Properties

Label 16.48.a.d.1.4
Level $16$
Weight $48$
Character 16.1
Self dual yes
Analytic conductor $223.852$
Analytic rank $1$
Dimension $4$
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] = [16,48,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 48, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 48);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(223.852260248\)
Analytic rank: \(1\)
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: \( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(129356.\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.21918e11 q^{3} -3.08341e16 q^{5} -1.11073e20 q^{7} +2.26588e22 q^{9} +O(q^{10})\) \(q+2.21918e11 q^{3} -3.08341e16 q^{5} -1.11073e20 q^{7} +2.26588e22 q^{9} +1.19449e24 q^{11} -2.71503e24 q^{13} -6.84264e27 q^{15} +1.17417e29 q^{17} +1.65620e30 q^{19} -2.46491e31 q^{21} -5.41759e31 q^{23} +2.40198e32 q^{25} -8.72142e32 q^{27} -1.03639e34 q^{29} +3.80854e34 q^{31} +2.65078e35 q^{33} +3.42483e36 q^{35} +8.41184e36 q^{37} -6.02513e35 q^{39} +7.68510e37 q^{41} +2.17109e38 q^{43} -6.98663e38 q^{45} -2.73873e39 q^{47} +7.09387e39 q^{49} +2.60570e40 q^{51} +8.29394e39 q^{53} -3.68309e40 q^{55} +3.67541e41 q^{57} -7.09596e40 q^{59} -6.43092e41 q^{61} -2.51678e42 q^{63} +8.37153e40 q^{65} -1.19612e43 q^{67} -1.20226e43 q^{69} -2.64034e43 q^{71} +3.87700e43 q^{73} +5.33042e43 q^{75} -1.32675e44 q^{77} +1.22812e44 q^{79} -7.96015e44 q^{81} -1.65007e45 q^{83} -3.62045e45 q^{85} -2.29995e45 q^{87} -6.00200e45 q^{89} +3.01566e44 q^{91} +8.45183e45 q^{93} -5.10674e46 q^{95} -1.75152e46 q^{97} +2.70656e46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) \) 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
\(3\) 2.21918e11 1.36095 0.680476 0.732770i \(-0.261773\pi\)
0.680476 + 0.732770i \(0.261773\pi\)
\(4\) 0 0
\(5\) −3.08341e16 −1.15674 −0.578370 0.815774i \(-0.696311\pi\)
−0.578370 + 0.815774i \(0.696311\pi\)
\(6\) 0 0
\(7\) −1.11073e20 −1.53393 −0.766963 0.641691i \(-0.778233\pi\)
−0.766963 + 0.641691i \(0.778233\pi\)
\(8\) 0 0
\(9\) 2.26588e22 0.852193
\(10\) 0 0
\(11\) 1.19449e24 0.402210 0.201105 0.979570i \(-0.435547\pi\)
0.201105 + 0.979570i \(0.435547\pi\)
\(12\) 0 0
\(13\) −2.71503e24 −0.0180345 −0.00901727 0.999959i \(-0.502870\pi\)
−0.00901727 + 0.999959i \(0.502870\pi\)
\(14\) 0 0
\(15\) −6.84264e27 −1.57427
\(16\) 0 0
\(17\) 1.17417e29 1.42620 0.713102 0.701060i \(-0.247289\pi\)
0.713102 + 0.701060i \(0.247289\pi\)
\(18\) 0 0
\(19\) 1.65620e30 1.47368 0.736840 0.676067i \(-0.236317\pi\)
0.736840 + 0.676067i \(0.236317\pi\)
\(20\) 0 0
\(21\) −2.46491e31 −2.08760
\(22\) 0 0
\(23\) −5.41759e31 −0.541006 −0.270503 0.962719i \(-0.587190\pi\)
−0.270503 + 0.962719i \(0.587190\pi\)
\(24\) 0 0
\(25\) 2.40198e32 0.338048
\(26\) 0 0
\(27\) −8.72142e32 −0.201159
\(28\) 0 0
\(29\) −1.03639e34 −0.445833 −0.222916 0.974838i \(-0.571558\pi\)
−0.222916 + 0.974838i \(0.571558\pi\)
\(30\) 0 0
\(31\) 3.80854e34 0.341789 0.170895 0.985289i \(-0.445334\pi\)
0.170895 + 0.985289i \(0.445334\pi\)
\(32\) 0 0
\(33\) 2.65078e35 0.547389
\(34\) 0 0
\(35\) 3.42483e36 1.77435
\(36\) 0 0
\(37\) 8.41184e36 1.18073 0.590365 0.807136i \(-0.298984\pi\)
0.590365 + 0.807136i \(0.298984\pi\)
\(38\) 0 0
\(39\) −6.02513e35 −0.0245442
\(40\) 0 0
\(41\) 7.68510e37 0.966560 0.483280 0.875466i \(-0.339445\pi\)
0.483280 + 0.875466i \(0.339445\pi\)
\(42\) 0 0
\(43\) 2.17109e38 0.891597 0.445799 0.895133i \(-0.352920\pi\)
0.445799 + 0.895133i \(0.352920\pi\)
\(44\) 0 0
\(45\) −6.98663e38 −0.985766
\(46\) 0 0
\(47\) −2.73873e39 −1.39075 −0.695375 0.718647i \(-0.744762\pi\)
−0.695375 + 0.718647i \(0.744762\pi\)
\(48\) 0 0
\(49\) 7.09387e39 1.35293
\(50\) 0 0
\(51\) 2.60570e40 1.94100
\(52\) 0 0
\(53\) 8.29394e39 0.250195 0.125097 0.992144i \(-0.460076\pi\)
0.125097 + 0.992144i \(0.460076\pi\)
\(54\) 0 0
\(55\) −3.68309e40 −0.465253
\(56\) 0 0
\(57\) 3.67541e41 2.00561
\(58\) 0 0
\(59\) −7.09596e40 −0.172182 −0.0860912 0.996287i \(-0.527438\pi\)
−0.0860912 + 0.996287i \(0.527438\pi\)
\(60\) 0 0
\(61\) −6.43092e41 −0.712889 −0.356444 0.934317i \(-0.616011\pi\)
−0.356444 + 0.934317i \(0.616011\pi\)
\(62\) 0 0
\(63\) −2.51678e42 −1.30720
\(64\) 0 0
\(65\) 8.37153e40 0.0208613
\(66\) 0 0
\(67\) −1.19612e43 −1.46223 −0.731114 0.682255i \(-0.760999\pi\)
−0.731114 + 0.682255i \(0.760999\pi\)
\(68\) 0 0
\(69\) −1.20226e43 −0.736283
\(70\) 0 0
\(71\) −2.64034e43 −0.826205 −0.413103 0.910684i \(-0.635555\pi\)
−0.413103 + 0.910684i \(0.635555\pi\)
\(72\) 0 0
\(73\) 3.87700e43 0.631552 0.315776 0.948834i \(-0.397735\pi\)
0.315776 + 0.948834i \(0.397735\pi\)
\(74\) 0 0
\(75\) 5.33042e43 0.460068
\(76\) 0 0
\(77\) −1.32675e44 −0.616961
\(78\) 0 0
\(79\) 1.22812e44 0.312612 0.156306 0.987709i \(-0.450041\pi\)
0.156306 + 0.987709i \(0.450041\pi\)
\(80\) 0 0
\(81\) −7.96015e44 −1.12596
\(82\) 0 0
\(83\) −1.65007e45 −1.31573 −0.657867 0.753134i \(-0.728541\pi\)
−0.657867 + 0.753134i \(0.728541\pi\)
\(84\) 0 0
\(85\) −3.62045e45 −1.64975
\(86\) 0 0
\(87\) −2.29995e45 −0.606757
\(88\) 0 0
\(89\) −6.00200e45 −0.928177 −0.464089 0.885789i \(-0.653618\pi\)
−0.464089 + 0.885789i \(0.653618\pi\)
\(90\) 0 0
\(91\) 3.01566e44 0.0276637
\(92\) 0 0
\(93\) 8.45183e45 0.465159
\(94\) 0 0
\(95\) −5.10674e46 −1.70466
\(96\) 0 0
\(97\) −1.75152e46 −0.358328 −0.179164 0.983819i \(-0.557339\pi\)
−0.179164 + 0.983819i \(0.557339\pi\)
\(98\) 0 0
\(99\) 2.70656e46 0.342761
\(100\) 0 0
\(101\) −2.93699e46 −0.232461 −0.116231 0.993222i \(-0.537081\pi\)
−0.116231 + 0.993222i \(0.537081\pi\)
\(102\) 0 0
\(103\) 2.42094e46 0.120868 0.0604338 0.998172i \(-0.480752\pi\)
0.0604338 + 0.998172i \(0.480752\pi\)
\(104\) 0 0
\(105\) 7.60032e47 2.41481
\(106\) 0 0
\(107\) 3.58808e47 0.731718 0.365859 0.930670i \(-0.380775\pi\)
0.365859 + 0.930670i \(0.380775\pi\)
\(108\) 0 0
\(109\) −2.00272e47 −0.264301 −0.132150 0.991230i \(-0.542188\pi\)
−0.132150 + 0.991230i \(0.542188\pi\)
\(110\) 0 0
\(111\) 1.86674e48 1.60692
\(112\) 0 0
\(113\) −4.99882e47 −0.282829 −0.141414 0.989950i \(-0.545165\pi\)
−0.141414 + 0.989950i \(0.545165\pi\)
\(114\) 0 0
\(115\) 1.67046e48 0.625803
\(116\) 0 0
\(117\) −6.15192e46 −0.0153689
\(118\) 0 0
\(119\) −1.30419e49 −2.18769
\(120\) 0 0
\(121\) −7.39295e48 −0.838227
\(122\) 0 0
\(123\) 1.70546e49 1.31544
\(124\) 0 0
\(125\) 1.45027e49 0.765706
\(126\) 0 0
\(127\) −8.93451e47 −0.0324850 −0.0162425 0.999868i \(-0.505170\pi\)
−0.0162425 + 0.999868i \(0.505170\pi\)
\(128\) 0 0
\(129\) 4.81804e49 1.21342
\(130\) 0 0
\(131\) −8.86188e49 −1.55471 −0.777357 0.629060i \(-0.783440\pi\)
−0.777357 + 0.629060i \(0.783440\pi\)
\(132\) 0 0
\(133\) −1.83959e50 −2.26052
\(134\) 0 0
\(135\) 2.68917e49 0.232688
\(136\) 0 0
\(137\) 1.56850e50 0.960619 0.480310 0.877099i \(-0.340524\pi\)
0.480310 + 0.877099i \(0.340524\pi\)
\(138\) 0 0
\(139\) −2.09943e50 −0.914648 −0.457324 0.889300i \(-0.651192\pi\)
−0.457324 + 0.889300i \(0.651192\pi\)
\(140\) 0 0
\(141\) −6.07773e50 −1.89275
\(142\) 0 0
\(143\) −3.24306e48 −0.00725368
\(144\) 0 0
\(145\) 3.19563e50 0.515713
\(146\) 0 0
\(147\) 1.57426e51 1.84128
\(148\) 0 0
\(149\) 1.17929e51 1.00402 0.502009 0.864862i \(-0.332594\pi\)
0.502009 + 0.864862i \(0.332594\pi\)
\(150\) 0 0
\(151\) 1.05626e51 0.657377 0.328689 0.944438i \(-0.393393\pi\)
0.328689 + 0.944438i \(0.393393\pi\)
\(152\) 0 0
\(153\) 2.66053e51 1.21540
\(154\) 0 0
\(155\) −1.17433e51 −0.395361
\(156\) 0 0
\(157\) −2.43063e51 −0.605447 −0.302724 0.953078i \(-0.597896\pi\)
−0.302724 + 0.953078i \(0.597896\pi\)
\(158\) 0 0
\(159\) 1.84057e51 0.340503
\(160\) 0 0
\(161\) 6.01748e51 0.829863
\(162\) 0 0
\(163\) −8.76362e51 −0.904222 −0.452111 0.891962i \(-0.649329\pi\)
−0.452111 + 0.891962i \(0.649329\pi\)
\(164\) 0 0
\(165\) −8.17344e51 −0.633187
\(166\) 0 0
\(167\) −2.63045e52 −1.53530 −0.767648 0.640871i \(-0.778573\pi\)
−0.767648 + 0.640871i \(0.778573\pi\)
\(168\) 0 0
\(169\) −2.26567e52 −0.999675
\(170\) 0 0
\(171\) 3.75275e52 1.25586
\(172\) 0 0
\(173\) −3.14535e52 −0.800913 −0.400456 0.916316i \(-0.631148\pi\)
−0.400456 + 0.916316i \(0.631148\pi\)
\(174\) 0 0
\(175\) −2.66795e52 −0.518541
\(176\) 0 0
\(177\) −1.57472e52 −0.234332
\(178\) 0 0
\(179\) 5.75744e52 0.657935 0.328968 0.944341i \(-0.393299\pi\)
0.328968 + 0.944341i \(0.393299\pi\)
\(180\) 0 0
\(181\) 1.66382e53 1.46440 0.732199 0.681091i \(-0.238494\pi\)
0.732199 + 0.681091i \(0.238494\pi\)
\(182\) 0 0
\(183\) −1.42714e53 −0.970208
\(184\) 0 0
\(185\) −2.59371e53 −1.36580
\(186\) 0 0
\(187\) 1.40253e53 0.573634
\(188\) 0 0
\(189\) 9.68714e52 0.308563
\(190\) 0 0
\(191\) 1.33768e53 0.332713 0.166356 0.986066i \(-0.446800\pi\)
0.166356 + 0.986066i \(0.446800\pi\)
\(192\) 0 0
\(193\) 6.53394e53 1.27227 0.636134 0.771579i \(-0.280533\pi\)
0.636134 + 0.771579i \(0.280533\pi\)
\(194\) 0 0
\(195\) 1.85779e52 0.0283912
\(196\) 0 0
\(197\) 9.89833e53 1.19016 0.595081 0.803666i \(-0.297120\pi\)
0.595081 + 0.803666i \(0.297120\pi\)
\(198\) 0 0
\(199\) −1.17074e54 −1.11023 −0.555117 0.831772i \(-0.687327\pi\)
−0.555117 + 0.831772i \(0.687327\pi\)
\(200\) 0 0
\(201\) −2.65440e54 −1.99002
\(202\) 0 0
\(203\) 1.15115e54 0.683875
\(204\) 0 0
\(205\) −2.36963e54 −1.11806
\(206\) 0 0
\(207\) −1.22756e54 −0.461041
\(208\) 0 0
\(209\) 1.97831e54 0.592729
\(210\) 0 0
\(211\) −1.13707e54 −0.272364 −0.136182 0.990684i \(-0.543483\pi\)
−0.136182 + 0.990684i \(0.543483\pi\)
\(212\) 0 0
\(213\) −5.85938e54 −1.12443
\(214\) 0 0
\(215\) −6.69435e54 −1.03135
\(216\) 0 0
\(217\) −4.23026e54 −0.524280
\(218\) 0 0
\(219\) 8.60377e54 0.859512
\(220\) 0 0
\(221\) −3.18791e53 −0.0257210
\(222\) 0 0
\(223\) −2.34179e55 −1.52892 −0.764459 0.644672i \(-0.776994\pi\)
−0.764459 + 0.644672i \(0.776994\pi\)
\(224\) 0 0
\(225\) 5.44259e54 0.288082
\(226\) 0 0
\(227\) −3.06780e55 −1.31892 −0.659462 0.751738i \(-0.729216\pi\)
−0.659462 + 0.751738i \(0.729216\pi\)
\(228\) 0 0
\(229\) −4.77257e55 −1.66962 −0.834811 0.550536i \(-0.814423\pi\)
−0.834811 + 0.550536i \(0.814423\pi\)
\(230\) 0 0
\(231\) −2.94430e55 −0.839655
\(232\) 0 0
\(233\) 1.85272e55 0.431464 0.215732 0.976453i \(-0.430786\pi\)
0.215732 + 0.976453i \(0.430786\pi\)
\(234\) 0 0
\(235\) 8.44462e55 1.60874
\(236\) 0 0
\(237\) 2.72543e55 0.425450
\(238\) 0 0
\(239\) 6.05119e55 0.775336 0.387668 0.921799i \(-0.373281\pi\)
0.387668 + 0.921799i \(0.373281\pi\)
\(240\) 0 0
\(241\) 3.35331e55 0.353242 0.176621 0.984279i \(-0.443483\pi\)
0.176621 + 0.984279i \(0.443483\pi\)
\(242\) 0 0
\(243\) −1.53461e56 −1.33122
\(244\) 0 0
\(245\) −2.18733e56 −1.56499
\(246\) 0 0
\(247\) −4.49663e54 −0.0265771
\(248\) 0 0
\(249\) −3.66181e56 −1.79065
\(250\) 0 0
\(251\) −3.39203e56 −1.37444 −0.687222 0.726448i \(-0.741170\pi\)
−0.687222 + 0.726448i \(0.741170\pi\)
\(252\) 0 0
\(253\) −6.47124e55 −0.217598
\(254\) 0 0
\(255\) −8.03443e56 −2.24523
\(256\) 0 0
\(257\) −7.45663e56 −1.73426 −0.867130 0.498082i \(-0.834038\pi\)
−0.867130 + 0.498082i \(0.834038\pi\)
\(258\) 0 0
\(259\) −9.34329e56 −1.81115
\(260\) 0 0
\(261\) −2.34834e56 −0.379935
\(262\) 0 0
\(263\) 1.28505e57 1.73764 0.868819 0.495129i \(-0.164879\pi\)
0.868819 + 0.495129i \(0.164879\pi\)
\(264\) 0 0
\(265\) −2.55736e56 −0.289410
\(266\) 0 0
\(267\) −1.33195e57 −1.26321
\(268\) 0 0
\(269\) 1.52143e57 1.21081 0.605403 0.795919i \(-0.293012\pi\)
0.605403 + 0.795919i \(0.293012\pi\)
\(270\) 0 0
\(271\) −2.08539e56 −0.139447 −0.0697235 0.997566i \(-0.522212\pi\)
−0.0697235 + 0.997566i \(0.522212\pi\)
\(272\) 0 0
\(273\) 6.69230e55 0.0376490
\(274\) 0 0
\(275\) 2.86913e56 0.135967
\(276\) 0 0
\(277\) −4.91573e57 −1.96478 −0.982388 0.186853i \(-0.940171\pi\)
−0.982388 + 0.186853i \(0.940171\pi\)
\(278\) 0 0
\(279\) 8.62969e56 0.291270
\(280\) 0 0
\(281\) 2.99077e57 0.853462 0.426731 0.904379i \(-0.359665\pi\)
0.426731 + 0.904379i \(0.359665\pi\)
\(282\) 0 0
\(283\) 2.40523e57 0.580999 0.290499 0.956875i \(-0.406179\pi\)
0.290499 + 0.956875i \(0.406179\pi\)
\(284\) 0 0
\(285\) −1.13328e58 −2.31997
\(286\) 0 0
\(287\) −8.53607e57 −1.48263
\(288\) 0 0
\(289\) 7.00882e57 1.03406
\(290\) 0 0
\(291\) −3.88695e57 −0.487667
\(292\) 0 0
\(293\) 7.67134e57 0.819374 0.409687 0.912226i \(-0.365638\pi\)
0.409687 + 0.912226i \(0.365638\pi\)
\(294\) 0 0
\(295\) 2.18797e57 0.199170
\(296\) 0 0
\(297\) −1.04176e57 −0.0809081
\(298\) 0 0
\(299\) 1.47089e56 0.00975679
\(300\) 0 0
\(301\) −2.41149e58 −1.36764
\(302\) 0 0
\(303\) −6.51772e57 −0.316369
\(304\) 0 0
\(305\) 1.98291e58 0.824627
\(306\) 0 0
\(307\) −1.66078e58 −0.592325 −0.296162 0.955138i \(-0.595707\pi\)
−0.296162 + 0.955138i \(0.595707\pi\)
\(308\) 0 0
\(309\) 5.37251e57 0.164495
\(310\) 0 0
\(311\) 2.17917e58 0.573352 0.286676 0.958028i \(-0.407450\pi\)
0.286676 + 0.958028i \(0.407450\pi\)
\(312\) 0 0
\(313\) 4.52774e58 1.02468 0.512342 0.858782i \(-0.328778\pi\)
0.512342 + 0.858782i \(0.328778\pi\)
\(314\) 0 0
\(315\) 7.76026e58 1.51209
\(316\) 0 0
\(317\) −6.03868e58 −1.01403 −0.507014 0.861938i \(-0.669251\pi\)
−0.507014 + 0.861938i \(0.669251\pi\)
\(318\) 0 0
\(319\) −1.23796e58 −0.179319
\(320\) 0 0
\(321\) 7.96260e58 0.995833
\(322\) 0 0
\(323\) 1.94466e59 2.10177
\(324\) 0 0
\(325\) −6.52143e56 −0.00609655
\(326\) 0 0
\(327\) −4.44440e58 −0.359701
\(328\) 0 0
\(329\) 3.04199e59 2.13331
\(330\) 0 0
\(331\) 2.03088e57 0.0123517 0.00617585 0.999981i \(-0.498034\pi\)
0.00617585 + 0.999981i \(0.498034\pi\)
\(332\) 0 0
\(333\) 1.90602e59 1.00621
\(334\) 0 0
\(335\) 3.68811e59 1.69142
\(336\) 0 0
\(337\) −1.81636e59 −0.724268 −0.362134 0.932126i \(-0.617952\pi\)
−0.362134 + 0.932126i \(0.617952\pi\)
\(338\) 0 0
\(339\) −1.10933e59 −0.384917
\(340\) 0 0
\(341\) 4.54925e58 0.137471
\(342\) 0 0
\(343\) −2.05545e59 −0.541370
\(344\) 0 0
\(345\) 3.70706e59 0.851689
\(346\) 0 0
\(347\) −1.01600e59 −0.203776 −0.101888 0.994796i \(-0.532488\pi\)
−0.101888 + 0.994796i \(0.532488\pi\)
\(348\) 0 0
\(349\) −4.81199e59 −0.843193 −0.421596 0.906784i \(-0.638530\pi\)
−0.421596 + 0.906784i \(0.638530\pi\)
\(350\) 0 0
\(351\) 2.36789e57 0.00362781
\(352\) 0 0
\(353\) 1.19790e60 1.60588 0.802941 0.596059i \(-0.203268\pi\)
0.802941 + 0.596059i \(0.203268\pi\)
\(354\) 0 0
\(355\) 8.14123e59 0.955705
\(356\) 0 0
\(357\) −2.89423e60 −2.97735
\(358\) 0 0
\(359\) 8.62063e58 0.0777713 0.0388857 0.999244i \(-0.487619\pi\)
0.0388857 + 0.999244i \(0.487619\pi\)
\(360\) 0 0
\(361\) 1.47995e60 1.17173
\(362\) 0 0
\(363\) −1.64063e60 −1.14079
\(364\) 0 0
\(365\) −1.19544e60 −0.730542
\(366\) 0 0
\(367\) 2.19138e60 1.17778 0.588892 0.808212i \(-0.299565\pi\)
0.588892 + 0.808212i \(0.299565\pi\)
\(368\) 0 0
\(369\) 1.74135e60 0.823696
\(370\) 0 0
\(371\) −9.21233e59 −0.383780
\(372\) 0 0
\(373\) 1.32206e60 0.485393 0.242696 0.970102i \(-0.421968\pi\)
0.242696 + 0.970102i \(0.421968\pi\)
\(374\) 0 0
\(375\) 3.21840e60 1.04209
\(376\) 0 0
\(377\) 2.81384e58 0.00804039
\(378\) 0 0
\(379\) −6.08671e60 −1.53589 −0.767947 0.640513i \(-0.778722\pi\)
−0.767947 + 0.640513i \(0.778722\pi\)
\(380\) 0 0
\(381\) −1.98273e59 −0.0442106
\(382\) 0 0
\(383\) −3.42858e60 −0.675995 −0.337997 0.941147i \(-0.609750\pi\)
−0.337997 + 0.941147i \(0.609750\pi\)
\(384\) 0 0
\(385\) 4.09092e60 0.713664
\(386\) 0 0
\(387\) 4.91942e60 0.759813
\(388\) 0 0
\(389\) −5.62159e60 −0.769208 −0.384604 0.923082i \(-0.625662\pi\)
−0.384604 + 0.923082i \(0.625662\pi\)
\(390\) 0 0
\(391\) −6.36118e60 −0.771585
\(392\) 0 0
\(393\) −1.96661e61 −2.11589
\(394\) 0 0
\(395\) −3.78680e60 −0.361611
\(396\) 0 0
\(397\) −1.94362e61 −1.64829 −0.824146 0.566377i \(-0.808345\pi\)
−0.824146 + 0.566377i \(0.808345\pi\)
\(398\) 0 0
\(399\) −4.08238e61 −3.07646
\(400\) 0 0
\(401\) −2.48633e61 −1.66597 −0.832984 0.553296i \(-0.813370\pi\)
−0.832984 + 0.553296i \(0.813370\pi\)
\(402\) 0 0
\(403\) −1.03403e59 −0.00616401
\(404\) 0 0
\(405\) 2.45444e61 1.30244
\(406\) 0 0
\(407\) 1.00478e61 0.474902
\(408\) 0 0
\(409\) 1.77202e61 0.746397 0.373199 0.927751i \(-0.378261\pi\)
0.373199 + 0.927751i \(0.378261\pi\)
\(410\) 0 0
\(411\) 3.48079e61 1.30736
\(412\) 0 0
\(413\) 7.88170e60 0.264115
\(414\) 0 0
\(415\) 5.08785e61 1.52196
\(416\) 0 0
\(417\) −4.65902e61 −1.24479
\(418\) 0 0
\(419\) −6.32251e61 −1.50959 −0.754797 0.655958i \(-0.772265\pi\)
−0.754797 + 0.655958i \(0.772265\pi\)
\(420\) 0 0
\(421\) 8.68512e61 1.85416 0.927078 0.374868i \(-0.122312\pi\)
0.927078 + 0.374868i \(0.122312\pi\)
\(422\) 0 0
\(423\) −6.20563e61 −1.18519
\(424\) 0 0
\(425\) 2.82033e61 0.482126
\(426\) 0 0
\(427\) 7.14301e61 1.09352
\(428\) 0 0
\(429\) −7.19694e59 −0.00987192
\(430\) 0 0
\(431\) −6.27735e61 −0.771898 −0.385949 0.922520i \(-0.626126\pi\)
−0.385949 + 0.922520i \(0.626126\pi\)
\(432\) 0 0
\(433\) −6.16297e61 −0.679711 −0.339855 0.940478i \(-0.610378\pi\)
−0.339855 + 0.940478i \(0.610378\pi\)
\(434\) 0 0
\(435\) 7.09167e61 0.701861
\(436\) 0 0
\(437\) −8.97261e61 −0.797269
\(438\) 0 0
\(439\) 1.66474e62 1.32871 0.664353 0.747419i \(-0.268707\pi\)
0.664353 + 0.747419i \(0.268707\pi\)
\(440\) 0 0
\(441\) 1.60739e62 1.15296
\(442\) 0 0
\(443\) −1.29305e62 −0.833929 −0.416965 0.908923i \(-0.636906\pi\)
−0.416965 + 0.908923i \(0.636906\pi\)
\(444\) 0 0
\(445\) 1.85066e62 1.07366
\(446\) 0 0
\(447\) 2.61705e62 1.36642
\(448\) 0 0
\(449\) −1.47587e62 −0.693839 −0.346920 0.937895i \(-0.612772\pi\)
−0.346920 + 0.937895i \(0.612772\pi\)
\(450\) 0 0
\(451\) 9.17975e61 0.388760
\(452\) 0 0
\(453\) 2.34404e62 0.894659
\(454\) 0 0
\(455\) −9.29851e60 −0.0319997
\(456\) 0 0
\(457\) −1.90700e62 −0.591998 −0.295999 0.955188i \(-0.595653\pi\)
−0.295999 + 0.955188i \(0.595653\pi\)
\(458\) 0 0
\(459\) −1.02404e62 −0.286893
\(460\) 0 0
\(461\) 2.19785e61 0.0555939 0.0277970 0.999614i \(-0.491151\pi\)
0.0277970 + 0.999614i \(0.491151\pi\)
\(462\) 0 0
\(463\) 1.68005e62 0.383857 0.191929 0.981409i \(-0.438526\pi\)
0.191929 + 0.981409i \(0.438526\pi\)
\(464\) 0 0
\(465\) −2.60604e62 −0.538068
\(466\) 0 0
\(467\) −1.37225e62 −0.256145 −0.128073 0.991765i \(-0.540879\pi\)
−0.128073 + 0.991765i \(0.540879\pi\)
\(468\) 0 0
\(469\) 1.32856e63 2.24295
\(470\) 0 0
\(471\) −5.39400e62 −0.823985
\(472\) 0 0
\(473\) 2.59334e62 0.358610
\(474\) 0 0
\(475\) 3.97815e62 0.498175
\(476\) 0 0
\(477\) 1.87931e62 0.213214
\(478\) 0 0
\(479\) 1.62356e63 1.66949 0.834745 0.550637i \(-0.185615\pi\)
0.834745 + 0.550637i \(0.185615\pi\)
\(480\) 0 0
\(481\) −2.28384e61 −0.0212939
\(482\) 0 0
\(483\) 1.33539e63 1.12940
\(484\) 0 0
\(485\) 5.40066e62 0.414492
\(486\) 0 0
\(487\) −1.09889e62 −0.0765640 −0.0382820 0.999267i \(-0.512189\pi\)
−0.0382820 + 0.999267i \(0.512189\pi\)
\(488\) 0 0
\(489\) −1.94481e63 −1.23060
\(490\) 0 0
\(491\) −2.66145e63 −1.53004 −0.765020 0.644007i \(-0.777271\pi\)
−0.765020 + 0.644007i \(0.777271\pi\)
\(492\) 0 0
\(493\) −1.21690e63 −0.635849
\(494\) 0 0
\(495\) −8.34544e62 −0.396485
\(496\) 0 0
\(497\) 2.93270e63 1.26734
\(498\) 0 0
\(499\) 2.22266e63 0.873997 0.436999 0.899462i \(-0.356041\pi\)
0.436999 + 0.899462i \(0.356041\pi\)
\(500\) 0 0
\(501\) −5.83743e63 −2.08947
\(502\) 0 0
\(503\) −3.07975e63 −1.00385 −0.501924 0.864912i \(-0.667374\pi\)
−0.501924 + 0.864912i \(0.667374\pi\)
\(504\) 0 0
\(505\) 9.05595e62 0.268897
\(506\) 0 0
\(507\) −5.02793e63 −1.36051
\(508\) 0 0
\(509\) −5.72771e63 −1.41291 −0.706453 0.707760i \(-0.749706\pi\)
−0.706453 + 0.707760i \(0.749706\pi\)
\(510\) 0 0
\(511\) −4.30631e63 −0.968754
\(512\) 0 0
\(513\) −1.44444e63 −0.296443
\(514\) 0 0
\(515\) −7.46476e62 −0.139812
\(516\) 0 0
\(517\) −3.27137e63 −0.559374
\(518\) 0 0
\(519\) −6.98009e63 −1.09000
\(520\) 0 0
\(521\) 1.17638e64 1.67826 0.839132 0.543928i \(-0.183064\pi\)
0.839132 + 0.543928i \(0.183064\pi\)
\(522\) 0 0
\(523\) 6.38331e63 0.832254 0.416127 0.909306i \(-0.363387\pi\)
0.416127 + 0.909306i \(0.363387\pi\)
\(524\) 0 0
\(525\) −5.92066e63 −0.705710
\(526\) 0 0
\(527\) 4.47188e63 0.487461
\(528\) 0 0
\(529\) −7.09283e63 −0.707313
\(530\) 0 0
\(531\) −1.60786e63 −0.146733
\(532\) 0 0
\(533\) −2.08652e62 −0.0174315
\(534\) 0 0
\(535\) −1.10635e64 −0.846408
\(536\) 0 0
\(537\) 1.27768e64 0.895419
\(538\) 0 0
\(539\) 8.47354e63 0.544163
\(540\) 0 0
\(541\) −7.64596e62 −0.0450087 −0.0225043 0.999747i \(-0.507164\pi\)
−0.0225043 + 0.999747i \(0.507164\pi\)
\(542\) 0 0
\(543\) 3.69231e64 1.99298
\(544\) 0 0
\(545\) 6.17521e63 0.305727
\(546\) 0 0
\(547\) 1.77108e64 0.804519 0.402259 0.915526i \(-0.368225\pi\)
0.402259 + 0.915526i \(0.368225\pi\)
\(548\) 0 0
\(549\) −1.45717e64 −0.607519
\(550\) 0 0
\(551\) −1.71648e64 −0.657015
\(552\) 0 0
\(553\) −1.36411e64 −0.479523
\(554\) 0 0
\(555\) −5.75592e64 −1.85879
\(556\) 0 0
\(557\) 1.48786e64 0.441535 0.220767 0.975327i \(-0.429144\pi\)
0.220767 + 0.975327i \(0.429144\pi\)
\(558\) 0 0
\(559\) −5.89456e62 −0.0160795
\(560\) 0 0
\(561\) 3.11247e64 0.780689
\(562\) 0 0
\(563\) −4.11252e64 −0.948770 −0.474385 0.880317i \(-0.657330\pi\)
−0.474385 + 0.880317i \(0.657330\pi\)
\(564\) 0 0
\(565\) 1.54134e64 0.327160
\(566\) 0 0
\(567\) 8.84157e64 1.72714
\(568\) 0 0
\(569\) −5.04135e64 −0.906586 −0.453293 0.891362i \(-0.649751\pi\)
−0.453293 + 0.891362i \(0.649751\pi\)
\(570\) 0 0
\(571\) −1.68666e64 −0.279305 −0.139652 0.990201i \(-0.544598\pi\)
−0.139652 + 0.990201i \(0.544598\pi\)
\(572\) 0 0
\(573\) 2.96856e64 0.452806
\(574\) 0 0
\(575\) −1.30129e64 −0.182886
\(576\) 0 0
\(577\) −9.79059e63 −0.126817 −0.0634084 0.997988i \(-0.520197\pi\)
−0.0634084 + 0.997988i \(0.520197\pi\)
\(578\) 0 0
\(579\) 1.45000e65 1.73150
\(580\) 0 0
\(581\) 1.83279e65 2.01824
\(582\) 0 0
\(583\) 9.90700e63 0.100631
\(584\) 0 0
\(585\) 1.89689e63 0.0177778
\(586\) 0 0
\(587\) −2.78831e64 −0.241183 −0.120591 0.992702i \(-0.538479\pi\)
−0.120591 + 0.992702i \(0.538479\pi\)
\(588\) 0 0
\(589\) 6.30770e64 0.503688
\(590\) 0 0
\(591\) 2.19662e65 1.61975
\(592\) 0 0
\(593\) 1.49430e65 1.01778 0.508889 0.860832i \(-0.330056\pi\)
0.508889 + 0.860832i \(0.330056\pi\)
\(594\) 0 0
\(595\) 4.02134e65 2.53059
\(596\) 0 0
\(597\) −2.59809e65 −1.51098
\(598\) 0 0
\(599\) −2.99285e65 −1.60899 −0.804496 0.593957i \(-0.797565\pi\)
−0.804496 + 0.593957i \(0.797565\pi\)
\(600\) 0 0
\(601\) −1.56472e65 −0.777831 −0.388916 0.921273i \(-0.627150\pi\)
−0.388916 + 0.921273i \(0.627150\pi\)
\(602\) 0 0
\(603\) −2.71025e65 −1.24610
\(604\) 0 0
\(605\) 2.27955e65 0.969611
\(606\) 0 0
\(607\) −8.61159e64 −0.338961 −0.169480 0.985534i \(-0.554209\pi\)
−0.169480 + 0.985534i \(0.554209\pi\)
\(608\) 0 0
\(609\) 2.55462e65 0.930721
\(610\) 0 0
\(611\) 7.43572e63 0.0250816
\(612\) 0 0
\(613\) −1.58806e65 −0.496072 −0.248036 0.968751i \(-0.579785\pi\)
−0.248036 + 0.968751i \(0.579785\pi\)
\(614\) 0 0
\(615\) −5.25864e65 −1.52163
\(616\) 0 0
\(617\) −6.01082e65 −1.61151 −0.805753 0.592251i \(-0.798239\pi\)
−0.805753 + 0.592251i \(0.798239\pi\)
\(618\) 0 0
\(619\) −3.62211e65 −0.899978 −0.449989 0.893034i \(-0.648572\pi\)
−0.449989 + 0.893034i \(0.648572\pi\)
\(620\) 0 0
\(621\) 4.72491e64 0.108828
\(622\) 0 0
\(623\) 6.66660e65 1.42376
\(624\) 0 0
\(625\) −6.17847e65 −1.22377
\(626\) 0 0
\(627\) 4.39022e65 0.806677
\(628\) 0 0
\(629\) 9.87695e65 1.68396
\(630\) 0 0
\(631\) −9.92614e65 −1.57069 −0.785345 0.619059i \(-0.787514\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(632\) 0 0
\(633\) −2.52336e65 −0.370675
\(634\) 0 0
\(635\) 2.75487e64 0.0375768
\(636\) 0 0
\(637\) −1.92601e64 −0.0243995
\(638\) 0 0
\(639\) −5.98268e65 −0.704086
\(640\) 0 0
\(641\) −1.70256e66 −1.86183 −0.930915 0.365236i \(-0.880988\pi\)
−0.930915 + 0.365236i \(0.880988\pi\)
\(642\) 0 0
\(643\) −8.06741e65 −0.819929 −0.409964 0.912102i \(-0.634459\pi\)
−0.409964 + 0.912102i \(0.634459\pi\)
\(644\) 0 0
\(645\) −1.48560e66 −1.40361
\(646\) 0 0
\(647\) 2.90914e65 0.255573 0.127786 0.991802i \(-0.459213\pi\)
0.127786 + 0.991802i \(0.459213\pi\)
\(648\) 0 0
\(649\) −8.47603e64 −0.0692536
\(650\) 0 0
\(651\) −9.38770e65 −0.713520
\(652\) 0 0
\(653\) 3.73213e65 0.263935 0.131967 0.991254i \(-0.457871\pi\)
0.131967 + 0.991254i \(0.457871\pi\)
\(654\) 0 0
\(655\) 2.73248e66 1.79840
\(656\) 0 0
\(657\) 8.78482e65 0.538204
\(658\) 0 0
\(659\) −7.84754e65 −0.447637 −0.223819 0.974631i \(-0.571852\pi\)
−0.223819 + 0.974631i \(0.571852\pi\)
\(660\) 0 0
\(661\) 5.69856e65 0.302713 0.151356 0.988479i \(-0.451636\pi\)
0.151356 + 0.988479i \(0.451636\pi\)
\(662\) 0 0
\(663\) −7.07454e64 −0.0350050
\(664\) 0 0
\(665\) 5.67221e66 2.61483
\(666\) 0 0
\(667\) 5.61476e65 0.241198
\(668\) 0 0
\(669\) −5.19686e66 −2.08079
\(670\) 0 0
\(671\) −7.68164e65 −0.286731
\(672\) 0 0
\(673\) 4.43578e66 1.54389 0.771945 0.635689i \(-0.219284\pi\)
0.771945 + 0.635689i \(0.219284\pi\)
\(674\) 0 0
\(675\) −2.09487e65 −0.0680013
\(676\) 0 0
\(677\) 3.52339e66 1.06691 0.533454 0.845829i \(-0.320894\pi\)
0.533454 + 0.845829i \(0.320894\pi\)
\(678\) 0 0
\(679\) 1.94547e66 0.549648
\(680\) 0 0
\(681\) −6.80801e66 −1.79499
\(682\) 0 0
\(683\) 3.36076e66 0.827087 0.413543 0.910484i \(-0.364291\pi\)
0.413543 + 0.910484i \(0.364291\pi\)
\(684\) 0 0
\(685\) −4.83633e66 −1.11119
\(686\) 0 0
\(687\) −1.05912e67 −2.27228
\(688\) 0 0
\(689\) −2.25183e64 −0.00451214
\(690\) 0 0
\(691\) −2.04623e66 −0.383020 −0.191510 0.981491i \(-0.561338\pi\)
−0.191510 + 0.981491i \(0.561338\pi\)
\(692\) 0 0
\(693\) −3.00626e66 −0.525770
\(694\) 0 0
\(695\) 6.47341e66 1.05801
\(696\) 0 0
\(697\) 9.02362e66 1.37851
\(698\) 0 0
\(699\) 4.11151e66 0.587203
\(700\) 0 0
\(701\) −1.21798e67 −1.62655 −0.813275 0.581879i \(-0.802318\pi\)
−0.813275 + 0.581879i \(0.802318\pi\)
\(702\) 0 0
\(703\) 1.39317e67 1.74002
\(704\) 0 0
\(705\) 1.87401e67 2.18942
\(706\) 0 0
\(707\) 3.26221e66 0.356578
\(708\) 0 0
\(709\) 1.52742e67 1.56233 0.781166 0.624323i \(-0.214625\pi\)
0.781166 + 0.624323i \(0.214625\pi\)
\(710\) 0 0
\(711\) 2.78278e66 0.266405
\(712\) 0 0
\(713\) −2.06331e66 −0.184910
\(714\) 0 0
\(715\) 9.99969e64 0.00839063
\(716\) 0 0
\(717\) 1.34287e67 1.05520
\(718\) 0 0
\(719\) −5.76158e66 −0.424045 −0.212023 0.977265i \(-0.568005\pi\)
−0.212023 + 0.977265i \(0.568005\pi\)
\(720\) 0 0
\(721\) −2.68901e66 −0.185402
\(722\) 0 0
\(723\) 7.44159e66 0.480746
\(724\) 0 0
\(725\) −2.48940e66 −0.150713
\(726\) 0 0
\(727\) 7.68143e65 0.0435896 0.0217948 0.999762i \(-0.493062\pi\)
0.0217948 + 0.999762i \(0.493062\pi\)
\(728\) 0 0
\(729\) −1.28906e67 −0.685768
\(730\) 0 0
\(731\) 2.54923e67 1.27160
\(732\) 0 0
\(733\) 3.73856e67 1.74889 0.874443 0.485129i \(-0.161227\pi\)
0.874443 + 0.485129i \(0.161227\pi\)
\(734\) 0 0
\(735\) −4.85408e67 −2.12988
\(736\) 0 0
\(737\) −1.42874e67 −0.588124
\(738\) 0 0
\(739\) 1.50450e67 0.581095 0.290548 0.956861i \(-0.406163\pi\)
0.290548 + 0.956861i \(0.406163\pi\)
\(740\) 0 0
\(741\) −9.97882e65 −0.0361702
\(742\) 0 0
\(743\) −5.97073e66 −0.203138 −0.101569 0.994829i \(-0.532386\pi\)
−0.101569 + 0.994829i \(0.532386\pi\)
\(744\) 0 0
\(745\) −3.63622e67 −1.16139
\(746\) 0 0
\(747\) −3.73887e67 −1.12126
\(748\) 0 0
\(749\) −3.98539e67 −1.12240
\(750\) 0 0
\(751\) 1.14848e67 0.303797 0.151898 0.988396i \(-0.451461\pi\)
0.151898 + 0.988396i \(0.451461\pi\)
\(752\) 0 0
\(753\) −7.52752e67 −1.87055
\(754\) 0 0
\(755\) −3.25690e67 −0.760415
\(756\) 0 0
\(757\) 3.51013e66 0.0770141 0.0385070 0.999258i \(-0.487740\pi\)
0.0385070 + 0.999258i \(0.487740\pi\)
\(758\) 0 0
\(759\) −1.43608e67 −0.296141
\(760\) 0 0
\(761\) 2.99687e67 0.580937 0.290468 0.956885i \(-0.406189\pi\)
0.290468 + 0.956885i \(0.406189\pi\)
\(762\) 0 0
\(763\) 2.22448e67 0.405418
\(764\) 0 0
\(765\) −8.20350e67 −1.40590
\(766\) 0 0
\(767\) 1.92657e65 0.00310523
\(768\) 0 0
\(769\) 1.28547e68 1.94892 0.974461 0.224557i \(-0.0720935\pi\)
0.974461 + 0.224557i \(0.0720935\pi\)
\(770\) 0 0
\(771\) −1.65476e68 −2.36025
\(772\) 0 0
\(773\) 6.24752e67 0.838474 0.419237 0.907877i \(-0.362298\pi\)
0.419237 + 0.907877i \(0.362298\pi\)
\(774\) 0 0
\(775\) 9.14802e66 0.115541
\(776\) 0 0
\(777\) −2.07344e68 −2.46489
\(778\) 0 0
\(779\) 1.27281e68 1.42440
\(780\) 0 0
\(781\) −3.15385e67 −0.332308
\(782\) 0 0
\(783\) 9.03883e66 0.0896831
\(784\) 0 0
\(785\) 7.49461e67 0.700345
\(786\) 0 0
\(787\) 3.00261e67 0.264297 0.132149 0.991230i \(-0.457812\pi\)
0.132149 + 0.991230i \(0.457812\pi\)
\(788\) 0 0
\(789\) 2.85175e68 2.36484
\(790\) 0 0
\(791\) 5.55233e67 0.433839
\(792\) 0 0
\(793\) 1.74601e66 0.0128566
\(794\) 0 0
\(795\) −5.67524e67 −0.393874
\(796\) 0 0
\(797\) −2.46726e67 −0.161415 −0.0807077 0.996738i \(-0.525718\pi\)
−0.0807077 + 0.996738i \(0.525718\pi\)
\(798\) 0 0
\(799\) −3.21574e68 −1.98349
\(800\) 0 0
\(801\) −1.35998e68 −0.790986
\(802\) 0 0
\(803\) 4.63103e67 0.254017
\(804\) 0 0
\(805\) −1.85543e68 −0.959936
\(806\) 0 0
\(807\) 3.37634e68 1.64785
\(808\) 0 0
\(809\) −2.87119e68 −1.32212 −0.661061 0.750332i \(-0.729894\pi\)
−0.661061 + 0.750332i \(0.729894\pi\)
\(810\) 0 0
\(811\) −4.03047e67 −0.175132 −0.0875661 0.996159i \(-0.527909\pi\)
−0.0875661 + 0.996159i \(0.527909\pi\)
\(812\) 0 0
\(813\) −4.62786e67 −0.189781
\(814\) 0 0
\(815\) 2.70218e68 1.04595
\(816\) 0 0
\(817\) 3.59575e68 1.31393
\(818\) 0 0
\(819\) 6.83312e66 0.0235748
\(820\) 0 0
\(821\) 2.55050e68 0.830923 0.415462 0.909611i \(-0.363620\pi\)
0.415462 + 0.909611i \(0.363620\pi\)
\(822\) 0 0
\(823\) −2.96708e68 −0.912922 −0.456461 0.889743i \(-0.650883\pi\)
−0.456461 + 0.889743i \(0.650883\pi\)
\(824\) 0 0
\(825\) 6.36712e67 0.185044
\(826\) 0 0
\(827\) 5.78282e68 1.58767 0.793835 0.608133i \(-0.208081\pi\)
0.793835 + 0.608133i \(0.208081\pi\)
\(828\) 0 0
\(829\) −1.50860e68 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(830\) 0 0
\(831\) −1.09089e69 −2.67397
\(832\) 0 0
\(833\) 8.32942e68 1.92956
\(834\) 0 0
\(835\) 8.11074e68 1.77594
\(836\) 0 0
\(837\) −3.32159e67 −0.0687539
\(838\) 0 0
\(839\) −5.78467e67 −0.113207 −0.0566034 0.998397i \(-0.518027\pi\)
−0.0566034 + 0.998397i \(0.518027\pi\)
\(840\) 0 0
\(841\) −4.32977e68 −0.801233
\(842\) 0 0
\(843\) 6.63705e68 1.16152
\(844\) 0 0
\(845\) 6.98598e68 1.15636
\(846\) 0 0
\(847\) 8.21157e68 1.28578
\(848\) 0 0
\(849\) 5.33764e68 0.790712
\(850\) 0 0
\(851\) −4.55719e68 −0.638782
\(852\) 0 0
\(853\) −6.70885e68 −0.889910 −0.444955 0.895553i \(-0.646780\pi\)
−0.444955 + 0.895553i \(0.646780\pi\)
\(854\) 0 0
\(855\) −1.15713e69 −1.45270
\(856\) 0 0
\(857\) 1.18679e69 1.41034 0.705168 0.709040i \(-0.250871\pi\)
0.705168 + 0.709040i \(0.250871\pi\)
\(858\) 0 0
\(859\) 1.90685e67 0.0214525 0.0107262 0.999942i \(-0.496586\pi\)
0.0107262 + 0.999942i \(0.496586\pi\)
\(860\) 0 0
\(861\) −1.89431e69 −2.01779
\(862\) 0 0
\(863\) 8.52667e68 0.860054 0.430027 0.902816i \(-0.358504\pi\)
0.430027 + 0.902816i \(0.358504\pi\)
\(864\) 0 0
\(865\) 9.69839e68 0.926448
\(866\) 0 0
\(867\) 1.55538e69 1.40731
\(868\) 0 0
\(869\) 1.46698e68 0.125736
\(870\) 0 0
\(871\) 3.24749e67 0.0263706
\(872\) 0 0
\(873\) −3.96874e68 −0.305364
\(874\) 0 0
\(875\) −1.61085e69 −1.17454
\(876\) 0 0
\(877\) 7.64441e67 0.0528267 0.0264134 0.999651i \(-0.491591\pi\)
0.0264134 + 0.999651i \(0.491591\pi\)
\(878\) 0 0
\(879\) 1.70241e69 1.11513
\(880\) 0 0
\(881\) −9.82470e68 −0.610078 −0.305039 0.952340i \(-0.598670\pi\)
−0.305039 + 0.952340i \(0.598670\pi\)
\(882\) 0 0
\(883\) 1.96558e68 0.115722 0.0578609 0.998325i \(-0.481572\pi\)
0.0578609 + 0.998325i \(0.481572\pi\)
\(884\) 0 0
\(885\) 4.85551e68 0.271062
\(886\) 0 0
\(887\) −3.52341e69 −1.86534 −0.932670 0.360731i \(-0.882527\pi\)
−0.932670 + 0.360731i \(0.882527\pi\)
\(888\) 0 0
\(889\) 9.92383e67 0.0498297
\(890\) 0 0
\(891\) −9.50829e68 −0.452873
\(892\) 0 0
\(893\) −4.53588e69 −2.04952
\(894\) 0 0
\(895\) −1.77525e69 −0.761061
\(896\) 0 0
\(897\) 3.26417e67 0.0132785
\(898\) 0 0
\(899\) −3.94715e68 −0.152381
\(900\) 0 0
\(901\) 9.73850e68 0.356829
\(902\) 0 0
\(903\) −5.35154e69 −1.86130
\(904\) 0 0
\(905\) −5.13022e69 −1.69393
\(906\) 0 0
\(907\) −1.85786e69 −0.582428 −0.291214 0.956658i \(-0.594059\pi\)
−0.291214 + 0.956658i \(0.594059\pi\)
\(908\) 0 0
\(909\) −6.65487e68 −0.198102
\(910\) 0 0
\(911\) −1.88130e69 −0.531835 −0.265918 0.963996i \(-0.585675\pi\)
−0.265918 + 0.963996i \(0.585675\pi\)
\(912\) 0 0
\(913\) −1.97099e69 −0.529202
\(914\) 0 0
\(915\) 4.40044e69 1.12228
\(916\) 0 0
\(917\) 9.84316e69 2.38482
\(918\) 0 0
\(919\) 3.45864e69 0.796143 0.398071 0.917354i \(-0.369680\pi\)
0.398071 + 0.917354i \(0.369680\pi\)
\(920\) 0 0
\(921\) −3.68557e69 −0.806126
\(922\) 0 0
\(923\) 7.16858e67 0.0149002
\(924\) 0 0
\(925\) 2.02051e69 0.399144
\(926\) 0 0
\(927\) 5.48557e68 0.103002
\(928\) 0 0
\(929\) −3.44629e69 −0.615153 −0.307576 0.951523i \(-0.599518\pi\)
−0.307576 + 0.951523i \(0.599518\pi\)
\(930\) 0 0
\(931\) 1.17489e70 1.99379
\(932\) 0 0
\(933\) 4.83597e69 0.780306
\(934\) 0 0
\(935\) −4.32458e69 −0.663546
\(936\) 0 0
\(937\) 2.88144e69 0.420464 0.210232 0.977652i \(-0.432578\pi\)
0.210232 + 0.977652i \(0.432578\pi\)
\(938\) 0 0
\(939\) 1.00479e70 1.39455
\(940\) 0 0
\(941\) 3.65396e69 0.482400 0.241200 0.970475i \(-0.422459\pi\)
0.241200 + 0.970475i \(0.422459\pi\)
\(942\) 0 0
\(943\) −4.16347e69 −0.522914
\(944\) 0 0
\(945\) −2.98694e69 −0.356927
\(946\) 0 0
\(947\) 2.53356e69 0.288075 0.144038 0.989572i \(-0.453991\pi\)
0.144038 + 0.989572i \(0.453991\pi\)
\(948\) 0 0
\(949\) −1.05262e68 −0.0113898
\(950\) 0 0
\(951\) −1.34009e70 −1.38004
\(952\) 0 0
\(953\) −4.10317e69 −0.402195 −0.201097 0.979571i \(-0.564451\pi\)
−0.201097 + 0.979571i \(0.564451\pi\)
\(954\) 0 0
\(955\) −4.12463e69 −0.384862
\(956\) 0 0
\(957\) −2.74725e69 −0.244044
\(958\) 0 0
\(959\) −1.74218e70 −1.47352
\(960\) 0 0
\(961\) −1.09660e70 −0.883180
\(962\) 0 0
\(963\) 8.13016e69 0.623565
\(964\) 0 0
\(965\) −2.01468e70 −1.47168
\(966\) 0 0
\(967\) 2.82974e69 0.196890 0.0984451 0.995142i \(-0.468613\pi\)
0.0984451 + 0.995142i \(0.468613\pi\)
\(968\) 0 0
\(969\) 4.31556e70 2.86041
\(970\) 0 0
\(971\) 2.13025e70 1.34517 0.672587 0.740018i \(-0.265183\pi\)
0.672587 + 0.740018i \(0.265183\pi\)
\(972\) 0 0
\(973\) 2.33190e70 1.40300
\(974\) 0 0
\(975\) −1.44722e68 −0.00829711
\(976\) 0 0
\(977\) 2.95878e70 1.61655 0.808277 0.588802i \(-0.200400\pi\)
0.808277 + 0.588802i \(0.200400\pi\)
\(978\) 0 0
\(979\) −7.16931e69 −0.373322
\(980\) 0 0
\(981\) −4.53793e69 −0.225235
\(982\) 0 0
\(983\) 5.99760e69 0.283772 0.141886 0.989883i \(-0.454683\pi\)
0.141886 + 0.989883i \(0.454683\pi\)
\(984\) 0 0
\(985\) −3.05206e70 −1.37671
\(986\) 0 0
\(987\) 6.75072e70 2.90333
\(988\) 0 0
\(989\) −1.17621e70 −0.482359
\(990\) 0 0
\(991\) −3.56295e70 −1.39341 −0.696706 0.717357i \(-0.745352\pi\)
−0.696706 + 0.717357i \(0.745352\pi\)
\(992\) 0 0
\(993\) 4.50689e68 0.0168101
\(994\) 0 0
\(995\) 3.60988e70 1.28425
\(996\) 0 0
\(997\) −8.80387e69 −0.298770 −0.149385 0.988779i \(-0.547729\pi\)
−0.149385 + 0.988779i \(0.547729\pi\)
\(998\) 0 0
\(999\) −7.33632e69 −0.237514
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 16.48.a.d.1.4 4
4.3 odd 2 1.48.a.a.1.3 4
12.11 even 2 9.48.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.48.a.a.1.3 4 4.3 odd 2
9.48.a.c.1.2 4 12.11 even 2
16.48.a.d.1.4 4 1.1 even 1 trivial