Properties

Label 50.10.b.d.49.2
Level $50$
Weight $10$
Character 50.49
Analytic conductor $25.752$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,10,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.7517918082\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.10.b.d.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000i q^{2} -174.000i q^{3} -256.000 q^{4} +2784.00 q^{6} +4658.00i q^{7} -4096.00i q^{8} -10593.0 q^{9} +O(q^{10})\) \(q+16.0000i q^{2} -174.000i q^{3} -256.000 q^{4} +2784.00 q^{6} +4658.00i q^{7} -4096.00i q^{8} -10593.0 q^{9} +28992.0 q^{11} +44544.0i q^{12} +164446. i q^{13} -74528.0 q^{14} +65536.0 q^{16} -594822. i q^{17} -169488. i q^{18} +295780. q^{19} +810492. q^{21} +463872. i q^{22} -2.54453e6i q^{23} -712704. q^{24} -2.63114e6 q^{26} -1.58166e6i q^{27} -1.19245e6i q^{28} +3.72297e6 q^{29} +2.33577e6 q^{31} +1.04858e6i q^{32} -5.04461e6i q^{33} +9.51715e6 q^{34} +2.71181e6 q^{36} +1.08404e7i q^{37} +4.73248e6i q^{38} +2.86136e7 q^{39} +2.15939e7 q^{41} +1.29679e7i q^{42} -1.08323e7i q^{43} -7.42195e6 q^{44} +4.07125e7 q^{46} +5.17214e6i q^{47} -1.14033e7i q^{48} +1.86566e7 q^{49} -1.03499e8 q^{51} -4.20982e7i q^{52} -9.81797e7i q^{53} +2.53066e7 q^{54} +1.90792e7 q^{56} -5.14657e7i q^{57} +5.95675e7i q^{58} -1.61629e7 q^{59} -4.39282e7 q^{61} +3.73724e7i q^{62} -4.93422e7i q^{63} -1.67772e7 q^{64} +8.07137e7 q^{66} -8.15574e7i q^{67} +1.52274e8i q^{68} -4.42749e8 q^{69} +1.61308e8 q^{71} +4.33889e7i q^{72} +2.47148e8i q^{73} -1.73447e8 q^{74} -7.57197e7 q^{76} +1.35045e8i q^{77} +4.57818e8i q^{78} +5.83346e8 q^{79} -4.83711e8 q^{81} +3.45502e8i q^{82} +1.45718e7i q^{83} -2.07486e8 q^{84} +1.73317e8 q^{86} -6.47797e8i q^{87} -1.18751e8i q^{88} -4.70134e8 q^{89} -7.65989e8 q^{91} +6.51401e8i q^{92} -4.06424e8i q^{93} -8.27542e7 q^{94} +1.82452e8 q^{96} -1.17838e8i q^{97} +2.98506e8i q^{98} -3.07112e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{4} + 5568 q^{6} - 21186 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 512 q^{4} + 5568 q^{6} - 21186 q^{9} + 57984 q^{11} - 149056 q^{14} + 131072 q^{16} + 591560 q^{19} + 1620984 q^{21} - 1425408 q^{24} - 5262272 q^{26} + 7445940 q^{29} + 4671544 q^{31} + 19034304 q^{34} + 5423616 q^{36} + 57227208 q^{39} + 43187724 q^{41} - 14843904 q^{44} + 81425088 q^{46} + 37313286 q^{49} - 206998056 q^{51} + 50613120 q^{54} + 38158336 q^{56} - 32325720 q^{59} - 87856316 q^{61} - 33554432 q^{64} + 161427456 q^{66} - 885497832 q^{69} + 322615464 q^{71} - 346893376 q^{74} - 151439360 q^{76} + 1166691440 q^{79} - 967421718 q^{81} - 414971904 q^{84} + 346633408 q^{86} - 940267380 q^{89} - 1531978936 q^{91} - 165508416 q^{94} + 364904448 q^{96} - 614224512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000i 0.707107i
\(3\) − 174.000i − 1.24023i −0.784509 0.620117i \(-0.787085\pi\)
0.784509 0.620117i \(-0.212915\pi\)
\(4\) −256.000 −0.500000
\(5\) 0 0
\(6\) 2784.00 0.876978
\(7\) 4658.00i 0.733261i 0.930367 + 0.366630i \(0.119489\pi\)
−0.930367 + 0.366630i \(0.880511\pi\)
\(8\) − 4096.00i − 0.353553i
\(9\) −10593.0 −0.538180
\(10\) 0 0
\(11\) 28992.0 0.597051 0.298525 0.954402i \(-0.403505\pi\)
0.298525 + 0.954402i \(0.403505\pi\)
\(12\) 44544.0i 0.620117i
\(13\) 164446.i 1.59690i 0.602060 + 0.798451i \(0.294347\pi\)
−0.602060 + 0.798451i \(0.705653\pi\)
\(14\) −74528.0 −0.518493
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) − 594822.i − 1.72730i −0.504095 0.863648i \(-0.668174\pi\)
0.504095 0.863648i \(-0.331826\pi\)
\(18\) − 169488.i − 0.380551i
\(19\) 295780. 0.520688 0.260344 0.965516i \(-0.416164\pi\)
0.260344 + 0.965516i \(0.416164\pi\)
\(20\) 0 0
\(21\) 810492. 0.909415
\(22\) 463872.i 0.422178i
\(23\) − 2.54453e6i − 1.89598i −0.318305 0.947988i \(-0.603114\pi\)
0.318305 0.947988i \(-0.396886\pi\)
\(24\) −712704. −0.438489
\(25\) 0 0
\(26\) −2.63114e6 −1.12918
\(27\) − 1.58166e6i − 0.572765i
\(28\) − 1.19245e6i − 0.366630i
\(29\) 3.72297e6 0.977459 0.488729 0.872435i \(-0.337461\pi\)
0.488729 + 0.872435i \(0.337461\pi\)
\(30\) 0 0
\(31\) 2.33577e6 0.454258 0.227129 0.973865i \(-0.427066\pi\)
0.227129 + 0.973865i \(0.427066\pi\)
\(32\) 1.04858e6i 0.176777i
\(33\) − 5.04461e6i − 0.740482i
\(34\) 9.51715e6 1.22138
\(35\) 0 0
\(36\) 2.71181e6 0.269090
\(37\) 1.08404e7i 0.950907i 0.879741 + 0.475454i \(0.157716\pi\)
−0.879741 + 0.475454i \(0.842284\pi\)
\(38\) 4.73248e6i 0.368182i
\(39\) 2.86136e7 1.98053
\(40\) 0 0
\(41\) 2.15939e7 1.19345 0.596723 0.802447i \(-0.296469\pi\)
0.596723 + 0.802447i \(0.296469\pi\)
\(42\) 1.29679e7i 0.643053i
\(43\) − 1.08323e7i − 0.483184i −0.970378 0.241592i \(-0.922330\pi\)
0.970378 0.241592i \(-0.0776695\pi\)
\(44\) −7.42195e6 −0.298525
\(45\) 0 0
\(46\) 4.07125e7 1.34066
\(47\) 5.17214e6i 0.154607i 0.997008 + 0.0773036i \(0.0246311\pi\)
−0.997008 + 0.0773036i \(0.975369\pi\)
\(48\) − 1.14033e7i − 0.310058i
\(49\) 1.86566e7 0.462329
\(50\) 0 0
\(51\) −1.03499e8 −2.14225
\(52\) − 4.20982e7i − 0.798451i
\(53\) − 9.81797e7i − 1.70915i −0.519328 0.854575i \(-0.673818\pi\)
0.519328 0.854575i \(-0.326182\pi\)
\(54\) 2.53066e7 0.405006
\(55\) 0 0
\(56\) 1.90792e7 0.259247
\(57\) − 5.14657e7i − 0.645775i
\(58\) 5.95675e7i 0.691168i
\(59\) −1.61629e7 −0.173654 −0.0868269 0.996223i \(-0.527673\pi\)
−0.0868269 + 0.996223i \(0.527673\pi\)
\(60\) 0 0
\(61\) −4.39282e7 −0.406218 −0.203109 0.979156i \(-0.565105\pi\)
−0.203109 + 0.979156i \(0.565105\pi\)
\(62\) 3.73724e7i 0.321209i
\(63\) − 4.93422e7i − 0.394626i
\(64\) −1.67772e7 −0.125000
\(65\) 0 0
\(66\) 8.07137e7 0.523600
\(67\) − 8.15574e7i − 0.494455i −0.968957 0.247228i \(-0.920480\pi\)
0.968957 0.247228i \(-0.0795195\pi\)
\(68\) 1.52274e8i 0.863648i
\(69\) −4.42749e8 −2.35145
\(70\) 0 0
\(71\) 1.61308e8 0.753343 0.376671 0.926347i \(-0.377069\pi\)
0.376671 + 0.926347i \(0.377069\pi\)
\(72\) 4.33889e7i 0.190275i
\(73\) 2.47148e8i 1.01860i 0.860589 + 0.509301i \(0.170096\pi\)
−0.860589 + 0.509301i \(0.829904\pi\)
\(74\) −1.73447e8 −0.672393
\(75\) 0 0
\(76\) −7.57197e7 −0.260344
\(77\) 1.35045e8i 0.437794i
\(78\) 4.57818e8i 1.40045i
\(79\) 5.83346e8 1.68502 0.842508 0.538684i \(-0.181078\pi\)
0.842508 + 0.538684i \(0.181078\pi\)
\(80\) 0 0
\(81\) −4.83711e8 −1.24854
\(82\) 3.45502e8i 0.843894i
\(83\) 1.45718e7i 0.0337024i 0.999858 + 0.0168512i \(0.00536416\pi\)
−0.999858 + 0.0168512i \(0.994636\pi\)
\(84\) −2.07486e8 −0.454707
\(85\) 0 0
\(86\) 1.73317e8 0.341663
\(87\) − 6.47797e8i − 1.21228i
\(88\) − 1.18751e8i − 0.211089i
\(89\) −4.70134e8 −0.794267 −0.397133 0.917761i \(-0.629995\pi\)
−0.397133 + 0.917761i \(0.629995\pi\)
\(90\) 0 0
\(91\) −7.65989e8 −1.17095
\(92\) 6.51401e8i 0.947988i
\(93\) − 4.06424e8i − 0.563386i
\(94\) −8.27542e7 −0.109324
\(95\) 0 0
\(96\) 1.82452e8 0.219244
\(97\) − 1.17838e8i − 0.135149i −0.997714 0.0675747i \(-0.978474\pi\)
0.997714 0.0675747i \(-0.0215261\pi\)
\(98\) 2.98506e8i 0.326916i
\(99\) −3.07112e8 −0.321321
\(100\) 0 0
\(101\) −8.60927e7 −0.0823228 −0.0411614 0.999153i \(-0.513106\pi\)
−0.0411614 + 0.999153i \(0.513106\pi\)
\(102\) − 1.65598e9i − 1.51480i
\(103\) − 1.92872e9i − 1.68850i −0.535947 0.844252i \(-0.680045\pi\)
0.535947 0.844252i \(-0.319955\pi\)
\(104\) 6.73571e8 0.564590
\(105\) 0 0
\(106\) 1.57087e9 1.20855
\(107\) 1.39685e9i 1.03020i 0.857130 + 0.515100i \(0.172245\pi\)
−0.857130 + 0.515100i \(0.827755\pi\)
\(108\) 4.04905e8i 0.286382i
\(109\) 6.04327e8 0.410065 0.205033 0.978755i \(-0.434270\pi\)
0.205033 + 0.978755i \(0.434270\pi\)
\(110\) 0 0
\(111\) 1.88623e9 1.17935
\(112\) 3.05267e8i 0.183315i
\(113\) 1.68580e9i 0.972643i 0.873780 + 0.486322i \(0.161662\pi\)
−0.873780 + 0.486322i \(0.838338\pi\)
\(114\) 8.23452e8 0.456632
\(115\) 0 0
\(116\) −9.53080e8 −0.488729
\(117\) − 1.74198e9i − 0.859421i
\(118\) − 2.58606e8i − 0.122792i
\(119\) 2.77068e9 1.26656
\(120\) 0 0
\(121\) −1.51741e9 −0.643531
\(122\) − 7.02851e8i − 0.287239i
\(123\) − 3.75733e9i − 1.48015i
\(124\) −5.97958e8 −0.227129
\(125\) 0 0
\(126\) 7.89475e8 0.279043
\(127\) 3.70716e9i 1.26452i 0.774758 + 0.632258i \(0.217872\pi\)
−0.774758 + 0.632258i \(0.782128\pi\)
\(128\) − 2.68435e8i − 0.0883883i
\(129\) −1.88482e9 −0.599261
\(130\) 0 0
\(131\) 2.54080e9 0.753789 0.376895 0.926256i \(-0.376992\pi\)
0.376895 + 0.926256i \(0.376992\pi\)
\(132\) 1.29142e9i 0.370241i
\(133\) 1.37774e9i 0.381800i
\(134\) 1.30492e9 0.349633
\(135\) 0 0
\(136\) −2.43639e9 −0.610692
\(137\) − 1.15390e9i − 0.279849i −0.990162 0.139925i \(-0.955314\pi\)
0.990162 0.139925i \(-0.0446860\pi\)
\(138\) − 7.08398e9i − 1.66273i
\(139\) 5.62721e9 1.27858 0.639288 0.768968i \(-0.279229\pi\)
0.639288 + 0.768968i \(0.279229\pi\)
\(140\) 0 0
\(141\) 8.99952e8 0.191749
\(142\) 2.58092e9i 0.532694i
\(143\) 4.76762e9i 0.953431i
\(144\) −6.94223e8 −0.134545
\(145\) 0 0
\(146\) −3.95437e9 −0.720260
\(147\) − 3.24626e9i − 0.573396i
\(148\) − 2.77515e9i − 0.475454i
\(149\) 2.13688e9 0.355174 0.177587 0.984105i \(-0.443171\pi\)
0.177587 + 0.984105i \(0.443171\pi\)
\(150\) 0 0
\(151\) −9.67515e6 −0.00151447 −0.000757236 1.00000i \(-0.500241\pi\)
−0.000757236 1.00000i \(0.500241\pi\)
\(152\) − 1.21151e9i − 0.184091i
\(153\) 6.30095e9i 0.929597i
\(154\) −2.16072e9 −0.309567
\(155\) 0 0
\(156\) −7.32508e9 −0.990266
\(157\) − 6.88488e8i − 0.0904373i −0.998977 0.0452187i \(-0.985602\pi\)
0.998977 0.0452187i \(-0.0143985\pi\)
\(158\) 9.33353e9i 1.19149i
\(159\) −1.70833e10 −2.11975
\(160\) 0 0
\(161\) 1.18524e10 1.39024
\(162\) − 7.73937e9i − 0.882853i
\(163\) 1.43082e10i 1.58759i 0.608182 + 0.793797i \(0.291899\pi\)
−0.608182 + 0.793797i \(0.708101\pi\)
\(164\) −5.52803e9 −0.596723
\(165\) 0 0
\(166\) −2.33149e8 −0.0238312
\(167\) 9.98735e9i 0.993633i 0.867856 + 0.496817i \(0.165498\pi\)
−0.867856 + 0.496817i \(0.834502\pi\)
\(168\) − 3.31978e9i − 0.321527i
\(169\) −1.64380e10 −1.55010
\(170\) 0 0
\(171\) −3.13320e9 −0.280224
\(172\) 2.77307e9i 0.241592i
\(173\) − 3.51396e9i − 0.298256i −0.988818 0.149128i \(-0.952353\pi\)
0.988818 0.149128i \(-0.0476466\pi\)
\(174\) 1.03647e10 0.857210
\(175\) 0 0
\(176\) 1.90002e9 0.149263
\(177\) 2.81234e9i 0.215371i
\(178\) − 7.52214e9i − 0.561631i
\(179\) −1.19502e9 −0.0870038 −0.0435019 0.999053i \(-0.513851\pi\)
−0.0435019 + 0.999053i \(0.513851\pi\)
\(180\) 0 0
\(181\) −9.12053e9 −0.631635 −0.315818 0.948820i \(-0.602279\pi\)
−0.315818 + 0.948820i \(0.602279\pi\)
\(182\) − 1.22558e10i − 0.827983i
\(183\) 7.64350e9i 0.503805i
\(184\) −1.04224e10 −0.670329
\(185\) 0 0
\(186\) 6.50279e9 0.398374
\(187\) − 1.72451e10i − 1.03128i
\(188\) − 1.32407e9i − 0.0773036i
\(189\) 7.36737e9 0.419986
\(190\) 0 0
\(191\) −9.37431e9 −0.509670 −0.254835 0.966985i \(-0.582021\pi\)
−0.254835 + 0.966985i \(0.582021\pi\)
\(192\) 2.91924e9i 0.155029i
\(193\) − 2.40000e10i − 1.24510i −0.782580 0.622550i \(-0.786097\pi\)
0.782580 0.622550i \(-0.213903\pi\)
\(194\) 1.88542e9 0.0955651
\(195\) 0 0
\(196\) −4.77610e9 −0.231165
\(197\) − 5.56124e8i − 0.0263071i −0.999913 0.0131536i \(-0.995813\pi\)
0.999913 0.0131536i \(-0.00418703\pi\)
\(198\) − 4.91380e9i − 0.227208i
\(199\) 2.51255e10 1.13573 0.567866 0.823121i \(-0.307769\pi\)
0.567866 + 0.823121i \(0.307769\pi\)
\(200\) 0 0
\(201\) −1.41910e10 −0.613240
\(202\) − 1.37748e9i − 0.0582110i
\(203\) 1.73416e10i 0.716732i
\(204\) 2.64958e10 1.07113
\(205\) 0 0
\(206\) 3.08595e10 1.19395
\(207\) 2.69542e10i 1.02038i
\(208\) 1.07771e10i 0.399225i
\(209\) 8.57525e9 0.310877
\(210\) 0 0
\(211\) −1.63915e10 −0.569309 −0.284654 0.958630i \(-0.591879\pi\)
−0.284654 + 0.958630i \(0.591879\pi\)
\(212\) 2.51340e10i 0.854575i
\(213\) − 2.80675e10i − 0.934321i
\(214\) −2.23495e10 −0.728461
\(215\) 0 0
\(216\) −6.47848e9 −0.202503
\(217\) 1.08800e10i 0.333090i
\(218\) 9.66924e9i 0.289960i
\(219\) 4.30037e10 1.26330
\(220\) 0 0
\(221\) 9.78161e10 2.75832
\(222\) 3.01797e10i 0.833925i
\(223\) − 4.65257e10i − 1.25986i −0.776654 0.629928i \(-0.783084\pi\)
0.776654 0.629928i \(-0.216916\pi\)
\(224\) −4.88427e9 −0.129623
\(225\) 0 0
\(226\) −2.69728e10 −0.687763
\(227\) − 1.91415e9i − 0.0478475i −0.999714 0.0239237i \(-0.992384\pi\)
0.999714 0.0239237i \(-0.00761589\pi\)
\(228\) 1.31752e10i 0.322887i
\(229\) 1.45825e10 0.350406 0.175203 0.984532i \(-0.443942\pi\)
0.175203 + 0.984532i \(0.443942\pi\)
\(230\) 0 0
\(231\) 2.34978e10 0.542966
\(232\) − 1.52493e10i − 0.345584i
\(233\) − 4.12790e10i − 0.917545i −0.888554 0.458773i \(-0.848289\pi\)
0.888554 0.458773i \(-0.151711\pi\)
\(234\) 2.78716e10 0.607702
\(235\) 0 0
\(236\) 4.13769e9 0.0868269
\(237\) − 1.01502e11i − 2.08981i
\(238\) 4.43309e10i 0.895592i
\(239\) −3.65502e10 −0.724602 −0.362301 0.932061i \(-0.618009\pi\)
−0.362301 + 0.932061i \(0.618009\pi\)
\(240\) 0 0
\(241\) −8.66070e10 −1.65377 −0.826887 0.562368i \(-0.809890\pi\)
−0.826887 + 0.562368i \(0.809890\pi\)
\(242\) − 2.42786e10i − 0.455045i
\(243\) 5.30339e10i 0.975720i
\(244\) 1.12456e10 0.203109
\(245\) 0 0
\(246\) 6.01173e10 1.04663
\(247\) 4.86398e10i 0.831488i
\(248\) − 9.56732e9i − 0.160605i
\(249\) 2.53549e9 0.0417989
\(250\) 0 0
\(251\) −7.20769e10 −1.14621 −0.573105 0.819482i \(-0.694261\pi\)
−0.573105 + 0.819482i \(0.694261\pi\)
\(252\) 1.26316e10i 0.197313i
\(253\) − 7.37711e10i − 1.13199i
\(254\) −5.93145e10 −0.894148
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) − 1.12729e11i − 1.61190i −0.591986 0.805948i \(-0.701656\pi\)
0.591986 0.805948i \(-0.298344\pi\)
\(258\) − 3.01571e10i − 0.423741i
\(259\) −5.04947e10 −0.697263
\(260\) 0 0
\(261\) −3.94374e10 −0.526049
\(262\) 4.06528e10i 0.533009i
\(263\) − 5.55225e10i − 0.715596i −0.933799 0.357798i \(-0.883528\pi\)
0.933799 0.357798i \(-0.116472\pi\)
\(264\) −2.06627e10 −0.261800
\(265\) 0 0
\(266\) −2.20439e10 −0.269973
\(267\) 8.18033e10i 0.985076i
\(268\) 2.08787e10i 0.247228i
\(269\) −2.79726e10 −0.325722 −0.162861 0.986649i \(-0.552072\pi\)
−0.162861 + 0.986649i \(0.552072\pi\)
\(270\) 0 0
\(271\) 3.32884e10 0.374914 0.187457 0.982273i \(-0.439975\pi\)
0.187457 + 0.982273i \(0.439975\pi\)
\(272\) − 3.89823e10i − 0.431824i
\(273\) 1.33282e11i 1.45225i
\(274\) 1.84623e10 0.197883
\(275\) 0 0
\(276\) 1.13344e11 1.17573
\(277\) 5.46240e10i 0.557474i 0.960367 + 0.278737i \(0.0899158\pi\)
−0.960367 + 0.278737i \(0.910084\pi\)
\(278\) 9.00353e10i 0.904090i
\(279\) −2.47428e10 −0.244473
\(280\) 0 0
\(281\) −8.37818e10 −0.801625 −0.400813 0.916160i \(-0.631272\pi\)
−0.400813 + 0.916160i \(0.631272\pi\)
\(282\) 1.43992e10i 0.135587i
\(283\) 8.36086e10i 0.774840i 0.921903 + 0.387420i \(0.126634\pi\)
−0.921903 + 0.387420i \(0.873366\pi\)
\(284\) −4.12948e10 −0.376671
\(285\) 0 0
\(286\) −7.62819e10 −0.674178
\(287\) 1.00584e11i 0.875107i
\(288\) − 1.11076e10i − 0.0951377i
\(289\) −2.35225e11 −1.98355
\(290\) 0 0
\(291\) −2.05039e10 −0.167617
\(292\) − 6.32699e10i − 0.509301i
\(293\) 2.28547e11i 1.81164i 0.423666 + 0.905819i \(0.360743\pi\)
−0.423666 + 0.905819i \(0.639257\pi\)
\(294\) 5.19401e10 0.405452
\(295\) 0 0
\(296\) 4.44024e10 0.336197
\(297\) − 4.58555e10i − 0.341969i
\(298\) 3.41900e10i 0.251146i
\(299\) 4.18438e11 3.02769
\(300\) 0 0
\(301\) 5.04568e10 0.354300
\(302\) − 1.54802e8i − 0.00107089i
\(303\) 1.49801e10i 0.102100i
\(304\) 1.93842e10 0.130172
\(305\) 0 0
\(306\) −1.00815e11 −0.657324
\(307\) 1.95064e11i 1.25330i 0.779302 + 0.626648i \(0.215574\pi\)
−0.779302 + 0.626648i \(0.784426\pi\)
\(308\) − 3.45715e10i − 0.218897i
\(309\) −3.35598e11 −2.09414
\(310\) 0 0
\(311\) 2.15637e11 1.30708 0.653540 0.756892i \(-0.273283\pi\)
0.653540 + 0.756892i \(0.273283\pi\)
\(312\) − 1.17201e11i − 0.700224i
\(313\) − 1.91755e11i − 1.12927i −0.825341 0.564635i \(-0.809017\pi\)
0.825341 0.564635i \(-0.190983\pi\)
\(314\) 1.10158e10 0.0639488
\(315\) 0 0
\(316\) −1.49337e11 −0.842508
\(317\) − 3.38886e11i − 1.88489i −0.334358 0.942446i \(-0.608519\pi\)
0.334358 0.942446i \(-0.391481\pi\)
\(318\) − 2.73332e11i − 1.49889i
\(319\) 1.07936e11 0.583592
\(320\) 0 0
\(321\) 2.43051e11 1.27769
\(322\) 1.89639e11i 0.983052i
\(323\) − 1.75936e11i − 0.899383i
\(324\) 1.23830e11 0.624271
\(325\) 0 0
\(326\) −2.28931e11 −1.12260
\(327\) − 1.05153e11i − 0.508577i
\(328\) − 8.84485e10i − 0.421947i
\(329\) −2.40918e10 −0.113367
\(330\) 0 0
\(331\) −1.78427e11 −0.817022 −0.408511 0.912753i \(-0.633952\pi\)
−0.408511 + 0.912753i \(0.633952\pi\)
\(332\) − 3.73038e9i − 0.0168512i
\(333\) − 1.14833e11i − 0.511760i
\(334\) −1.59798e11 −0.702605
\(335\) 0 0
\(336\) 5.31164e10 0.227354
\(337\) 2.64693e11i 1.11791i 0.829196 + 0.558957i \(0.188798\pi\)
−0.829196 + 0.558957i \(0.811202\pi\)
\(338\) − 2.63008e11i − 1.09608i
\(339\) 2.93330e11 1.20631
\(340\) 0 0
\(341\) 6.77187e10 0.271215
\(342\) − 5.01312e10i − 0.198148i
\(343\) 2.74870e11i 1.07227i
\(344\) −4.43691e10 −0.170831
\(345\) 0 0
\(346\) 5.62233e10 0.210899
\(347\) − 8.11912e10i − 0.300626i −0.988638 0.150313i \(-0.951972\pi\)
0.988638 0.150313i \(-0.0480281\pi\)
\(348\) 1.65836e11i 0.606139i
\(349\) 2.26688e11 0.817926 0.408963 0.912551i \(-0.365890\pi\)
0.408963 + 0.912551i \(0.365890\pi\)
\(350\) 0 0
\(351\) 2.60098e11 0.914649
\(352\) 3.04003e10i 0.105545i
\(353\) − 3.28733e10i − 0.112683i −0.998412 0.0563413i \(-0.982056\pi\)
0.998412 0.0563413i \(-0.0179435\pi\)
\(354\) −4.49974e10 −0.152290
\(355\) 0 0
\(356\) 1.20354e11 0.397133
\(357\) − 4.82098e11i − 1.57083i
\(358\) − 1.91204e10i − 0.0615210i
\(359\) 3.12096e11 0.991661 0.495831 0.868419i \(-0.334864\pi\)
0.495831 + 0.868419i \(0.334864\pi\)
\(360\) 0 0
\(361\) −2.35202e11 −0.728884
\(362\) − 1.45928e11i − 0.446634i
\(363\) 2.64030e11i 0.798129i
\(364\) 1.96093e11 0.585473
\(365\) 0 0
\(366\) −1.22296e11 −0.356244
\(367\) 6.28209e11i 1.80762i 0.427934 + 0.903810i \(0.359241\pi\)
−0.427934 + 0.903810i \(0.640759\pi\)
\(368\) − 1.66759e11i − 0.473994i
\(369\) −2.28744e11 −0.642289
\(370\) 0 0
\(371\) 4.57321e11 1.25325
\(372\) 1.04045e11i 0.281693i
\(373\) 9.84770e10i 0.263418i 0.991288 + 0.131709i \(0.0420464\pi\)
−0.991288 + 0.131709i \(0.957954\pi\)
\(374\) 2.75921e11 0.729227
\(375\) 0 0
\(376\) 2.11851e10 0.0546619
\(377\) 6.12228e11i 1.56091i
\(378\) 1.17878e11i 0.296975i
\(379\) −2.89024e11 −0.719544 −0.359772 0.933040i \(-0.617145\pi\)
−0.359772 + 0.933040i \(0.617145\pi\)
\(380\) 0 0
\(381\) 6.45046e11 1.56830
\(382\) − 1.49989e11i − 0.360391i
\(383\) − 2.01350e11i − 0.478141i −0.971002 0.239071i \(-0.923157\pi\)
0.971002 0.239071i \(-0.0768427\pi\)
\(384\) −4.67078e10 −0.109622
\(385\) 0 0
\(386\) 3.84001e11 0.880418
\(387\) 1.14746e11i 0.260040i
\(388\) 3.01666e10i 0.0675747i
\(389\) −3.93769e11 −0.871904 −0.435952 0.899970i \(-0.643588\pi\)
−0.435952 + 0.899970i \(0.643588\pi\)
\(390\) 0 0
\(391\) −1.51354e12 −3.27491
\(392\) − 7.64176e10i − 0.163458i
\(393\) − 4.42099e11i − 0.934875i
\(394\) 8.89798e9 0.0186019
\(395\) 0 0
\(396\) 7.86207e10 0.160660
\(397\) − 5.15625e11i − 1.04178i −0.853623 0.520891i \(-0.825600\pi\)
0.853623 0.520891i \(-0.174400\pi\)
\(398\) 4.02008e11i 0.803084i
\(399\) 2.39727e11 0.473521
\(400\) 0 0
\(401\) −8.43309e11 −1.62869 −0.814343 0.580384i \(-0.802902\pi\)
−0.814343 + 0.580384i \(0.802902\pi\)
\(402\) − 2.27056e11i − 0.433626i
\(403\) 3.84108e11i 0.725406i
\(404\) 2.20397e10 0.0411614
\(405\) 0 0
\(406\) −2.77466e11 −0.506806
\(407\) 3.14285e11i 0.567740i
\(408\) 4.23932e11i 0.757400i
\(409\) −5.64549e11 −0.997577 −0.498789 0.866724i \(-0.666222\pi\)
−0.498789 + 0.866724i \(0.666222\pi\)
\(410\) 0 0
\(411\) −2.00778e11 −0.347078
\(412\) 4.93753e11i 0.844252i
\(413\) − 7.52866e10i − 0.127333i
\(414\) −4.31268e11 −0.721516
\(415\) 0 0
\(416\) −1.72434e11 −0.282295
\(417\) − 9.79134e11i − 1.58573i
\(418\) 1.37204e11i 0.219823i
\(419\) −6.92475e11 −1.09759 −0.548796 0.835956i \(-0.684914\pi\)
−0.548796 + 0.835956i \(0.684914\pi\)
\(420\) 0 0
\(421\) 4.08125e11 0.633176 0.316588 0.948563i \(-0.397463\pi\)
0.316588 + 0.948563i \(0.397463\pi\)
\(422\) − 2.62264e11i − 0.402562i
\(423\) − 5.47885e10i − 0.0832065i
\(424\) −4.02144e11 −0.604276
\(425\) 0 0
\(426\) 4.49081e11 0.660665
\(427\) − 2.04617e11i − 0.297863i
\(428\) − 3.57592e11i − 0.515100i
\(429\) 8.29566e11 1.18248
\(430\) 0 0
\(431\) 4.41055e11 0.615665 0.307833 0.951441i \(-0.400396\pi\)
0.307833 + 0.951441i \(0.400396\pi\)
\(432\) − 1.03656e11i − 0.143191i
\(433\) 7.15390e10i 0.0978019i 0.998804 + 0.0489009i \(0.0155719\pi\)
−0.998804 + 0.0489009i \(0.984428\pi\)
\(434\) −1.74080e11 −0.235530
\(435\) 0 0
\(436\) −1.54708e11 −0.205033
\(437\) − 7.52622e11i − 0.987212i
\(438\) 6.88060e11i 0.893291i
\(439\) −4.74967e11 −0.610342 −0.305171 0.952298i \(-0.598714\pi\)
−0.305171 + 0.952298i \(0.598714\pi\)
\(440\) 0 0
\(441\) −1.97630e11 −0.248816
\(442\) 1.56506e12i 1.95043i
\(443\) 1.97072e11i 0.243113i 0.992585 + 0.121556i \(0.0387885\pi\)
−0.992585 + 0.121556i \(0.961211\pi\)
\(444\) −4.82876e11 −0.589674
\(445\) 0 0
\(446\) 7.44411e11 0.890853
\(447\) − 3.71817e11i − 0.440499i
\(448\) − 7.81483e10i − 0.0916576i
\(449\) 6.20156e11 0.720100 0.360050 0.932933i \(-0.382760\pi\)
0.360050 + 0.932933i \(0.382760\pi\)
\(450\) 0 0
\(451\) 6.26049e11 0.712548
\(452\) − 4.31565e11i − 0.486322i
\(453\) 1.68348e9i 0.00187830i
\(454\) 3.06264e10 0.0338333
\(455\) 0 0
\(456\) −2.10804e11 −0.228316
\(457\) 7.88006e11i 0.845097i 0.906340 + 0.422549i \(0.138864\pi\)
−0.906340 + 0.422549i \(0.861136\pi\)
\(458\) 2.33319e11i 0.247774i
\(459\) −9.40806e11 −0.989334
\(460\) 0 0
\(461\) 1.53496e10 0.0158286 0.00791429 0.999969i \(-0.497481\pi\)
0.00791429 + 0.999969i \(0.497481\pi\)
\(462\) 3.75965e11i 0.383935i
\(463\) − 1.94494e11i − 0.196694i −0.995152 0.0983471i \(-0.968644\pi\)
0.995152 0.0983471i \(-0.0313555\pi\)
\(464\) 2.43989e11 0.244365
\(465\) 0 0
\(466\) 6.60464e11 0.648802
\(467\) 1.06503e11i 0.103618i 0.998657 + 0.0518089i \(0.0164987\pi\)
−0.998657 + 0.0518089i \(0.983501\pi\)
\(468\) 4.45946e11i 0.429710i
\(469\) 3.79894e11 0.362564
\(470\) 0 0
\(471\) −1.19797e11 −0.112163
\(472\) 6.62031e10i 0.0613959i
\(473\) − 3.14050e11i − 0.288485i
\(474\) 1.62403e12 1.47772
\(475\) 0 0
\(476\) −7.09294e11 −0.633279
\(477\) 1.04002e12i 0.919831i
\(478\) − 5.84803e11i − 0.512371i
\(479\) 8.31146e11 0.721386 0.360693 0.932685i \(-0.382540\pi\)
0.360693 + 0.932685i \(0.382540\pi\)
\(480\) 0 0
\(481\) −1.78266e12 −1.51851
\(482\) − 1.38571e12i − 1.16939i
\(483\) − 2.06232e12i − 1.72423i
\(484\) 3.88457e11 0.321765
\(485\) 0 0
\(486\) −8.48542e11 −0.689938
\(487\) − 1.38566e12i − 1.11629i −0.829743 0.558146i \(-0.811513\pi\)
0.829743 0.558146i \(-0.188487\pi\)
\(488\) 1.79930e11i 0.143620i
\(489\) 2.48962e12 1.96899
\(490\) 0 0
\(491\) 3.92804e11 0.305007 0.152503 0.988303i \(-0.451266\pi\)
0.152503 + 0.988303i \(0.451266\pi\)
\(492\) 9.61877e11i 0.740076i
\(493\) − 2.21450e12i − 1.68836i
\(494\) −7.78237e11 −0.587951
\(495\) 0 0
\(496\) 1.53077e11 0.113565
\(497\) 7.51371e11i 0.552397i
\(498\) 4.05679e10i 0.0295563i
\(499\) 7.52029e11 0.542978 0.271489 0.962442i \(-0.412484\pi\)
0.271489 + 0.962442i \(0.412484\pi\)
\(500\) 0 0
\(501\) 1.73780e12 1.23234
\(502\) − 1.15323e12i − 0.810493i
\(503\) 1.83429e12i 1.27765i 0.769351 + 0.638826i \(0.220580\pi\)
−0.769351 + 0.638826i \(0.779420\pi\)
\(504\) −2.02106e11 −0.139521
\(505\) 0 0
\(506\) 1.18034e12 0.800441
\(507\) 2.86021e12i 1.92248i
\(508\) − 9.49033e11i − 0.632258i
\(509\) 6.19864e11 0.409323 0.204662 0.978833i \(-0.434391\pi\)
0.204662 + 0.978833i \(0.434391\pi\)
\(510\) 0 0
\(511\) −1.15122e12 −0.746900
\(512\) 6.87195e10i 0.0441942i
\(513\) − 4.67823e11i − 0.298232i
\(514\) 1.80367e12 1.13978
\(515\) 0 0
\(516\) 4.82514e11 0.299630
\(517\) 1.49951e11i 0.0923083i
\(518\) − 8.07915e11i − 0.493039i
\(519\) −6.11428e11 −0.369907
\(520\) 0 0
\(521\) 5.25683e11 0.312575 0.156287 0.987712i \(-0.450047\pi\)
0.156287 + 0.987712i \(0.450047\pi\)
\(522\) − 6.30999e11i − 0.371973i
\(523\) − 1.68426e11i − 0.0984353i −0.998788 0.0492177i \(-0.984327\pi\)
0.998788 0.0492177i \(-0.0156728\pi\)
\(524\) −6.50445e11 −0.376895
\(525\) 0 0
\(526\) 8.88360e11 0.506003
\(527\) − 1.38937e12i − 0.784639i
\(528\) − 3.30603e11i − 0.185121i
\(529\) −4.67350e12 −2.59473
\(530\) 0 0
\(531\) 1.71213e11 0.0934570
\(532\) − 3.52702e11i − 0.190900i
\(533\) 3.55102e12i 1.90582i
\(534\) −1.30885e12 −0.696554
\(535\) 0 0
\(536\) −3.34059e11 −0.174816
\(537\) 2.07934e11i 0.107905i
\(538\) − 4.47561e11i − 0.230320i
\(539\) 5.40893e11 0.276034
\(540\) 0 0
\(541\) 2.20612e12 1.10724 0.553620 0.832769i \(-0.313246\pi\)
0.553620 + 0.832769i \(0.313246\pi\)
\(542\) 5.32615e11i 0.265104i
\(543\) 1.58697e12i 0.783375i
\(544\) 6.23716e11 0.305346
\(545\) 0 0
\(546\) −2.13251e12 −1.02689
\(547\) − 3.51639e12i − 1.67940i −0.543049 0.839701i \(-0.682730\pi\)
0.543049 0.839701i \(-0.317270\pi\)
\(548\) 2.95397e11i 0.139925i
\(549\) 4.65331e11 0.218618
\(550\) 0 0
\(551\) 1.10118e12 0.508951
\(552\) 1.81350e12i 0.831365i
\(553\) 2.71722e12i 1.23556i
\(554\) −8.73985e11 −0.394194
\(555\) 0 0
\(556\) −1.44057e12 −0.639288
\(557\) 6.29996e11i 0.277325i 0.990340 + 0.138663i \(0.0442803\pi\)
−0.990340 + 0.138663i \(0.955720\pi\)
\(558\) − 3.95885e11i − 0.172868i
\(559\) 1.78133e12 0.771597
\(560\) 0 0
\(561\) −3.00064e12 −1.27903
\(562\) − 1.34051e12i − 0.566835i
\(563\) − 4.02619e11i − 0.168891i −0.996428 0.0844455i \(-0.973088\pi\)
0.996428 0.0844455i \(-0.0269119\pi\)
\(564\) −2.30388e11 −0.0958746
\(565\) 0 0
\(566\) −1.33774e12 −0.547894
\(567\) − 2.25313e12i − 0.915507i
\(568\) − 6.60716e11i − 0.266347i
\(569\) −2.55482e12 −1.02177 −0.510887 0.859648i \(-0.670683\pi\)
−0.510887 + 0.859648i \(0.670683\pi\)
\(570\) 0 0
\(571\) 1.16243e12 0.457620 0.228810 0.973471i \(-0.426516\pi\)
0.228810 + 0.973471i \(0.426516\pi\)
\(572\) − 1.22051e12i − 0.476716i
\(573\) 1.63113e12i 0.632110i
\(574\) −1.60935e12 −0.618794
\(575\) 0 0
\(576\) 1.77721e11 0.0672725
\(577\) 4.66332e12i 1.75148i 0.482787 + 0.875738i \(0.339625\pi\)
−0.482787 + 0.875738i \(0.660375\pi\)
\(578\) − 3.76361e12i − 1.40258i
\(579\) −4.17601e12 −1.54421
\(580\) 0 0
\(581\) −6.78754e10 −0.0247127
\(582\) − 3.28062e11i − 0.118523i
\(583\) − 2.84643e12i − 1.02045i
\(584\) 1.01232e12 0.360130
\(585\) 0 0
\(586\) −3.65675e12 −1.28102
\(587\) 3.88422e12i 1.35030i 0.737678 + 0.675152i \(0.235922\pi\)
−0.737678 + 0.675152i \(0.764078\pi\)
\(588\) 8.31042e11i 0.286698i
\(589\) 6.90875e11 0.236527
\(590\) 0 0
\(591\) −9.67655e10 −0.0326270
\(592\) 7.10438e11i 0.237727i
\(593\) 2.01341e12i 0.668631i 0.942461 + 0.334315i \(0.108505\pi\)
−0.942461 + 0.334315i \(0.891495\pi\)
\(594\) 7.33688e11 0.241809
\(595\) 0 0
\(596\) −5.47041e11 −0.177587
\(597\) − 4.37184e12i − 1.40857i
\(598\) 6.69502e12i 2.14090i
\(599\) −2.10578e12 −0.668333 −0.334166 0.942514i \(-0.608455\pi\)
−0.334166 + 0.942514i \(0.608455\pi\)
\(600\) 0 0
\(601\) 4.74058e12 1.48216 0.741082 0.671415i \(-0.234313\pi\)
0.741082 + 0.671415i \(0.234313\pi\)
\(602\) 8.07309e11i 0.250528i
\(603\) 8.63938e11i 0.266106i
\(604\) 2.47684e9 0.000757236 0
\(605\) 0 0
\(606\) −2.39682e11 −0.0721953
\(607\) − 4.01476e12i − 1.20036i −0.799866 0.600179i \(-0.795096\pi\)
0.799866 0.600179i \(-0.204904\pi\)
\(608\) 3.10148e11i 0.0920455i
\(609\) 3.01744e12 0.888915
\(610\) 0 0
\(611\) −8.50537e11 −0.246893
\(612\) − 1.61304e12i − 0.464798i
\(613\) − 1.30399e12i − 0.372993i −0.982456 0.186496i \(-0.940287\pi\)
0.982456 0.186496i \(-0.0597133\pi\)
\(614\) −3.12102e12 −0.886214
\(615\) 0 0
\(616\) 5.53143e11 0.154783
\(617\) 5.10164e12i 1.41719i 0.705618 + 0.708593i \(0.250670\pi\)
−0.705618 + 0.708593i \(0.749330\pi\)
\(618\) − 5.36956e12i − 1.48078i
\(619\) 4.50630e11 0.123371 0.0616854 0.998096i \(-0.480352\pi\)
0.0616854 + 0.998096i \(0.480352\pi\)
\(620\) 0 0
\(621\) −4.02459e12 −1.08595
\(622\) 3.45020e12i 0.924246i
\(623\) − 2.18988e12i − 0.582404i
\(624\) 1.87522e12 0.495133
\(625\) 0 0
\(626\) 3.06808e12 0.798514
\(627\) − 1.49209e12i − 0.385560i
\(628\) 1.76253e11i 0.0452187i
\(629\) 6.44812e12 1.64250
\(630\) 0 0
\(631\) 7.97105e10 0.0200163 0.0100081 0.999950i \(-0.496814\pi\)
0.0100081 + 0.999950i \(0.496814\pi\)
\(632\) − 2.38938e12i − 0.595743i
\(633\) 2.85212e12i 0.706076i
\(634\) 5.42217e12 1.33282
\(635\) 0 0
\(636\) 4.37332e12 1.05987
\(637\) 3.06801e12i 0.738294i
\(638\) 1.72698e12i 0.412662i
\(639\) −1.70873e12 −0.405434
\(640\) 0 0
\(641\) −3.12648e12 −0.731468 −0.365734 0.930719i \(-0.619182\pi\)
−0.365734 + 0.930719i \(0.619182\pi\)
\(642\) 3.88882e12i 0.903462i
\(643\) 5.94982e12i 1.37263i 0.727303 + 0.686316i \(0.240773\pi\)
−0.727303 + 0.686316i \(0.759227\pi\)
\(644\) −3.03422e12 −0.695122
\(645\) 0 0
\(646\) 2.81498e12 0.635960
\(647\) 3.56187e12i 0.799115i 0.916708 + 0.399557i \(0.130836\pi\)
−0.916708 + 0.399557i \(0.869164\pi\)
\(648\) 1.98128e12i 0.441426i
\(649\) −4.68594e11 −0.103680
\(650\) 0 0
\(651\) 1.89312e12 0.413109
\(652\) − 3.66289e12i − 0.793797i
\(653\) 1.37883e11i 0.0296757i 0.999890 + 0.0148379i \(0.00472321\pi\)
−0.999890 + 0.0148379i \(0.995277\pi\)
\(654\) 1.68245e12 0.359618
\(655\) 0 0
\(656\) 1.41518e12 0.298362
\(657\) − 2.61804e12i − 0.548191i
\(658\) − 3.85469e11i − 0.0801628i
\(659\) 4.16654e12 0.860581 0.430290 0.902691i \(-0.358411\pi\)
0.430290 + 0.902691i \(0.358411\pi\)
\(660\) 0 0
\(661\) −3.29083e12 −0.670500 −0.335250 0.942129i \(-0.608821\pi\)
−0.335250 + 0.942129i \(0.608821\pi\)
\(662\) − 2.85483e12i − 0.577722i
\(663\) − 1.70200e13i − 3.42097i
\(664\) 5.96860e10 0.0119156
\(665\) 0 0
\(666\) 1.83732e12 0.361869
\(667\) − 9.47322e12i − 1.85324i
\(668\) − 2.55676e12i − 0.496817i
\(669\) −8.09547e12 −1.56252
\(670\) 0 0
\(671\) −1.27357e12 −0.242532
\(672\) 8.49862e11i 0.160763i
\(673\) − 5.74732e12i − 1.07993i −0.841686 0.539967i \(-0.818437\pi\)
0.841686 0.539967i \(-0.181563\pi\)
\(674\) −4.23510e12 −0.790485
\(675\) 0 0
\(676\) 4.20812e12 0.775048
\(677\) 9.31245e12i 1.70379i 0.523716 + 0.851893i \(0.324545\pi\)
−0.523716 + 0.851893i \(0.675455\pi\)
\(678\) 4.69327e12i 0.852987i
\(679\) 5.48892e11 0.0990998
\(680\) 0 0
\(681\) −3.33062e11 −0.0593421
\(682\) 1.08350e12i 0.191778i
\(683\) − 9.19967e12i − 1.61763i −0.588063 0.808815i \(-0.700109\pi\)
0.588063 0.808815i \(-0.299891\pi\)
\(684\) 8.02099e11 0.140112
\(685\) 0 0
\(686\) −4.39792e12 −0.758208
\(687\) − 2.53735e12i − 0.434585i
\(688\) − 7.09905e11i − 0.120796i
\(689\) 1.61453e13 2.72934
\(690\) 0 0
\(691\) 1.03125e13 1.72074 0.860369 0.509672i \(-0.170233\pi\)
0.860369 + 0.509672i \(0.170233\pi\)
\(692\) 8.99573e11i 0.149128i
\(693\) − 1.43053e12i − 0.235612i
\(694\) 1.29906e12 0.212574
\(695\) 0 0
\(696\) −2.65338e12 −0.428605
\(697\) − 1.28445e13i − 2.06144i
\(698\) 3.62701e12i 0.578361i
\(699\) −7.18255e12 −1.13797
\(700\) 0 0
\(701\) −9.52566e12 −1.48992 −0.744961 0.667108i \(-0.767532\pi\)
−0.744961 + 0.667108i \(0.767532\pi\)
\(702\) 4.16156e12i 0.646754i
\(703\) 3.20638e12i 0.495126i
\(704\) −4.86405e11 −0.0746313
\(705\) 0 0
\(706\) 5.25973e11 0.0796787
\(707\) − 4.01020e11i − 0.0603641i
\(708\) − 7.19958e11i − 0.107686i
\(709\) 8.38537e12 1.24628 0.623138 0.782112i \(-0.285858\pi\)
0.623138 + 0.782112i \(0.285858\pi\)
\(710\) 0 0
\(711\) −6.17938e12 −0.906842
\(712\) 1.92567e12i 0.280816i
\(713\) − 5.94345e12i − 0.861263i
\(714\) 7.71358e12 1.11074
\(715\) 0 0
\(716\) 3.05926e11 0.0435019
\(717\) 6.35974e12i 0.898676i
\(718\) 4.99354e12i 0.701211i
\(719\) −6.94013e12 −0.968473 −0.484236 0.874937i \(-0.660903\pi\)
−0.484236 + 0.874937i \(0.660903\pi\)
\(720\) 0 0
\(721\) 8.98398e12 1.23811
\(722\) − 3.76323e12i − 0.515399i
\(723\) 1.50696e13i 2.05107i
\(724\) 2.33485e12 0.315818
\(725\) 0 0
\(726\) −4.22447e12 −0.564362
\(727\) 7.74984e12i 1.02893i 0.857510 + 0.514467i \(0.172010\pi\)
−0.857510 + 0.514467i \(0.827990\pi\)
\(728\) 3.13749e12i 0.413992i
\(729\) −2.92986e11 −0.0384214
\(730\) 0 0
\(731\) −6.44329e12 −0.834602
\(732\) − 1.95674e12i − 0.251902i
\(733\) − 4.12555e12i − 0.527854i −0.964543 0.263927i \(-0.914982\pi\)
0.964543 0.263927i \(-0.0850178\pi\)
\(734\) −1.00513e13 −1.27818
\(735\) 0 0
\(736\) 2.66814e12 0.335164
\(737\) − 2.36451e12i − 0.295215i
\(738\) − 3.65990e12i − 0.454167i
\(739\) −3.33931e12 −0.411867 −0.205933 0.978566i \(-0.566023\pi\)
−0.205933 + 0.978566i \(0.566023\pi\)
\(740\) 0 0
\(741\) 8.46333e12 1.03124
\(742\) 7.31713e12i 0.886183i
\(743\) 1.26471e13i 1.52245i 0.648489 + 0.761224i \(0.275401\pi\)
−0.648489 + 0.761224i \(0.724599\pi\)
\(744\) −1.66471e12 −0.199187
\(745\) 0 0
\(746\) −1.57563e12 −0.186265
\(747\) − 1.54359e11i − 0.0181380i
\(748\) 4.41474e12i 0.515642i
\(749\) −6.50651e12 −0.755405
\(750\) 0 0
\(751\) 1.56552e12 0.179588 0.0897941 0.995960i \(-0.471379\pi\)
0.0897941 + 0.995960i \(0.471379\pi\)
\(752\) 3.38961e11i 0.0386518i
\(753\) 1.25414e13i 1.42157i
\(754\) −9.79564e12 −1.10373
\(755\) 0 0
\(756\) −1.88605e12 −0.209993
\(757\) 9.54781e11i 0.105675i 0.998603 + 0.0528375i \(0.0168265\pi\)
−0.998603 + 0.0528375i \(0.983173\pi\)
\(758\) − 4.62438e12i − 0.508794i
\(759\) −1.28362e13 −1.40394
\(760\) 0 0
\(761\) 5.27330e12 0.569969 0.284985 0.958532i \(-0.408012\pi\)
0.284985 + 0.958532i \(0.408012\pi\)
\(762\) 1.03207e13i 1.10895i
\(763\) 2.81496e12i 0.300685i
\(764\) 2.39982e12 0.254835
\(765\) 0 0
\(766\) 3.22159e12 0.338097
\(767\) − 2.65792e12i − 0.277308i
\(768\) − 7.47324e11i − 0.0775146i
\(769\) 6.96923e12 0.718648 0.359324 0.933213i \(-0.383007\pi\)
0.359324 + 0.933213i \(0.383007\pi\)
\(770\) 0 0
\(771\) −1.96149e13 −1.99913
\(772\) 6.14401e12i 0.622550i
\(773\) 3.79383e12i 0.382182i 0.981572 + 0.191091i \(0.0612025\pi\)
−0.981572 + 0.191091i \(0.938797\pi\)
\(774\) −1.83594e12 −0.183876
\(775\) 0 0
\(776\) −4.82666e11 −0.0477825
\(777\) 8.78607e12i 0.864769i
\(778\) − 6.30031e12i − 0.616530i
\(779\) 6.38703e12 0.621413
\(780\) 0 0
\(781\) 4.67663e12 0.449784
\(782\) − 2.42167e13i − 2.31571i
\(783\) − 5.88847e12i − 0.559854i
\(784\) 1.22268e12 0.115582
\(785\) 0 0
\(786\) 7.07359e12 0.661056
\(787\) − 2.05939e13i − 1.91361i −0.290736 0.956803i \(-0.593900\pi\)
0.290736 0.956803i \(-0.406100\pi\)
\(788\) 1.42368e11i 0.0131536i
\(789\) −9.66092e12 −0.887507
\(790\) 0 0
\(791\) −7.85247e12 −0.713201
\(792\) 1.25793e12i 0.113604i
\(793\) − 7.22381e12i − 0.648690i
\(794\) 8.25000e12 0.736651
\(795\) 0 0
\(796\) −6.43213e12 −0.567866
\(797\) 1.45507e12i 0.127738i 0.997958 + 0.0638690i \(0.0203440\pi\)
−0.997958 + 0.0638690i \(0.979656\pi\)
\(798\) 3.83564e12i 0.334830i
\(799\) 3.07650e12 0.267052
\(800\) 0 0
\(801\) 4.98013e12 0.427459
\(802\) − 1.34929e13i − 1.15165i
\(803\) 7.16531e12i 0.608156i
\(804\) 3.63289e12 0.306620
\(805\) 0 0
\(806\) −6.14573e12 −0.512939
\(807\) 4.86722e12i 0.403971i
\(808\) 3.52636e11i 0.0291055i
\(809\) −1.31749e13 −1.08138 −0.540690 0.841222i \(-0.681837\pi\)
−0.540690 + 0.841222i \(0.681837\pi\)
\(810\) 0 0
\(811\) 9.97833e12 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(812\) − 4.43945e12i − 0.358366i
\(813\) − 5.79219e12i − 0.464981i
\(814\) −5.02857e12 −0.401453
\(815\) 0 0
\(816\) −6.78291e12 −0.535563
\(817\) − 3.20398e12i − 0.251588i
\(818\) − 9.03278e12i − 0.705394i
\(819\) 8.11413e12 0.630179
\(820\) 0 0
\(821\) −1.80913e13 −1.38972 −0.694859 0.719146i \(-0.744533\pi\)
−0.694859 + 0.719146i \(0.744533\pi\)
\(822\) − 3.21245e12i − 0.245422i
\(823\) − 1.39801e12i − 0.106221i −0.998589 0.0531107i \(-0.983086\pi\)
0.998589 0.0531107i \(-0.0169136\pi\)
\(824\) −7.90004e12 −0.596976
\(825\) 0 0
\(826\) 1.20459e12 0.0900383
\(827\) 2.26201e13i 1.68159i 0.541354 + 0.840794i \(0.317912\pi\)
−0.541354 + 0.840794i \(0.682088\pi\)
\(828\) − 6.90029e12i − 0.510189i
\(829\) −1.72646e13 −1.26959 −0.634793 0.772682i \(-0.718915\pi\)
−0.634793 + 0.772682i \(0.718915\pi\)
\(830\) 0 0
\(831\) 9.50458e12 0.691398
\(832\) − 2.75895e12i − 0.199613i
\(833\) − 1.10974e13i − 0.798579i
\(834\) 1.56662e13 1.12128
\(835\) 0 0
\(836\) −2.19526e12 −0.155439
\(837\) − 3.69440e12i − 0.260183i
\(838\) − 1.10796e13i − 0.776115i
\(839\) 1.76607e12 0.123049 0.0615245 0.998106i \(-0.480404\pi\)
0.0615245 + 0.998106i \(0.480404\pi\)
\(840\) 0 0
\(841\) −6.46640e11 −0.0445739
\(842\) 6.53001e12i 0.447723i
\(843\) 1.45780e13i 0.994203i
\(844\) 4.19622e12 0.284654
\(845\) 0 0
\(846\) 8.76615e11 0.0588359
\(847\) − 7.06810e12i − 0.471876i
\(848\) − 6.43430e12i − 0.427287i
\(849\) 1.45479e13 0.960983
\(850\) 0 0
\(851\) 2.75838e13 1.80290
\(852\) 7.18529e12i 0.467161i
\(853\) 2.37844e13i 1.53823i 0.639109 + 0.769116i \(0.279303\pi\)
−0.639109 + 0.769116i \(0.720697\pi\)
\(854\) 3.27388e12 0.210621
\(855\) 0 0
\(856\) 5.72148e12 0.364230
\(857\) 7.61941e12i 0.482511i 0.970462 + 0.241256i \(0.0775593\pi\)
−0.970462 + 0.241256i \(0.922441\pi\)
\(858\) 1.32730e13i 0.836138i
\(859\) 1.27695e13 0.800212 0.400106 0.916469i \(-0.368973\pi\)
0.400106 + 0.916469i \(0.368973\pi\)
\(860\) 0 0
\(861\) 1.75017e13 1.08534
\(862\) 7.05687e12i 0.435341i
\(863\) − 2.63038e13i − 1.61425i −0.590382 0.807124i \(-0.701023\pi\)
0.590382 0.807124i \(-0.298977\pi\)
\(864\) 1.65849e12 0.101251
\(865\) 0 0
\(866\) −1.14462e12 −0.0691564
\(867\) 4.09292e13i 2.46007i
\(868\) − 2.78529e12i − 0.166545i
\(869\) 1.69124e13 1.00604
\(870\) 0 0
\(871\) 1.34118e13 0.789596
\(872\) − 2.47532e12i − 0.144980i
\(873\) 1.24826e12i 0.0727347i
\(874\) 1.20420e13 0.698065
\(875\) 0 0
\(876\) −1.10090e13 −0.631652
\(877\) 4.76832e11i 0.0272187i 0.999907 + 0.0136093i \(0.00433212\pi\)
−0.999907 + 0.0136093i \(0.995668\pi\)
\(878\) − 7.59947e12i − 0.431577i
\(879\) 3.97672e13 2.24685
\(880\) 0 0
\(881\) 1.74839e13 0.977791 0.488896 0.872342i \(-0.337400\pi\)
0.488896 + 0.872342i \(0.337400\pi\)
\(882\) − 3.16208e12i − 0.175940i
\(883\) 6.48321e12i 0.358894i 0.983768 + 0.179447i \(0.0574309\pi\)
−0.983768 + 0.179447i \(0.942569\pi\)
\(884\) −2.50409e13 −1.37916
\(885\) 0 0
\(886\) −3.15315e12 −0.171907
\(887\) − 9.92450e12i − 0.538335i −0.963093 0.269167i \(-0.913252\pi\)
0.963093 0.269167i \(-0.0867485\pi\)
\(888\) − 7.72601e12i − 0.416962i
\(889\) −1.72679e13 −0.927220
\(890\) 0 0
\(891\) −1.40237e13 −0.745443
\(892\) 1.19106e13i 0.629928i
\(893\) 1.52981e12i 0.0805021i
\(894\) 5.94907e12 0.311480
\(895\) 0 0
\(896\) 1.25037e12 0.0648117
\(897\) − 7.28083e13i − 3.75504i
\(898\) 9.92250e12i 0.509187i
\(899\) 8.69601e12 0.444019
\(900\) 0 0
\(901\) −5.83994e13 −2.95221
\(902\) 1.00168e13i 0.503847i
\(903\) − 8.77949e12i − 0.439414i
\(904\) 6.90505e12 0.343881
\(905\) 0 0
\(906\) −2.69356e10 −0.00132816
\(907\) 1.40768e12i 0.0690671i 0.999404 + 0.0345335i \(0.0109946\pi\)
−0.999404 + 0.0345335i \(0.989005\pi\)
\(908\) 4.90022e11i 0.0239237i
\(909\) 9.11980e11 0.0443045
\(910\) 0 0
\(911\) 6.91343e12 0.332553 0.166277 0.986079i \(-0.446826\pi\)
0.166277 + 0.986079i \(0.446826\pi\)
\(912\) − 3.37286e12i − 0.161444i
\(913\) 4.22465e11i 0.0201221i
\(914\) −1.26081e13 −0.597574
\(915\) 0 0
\(916\) −3.73311e12 −0.175203
\(917\) 1.18351e13i 0.552724i
\(918\) − 1.50529e13i − 0.699565i
\(919\) 8.50189e12 0.393184 0.196592 0.980485i \(-0.437013\pi\)
0.196592 + 0.980485i \(0.437013\pi\)
\(920\) 0 0
\(921\) 3.39411e13 1.55438
\(922\) 2.45593e11i 0.0111925i
\(923\) 2.65264e13i 1.20301i
\(924\) −6.01543e12 −0.271483
\(925\) 0 0
\(926\) 3.11190e12 0.139084
\(927\) 2.04309e13i 0.908719i
\(928\) 3.90382e12i 0.172792i
\(929\) −8.90639e12 −0.392311 −0.196156 0.980573i \(-0.562846\pi\)
−0.196156 + 0.980573i \(0.562846\pi\)
\(930\) 0 0
\(931\) 5.51826e12 0.240729
\(932\) 1.05674e13i 0.458773i
\(933\) − 3.75209e13i − 1.62109i
\(934\) −1.70404e12 −0.0732689
\(935\) 0 0
\(936\) −7.13514e12 −0.303851
\(937\) 7.40064e12i 0.313647i 0.987627 + 0.156824i \(0.0501254\pi\)
−0.987627 + 0.156824i \(0.949875\pi\)
\(938\) 6.07831e12i 0.256372i
\(939\) −3.33654e13 −1.40056
\(940\) 0 0
\(941\) −8.13867e12 −0.338377 −0.169188 0.985584i \(-0.554115\pi\)
−0.169188 + 0.985584i \(0.554115\pi\)
\(942\) − 1.91675e12i − 0.0793115i
\(943\) − 5.49463e13i − 2.26275i
\(944\) −1.05925e12 −0.0434134
\(945\) 0 0
\(946\) 5.02480e12 0.203990
\(947\) − 1.16975e13i − 0.472627i −0.971677 0.236313i \(-0.924061\pi\)
0.971677 0.236313i \(-0.0759392\pi\)
\(948\) 2.59846e13i 1.04491i
\(949\) −4.06425e13 −1.62661
\(950\) 0 0
\(951\) −5.89661e13 −2.33771
\(952\) − 1.13487e13i − 0.447796i
\(953\) 2.50410e13i 0.983406i 0.870763 + 0.491703i \(0.163625\pi\)
−0.870763 + 0.491703i \(0.836375\pi\)
\(954\) −1.66403e13 −0.650418
\(955\) 0 0
\(956\) 9.35686e12 0.362301
\(957\) − 1.87809e13i − 0.723791i
\(958\) 1.32983e13i 0.510097i
\(959\) 5.37485e12 0.205202
\(960\) 0 0
\(961\) −2.09838e13 −0.793649
\(962\) − 2.85226e13i − 1.07375i
\(963\) − 1.47968e13i − 0.554433i
\(964\) 2.21714e13 0.826887
\(965\) 0 0
\(966\) 3.29972e13 1.21921
\(967\) 2.68904e13i 0.988960i 0.869189 + 0.494480i \(0.164641\pi\)
−0.869189 + 0.494480i \(0.835359\pi\)
\(968\) 6.21532e12i 0.227522i
\(969\) −3.06129e13 −1.11544
\(970\) 0 0
\(971\) 3.28780e13 1.18691 0.593456 0.804866i \(-0.297763\pi\)
0.593456 + 0.804866i \(0.297763\pi\)
\(972\) − 1.35767e13i − 0.487860i
\(973\) 2.62115e13i 0.937529i
\(974\) 2.21706e13 0.789338
\(975\) 0 0
\(976\) −2.87888e12 −0.101554
\(977\) − 8.13505e11i − 0.0285650i −0.999898 0.0142825i \(-0.995454\pi\)
0.999898 0.0142825i \(-0.00454642\pi\)
\(978\) 3.98339e13i 1.39229i
\(979\) −1.36301e13 −0.474217
\(980\) 0 0
\(981\) −6.40164e12 −0.220689
\(982\) 6.28487e12i 0.215672i
\(983\) 4.13240e13i 1.41160i 0.708412 + 0.705800i \(0.249412\pi\)
−0.708412 + 0.705800i \(0.750588\pi\)
\(984\) −1.53900e13 −0.523313
\(985\) 0 0
\(986\) 3.54321e13 1.19385
\(987\) 4.19198e12i 0.140602i
\(988\) − 1.24518e13i − 0.415744i
\(989\) −2.75631e13 −0.916105
\(990\) 0 0
\(991\) −3.61227e13 −1.18973 −0.594866 0.803825i \(-0.702795\pi\)
−0.594866 + 0.803825i \(0.702795\pi\)
\(992\) 2.44923e12i 0.0803023i
\(993\) 3.10462e13i 1.01330i
\(994\) −1.20219e13 −0.390603
\(995\) 0 0
\(996\) −6.49086e11 −0.0208995
\(997\) − 3.07166e13i − 0.984565i −0.870435 0.492283i \(-0.836163\pi\)
0.870435 0.492283i \(-0.163837\pi\)
\(998\) 1.20325e13i 0.383944i
\(999\) 1.71459e13 0.544646
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 50.10.b.d.49.2 2
4.3 odd 2 400.10.c.c.49.2 2
5.2 odd 4 50.10.a.a.1.1 1
5.3 odd 4 10.10.a.c.1.1 1
5.4 even 2 inner 50.10.b.d.49.1 2
15.8 even 4 90.10.a.e.1.1 1
20.3 even 4 80.10.a.a.1.1 1
20.7 even 4 400.10.a.j.1.1 1
20.19 odd 2 400.10.c.c.49.1 2
40.3 even 4 320.10.a.i.1.1 1
40.13 odd 4 320.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.10.a.c.1.1 1 5.3 odd 4
50.10.a.a.1.1 1 5.2 odd 4
50.10.b.d.49.1 2 5.4 even 2 inner
50.10.b.d.49.2 2 1.1 even 1 trivial
80.10.a.a.1.1 1 20.3 even 4
90.10.a.e.1.1 1 15.8 even 4
320.10.a.b.1.1 1 40.13 odd 4
320.10.a.i.1.1 1 40.3 even 4
400.10.a.j.1.1 1 20.7 even 4
400.10.c.c.49.1 2 20.19 odd 2
400.10.c.c.49.2 2 4.3 odd 2