27764e13i − 0.298784i
\(188\) 0 0
\(189\) −5.30257e11 −0.0116336
\(190\) 0 0
\(191\) 6.52191e13i 1.34330i 0.740866 + 0.671652i \(0.234415\pi\)
−0.740866 + 0.671652i \(0.765585\pi\)
\(192\) 0 0
\(193\) −1.62613e13 −0.314638 −0.157319 0.987548i \(-0.550285\pi\)
−0.157319 + 0.987548i \(0.550285\pi\)
\(194\) 0 0
\(195\) − 7.67015e13i − 1.39507i
\(196\) 0 0
\(197\) −8.69576e13 −1.48768 −0.743841 0.668356i \(-0.766998\pi\)
−0.743841 + 0.668356i \(0.766998\pi\)
\(198\) 0 0
\(199\) − 1.03084e13i − 0.165986i −0.996550 0.0829932i \(-0.973552\pi\)
0.996550 0.0829932i \(-0.0264480\pi\)
\(200\) 0 0
\(201\) 6.58874e13 0.999139
\(202\) 0 0
\(203\) − 2.60805e12i − 0.0372683i
\(204\) 0 0
\(205\) −1.20517e14 −1.62378
\(206\) 0 0
\(207\) − 3.82095e13i − 0.485678i
\(208\) 0 0
\(209\) 1.21749e12 0.0146078
\(210\) 0 0
\(211\) − 1.21337e14i − 1.37499i −0.726189 0.687495i \(-0.758710\pi\)
0.726189 0.687495i \(-0.241290\pi\)
\(212\) 0 0
\(213\) 1.75417e13 0.187843
\(214\) 0 0
\(215\) 1.03545e14i 1.04833i
\(216\) 0 0
\(217\) 7.38627e12 0.0707402
\(218\) 0 0
\(219\) − 4.21339e13i − 0.381915i
\(220\) 0 0
\(221\) −5.80137e13 −0.497940
\(222\) 0 0
\(223\) 1.52486e14i 1.23994i 0.784624 + 0.619972i \(0.212856\pi\)
−0.784624 + 0.619972i \(0.787144\pi\)
\(224\) 0 0
\(225\) −3.70212e13 −0.285335
\(226\) 0 0
\(227\) − 2.60931e14i − 1.90709i −0.301249 0.953545i \(-0.597404\pi\)
0.301249 0.953545i \(-0.402596\pi\)
\(228\) 0 0
\(229\) 1.11017e14 0.769798 0.384899 0.922959i \(-0.374236\pi\)
0.384899 + 0.922959i \(0.374236\pi\)
\(230\) 0 0
\(231\) − 5.64360e12i − 0.0371436i
\(232\) 0 0
\(233\) −1.48696e14 −0.929318 −0.464659 0.885490i \(-0.653823\pi\)
−0.464659 + 0.885490i \(0.653823\pi\)
\(234\) 0 0
\(235\) − 4.02383e14i − 2.38909i
\(236\) 0 0
\(237\) −8.37887e13 −0.472819
\(238\) 0 0
\(239\) 3.25593e14i 1.74698i 0.486841 + 0.873491i \(0.338149\pi\)
−0.486841 + 0.873491i \(0.661851\pi\)
\(240\) 0 0
\(241\) −1.46403e14 −0.747220 −0.373610 0.927586i \(-0.621880\pi\)
−0.373610 + 0.927586i \(0.621880\pi\)
\(242\) 0 0
\(243\) − 1.32079e13i − 0.0641500i
\(244\) 0 0
\(245\) −2.93559e14 −1.35737
\(246\) 0 0
\(247\) − 5.52825e12i − 0.0243448i
\(248\) 0 0
\(249\) 9.70949e13 0.407381
\(250\) 0 0
\(251\) 4.56243e14i 1.82454i 0.409589 + 0.912270i \(0.365672\pi\)
−0.409589 + 0.912270i \(0.634328\pi\)
\(252\) 0 0
\(253\) 4.06669e14 1.55067
\(254\) 0 0
\(255\) 6.07127e13i 0.220820i
\(256\) 0 0
\(257\) 4.70340e14 1.63235 0.816175 0.577804i \(-0.196090\pi\)
0.816175 + 0.577804i \(0.196090\pi\)
\(258\) 0 0
\(259\) 2.50924e13i 0.0831274i
\(260\) 0 0
\(261\) 6.49627e13 0.205504
\(262\) 0 0
\(263\) − 1.42061e14i − 0.429280i −0.976693 0.214640i \(-0.931142\pi\)
0.976693 0.214640i \(-0.0688578\pi\)
\(264\) 0 0
\(265\) −3.43594e13 −0.0992135
\(266\) 0 0
\(267\) 1.99212e14i 0.549854i
\(268\) 0 0
\(269\) 2.94994e14 0.778573 0.389286 0.921117i \(-0.372722\pi\)
0.389286 + 0.921117i \(0.372722\pi\)
\(270\) 0 0
\(271\) 4.70773e14i 1.18849i 0.804284 + 0.594246i \(0.202549\pi\)
−0.804284 + 0.594246i \(0.797451\pi\)
\(272\) 0 0
\(273\) −2.56259e13 −0.0619019
\(274\) 0 0
\(275\) − 3.94022e14i − 0.911012i
\(276\) 0 0
\(277\) 4.15318e14 0.919395 0.459698 0.888075i \(-0.347958\pi\)
0.459698 + 0.888075i \(0.347958\pi\)
\(278\) 0 0
\(279\) 1.83981e14i 0.390075i
\(280\) 0 0
\(281\) 6.73345e14 1.36773 0.683864 0.729609i \(-0.260298\pi\)
0.683864 + 0.729609i \(0.260298\pi\)
\(282\) 0 0
\(283\) 3.70518e14i 0.721259i 0.932709 + 0.360629i \(0.117438\pi\)
−0.932709 + 0.360629i \(0.882562\pi\)
\(284\) 0 0
\(285\) −5.78544e12 −0.0107961
\(286\) 0 0
\(287\) 4.02647e13i 0.0720499i
\(288\) 0 0
\(289\) −5.36702e14 −0.921183
\(290\) 0 0
\(291\) − 3.91049e14i − 0.643981i
\(292\) 0 0
\(293\) 2.56469e14 0.405350 0.202675 0.979246i \(-0.435037\pi\)
0.202675 + 0.979246i \(0.435037\pi\)
\(294\) 0 0
\(295\) 2.73268e14i 0.414626i
\(296\) 0 0
\(297\) 1.40574e14 0.204817
\(298\) 0 0
\(299\) − 1.84657e15i − 2.58427i
\(300\) 0 0
\(301\) 3.45944e13 0.0465165
\(302\) 0 0
\(303\) 1.81190e14i 0.234142i
\(304\) 0 0
\(305\) −8.23627e14 −1.02313
\(306\) 0 0
\(307\) 8.24545e14i 0.984882i 0.870346 + 0.492441i \(0.163895\pi\)
−0.870346 + 0.492441i \(0.836105\pi\)
\(308\) 0 0
\(309\) −5.23793e14 −0.601740
\(310\) 0 0
\(311\) 7.97009e14i 0.880848i 0.897790 + 0.440424i \(0.145172\pi\)
−0.897790 + 0.440424i \(0.854828\pi\)
\(312\) 0 0
\(313\) 1.79978e14 0.191406 0.0957028 0.995410i \(-0.469490\pi\)
0.0957028 + 0.995410i \(0.469490\pi\)
\(314\) 0 0
\(315\) 2.68181e13i 0.0274515i
\(316\) 0 0
\(317\) 1.63990e15 1.61608 0.808039 0.589130i \(-0.200529\pi\)
0.808039 + 0.589130i \(0.200529\pi\)
\(318\) 0 0
\(319\) 6.91408e14i 0.656131i
\(320\) 0 0
\(321\) 1.43303e14 0.130986
\(322\) 0 0
\(323\) 4.37586e12i 0.00385343i
\(324\) 0 0
\(325\) −1.78914e15 −1.51825
\(326\) 0 0
\(327\) 1.39688e15i 1.14254i
\(328\) 0 0
\(329\) −1.34436e14 −0.106008
\(330\) 0 0
\(331\) − 2.09770e15i − 1.59505i −0.603283 0.797527i \(-0.706141\pi\)
0.603283 0.797527i \(-0.293859\pi\)
\(332\) 0 0
\(333\) −6.25016e14 −0.458380
\(334\) 0 0
\(335\) − 3.33231e15i − 2.35763i
\(336\) 0 0
\(337\) −1.36274e15 −0.930321 −0.465160 0.885226i \(-0.654003\pi\)
−0.465160 + 0.885226i \(0.654003\pi\)
\(338\) 0 0
\(339\) 1.62910e15i 1.07337i
\(340\) 0 0
\(341\) −1.95814e15 −1.24542
\(342\) 0 0
\(343\) 1.96516e14i 0.120679i
\(344\) 0 0
\(345\) −1.93247e15 −1.14604
\(346\) 0 0
\(347\) − 2.21109e15i − 1.26657i −0.773917 0.633287i \(-0.781705\pi\)
0.773917 0.633287i \(-0.218295\pi\)
\(348\) 0 0
\(349\) −8.25554e14 −0.456870 −0.228435 0.973559i \(-0.573361\pi\)
−0.228435 + 0.973559i \(0.573361\pi\)
\(350\) 0 0
\(351\) − 6.38305e14i − 0.341339i
\(352\) 0 0
\(353\) 9.06854e14 0.468694 0.234347 0.972153i \(-0.424705\pi\)
0.234347 + 0.972153i \(0.424705\pi\)
\(354\) 0 0
\(355\) − 8.87187e14i − 0.443246i
\(356\) 0 0
\(357\) 2.02841e13 0.00979820
\(358\) 0 0
\(359\) 5.13417e14i 0.239830i 0.992784 + 0.119915i \(0.0382622\pi\)
−0.992784 + 0.119915i \(0.961738\pi\)
\(360\) 0 0
\(361\) 2.21290e15 0.999812
\(362\) 0 0
\(363\) 1.75222e14i 0.0765858i
\(364\) 0 0
\(365\) −2.13096e15 −0.901192
\(366\) 0 0
\(367\) − 6.68311e14i − 0.273516i −0.990605 0.136758i \(-0.956332\pi\)
0.990605 0.136758i \(-0.0436682\pi\)
\(368\) 0 0
\(369\) −1.00294e15 −0.397297
\(370\) 0 0
\(371\) 1.14795e13i 0.00440229i
\(372\) 0 0
\(373\) 9.11568e13 0.0338483 0.0169241 0.999857i \(-0.494613\pi\)
0.0169241 + 0.999857i \(0.494613\pi\)
\(374\) 0 0
\(375\) − 3.14967e14i − 0.113260i
\(376\) 0 0
\(377\) 3.13948e15 1.09348
\(378\) 0 0
\(379\) 3.53114e15i 1.19146i 0.803184 + 0.595731i \(0.203137\pi\)
−0.803184 + 0.595731i \(0.796863\pi\)
\(380\) 0 0
\(381\) 3.46649e15 1.13329
\(382\) 0 0
\(383\) − 2.41943e15i − 0.766515i −0.923642 0.383257i \(-0.874802\pi\)
0.923642 0.383257i \(-0.125198\pi\)
\(384\) 0 0
\(385\) −2.85430e14 −0.0876465
\(386\) 0 0
\(387\) 8.61697e14i 0.256501i
\(388\) 0 0
\(389\) 4.88362e15 1.40943 0.704717 0.709489i \(-0.251074\pi\)
0.704717 + 0.709489i \(0.251074\pi\)
\(390\) 0 0
\(391\) 1.46164e15i 0.409053i
\(392\) 0 0
\(393\) −6.80125e14 −0.184601
\(394\) 0 0
\(395\) 4.23768e15i 1.11569i
\(396\) 0 0
\(397\) −3.63632e15 −0.928792 −0.464396 0.885628i \(-0.653729\pi\)
−0.464396 + 0.885628i \(0.653729\pi\)
\(398\) 0 0
\(399\) 1.93291e12i 0 0.000479043i
\(400\) 0 0
\(401\) 2.72304e14 0.0654919 0.0327459 0.999464i \(-0.489575\pi\)
0.0327459 + 0.999464i \(0.489575\pi\)
\(402\) 0 0
\(403\) 8.89134e15i 2.07557i
\(404\) 0 0
\(405\) −6.68001e14 −0.151373
\(406\) 0 0
\(407\) − 6.65214e15i − 1.46351i
\(408\) 0 0
\(409\) 1.47132e15 0.314317 0.157158 0.987573i \(-0.449767\pi\)
0.157158 + 0.987573i \(0.449767\pi\)
\(410\) 0 0
\(411\) 1.80857e14i 0.0375218i
\(412\) 0 0
\(413\) 9.12988e13 0.0183977
\(414\) 0 0
\(415\) − 4.91065e15i − 0.961281i
\(416\) 0 0
\(417\) 7.16906e14 0.136347
\(418\) 0 0
\(419\) 6.30146e15i 1.16455i 0.812993 + 0.582273i \(0.197837\pi\)
−0.812993 + 0.582273i \(0.802163\pi\)
\(420\) 0 0
\(421\) 3.25314e15 0.584266 0.292133 0.956378i \(-0.405635\pi\)
0.292133 + 0.956378i \(0.405635\pi\)
\(422\) 0 0
\(423\) − 3.34860e15i − 0.584550i
\(424\) 0 0
\(425\) 1.41618e15 0.240318
\(426\) 0 0
\(427\) 2.75174e14i 0.0453983i
\(428\) 0 0
\(429\) 6.79358e15 1.08982
\(430\) 0 0
\(431\) 1.03071e16i 1.60795i 0.594660 + 0.803977i \(0.297287\pi\)
−0.594660 + 0.803977i \(0.702713\pi\)
\(432\) 0 0
\(433\) 9.64406e15 1.46330 0.731648 0.681682i \(-0.238751\pi\)
0.731648 + 0.681682i \(0.238751\pi\)
\(434\) 0 0
\(435\) − 3.28554e15i − 0.484921i
\(436\) 0 0
\(437\) −1.39283e14 −0.0199990
\(438\) 0 0
\(439\) − 3.83012e15i − 0.535087i −0.963546 0.267544i \(-0.913788\pi\)
0.963546 0.267544i \(-0.0862120\pi\)
\(440\) 0 0
\(441\) −2.44298e15 −0.332115
\(442\) 0 0
\(443\) 6.98519e15i 0.924179i 0.886833 + 0.462089i \(0.152900\pi\)
−0.886833 + 0.462089i \(0.847100\pi\)
\(444\) 0 0
\(445\) 1.00753e16 1.29747
\(446\) 0 0
\(447\) 2.42187e15i 0.303602i
\(448\) 0 0
\(449\) −6.54024e15 −0.798207 −0.399103 0.916906i \(-0.630679\pi\)
−0.399103 + 0.916906i \(0.630679\pi\)
\(450\) 0 0
\(451\) − 1.06744e16i − 1.26848i
\(452\) 0 0
\(453\) −7.29969e15 −0.844725
\(454\) 0 0
\(455\) 1.29605e15i 0.146068i
\(456\) 0 0
\(457\) −3.93711e15 −0.432196 −0.216098 0.976372i \(-0.569333\pi\)
−0.216098 + 0.976372i \(0.569333\pi\)
\(458\) 0 0
\(459\) 5.05247e14i 0.0540291i
\(460\) 0 0
\(461\) −6.78296e15 −0.706665 −0.353333 0.935498i \(-0.614952\pi\)
−0.353333 + 0.935498i \(0.614952\pi\)
\(462\) 0 0
\(463\) − 1.21932e16i − 1.23774i −0.785492 0.618872i \(-0.787590\pi\)
0.785492 0.618872i \(-0.212410\pi\)
\(464\) 0 0
\(465\) 9.30499e15 0.920446
\(466\) 0 0
\(467\) − 9.69994e14i − 0.0935121i −0.998906 0.0467560i \(-0.985112\pi\)
0.998906 0.0467560i \(-0.0148883\pi\)
\(468\) 0 0
\(469\) −1.11332e15 −0.104613
\(470\) 0 0
\(471\) − 4.97750e15i − 0.455917i
\(472\) 0 0
\(473\) −9.17117e15 −0.818951
\(474\) 0 0
\(475\) 1.34951e14i 0.0117494i
\(476\) 0 0
\(477\) −2.85937e14 −0.0242750
\(478\) 0 0
\(479\) 6.76625e15i 0.560190i 0.959972 + 0.280095i \(0.0903659\pi\)
−0.959972 + 0.280095i \(0.909634\pi\)
\(480\) 0 0
\(481\) −3.02054e16 −2.43901
\(482\) 0 0
\(483\) 6.45639e14i 0.0508518i
\(484\) 0 0
\(485\) −1.97776e16 −1.51958
\(486\) 0 0
\(487\) − 1.15311e16i − 0.864367i −0.901786 0.432184i \(-0.857743\pi\)
0.901786 0.432184i \(-0.142257\pi\)
\(488\) 0 0
\(489\) −7.25385e15 −0.530537
\(490\) 0 0
\(491\) 1.32213e16i 0.943596i 0.881707 + 0.471798i \(0.156395\pi\)
−0.881707 + 0.471798i \(0.843605\pi\)
\(492\) 0 0
\(493\) −2.48504e15 −0.173082
\(494\) 0 0
\(495\) − 7.10964e15i − 0.483299i
\(496\) 0 0
\(497\) −2.96409e14 −0.0196677
\(498\) 0 0
\(499\) 4.97689e15i 0.322370i 0.986924 + 0.161185i \(0.0515316\pi\)
−0.986924 + 0.161185i \(0.948468\pi\)
\(500\) 0 0
\(501\) 1.36892e16 0.865672
\(502\) 0 0
\(503\) − 1.01640e16i − 0.627562i −0.949495 0.313781i \(-0.898404\pi\)
0.949495 0.313781i \(-0.101596\pi\)
\(504\) 0 0
\(505\) 9.16383e15 0.552496
\(506\) 0 0
\(507\) − 2.10417e16i − 1.23889i
\(508\) 0 0
\(509\) −1.39231e16 −0.800626 −0.400313 0.916379i \(-0.631099\pi\)
−0.400313 + 0.916379i \(0.631099\pi\)
\(510\) 0 0
\(511\) 7.11951e14i 0.0399876i
\(512\) 0 0
\(513\) −4.81460e13 −0.00264153
\(514\) 0 0
\(515\) 2.64913e16i 1.41991i
\(516\) 0 0
\(517\) 3.56397e16 1.86634
\(518\) 0 0
\(519\) − 1.24147e16i − 0.635230i
\(520\) 0 0
\(521\) 1.75513e16 0.877572 0.438786 0.898592i \(-0.355409\pi\)
0.438786 + 0.898592i \(0.355409\pi\)
\(522\) 0 0
\(523\) 1.65340e16i 0.807919i 0.914777 + 0.403959i \(0.132366\pi\)
−0.914777 + 0.403959i \(0.867634\pi\)
\(524\) 0 0
\(525\) 6.25559e14 0.0298753
\(526\) 0 0
\(527\) − 7.03790e15i − 0.328533i
\(528\) 0 0
\(529\) −2.46091e16 −1.12295
\(530\) 0 0
\(531\) 2.27412e15i 0.101449i
\(532\) 0 0
\(533\) −4.84693e16 −2.11399
\(534\) 0 0
\(535\) − 7.24767e15i − 0.309083i
\(536\) 0 0
\(537\) 9.14697e15 0.381445
\(538\) 0 0
\(539\) − 2.60010e16i − 1.06037i
\(540\) 0 0
\(541\) −3.40764e16 −1.35916 −0.679580 0.733602i \(-0.737838\pi\)
−0.679580 + 0.733602i \(0.737838\pi\)
\(542\) 0 0
\(543\) 5.25211e14i 0.0204897i
\(544\) 0 0
\(545\) 7.06482e16 2.69601
\(546\) 0 0
\(547\) − 3.02110e16i − 1.12782i −0.825835 0.563912i \(-0.809296\pi\)
0.825835 0.563912i \(-0.190704\pi\)
\(548\) 0 0
\(549\) −6.85417e15 −0.250335
\(550\) 0 0
\(551\) − 2.36805e14i − 0.00846215i
\(552\) 0 0
\(553\) 1.41581e15 0.0495054
\(554\) 0 0
\(555\) 3.16107e16i 1.08162i
\(556\) 0 0
\(557\) 1.86821e16 0.625596 0.312798 0.949820i \(-0.398734\pi\)
0.312798 + 0.949820i \(0.398734\pi\)
\(558\) 0 0
\(559\) 4.16436e16i 1.36483i
\(560\) 0 0
\(561\) −5.37742e15 −0.172503
\(562\) 0 0
\(563\) 1.77453e16i 0.557228i 0.960403 + 0.278614i \(0.0898750\pi\)
−0.960403 + 0.278614i \(0.910125\pi\)
\(564\) 0 0
\(565\) 8.23931e16 2.53280
\(566\) 0 0
\(567\) 2.23179e14i 0.00671668i
\(568\) 0 0
\(569\) −2.04279e16 −0.601935 −0.300968 0.953634i \(-0.597310\pi\)
−0.300968 + 0.953634i \(0.597310\pi\)
\(570\) 0 0
\(571\) − 5.67402e15i − 0.163709i −0.996644 0.0818547i \(-0.973916\pi\)
0.996644 0.0818547i \(-0.0260844\pi\)
\(572\) 0 0
\(573\) 2.74500e16 0.775557
\(574\) 0 0
\(575\) 4.50768e16i 1.24723i
\(576\) 0 0
\(577\) 1.70028e16 0.460750 0.230375 0.973102i \(-0.426005\pi\)
0.230375 + 0.973102i \(0.426005\pi\)
\(578\) 0 0
\(579\) 6.84418e15i 0.181656i
\(580\) 0 0
\(581\) −1.64065e15 −0.0426538
\(582\) 0 0
\(583\) − 3.04327e15i − 0.0775049i
\(584\) 0 0
\(585\) −3.22828e16 −0.805445
\(586\) 0 0
\(587\) 3.27992e16i 0.801742i 0.916134 + 0.400871i \(0.131292\pi\)
−0.916134 + 0.400871i \(0.868708\pi\)
\(588\) 0 0
\(589\) 6.70655e14 0.0160623
\(590\) 0 0
\(591\) 3.65995e16i 0.858914i
\(592\) 0 0
\(593\) 6.54880e16 1.50603 0.753015 0.658003i \(-0.228599\pi\)
0.753015 + 0.658003i \(0.228599\pi\)
\(594\) 0 0
\(595\) − 1.02588e15i − 0.0231205i
\(596\) 0 0
\(597\) −4.33868e15 −0.0958322
\(598\) 0 0
\(599\) 1.46157e16i 0.316417i 0.987406 + 0.158209i \(0.0505718\pi\)
−0.987406 + 0.158209i \(0.949428\pi\)
\(600\) 0 0
\(601\) 3.63481e16 0.771320 0.385660 0.922641i \(-0.373974\pi\)
0.385660 + 0.922641i \(0.373974\pi\)
\(602\) 0 0
\(603\) − 2.77312e16i − 0.576853i
\(604\) 0 0
\(605\) 8.86197e15 0.180717
\(606\) 0 0
\(607\) 4.19339e16i 0.838364i 0.907902 + 0.419182i \(0.137683\pi\)
−0.907902 + 0.419182i \(0.862317\pi\)
\(608\) 0 0
\(609\) −1.09770e15 −0.0215169
\(610\) 0 0
\(611\) − 1.61829e17i − 3.11036i
\(612\) 0 0
\(613\) −2.03819e16 −0.384134 −0.192067 0.981382i \(-0.561519\pi\)
−0.192067 + 0.981382i \(0.561519\pi\)
\(614\) 0 0
\(615\) 5.07243e16i 0.937487i
\(616\) 0 0
\(617\) −3.74912e16 −0.679545 −0.339772 0.940508i \(-0.610350\pi\)
−0.339772 + 0.940508i \(0.610350\pi\)
\(618\) 0 0
\(619\) − 2.18233e16i − 0.387951i −0.981006 0.193975i \(-0.937862\pi\)
0.981006 0.193975i \(-0.0621382\pi\)
\(620\) 0 0
\(621\) −1.60819e16 −0.280407
\(622\) 0 0
\(623\) − 3.36615e15i − 0.0575712i
\(624\) 0 0
\(625\) −6.69516e16 −1.12326
\(626\) 0 0
\(627\) − 5.12426e14i − 0.00843384i
\(628\) 0 0
\(629\) 2.39089e16 0.386062
\(630\) 0 0
\(631\) 4.54812e16i 0.720537i 0.932849 + 0.360268i \(0.117315\pi\)
−0.932849 + 0.360268i \(0.882685\pi\)
\(632\) 0 0
\(633\) −5.10695e16 −0.793851
\(634\) 0 0
\(635\) − 1.75320e17i − 2.67418i
\(636\) 0 0
\(637\) −1.18063e17 −1.76717
\(638\) 0 0
\(639\) − 7.38311e15i − 0.108451i
\(640\) 0 0
\(641\) 7.57002e16 1.09131 0.545656 0.838009i \(-0.316281\pi\)
0.545656 + 0.838009i \(0.316281\pi\)
\(642\) 0 0
\(643\) 5.74737e16i 0.813212i 0.913604 + 0.406606i \(0.133288\pi\)
−0.913604 + 0.406606i \(0.866712\pi\)
\(644\) 0 0
\(645\) 4.35810e16 0.605255
\(646\) 0 0
\(647\) − 5.03250e16i − 0.686053i −0.939326 0.343027i \(-0.888548\pi\)
0.939326 0.343027i \(-0.111452\pi\)
\(648\) 0 0
\(649\) −2.42038e16 −0.323903
\(650\) 0 0
\(651\) − 3.10879e15i − 0.0408419i
\(652\) 0 0
\(653\) 1.84738e15 0.0238275 0.0119137 0.999929i \(-0.496208\pi\)
0.0119137 + 0.999929i \(0.496208\pi\)
\(654\) 0 0
\(655\) 3.43979e16i 0.435596i
\(656\) 0 0
\(657\) −1.77337e16 −0.220499
\(658\) 0 0
\(659\) 8.93078e16i 1.09038i 0.838313 + 0.545189i \(0.183542\pi\)
−0.838313 + 0.545189i \(0.816458\pi\)
\(660\) 0 0
\(661\) 1.13618e16 0.136219 0.0681095 0.997678i \(-0.478303\pi\)
0.0681095 + 0.997678i \(0.478303\pi\)
\(662\) 0 0
\(663\) 2.44173e16i 0.287486i
\(664\) 0 0
\(665\) 9.77586e13 0.00113038
\(666\) 0 0
\(667\) − 7.90984e16i − 0.898282i
\(668\) 0 0
\(669\) 6.41797e16 0.715881
\(670\) 0 0
\(671\) − 7.29500e16i − 0.799264i
\(672\) 0 0
\(673\) −1.40873e17 −1.51614 −0.758069 0.652174i \(-0.773857\pi\)
−0.758069 + 0.652174i \(0.773857\pi\)
\(674\) 0 0
\(675\) 1.55818e16i 0.164738i
\(676\) 0 0
\(677\) −1.06344e17 −1.10454 −0.552268 0.833667i \(-0.686238\pi\)
−0.552268 + 0.833667i \(0.686238\pi\)
\(678\) 0 0
\(679\) 6.60769e15i 0.0674265i
\(680\) 0 0
\(681\) −1.09823e17 −1.10106
\(682\) 0 0
\(683\) − 7.82367e16i − 0.770702i −0.922770 0.385351i \(-0.874080\pi\)
0.922770 0.385351i \(-0.125920\pi\)
\(684\) 0 0
\(685\) 9.14697e15 0.0885389
\(686\) 0 0
\(687\) − 4.67258e16i − 0.444443i
\(688\) 0 0
\(689\) −1.38186e16 −0.129166
\(690\) 0 0
\(691\) 1.63249e17i 1.49963i 0.661650 + 0.749813i \(0.269857\pi\)
−0.661650 + 0.749813i \(0.730143\pi\)
\(692\) 0 0
\(693\) −2.37533e15 −0.0214449
\(694\) 0 0
\(695\) − 3.62581e16i − 0.321733i
\(696\) 0 0
\(697\) 3.83657e16 0.334616
\(698\) 0 0
\(699\) 6.25845e16i 0.536542i
\(700\) 0 0
\(701\) −1.46995e17 −1.23878 −0.619391 0.785083i \(-0.712620\pi\)
−0.619391 + 0.785083i \(0.712620\pi\)
\(702\) 0 0
\(703\) 2.27833e15i 0.0188749i
\(704\) 0 0
\(705\) −1.69358e17 −1.37934
\(706\) 0 0
\(707\) − 3.06163e15i − 0.0245153i
\(708\) 0 0
\(709\) 5.55937e16 0.437672 0.218836 0.975762i \(-0.429774\pi\)
0.218836 + 0.975762i \(0.429774\pi\)
\(710\) 0 0
\(711\) 3.52657e16i 0.272982i
\(712\) 0 0
\(713\) 2.24015e17 1.70506
\(714\) 0 0
\(715\) − 3.43590e17i − 2.57161i
\(716\) 0 0
\(717\) 1.37038e17 1.00862
\(718\) 0 0
\(719\) 1.11386e16i 0.0806229i 0.999187 + 0.0403115i \(0.0128350\pi\)
−0.999187 + 0.0403115i \(0.987165\pi\)
\(720\) 0 0
\(721\) 8.85072e15 0.0630038
\(722\) 0 0
\(723\) 6.16194e16i 0.431408i
\(724\) 0 0
\(725\) −7.66384e16 −0.527738
\(726\) 0 0
\(727\) 4.35822e16i 0.295191i 0.989048 + 0.147595i \(0.0471533\pi\)
−0.989048 + 0.147595i \(0.952847\pi\)
\(728\) 0 0
\(729\) −5.55906e15 −0.0370370
\(730\) 0 0
\(731\) − 3.29628e16i − 0.216033i
\(732\) 0 0
\(733\) 1.02849e17 0.663095 0.331547 0.943439i \(-0.392429\pi\)
0.331547 + 0.943439i \(0.392429\pi\)
\(734\) 0 0
\(735\) 1.23556e17i 0.783680i
\(736\) 0 0
\(737\) 2.95148e17 1.84177
\(738\) 0 0
\(739\) 2.89187e16i 0.177546i 0.996052 + 0.0887732i \(0.0282946\pi\)
−0.996052 + 0.0887732i \(0.971705\pi\)
\(740\) 0 0
\(741\) −2.32677e15 −0.0140555
\(742\) 0 0
\(743\) 1.84464e17i 1.09642i 0.836339 + 0.548212i \(0.184691\pi\)
−0.836339 + 0.548212i \(0.815309\pi\)
\(744\) 0 0
\(745\) 1.22488e17 0.716399
\(746\) 0 0
\(747\) − 4.08661e16i − 0.235201i
\(748\) 0 0
\(749\) −2.42144e15 −0.0137146
\(750\) 0 0
\(751\) 1.07917e17i 0.601522i 0.953700 + 0.300761i \(0.0972406\pi\)
−0.953700 + 0.300761i \(0.902759\pi\)
\(752\) 0 0
\(753\) 1.92027e17 1.05340
\(754\) 0 0
\(755\) 3.69188e17i 1.99327i
\(756\) 0 0
\(757\) 2.20280e17 1.17058 0.585288 0.810825i \(-0.300981\pi\)
0.585288 + 0.810825i \(0.300981\pi\)
\(758\) 0 0
\(759\) − 1.71162e17i − 0.895277i
\(760\) 0 0
\(761\) 1.15648e17 0.595430 0.297715 0.954655i \(-0.403775\pi\)
0.297715 + 0.954655i \(0.403775\pi\)
\(762\) 0 0
\(763\) − 2.36035e16i − 0.119627i
\(764\) 0 0
\(765\) 2.55533e16 0.127491
\(766\) 0 0
\(767\) 1.09902e17i 0.539803i
\(768\) 0 0
\(769\) −9.40988e14 −0.00455016 −0.00227508 0.999997i \(-0.500724\pi\)
−0.00227508 + 0.999997i \(0.500724\pi\)
\(770\) 0 0
\(771\) − 1.97961e17i − 0.942438i
\(772\) 0 0
\(773\) −2.12581e17 −0.996434 −0.498217 0.867052i \(-0.666012\pi\)
−0.498217 + 0.867052i \(0.666012\pi\)
\(774\) 0 0
\(775\) − 2.17048e17i − 1.00172i
\(776\) 0 0
\(777\) 1.05611e16 0.0479936
\(778\) 0 0
\(779\) 3.65594e15i 0.0163597i
\(780\) 0 0
\(781\) 7.85796e16 0.346261
\(782\) 0 0
\(783\) − 2.73421e16i − 0.118648i
\(784\) 0 0
\(785\) −2.51741e17 −1.07581
\(786\) 0 0
\(787\) 3.88757e17i 1.63617i 0.575095 + 0.818087i \(0.304965\pi\)
−0.575095 + 0.818087i \(0.695035\pi\)
\(788\) 0 0
\(789\) −5.97919e16 −0.247845
\(790\) 0 0
\(791\) − 2.75275e16i − 0.112385i
\(792\) 0 0
\(793\) −3.31245e17 −1.33202
\(794\) 0 0
\(795\) 1.44615e16i 0.0572809i
\(796\) 0 0
\(797\) −2.11775e16 −0.0826276 −0.0413138 0.999146i \(-0.513154\pi\)
−0.0413138 + 0.999146i \(0.513154\pi\)
\(798\) 0 0
\(799\) 1.28095e17i 0.492326i
\(800\) 0 0
\(801\) 8.38459e16 0.317458
\(802\) 0 0
\(803\) − 1.88742e17i − 0.704005i
\(804\) 0 0
\(805\) 3.26537e16 0.119993
\(806\) 0 0
\(807\) − 1.24159e17i − 0.449509i
\(808\) 0 0
\(809\) 2.74837e17 0.980357 0.490179 0.871622i \(-0.336932\pi\)
0.490179 + 0.871622i \(0.336932\pi\)
\(810\) 0 0
\(811\) 3.56743e17i 1.25381i 0.779098 + 0.626903i \(0.215678\pi\)
−0.779098 + 0.626903i \(0.784322\pi\)
\(812\) 0 0
\(813\) 1.98143e17 0.686176
\(814\) 0 0
\(815\) 3.66869e17i 1.25189i
\(816\) 0 0
\(817\) 3.14109e15 0.0105620
\(818\) 0 0
\(819\) 1.07857e16i 0.0357391i
\(820\) 0 0
\(821\) −1.70191e17 −0.555748 −0.277874 0.960617i \(-0.589630\pi\)
−0.277874 + 0.960617i \(0.589630\pi\)
\(822\) 0 0
\(823\) − 4.53033e17i − 1.45791i −0.684562 0.728955i \(-0.740006\pi\)
0.684562 0.728955i \(-0.259994\pi\)
\(824\) 0 0
\(825\) −1.65839e17 −0.525973
\(826\) 0 0
\(827\) 4.07186e17i 1.27280i 0.771360 + 0.636399i \(0.219577\pi\)
−0.771360 + 0.636399i \(0.780423\pi\)
\(828\) 0 0
\(829\) 2.99377e17 0.922342 0.461171 0.887311i \(-0.347429\pi\)
0.461171 + 0.887311i \(0.347429\pi\)
\(830\) 0 0
\(831\) − 1.74803e17i − 0.530813i
\(832\) 0 0
\(833\) 9.34523e16 0.279718
\(834\) 0 0
\(835\) − 6.92344e17i − 2.04269i
\(836\) 0 0
\(837\) 7.74355e16 0.225210
\(838\) 0 0
\(839\) − 6.54448e17i − 1.87631i −0.346222 0.938153i \(-0.612536\pi\)
0.346222 0.938153i \(-0.387464\pi\)
\(840\) 0 0
\(841\) −2.19334e17 −0.619911
\(842\) 0 0
\(843\) − 2.83403e17i − 0.789659i
\(844\) 0 0
\(845\) −1.06420e18 −2.92337
\(846\) 0 0
\(847\) − 2.96078e15i − 0.00801873i
\(848\) 0 0
\(849\) 1.55947e17 0.416419
\(850\) 0 0
\(851\) 7.61017e17i 2.00363i
\(852\) 0 0
\(853\) 9.94589e16 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(854\) 0 0
\(855\) 2.43502e15i 0.00623313i
\(856\) 0 0
\(857\) 1.81107e17 0.457142 0.228571 0.973527i \(-0.426595\pi\)
0.228571 + 0.973527i \(0.426595\pi\)
\(858\) 0 0
\(859\) − 2.11062e17i − 0.525353i −0.964884 0.262677i \(-0.915395\pi\)
0.964884 0.262677i \(-0.0846052\pi\)
\(860\) 0 0
\(861\) 1.69470e16 0.0415980
\(862\) 0 0
\(863\) 3.64870e17i 0.883229i 0.897205 + 0.441614i \(0.145594\pi\)
−0.897205 + 0.441614i \(0.854406\pi\)
\(864\) 0 0
\(865\) −6.27881e17 −1.49893
\(866\) 0 0
\(867\) 2.25892e17i 0.531845i
\(868\) 0 0
\(869\) −3.75338e17 −0.871573
\(870\) 0 0
\(871\) − 1.34018e18i − 3.06940i
\(872\) 0 0
\(873\) −1.64588e17 −0.371803
\(874\) 0 0
\(875\) 5.32211e15i 0.0118587i
\(876\) 0 0
\(877\) 2.20060e17 0.483663 0.241832 0.970318i \(-0.422252\pi\)
0.241832 + 0.970318i \(0.422252\pi\)
\(878\) 0 0
\(879\) − 1.07945e17i − 0.234029i
\(880\) 0 0
\(881\) −5.23854e17 −1.12035 −0.560176 0.828373i \(-0.689267\pi\)
−0.560176 + 0.828373i \(0.689267\pi\)
\(882\) 0 0
\(883\) 5.19490e17i 1.09601i 0.836476 + 0.548003i \(0.184612\pi\)
−0.836476 + 0.548003i \(0.815388\pi\)
\(884\) 0 0
\(885\) 1.15015e17 0.239385
\(886\) 0 0
\(887\) − 6.43189e17i − 1.32068i −0.750968 0.660339i \(-0.770413\pi\)
0.750968 0.660339i \(-0.229587\pi\)
\(888\) 0 0
\(889\) −5.85745e16 −0.118658
\(890\) 0 0
\(891\) − 5.91659e16i − 0.118251i
\(892\) 0 0
\(893\) −1.22065e16 −0.0240703
\(894\) 0 0
\(895\) − 4.62615e17i − 0.900081i
\(896\) 0 0
\(897\) −7.77198e17 −1.49203
\(898\) 0 0
\(899\) 3.80864e17i 0.721459i
\(900\) 0 0
\(901\) 1.09380e16 0.0204452
\(902\) 0 0
\(903\) − 1.45604e16i − 0.0268563i
\(904\) 0 0
\(905\) 2.65630e16 0.0483487
\(906\) 0 0
\(907\) 8.20905e17i 1.47452i 0.675612 + 0.737258i \(0.263880\pi\)
−0.675612 + 0.737258i \(0.736120\pi\)
\(908\) 0 0
\(909\) 7.62608e16 0.135182
\(910\) 0 0
\(911\) 7.92986e16i 0.138725i 0.997592 + 0.0693625i \(0.0220965\pi\)
−0.997592 + 0.0693625i \(0.977903\pi\)
\(912\) 0 0
\(913\) 4.34944e17 0.750946
\(914\) 0 0
\(915\) 3.46655e17i 0.590706i
\(916\) 0 0
\(917\) 1.14923e16 0.0193282
\(918\) 0 0
\(919\) 7.38065e17i 1.22518i 0.790399 + 0.612592i \(0.209873\pi\)
−0.790399 + 0.612592i \(0.790127\pi\)
\(920\) 0 0
\(921\) 3.47041e17 0.568622
\(922\) 0 0
\(923\) − 3.56807e17i − 0.577063i
\(924\) 0 0
\(925\) 7.37349e17 1.17713
\(926\) 0 0
\(927\) 2.20459e17i 0.347415i
\(928\) 0 0
\(929\) 1.11243e18 1.73052 0.865261 0.501322i \(-0.167153\pi\)
0.865261 + 0.501322i \(0.167153\pi\)
\(930\) 0 0
\(931\) 8.90526e15i 0.0136757i
\(932\) 0 0
\(933\) 3.35452e17 0.508558
\(934\) 0 0
\(935\) 2.71967e17i 0.407049i
\(936\) 0 0
\(937\) −3.87919e17 −0.573197 −0.286598 0.958051i \(-0.592525\pi\)
−0.286598 + 0.958051i \(0.592525\pi\)
\(938\) 0 0
\(939\) − 7.57508e16i − 0.110508i
\(940\) 0 0
\(941\) 3.15087e17 0.453829 0.226914 0.973915i \(-0.427136\pi\)
0.226914 + 0.973915i \(0.427136\pi\)
\(942\) 0 0
\(943\) 1.22117e18i 1.73663i
\(944\) 0 0
\(945\) 1.12874e16 0.0158491
\(946\) 0 0
\(947\) − 3.79195e16i − 0.0525730i −0.999654 0.0262865i \(-0.991632\pi\)
0.999654 0.0262865i \(-0.00836821\pi\)
\(948\) 0 0
\(949\) −8.57023e17 −1.17326
\(950\) 0 0
\(951\) − 6.90215e17i − 0.933043i
\(952\) 0 0
\(953\) 6.56690e17 0.876603 0.438302 0.898828i \(-0.355580\pi\)
0.438302 + 0.898828i \(0.355580\pi\)
\(954\) 0 0
\(955\) − 1.38830e18i − 1.83005i
\(956\) 0 0
\(957\) 2.91006e17 0.378817
\(958\) 0 0
\(959\) − 3.05600e15i − 0.00392864i
\(960\) 0 0
\(961\) −2.90984e17 −0.369427
\(962\) 0 0
\(963\) − 6.03146e16i − 0.0756249i
\(964\) 0 0
\(965\) 3.46150e17 0.428647
\(966\) 0 0
\(967\) 6.24136e17i 0.763345i 0.924298 + 0.381672i \(0.124652\pi\)
−0.924298 + 0.381672i \(0.875348\pi\)
\(968\) 0 0
\(969\) 1.84175e15 0.00222478
\(970\) 0 0
\(971\) − 9.04996e17i − 1.07977i −0.841739 0.539885i \(-0.818468\pi\)
0.841739 0.539885i \(-0.181532\pi\)
\(972\) 0 0
\(973\) −1.21138e16 −0.0142759
\(974\) 0 0
\(975\) 7.53027e17i 0.876562i
\(976\) 0 0
\(977\) −1.37695e17 −0.158326 −0.0791628 0.996862i \(-0.525225\pi\)
−0.0791628 + 0.996862i \(0.525225\pi\)
\(978\) 0 0
\(979\) 8.92384e17i 1.01357i
\(980\) 0 0
\(981\) 5.87930e17 0.659647
\(982\) 0 0
\(983\) − 1.06312e18i − 1.17831i −0.808018 0.589157i \(-0.799460\pi\)
0.808018 0.589157i \(-0.200540\pi\)
\(984\) 0 0
\(985\) 1.85105e18 2.02675
\(986\) 0 0
\(987\) 5.65825e16i 0.0612039i
\(988\) 0 0
\(989\) 1.04920e18 1.12119
\(990\) 0 0
\(991\) 9.28788e17i 0.980561i 0.871565 + 0.490281i \(0.163106\pi\)
−0.871565 + 0.490281i \(0.836894\pi\)
\(992\) 0 0
\(993\) −8.82898e17 −0.920905
\(994\) 0 0
\(995\) 2.19432e17i 0.226132i
\(996\) 0 0
\(997\) −9.52810e17 −0.970142 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(998\) 0 0
\(999\) 2.63062e17i 0.264646i
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 48.13.g.a.31.1 4
3.2 odd 2 144.13.g.h.127.4 4
4.3 odd 2 inner 48.13.g.a.31.3 yes 4
8.3 odd 2 192.13.g.c.127.2 4
8.5 even 2 192.13.g.c.127.4 4
12.11 even 2 144.13.g.h.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.13.g.a.31.1 4 1.1 even 1 trivial
48.13.g.a.31.3 yes 4 4.3 odd 2 inner
144.13.g.h.127.3 4 12.11 even 2
144.13.g.h.127.4 4 3.2 odd 2
192.13.g.c.127.2 4 8.3 odd 2
192.13.g.c.127.4 4 8.5 even 2