Properties

Label 10.28.a.a.1.2
Level $10$
Weight $28$
Character 10.1
Self dual yes
Analytic conductor $46.186$
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] = [10,28,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(46.1855574838\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 19551870 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4421.25\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8192.00 q^{2} +186813. q^{3} +6.71089e7 q^{4} +1.22070e9 q^{5} -1.53037e9 q^{6} +1.52169e11 q^{7} -5.49756e11 q^{8} -7.59070e12 q^{9} -1.00000e13 q^{10} +5.20477e13 q^{11} +1.25368e13 q^{12} -9.66570e14 q^{13} -1.24656e15 q^{14} +2.28043e14 q^{15} +4.50360e15 q^{16} -8.84096e15 q^{17} +6.21830e16 q^{18} -4.96067e16 q^{19} +8.19200e16 q^{20} +2.84270e16 q^{21} -4.26375e17 q^{22} +2.39358e18 q^{23} -1.02701e17 q^{24} +1.49012e18 q^{25} +7.91814e18 q^{26} -2.84260e18 q^{27} +1.02119e19 q^{28} +5.37225e18 q^{29} -1.86813e18 q^{30} +1.60230e20 q^{31} -3.68935e19 q^{32} +9.72319e18 q^{33} +7.24251e19 q^{34} +1.85753e20 q^{35} -5.09403e20 q^{36} -2.24843e21 q^{37} +4.06378e20 q^{38} -1.80568e20 q^{39} -6.71089e20 q^{40} -4.36695e21 q^{41} -2.32874e20 q^{42} -5.19463e21 q^{43} +3.49286e21 q^{44} -9.26599e21 q^{45} -1.96082e22 q^{46} +1.42831e21 q^{47} +8.41330e20 q^{48} -4.25571e22 q^{49} -1.22070e22 q^{50} -1.65160e21 q^{51} -6.48654e22 q^{52} -2.24935e23 q^{53} +2.32866e22 q^{54} +6.35348e22 q^{55} -8.36555e22 q^{56} -9.26717e21 q^{57} -4.40094e22 q^{58} -1.17699e24 q^{59} +1.53037e22 q^{60} -2.05590e24 q^{61} -1.31261e24 q^{62} -1.15507e24 q^{63} +3.02231e23 q^{64} -1.17989e24 q^{65} -7.96523e22 q^{66} +3.81110e24 q^{67} -5.93307e23 q^{68} +4.47151e23 q^{69} -1.52169e24 q^{70} -3.12654e24 q^{71} +4.17303e24 q^{72} +1.49753e25 q^{73} +1.84191e25 q^{74} +2.78373e23 q^{75} -3.32905e24 q^{76} +7.92003e24 q^{77} +1.47921e24 q^{78} +1.41656e25 q^{79} +5.49756e24 q^{80} +5.73526e25 q^{81} +3.57740e25 q^{82} -7.95957e24 q^{83} +1.90771e24 q^{84} -1.07922e25 q^{85} +4.25544e25 q^{86} +1.00360e24 q^{87} -2.86135e25 q^{88} -1.05141e26 q^{89} +7.59070e25 q^{90} -1.47081e26 q^{91} +1.60630e26 q^{92} +2.99330e25 q^{93} -1.17007e25 q^{94} -6.05551e25 q^{95} -6.89218e24 q^{96} -2.67236e26 q^{97} +3.48628e26 q^{98} -3.95079e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16384 q^{2} - 3340644 q^{3} + 134217728 q^{4} + 2441406250 q^{5} + 27366555648 q^{6} - 58706842292 q^{7} - 1099511627776 q^{8} - 2773343978406 q^{9} - 20000000000000 q^{10} - 148584207397296 q^{11}+ \cdots - 13\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8192.00 −0.707107
\(3\) 186813. 0.0676503 0.0338252 0.999428i \(-0.489231\pi\)
0.0338252 + 0.999428i \(0.489231\pi\)
\(4\) 6.71089e7 0.500000
\(5\) 1.22070e9 0.447214
\(6\) −1.53037e9 −0.0478360
\(7\) 1.52169e11 0.593610 0.296805 0.954938i \(-0.404079\pi\)
0.296805 + 0.954938i \(0.404079\pi\)
\(8\) −5.49756e11 −0.353553
\(9\) −7.59070e12 −0.995423
\(10\) −1.00000e13 −0.316228
\(11\) 5.20477e13 0.454570 0.227285 0.973828i \(-0.427015\pi\)
0.227285 + 0.973828i \(0.427015\pi\)
\(12\) 1.25368e13 0.0338252
\(13\) −9.66570e14 −0.885111 −0.442556 0.896741i \(-0.645928\pi\)
−0.442556 + 0.896741i \(0.645928\pi\)
\(14\) −1.24656e15 −0.419746
\(15\) 2.28043e14 0.0302541
\(16\) 4.50360e15 0.250000
\(17\) −8.84096e15 −0.216491 −0.108245 0.994124i \(-0.534523\pi\)
−0.108245 + 0.994124i \(0.534523\pi\)
\(18\) 6.21830e16 0.703871
\(19\) −4.96067e16 −0.270624 −0.135312 0.990803i \(-0.543204\pi\)
−0.135312 + 0.990803i \(0.543204\pi\)
\(20\) 8.19200e16 0.223607
\(21\) 2.84270e16 0.0401579
\(22\) −4.26375e17 −0.321429
\(23\) 2.39358e18 0.990197 0.495098 0.868837i \(-0.335132\pi\)
0.495098 + 0.868837i \(0.335132\pi\)
\(24\) −1.02701e17 −0.0239180
\(25\) 1.49012e18 0.200000
\(26\) 7.91814e18 0.625868
\(27\) −2.84260e18 −0.134991
\(28\) 1.02119e19 0.296805
\(29\) 5.37225e18 0.0972261 0.0486131 0.998818i \(-0.484520\pi\)
0.0486131 + 0.998818i \(0.484520\pi\)
\(30\) −1.86813e18 −0.0213929
\(31\) 1.60230e20 1.17859 0.589293 0.807919i \(-0.299406\pi\)
0.589293 + 0.807919i \(0.299406\pi\)
\(32\) −3.68935e19 −0.176777
\(33\) 9.72319e18 0.0307518
\(34\) 7.24251e19 0.153082
\(35\) 1.85753e20 0.265470
\(36\) −5.09403e20 −0.497712
\(37\) −2.24843e21 −1.51759 −0.758797 0.651327i \(-0.774213\pi\)
−0.758797 + 0.651327i \(0.774213\pi\)
\(38\) 4.06378e20 0.191360
\(39\) −1.80568e20 −0.0598781
\(40\) −6.71089e20 −0.158114
\(41\) −4.36695e21 −0.737218 −0.368609 0.929585i \(-0.620166\pi\)
−0.368609 + 0.929585i \(0.620166\pi\)
\(42\) −2.32874e20 −0.0283959
\(43\) −5.19463e21 −0.461031 −0.230516 0.973069i \(-0.574041\pi\)
−0.230516 + 0.973069i \(0.574041\pi\)
\(44\) 3.49286e21 0.227285
\(45\) −9.26599e21 −0.445167
\(46\) −1.96082e22 −0.700175
\(47\) 1.42831e21 0.0381507 0.0190753 0.999818i \(-0.493928\pi\)
0.0190753 + 0.999818i \(0.493928\pi\)
\(48\) 8.41330e20 0.0169126
\(49\) −4.25571e22 −0.647627
\(50\) −1.22070e22 −0.141421
\(51\) −1.65160e21 −0.0146457
\(52\) −6.48654e22 −0.442556
\(53\) −2.24935e23 −1.18668 −0.593338 0.804953i \(-0.702190\pi\)
−0.593338 + 0.804953i \(0.702190\pi\)
\(54\) 2.32866e22 0.0954531
\(55\) 6.35348e22 0.203290
\(56\) −8.36555e22 −0.209873
\(57\) −9.26717e21 −0.0183078
\(58\) −4.40094e22 −0.0687493
\(59\) −1.17699e24 −1.45972 −0.729859 0.683598i \(-0.760414\pi\)
−0.729859 + 0.683598i \(0.760414\pi\)
\(60\) 1.53037e22 0.0151271
\(61\) −2.05590e24 −1.62574 −0.812868 0.582448i \(-0.802095\pi\)
−0.812868 + 0.582448i \(0.802095\pi\)
\(62\) −1.31261e24 −0.833386
\(63\) −1.15507e24 −0.590893
\(64\) 3.02231e23 0.125000
\(65\) −1.17989e24 −0.395834
\(66\) −7.96523e22 −0.0217448
\(67\) 3.81110e24 0.849260 0.424630 0.905367i \(-0.360404\pi\)
0.424630 + 0.905367i \(0.360404\pi\)
\(68\) −5.93307e23 −0.108245
\(69\) 4.47151e23 0.0669871
\(70\) −1.52169e24 −0.187716
\(71\) −3.12654e24 −0.318476 −0.159238 0.987240i \(-0.550904\pi\)
−0.159238 + 0.987240i \(0.550904\pi\)
\(72\) 4.17303e24 0.351935
\(73\) 1.49753e25 1.04838 0.524189 0.851602i \(-0.324369\pi\)
0.524189 + 0.851602i \(0.324369\pi\)
\(74\) 1.84191e25 1.07310
\(75\) 2.78373e23 0.0135301
\(76\) −3.32905e24 −0.135312
\(77\) 7.92003e24 0.269837
\(78\) 1.47921e24 0.0423402
\(79\) 1.41656e25 0.341404 0.170702 0.985323i \(-0.445397\pi\)
0.170702 + 0.985323i \(0.445397\pi\)
\(80\) 5.49756e24 0.111803
\(81\) 5.73526e25 0.986291
\(82\) 3.57740e25 0.521292
\(83\) −7.95957e24 −0.0984771 −0.0492385 0.998787i \(-0.515679\pi\)
−0.0492385 + 0.998787i \(0.515679\pi\)
\(84\) 1.90771e24 0.0200790
\(85\) −1.07922e25 −0.0968175
\(86\) 4.25544e25 0.325998
\(87\) 1.00360e24 0.00657738
\(88\) −2.86135e25 −0.160715
\(89\) −1.05141e26 −0.506998 −0.253499 0.967336i \(-0.581581\pi\)
−0.253499 + 0.967336i \(0.581581\pi\)
\(90\) 7.59070e25 0.314781
\(91\) −1.47081e26 −0.525411
\(92\) 1.60630e26 0.495098
\(93\) 2.99330e25 0.0797317
\(94\) −1.17007e25 −0.0269766
\(95\) −6.05551e25 −0.121027
\(96\) −6.89218e24 −0.0119590
\(97\) −2.67236e26 −0.403159 −0.201580 0.979472i \(-0.564608\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(98\) 3.48628e26 0.457942
\(99\) −3.95079e26 −0.452489
\(100\) 1.00000e26 0.100000
\(101\) −4.51294e26 −0.394567 −0.197283 0.980346i \(-0.563212\pi\)
−0.197283 + 0.980346i \(0.563212\pi\)
\(102\) 1.35299e25 0.0103560
\(103\) −2.25718e27 −1.51448 −0.757240 0.653137i \(-0.773453\pi\)
−0.757240 + 0.653137i \(0.773453\pi\)
\(104\) 5.31377e26 0.312934
\(105\) 3.47010e25 0.0179592
\(106\) 1.84267e27 0.839107
\(107\) −2.71910e27 −1.09080 −0.545399 0.838177i \(-0.683622\pi\)
−0.545399 + 0.838177i \(0.683622\pi\)
\(108\) −1.90764e26 −0.0674955
\(109\) −4.53116e27 −1.41564 −0.707818 0.706395i \(-0.750320\pi\)
−0.707818 + 0.706395i \(0.750320\pi\)
\(110\) −5.20477e26 −0.143748
\(111\) −4.20035e26 −0.102666
\(112\) 6.85306e26 0.148403
\(113\) −2.52001e27 −0.483998 −0.241999 0.970276i \(-0.577803\pi\)
−0.241999 + 0.970276i \(0.577803\pi\)
\(114\) 7.59167e25 0.0129456
\(115\) 2.92185e27 0.442829
\(116\) 3.60525e26 0.0486131
\(117\) 7.33694e27 0.881061
\(118\) 9.64187e27 1.03218
\(119\) −1.34532e27 −0.128511
\(120\) −1.25368e26 −0.0106965
\(121\) −1.04010e28 −0.793366
\(122\) 1.68420e28 1.14957
\(123\) −8.15802e26 −0.0498730
\(124\) 1.07529e28 0.589293
\(125\) 1.81899e27 0.0894427
\(126\) 9.46229e27 0.417825
\(127\) 4.15616e28 1.64946 0.824730 0.565526i \(-0.191327\pi\)
0.824730 + 0.565526i \(0.191327\pi\)
\(128\) −2.47588e27 −0.0883883
\(129\) −9.70423e26 −0.0311889
\(130\) 9.66570e27 0.279897
\(131\) −5.42612e27 −0.141686 −0.0708430 0.997487i \(-0.522569\pi\)
−0.0708430 + 0.997487i \(0.522569\pi\)
\(132\) 6.52512e26 0.0153759
\(133\) −7.54858e27 −0.160645
\(134\) −3.12205e28 −0.600517
\(135\) −3.46997e27 −0.0603698
\(136\) 4.86037e27 0.0765410
\(137\) −2.10760e28 −0.300650 −0.150325 0.988637i \(-0.548032\pi\)
−0.150325 + 0.988637i \(0.548032\pi\)
\(138\) −3.66306e27 −0.0473671
\(139\) −7.38363e28 −0.866104 −0.433052 0.901369i \(-0.642563\pi\)
−0.433052 + 0.901369i \(0.642563\pi\)
\(140\) 1.24656e28 0.132735
\(141\) 2.66827e26 0.00258091
\(142\) 2.56126e28 0.225196
\(143\) −5.03078e28 −0.402345
\(144\) −3.41855e28 −0.248856
\(145\) 6.55792e27 0.0434808
\(146\) −1.22678e29 −0.741315
\(147\) −7.95022e27 −0.0438122
\(148\) −1.50889e29 −0.758797
\(149\) −1.91029e29 −0.877170 −0.438585 0.898690i \(-0.644520\pi\)
−0.438585 + 0.898690i \(0.644520\pi\)
\(150\) −2.28043e27 −0.00956720
\(151\) −8.87996e27 −0.0340583 −0.0170291 0.999855i \(-0.505421\pi\)
−0.0170291 + 0.999855i \(0.505421\pi\)
\(152\) 2.72716e28 0.0956802
\(153\) 6.71091e28 0.215500
\(154\) −6.48809e28 −0.190804
\(155\) 1.95593e29 0.527080
\(156\) −1.21177e28 −0.0299390
\(157\) 6.26636e29 1.42027 0.710133 0.704067i \(-0.248635\pi\)
0.710133 + 0.704067i \(0.248635\pi\)
\(158\) −1.16044e29 −0.241409
\(159\) −4.20207e28 −0.0802790
\(160\) −4.50360e28 −0.0790569
\(161\) 3.64227e29 0.587791
\(162\) −4.69832e29 −0.697413
\(163\) 7.21779e28 0.0985988 0.0492994 0.998784i \(-0.484301\pi\)
0.0492994 + 0.998784i \(0.484301\pi\)
\(164\) −2.93061e29 −0.368609
\(165\) 1.18691e28 0.0137526
\(166\) 6.52048e28 0.0696338
\(167\) 6.72685e29 0.662429 0.331214 0.943556i \(-0.392542\pi\)
0.331214 + 0.943556i \(0.392542\pi\)
\(168\) −1.56279e28 −0.0141980
\(169\) −2.58277e29 −0.216578
\(170\) 8.84096e28 0.0684603
\(171\) 3.76550e29 0.269386
\(172\) −3.48605e29 −0.230516
\(173\) 8.69859e29 0.531896 0.265948 0.963987i \(-0.414315\pi\)
0.265948 + 0.963987i \(0.414315\pi\)
\(174\) −8.22153e27 −0.00465091
\(175\) 2.26749e29 0.118722
\(176\) 2.34402e29 0.113642
\(177\) −2.19876e29 −0.0987504
\(178\) 8.61314e29 0.358502
\(179\) 3.70782e30 1.43088 0.715439 0.698675i \(-0.246227\pi\)
0.715439 + 0.698675i \(0.246227\pi\)
\(180\) −6.21830e29 −0.222583
\(181\) 1.58156e30 0.525321 0.262661 0.964888i \(-0.415400\pi\)
0.262661 + 0.964888i \(0.415400\pi\)
\(182\) 1.20489e30 0.371522
\(183\) −3.84069e29 −0.109982
\(184\) −1.31588e30 −0.350087
\(185\) −2.74466e30 −0.678689
\(186\) −2.45212e29 −0.0563788
\(187\) −4.60152e29 −0.0984101
\(188\) 9.58525e28 0.0190753
\(189\) −4.32554e29 −0.0801320
\(190\) 4.96067e29 0.0855789
\(191\) −8.65494e30 −1.39096 −0.695478 0.718547i \(-0.744807\pi\)
−0.695478 + 0.718547i \(0.744807\pi\)
\(192\) 5.64607e28 0.00845629
\(193\) 2.56974e30 0.358811 0.179405 0.983775i \(-0.442583\pi\)
0.179405 + 0.983775i \(0.442583\pi\)
\(194\) 2.18920e30 0.285077
\(195\) −2.20419e29 −0.0267783
\(196\) −2.85596e30 −0.323814
\(197\) −1.43330e31 −1.51720 −0.758598 0.651559i \(-0.774115\pi\)
−0.758598 + 0.651559i \(0.774115\pi\)
\(198\) 3.23648e30 0.319958
\(199\) −1.47927e31 −1.36626 −0.683128 0.730299i \(-0.739381\pi\)
−0.683128 + 0.730299i \(0.739381\pi\)
\(200\) −8.19200e29 −0.0707107
\(201\) 7.11963e29 0.0574527
\(202\) 3.69700e30 0.279001
\(203\) 8.17487e29 0.0577144
\(204\) −1.10837e29 −0.00732283
\(205\) −5.33074e30 −0.329694
\(206\) 1.84908e31 1.07090
\(207\) −1.81689e31 −0.985665
\(208\) −4.35304e30 −0.221278
\(209\) −2.58192e30 −0.123018
\(210\) −2.84270e29 −0.0126990
\(211\) 2.00442e31 0.839800 0.419900 0.907570i \(-0.362065\pi\)
0.419900 + 0.907570i \(0.362065\pi\)
\(212\) −1.50951e31 −0.593338
\(213\) −5.84078e29 −0.0215450
\(214\) 2.22749e31 0.771310
\(215\) −6.34110e30 −0.206179
\(216\) 1.56274e30 0.0477265
\(217\) 2.43820e31 0.699621
\(218\) 3.71192e31 1.00101
\(219\) 2.79758e30 0.0709231
\(220\) 4.26375e30 0.101645
\(221\) 8.54540e30 0.191618
\(222\) 3.44093e30 0.0725956
\(223\) 9.30611e31 1.84779 0.923894 0.382647i \(-0.124988\pi\)
0.923894 + 0.382647i \(0.124988\pi\)
\(224\) −5.61403e30 −0.104936
\(225\) −1.13110e31 −0.199085
\(226\) 2.06439e31 0.342238
\(227\) 2.62678e31 0.410274 0.205137 0.978733i \(-0.434236\pi\)
0.205137 + 0.978733i \(0.434236\pi\)
\(228\) −6.21909e29 −0.00915391
\(229\) −1.81385e31 −0.251665 −0.125832 0.992052i \(-0.540160\pi\)
−0.125832 + 0.992052i \(0.540160\pi\)
\(230\) −2.39358e31 −0.313128
\(231\) 1.47956e30 0.0182546
\(232\) −2.95342e30 −0.0343746
\(233\) −1.08135e31 −0.118757 −0.0593786 0.998236i \(-0.518912\pi\)
−0.0593786 + 0.998236i \(0.518912\pi\)
\(234\) −6.01042e31 −0.623004
\(235\) 1.74355e30 0.0170615
\(236\) −7.89862e31 −0.729859
\(237\) 2.64631e30 0.0230961
\(238\) 1.10208e31 0.0908710
\(239\) −2.49336e32 −1.94273 −0.971367 0.237586i \(-0.923644\pi\)
−0.971367 + 0.237586i \(0.923644\pi\)
\(240\) 1.02701e30 0.00756354
\(241\) 1.52145e32 1.05932 0.529662 0.848209i \(-0.322319\pi\)
0.529662 + 0.848209i \(0.322319\pi\)
\(242\) 8.52052e31 0.560995
\(243\) 3.23907e31 0.201714
\(244\) −1.37969e32 −0.812868
\(245\) −5.19496e31 −0.289628
\(246\) 6.68305e30 0.0352655
\(247\) 4.79483e31 0.239533
\(248\) −8.80874e31 −0.416693
\(249\) −1.48695e30 −0.00666201
\(250\) −1.49012e31 −0.0632456
\(251\) −1.40439e32 −0.564799 −0.282400 0.959297i \(-0.591130\pi\)
−0.282400 + 0.959297i \(0.591130\pi\)
\(252\) −7.75151e31 −0.295447
\(253\) 1.24580e32 0.450114
\(254\) −3.40473e32 −1.16634
\(255\) −2.01612e30 −0.00654974
\(256\) 2.02824e31 0.0625000
\(257\) 4.52958e32 1.32422 0.662111 0.749406i \(-0.269661\pi\)
0.662111 + 0.749406i \(0.269661\pi\)
\(258\) 7.94971e30 0.0220539
\(259\) −3.42140e32 −0.900859
\(260\) −7.91814e31 −0.197917
\(261\) −4.07791e31 −0.0967812
\(262\) 4.44508e31 0.100187
\(263\) 7.58089e32 1.62300 0.811499 0.584354i \(-0.198652\pi\)
0.811499 + 0.584354i \(0.198652\pi\)
\(264\) −5.34538e30 −0.0108724
\(265\) −2.74579e32 −0.530698
\(266\) 6.18380e31 0.113593
\(267\) −1.96417e31 −0.0342986
\(268\) 2.55759e32 0.424630
\(269\) −1.08950e33 −1.72016 −0.860082 0.510156i \(-0.829588\pi\)
−0.860082 + 0.510156i \(0.829588\pi\)
\(270\) 2.84260e31 0.0426879
\(271\) 9.86275e32 1.40900 0.704502 0.709702i \(-0.251170\pi\)
0.704502 + 0.709702i \(0.251170\pi\)
\(272\) −3.98161e31 −0.0541226
\(273\) −2.74767e31 −0.0355442
\(274\) 1.72655e32 0.212592
\(275\) 7.75572e31 0.0909140
\(276\) 3.00078e31 0.0334936
\(277\) 5.21841e32 0.554704 0.277352 0.960768i \(-0.410543\pi\)
0.277352 + 0.960768i \(0.410543\pi\)
\(278\) 6.04867e32 0.612428
\(279\) −1.21626e33 −1.17319
\(280\) −1.02119e32 −0.0938580
\(281\) 1.10443e33 0.967396 0.483698 0.875235i \(-0.339293\pi\)
0.483698 + 0.875235i \(0.339293\pi\)
\(282\) −2.18585e30 −0.00182498
\(283\) 1.51510e33 1.20594 0.602968 0.797765i \(-0.293985\pi\)
0.602968 + 0.797765i \(0.293985\pi\)
\(284\) −2.09819e32 −0.159238
\(285\) −1.13125e31 −0.00818751
\(286\) 4.12121e32 0.284501
\(287\) −6.64512e32 −0.437620
\(288\) 2.80047e32 0.175968
\(289\) −1.58955e33 −0.953132
\(290\) −5.37225e31 −0.0307456
\(291\) −4.99231e31 −0.0272739
\(292\) 1.00498e33 0.524189
\(293\) 1.40575e33 0.700156 0.350078 0.936721i \(-0.386155\pi\)
0.350078 + 0.936721i \(0.386155\pi\)
\(294\) 6.51282e31 0.0309799
\(295\) −1.43675e33 −0.652806
\(296\) 1.23609e33 0.536550
\(297\) −1.47951e32 −0.0613628
\(298\) 1.56491e33 0.620253
\(299\) −2.31356e33 −0.876434
\(300\) 1.86813e31 0.00676503
\(301\) −7.90459e32 −0.273673
\(302\) 7.27447e31 0.0240828
\(303\) −8.43075e31 −0.0266926
\(304\) −2.23409e32 −0.0676561
\(305\) −2.50965e33 −0.727051
\(306\) −5.49757e32 −0.152381
\(307\) −4.49630e33 −1.19258 −0.596290 0.802769i \(-0.703359\pi\)
−0.596290 + 0.802769i \(0.703359\pi\)
\(308\) 5.31504e32 0.134919
\(309\) −4.21670e32 −0.102455
\(310\) −1.60230e33 −0.372702
\(311\) −2.75632e33 −0.613853 −0.306927 0.951733i \(-0.599301\pi\)
−0.306927 + 0.951733i \(0.599301\pi\)
\(312\) 9.92681e31 0.0211701
\(313\) 3.75694e33 0.767335 0.383668 0.923471i \(-0.374661\pi\)
0.383668 + 0.923471i \(0.374661\pi\)
\(314\) −5.13340e33 −1.00428
\(315\) −1.40999e33 −0.264256
\(316\) 9.50635e32 0.170702
\(317\) −1.58056e33 −0.271964 −0.135982 0.990711i \(-0.543419\pi\)
−0.135982 + 0.990711i \(0.543419\pi\)
\(318\) 3.44234e32 0.0567658
\(319\) 2.79613e32 0.0441961
\(320\) 3.68935e32 0.0559017
\(321\) −5.07963e32 −0.0737928
\(322\) −2.98375e33 −0.415631
\(323\) 4.38571e32 0.0585876
\(324\) 3.84887e33 0.493146
\(325\) −1.44030e33 −0.177022
\(326\) −5.91281e32 −0.0697199
\(327\) −8.46479e32 −0.0957682
\(328\) 2.40075e33 0.260646
\(329\) 2.17344e32 0.0226466
\(330\) −9.72319e31 −0.00972457
\(331\) 1.82677e34 1.75390 0.876952 0.480578i \(-0.159573\pi\)
0.876952 + 0.480578i \(0.159573\pi\)
\(332\) −5.34158e32 −0.0492385
\(333\) 1.70671e34 1.51065
\(334\) −5.51063e33 −0.468408
\(335\) 4.65222e33 0.379801
\(336\) 1.28024e32 0.0100395
\(337\) 1.86275e34 1.40330 0.701650 0.712521i \(-0.252447\pi\)
0.701650 + 0.712521i \(0.252447\pi\)
\(338\) 2.11580e33 0.153144
\(339\) −4.70771e32 −0.0327426
\(340\) −7.24251e32 −0.0484088
\(341\) 8.33961e33 0.535750
\(342\) −3.08469e33 −0.190485
\(343\) −1.64752e34 −0.978048
\(344\) 2.85578e33 0.162999
\(345\) 5.45838e32 0.0299576
\(346\) −7.12589e33 −0.376107
\(347\) 9.97856e33 0.506547 0.253274 0.967395i \(-0.418493\pi\)
0.253274 + 0.967395i \(0.418493\pi\)
\(348\) 6.73508e31 0.00328869
\(349\) −1.32833e33 −0.0623969 −0.0311984 0.999513i \(-0.509932\pi\)
−0.0311984 + 0.999513i \(0.509932\pi\)
\(350\) −1.85753e33 −0.0839491
\(351\) 2.74757e33 0.119482
\(352\) −1.92022e33 −0.0803573
\(353\) 5.06941e32 0.0204173 0.0102087 0.999948i \(-0.496750\pi\)
0.0102087 + 0.999948i \(0.496750\pi\)
\(354\) 1.80122e33 0.0698271
\(355\) −3.81658e33 −0.142427
\(356\) −7.05588e33 −0.253499
\(357\) −2.51322e32 −0.00869381
\(358\) −3.03745e34 −1.01178
\(359\) −1.29578e34 −0.415677 −0.207839 0.978163i \(-0.566643\pi\)
−0.207839 + 0.978163i \(0.566643\pi\)
\(360\) 5.09403e33 0.157390
\(361\) −3.11398e34 −0.926762
\(362\) −1.29561e34 −0.371458
\(363\) −1.94305e33 −0.0536715
\(364\) −9.87047e33 −0.262705
\(365\) 1.82804e34 0.468849
\(366\) 3.14630e33 0.0777687
\(367\) −2.12850e34 −0.507087 −0.253543 0.967324i \(-0.581596\pi\)
−0.253543 + 0.967324i \(0.581596\pi\)
\(368\) 1.07797e34 0.247549
\(369\) 3.31482e34 0.733844
\(370\) 2.24843e34 0.479905
\(371\) −3.42280e34 −0.704423
\(372\) 2.00877e33 0.0398659
\(373\) −4.23988e34 −0.811492 −0.405746 0.913986i \(-0.632988\pi\)
−0.405746 + 0.913986i \(0.632988\pi\)
\(374\) 3.76956e33 0.0695864
\(375\) 3.39811e32 0.00605083
\(376\) −7.85224e32 −0.0134883
\(377\) −5.19265e33 −0.0860559
\(378\) 3.54348e33 0.0566619
\(379\) −2.53814e34 −0.391639 −0.195820 0.980640i \(-0.562737\pi\)
−0.195820 + 0.980640i \(0.562737\pi\)
\(380\) −4.06378e33 −0.0605134
\(381\) 7.76424e33 0.111587
\(382\) 7.09012e34 0.983555
\(383\) −1.37667e35 −1.84351 −0.921755 0.387772i \(-0.873245\pi\)
−0.921755 + 0.387772i \(0.873245\pi\)
\(384\) −4.62526e32 −0.00597950
\(385\) 9.66800e33 0.120675
\(386\) −2.10513e34 −0.253717
\(387\) 3.94308e34 0.458921
\(388\) −1.79339e34 −0.201580
\(389\) 1.31877e35 1.43169 0.715844 0.698260i \(-0.246042\pi\)
0.715844 + 0.698260i \(0.246042\pi\)
\(390\) 1.80568e33 0.0189351
\(391\) −2.11615e34 −0.214368
\(392\) 2.33960e34 0.228971
\(393\) −1.01367e33 −0.00958510
\(394\) 1.17416e35 1.07282
\(395\) 1.72920e34 0.152680
\(396\) −2.65133e34 −0.226245
\(397\) 6.51145e34 0.537039 0.268520 0.963274i \(-0.413466\pi\)
0.268520 + 0.963274i \(0.413466\pi\)
\(398\) 1.21182e35 0.966089
\(399\) −1.41017e33 −0.0108677
\(400\) 6.71089e33 0.0500000
\(401\) −9.62542e34 −0.693379 −0.346690 0.937980i \(-0.612694\pi\)
−0.346690 + 0.937980i \(0.612694\pi\)
\(402\) −5.83240e33 −0.0406252
\(403\) −1.54874e35 −1.04318
\(404\) −3.02858e34 −0.197283
\(405\) 7.00105e34 0.441083
\(406\) −6.69685e33 −0.0408102
\(407\) −1.17026e35 −0.689852
\(408\) 9.07979e32 0.00517802
\(409\) 3.10105e35 1.71098 0.855489 0.517821i \(-0.173257\pi\)
0.855489 + 0.517821i \(0.173257\pi\)
\(410\) 4.36695e34 0.233129
\(411\) −3.93728e33 −0.0203391
\(412\) −1.51477e35 −0.757240
\(413\) −1.79100e35 −0.866503
\(414\) 1.48840e35 0.696970
\(415\) −9.71627e33 −0.0440403
\(416\) 3.56601e34 0.156467
\(417\) −1.37936e34 −0.0585922
\(418\) 2.11511e34 0.0869866
\(419\) 2.11087e35 0.840568 0.420284 0.907393i \(-0.361931\pi\)
0.420284 + 0.907393i \(0.361931\pi\)
\(420\) 2.32874e33 0.00897958
\(421\) 4.74937e35 1.77349 0.886743 0.462263i \(-0.152962\pi\)
0.886743 + 0.462263i \(0.152962\pi\)
\(422\) −1.64202e35 −0.593828
\(423\) −1.08419e34 −0.0379761
\(424\) 1.23659e35 0.419553
\(425\) −1.31741e34 −0.0432981
\(426\) 4.78477e33 0.0152346
\(427\) −3.12844e35 −0.965054
\(428\) −1.82476e35 −0.545399
\(429\) −9.39814e33 −0.0272188
\(430\) 5.19463e34 0.145791
\(431\) −4.97520e35 −1.35322 −0.676608 0.736343i \(-0.736551\pi\)
−0.676608 + 0.736343i \(0.736551\pi\)
\(432\) −1.28019e34 −0.0337478
\(433\) 1.48404e35 0.379191 0.189595 0.981862i \(-0.439282\pi\)
0.189595 + 0.981862i \(0.439282\pi\)
\(434\) −1.99737e35 −0.494706
\(435\) 1.22510e33 0.00294149
\(436\) −3.04081e35 −0.707818
\(437\) −1.18737e35 −0.267971
\(438\) −2.29178e34 −0.0501502
\(439\) 3.88897e35 0.825209 0.412604 0.910910i \(-0.364619\pi\)
0.412604 + 0.910910i \(0.364619\pi\)
\(440\) −3.49286e34 −0.0718738
\(441\) 3.23038e35 0.644663
\(442\) −7.00039e34 −0.135495
\(443\) −7.59267e35 −1.42542 −0.712712 0.701457i \(-0.752533\pi\)
−0.712712 + 0.701457i \(0.752533\pi\)
\(444\) −2.81881e34 −0.0513329
\(445\) −1.28346e35 −0.226736
\(446\) −7.62356e35 −1.30658
\(447\) −3.56866e34 −0.0593408
\(448\) 4.59901e34 0.0742013
\(449\) 4.69834e33 0.00735562 0.00367781 0.999993i \(-0.498829\pi\)
0.00367781 + 0.999993i \(0.498829\pi\)
\(450\) 9.26599e34 0.140774
\(451\) −2.27290e35 −0.335117
\(452\) −1.69115e35 −0.241999
\(453\) −1.65889e33 −0.00230405
\(454\) −2.15186e35 −0.290108
\(455\) −1.79543e35 −0.234971
\(456\) 5.09468e33 0.00647279
\(457\) 1.07406e36 1.32483 0.662414 0.749138i \(-0.269532\pi\)
0.662414 + 0.749138i \(0.269532\pi\)
\(458\) 1.48591e35 0.177954
\(459\) 2.51313e34 0.0292243
\(460\) 1.96082e35 0.221415
\(461\) 9.20062e35 1.00892 0.504458 0.863436i \(-0.331692\pi\)
0.504458 + 0.863436i \(0.331692\pi\)
\(462\) −1.21206e34 −0.0129079
\(463\) 5.38671e35 0.557161 0.278580 0.960413i \(-0.410136\pi\)
0.278580 + 0.960413i \(0.410136\pi\)
\(464\) 2.41944e34 0.0243065
\(465\) 3.65394e34 0.0356571
\(466\) 8.85839e34 0.0839740
\(467\) −4.52515e35 −0.416731 −0.208365 0.978051i \(-0.566814\pi\)
−0.208365 + 0.978051i \(0.566814\pi\)
\(468\) 4.92374e35 0.440530
\(469\) 5.79929e35 0.504129
\(470\) −1.42831e34 −0.0120643
\(471\) 1.17064e35 0.0960815
\(472\) 6.47055e35 0.516088
\(473\) −2.70369e35 −0.209571
\(474\) −2.16786e34 −0.0163314
\(475\) −7.39198e34 −0.0541249
\(476\) −9.02826e34 −0.0642555
\(477\) 1.70741e36 1.18125
\(478\) 2.04256e36 1.37372
\(479\) −2.10559e36 −1.37672 −0.688359 0.725370i \(-0.741669\pi\)
−0.688359 + 0.725370i \(0.741669\pi\)
\(480\) −8.41330e33 −0.00534823
\(481\) 2.17326e36 1.34324
\(482\) −1.24638e36 −0.749055
\(483\) 6.80423e34 0.0397642
\(484\) −6.98001e35 −0.396683
\(485\) −3.26216e35 −0.180298
\(486\) −2.65345e35 −0.142633
\(487\) 1.39900e36 0.731437 0.365718 0.930726i \(-0.380823\pi\)
0.365718 + 0.930726i \(0.380823\pi\)
\(488\) 1.13025e36 0.574785
\(489\) 1.34838e34 0.00667024
\(490\) 4.25571e35 0.204798
\(491\) −1.78697e36 −0.836597 −0.418298 0.908310i \(-0.637373\pi\)
−0.418298 + 0.908310i \(0.637373\pi\)
\(492\) −5.47475e34 −0.0249365
\(493\) −4.74958e34 −0.0210485
\(494\) −3.92793e35 −0.169375
\(495\) −4.82274e35 −0.202359
\(496\) 7.21612e35 0.294647
\(497\) −4.75761e35 −0.189050
\(498\) 1.21811e34 0.00471075
\(499\) −5.03865e36 −1.89652 −0.948259 0.317499i \(-0.897157\pi\)
−0.948259 + 0.317499i \(0.897157\pi\)
\(500\) 1.22070e35 0.0447214
\(501\) 1.25666e35 0.0448135
\(502\) 1.15048e36 0.399373
\(503\) −5.02862e36 −1.69934 −0.849671 0.527313i \(-0.823199\pi\)
−0.849671 + 0.527313i \(0.823199\pi\)
\(504\) 6.35004e35 0.208912
\(505\) −5.50896e35 −0.176456
\(506\) −1.02056e36 −0.318278
\(507\) −4.82494e34 −0.0146516
\(508\) 2.78915e36 0.824730
\(509\) −5.99564e36 −1.72642 −0.863209 0.504847i \(-0.831549\pi\)
−0.863209 + 0.504847i \(0.831549\pi\)
\(510\) 1.65160e34 0.00463136
\(511\) 2.27877e36 0.622327
\(512\) −1.66153e35 −0.0441942
\(513\) 1.41012e35 0.0365319
\(514\) −3.71063e36 −0.936366
\(515\) −2.75535e36 −0.677296
\(516\) −6.51240e34 −0.0155944
\(517\) 7.43405e34 0.0173422
\(518\) 2.80281e36 0.637004
\(519\) 1.62501e35 0.0359829
\(520\) 6.48654e35 0.139948
\(521\) 5.03095e36 1.05765 0.528824 0.848732i \(-0.322633\pi\)
0.528824 + 0.848732i \(0.322633\pi\)
\(522\) 3.34062e35 0.0684346
\(523\) 3.80823e36 0.760239 0.380119 0.924937i \(-0.375883\pi\)
0.380119 + 0.924937i \(0.375883\pi\)
\(524\) −3.64141e35 −0.0708430
\(525\) 4.23596e34 0.00803158
\(526\) −6.21027e36 −1.14763
\(527\) −1.41659e36 −0.255153
\(528\) 4.37893e34 0.00768795
\(529\) −1.14003e35 −0.0195103
\(530\) 2.24935e36 0.375260
\(531\) 8.93414e36 1.45304
\(532\) −5.06577e35 −0.0803227
\(533\) 4.22096e36 0.652520
\(534\) 1.60904e35 0.0242528
\(535\) −3.31921e36 −0.487820
\(536\) −2.09517e36 −0.300259
\(537\) 6.92669e35 0.0967994
\(538\) 8.92515e36 1.21634
\(539\) −2.21500e36 −0.294392
\(540\) −2.32866e35 −0.0301849
\(541\) 8.39039e35 0.106076 0.0530382 0.998592i \(-0.483109\pi\)
0.0530382 + 0.998592i \(0.483109\pi\)
\(542\) −8.07956e36 −0.996317
\(543\) 2.95456e35 0.0355381
\(544\) 3.26174e35 0.0382705
\(545\) −5.53120e36 −0.633092
\(546\) 2.25089e35 0.0251336
\(547\) 5.66797e36 0.617445 0.308722 0.951152i \(-0.400099\pi\)
0.308722 + 0.951152i \(0.400099\pi\)
\(548\) −1.41439e36 −0.150325
\(549\) 1.56058e37 1.61830
\(550\) −6.35348e35 −0.0642859
\(551\) −2.66500e35 −0.0263118
\(552\) −2.45824e35 −0.0236835
\(553\) 2.15555e36 0.202661
\(554\) −4.27492e36 −0.392235
\(555\) −5.12738e35 −0.0459135
\(556\) −4.95507e36 −0.433052
\(557\) 9.07435e36 0.774053 0.387026 0.922069i \(-0.373502\pi\)
0.387026 + 0.922069i \(0.373502\pi\)
\(558\) 9.96359e36 0.829572
\(559\) 5.02097e36 0.408064
\(560\) 8.36555e35 0.0663676
\(561\) −8.59623e34 −0.00665747
\(562\) −9.04752e36 −0.684052
\(563\) −1.68489e37 −1.24368 −0.621840 0.783145i \(-0.713614\pi\)
−0.621840 + 0.783145i \(0.713614\pi\)
\(564\) 1.79065e34 0.00129045
\(565\) −3.07619e36 −0.216451
\(566\) −1.24117e37 −0.852726
\(567\) 8.72726e36 0.585472
\(568\) 1.71883e36 0.112598
\(569\) −3.14141e36 −0.200960 −0.100480 0.994939i \(-0.532038\pi\)
−0.100480 + 0.994939i \(0.532038\pi\)
\(570\) 9.26717e34 0.00578944
\(571\) 2.78070e37 1.69655 0.848274 0.529557i \(-0.177642\pi\)
0.848274 + 0.529557i \(0.177642\pi\)
\(572\) −3.37610e36 −0.201172
\(573\) −1.61685e36 −0.0940986
\(574\) 5.44368e36 0.309444
\(575\) 3.56671e36 0.198039
\(576\) −2.29415e36 −0.124428
\(577\) −1.74113e37 −0.922482 −0.461241 0.887275i \(-0.652596\pi\)
−0.461241 + 0.887275i \(0.652596\pi\)
\(578\) 1.30216e37 0.673966
\(579\) 4.80060e35 0.0242737
\(580\) 4.40094e35 0.0217404
\(581\) −1.21120e36 −0.0584570
\(582\) 4.08970e35 0.0192855
\(583\) −1.17073e37 −0.539427
\(584\) −8.23277e36 −0.370657
\(585\) 8.95622e36 0.394022
\(586\) −1.15159e37 −0.495085
\(587\) −2.14558e37 −0.901426 −0.450713 0.892669i \(-0.648830\pi\)
−0.450713 + 0.892669i \(0.648830\pi\)
\(588\) −5.33530e35 −0.0219061
\(589\) −7.94849e36 −0.318954
\(590\) 1.17699e37 0.461603
\(591\) −2.67758e36 −0.102639
\(592\) −1.01260e37 −0.379398
\(593\) −4.77291e37 −1.74801 −0.874007 0.485913i \(-0.838487\pi\)
−0.874007 + 0.485913i \(0.838487\pi\)
\(594\) 1.21201e36 0.0433901
\(595\) −1.64223e36 −0.0574719
\(596\) −1.28197e37 −0.438585
\(597\) −2.76347e36 −0.0924276
\(598\) 1.89527e37 0.619733
\(599\) 3.32612e37 1.06335 0.531674 0.846949i \(-0.321563\pi\)
0.531674 + 0.846949i \(0.321563\pi\)
\(600\) −1.53037e35 −0.00478360
\(601\) −1.18443e37 −0.361996 −0.180998 0.983483i \(-0.557933\pi\)
−0.180998 + 0.983483i \(0.557933\pi\)
\(602\) 6.47544e36 0.193516
\(603\) −2.89289e37 −0.845373
\(604\) −5.95924e35 −0.0170291
\(605\) −1.26966e37 −0.354804
\(606\) 6.90647e35 0.0188745
\(607\) 1.62584e37 0.434540 0.217270 0.976112i \(-0.430285\pi\)
0.217270 + 0.976112i \(0.430285\pi\)
\(608\) 1.83016e36 0.0478401
\(609\) 1.52717e35 0.00390440
\(610\) 2.05590e37 0.514103
\(611\) −1.38056e36 −0.0337676
\(612\) 4.50361e36 0.107750
\(613\) −6.57104e37 −1.53786 −0.768932 0.639331i \(-0.779212\pi\)
−0.768932 + 0.639331i \(0.779212\pi\)
\(614\) 3.68337e37 0.843281
\(615\) −9.95852e35 −0.0223039
\(616\) −4.35408e36 −0.0954019
\(617\) 4.95910e37 1.06305 0.531524 0.847043i \(-0.321619\pi\)
0.531524 + 0.847043i \(0.321619\pi\)
\(618\) 3.45432e36 0.0724466
\(619\) 4.50080e37 0.923561 0.461781 0.886994i \(-0.347211\pi\)
0.461781 + 0.886994i \(0.347211\pi\)
\(620\) 1.31261e37 0.263540
\(621\) −6.80398e36 −0.133668
\(622\) 2.25798e37 0.434060
\(623\) −1.59991e37 −0.300959
\(624\) −8.13204e35 −0.0149695
\(625\) 2.22045e36 0.0400000
\(626\) −3.07768e37 −0.542588
\(627\) −4.82335e35 −0.00832218
\(628\) 4.20528e37 0.710133
\(629\) 1.98783e37 0.328545
\(630\) 1.15507e37 0.186857
\(631\) 3.74258e37 0.592617 0.296309 0.955092i \(-0.404244\pi\)
0.296309 + 0.955092i \(0.404244\pi\)
\(632\) −7.78760e36 −0.120704
\(633\) 3.74452e36 0.0568127
\(634\) 1.29480e37 0.192308
\(635\) 5.07344e37 0.737661
\(636\) −2.81996e36 −0.0401395
\(637\) 4.11344e37 0.573222
\(638\) −2.29059e36 −0.0312513
\(639\) 2.37326e37 0.317018
\(640\) −3.02231e36 −0.0395285
\(641\) −1.44495e37 −0.185042 −0.0925211 0.995711i \(-0.529493\pi\)
−0.0925211 + 0.995711i \(0.529493\pi\)
\(642\) 4.16123e36 0.0521794
\(643\) −2.85263e37 −0.350265 −0.175133 0.984545i \(-0.556035\pi\)
−0.175133 + 0.984545i \(0.556035\pi\)
\(644\) 2.44429e37 0.293895
\(645\) −1.18460e36 −0.0139481
\(646\) −3.59277e36 −0.0414277
\(647\) 1.04680e38 1.18210 0.591050 0.806635i \(-0.298713\pi\)
0.591050 + 0.806635i \(0.298713\pi\)
\(648\) −3.15299e37 −0.348707
\(649\) −6.12594e37 −0.663544
\(650\) 1.17989e37 0.125174
\(651\) 4.55487e36 0.0473296
\(652\) 4.84378e36 0.0492994
\(653\) −1.52185e38 −1.51720 −0.758600 0.651556i \(-0.774116\pi\)
−0.758600 + 0.651556i \(0.774116\pi\)
\(654\) 6.93435e36 0.0677183
\(655\) −6.62368e36 −0.0633639
\(656\) −1.96670e37 −0.184304
\(657\) −1.13673e38 −1.04358
\(658\) −1.78048e36 −0.0160136
\(659\) 1.20125e38 1.05847 0.529237 0.848474i \(-0.322478\pi\)
0.529237 + 0.848474i \(0.322478\pi\)
\(660\) 7.96523e35 0.00687631
\(661\) 1.16495e36 0.00985346 0.00492673 0.999988i \(-0.498432\pi\)
0.00492673 + 0.999988i \(0.498432\pi\)
\(662\) −1.49649e38 −1.24020
\(663\) 1.59639e36 0.0129630
\(664\) 4.37582e36 0.0348169
\(665\) −9.21458e36 −0.0718428
\(666\) −1.39814e38 −1.06819
\(667\) 1.28589e37 0.0962730
\(668\) 4.51431e37 0.331214
\(669\) 1.73850e37 0.125004
\(670\) −3.81110e37 −0.268560
\(671\) −1.07005e38 −0.739011
\(672\) −1.04877e36 −0.00709898
\(673\) −7.30965e37 −0.484946 −0.242473 0.970158i \(-0.577959\pi\)
−0.242473 + 0.970158i \(0.577959\pi\)
\(674\) −1.52596e38 −0.992283
\(675\) −4.23580e36 −0.0269982
\(676\) −1.73326e37 −0.108289
\(677\) −1.69952e38 −1.04083 −0.520415 0.853913i \(-0.674223\pi\)
−0.520415 + 0.853913i \(0.674223\pi\)
\(678\) 3.85655e36 0.0231525
\(679\) −4.06649e37 −0.239319
\(680\) 5.93307e36 0.0342302
\(681\) 4.90716e36 0.0277552
\(682\) −6.83181e37 −0.378832
\(683\) 3.18220e38 1.73001 0.865004 0.501765i \(-0.167316\pi\)
0.865004 + 0.501765i \(0.167316\pi\)
\(684\) 2.52698e37 0.134693
\(685\) −2.57276e37 −0.134455
\(686\) 1.34965e38 0.691584
\(687\) −3.38850e36 −0.0170252
\(688\) −2.33945e37 −0.115258
\(689\) 2.17415e38 1.05034
\(690\) −4.47151e36 −0.0211832
\(691\) 2.86475e38 1.33086 0.665432 0.746459i \(-0.268248\pi\)
0.665432 + 0.746459i \(0.268248\pi\)
\(692\) 5.83753e37 0.265948
\(693\) −6.01185e37 −0.268602
\(694\) −8.17443e37 −0.358183
\(695\) −9.01322e37 −0.387334
\(696\) −5.51738e35 −0.00232545
\(697\) 3.86080e37 0.159601
\(698\) 1.08817e37 0.0441213
\(699\) −2.02009e36 −0.00803396
\(700\) 1.52169e37 0.0593610
\(701\) 3.03977e38 1.16318 0.581590 0.813482i \(-0.302431\pi\)
0.581590 + 0.813482i \(0.302431\pi\)
\(702\) −2.25081e37 −0.0844866
\(703\) 1.11537e38 0.410698
\(704\) 1.57305e37 0.0568212
\(705\) 3.25717e35 0.00115422
\(706\) −4.15286e36 −0.0144372
\(707\) −6.86727e37 −0.234219
\(708\) −1.47556e37 −0.0493752
\(709\) −2.29087e38 −0.752103 −0.376051 0.926599i \(-0.622718\pi\)
−0.376051 + 0.926599i \(0.622718\pi\)
\(710\) 3.12654e37 0.100711
\(711\) −1.07527e38 −0.339841
\(712\) 5.78018e37 0.179251
\(713\) 3.83523e38 1.16703
\(714\) 2.05883e36 0.00614745
\(715\) −6.14108e37 −0.179934
\(716\) 2.48828e38 0.715439
\(717\) −4.65792e37 −0.131427
\(718\) 1.06150e38 0.293928
\(719\) −8.47935e37 −0.230421 −0.115211 0.993341i \(-0.536754\pi\)
−0.115211 + 0.993341i \(0.536754\pi\)
\(720\) −4.17303e37 −0.111292
\(721\) −3.43472e38 −0.899010
\(722\) 2.55097e38 0.655320
\(723\) 2.84227e37 0.0716636
\(724\) 1.06137e38 0.262661
\(725\) 8.00527e36 0.0194452
\(726\) 1.59174e37 0.0379515
\(727\) 6.68474e37 0.156448 0.0782240 0.996936i \(-0.475075\pi\)
0.0782240 + 0.996936i \(0.475075\pi\)
\(728\) 8.08589e37 0.185761
\(729\) −4.31297e38 −0.972645
\(730\) −1.49753e38 −0.331526
\(731\) 4.59255e37 0.0998089
\(732\) −2.57745e37 −0.0549908
\(733\) 6.44770e38 1.35052 0.675260 0.737580i \(-0.264031\pi\)
0.675260 + 0.737580i \(0.264031\pi\)
\(734\) 1.74367e38 0.358564
\(735\) −9.70485e36 −0.0195934
\(736\) −8.83074e37 −0.175044
\(737\) 1.98359e38 0.386048
\(738\) −2.71550e38 −0.518906
\(739\) −9.08269e38 −1.70418 −0.852088 0.523398i \(-0.824664\pi\)
−0.852088 + 0.523398i \(0.824664\pi\)
\(740\) −1.84191e38 −0.339344
\(741\) 8.95737e36 0.0162045
\(742\) 2.80396e38 0.498102
\(743\) 2.09598e38 0.365627 0.182813 0.983148i \(-0.441480\pi\)
0.182813 + 0.983148i \(0.441480\pi\)
\(744\) −1.64559e37 −0.0281894
\(745\) −2.33189e38 −0.392282
\(746\) 3.47331e38 0.573812
\(747\) 6.04187e37 0.0980264
\(748\) −3.08803e37 −0.0492050
\(749\) −4.13761e38 −0.647509
\(750\) −2.78373e36 −0.00427858
\(751\) −5.96152e38 −0.899949 −0.449975 0.893041i \(-0.648567\pi\)
−0.449975 + 0.893041i \(0.648567\pi\)
\(752\) 6.43255e36 0.00953767
\(753\) −2.62359e37 −0.0382088
\(754\) 4.25382e37 0.0608507
\(755\) −1.08398e37 −0.0152313
\(756\) −2.90282e37 −0.0400660
\(757\) −7.52761e38 −1.02062 −0.510309 0.859991i \(-0.670469\pi\)
−0.510309 + 0.859991i \(0.670469\pi\)
\(758\) 2.07924e38 0.276931
\(759\) 2.32732e37 0.0304503
\(760\) 3.32905e37 0.0427895
\(761\) −4.68849e37 −0.0592025 −0.0296013 0.999562i \(-0.509424\pi\)
−0.0296013 + 0.999562i \(0.509424\pi\)
\(762\) −6.36047e37 −0.0789036
\(763\) −6.89500e38 −0.840336
\(764\) −5.80823e38 −0.695478
\(765\) 8.19202e37 0.0963744
\(766\) 1.12777e39 1.30356
\(767\) 1.13764e39 1.29201
\(768\) 3.78902e36 0.00422814
\(769\) −1.28141e39 −1.40502 −0.702510 0.711674i \(-0.747937\pi\)
−0.702510 + 0.711674i \(0.747937\pi\)
\(770\) −7.92003e37 −0.0853300
\(771\) 8.46184e37 0.0895840
\(772\) 1.72452e38 0.179405
\(773\) 1.05565e39 1.07918 0.539591 0.841927i \(-0.318579\pi\)
0.539591 + 0.841927i \(0.318579\pi\)
\(774\) −3.23017e38 −0.324506
\(775\) 2.38761e38 0.235717
\(776\) 1.46915e38 0.142538
\(777\) −6.39161e37 −0.0609434
\(778\) −1.08033e39 −1.01236
\(779\) 2.16630e38 0.199509
\(780\) −1.47921e37 −0.0133891
\(781\) −1.62729e38 −0.144769
\(782\) 1.73355e38 0.151581
\(783\) −1.52711e37 −0.0131247
\(784\) −1.91660e38 −0.161907
\(785\) 7.64936e38 0.635162
\(786\) 8.30398e36 0.00677769
\(787\) −2.14383e39 −1.72001 −0.860006 0.510283i \(-0.829540\pi\)
−0.860006 + 0.510283i \(0.829540\pi\)
\(788\) −9.61868e38 −0.758598
\(789\) 1.41621e38 0.109796
\(790\) −1.41656e38 −0.107961
\(791\) −3.83466e38 −0.287306
\(792\) 2.17197e38 0.159979
\(793\) 1.98717e39 1.43896
\(794\) −5.33418e38 −0.379744
\(795\) −5.12948e37 −0.0359019
\(796\) −9.92724e38 −0.683128
\(797\) 2.60844e39 1.76479 0.882395 0.470510i \(-0.155930\pi\)
0.882395 + 0.470510i \(0.155930\pi\)
\(798\) 1.15521e37 0.00768463
\(799\) −1.26277e37 −0.00825927
\(800\) −5.49756e37 −0.0353553
\(801\) 7.98092e38 0.504678
\(802\) 7.88514e38 0.490293
\(803\) 7.79432e38 0.476561
\(804\) 4.77790e37 0.0287263
\(805\) 4.44613e38 0.262868
\(806\) 1.26872e39 0.737639
\(807\) −2.03532e38 −0.116370
\(808\) 2.48101e38 0.139500
\(809\) 3.01667e39 1.66810 0.834051 0.551688i \(-0.186016\pi\)
0.834051 + 0.551688i \(0.186016\pi\)
\(810\) −5.73526e38 −0.311893
\(811\) −6.69658e38 −0.358155 −0.179078 0.983835i \(-0.557311\pi\)
−0.179078 + 0.983835i \(0.557311\pi\)
\(812\) 5.48606e37 0.0288572
\(813\) 1.84249e38 0.0953196
\(814\) 9.58673e38 0.487799
\(815\) 8.81078e37 0.0440947
\(816\) −7.43817e36 −0.00366141
\(817\) 2.57688e38 0.124766
\(818\) −2.54038e39 −1.20984
\(819\) 1.11645e39 0.523006
\(820\) −3.57740e38 −0.164847
\(821\) −1.05326e39 −0.477422 −0.238711 0.971091i \(-0.576725\pi\)
−0.238711 + 0.971091i \(0.576725\pi\)
\(822\) 3.22542e37 0.0143819
\(823\) 2.68231e38 0.117655 0.0588276 0.998268i \(-0.481264\pi\)
0.0588276 + 0.998268i \(0.481264\pi\)
\(824\) 1.24090e39 0.535449
\(825\) 1.44887e37 0.00615036
\(826\) 1.46719e39 0.612710
\(827\) −2.78651e39 −1.14482 −0.572409 0.819968i \(-0.693991\pi\)
−0.572409 + 0.819968i \(0.693991\pi\)
\(828\) −1.21930e39 −0.492833
\(829\) 3.50578e39 1.39412 0.697058 0.717015i \(-0.254492\pi\)
0.697058 + 0.717015i \(0.254492\pi\)
\(830\) 7.95957e37 0.0311412
\(831\) 9.74867e37 0.0375259
\(832\) −2.92128e38 −0.110639
\(833\) 3.76246e38 0.140205
\(834\) 1.12997e38 0.0414310
\(835\) 8.21148e38 0.296247
\(836\) −1.73270e38 −0.0615088
\(837\) −4.55470e38 −0.159099
\(838\) −1.72923e39 −0.594371
\(839\) −3.34496e39 −1.13137 −0.565685 0.824621i \(-0.691388\pi\)
−0.565685 + 0.824621i \(0.691388\pi\)
\(840\) −1.90771e37 −0.00634952
\(841\) −3.02427e39 −0.990547
\(842\) −3.89068e39 −1.25404
\(843\) 2.06322e38 0.0654446
\(844\) 1.34514e39 0.419900
\(845\) −3.15279e38 −0.0968566
\(846\) 8.88168e37 0.0268532
\(847\) −1.58271e39 −0.470950
\(848\) −1.01302e39 −0.296669
\(849\) 2.83040e38 0.0815820
\(850\) 1.07922e38 0.0306164
\(851\) −5.38178e39 −1.50272
\(852\) −3.91968e37 −0.0107725
\(853\) −5.83249e39 −1.57776 −0.788882 0.614545i \(-0.789340\pi\)
−0.788882 + 0.614545i \(0.789340\pi\)
\(854\) 2.56282e39 0.682396
\(855\) 4.59655e38 0.120473
\(856\) 1.49484e39 0.385655
\(857\) −3.39841e39 −0.863048 −0.431524 0.902102i \(-0.642024\pi\)
−0.431524 + 0.902102i \(0.642024\pi\)
\(858\) 7.69895e37 0.0192466
\(859\) 4.28955e39 1.05561 0.527805 0.849365i \(-0.323015\pi\)
0.527805 + 0.849365i \(0.323015\pi\)
\(860\) −4.25544e38 −0.103090
\(861\) −1.24139e38 −0.0296051
\(862\) 4.07568e39 0.956869
\(863\) −1.58857e39 −0.367165 −0.183583 0.983004i \(-0.558770\pi\)
−0.183583 + 0.983004i \(0.558770\pi\)
\(864\) 1.04873e38 0.0238633
\(865\) 1.06184e39 0.237871
\(866\) −1.21572e39 −0.268128
\(867\) −2.96948e38 −0.0644797
\(868\) 1.63625e39 0.349810
\(869\) 7.37286e38 0.155192
\(870\) −1.00360e37 −0.00207995
\(871\) −3.68369e39 −0.751689
\(872\) 2.49103e39 0.500503
\(873\) 2.02851e39 0.401314
\(874\) 9.72697e38 0.189484
\(875\) 2.76793e38 0.0530941
\(876\) 1.87743e38 0.0354615
\(877\) 2.78657e39 0.518293 0.259147 0.965838i \(-0.416559\pi\)
0.259147 + 0.965838i \(0.416559\pi\)
\(878\) −3.18585e39 −0.583511
\(879\) 2.62612e38 0.0473657
\(880\) 2.86135e38 0.0508224
\(881\) 1.86266e39 0.325806 0.162903 0.986642i \(-0.447914\pi\)
0.162903 + 0.986642i \(0.447914\pi\)
\(882\) −2.64633e39 −0.455846
\(883\) −1.12897e39 −0.191520 −0.0957601 0.995404i \(-0.530528\pi\)
−0.0957601 + 0.995404i \(0.530528\pi\)
\(884\) 5.73472e38 0.0958091
\(885\) −2.68403e38 −0.0441625
\(886\) 6.21991e39 1.00793
\(887\) −3.48784e39 −0.556656 −0.278328 0.960486i \(-0.589780\pi\)
−0.278328 + 0.960486i \(0.589780\pi\)
\(888\) 2.30917e38 0.0362978
\(889\) 6.32437e39 0.979136
\(890\) 1.05141e39 0.160327
\(891\) 2.98507e39 0.448338
\(892\) 6.24522e39 0.923894
\(893\) −7.08540e37 −0.0103245
\(894\) 2.92345e38 0.0419603
\(895\) 4.52615e39 0.639908
\(896\) −3.76751e38 −0.0524682
\(897\) −4.32202e38 −0.0592911
\(898\) −3.84888e37 −0.00520121
\(899\) 8.60796e38 0.114589
\(900\) −7.59070e38 −0.0995423
\(901\) 1.98864e39 0.256904
\(902\) 1.86196e39 0.236963
\(903\) −1.47668e38 −0.0185140
\(904\) 1.38539e39 0.171119
\(905\) 1.93062e39 0.234931
\(906\) 1.35896e37 0.00162921
\(907\) −3.51472e39 −0.415138 −0.207569 0.978220i \(-0.566555\pi\)
−0.207569 + 0.978220i \(0.566555\pi\)
\(908\) 1.76280e39 0.205137
\(909\) 3.42563e39 0.392761
\(910\) 1.47081e39 0.166150
\(911\) 9.61927e38 0.107064 0.0535321 0.998566i \(-0.482952\pi\)
0.0535321 + 0.998566i \(0.482952\pi\)
\(912\) −4.17356e37 −0.00457696
\(913\) −4.14277e38 −0.0447647
\(914\) −8.79869e39 −0.936795
\(915\) −4.68835e38 −0.0491853
\(916\) −1.21725e39 −0.125832
\(917\) −8.25684e38 −0.0841062
\(918\) −2.05876e38 −0.0206647
\(919\) −4.28853e39 −0.424179 −0.212089 0.977250i \(-0.568027\pi\)
−0.212089 + 0.977250i \(0.568027\pi\)
\(920\) −1.60630e39 −0.156564
\(921\) −8.39967e38 −0.0806784
\(922\) −7.53715e39 −0.713411
\(923\) 3.02202e39 0.281886
\(924\) 9.92918e37 0.00912729
\(925\) −3.35042e39 −0.303519
\(926\) −4.41279e39 −0.393972
\(927\) 1.71336e40 1.50755
\(928\) −1.98201e38 −0.0171873
\(929\) −2.10447e40 −1.79858 −0.899290 0.437353i \(-0.855916\pi\)
−0.899290 + 0.437353i \(0.855916\pi\)
\(930\) −2.99330e38 −0.0252134
\(931\) 2.11112e39 0.175264
\(932\) −7.25679e38 −0.0593786
\(933\) −5.14916e38 −0.0415274
\(934\) 3.70700e39 0.294673
\(935\) −5.61709e38 −0.0440103
\(936\) −4.03352e39 −0.311502
\(937\) 1.07956e40 0.821795 0.410897 0.911682i \(-0.365215\pi\)
0.410897 + 0.911682i \(0.365215\pi\)
\(938\) −4.75078e39 −0.356473
\(939\) 7.01844e38 0.0519105
\(940\) 1.17007e38 0.00853076
\(941\) −6.73545e39 −0.484069 −0.242034 0.970268i \(-0.577815\pi\)
−0.242034 + 0.970268i \(0.577815\pi\)
\(942\) −9.58985e38 −0.0679399
\(943\) −1.04526e40 −0.729991
\(944\) −5.30067e39 −0.364930
\(945\) −5.28020e38 −0.0358361
\(946\) 2.21486e39 0.148189
\(947\) 1.10409e40 0.728249 0.364124 0.931350i \(-0.381368\pi\)
0.364124 + 0.931350i \(0.381368\pi\)
\(948\) 1.77591e38 0.0115480
\(949\) −1.44747e40 −0.927930
\(950\) 6.05551e38 0.0382721
\(951\) −2.95270e38 −0.0183985
\(952\) 7.39595e38 0.0454355
\(953\) 1.05932e40 0.641612 0.320806 0.947145i \(-0.396046\pi\)
0.320806 + 0.947145i \(0.396046\pi\)
\(954\) −1.39871e40 −0.835267
\(955\) −1.05651e40 −0.622055
\(956\) −1.67326e40 −0.971367
\(957\) 5.22354e37 0.00298988
\(958\) 1.72490e40 0.973487
\(959\) −3.20711e39 −0.178469
\(960\) 6.89218e37 0.00378177
\(961\) 7.19098e39 0.389065
\(962\) −1.78034e40 −0.949814
\(963\) 2.06399e40 1.08581
\(964\) 1.02103e40 0.529662
\(965\) 3.13689e39 0.160465
\(966\) −5.57402e38 −0.0281176
\(967\) 9.48098e39 0.471624 0.235812 0.971799i \(-0.424225\pi\)
0.235812 + 0.971799i \(0.424225\pi\)
\(968\) 5.71803e39 0.280497
\(969\) 8.19307e37 0.00396347
\(970\) 2.67236e39 0.127490
\(971\) 1.33762e40 0.629324 0.314662 0.949204i \(-0.398109\pi\)
0.314662 + 0.949204i \(0.398109\pi\)
\(972\) 2.17370e39 0.100857
\(973\) −1.12356e40 −0.514128
\(974\) −1.14606e40 −0.517204
\(975\) −2.69067e38 −0.0119756
\(976\) −9.25897e39 −0.406434
\(977\) −2.56073e40 −1.10863 −0.554317 0.832306i \(-0.687020\pi\)
−0.554317 + 0.832306i \(0.687020\pi\)
\(978\) −1.10459e38 −0.00471657
\(979\) −5.47234e39 −0.230466
\(980\) −3.48628e39 −0.144814
\(981\) 3.43947e40 1.40916
\(982\) 1.46388e40 0.591563
\(983\) 1.03636e39 0.0413085 0.0206543 0.999787i \(-0.493425\pi\)
0.0206543 + 0.999787i \(0.493425\pi\)
\(984\) 4.48492e38 0.0176328
\(985\) −1.74963e40 −0.678511
\(986\) 3.89086e38 0.0148836
\(987\) 4.06027e37 0.00153205
\(988\) 3.21776e39 0.119766
\(989\) −1.24337e40 −0.456511
\(990\) 3.95079e39 0.143090
\(991\) 4.36938e40 1.56108 0.780541 0.625104i \(-0.214944\pi\)
0.780541 + 0.625104i \(0.214944\pi\)
\(992\) −5.91145e39 −0.208347
\(993\) 3.41264e39 0.118652
\(994\) 3.89743e39 0.133679
\(995\) −1.80575e40 −0.611008
\(996\) −9.97875e37 −0.00333100
\(997\) −3.43978e40 −1.13278 −0.566391 0.824137i \(-0.691661\pi\)
−0.566391 + 0.824137i \(0.691661\pi\)
\(998\) 4.12766e40 1.34104
\(999\) 6.39138e39 0.204862
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 10.28.a.a.1.2 2
5.2 odd 4 50.28.b.e.49.1 4
5.3 odd 4 50.28.b.e.49.4 4
5.4 even 2 50.28.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.28.a.a.1.2 2 1.1 even 1 trivial
50.28.a.e.1.1 2 5.4 even 2
50.28.b.e.49.1 4 5.2 odd 4
50.28.b.e.49.4 4 5.3 odd 4