Properties

Label 48.22.a.i.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $1$
Dimension $2$
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] = [48,22,Mod(1,48)]
 
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([0, 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("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 797544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(893.553\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} -1.13060e7 q^{5} +8.09970e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} -1.13060e7 q^{5} +8.09970e8 q^{7} +3.48678e9 q^{9} +5.57516e10 q^{11} -2.33461e11 q^{13} -6.67607e11 q^{15} -9.51155e12 q^{17} -1.41689e13 q^{19} +4.78279e13 q^{21} -1.44892e14 q^{23} -3.49012e14 q^{25} +2.05891e14 q^{27} +1.75961e15 q^{29} -1.79660e15 q^{31} +3.29208e15 q^{33} -9.15751e15 q^{35} +5.36600e16 q^{37} -1.37856e16 q^{39} +6.45090e16 q^{41} -1.20122e17 q^{43} -3.94215e16 q^{45} -6.12446e17 q^{47} +9.75063e16 q^{49} -5.61648e17 q^{51} -3.70894e17 q^{53} -6.30327e17 q^{55} -8.36658e17 q^{57} +4.74196e18 q^{59} -8.07022e18 q^{61} +2.82419e18 q^{63} +2.63950e18 q^{65} -2.31777e17 q^{67} -8.55572e18 q^{69} +2.36832e19 q^{71} -2.16445e19 q^{73} -2.06088e19 q^{75} +4.51572e19 q^{77} -2.92132e19 q^{79} +1.21577e19 q^{81} -1.33727e20 q^{83} +1.07537e20 q^{85} +1.03903e20 q^{87} +4.98227e20 q^{89} -1.89096e20 q^{91} -1.06088e20 q^{93} +1.60193e20 q^{95} -6.53695e20 q^{97} +1.94394e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9} + 76050855288 q^{11} - 809326043300 q^{13} + 1702258667100 q^{15} + 831385897668 q^{17} - 29817568652920 q^{19} - 30095487768672 q^{21} - 62680641406128 q^{23} + 784879798238750 q^{25} + 411782264189298 q^{27} - 21\!\cdots\!52 q^{29}+ \cdots + 26\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) −1.13060e7 −0.517754 −0.258877 0.965910i \(-0.583352\pi\)
−0.258877 + 0.965910i \(0.583352\pi\)
\(6\) 0 0
\(7\) 8.09970e8 1.08378 0.541888 0.840451i \(-0.317710\pi\)
0.541888 + 0.840451i \(0.317710\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 5.57516e10 0.648088 0.324044 0.946042i \(-0.394957\pi\)
0.324044 + 0.946042i \(0.394957\pi\)
\(12\) 0 0
\(13\) −2.33461e11 −0.469687 −0.234844 0.972033i \(-0.575458\pi\)
−0.234844 + 0.972033i \(0.575458\pi\)
\(14\) 0 0
\(15\) −6.67607e11 −0.298925
\(16\) 0 0
\(17\) −9.51155e12 −1.14429 −0.572147 0.820151i \(-0.693889\pi\)
−0.572147 + 0.820151i \(0.693889\pi\)
\(18\) 0 0
\(19\) −1.41689e13 −0.530179 −0.265090 0.964224i \(-0.585402\pi\)
−0.265090 + 0.964224i \(0.585402\pi\)
\(20\) 0 0
\(21\) 4.78279e13 0.625719
\(22\) 0 0
\(23\) −1.44892e14 −0.729293 −0.364646 0.931146i \(-0.618810\pi\)
−0.364646 + 0.931146i \(0.618810\pi\)
\(24\) 0 0
\(25\) −3.49012e14 −0.731931
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) 1.75961e15 0.776669 0.388335 0.921518i \(-0.373050\pi\)
0.388335 + 0.921518i \(0.373050\pi\)
\(30\) 0 0
\(31\) −1.79660e15 −0.393690 −0.196845 0.980435i \(-0.563070\pi\)
−0.196845 + 0.980435i \(0.563070\pi\)
\(32\) 0 0
\(33\) 3.29208e15 0.374174
\(34\) 0 0
\(35\) −9.15751e15 −0.561130
\(36\) 0 0
\(37\) 5.36600e16 1.83456 0.917281 0.398240i \(-0.130379\pi\)
0.917281 + 0.398240i \(0.130379\pi\)
\(38\) 0 0
\(39\) −1.37856e16 −0.271174
\(40\) 0 0
\(41\) 6.45090e16 0.750568 0.375284 0.926910i \(-0.377545\pi\)
0.375284 + 0.926910i \(0.377545\pi\)
\(42\) 0 0
\(43\) −1.20122e17 −0.847624 −0.423812 0.905750i \(-0.639308\pi\)
−0.423812 + 0.905750i \(0.639308\pi\)
\(44\) 0 0
\(45\) −3.94215e16 −0.172585
\(46\) 0 0
\(47\) −6.12446e17 −1.69840 −0.849201 0.528070i \(-0.822916\pi\)
−0.849201 + 0.528070i \(0.822916\pi\)
\(48\) 0 0
\(49\) 9.75063e16 0.174572
\(50\) 0 0
\(51\) −5.61648e17 −0.660659
\(52\) 0 0
\(53\) −3.70894e17 −0.291308 −0.145654 0.989336i \(-0.546529\pi\)
−0.145654 + 0.989336i \(0.546529\pi\)
\(54\) 0 0
\(55\) −6.30327e17 −0.335550
\(56\) 0 0
\(57\) −8.36658e17 −0.306099
\(58\) 0 0
\(59\) 4.74196e18 1.20785 0.603923 0.797043i \(-0.293604\pi\)
0.603923 + 0.797043i \(0.293604\pi\)
\(60\) 0 0
\(61\) −8.07022e18 −1.44851 −0.724256 0.689532i \(-0.757816\pi\)
−0.724256 + 0.689532i \(0.757816\pi\)
\(62\) 0 0
\(63\) 2.82419e18 0.361259
\(64\) 0 0
\(65\) 2.63950e18 0.243182
\(66\) 0 0
\(67\) −2.31777e17 −0.0155340 −0.00776702 0.999970i \(-0.502472\pi\)
−0.00776702 + 0.999970i \(0.502472\pi\)
\(68\) 0 0
\(69\) −8.55572e18 −0.421057
\(70\) 0 0
\(71\) 2.36832e19 0.863430 0.431715 0.902010i \(-0.357909\pi\)
0.431715 + 0.902010i \(0.357909\pi\)
\(72\) 0 0
\(73\) −2.16445e19 −0.589466 −0.294733 0.955580i \(-0.595231\pi\)
−0.294733 + 0.955580i \(0.595231\pi\)
\(74\) 0 0
\(75\) −2.06088e19 −0.422580
\(76\) 0 0
\(77\) 4.51572e19 0.702383
\(78\) 0 0
\(79\) −2.92132e19 −0.347133 −0.173566 0.984822i \(-0.555529\pi\)
−0.173566 + 0.984822i \(0.555529\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) −1.33727e20 −0.946017 −0.473008 0.881058i \(-0.656832\pi\)
−0.473008 + 0.881058i \(0.656832\pi\)
\(84\) 0 0
\(85\) 1.07537e20 0.592463
\(86\) 0 0
\(87\) 1.03903e20 0.448410
\(88\) 0 0
\(89\) 4.98227e20 1.69368 0.846842 0.531845i \(-0.178501\pi\)
0.846842 + 0.531845i \(0.178501\pi\)
\(90\) 0 0
\(91\) −1.89096e20 −0.509036
\(92\) 0 0
\(93\) −1.06088e20 −0.227297
\(94\) 0 0
\(95\) 1.60193e20 0.274502
\(96\) 0 0
\(97\) −6.53695e20 −0.900061 −0.450031 0.893013i \(-0.648587\pi\)
−0.450031 + 0.893013i \(0.648587\pi\)
\(98\) 0 0
\(99\) 1.94394e20 0.216029
\(100\) 0 0
\(101\) −6.51751e20 −0.587093 −0.293547 0.955945i \(-0.594836\pi\)
−0.293547 + 0.955945i \(0.594836\pi\)
\(102\) 0 0
\(103\) 1.29509e21 0.949531 0.474765 0.880112i \(-0.342533\pi\)
0.474765 + 0.880112i \(0.342533\pi\)
\(104\) 0 0
\(105\) −5.40742e20 −0.323968
\(106\) 0 0
\(107\) −3.07189e21 −1.50965 −0.754825 0.655927i \(-0.772278\pi\)
−0.754825 + 0.655927i \(0.772278\pi\)
\(108\) 0 0
\(109\) −2.99895e21 −1.21337 −0.606683 0.794944i \(-0.707500\pi\)
−0.606683 + 0.794944i \(0.707500\pi\)
\(110\) 0 0
\(111\) 3.16857e21 1.05919
\(112\) 0 0
\(113\) 1.29863e21 0.359884 0.179942 0.983677i \(-0.442409\pi\)
0.179942 + 0.983677i \(0.442409\pi\)
\(114\) 0 0
\(115\) 1.63815e21 0.377594
\(116\) 0 0
\(117\) −8.14028e20 −0.156562
\(118\) 0 0
\(119\) −7.70408e21 −1.24016
\(120\) 0 0
\(121\) −4.29201e21 −0.579982
\(122\) 0 0
\(123\) 3.80919e21 0.433340
\(124\) 0 0
\(125\) 9.33704e21 0.896714
\(126\) 0 0
\(127\) −1.95827e22 −1.59197 −0.795984 0.605318i \(-0.793046\pi\)
−0.795984 + 0.605318i \(0.793046\pi\)
\(128\) 0 0
\(129\) −7.09307e21 −0.489376
\(130\) 0 0
\(131\) −3.35956e22 −1.97212 −0.986062 0.166381i \(-0.946792\pi\)
−0.986062 + 0.166381i \(0.946792\pi\)
\(132\) 0 0
\(133\) −1.14764e22 −0.574596
\(134\) 0 0
\(135\) −2.32780e21 −0.0996418
\(136\) 0 0
\(137\) 3.74398e22 1.37331 0.686653 0.726985i \(-0.259079\pi\)
0.686653 + 0.726985i \(0.259079\pi\)
\(138\) 0 0
\(139\) 5.10058e22 1.60681 0.803405 0.595433i \(-0.203020\pi\)
0.803405 + 0.595433i \(0.203020\pi\)
\(140\) 0 0
\(141\) −3.61643e22 −0.980572
\(142\) 0 0
\(143\) −1.30158e22 −0.304399
\(144\) 0 0
\(145\) −1.98941e22 −0.402124
\(146\) 0 0
\(147\) 5.75765e21 0.100789
\(148\) 0 0
\(149\) −4.22264e22 −0.641399 −0.320699 0.947181i \(-0.603918\pi\)
−0.320699 + 0.947181i \(0.603918\pi\)
\(150\) 0 0
\(151\) −1.00887e22 −0.133223 −0.0666114 0.997779i \(-0.521219\pi\)
−0.0666114 + 0.997779i \(0.521219\pi\)
\(152\) 0 0
\(153\) −3.31647e22 −0.381432
\(154\) 0 0
\(155\) 2.03124e22 0.203835
\(156\) 0 0
\(157\) −1.38867e23 −1.21802 −0.609009 0.793163i \(-0.708433\pi\)
−0.609009 + 0.793163i \(0.708433\pi\)
\(158\) 0 0
\(159\) −2.19009e22 −0.168187
\(160\) 0 0
\(161\) −1.17358e23 −0.790390
\(162\) 0 0
\(163\) −2.13956e23 −1.26577 −0.632885 0.774246i \(-0.718129\pi\)
−0.632885 + 0.774246i \(0.718129\pi\)
\(164\) 0 0
\(165\) −3.72202e22 −0.193730
\(166\) 0 0
\(167\) −3.67329e23 −1.68474 −0.842368 0.538902i \(-0.818839\pi\)
−0.842368 + 0.538902i \(0.818839\pi\)
\(168\) 0 0
\(169\) −1.92561e23 −0.779394
\(170\) 0 0
\(171\) −4.94038e22 −0.176726
\(172\) 0 0
\(173\) −3.45906e23 −1.09515 −0.547576 0.836756i \(-0.684449\pi\)
−0.547576 + 0.836756i \(0.684449\pi\)
\(174\) 0 0
\(175\) −2.82689e23 −0.793250
\(176\) 0 0
\(177\) 2.80008e23 0.697350
\(178\) 0 0
\(179\) −1.89936e23 −0.420388 −0.210194 0.977660i \(-0.567410\pi\)
−0.210194 + 0.977660i \(0.567410\pi\)
\(180\) 0 0
\(181\) 7.45122e23 1.46758 0.733791 0.679376i \(-0.237749\pi\)
0.733791 + 0.679376i \(0.237749\pi\)
\(182\) 0 0
\(183\) −4.76538e23 −0.836298
\(184\) 0 0
\(185\) −6.06679e23 −0.949852
\(186\) 0 0
\(187\) −5.30284e23 −0.741604
\(188\) 0 0
\(189\) 1.66766e23 0.208573
\(190\) 0 0
\(191\) 7.67578e23 0.859552 0.429776 0.902935i \(-0.358592\pi\)
0.429776 + 0.902935i \(0.358592\pi\)
\(192\) 0 0
\(193\) −7.52907e23 −0.755770 −0.377885 0.925852i \(-0.623349\pi\)
−0.377885 + 0.925852i \(0.623349\pi\)
\(194\) 0 0
\(195\) 1.55860e23 0.140401
\(196\) 0 0
\(197\) −1.80684e24 −1.46226 −0.731130 0.682238i \(-0.761007\pi\)
−0.731130 + 0.682238i \(0.761007\pi\)
\(198\) 0 0
\(199\) −3.86235e23 −0.281121 −0.140561 0.990072i \(-0.544891\pi\)
−0.140561 + 0.990072i \(0.544891\pi\)
\(200\) 0 0
\(201\) −1.36862e22 −0.00896858
\(202\) 0 0
\(203\) 1.42523e24 0.841736
\(204\) 0 0
\(205\) −7.29338e23 −0.388609
\(206\) 0 0
\(207\) −5.05207e23 −0.243098
\(208\) 0 0
\(209\) −7.89938e23 −0.343603
\(210\) 0 0
\(211\) −1.97752e23 −0.0778315 −0.0389158 0.999242i \(-0.512390\pi\)
−0.0389158 + 0.999242i \(0.512390\pi\)
\(212\) 0 0
\(213\) 1.39847e24 0.498502
\(214\) 0 0
\(215\) 1.35810e24 0.438860
\(216\) 0 0
\(217\) −1.45520e24 −0.426672
\(218\) 0 0
\(219\) −1.27809e24 −0.340328
\(220\) 0 0
\(221\) 2.22058e24 0.537461
\(222\) 0 0
\(223\) −3.85195e24 −0.848164 −0.424082 0.905624i \(-0.639403\pi\)
−0.424082 + 0.905624i \(0.639403\pi\)
\(224\) 0 0
\(225\) −1.21693e24 −0.243977
\(226\) 0 0
\(227\) 2.39534e24 0.437619 0.218809 0.975768i \(-0.429783\pi\)
0.218809 + 0.975768i \(0.429783\pi\)
\(228\) 0 0
\(229\) 5.70738e24 0.950964 0.475482 0.879726i \(-0.342274\pi\)
0.475482 + 0.879726i \(0.342274\pi\)
\(230\) 0 0
\(231\) 2.66648e24 0.405521
\(232\) 0 0
\(233\) 1.29921e25 1.80486 0.902430 0.430836i \(-0.141781\pi\)
0.902430 + 0.430836i \(0.141781\pi\)
\(234\) 0 0
\(235\) 6.92431e24 0.879354
\(236\) 0 0
\(237\) −1.72501e24 −0.200417
\(238\) 0 0
\(239\) −1.61680e25 −1.71980 −0.859901 0.510460i \(-0.829475\pi\)
−0.859901 + 0.510460i \(0.829475\pi\)
\(240\) 0 0
\(241\) 3.12483e24 0.304542 0.152271 0.988339i \(-0.451341\pi\)
0.152271 + 0.988339i \(0.451341\pi\)
\(242\) 0 0
\(243\) 7.17898e23 0.0641500
\(244\) 0 0
\(245\) −1.10241e24 −0.0903852
\(246\) 0 0
\(247\) 3.30788e24 0.249019
\(248\) 0 0
\(249\) −7.89644e24 −0.546183
\(250\) 0 0
\(251\) 1.13803e25 0.723736 0.361868 0.932229i \(-0.382139\pi\)
0.361868 + 0.932229i \(0.382139\pi\)
\(252\) 0 0
\(253\) −8.07796e24 −0.472646
\(254\) 0 0
\(255\) 6.34998e24 0.342059
\(256\) 0 0
\(257\) 1.24576e25 0.618211 0.309105 0.951028i \(-0.399970\pi\)
0.309105 + 0.951028i \(0.399970\pi\)
\(258\) 0 0
\(259\) 4.34630e25 1.98826
\(260\) 0 0
\(261\) 6.13536e24 0.258890
\(262\) 0 0
\(263\) −3.40375e24 −0.132563 −0.0662814 0.997801i \(-0.521114\pi\)
−0.0662814 + 0.997801i \(0.521114\pi\)
\(264\) 0 0
\(265\) 4.19332e24 0.150826
\(266\) 0 0
\(267\) 2.94198e25 0.977848
\(268\) 0 0
\(269\) 1.28652e25 0.395383 0.197691 0.980264i \(-0.436656\pi\)
0.197691 + 0.980264i \(0.436656\pi\)
\(270\) 0 0
\(271\) 4.50882e25 1.28199 0.640996 0.767545i \(-0.278522\pi\)
0.640996 + 0.767545i \(0.278522\pi\)
\(272\) 0 0
\(273\) −1.11660e25 −0.293892
\(274\) 0 0
\(275\) −1.94580e25 −0.474356
\(276\) 0 0
\(277\) 4.61059e25 1.04164 0.520821 0.853666i \(-0.325626\pi\)
0.520821 + 0.853666i \(0.325626\pi\)
\(278\) 0 0
\(279\) −6.26437e24 −0.131230
\(280\) 0 0
\(281\) 5.36154e25 1.04201 0.521006 0.853553i \(-0.325557\pi\)
0.521006 + 0.853553i \(0.325557\pi\)
\(282\) 0 0
\(283\) −5.70275e25 −1.02879 −0.514395 0.857553i \(-0.671983\pi\)
−0.514395 + 0.857553i \(0.671983\pi\)
\(284\) 0 0
\(285\) 9.45925e24 0.158484
\(286\) 0 0
\(287\) 5.22504e25 0.813448
\(288\) 0 0
\(289\) 2.13777e25 0.309410
\(290\) 0 0
\(291\) −3.86000e25 −0.519651
\(292\) 0 0
\(293\) −2.56248e25 −0.321033 −0.160517 0.987033i \(-0.551316\pi\)
−0.160517 + 0.987033i \(0.551316\pi\)
\(294\) 0 0
\(295\) −5.36125e25 −0.625367
\(296\) 0 0
\(297\) 1.14788e25 0.124725
\(298\) 0 0
\(299\) 3.38266e25 0.342540
\(300\) 0 0
\(301\) −9.72951e25 −0.918635
\(302\) 0 0
\(303\) −3.84852e25 −0.338958
\(304\) 0 0
\(305\) 9.12417e25 0.749972
\(306\) 0 0
\(307\) −1.95148e26 −1.49766 −0.748828 0.662765i \(-0.769383\pi\)
−0.748828 + 0.662765i \(0.769383\pi\)
\(308\) 0 0
\(309\) 7.64737e25 0.548212
\(310\) 0 0
\(311\) 1.09220e26 0.731677 0.365838 0.930678i \(-0.380782\pi\)
0.365838 + 0.930678i \(0.380782\pi\)
\(312\) 0 0
\(313\) −9.63701e25 −0.603569 −0.301784 0.953376i \(-0.597582\pi\)
−0.301784 + 0.953376i \(0.597582\pi\)
\(314\) 0 0
\(315\) −3.19303e25 −0.187043
\(316\) 0 0
\(317\) −3.07372e26 −1.68477 −0.842387 0.538874i \(-0.818850\pi\)
−0.842387 + 0.538874i \(0.818850\pi\)
\(318\) 0 0
\(319\) 9.81008e25 0.503350
\(320\) 0 0
\(321\) −1.81392e26 −0.871596
\(322\) 0 0
\(323\) 1.34768e26 0.606681
\(324\) 0 0
\(325\) 8.14806e25 0.343779
\(326\) 0 0
\(327\) −1.77085e26 −0.700537
\(328\) 0 0
\(329\) −4.96063e26 −1.84069
\(330\) 0 0
\(331\) −6.07487e25 −0.211516 −0.105758 0.994392i \(-0.533727\pi\)
−0.105758 + 0.994392i \(0.533727\pi\)
\(332\) 0 0
\(333\) 1.87101e26 0.611521
\(334\) 0 0
\(335\) 2.62046e24 0.00804281
\(336\) 0 0
\(337\) −1.45000e26 −0.418075 −0.209037 0.977908i \(-0.567033\pi\)
−0.209037 + 0.977908i \(0.567033\pi\)
\(338\) 0 0
\(339\) 7.66829e25 0.207779
\(340\) 0 0
\(341\) −1.00164e26 −0.255146
\(342\) 0 0
\(343\) −3.73428e26 −0.894580
\(344\) 0 0
\(345\) 9.67309e25 0.218004
\(346\) 0 0
\(347\) −2.56465e26 −0.543961 −0.271981 0.962303i \(-0.587679\pi\)
−0.271981 + 0.962303i \(0.587679\pi\)
\(348\) 0 0
\(349\) 1.11631e26 0.222903 0.111452 0.993770i \(-0.464450\pi\)
0.111452 + 0.993770i \(0.464450\pi\)
\(350\) 0 0
\(351\) −4.80675e25 −0.0903914
\(352\) 0 0
\(353\) 9.82297e26 1.74024 0.870118 0.492843i \(-0.164042\pi\)
0.870118 + 0.492843i \(0.164042\pi\)
\(354\) 0 0
\(355\) −2.67762e26 −0.447044
\(356\) 0 0
\(357\) −4.54918e26 −0.716006
\(358\) 0 0
\(359\) −4.96503e26 −0.736937 −0.368468 0.929640i \(-0.620118\pi\)
−0.368468 + 0.929640i \(0.620118\pi\)
\(360\) 0 0
\(361\) −5.13452e26 −0.718910
\(362\) 0 0
\(363\) −2.53439e26 −0.334852
\(364\) 0 0
\(365\) 2.44713e26 0.305198
\(366\) 0 0
\(367\) 3.07854e26 0.362536 0.181268 0.983434i \(-0.441980\pi\)
0.181268 + 0.983434i \(0.441980\pi\)
\(368\) 0 0
\(369\) 2.24929e26 0.250189
\(370\) 0 0
\(371\) −3.00413e26 −0.315713
\(372\) 0 0
\(373\) 2.47938e26 0.246264 0.123132 0.992390i \(-0.460706\pi\)
0.123132 + 0.992390i \(0.460706\pi\)
\(374\) 0 0
\(375\) 5.51343e26 0.517718
\(376\) 0 0
\(377\) −4.10799e26 −0.364792
\(378\) 0 0
\(379\) −2.02616e26 −0.170201 −0.0851005 0.996372i \(-0.527121\pi\)
−0.0851005 + 0.996372i \(0.527121\pi\)
\(380\) 0 0
\(381\) −1.15634e27 −0.919123
\(382\) 0 0
\(383\) −3.73346e25 −0.0280882 −0.0140441 0.999901i \(-0.504471\pi\)
−0.0140441 + 0.999901i \(0.504471\pi\)
\(384\) 0 0
\(385\) −5.10546e26 −0.363662
\(386\) 0 0
\(387\) −4.18839e26 −0.282541
\(388\) 0 0
\(389\) 1.25917e27 0.804664 0.402332 0.915494i \(-0.368200\pi\)
0.402332 + 0.915494i \(0.368200\pi\)
\(390\) 0 0
\(391\) 1.37815e27 0.834526
\(392\) 0 0
\(393\) −1.98379e27 −1.13861
\(394\) 0 0
\(395\) 3.30285e26 0.179729
\(396\) 0 0
\(397\) −3.27047e27 −1.68776 −0.843878 0.536536i \(-0.819733\pi\)
−0.843878 + 0.536536i \(0.819733\pi\)
\(398\) 0 0
\(399\) −6.77669e26 −0.331743
\(400\) 0 0
\(401\) 2.91653e27 1.35472 0.677362 0.735650i \(-0.263123\pi\)
0.677362 + 0.735650i \(0.263123\pi\)
\(402\) 0 0
\(403\) 4.19437e26 0.184911
\(404\) 0 0
\(405\) −1.37454e26 −0.0575282
\(406\) 0 0
\(407\) 2.99163e27 1.18896
\(408\) 0 0
\(409\) 1.31079e27 0.494808 0.247404 0.968912i \(-0.420423\pi\)
0.247404 + 0.968912i \(0.420423\pi\)
\(410\) 0 0
\(411\) 2.21078e27 0.792879
\(412\) 0 0
\(413\) 3.84084e27 1.30903
\(414\) 0 0
\(415\) 1.51191e27 0.489804
\(416\) 0 0
\(417\) 3.01184e27 0.927692
\(418\) 0 0
\(419\) 4.30721e27 1.26168 0.630840 0.775913i \(-0.282710\pi\)
0.630840 + 0.775913i \(0.282710\pi\)
\(420\) 0 0
\(421\) 2.00292e27 0.558087 0.279044 0.960278i \(-0.409983\pi\)
0.279044 + 0.960278i \(0.409983\pi\)
\(422\) 0 0
\(423\) −2.13547e27 −0.566134
\(424\) 0 0
\(425\) 3.31965e27 0.837544
\(426\) 0 0
\(427\) −6.53664e27 −1.56986
\(428\) 0 0
\(429\) −7.68571e26 −0.175745
\(430\) 0 0
\(431\) −6.73790e26 −0.146728 −0.0733640 0.997305i \(-0.523373\pi\)
−0.0733640 + 0.997305i \(0.523373\pi\)
\(432\) 0 0
\(433\) 5.69651e27 1.18164 0.590821 0.806802i \(-0.298804\pi\)
0.590821 + 0.806802i \(0.298804\pi\)
\(434\) 0 0
\(435\) −1.17472e27 −0.232166
\(436\) 0 0
\(437\) 2.05296e27 0.386656
\(438\) 0 0
\(439\) −1.20501e27 −0.216328 −0.108164 0.994133i \(-0.534497\pi\)
−0.108164 + 0.994133i \(0.534497\pi\)
\(440\) 0 0
\(441\) 3.39984e26 0.0581906
\(442\) 0 0
\(443\) 9.12315e27 1.48904 0.744519 0.667602i \(-0.232679\pi\)
0.744519 + 0.667602i \(0.232679\pi\)
\(444\) 0 0
\(445\) −5.63295e27 −0.876911
\(446\) 0 0
\(447\) −2.49343e27 −0.370312
\(448\) 0 0
\(449\) −4.91421e27 −0.696414 −0.348207 0.937418i \(-0.613209\pi\)
−0.348207 + 0.937418i \(0.613209\pi\)
\(450\) 0 0
\(451\) 3.59648e27 0.486434
\(452\) 0 0
\(453\) −5.95730e26 −0.0769162
\(454\) 0 0
\(455\) 2.13792e27 0.263555
\(456\) 0 0
\(457\) 4.25031e27 0.500380 0.250190 0.968197i \(-0.419507\pi\)
0.250190 + 0.968197i \(0.419507\pi\)
\(458\) 0 0
\(459\) −1.95834e27 −0.220220
\(460\) 0 0
\(461\) −1.47146e27 −0.158084 −0.0790421 0.996871i \(-0.525186\pi\)
−0.0790421 + 0.996871i \(0.525186\pi\)
\(462\) 0 0
\(463\) −1.91401e27 −0.196491 −0.0982456 0.995162i \(-0.531323\pi\)
−0.0982456 + 0.995162i \(0.531323\pi\)
\(464\) 0 0
\(465\) 1.19943e27 0.117684
\(466\) 0 0
\(467\) 1.92082e28 1.80161 0.900804 0.434225i \(-0.142978\pi\)
0.900804 + 0.434225i \(0.142978\pi\)
\(468\) 0 0
\(469\) −1.87732e26 −0.0168354
\(470\) 0 0
\(471\) −8.19998e27 −0.703223
\(472\) 0 0
\(473\) −6.69698e27 −0.549335
\(474\) 0 0
\(475\) 4.94511e27 0.388055
\(476\) 0 0
\(477\) −1.29323e27 −0.0971028
\(478\) 0 0
\(479\) 1.32440e28 0.951692 0.475846 0.879529i \(-0.342142\pi\)
0.475846 + 0.879529i \(0.342142\pi\)
\(480\) 0 0
\(481\) −1.25275e28 −0.861671
\(482\) 0 0
\(483\) −6.92988e27 −0.456332
\(484\) 0 0
\(485\) 7.39067e27 0.466010
\(486\) 0 0
\(487\) −2.28732e28 −1.38125 −0.690625 0.723213i \(-0.742664\pi\)
−0.690625 + 0.723213i \(0.742664\pi\)
\(488\) 0 0
\(489\) −1.26339e28 −0.730792
\(490\) 0 0
\(491\) −1.73094e28 −0.959238 −0.479619 0.877477i \(-0.659225\pi\)
−0.479619 + 0.877477i \(0.659225\pi\)
\(492\) 0 0
\(493\) −1.67366e28 −0.888739
\(494\) 0 0
\(495\) −2.19781e27 −0.111850
\(496\) 0 0
\(497\) 1.91827e28 0.935766
\(498\) 0 0
\(499\) −2.15871e28 −1.00958 −0.504788 0.863243i \(-0.668429\pi\)
−0.504788 + 0.863243i \(0.668429\pi\)
\(500\) 0 0
\(501\) −2.16904e28 −0.972683
\(502\) 0 0
\(503\) 1.63654e27 0.0703821 0.0351911 0.999381i \(-0.488796\pi\)
0.0351911 + 0.999381i \(0.488796\pi\)
\(504\) 0 0
\(505\) 7.36868e27 0.303970
\(506\) 0 0
\(507\) −1.13705e28 −0.449983
\(508\) 0 0
\(509\) 2.18700e28 0.830447 0.415223 0.909720i \(-0.363703\pi\)
0.415223 + 0.909720i \(0.363703\pi\)
\(510\) 0 0
\(511\) −1.75314e28 −0.638849
\(512\) 0 0
\(513\) −2.91725e27 −0.102033
\(514\) 0 0
\(515\) −1.46423e28 −0.491623
\(516\) 0 0
\(517\) −3.41448e28 −1.10071
\(518\) 0 0
\(519\) −2.04254e28 −0.632286
\(520\) 0 0
\(521\) 6.08574e28 1.80933 0.904664 0.426125i \(-0.140122\pi\)
0.904664 + 0.426125i \(0.140122\pi\)
\(522\) 0 0
\(523\) −8.60789e27 −0.245827 −0.122913 0.992417i \(-0.539224\pi\)
−0.122913 + 0.992417i \(0.539224\pi\)
\(524\) 0 0
\(525\) −1.66925e28 −0.457983
\(526\) 0 0
\(527\) 1.70885e28 0.450498
\(528\) 0 0
\(529\) −1.84779e28 −0.468132
\(530\) 0 0
\(531\) 1.65342e28 0.402615
\(532\) 0 0
\(533\) −1.50603e28 −0.352532
\(534\) 0 0
\(535\) 3.47307e28 0.781627
\(536\) 0 0
\(537\) −1.12155e28 −0.242711
\(538\) 0 0
\(539\) 5.43613e27 0.113138
\(540\) 0 0
\(541\) −5.28529e28 −1.05803 −0.529015 0.848613i \(-0.677438\pi\)
−0.529015 + 0.848613i \(0.677438\pi\)
\(542\) 0 0
\(543\) 4.39987e28 0.847308
\(544\) 0 0
\(545\) 3.39061e28 0.628225
\(546\) 0 0
\(547\) 4.42072e27 0.0788181 0.0394091 0.999223i \(-0.487452\pi\)
0.0394091 + 0.999223i \(0.487452\pi\)
\(548\) 0 0
\(549\) −2.81391e28 −0.482837
\(550\) 0 0
\(551\) −2.49316e28 −0.411774
\(552\) 0 0
\(553\) −2.36619e28 −0.376214
\(554\) 0 0
\(555\) −3.58238e28 −0.548397
\(556\) 0 0
\(557\) −3.85988e28 −0.568976 −0.284488 0.958680i \(-0.591824\pi\)
−0.284488 + 0.958680i \(0.591824\pi\)
\(558\) 0 0
\(559\) 2.80437e28 0.398118
\(560\) 0 0
\(561\) −3.13128e28 −0.428165
\(562\) 0 0
\(563\) 4.50679e28 0.593648 0.296824 0.954932i \(-0.404072\pi\)
0.296824 + 0.954932i \(0.404072\pi\)
\(564\) 0 0
\(565\) −1.46823e28 −0.186331
\(566\) 0 0
\(567\) 9.84735e27 0.120420
\(568\) 0 0
\(569\) 1.08302e29 1.27632 0.638160 0.769904i \(-0.279696\pi\)
0.638160 + 0.769904i \(0.279696\pi\)
\(570\) 0 0
\(571\) −1.40701e29 −1.59815 −0.799076 0.601231i \(-0.794677\pi\)
−0.799076 + 0.601231i \(0.794677\pi\)
\(572\) 0 0
\(573\) 4.53247e28 0.496263
\(574\) 0 0
\(575\) 5.05690e28 0.533792
\(576\) 0 0
\(577\) 3.38612e28 0.344633 0.172316 0.985042i \(-0.444875\pi\)
0.172316 + 0.985042i \(0.444875\pi\)
\(578\) 0 0
\(579\) −4.44584e28 −0.436344
\(580\) 0 0
\(581\) −1.08315e29 −1.02527
\(582\) 0 0
\(583\) −2.06779e28 −0.188794
\(584\) 0 0
\(585\) 9.20338e27 0.0810608
\(586\) 0 0
\(587\) 6.58705e28 0.559746 0.279873 0.960037i \(-0.409708\pi\)
0.279873 + 0.960037i \(0.409708\pi\)
\(588\) 0 0
\(589\) 2.54559e28 0.208726
\(590\) 0 0
\(591\) −1.06692e29 −0.844237
\(592\) 0 0
\(593\) 2.36861e29 1.80892 0.904459 0.426561i \(-0.140275\pi\)
0.904459 + 0.426561i \(0.140275\pi\)
\(594\) 0 0
\(595\) 8.71022e28 0.642097
\(596\) 0 0
\(597\) −2.28068e28 −0.162306
\(598\) 0 0
\(599\) 1.61602e29 1.11037 0.555183 0.831728i \(-0.312648\pi\)
0.555183 + 0.831728i \(0.312648\pi\)
\(600\) 0 0
\(601\) 1.80624e29 1.19837 0.599187 0.800609i \(-0.295490\pi\)
0.599187 + 0.800609i \(0.295490\pi\)
\(602\) 0 0
\(603\) −8.08156e26 −0.00517801
\(604\) 0 0
\(605\) 4.85254e28 0.300288
\(606\) 0 0
\(607\) −1.81082e29 −1.08242 −0.541208 0.840888i \(-0.682033\pi\)
−0.541208 + 0.840888i \(0.682033\pi\)
\(608\) 0 0
\(609\) 8.41583e28 0.485977
\(610\) 0 0
\(611\) 1.42982e29 0.797717
\(612\) 0 0
\(613\) 7.75920e27 0.0418295 0.0209147 0.999781i \(-0.493342\pi\)
0.0209147 + 0.999781i \(0.493342\pi\)
\(614\) 0 0
\(615\) −4.30667e28 −0.224364
\(616\) 0 0
\(617\) 1.97354e29 0.993692 0.496846 0.867839i \(-0.334491\pi\)
0.496846 + 0.867839i \(0.334491\pi\)
\(618\) 0 0
\(619\) 2.26065e29 1.10022 0.550112 0.835091i \(-0.314585\pi\)
0.550112 + 0.835091i \(0.314585\pi\)
\(620\) 0 0
\(621\) −2.98320e28 −0.140352
\(622\) 0 0
\(623\) 4.03549e29 1.83557
\(624\) 0 0
\(625\) 6.08574e28 0.267654
\(626\) 0 0
\(627\) −4.66451e28 −0.198379
\(628\) 0 0
\(629\) −5.10390e29 −2.09928
\(630\) 0 0
\(631\) −6.50393e28 −0.258742 −0.129371 0.991596i \(-0.541296\pi\)
−0.129371 + 0.991596i \(0.541296\pi\)
\(632\) 0 0
\(633\) −1.16771e28 −0.0449360
\(634\) 0 0
\(635\) 2.21402e29 0.824248
\(636\) 0 0
\(637\) −2.27639e28 −0.0819941
\(638\) 0 0
\(639\) 8.25781e28 0.287810
\(640\) 0 0
\(641\) −1.44790e29 −0.488349 −0.244175 0.969731i \(-0.578517\pi\)
−0.244175 + 0.969731i \(0.578517\pi\)
\(642\) 0 0
\(643\) 1.95085e29 0.636809 0.318404 0.947955i \(-0.396853\pi\)
0.318404 + 0.947955i \(0.396853\pi\)
\(644\) 0 0
\(645\) 8.01942e28 0.253376
\(646\) 0 0
\(647\) −2.01626e29 −0.616667 −0.308334 0.951278i \(-0.599771\pi\)
−0.308334 + 0.951278i \(0.599771\pi\)
\(648\) 0 0
\(649\) 2.64372e29 0.782791
\(650\) 0 0
\(651\) −8.59279e28 −0.246339
\(652\) 0 0
\(653\) 2.81684e29 0.781941 0.390971 0.920403i \(-0.372139\pi\)
0.390971 + 0.920403i \(0.372139\pi\)
\(654\) 0 0
\(655\) 3.79832e29 1.02107
\(656\) 0 0
\(657\) −7.54699e28 −0.196489
\(658\) 0 0
\(659\) −5.54764e29 −1.39898 −0.699489 0.714643i \(-0.746589\pi\)
−0.699489 + 0.714643i \(0.746589\pi\)
\(660\) 0 0
\(661\) −6.76779e29 −1.65322 −0.826612 0.562772i \(-0.809735\pi\)
−0.826612 + 0.562772i \(0.809735\pi\)
\(662\) 0 0
\(663\) 1.31123e29 0.310303
\(664\) 0 0
\(665\) 1.29752e29 0.297499
\(666\) 0 0
\(667\) −2.54953e29 −0.566419
\(668\) 0 0
\(669\) −2.27454e29 −0.489688
\(670\) 0 0
\(671\) −4.49927e29 −0.938763
\(672\) 0 0
\(673\) 7.48648e29 1.51398 0.756989 0.653428i \(-0.226670\pi\)
0.756989 + 0.653428i \(0.226670\pi\)
\(674\) 0 0
\(675\) −7.18584e28 −0.140860
\(676\) 0 0
\(677\) −7.29954e28 −0.138712 −0.0693561 0.997592i \(-0.522094\pi\)
−0.0693561 + 0.997592i \(0.522094\pi\)
\(678\) 0 0
\(679\) −5.29474e29 −0.975465
\(680\) 0 0
\(681\) 1.41442e29 0.252659
\(682\) 0 0
\(683\) −6.79117e29 −1.17632 −0.588162 0.808743i \(-0.700148\pi\)
−0.588162 + 0.808743i \(0.700148\pi\)
\(684\) 0 0
\(685\) −4.23294e29 −0.711035
\(686\) 0 0
\(687\) 3.37015e29 0.549039
\(688\) 0 0
\(689\) 8.65892e28 0.136824
\(690\) 0 0
\(691\) 8.63963e28 0.132427 0.0662134 0.997805i \(-0.478908\pi\)
0.0662134 + 0.997805i \(0.478908\pi\)
\(692\) 0 0
\(693\) 1.57453e29 0.234128
\(694\) 0 0
\(695\) −5.76671e29 −0.831932
\(696\) 0 0
\(697\) −6.13581e29 −0.858871
\(698\) 0 0
\(699\) 7.67173e29 1.04204
\(700\) 0 0
\(701\) −5.58498e29 −0.736178 −0.368089 0.929791i \(-0.619988\pi\)
−0.368089 + 0.929791i \(0.619988\pi\)
\(702\) 0 0
\(703\) −7.60302e29 −0.972647
\(704\) 0 0
\(705\) 4.08873e29 0.507695
\(706\) 0 0
\(707\) −5.27899e29 −0.636278
\(708\) 0 0
\(709\) −1.18180e30 −1.38279 −0.691397 0.722475i \(-0.743004\pi\)
−0.691397 + 0.722475i \(0.743004\pi\)
\(710\) 0 0
\(711\) −1.01860e29 −0.115711
\(712\) 0 0
\(713\) 2.60313e29 0.287115
\(714\) 0 0
\(715\) 1.47157e29 0.157604
\(716\) 0 0
\(717\) −9.54705e29 −0.992929
\(718\) 0 0
\(719\) −8.23046e29 −0.831324 −0.415662 0.909519i \(-0.636450\pi\)
−0.415662 + 0.909519i \(0.636450\pi\)
\(720\) 0 0
\(721\) 1.04898e30 1.02908
\(722\) 0 0
\(723\) 1.84518e29 0.175827
\(724\) 0 0
\(725\) −6.14123e29 −0.568468
\(726\) 0 0
\(727\) 1.96290e30 1.76518 0.882588 0.470148i \(-0.155799\pi\)
0.882588 + 0.470148i \(0.155799\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 1.14254e30 0.969931
\(732\) 0 0
\(733\) −6.34287e29 −0.523232 −0.261616 0.965172i \(-0.584255\pi\)
−0.261616 + 0.965172i \(0.584255\pi\)
\(734\) 0 0
\(735\) −6.50959e28 −0.0521839
\(736\) 0 0
\(737\) −1.29219e28 −0.0100674
\(738\) 0 0
\(739\) 1.33383e30 1.01003 0.505013 0.863112i \(-0.331488\pi\)
0.505013 + 0.863112i \(0.331488\pi\)
\(740\) 0 0
\(741\) 1.95327e29 0.143771
\(742\) 0 0
\(743\) −4.74142e29 −0.339255 −0.169627 0.985508i \(-0.554256\pi\)
−0.169627 + 0.985508i \(0.554256\pi\)
\(744\) 0 0
\(745\) 4.77411e29 0.332087
\(746\) 0 0
\(747\) −4.66277e29 −0.315339
\(748\) 0 0
\(749\) −2.48814e30 −1.63612
\(750\) 0 0
\(751\) −1.47957e30 −0.946056 −0.473028 0.881047i \(-0.656839\pi\)
−0.473028 + 0.881047i \(0.656839\pi\)
\(752\) 0 0
\(753\) 6.71996e29 0.417849
\(754\) 0 0
\(755\) 1.14063e29 0.0689766
\(756\) 0 0
\(757\) 4.39255e29 0.258351 0.129175 0.991622i \(-0.458767\pi\)
0.129175 + 0.991622i \(0.458767\pi\)
\(758\) 0 0
\(759\) −4.76995e29 −0.272882
\(760\) 0 0
\(761\) −1.00062e30 −0.556838 −0.278419 0.960460i \(-0.589810\pi\)
−0.278419 + 0.960460i \(0.589810\pi\)
\(762\) 0 0
\(763\) −2.42906e30 −1.31502
\(764\) 0 0
\(765\) 3.74960e29 0.197488
\(766\) 0 0
\(767\) −1.10706e30 −0.567310
\(768\) 0 0
\(769\) 2.87704e30 1.43456 0.717280 0.696785i \(-0.245387\pi\)
0.717280 + 0.696785i \(0.245387\pi\)
\(770\) 0 0
\(771\) 7.35609e29 0.356924
\(772\) 0 0
\(773\) −9.54616e29 −0.450759 −0.225379 0.974271i \(-0.572362\pi\)
−0.225379 + 0.974271i \(0.572362\pi\)
\(774\) 0 0
\(775\) 6.27036e29 0.288154
\(776\) 0 0
\(777\) 2.56645e30 1.14792
\(778\) 0 0
\(779\) −9.14021e29 −0.397936
\(780\) 0 0
\(781\) 1.32038e30 0.559579
\(782\) 0 0
\(783\) 3.62287e29 0.149470
\(784\) 0 0
\(785\) 1.57003e30 0.630634
\(786\) 0 0
\(787\) −7.94377e29 −0.310665 −0.155332 0.987862i \(-0.549645\pi\)
−0.155332 + 0.987862i \(0.549645\pi\)
\(788\) 0 0
\(789\) −2.00988e29 −0.0765352
\(790\) 0 0
\(791\) 1.05185e30 0.390034
\(792\) 0 0
\(793\) 1.88408e30 0.680347
\(794\) 0 0
\(795\) 2.47611e29 0.0870795
\(796\) 0 0
\(797\) −3.52810e30 −1.20845 −0.604225 0.796814i \(-0.706517\pi\)
−0.604225 + 0.796814i \(0.706517\pi\)
\(798\) 0 0
\(799\) 5.82531e30 1.94347
\(800\) 0 0
\(801\) 1.73721e30 0.564561
\(802\) 0 0
\(803\) −1.20672e30 −0.382026
\(804\) 0 0
\(805\) 1.32685e30 0.409228
\(806\) 0 0
\(807\) 7.59677e29 0.228274
\(808\) 0 0
\(809\) −4.39629e30 −1.28714 −0.643572 0.765385i \(-0.722548\pi\)
−0.643572 + 0.765385i \(0.722548\pi\)
\(810\) 0 0
\(811\) 7.09120e29 0.202302 0.101151 0.994871i \(-0.467747\pi\)
0.101151 + 0.994871i \(0.467747\pi\)
\(812\) 0 0
\(813\) 2.66241e30 0.740158
\(814\) 0 0
\(815\) 2.41898e30 0.655357
\(816\) 0 0
\(817\) 1.70199e30 0.449392
\(818\) 0 0
\(819\) −6.59338e29 −0.169679
\(820\) 0 0
\(821\) 1.18582e30 0.297451 0.148725 0.988879i \(-0.452483\pi\)
0.148725 + 0.988879i \(0.452483\pi\)
\(822\) 0 0
\(823\) 2.11669e30 0.517559 0.258780 0.965936i \(-0.416680\pi\)
0.258780 + 0.965936i \(0.416680\pi\)
\(824\) 0 0
\(825\) −1.14897e30 −0.273869
\(826\) 0 0
\(827\) −2.22718e29 −0.0517544 −0.0258772 0.999665i \(-0.508238\pi\)
−0.0258772 + 0.999665i \(0.508238\pi\)
\(828\) 0 0
\(829\) 9.55590e29 0.216495 0.108248 0.994124i \(-0.465476\pi\)
0.108248 + 0.994124i \(0.465476\pi\)
\(830\) 0 0
\(831\) 2.72251e30 0.601392
\(832\) 0 0
\(833\) −9.27437e29 −0.199761
\(834\) 0 0
\(835\) 4.15301e30 0.872279
\(836\) 0 0
\(837\) −3.69905e29 −0.0757657
\(838\) 0 0
\(839\) 4.20883e30 0.840738 0.420369 0.907353i \(-0.361901\pi\)
0.420369 + 0.907353i \(0.361901\pi\)
\(840\) 0 0
\(841\) −2.03663e30 −0.396785
\(842\) 0 0
\(843\) 3.16593e30 0.601606
\(844\) 0 0
\(845\) 2.17709e30 0.403534
\(846\) 0 0
\(847\) −3.47640e30 −0.628570
\(848\) 0 0
\(849\) −3.36742e30 −0.593972
\(850\) 0 0
\(851\) −7.77490e30 −1.33793
\(852\) 0 0
\(853\) −5.44490e30 −0.914166 −0.457083 0.889424i \(-0.651106\pi\)
−0.457083 + 0.889424i \(0.651106\pi\)
\(854\) 0 0
\(855\) 5.58559e29 0.0915008
\(856\) 0 0
\(857\) 8.17339e30 1.30648 0.653242 0.757150i \(-0.273409\pi\)
0.653242 + 0.757150i \(0.273409\pi\)
\(858\) 0 0
\(859\) 4.15256e30 0.647721 0.323861 0.946105i \(-0.395019\pi\)
0.323861 + 0.946105i \(0.395019\pi\)
\(860\) 0 0
\(861\) 3.08533e30 0.469644
\(862\) 0 0
\(863\) −8.90281e30 −1.32256 −0.661278 0.750141i \(-0.729986\pi\)
−0.661278 + 0.750141i \(0.729986\pi\)
\(864\) 0 0
\(865\) 3.91081e30 0.567019
\(866\) 0 0
\(867\) 1.26233e30 0.178638
\(868\) 0 0
\(869\) −1.62869e30 −0.224973
\(870\) 0 0
\(871\) 5.41108e28 0.00729614
\(872\) 0 0
\(873\) −2.27929e30 −0.300020
\(874\) 0 0
\(875\) 7.56272e30 0.971838
\(876\) 0 0
\(877\) 6.40744e30 0.803876 0.401938 0.915667i \(-0.368337\pi\)
0.401938 + 0.915667i \(0.368337\pi\)
\(878\) 0 0
\(879\) −1.51312e30 −0.185349
\(880\) 0 0
\(881\) 1.51950e31 1.81741 0.908704 0.417440i \(-0.137073\pi\)
0.908704 + 0.417440i \(0.137073\pi\)
\(882\) 0 0
\(883\) −1.28784e31 −1.50409 −0.752046 0.659111i \(-0.770933\pi\)
−0.752046 + 0.659111i \(0.770933\pi\)
\(884\) 0 0
\(885\) −3.16576e30 −0.361056
\(886\) 0 0
\(887\) −5.05037e30 −0.562503 −0.281252 0.959634i \(-0.590750\pi\)
−0.281252 + 0.959634i \(0.590750\pi\)
\(888\) 0 0
\(889\) −1.58614e31 −1.72534
\(890\) 0 0
\(891\) 6.77809e29 0.0720098
\(892\) 0 0
\(893\) 8.67768e30 0.900457
\(894\) 0 0
\(895\) 2.14741e30 0.217657
\(896\) 0 0
\(897\) 1.99743e30 0.197765
\(898\) 0 0
\(899\) −3.16131e30 −0.305767
\(900\) 0 0
\(901\) 3.52778e30 0.333343
\(902\) 0 0
\(903\) −5.74518e30 −0.530374
\(904\) 0 0
\(905\) −8.42433e30 −0.759846
\(906\) 0 0
\(907\) 4.07169e30 0.358838 0.179419 0.983773i \(-0.442578\pi\)
0.179419 + 0.983773i \(0.442578\pi\)
\(908\) 0 0
\(909\) −2.27251e30 −0.195698
\(910\) 0 0
\(911\) −1.43720e31 −1.20941 −0.604705 0.796450i \(-0.706709\pi\)
−0.604705 + 0.796450i \(0.706709\pi\)
\(912\) 0 0
\(913\) −7.45549e30 −0.613103
\(914\) 0 0
\(915\) 5.38773e30 0.432997
\(916\) 0 0
\(917\) −2.72115e31 −2.13734
\(918\) 0 0
\(919\) −3.79023e30 −0.290973 −0.145487 0.989360i \(-0.546475\pi\)
−0.145487 + 0.989360i \(0.546475\pi\)
\(920\) 0 0
\(921\) −1.15233e31 −0.864672
\(922\) 0 0
\(923\) −5.52909e30 −0.405542
\(924\) 0 0
\(925\) −1.87280e31 −1.34277
\(926\) 0 0
\(927\) 4.51570e30 0.316510
\(928\) 0 0
\(929\) −2.28927e31 −1.56867 −0.784336 0.620337i \(-0.786996\pi\)
−0.784336 + 0.620337i \(0.786996\pi\)
\(930\) 0 0
\(931\) −1.38156e30 −0.0925543
\(932\) 0 0
\(933\) 6.44935e30 0.422434
\(934\) 0 0
\(935\) 5.99539e30 0.383968
\(936\) 0 0
\(937\) 9.72458e30 0.608983 0.304491 0.952515i \(-0.401514\pi\)
0.304491 + 0.952515i \(0.401514\pi\)
\(938\) 0 0
\(939\) −5.69056e30 −0.348471
\(940\) 0 0
\(941\) 1.55337e31 0.930217 0.465108 0.885254i \(-0.346015\pi\)
0.465108 + 0.885254i \(0.346015\pi\)
\(942\) 0 0
\(943\) −9.34683e30 −0.547384
\(944\) 0 0
\(945\) −1.88545e30 −0.107989
\(946\) 0 0
\(947\) 1.82914e31 1.02464 0.512320 0.858794i \(-0.328786\pi\)
0.512320 + 0.858794i \(0.328786\pi\)
\(948\) 0 0
\(949\) 5.05315e30 0.276865
\(950\) 0 0
\(951\) −1.81500e31 −0.972704
\(952\) 0 0
\(953\) −1.25691e31 −0.658915 −0.329458 0.944170i \(-0.606866\pi\)
−0.329458 + 0.944170i \(0.606866\pi\)
\(954\) 0 0
\(955\) −8.67823e30 −0.445037
\(956\) 0 0
\(957\) 5.79275e30 0.290609
\(958\) 0 0
\(959\) 3.03251e31 1.48836
\(960\) 0 0
\(961\) −1.75977e31 −0.845008
\(962\) 0 0
\(963\) −1.07110e31 −0.503216
\(964\) 0 0
\(965\) 8.51236e30 0.391303
\(966\) 0 0
\(967\) −3.37160e30 −0.151655 −0.0758277 0.997121i \(-0.524160\pi\)
−0.0758277 + 0.997121i \(0.524160\pi\)
\(968\) 0 0
\(969\) 7.95792e30 0.350268
\(970\) 0 0
\(971\) 6.40435e30 0.275850 0.137925 0.990443i \(-0.455957\pi\)
0.137925 + 0.990443i \(0.455957\pi\)
\(972\) 0 0
\(973\) 4.13132e31 1.74142
\(974\) 0 0
\(975\) 4.81135e30 0.198481
\(976\) 0 0
\(977\) 1.82208e31 0.735657 0.367829 0.929894i \(-0.380101\pi\)
0.367829 + 0.929894i \(0.380101\pi\)
\(978\) 0 0
\(979\) 2.77770e31 1.09766
\(980\) 0 0
\(981\) −1.04567e31 −0.404455
\(982\) 0 0
\(983\) 2.86098e30 0.108319 0.0541594 0.998532i \(-0.482752\pi\)
0.0541594 + 0.998532i \(0.482752\pi\)
\(984\) 0 0
\(985\) 2.04281e31 0.757091
\(986\) 0 0
\(987\) −2.92920e31 −1.06272
\(988\) 0 0
\(989\) 1.74047e31 0.618166
\(990\) 0 0
\(991\) −3.44835e31 −1.19905 −0.599526 0.800356i \(-0.704644\pi\)
−0.599526 + 0.800356i \(0.704644\pi\)
\(992\) 0 0
\(993\) −3.58715e30 −0.122119
\(994\) 0 0
\(995\) 4.36676e30 0.145552
\(996\) 0 0
\(997\) −1.59375e31 −0.520142 −0.260071 0.965589i \(-0.583746\pi\)
−0.260071 + 0.965589i \(0.583746\pi\)
\(998\) 0 0
\(999\) 1.10481e31 0.353062
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 48.22.a.i.1.1 2
4.3 odd 2 12.22.a.b.1.1 2
12.11 even 2 36.22.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.22.a.b.1.1 2 4.3 odd 2
36.22.a.b.1.2 2 12.11 even 2
48.22.a.i.1.1 2 1.1 even 1 trivial