Properties

Label 80.18.a.d.1.1
Level $80$
Weight $18$
Character 80.1
Self dual yes
Analytic conductor $146.578$
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] = [80,18,Mod(1,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(146.577669876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{83281}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 20820 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(144.792\) of defining polynomial
Character \(\chi\) \(=\) 80.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8003.53 q^{3} +390625. q^{5} -2.54694e7 q^{7} -6.50836e7 q^{9} +2.81089e8 q^{11} -1.52610e9 q^{13} -3.12638e9 q^{15} +5.46901e10 q^{17} -6.88947e8 q^{19} +2.03845e11 q^{21} -3.91035e11 q^{23} +1.52588e11 q^{25} +1.55448e12 q^{27} +5.12259e12 q^{29} +7.31204e10 q^{31} -2.24970e12 q^{33} -9.94897e12 q^{35} -6.81637e12 q^{37} +1.22142e13 q^{39} -5.76386e13 q^{41} -7.57081e13 q^{43} -2.54233e13 q^{45} +4.60351e13 q^{47} +4.16059e14 q^{49} -4.37714e14 q^{51} +6.58261e14 q^{53} +1.09800e14 q^{55} +5.51401e12 q^{57} +2.98287e14 q^{59} +8.50623e14 q^{61} +1.65764e15 q^{63} -5.96131e14 q^{65} +6.12967e15 q^{67} +3.12966e15 q^{69} -5.41472e14 q^{71} -7.16849e15 q^{73} -1.22124e15 q^{75} -7.15916e15 q^{77} -5.45373e15 q^{79} -4.03640e15 q^{81} +3.64723e15 q^{83} +2.13633e16 q^{85} -4.09988e16 q^{87} +6.81792e14 q^{89} +3.88687e16 q^{91} -5.85222e14 q^{93} -2.69120e14 q^{95} -1.20373e17 q^{97} -1.82943e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1308 q^{3} + 781250 q^{5} - 603844 q^{7} - 107519094 q^{9} + 471481296 q^{11} - 1541834228 q^{13} + 510937500 q^{15} + 32139900564 q^{17} - 128672529400 q^{19} + 435381246624 q^{21} - 650359859292 q^{23}+ \cdots - 26\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −8003.53 −0.704289 −0.352145 0.935946i \(-0.614548\pi\)
−0.352145 + 0.935946i \(0.614548\pi\)
\(4\) 0 0
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) −2.54694e7 −1.66988 −0.834939 0.550342i \(-0.814497\pi\)
−0.834939 + 0.550342i \(0.814497\pi\)
\(8\) 0 0
\(9\) −6.50836e7 −0.503976
\(10\) 0 0
\(11\) 2.81089e8 0.395372 0.197686 0.980265i \(-0.436657\pi\)
0.197686 + 0.980265i \(0.436657\pi\)
\(12\) 0 0
\(13\) −1.52610e9 −0.518876 −0.259438 0.965760i \(-0.583537\pi\)
−0.259438 + 0.965760i \(0.583537\pi\)
\(14\) 0 0
\(15\) −3.12638e9 −0.314968
\(16\) 0 0
\(17\) 5.46901e10 1.90148 0.950742 0.309984i \(-0.100324\pi\)
0.950742 + 0.309984i \(0.100324\pi\)
\(18\) 0 0
\(19\) −6.88947e8 −0.00930637 −0.00465318 0.999989i \(-0.501481\pi\)
−0.00465318 + 0.999989i \(0.501481\pi\)
\(20\) 0 0
\(21\) 2.03845e11 1.17608
\(22\) 0 0
\(23\) −3.91035e11 −1.04119 −0.520594 0.853804i \(-0.674290\pi\)
−0.520594 + 0.853804i \(0.674290\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) 0 0
\(27\) 1.55448e12 1.05923
\(28\) 0 0
\(29\) 5.12259e12 1.90154 0.950772 0.309892i \(-0.100293\pi\)
0.950772 + 0.309892i \(0.100293\pi\)
\(30\) 0 0
\(31\) 7.31204e10 0.0153980 0.00769900 0.999970i \(-0.497549\pi\)
0.00769900 + 0.999970i \(0.497549\pi\)
\(32\) 0 0
\(33\) −2.24970e12 −0.278456
\(34\) 0 0
\(35\) −9.94897e12 −0.746792
\(36\) 0 0
\(37\) −6.81637e12 −0.319035 −0.159517 0.987195i \(-0.550994\pi\)
−0.159517 + 0.987195i \(0.550994\pi\)
\(38\) 0 0
\(39\) 1.22142e13 0.365439
\(40\) 0 0
\(41\) −5.76386e13 −1.12733 −0.563665 0.826004i \(-0.690609\pi\)
−0.563665 + 0.826004i \(0.690609\pi\)
\(42\) 0 0
\(43\) −7.57081e13 −0.987781 −0.493890 0.869524i \(-0.664426\pi\)
−0.493890 + 0.869524i \(0.664426\pi\)
\(44\) 0 0
\(45\) −2.54233e13 −0.225385
\(46\) 0 0
\(47\) 4.60351e13 0.282006 0.141003 0.990009i \(-0.454967\pi\)
0.141003 + 0.990009i \(0.454967\pi\)
\(48\) 0 0
\(49\) 4.16059e14 1.78849
\(50\) 0 0
\(51\) −4.37714e14 −1.33919
\(52\) 0 0
\(53\) 6.58261e14 1.45229 0.726145 0.687541i \(-0.241310\pi\)
0.726145 + 0.687541i \(0.241310\pi\)
\(54\) 0 0
\(55\) 1.09800e14 0.176816
\(56\) 0 0
\(57\) 5.51401e12 0.00655437
\(58\) 0 0
\(59\) 2.98287e14 0.264479 0.132240 0.991218i \(-0.457783\pi\)
0.132240 + 0.991218i \(0.457783\pi\)
\(60\) 0 0
\(61\) 8.50623e14 0.568111 0.284056 0.958808i \(-0.408320\pi\)
0.284056 + 0.958808i \(0.408320\pi\)
\(62\) 0 0
\(63\) 1.65764e15 0.841580
\(64\) 0 0
\(65\) −5.96131e14 −0.232048
\(66\) 0 0
\(67\) 6.12967e15 1.84417 0.922085 0.386987i \(-0.126484\pi\)
0.922085 + 0.386987i \(0.126484\pi\)
\(68\) 0 0
\(69\) 3.12966e15 0.733298
\(70\) 0 0
\(71\) −5.41472e14 −0.0995131 −0.0497565 0.998761i \(-0.515845\pi\)
−0.0497565 + 0.998761i \(0.515845\pi\)
\(72\) 0 0
\(73\) −7.16849e15 −1.04036 −0.520180 0.854057i \(-0.674135\pi\)
−0.520180 + 0.854057i \(0.674135\pi\)
\(74\) 0 0
\(75\) −1.22124e15 −0.140858
\(76\) 0 0
\(77\) −7.15916e15 −0.660223
\(78\) 0 0
\(79\) −5.45373e15 −0.404448 −0.202224 0.979339i \(-0.564817\pi\)
−0.202224 + 0.979339i \(0.564817\pi\)
\(80\) 0 0
\(81\) −4.03640e15 −0.242031
\(82\) 0 0
\(83\) 3.64723e15 0.177746 0.0888728 0.996043i \(-0.471674\pi\)
0.0888728 + 0.996043i \(0.471674\pi\)
\(84\) 0 0
\(85\) 2.13633e16 0.850369
\(86\) 0 0
\(87\) −4.09988e16 −1.33924
\(88\) 0 0
\(89\) 6.81792e14 0.0183585 0.00917925 0.999958i \(-0.497078\pi\)
0.00917925 + 0.999958i \(0.497078\pi\)
\(90\) 0 0
\(91\) 3.88687e16 0.866460
\(92\) 0 0
\(93\) −5.85222e14 −0.0108446
\(94\) 0 0
\(95\) −2.69120e14 −0.00416193
\(96\) 0 0
\(97\) −1.20373e17 −1.55944 −0.779720 0.626129i \(-0.784638\pi\)
−0.779720 + 0.626129i \(0.784638\pi\)
\(98\) 0 0
\(99\) −1.82943e16 −0.199258
\(100\) 0 0
\(101\) 7.74092e16 0.711314 0.355657 0.934617i \(-0.384257\pi\)
0.355657 + 0.934617i \(0.384257\pi\)
\(102\) 0 0
\(103\) 1.39309e17 1.08359 0.541793 0.840512i \(-0.317746\pi\)
0.541793 + 0.840512i \(0.317746\pi\)
\(104\) 0 0
\(105\) 7.96270e16 0.525958
\(106\) 0 0
\(107\) −2.75662e17 −1.55101 −0.775505 0.631341i \(-0.782505\pi\)
−0.775505 + 0.631341i \(0.782505\pi\)
\(108\) 0 0
\(109\) 1.37550e17 0.661202 0.330601 0.943771i \(-0.392749\pi\)
0.330601 + 0.943771i \(0.392749\pi\)
\(110\) 0 0
\(111\) 5.45550e16 0.224693
\(112\) 0 0
\(113\) −3.90258e17 −1.38097 −0.690486 0.723345i \(-0.742604\pi\)
−0.690486 + 0.723345i \(0.742604\pi\)
\(114\) 0 0
\(115\) −1.52748e17 −0.465634
\(116\) 0 0
\(117\) 9.93239e16 0.261501
\(118\) 0 0
\(119\) −1.39292e18 −3.17525
\(120\) 0 0
\(121\) −4.26436e17 −0.843681
\(122\) 0 0
\(123\) 4.61313e17 0.793966
\(124\) 0 0
\(125\) 5.96046e16 0.0894427
\(126\) 0 0
\(127\) 7.37824e17 0.967433 0.483717 0.875225i \(-0.339287\pi\)
0.483717 + 0.875225i \(0.339287\pi\)
\(128\) 0 0
\(129\) 6.05933e17 0.695684
\(130\) 0 0
\(131\) 1.45408e18 1.46482 0.732409 0.680865i \(-0.238396\pi\)
0.732409 + 0.680865i \(0.238396\pi\)
\(132\) 0 0
\(133\) 1.75470e16 0.0155405
\(134\) 0 0
\(135\) 6.07217e17 0.473704
\(136\) 0 0
\(137\) −2.27534e18 −1.56647 −0.783236 0.621725i \(-0.786432\pi\)
−0.783236 + 0.621725i \(0.786432\pi\)
\(138\) 0 0
\(139\) −1.41598e17 −0.0861849 −0.0430924 0.999071i \(-0.513721\pi\)
−0.0430924 + 0.999071i \(0.513721\pi\)
\(140\) 0 0
\(141\) −3.68444e17 −0.198614
\(142\) 0 0
\(143\) −4.28969e17 −0.205149
\(144\) 0 0
\(145\) 2.00101e18 0.850396
\(146\) 0 0
\(147\) −3.32994e18 −1.25962
\(148\) 0 0
\(149\) −1.31089e18 −0.442063 −0.221032 0.975267i \(-0.570942\pi\)
−0.221032 + 0.975267i \(0.570942\pi\)
\(150\) 0 0
\(151\) 2.76548e18 0.832658 0.416329 0.909214i \(-0.363316\pi\)
0.416329 + 0.909214i \(0.363316\pi\)
\(152\) 0 0
\(153\) −3.55943e18 −0.958303
\(154\) 0 0
\(155\) 2.85627e16 0.00688619
\(156\) 0 0
\(157\) −3.81235e18 −0.824225 −0.412112 0.911133i \(-0.635209\pi\)
−0.412112 + 0.911133i \(0.635209\pi\)
\(158\) 0 0
\(159\) −5.26842e18 −1.02283
\(160\) 0 0
\(161\) 9.95942e18 1.73866
\(162\) 0 0
\(163\) −1.03773e19 −1.63114 −0.815570 0.578658i \(-0.803577\pi\)
−0.815570 + 0.578658i \(0.803577\pi\)
\(164\) 0 0
\(165\) −8.78791e17 −0.124529
\(166\) 0 0
\(167\) −1.21920e19 −1.55949 −0.779747 0.626095i \(-0.784652\pi\)
−0.779747 + 0.626095i \(0.784652\pi\)
\(168\) 0 0
\(169\) −6.32145e18 −0.730768
\(170\) 0 0
\(171\) 4.48391e16 0.00469019
\(172\) 0 0
\(173\) −3.88210e18 −0.367853 −0.183927 0.982940i \(-0.558881\pi\)
−0.183927 + 0.982940i \(0.558881\pi\)
\(174\) 0 0
\(175\) −3.88632e18 −0.333976
\(176\) 0 0
\(177\) −2.38735e18 −0.186270
\(178\) 0 0
\(179\) 1.62354e19 1.15136 0.575680 0.817675i \(-0.304737\pi\)
0.575680 + 0.817675i \(0.304737\pi\)
\(180\) 0 0
\(181\) −2.32195e19 −1.49825 −0.749127 0.662427i \(-0.769527\pi\)
−0.749127 + 0.662427i \(0.769527\pi\)
\(182\) 0 0
\(183\) −6.80799e18 −0.400115
\(184\) 0 0
\(185\) −2.66264e18 −0.142677
\(186\) 0 0
\(187\) 1.53728e19 0.751793
\(188\) 0 0
\(189\) −3.95915e19 −1.76879
\(190\) 0 0
\(191\) −4.47928e19 −1.82989 −0.914944 0.403580i \(-0.867766\pi\)
−0.914944 + 0.403580i \(0.867766\pi\)
\(192\) 0 0
\(193\) 3.61154e19 1.35038 0.675189 0.737645i \(-0.264062\pi\)
0.675189 + 0.737645i \(0.264062\pi\)
\(194\) 0 0
\(195\) 4.77116e18 0.163429
\(196\) 0 0
\(197\) −1.80503e19 −0.566920 −0.283460 0.958984i \(-0.591482\pi\)
−0.283460 + 0.958984i \(0.591482\pi\)
\(198\) 0 0
\(199\) 6.39636e18 0.184366 0.0921832 0.995742i \(-0.470615\pi\)
0.0921832 + 0.995742i \(0.470615\pi\)
\(200\) 0 0
\(201\) −4.90590e19 −1.29883
\(202\) 0 0
\(203\) −1.30469e20 −3.17535
\(204\) 0 0
\(205\) −2.25151e19 −0.504157
\(206\) 0 0
\(207\) 2.54500e19 0.524734
\(208\) 0 0
\(209\) −1.93655e17 −0.00367948
\(210\) 0 0
\(211\) −7.72101e18 −0.135292 −0.0676462 0.997709i \(-0.521549\pi\)
−0.0676462 + 0.997709i \(0.521549\pi\)
\(212\) 0 0
\(213\) 4.33369e18 0.0700860
\(214\) 0 0
\(215\) −2.95735e19 −0.441749
\(216\) 0 0
\(217\) −1.86233e18 −0.0257128
\(218\) 0 0
\(219\) 5.73733e19 0.732715
\(220\) 0 0
\(221\) −8.34623e19 −0.986634
\(222\) 0 0
\(223\) 1.08588e20 1.18902 0.594510 0.804088i \(-0.297346\pi\)
0.594510 + 0.804088i \(0.297346\pi\)
\(224\) 0 0
\(225\) −9.93097e18 −0.100795
\(226\) 0 0
\(227\) −1.95539e19 −0.184083 −0.0920415 0.995755i \(-0.529339\pi\)
−0.0920415 + 0.995755i \(0.529339\pi\)
\(228\) 0 0
\(229\) 1.20601e20 1.05378 0.526890 0.849933i \(-0.323358\pi\)
0.526890 + 0.849933i \(0.323358\pi\)
\(230\) 0 0
\(231\) 5.72986e19 0.464988
\(232\) 0 0
\(233\) 7.20388e19 0.543302 0.271651 0.962396i \(-0.412430\pi\)
0.271651 + 0.962396i \(0.412430\pi\)
\(234\) 0 0
\(235\) 1.79825e19 0.126117
\(236\) 0 0
\(237\) 4.36491e19 0.284849
\(238\) 0 0
\(239\) −1.70425e20 −1.03550 −0.517750 0.855532i \(-0.673230\pi\)
−0.517750 + 0.855532i \(0.673230\pi\)
\(240\) 0 0
\(241\) −1.71129e20 −0.968679 −0.484340 0.874880i \(-0.660940\pi\)
−0.484340 + 0.874880i \(0.660940\pi\)
\(242\) 0 0
\(243\) −1.68440e20 −0.888775
\(244\) 0 0
\(245\) 1.62523e20 0.799839
\(246\) 0 0
\(247\) 1.05140e18 0.00482885
\(248\) 0 0
\(249\) −2.91907e19 −0.125184
\(250\) 0 0
\(251\) −1.85831e20 −0.744546 −0.372273 0.928123i \(-0.621422\pi\)
−0.372273 + 0.928123i \(0.621422\pi\)
\(252\) 0 0
\(253\) −1.09916e20 −0.411657
\(254\) 0 0
\(255\) −1.70982e20 −0.598906
\(256\) 0 0
\(257\) 1.68273e19 0.0551547 0.0275773 0.999620i \(-0.491221\pi\)
0.0275773 + 0.999620i \(0.491221\pi\)
\(258\) 0 0
\(259\) 1.73609e20 0.532750
\(260\) 0 0
\(261\) −3.33396e20 −0.958333
\(262\) 0 0
\(263\) −2.77268e20 −0.746924 −0.373462 0.927645i \(-0.621829\pi\)
−0.373462 + 0.927645i \(0.621829\pi\)
\(264\) 0 0
\(265\) 2.57133e20 0.649484
\(266\) 0 0
\(267\) −5.45675e18 −0.0129297
\(268\) 0 0
\(269\) 9.40799e19 0.209220 0.104610 0.994513i \(-0.466641\pi\)
0.104610 + 0.994513i \(0.466641\pi\)
\(270\) 0 0
\(271\) 6.17628e20 1.28970 0.644849 0.764310i \(-0.276920\pi\)
0.644849 + 0.764310i \(0.276920\pi\)
\(272\) 0 0
\(273\) −3.11087e20 −0.610238
\(274\) 0 0
\(275\) 4.28907e19 0.0790744
\(276\) 0 0
\(277\) −6.67865e20 −1.15774 −0.578869 0.815420i \(-0.696506\pi\)
−0.578869 + 0.815420i \(0.696506\pi\)
\(278\) 0 0
\(279\) −4.75894e18 −0.00776023
\(280\) 0 0
\(281\) −1.13640e20 −0.174393 −0.0871963 0.996191i \(-0.527791\pi\)
−0.0871963 + 0.996191i \(0.527791\pi\)
\(282\) 0 0
\(283\) 3.76957e20 0.544637 0.272318 0.962207i \(-0.412210\pi\)
0.272318 + 0.962207i \(0.412210\pi\)
\(284\) 0 0
\(285\) 2.15391e18 0.00293121
\(286\) 0 0
\(287\) 1.46802e21 1.88250
\(288\) 0 0
\(289\) 2.16376e21 2.61564
\(290\) 0 0
\(291\) 9.63408e20 1.09830
\(292\) 0 0
\(293\) −3.57350e20 −0.384344 −0.192172 0.981361i \(-0.561553\pi\)
−0.192172 + 0.981361i \(0.561553\pi\)
\(294\) 0 0
\(295\) 1.16518e20 0.118279
\(296\) 0 0
\(297\) 4.36946e20 0.418792
\(298\) 0 0
\(299\) 5.96757e20 0.540247
\(300\) 0 0
\(301\) 1.92824e21 1.64947
\(302\) 0 0
\(303\) −6.19547e20 −0.500971
\(304\) 0 0
\(305\) 3.32275e20 0.254067
\(306\) 0 0
\(307\) −2.25658e21 −1.63220 −0.816102 0.577908i \(-0.803869\pi\)
−0.816102 + 0.577908i \(0.803869\pi\)
\(308\) 0 0
\(309\) −1.11497e21 −0.763158
\(310\) 0 0
\(311\) −5.76844e20 −0.373762 −0.186881 0.982383i \(-0.559838\pi\)
−0.186881 + 0.982383i \(0.559838\pi\)
\(312\) 0 0
\(313\) 2.63218e21 1.61506 0.807530 0.589827i \(-0.200804\pi\)
0.807530 + 0.589827i \(0.200804\pi\)
\(314\) 0 0
\(315\) 6.47515e20 0.376366
\(316\) 0 0
\(317\) −9.95998e20 −0.548599 −0.274299 0.961644i \(-0.588446\pi\)
−0.274299 + 0.961644i \(0.588446\pi\)
\(318\) 0 0
\(319\) 1.43990e21 0.751817
\(320\) 0 0
\(321\) 2.20627e21 1.09236
\(322\) 0 0
\(323\) −3.76786e19 −0.0176959
\(324\) 0 0
\(325\) −2.32864e20 −0.103775
\(326\) 0 0
\(327\) −1.10088e21 −0.465678
\(328\) 0 0
\(329\) −1.17249e21 −0.470915
\(330\) 0 0
\(331\) 3.56520e21 1.36002 0.680011 0.733202i \(-0.261975\pi\)
0.680011 + 0.733202i \(0.261975\pi\)
\(332\) 0 0
\(333\) 4.43634e20 0.160786
\(334\) 0 0
\(335\) 2.39440e21 0.824738
\(336\) 0 0
\(337\) 3.65283e20 0.119612 0.0598060 0.998210i \(-0.480952\pi\)
0.0598060 + 0.998210i \(0.480952\pi\)
\(338\) 0 0
\(339\) 3.12344e21 0.972605
\(340\) 0 0
\(341\) 2.05533e19 0.00608793
\(342\) 0 0
\(343\) −4.67180e21 −1.31669
\(344\) 0 0
\(345\) 1.22252e21 0.327941
\(346\) 0 0
\(347\) 2.06934e21 0.528483 0.264242 0.964457i \(-0.414878\pi\)
0.264242 + 0.964457i \(0.414878\pi\)
\(348\) 0 0
\(349\) 3.13803e20 0.0763204 0.0381602 0.999272i \(-0.487850\pi\)
0.0381602 + 0.999272i \(0.487850\pi\)
\(350\) 0 0
\(351\) −2.37228e21 −0.549611
\(352\) 0 0
\(353\) −6.87901e21 −1.51859 −0.759296 0.650746i \(-0.774456\pi\)
−0.759296 + 0.650746i \(0.774456\pi\)
\(354\) 0 0
\(355\) −2.11513e20 −0.0445036
\(356\) 0 0
\(357\) 1.11483e22 2.23629
\(358\) 0 0
\(359\) 9.50907e21 1.81901 0.909505 0.415693i \(-0.136461\pi\)
0.909505 + 0.415693i \(0.136461\pi\)
\(360\) 0 0
\(361\) −5.47991e21 −0.999913
\(362\) 0 0
\(363\) 3.41300e21 0.594196
\(364\) 0 0
\(365\) −2.80019e21 −0.465263
\(366\) 0 0
\(367\) −1.88008e20 −0.0298205 −0.0149103 0.999889i \(-0.504746\pi\)
−0.0149103 + 0.999889i \(0.504746\pi\)
\(368\) 0 0
\(369\) 3.75133e21 0.568147
\(370\) 0 0
\(371\) −1.67655e22 −2.42515
\(372\) 0 0
\(373\) 9.66745e20 0.133594 0.0667970 0.997767i \(-0.478722\pi\)
0.0667970 + 0.997767i \(0.478722\pi\)
\(374\) 0 0
\(375\) −4.77048e20 −0.0629936
\(376\) 0 0
\(377\) −7.81756e21 −0.986665
\(378\) 0 0
\(379\) −1.09478e22 −1.32097 −0.660484 0.750840i \(-0.729649\pi\)
−0.660484 + 0.750840i \(0.729649\pi\)
\(380\) 0 0
\(381\) −5.90520e21 −0.681353
\(382\) 0 0
\(383\) 9.48135e21 1.04636 0.523179 0.852223i \(-0.324746\pi\)
0.523179 + 0.852223i \(0.324746\pi\)
\(384\) 0 0
\(385\) −2.79655e21 −0.295261
\(386\) 0 0
\(387\) 4.92736e21 0.497818
\(388\) 0 0
\(389\) 8.68048e21 0.839406 0.419703 0.907661i \(-0.362134\pi\)
0.419703 + 0.907661i \(0.362134\pi\)
\(390\) 0 0
\(391\) −2.13857e22 −1.97980
\(392\) 0 0
\(393\) −1.16378e22 −1.03166
\(394\) 0 0
\(395\) −2.13036e21 −0.180875
\(396\) 0 0
\(397\) −5.61938e21 −0.457055 −0.228528 0.973537i \(-0.573391\pi\)
−0.228528 + 0.973537i \(0.573391\pi\)
\(398\) 0 0
\(399\) −1.40438e20 −0.0109450
\(400\) 0 0
\(401\) −4.97857e21 −0.371858 −0.185929 0.982563i \(-0.559530\pi\)
−0.185929 + 0.982563i \(0.559530\pi\)
\(402\) 0 0
\(403\) −1.11589e20 −0.00798965
\(404\) 0 0
\(405\) −1.57672e21 −0.108240
\(406\) 0 0
\(407\) −1.91600e21 −0.126137
\(408\) 0 0
\(409\) 2.65693e20 0.0167777 0.00838883 0.999965i \(-0.497330\pi\)
0.00838883 + 0.999965i \(0.497330\pi\)
\(410\) 0 0
\(411\) 1.82108e22 1.10325
\(412\) 0 0
\(413\) −7.59717e21 −0.441648
\(414\) 0 0
\(415\) 1.42470e21 0.0794903
\(416\) 0 0
\(417\) 1.13328e21 0.0606991
\(418\) 0 0
\(419\) −1.71242e22 −0.880626 −0.440313 0.897844i \(-0.645133\pi\)
−0.440313 + 0.897844i \(0.645133\pi\)
\(420\) 0 0
\(421\) 2.71839e22 1.34250 0.671249 0.741231i \(-0.265758\pi\)
0.671249 + 0.741231i \(0.265758\pi\)
\(422\) 0 0
\(423\) −2.99613e21 −0.142124
\(424\) 0 0
\(425\) 8.34504e21 0.380297
\(426\) 0 0
\(427\) −2.16648e22 −0.948677
\(428\) 0 0
\(429\) 3.43327e21 0.144484
\(430\) 0 0
\(431\) 2.34759e22 0.949656 0.474828 0.880079i \(-0.342510\pi\)
0.474828 + 0.880079i \(0.342510\pi\)
\(432\) 0 0
\(433\) −1.88732e22 −0.734003 −0.367002 0.930220i \(-0.619616\pi\)
−0.367002 + 0.930220i \(0.619616\pi\)
\(434\) 0 0
\(435\) −1.60152e22 −0.598925
\(436\) 0 0
\(437\) 2.69402e20 0.00968968
\(438\) 0 0
\(439\) −5.51895e22 −1.90945 −0.954724 0.297494i \(-0.903849\pi\)
−0.954724 + 0.297494i \(0.903849\pi\)
\(440\) 0 0
\(441\) −2.70786e22 −0.901359
\(442\) 0 0
\(443\) 1.20735e22 0.386723 0.193362 0.981128i \(-0.438061\pi\)
0.193362 + 0.981128i \(0.438061\pi\)
\(444\) 0 0
\(445\) 2.66325e20 0.00821017
\(446\) 0 0
\(447\) 1.04918e22 0.311341
\(448\) 0 0
\(449\) 2.57466e22 0.735573 0.367787 0.929910i \(-0.380116\pi\)
0.367787 + 0.929910i \(0.380116\pi\)
\(450\) 0 0
\(451\) −1.62016e22 −0.445714
\(452\) 0 0
\(453\) −2.21336e22 −0.586432
\(454\) 0 0
\(455\) 1.51831e22 0.387493
\(456\) 0 0
\(457\) −1.01069e22 −0.248503 −0.124252 0.992251i \(-0.539653\pi\)
−0.124252 + 0.992251i \(0.539653\pi\)
\(458\) 0 0
\(459\) 8.50144e22 2.01412
\(460\) 0 0
\(461\) 1.68599e21 0.0384942 0.0192471 0.999815i \(-0.493873\pi\)
0.0192471 + 0.999815i \(0.493873\pi\)
\(462\) 0 0
\(463\) 2.77945e22 0.611674 0.305837 0.952084i \(-0.401064\pi\)
0.305837 + 0.952084i \(0.401064\pi\)
\(464\) 0 0
\(465\) −2.28602e20 −0.00484987
\(466\) 0 0
\(467\) −8.26643e22 −1.69093 −0.845463 0.534034i \(-0.820676\pi\)
−0.845463 + 0.534034i \(0.820676\pi\)
\(468\) 0 0
\(469\) −1.56119e23 −3.07954
\(470\) 0 0
\(471\) 3.05123e22 0.580493
\(472\) 0 0
\(473\) −2.12807e22 −0.390541
\(474\) 0 0
\(475\) −1.05125e20 −0.00186127
\(476\) 0 0
\(477\) −4.28420e22 −0.731920
\(478\) 0 0
\(479\) −2.59191e22 −0.427335 −0.213667 0.976906i \(-0.568541\pi\)
−0.213667 + 0.976906i \(0.568541\pi\)
\(480\) 0 0
\(481\) 1.04024e22 0.165540
\(482\) 0 0
\(483\) −7.97106e22 −1.22452
\(484\) 0 0
\(485\) −4.70206e22 −0.697403
\(486\) 0 0
\(487\) 4.14828e22 0.594116 0.297058 0.954859i \(-0.403994\pi\)
0.297058 + 0.954859i \(0.403994\pi\)
\(488\) 0 0
\(489\) 8.30554e22 1.14879
\(490\) 0 0
\(491\) −3.08644e22 −0.412349 −0.206174 0.978515i \(-0.566101\pi\)
−0.206174 + 0.978515i \(0.566101\pi\)
\(492\) 0 0
\(493\) 2.80155e23 3.61575
\(494\) 0 0
\(495\) −7.14620e21 −0.0891109
\(496\) 0 0
\(497\) 1.37910e22 0.166175
\(498\) 0 0
\(499\) −3.24182e22 −0.377516 −0.188758 0.982024i \(-0.560446\pi\)
−0.188758 + 0.982024i \(0.560446\pi\)
\(500\) 0 0
\(501\) 9.75788e22 1.09833
\(502\) 0 0
\(503\) −6.39324e22 −0.695654 −0.347827 0.937559i \(-0.613080\pi\)
−0.347827 + 0.937559i \(0.613080\pi\)
\(504\) 0 0
\(505\) 3.02380e22 0.318109
\(506\) 0 0
\(507\) 5.05939e22 0.514672
\(508\) 0 0
\(509\) −9.33355e22 −0.918218 −0.459109 0.888380i \(-0.651831\pi\)
−0.459109 + 0.888380i \(0.651831\pi\)
\(510\) 0 0
\(511\) 1.82577e23 1.73728
\(512\) 0 0
\(513\) −1.07095e21 −0.00985763
\(514\) 0 0
\(515\) 5.44176e22 0.484594
\(516\) 0 0
\(517\) 1.29400e22 0.111497
\(518\) 0 0
\(519\) 3.10705e22 0.259075
\(520\) 0 0
\(521\) 3.30201e22 0.266476 0.133238 0.991084i \(-0.457463\pi\)
0.133238 + 0.991084i \(0.457463\pi\)
\(522\) 0 0
\(523\) −4.06002e22 −0.317149 −0.158575 0.987347i \(-0.550690\pi\)
−0.158575 + 0.987347i \(0.550690\pi\)
\(524\) 0 0
\(525\) 3.11043e22 0.235216
\(526\) 0 0
\(527\) 3.99896e21 0.0292790
\(528\) 0 0
\(529\) 1.18585e22 0.0840729
\(530\) 0 0
\(531\) −1.94136e22 −0.133291
\(532\) 0 0
\(533\) 8.79621e22 0.584944
\(534\) 0 0
\(535\) −1.07680e23 −0.693633
\(536\) 0 0
\(537\) −1.29940e23 −0.810890
\(538\) 0 0
\(539\) 1.16949e23 0.707121
\(540\) 0 0
\(541\) −4.20460e22 −0.246347 −0.123174 0.992385i \(-0.539307\pi\)
−0.123174 + 0.992385i \(0.539307\pi\)
\(542\) 0 0
\(543\) 1.85838e23 1.05520
\(544\) 0 0
\(545\) 5.37303e22 0.295699
\(546\) 0 0
\(547\) 2.01110e23 1.07285 0.536427 0.843947i \(-0.319774\pi\)
0.536427 + 0.843947i \(0.319774\pi\)
\(548\) 0 0
\(549\) −5.53616e22 −0.286315
\(550\) 0 0
\(551\) −3.52919e21 −0.0176965
\(552\) 0 0
\(553\) 1.38903e23 0.675380
\(554\) 0 0
\(555\) 2.13106e22 0.100486
\(556\) 0 0
\(557\) 1.46727e23 0.671030 0.335515 0.942035i \(-0.391090\pi\)
0.335515 + 0.942035i \(0.391090\pi\)
\(558\) 0 0
\(559\) 1.15538e23 0.512536
\(560\) 0 0
\(561\) −1.23036e23 −0.529480
\(562\) 0 0
\(563\) 3.39706e23 1.41834 0.709172 0.705036i \(-0.249069\pi\)
0.709172 + 0.705036i \(0.249069\pi\)
\(564\) 0 0
\(565\) −1.52445e23 −0.617590
\(566\) 0 0
\(567\) 1.02805e23 0.404163
\(568\) 0 0
\(569\) 2.72602e23 1.04010 0.520050 0.854136i \(-0.325913\pi\)
0.520050 + 0.854136i \(0.325913\pi\)
\(570\) 0 0
\(571\) 2.79882e22 0.103650 0.0518249 0.998656i \(-0.483496\pi\)
0.0518249 + 0.998656i \(0.483496\pi\)
\(572\) 0 0
\(573\) 3.58501e23 1.28877
\(574\) 0 0
\(575\) −5.96672e22 −0.208238
\(576\) 0 0
\(577\) 3.86678e23 1.31025 0.655127 0.755519i \(-0.272615\pi\)
0.655127 + 0.755519i \(0.272615\pi\)
\(578\) 0 0
\(579\) −2.89051e23 −0.951056
\(580\) 0 0
\(581\) −9.28926e22 −0.296814
\(582\) 0 0
\(583\) 1.85030e23 0.574195
\(584\) 0 0
\(585\) 3.87984e22 0.116947
\(586\) 0 0
\(587\) −3.64152e23 −1.06625 −0.533125 0.846036i \(-0.678983\pi\)
−0.533125 + 0.846036i \(0.678983\pi\)
\(588\) 0 0
\(589\) −5.03761e19 −0.000143299 0
\(590\) 0 0
\(591\) 1.44466e23 0.399276
\(592\) 0 0
\(593\) −6.86689e23 −1.84415 −0.922073 0.387017i \(-0.873506\pi\)
−0.922073 + 0.387017i \(0.873506\pi\)
\(594\) 0 0
\(595\) −5.44110e23 −1.42001
\(596\) 0 0
\(597\) −5.11935e22 −0.129847
\(598\) 0 0
\(599\) 3.82850e23 0.943846 0.471923 0.881640i \(-0.343560\pi\)
0.471923 + 0.881640i \(0.343560\pi\)
\(600\) 0 0
\(601\) −2.66402e23 −0.638417 −0.319209 0.947684i \(-0.603417\pi\)
−0.319209 + 0.947684i \(0.603417\pi\)
\(602\) 0 0
\(603\) −3.98941e23 −0.929418
\(604\) 0 0
\(605\) −1.66577e23 −0.377306
\(606\) 0 0
\(607\) 1.11277e23 0.245076 0.122538 0.992464i \(-0.460897\pi\)
0.122538 + 0.992464i \(0.460897\pi\)
\(608\) 0 0
\(609\) 1.04421e24 2.23636
\(610\) 0 0
\(611\) −7.02541e22 −0.146326
\(612\) 0 0
\(613\) −3.91229e23 −0.792532 −0.396266 0.918136i \(-0.629694\pi\)
−0.396266 + 0.918136i \(0.629694\pi\)
\(614\) 0 0
\(615\) 1.80200e23 0.355072
\(616\) 0 0
\(617\) −5.72626e23 −1.09761 −0.548804 0.835951i \(-0.684917\pi\)
−0.548804 + 0.835951i \(0.684917\pi\)
\(618\) 0 0
\(619\) −7.86155e23 −1.46601 −0.733007 0.680222i \(-0.761884\pi\)
−0.733007 + 0.680222i \(0.761884\pi\)
\(620\) 0 0
\(621\) −6.07855e23 −1.10286
\(622\) 0 0
\(623\) −1.73648e22 −0.0306565
\(624\) 0 0
\(625\) 2.32831e22 0.0400000
\(626\) 0 0
\(627\) 1.54993e21 0.00259142
\(628\) 0 0
\(629\) −3.72788e23 −0.606640
\(630\) 0 0
\(631\) −6.52712e23 −1.03389 −0.516943 0.856020i \(-0.672930\pi\)
−0.516943 + 0.856020i \(0.672930\pi\)
\(632\) 0 0
\(633\) 6.17954e22 0.0952850
\(634\) 0 0
\(635\) 2.88212e23 0.432649
\(636\) 0 0
\(637\) −6.34945e23 −0.928007
\(638\) 0 0
\(639\) 3.52410e22 0.0501522
\(640\) 0 0
\(641\) 7.10734e23 0.984950 0.492475 0.870327i \(-0.336092\pi\)
0.492475 + 0.870327i \(0.336092\pi\)
\(642\) 0 0
\(643\) −2.34278e23 −0.316183 −0.158092 0.987424i \(-0.550534\pi\)
−0.158092 + 0.987424i \(0.550534\pi\)
\(644\) 0 0
\(645\) 2.36692e23 0.311119
\(646\) 0 0
\(647\) −1.50806e23 −0.193078 −0.0965391 0.995329i \(-0.530777\pi\)
−0.0965391 + 0.995329i \(0.530777\pi\)
\(648\) 0 0
\(649\) 8.38450e22 0.104568
\(650\) 0 0
\(651\) 1.49052e22 0.0181092
\(652\) 0 0
\(653\) 6.74375e22 0.0798251 0.0399126 0.999203i \(-0.487292\pi\)
0.0399126 + 0.999203i \(0.487292\pi\)
\(654\) 0 0
\(655\) 5.68001e23 0.655086
\(656\) 0 0
\(657\) 4.66551e23 0.524317
\(658\) 0 0
\(659\) −7.11301e22 −0.0778981 −0.0389491 0.999241i \(-0.512401\pi\)
−0.0389491 + 0.999241i \(0.512401\pi\)
\(660\) 0 0
\(661\) 1.69805e24 1.81233 0.906164 0.422926i \(-0.138997\pi\)
0.906164 + 0.422926i \(0.138997\pi\)
\(662\) 0 0
\(663\) 6.67994e23 0.694876
\(664\) 0 0
\(665\) 6.85431e21 0.00694992
\(666\) 0 0
\(667\) −2.00311e24 −1.97986
\(668\) 0 0
\(669\) −8.69085e23 −0.837414
\(670\) 0 0
\(671\) 2.39101e23 0.224615
\(672\) 0 0
\(673\) −1.39848e24 −1.28094 −0.640469 0.767984i \(-0.721260\pi\)
−0.640469 + 0.767984i \(0.721260\pi\)
\(674\) 0 0
\(675\) 2.37194e23 0.211847
\(676\) 0 0
\(677\) −9.38639e23 −0.817514 −0.408757 0.912643i \(-0.634038\pi\)
−0.408757 + 0.912643i \(0.634038\pi\)
\(678\) 0 0
\(679\) 3.06582e24 2.60408
\(680\) 0 0
\(681\) 1.56500e23 0.129648
\(682\) 0 0
\(683\) −1.23254e24 −0.995922 −0.497961 0.867199i \(-0.665918\pi\)
−0.497961 + 0.867199i \(0.665918\pi\)
\(684\) 0 0
\(685\) −8.88807e23 −0.700547
\(686\) 0 0
\(687\) −9.65236e23 −0.742167
\(688\) 0 0
\(689\) −1.00457e24 −0.753559
\(690\) 0 0
\(691\) −3.83055e23 −0.280348 −0.140174 0.990127i \(-0.544766\pi\)
−0.140174 + 0.990127i \(0.544766\pi\)
\(692\) 0 0
\(693\) 4.65944e23 0.332737
\(694\) 0 0
\(695\) −5.53117e22 −0.0385431
\(696\) 0 0
\(697\) −3.15226e24 −2.14360
\(698\) 0 0
\(699\) −5.76565e23 −0.382642
\(700\) 0 0
\(701\) 1.63235e24 1.05733 0.528665 0.848831i \(-0.322693\pi\)
0.528665 + 0.848831i \(0.322693\pi\)
\(702\) 0 0
\(703\) 4.69611e21 0.00296906
\(704\) 0 0
\(705\) −1.43923e23 −0.0888227
\(706\) 0 0
\(707\) −1.97156e24 −1.18781
\(708\) 0 0
\(709\) −6.09431e23 −0.358453 −0.179226 0.983808i \(-0.557359\pi\)
−0.179226 + 0.983808i \(0.557359\pi\)
\(710\) 0 0
\(711\) 3.54948e23 0.203832
\(712\) 0 0
\(713\) −2.85926e22 −0.0160322
\(714\) 0 0
\(715\) −1.67566e23 −0.0917454
\(716\) 0 0
\(717\) 1.36400e24 0.729292
\(718\) 0 0
\(719\) 4.49148e23 0.234527 0.117264 0.993101i \(-0.462588\pi\)
0.117264 + 0.993101i \(0.462588\pi\)
\(720\) 0 0
\(721\) −3.54812e24 −1.80946
\(722\) 0 0
\(723\) 1.36964e24 0.682230
\(724\) 0 0
\(725\) 7.81644e23 0.380309
\(726\) 0 0
\(727\) 5.00713e23 0.237983 0.118992 0.992895i \(-0.462034\pi\)
0.118992 + 0.992895i \(0.462034\pi\)
\(728\) 0 0
\(729\) 1.86938e24 0.867986
\(730\) 0 0
\(731\) −4.14048e24 −1.87825
\(732\) 0 0
\(733\) 1.20329e24 0.533320 0.266660 0.963791i \(-0.414080\pi\)
0.266660 + 0.963791i \(0.414080\pi\)
\(734\) 0 0
\(735\) −1.30076e24 −0.563318
\(736\) 0 0
\(737\) 1.72298e24 0.729133
\(738\) 0 0
\(739\) 2.19426e24 0.907426 0.453713 0.891148i \(-0.350099\pi\)
0.453713 + 0.891148i \(0.350099\pi\)
\(740\) 0 0
\(741\) −8.41491e21 −0.00340091
\(742\) 0 0
\(743\) 9.89633e22 0.0390903 0.0195452 0.999809i \(-0.493778\pi\)
0.0195452 + 0.999809i \(0.493778\pi\)
\(744\) 0 0
\(745\) −5.12068e23 −0.197697
\(746\) 0 0
\(747\) −2.37375e23 −0.0895796
\(748\) 0 0
\(749\) 7.02094e24 2.59000
\(750\) 0 0
\(751\) 1.01856e24 0.367322 0.183661 0.982990i \(-0.441205\pi\)
0.183661 + 0.982990i \(0.441205\pi\)
\(752\) 0 0
\(753\) 1.48731e24 0.524376
\(754\) 0 0
\(755\) 1.08027e24 0.372376
\(756\) 0 0
\(757\) −1.43989e24 −0.485305 −0.242653 0.970113i \(-0.578018\pi\)
−0.242653 + 0.970113i \(0.578018\pi\)
\(758\) 0 0
\(759\) 8.79713e23 0.289925
\(760\) 0 0
\(761\) −7.15424e23 −0.230565 −0.115283 0.993333i \(-0.536777\pi\)
−0.115283 + 0.993333i \(0.536777\pi\)
\(762\) 0 0
\(763\) −3.50330e24 −1.10413
\(764\) 0 0
\(765\) −1.39040e24 −0.428566
\(766\) 0 0
\(767\) −4.55214e23 −0.137232
\(768\) 0 0
\(769\) 1.87177e24 0.551922 0.275961 0.961169i \(-0.411004\pi\)
0.275961 + 0.961169i \(0.411004\pi\)
\(770\) 0 0
\(771\) −1.34678e23 −0.0388449
\(772\) 0 0
\(773\) −1.86999e24 −0.527611 −0.263805 0.964576i \(-0.584978\pi\)
−0.263805 + 0.964576i \(0.584978\pi\)
\(774\) 0 0
\(775\) 1.11573e22 0.00307960
\(776\) 0 0
\(777\) −1.38948e24 −0.375210
\(778\) 0 0
\(779\) 3.97099e22 0.0104913
\(780\) 0 0
\(781\) −1.52202e23 −0.0393447
\(782\) 0 0
\(783\) 7.96294e24 2.01418
\(784\) 0 0
\(785\) −1.48920e24 −0.368604
\(786\) 0 0
\(787\) 3.29304e24 0.797648 0.398824 0.917027i \(-0.369418\pi\)
0.398824 + 0.917027i \(0.369418\pi\)
\(788\) 0 0
\(789\) 2.21913e24 0.526051
\(790\) 0 0
\(791\) 9.93963e24 2.30606
\(792\) 0 0
\(793\) −1.29813e24 −0.294779
\(794\) 0 0
\(795\) −2.05798e24 −0.457425
\(796\) 0 0
\(797\) −3.80089e24 −0.826969 −0.413485 0.910511i \(-0.635688\pi\)
−0.413485 + 0.910511i \(0.635688\pi\)
\(798\) 0 0
\(799\) 2.51767e24 0.536229
\(800\) 0 0
\(801\) −4.43735e22 −0.00925225
\(802\) 0 0
\(803\) −2.01498e24 −0.411329
\(804\) 0 0
\(805\) 3.89040e24 0.777552
\(806\) 0 0
\(807\) −7.52972e23 −0.147351
\(808\) 0 0
\(809\) 1.38713e24 0.265799 0.132900 0.991129i \(-0.457571\pi\)
0.132900 + 0.991129i \(0.457571\pi\)
\(810\) 0 0
\(811\) −2.43418e24 −0.456747 −0.228374 0.973574i \(-0.573341\pi\)
−0.228374 + 0.973574i \(0.573341\pi\)
\(812\) 0 0
\(813\) −4.94321e24 −0.908321
\(814\) 0 0
\(815\) −4.05365e24 −0.729468
\(816\) 0 0
\(817\) 5.21589e22 0.00919265
\(818\) 0 0
\(819\) −2.52972e24 −0.436675
\(820\) 0 0
\(821\) −7.07675e24 −1.19651 −0.598256 0.801305i \(-0.704140\pi\)
−0.598256 + 0.801305i \(0.704140\pi\)
\(822\) 0 0
\(823\) 5.31425e24 0.880124 0.440062 0.897968i \(-0.354957\pi\)
0.440062 + 0.897968i \(0.354957\pi\)
\(824\) 0 0
\(825\) −3.43278e23 −0.0556912
\(826\) 0 0
\(827\) −5.13487e24 −0.816080 −0.408040 0.912964i \(-0.633788\pi\)
−0.408040 + 0.912964i \(0.633788\pi\)
\(828\) 0 0
\(829\) 1.06738e25 1.66191 0.830954 0.556341i \(-0.187795\pi\)
0.830954 + 0.556341i \(0.187795\pi\)
\(830\) 0 0
\(831\) 5.34528e24 0.815383
\(832\) 0 0
\(833\) 2.27543e25 3.40079
\(834\) 0 0
\(835\) −4.76248e24 −0.697427
\(836\) 0 0
\(837\) 1.13664e23 0.0163101
\(838\) 0 0
\(839\) 6.47302e24 0.910186 0.455093 0.890444i \(-0.349606\pi\)
0.455093 + 0.890444i \(0.349606\pi\)
\(840\) 0 0
\(841\) 1.89837e25 2.61587
\(842\) 0 0
\(843\) 9.09523e23 0.122823
\(844\) 0 0
\(845\) −2.46931e24 −0.326809
\(846\) 0 0
\(847\) 1.08611e25 1.40885
\(848\) 0 0
\(849\) −3.01699e24 −0.383582
\(850\) 0 0
\(851\) 2.66544e24 0.332175
\(852\) 0 0
\(853\) 2.69306e24 0.328988 0.164494 0.986378i \(-0.447401\pi\)
0.164494 + 0.986378i \(0.447401\pi\)
\(854\) 0 0
\(855\) 1.75153e22 0.00209752
\(856\) 0 0
\(857\) −1.86785e24 −0.219283 −0.109641 0.993971i \(-0.534970\pi\)
−0.109641 + 0.993971i \(0.534970\pi\)
\(858\) 0 0
\(859\) 1.13124e25 1.30200 0.651000 0.759077i \(-0.274350\pi\)
0.651000 + 0.759077i \(0.274350\pi\)
\(860\) 0 0
\(861\) −1.17493e25 −1.32583
\(862\) 0 0
\(863\) 2.37061e24 0.262282 0.131141 0.991364i \(-0.458136\pi\)
0.131141 + 0.991364i \(0.458136\pi\)
\(864\) 0 0
\(865\) −1.51645e24 −0.164509
\(866\) 0 0
\(867\) −1.73178e25 −1.84217
\(868\) 0 0
\(869\) −1.53298e24 −0.159907
\(870\) 0 0
\(871\) −9.35446e24 −0.956895
\(872\) 0 0
\(873\) 7.83429e24 0.785921
\(874\) 0 0
\(875\) −1.51809e24 −0.149358
\(876\) 0 0
\(877\) −6.79534e24 −0.655714 −0.327857 0.944727i \(-0.606326\pi\)
−0.327857 + 0.944727i \(0.606326\pi\)
\(878\) 0 0
\(879\) 2.86007e24 0.270689
\(880\) 0 0
\(881\) −1.88402e25 −1.74900 −0.874501 0.485024i \(-0.838811\pi\)
−0.874501 + 0.485024i \(0.838811\pi\)
\(882\) 0 0
\(883\) −3.85491e24 −0.351033 −0.175517 0.984476i \(-0.556160\pi\)
−0.175517 + 0.984476i \(0.556160\pi\)
\(884\) 0 0
\(885\) −9.32557e23 −0.0833024
\(886\) 0 0
\(887\) −8.32727e24 −0.729712 −0.364856 0.931064i \(-0.618882\pi\)
−0.364856 + 0.931064i \(0.618882\pi\)
\(888\) 0 0
\(889\) −1.87919e25 −1.61550
\(890\) 0 0
\(891\) −1.13459e24 −0.0956924
\(892\) 0 0
\(893\) −3.17158e22 −0.00262445
\(894\) 0 0
\(895\) 6.34194e24 0.514904
\(896\) 0 0
\(897\) −4.77617e24 −0.380491
\(898\) 0 0
\(899\) 3.74565e23 0.0292799
\(900\) 0 0
\(901\) 3.60004e25 2.76151
\(902\) 0 0
\(903\) −1.54327e25 −1.16171
\(904\) 0 0
\(905\) −9.07012e24 −0.670039
\(906\) 0 0
\(907\) −1.92435e25 −1.39515 −0.697576 0.716511i \(-0.745738\pi\)
−0.697576 + 0.716511i \(0.745738\pi\)
\(908\) 0 0
\(909\) −5.03807e24 −0.358485
\(910\) 0 0
\(911\) 1.57682e25 1.10123 0.550613 0.834761i \(-0.314394\pi\)
0.550613 + 0.834761i \(0.314394\pi\)
\(912\) 0 0
\(913\) 1.02519e24 0.0702756
\(914\) 0 0
\(915\) −2.65937e24 −0.178937
\(916\) 0 0
\(917\) −3.70346e25 −2.44607
\(918\) 0 0
\(919\) −1.03058e25 −0.668189 −0.334094 0.942540i \(-0.608430\pi\)
−0.334094 + 0.942540i \(0.608430\pi\)
\(920\) 0 0
\(921\) 1.80606e25 1.14954
\(922\) 0 0
\(923\) 8.26339e23 0.0516349
\(924\) 0 0
\(925\) −1.04010e24 −0.0638070
\(926\) 0 0
\(927\) −9.06674e24 −0.546102
\(928\) 0 0
\(929\) 9.73688e24 0.575819 0.287910 0.957658i \(-0.407040\pi\)
0.287910 + 0.957658i \(0.407040\pi\)
\(930\) 0 0
\(931\) −2.86642e23 −0.0166444
\(932\) 0 0
\(933\) 4.61679e24 0.263236
\(934\) 0 0
\(935\) 6.00499e24 0.336212
\(936\) 0 0
\(937\) −3.22145e25 −1.77119 −0.885594 0.464459i \(-0.846249\pi\)
−0.885594 + 0.464459i \(0.846249\pi\)
\(938\) 0 0
\(939\) −2.10667e25 −1.13747
\(940\) 0 0
\(941\) −7.90374e24 −0.419103 −0.209552 0.977798i \(-0.567200\pi\)
−0.209552 + 0.977798i \(0.567200\pi\)
\(942\) 0 0
\(943\) 2.25387e25 1.17376
\(944\) 0 0
\(945\) −1.54654e25 −0.791028
\(946\) 0 0
\(947\) −2.36682e25 −1.18902 −0.594512 0.804086i \(-0.702655\pi\)
−0.594512 + 0.804086i \(0.702655\pi\)
\(948\) 0 0
\(949\) 1.09398e25 0.539818
\(950\) 0 0
\(951\) 7.97150e24 0.386372
\(952\) 0 0
\(953\) 9.09477e24 0.433014 0.216507 0.976281i \(-0.430534\pi\)
0.216507 + 0.976281i \(0.430534\pi\)
\(954\) 0 0
\(955\) −1.74972e25 −0.818351
\(956\) 0 0
\(957\) −1.15243e25 −0.529496
\(958\) 0 0
\(959\) 5.79516e25 2.61582
\(960\) 0 0
\(961\) −2.25448e25 −0.999763
\(962\) 0 0
\(963\) 1.79411e25 0.781673
\(964\) 0 0
\(965\) 1.41076e25 0.603907
\(966\) 0 0
\(967\) −3.62970e25 −1.52667 −0.763337 0.646001i \(-0.776440\pi\)
−0.763337 + 0.646001i \(0.776440\pi\)
\(968\) 0 0
\(969\) 3.01562e23 0.0124630
\(970\) 0 0
\(971\) 9.70396e24 0.394081 0.197040 0.980395i \(-0.436867\pi\)
0.197040 + 0.980395i \(0.436867\pi\)
\(972\) 0 0
\(973\) 3.60641e24 0.143918
\(974\) 0 0
\(975\) 1.86373e24 0.0730878
\(976\) 0 0
\(977\) 3.70141e25 1.42647 0.713236 0.700924i \(-0.247229\pi\)
0.713236 + 0.700924i \(0.247229\pi\)
\(978\) 0 0
\(979\) 1.91644e23 0.00725843
\(980\) 0 0
\(981\) −8.95223e24 −0.333230
\(982\) 0 0
\(983\) 6.43184e24 0.235305 0.117652 0.993055i \(-0.462463\pi\)
0.117652 + 0.993055i \(0.462463\pi\)
\(984\) 0 0
\(985\) −7.05090e24 −0.253534
\(986\) 0 0
\(987\) 9.38403e24 0.331661
\(988\) 0 0
\(989\) 2.96045e25 1.02847
\(990\) 0 0
\(991\) −5.17070e25 −1.76573 −0.882863 0.469631i \(-0.844387\pi\)
−0.882863 + 0.469631i \(0.844387\pi\)
\(992\) 0 0
\(993\) −2.85342e25 −0.957848
\(994\) 0 0
\(995\) 2.49858e24 0.0824512
\(996\) 0 0
\(997\) −2.77457e25 −0.900093 −0.450047 0.893005i \(-0.648593\pi\)
−0.450047 + 0.893005i \(0.648593\pi\)
\(998\) 0 0
\(999\) −1.05959e25 −0.337933
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 80.18.a.d.1.1 2
4.3 odd 2 10.18.a.c.1.2 2
12.11 even 2 90.18.a.k.1.2 2
20.3 even 4 50.18.b.d.49.4 4
20.7 even 4 50.18.b.d.49.1 4
20.19 odd 2 50.18.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.18.a.c.1.2 2 4.3 odd 2
50.18.a.f.1.1 2 20.19 odd 2
50.18.b.d.49.1 4 20.7 even 4
50.18.b.d.49.4 4 20.3 even 4
80.18.a.d.1.1 2 1.1 even 1 trivial
90.18.a.k.1.2 2 12.11 even 2