Properties

Label 5.10.a.a.1.1
Level $5$
Weight $10$
Character 5.1
Self dual yes
Analytic conductor $2.575$
Analytic rank $1$
Dimension $1$
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] = [5,10,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(2.57517918082\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -114.000 q^{3} -448.000 q^{4} -625.000 q^{5} +912.000 q^{6} +4242.00 q^{7} +7680.00 q^{8} -6687.00 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -114.000 q^{3} -448.000 q^{4} -625.000 q^{5} +912.000 q^{6} +4242.00 q^{7} +7680.00 q^{8} -6687.00 q^{9} +5000.00 q^{10} -46208.0 q^{11} +51072.0 q^{12} -115934. q^{13} -33936.0 q^{14} +71250.0 q^{15} +167936. q^{16} +494842. q^{17} +53496.0 q^{18} -1.00874e6 q^{19} +280000. q^{20} -483588. q^{21} +369664. q^{22} -532554. q^{23} -875520. q^{24} +390625. q^{25} +927472. q^{26} +3.00618e6 q^{27} -1.90042e6 q^{28} +4.19639e6 q^{29} -570000. q^{30} -3.36503e6 q^{31} -5.27565e6 q^{32} +5.26771e6 q^{33} -3.95874e6 q^{34} -2.65125e6 q^{35} +2.99578e6 q^{36} -1.49314e7 q^{37} +8.06992e6 q^{38} +1.32165e7 q^{39} -4.80000e6 q^{40} +1.10563e7 q^{41} +3.86870e6 q^{42} -6.39679e6 q^{43} +2.07012e7 q^{44} +4.17937e6 q^{45} +4.26043e6 q^{46} -3.55592e7 q^{47} -1.91447e7 q^{48} -2.23590e7 q^{49} -3.12500e6 q^{50} -5.64120e7 q^{51} +5.19384e7 q^{52} +3.97386e7 q^{53} -2.40494e7 q^{54} +2.88800e7 q^{55} +3.25786e7 q^{56} +1.14996e8 q^{57} -3.35711e7 q^{58} -8.51856e7 q^{59} -3.19200e7 q^{60} +4.57486e7 q^{61} +2.69202e7 q^{62} -2.83663e7 q^{63} -4.37780e7 q^{64} +7.24588e7 q^{65} -4.21417e7 q^{66} -4.52862e7 q^{67} -2.21689e8 q^{68} +6.07112e7 q^{69} +2.12100e7 q^{70} -1.89967e8 q^{71} -5.13562e7 q^{72} +4.12171e8 q^{73} +1.19451e8 q^{74} -4.45312e7 q^{75} +4.51916e8 q^{76} -1.96014e8 q^{77} -1.05732e8 q^{78} +9.50408e7 q^{79} -1.04960e8 q^{80} -2.11084e8 q^{81} -8.84501e7 q^{82} +2.61706e8 q^{83} +2.16647e8 q^{84} -3.09276e8 q^{85} +5.11744e7 q^{86} -4.78388e8 q^{87} -3.54877e8 q^{88} -1.99386e7 q^{89} -3.34350e7 q^{90} -4.91792e8 q^{91} +2.38584e8 q^{92} +3.83613e8 q^{93} +2.84473e8 q^{94} +6.30462e8 q^{95} +6.01424e8 q^{96} -1.95034e7 q^{97} +1.78872e8 q^{98} +3.08993e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −114.000 −0.812567 −0.406284 0.913747i \(-0.633175\pi\)
−0.406284 + 0.913747i \(0.633175\pi\)
\(4\) −448.000 −0.875000
\(5\) −625.000 −0.447214
\(6\) 912.000 0.287286
\(7\) 4242.00 0.667774 0.333887 0.942613i \(-0.391640\pi\)
0.333887 + 0.942613i \(0.391640\pi\)
\(8\) 7680.00 0.662913
\(9\) −6687.00 −0.339735
\(10\) 5000.00 0.158114
\(11\) −46208.0 −0.951590 −0.475795 0.879556i \(-0.657840\pi\)
−0.475795 + 0.879556i \(0.657840\pi\)
\(12\) 51072.0 0.710996
\(13\) −115934. −1.12581 −0.562906 0.826521i \(-0.690317\pi\)
−0.562906 + 0.826521i \(0.690317\pi\)
\(14\) −33936.0 −0.236094
\(15\) 71250.0 0.363391
\(16\) 167936. 0.640625
\(17\) 494842. 1.43697 0.718483 0.695545i \(-0.244837\pi\)
0.718483 + 0.695545i \(0.244837\pi\)
\(18\) 53496.0 0.120114
\(19\) −1.00874e6 −1.77578 −0.887888 0.460060i \(-0.847828\pi\)
−0.887888 + 0.460060i \(0.847828\pi\)
\(20\) 280000. 0.391312
\(21\) −483588. −0.542611
\(22\) 369664. 0.336438
\(23\) −532554. −0.396815 −0.198408 0.980120i \(-0.563577\pi\)
−0.198408 + 0.980120i \(0.563577\pi\)
\(24\) −875520. −0.538661
\(25\) 390625. 0.200000
\(26\) 927472. 0.398035
\(27\) 3.00618e6 1.08862
\(28\) −1.90042e6 −0.584302
\(29\) 4.19639e6 1.10175 0.550877 0.834586i \(-0.314293\pi\)
0.550877 + 0.834586i \(0.314293\pi\)
\(30\) −570000. −0.128478
\(31\) −3.36503e6 −0.654427 −0.327213 0.944950i \(-0.606110\pi\)
−0.327213 + 0.944950i \(0.606110\pi\)
\(32\) −5.27565e6 −0.889408
\(33\) 5.26771e6 0.773231
\(34\) −3.95874e6 −0.508044
\(35\) −2.65125e6 −0.298638
\(36\) 2.99578e6 0.297268
\(37\) −1.49314e7 −1.30976 −0.654880 0.755733i \(-0.727281\pi\)
−0.654880 + 0.755733i \(0.727281\pi\)
\(38\) 8.06992e6 0.627831
\(39\) 1.32165e7 0.914797
\(40\) −4.80000e6 −0.296464
\(41\) 1.10563e7 0.611056 0.305528 0.952183i \(-0.401167\pi\)
0.305528 + 0.952183i \(0.401167\pi\)
\(42\) 3.86870e6 0.191842
\(43\) −6.39679e6 −0.285335 −0.142667 0.989771i \(-0.545568\pi\)
−0.142667 + 0.989771i \(0.545568\pi\)
\(44\) 2.07012e7 0.832642
\(45\) 4.17937e6 0.151934
\(46\) 4.26043e6 0.140295
\(47\) −3.55592e7 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(48\) −1.91447e7 −0.520551
\(49\) −2.23590e7 −0.554078
\(50\) −3.12500e6 −0.0707107
\(51\) −5.64120e7 −1.16763
\(52\) 5.19384e7 0.985085
\(53\) 3.97386e7 0.691785 0.345892 0.938274i \(-0.387576\pi\)
0.345892 + 0.938274i \(0.387576\pi\)
\(54\) −2.40494e7 −0.384887
\(55\) 2.88800e7 0.425564
\(56\) 3.25786e7 0.442676
\(57\) 1.14996e8 1.44294
\(58\) −3.35711e7 −0.389529
\(59\) −8.51856e7 −0.915234 −0.457617 0.889149i \(-0.651297\pi\)
−0.457617 + 0.889149i \(0.651297\pi\)
\(60\) −3.19200e7 −0.317967
\(61\) 4.57486e7 0.423052 0.211526 0.977372i \(-0.432157\pi\)
0.211526 + 0.977372i \(0.432157\pi\)
\(62\) 2.69202e7 0.231375
\(63\) −2.83663e7 −0.226866
\(64\) −4.37780e7 −0.326172
\(65\) 7.24588e7 0.503478
\(66\) −4.21417e7 −0.273378
\(67\) −4.52862e7 −0.274555 −0.137277 0.990533i \(-0.543835\pi\)
−0.137277 + 0.990533i \(0.543835\pi\)
\(68\) −2.21689e8 −1.25735
\(69\) 6.07112e7 0.322439
\(70\) 2.12100e7 0.105584
\(71\) −1.89967e8 −0.887190 −0.443595 0.896227i \(-0.646297\pi\)
−0.443595 + 0.896227i \(0.646297\pi\)
\(72\) −5.13562e7 −0.225214
\(73\) 4.12171e8 1.69873 0.849365 0.527805i \(-0.176985\pi\)
0.849365 + 0.527805i \(0.176985\pi\)
\(74\) 1.19451e8 0.463070
\(75\) −4.45312e7 −0.162513
\(76\) 4.51916e8 1.55380
\(77\) −1.96014e8 −0.635447
\(78\) −1.05732e8 −0.323430
\(79\) 9.50408e7 0.274529 0.137265 0.990534i \(-0.456169\pi\)
0.137265 + 0.990534i \(0.456169\pi\)
\(80\) −1.04960e8 −0.286496
\(81\) −2.11084e8 −0.544845
\(82\) −8.84501e7 −0.216041
\(83\) 2.61706e8 0.605289 0.302645 0.953104i \(-0.402131\pi\)
0.302645 + 0.953104i \(0.402131\pi\)
\(84\) 2.16647e8 0.474785
\(85\) −3.09276e8 −0.642631
\(86\) 5.11744e7 0.100881
\(87\) −4.78388e8 −0.895249
\(88\) −3.54877e8 −0.630821
\(89\) −1.99386e7 −0.0336853 −0.0168426 0.999858i \(-0.505361\pi\)
−0.0168426 + 0.999858i \(0.505361\pi\)
\(90\) −3.34350e7 −0.0537168
\(91\) −4.91792e8 −0.751788
\(92\) 2.38584e8 0.347213
\(93\) 3.83613e8 0.531766
\(94\) 2.84473e8 0.375808
\(95\) 6.30462e8 0.794151
\(96\) 6.01424e8 0.722703
\(97\) −1.95034e7 −0.0223685 −0.0111842 0.999937i \(-0.503560\pi\)
−0.0111842 + 0.999937i \(0.503560\pi\)
\(98\) 1.78872e8 0.195896
\(99\) 3.08993e8 0.323288
\(100\) −1.75000e8 −0.175000
\(101\) −2.16672e8 −0.207184 −0.103592 0.994620i \(-0.533034\pi\)
−0.103592 + 0.994620i \(0.533034\pi\)
\(102\) 4.51296e8 0.412820
\(103\) −7.48234e8 −0.655043 −0.327522 0.944844i \(-0.606213\pi\)
−0.327522 + 0.944844i \(0.606213\pi\)
\(104\) −8.90373e8 −0.746315
\(105\) 3.02242e8 0.242663
\(106\) −3.17909e8 −0.244583
\(107\) 1.05711e9 0.779637 0.389819 0.920892i \(-0.372538\pi\)
0.389819 + 0.920892i \(0.372538\pi\)
\(108\) −1.34677e9 −0.952546
\(109\) −2.49659e9 −1.69406 −0.847029 0.531547i \(-0.821611\pi\)
−0.847029 + 0.531547i \(0.821611\pi\)
\(110\) −2.31040e8 −0.150460
\(111\) 1.70217e9 1.06427
\(112\) 7.12385e8 0.427793
\(113\) 1.11566e9 0.643695 0.321848 0.946791i \(-0.395696\pi\)
0.321848 + 0.946791i \(0.395696\pi\)
\(114\) −9.19971e8 −0.510155
\(115\) 3.32846e8 0.177461
\(116\) −1.87998e9 −0.964035
\(117\) 7.75251e8 0.382477
\(118\) 6.81485e8 0.323584
\(119\) 2.09912e9 0.959568
\(120\) 5.47200e8 0.240896
\(121\) −2.22768e8 −0.0944756
\(122\) −3.65989e8 −0.149572
\(123\) −1.26041e9 −0.496524
\(124\) 1.50753e9 0.572623
\(125\) −2.44141e8 −0.0894427
\(126\) 2.26930e8 0.0802093
\(127\) 3.12054e8 0.106442 0.0532210 0.998583i \(-0.483051\pi\)
0.0532210 + 0.998583i \(0.483051\pi\)
\(128\) 3.05136e9 1.00473
\(129\) 7.29235e8 0.231853
\(130\) −5.79670e8 −0.178006
\(131\) −2.81701e9 −0.835733 −0.417867 0.908508i \(-0.637222\pi\)
−0.417867 + 0.908508i \(0.637222\pi\)
\(132\) −2.35993e9 −0.676577
\(133\) −4.27908e9 −1.18582
\(134\) 3.62289e8 0.0970697
\(135\) −1.87886e9 −0.486848
\(136\) 3.80039e9 0.952583
\(137\) −7.15311e9 −1.73481 −0.867406 0.497600i \(-0.834215\pi\)
−0.867406 + 0.497600i \(0.834215\pi\)
\(138\) −4.85689e8 −0.113999
\(139\) 2.52323e9 0.573310 0.286655 0.958034i \(-0.407457\pi\)
0.286655 + 0.958034i \(0.407457\pi\)
\(140\) 1.18776e9 0.261308
\(141\) 4.05374e9 0.863715
\(142\) 1.51974e9 0.313669
\(143\) 5.35708e9 1.07131
\(144\) −1.12299e9 −0.217643
\(145\) −2.62274e9 −0.492720
\(146\) −3.29737e9 −0.600592
\(147\) 2.54893e9 0.450225
\(148\) 6.68925e9 1.14604
\(149\) 1.46531e9 0.243551 0.121775 0.992558i \(-0.461141\pi\)
0.121775 + 0.992558i \(0.461141\pi\)
\(150\) 3.56250e8 0.0574572
\(151\) 3.65515e9 0.572149 0.286075 0.958207i \(-0.407650\pi\)
0.286075 + 0.958207i \(0.407650\pi\)
\(152\) −7.74712e9 −1.17718
\(153\) −3.30901e9 −0.488187
\(154\) 1.56811e9 0.224665
\(155\) 2.10314e9 0.292669
\(156\) −5.92098e9 −0.800448
\(157\) 5.55191e9 0.729279 0.364639 0.931149i \(-0.381192\pi\)
0.364639 + 0.931149i \(0.381192\pi\)
\(158\) −7.60327e8 −0.0970607
\(159\) −4.53020e9 −0.562122
\(160\) 3.29728e9 0.397755
\(161\) −2.25909e9 −0.264983
\(162\) 1.68867e9 0.192632
\(163\) −1.67637e10 −1.86006 −0.930029 0.367485i \(-0.880219\pi\)
−0.930029 + 0.367485i \(0.880219\pi\)
\(164\) −4.95321e9 −0.534674
\(165\) −3.29232e9 −0.345799
\(166\) −2.09365e9 −0.214002
\(167\) 1.39549e10 1.38836 0.694182 0.719799i \(-0.255766\pi\)
0.694182 + 0.719799i \(0.255766\pi\)
\(168\) −3.71396e9 −0.359704
\(169\) 2.83619e9 0.267452
\(170\) 2.47421e9 0.227204
\(171\) 6.74544e9 0.603293
\(172\) 2.86576e9 0.249668
\(173\) 1.57000e10 1.33258 0.666289 0.745694i \(-0.267882\pi\)
0.666289 + 0.745694i \(0.267882\pi\)
\(174\) 3.82711e9 0.316518
\(175\) 1.65703e9 0.133555
\(176\) −7.75999e9 −0.609613
\(177\) 9.71116e9 0.743689
\(178\) 1.59509e8 0.0119095
\(179\) −2.51902e10 −1.83397 −0.916985 0.398921i \(-0.869385\pi\)
−0.916985 + 0.398921i \(0.869385\pi\)
\(180\) −1.87236e9 −0.132942
\(181\) 1.04482e10 0.723579 0.361789 0.932260i \(-0.382166\pi\)
0.361789 + 0.932260i \(0.382166\pi\)
\(182\) 3.93434e9 0.265797
\(183\) −5.21535e9 −0.343758
\(184\) −4.09001e9 −0.263054
\(185\) 9.33210e9 0.585742
\(186\) −3.06891e9 −0.188008
\(187\) −2.28657e10 −1.36740
\(188\) 1.59305e10 0.930078
\(189\) 1.27522e10 0.726955
\(190\) −5.04370e9 −0.280775
\(191\) −9.68625e9 −0.526630 −0.263315 0.964710i \(-0.584816\pi\)
−0.263315 + 0.964710i \(0.584816\pi\)
\(192\) 4.99070e9 0.265037
\(193\) −2.04431e10 −1.06057 −0.530285 0.847820i \(-0.677915\pi\)
−0.530285 + 0.847820i \(0.677915\pi\)
\(194\) 1.56027e8 0.00790845
\(195\) −8.26030e9 −0.409110
\(196\) 1.00169e10 0.484818
\(197\) −2.52431e10 −1.19411 −0.597056 0.802200i \(-0.703663\pi\)
−0.597056 + 0.802200i \(0.703663\pi\)
\(198\) −2.47194e9 −0.114300
\(199\) 8.06736e9 0.364664 0.182332 0.983237i \(-0.441635\pi\)
0.182332 + 0.983237i \(0.441635\pi\)
\(200\) 3.00000e9 0.132583
\(201\) 5.16262e9 0.223094
\(202\) 1.73338e9 0.0732506
\(203\) 1.78011e10 0.735723
\(204\) 2.52726e10 1.02168
\(205\) −6.91016e9 −0.273273
\(206\) 5.98587e9 0.231593
\(207\) 3.56119e9 0.134812
\(208\) −1.94695e10 −0.721223
\(209\) 4.66119e10 1.68981
\(210\) −2.41794e9 −0.0857944
\(211\) −8.04589e9 −0.279449 −0.139725 0.990190i \(-0.544622\pi\)
−0.139725 + 0.990190i \(0.544622\pi\)
\(212\) −1.78029e10 −0.605312
\(213\) 2.16563e10 0.720901
\(214\) −8.45687e9 −0.275643
\(215\) 3.99800e9 0.127605
\(216\) 2.30875e10 0.721663
\(217\) −1.42744e10 −0.437009
\(218\) 1.99727e10 0.598940
\(219\) −4.69875e10 −1.38033
\(220\) −1.29382e10 −0.372369
\(221\) −5.73690e10 −1.61775
\(222\) −1.36174e10 −0.376275
\(223\) 3.18618e10 0.862777 0.431388 0.902166i \(-0.358024\pi\)
0.431388 + 0.902166i \(0.358024\pi\)
\(224\) −2.23793e10 −0.593923
\(225\) −2.61211e9 −0.0679470
\(226\) −8.92531e9 −0.227581
\(227\) 3.02621e10 0.756454 0.378227 0.925713i \(-0.376534\pi\)
0.378227 + 0.925713i \(0.376534\pi\)
\(228\) −5.15184e10 −1.26257
\(229\) 2.06101e10 0.495245 0.247623 0.968857i \(-0.420351\pi\)
0.247623 + 0.968857i \(0.420351\pi\)
\(230\) −2.66277e9 −0.0627420
\(231\) 2.23456e10 0.516344
\(232\) 3.22283e10 0.730367
\(233\) 3.61735e9 0.0804061 0.0402031 0.999192i \(-0.487200\pi\)
0.0402031 + 0.999192i \(0.487200\pi\)
\(234\) −6.20201e9 −0.135226
\(235\) 2.22245e10 0.475364
\(236\) 3.81632e10 0.800830
\(237\) −1.08347e10 −0.223073
\(238\) −1.67930e10 −0.339259
\(239\) 2.21916e9 0.0439945 0.0219973 0.999758i \(-0.492997\pi\)
0.0219973 + 0.999758i \(0.492997\pi\)
\(240\) 1.19654e10 0.232797
\(241\) 2.57977e10 0.492611 0.246306 0.969192i \(-0.420783\pi\)
0.246306 + 0.969192i \(0.420783\pi\)
\(242\) 1.78215e9 0.0334022
\(243\) −3.51070e10 −0.645901
\(244\) −2.04954e10 −0.370171
\(245\) 1.39744e10 0.247791
\(246\) 1.00833e10 0.175548
\(247\) 1.16947e11 1.99919
\(248\) −2.58434e10 −0.433828
\(249\) −2.98345e10 −0.491838
\(250\) 1.95312e9 0.0316228
\(251\) −8.91848e10 −1.41827 −0.709135 0.705072i \(-0.750914\pi\)
−0.709135 + 0.705072i \(0.750914\pi\)
\(252\) 1.27081e10 0.198508
\(253\) 2.46083e10 0.377606
\(254\) −2.49643e9 −0.0376329
\(255\) 3.52575e10 0.522180
\(256\) −1.99649e9 −0.0290527
\(257\) −1.10058e11 −1.57371 −0.786853 0.617141i \(-0.788291\pi\)
−0.786853 + 0.617141i \(0.788291\pi\)
\(258\) −5.83388e9 −0.0819726
\(259\) −6.33388e10 −0.874623
\(260\) −3.24615e10 −0.440543
\(261\) −2.80613e10 −0.374304
\(262\) 2.25361e10 0.295476
\(263\) 2.06374e10 0.265983 0.132992 0.991117i \(-0.457542\pi\)
0.132992 + 0.991117i \(0.457542\pi\)
\(264\) 4.04560e10 0.512585
\(265\) −2.48366e10 −0.309376
\(266\) 3.42326e10 0.419249
\(267\) 2.27300e9 0.0273716
\(268\) 2.02882e10 0.240235
\(269\) −2.64890e10 −0.308446 −0.154223 0.988036i \(-0.549287\pi\)
−0.154223 + 0.988036i \(0.549287\pi\)
\(270\) 1.50309e10 0.172127
\(271\) 3.44004e10 0.387438 0.193719 0.981057i \(-0.437945\pi\)
0.193719 + 0.981057i \(0.437945\pi\)
\(272\) 8.31018e10 0.920556
\(273\) 5.60643e10 0.610878
\(274\) 5.72249e10 0.613349
\(275\) −1.80500e10 −0.190318
\(276\) −2.71986e10 −0.282134
\(277\) −1.06023e11 −1.08203 −0.541017 0.841012i \(-0.681960\pi\)
−0.541017 + 0.841012i \(0.681960\pi\)
\(278\) −2.01858e10 −0.202696
\(279\) 2.25019e10 0.222332
\(280\) −2.03616e10 −0.197971
\(281\) 1.00299e11 0.959662 0.479831 0.877361i \(-0.340698\pi\)
0.479831 + 0.877361i \(0.340698\pi\)
\(282\) −3.24300e10 −0.305369
\(283\) −2.49296e9 −0.0231035 −0.0115517 0.999933i \(-0.503677\pi\)
−0.0115517 + 0.999933i \(0.503677\pi\)
\(284\) 8.51054e10 0.776291
\(285\) −7.18727e10 −0.645301
\(286\) −4.28566e10 −0.378766
\(287\) 4.69007e10 0.408047
\(288\) 3.52783e10 0.302163
\(289\) 1.26281e11 1.06487
\(290\) 2.09820e10 0.174203
\(291\) 2.22338e9 0.0181759
\(292\) −1.84653e11 −1.48639
\(293\) −8.47068e10 −0.671451 −0.335725 0.941960i \(-0.608981\pi\)
−0.335725 + 0.941960i \(0.608981\pi\)
\(294\) −2.03914e10 −0.159179
\(295\) 5.32410e10 0.409305
\(296\) −1.14673e11 −0.868256
\(297\) −1.38910e11 −1.03592
\(298\) −1.17224e10 −0.0861083
\(299\) 6.17411e10 0.446739
\(300\) 1.99500e10 0.142199
\(301\) −2.71352e10 −0.190539
\(302\) −2.92412e10 −0.202285
\(303\) 2.47006e10 0.168351
\(304\) −1.69404e11 −1.13761
\(305\) −2.85929e10 −0.189195
\(306\) 2.64721e10 0.172600
\(307\) 6.35507e9 0.0408317 0.0204159 0.999792i \(-0.493501\pi\)
0.0204159 + 0.999792i \(0.493501\pi\)
\(308\) 8.78144e10 0.556016
\(309\) 8.52987e10 0.532267
\(310\) −1.68251e10 −0.103474
\(311\) 2.71253e11 1.64419 0.822096 0.569350i \(-0.192805\pi\)
0.822096 + 0.569350i \(0.192805\pi\)
\(312\) 1.01503e11 0.606431
\(313\) 1.63499e11 0.962865 0.481432 0.876483i \(-0.340117\pi\)
0.481432 + 0.876483i \(0.340117\pi\)
\(314\) −4.44153e10 −0.257839
\(315\) 1.77289e10 0.101458
\(316\) −4.25783e10 −0.240213
\(317\) 2.53606e11 1.41056 0.705282 0.708927i \(-0.250820\pi\)
0.705282 + 0.708927i \(0.250820\pi\)
\(318\) 3.62416e10 0.198740
\(319\) −1.93907e11 −1.04842
\(320\) 2.73613e10 0.145868
\(321\) −1.20510e11 −0.633508
\(322\) 1.80728e10 0.0936856
\(323\) −4.99167e11 −2.55173
\(324\) 9.45658e10 0.476740
\(325\) −4.52867e10 −0.225162
\(326\) 1.34110e11 0.657630
\(327\) 2.84611e11 1.37654
\(328\) 8.49121e10 0.405077
\(329\) −1.50842e11 −0.709808
\(330\) 2.63386e10 0.122259
\(331\) −2.40064e11 −1.09926 −0.549631 0.835408i \(-0.685232\pi\)
−0.549631 + 0.835408i \(0.685232\pi\)
\(332\) −1.17244e11 −0.529628
\(333\) 9.98460e10 0.444971
\(334\) −1.11639e11 −0.490861
\(335\) 2.83038e10 0.122785
\(336\) −8.12118e10 −0.347610
\(337\) −5.13812e10 −0.217005 −0.108502 0.994096i \(-0.534606\pi\)
−0.108502 + 0.994096i \(0.534606\pi\)
\(338\) −2.26895e10 −0.0945585
\(339\) −1.27186e11 −0.523046
\(340\) 1.38556e11 0.562302
\(341\) 1.55491e11 0.622746
\(342\) −5.39636e10 −0.213296
\(343\) −2.66027e11 −1.03777
\(344\) −4.91274e10 −0.189152
\(345\) −3.79445e10 −0.144199
\(346\) −1.25600e11 −0.471137
\(347\) 2.76560e11 1.02401 0.512007 0.858981i \(-0.328902\pi\)
0.512007 + 0.858981i \(0.328902\pi\)
\(348\) 2.14318e11 0.783343
\(349\) 4.66592e11 1.68354 0.841770 0.539837i \(-0.181514\pi\)
0.841770 + 0.539837i \(0.181514\pi\)
\(350\) −1.32562e10 −0.0472188
\(351\) −3.48518e11 −1.22559
\(352\) 2.43777e11 0.846352
\(353\) −1.00998e11 −0.346198 −0.173099 0.984904i \(-0.555378\pi\)
−0.173099 + 0.984904i \(0.555378\pi\)
\(354\) −7.76893e10 −0.262934
\(355\) 1.18730e11 0.396763
\(356\) 8.93251e9 0.0294746
\(357\) −2.39300e11 −0.779714
\(358\) 2.01521e11 0.648406
\(359\) −1.66447e11 −0.528874 −0.264437 0.964403i \(-0.585186\pi\)
−0.264437 + 0.964403i \(0.585186\pi\)
\(360\) 3.20976e10 0.100719
\(361\) 6.94869e11 2.15338
\(362\) −8.35852e10 −0.255824
\(363\) 2.53956e10 0.0767677
\(364\) 2.20323e11 0.657814
\(365\) −2.57607e11 −0.759695
\(366\) 4.17228e10 0.121537
\(367\) −2.97381e11 −0.855688 −0.427844 0.903853i \(-0.640727\pi\)
−0.427844 + 0.903853i \(0.640727\pi\)
\(368\) −8.94350e10 −0.254210
\(369\) −7.39332e10 −0.207597
\(370\) −7.46568e10 −0.207091
\(371\) 1.68571e11 0.461956
\(372\) −1.71859e11 −0.465295
\(373\) 1.95714e11 0.523517 0.261759 0.965133i \(-0.415698\pi\)
0.261759 + 0.965133i \(0.415698\pi\)
\(374\) 1.82925e11 0.483450
\(375\) 2.78320e10 0.0726782
\(376\) −2.73094e11 −0.704640
\(377\) −4.86504e11 −1.24037
\(378\) −1.02018e11 −0.257017
\(379\) −1.67009e11 −0.415781 −0.207890 0.978152i \(-0.566660\pi\)
−0.207890 + 0.978152i \(0.566660\pi\)
\(380\) −2.82447e11 −0.694882
\(381\) −3.55741e10 −0.0864912
\(382\) 7.74900e10 0.186192
\(383\) 7.20782e10 0.171163 0.0855814 0.996331i \(-0.472725\pi\)
0.0855814 + 0.996331i \(0.472725\pi\)
\(384\) −3.47855e11 −0.816408
\(385\) 1.22509e11 0.284181
\(386\) 1.63545e11 0.374968
\(387\) 4.27754e10 0.0969381
\(388\) 8.73750e9 0.0195724
\(389\) 7.92249e11 1.75424 0.877119 0.480272i \(-0.159462\pi\)
0.877119 + 0.480272i \(0.159462\pi\)
\(390\) 6.60824e10 0.144642
\(391\) −2.63530e11 −0.570210
\(392\) −1.71717e11 −0.367305
\(393\) 3.21139e11 0.679089
\(394\) 2.01945e11 0.422182
\(395\) −5.94005e10 −0.122773
\(396\) −1.38429e11 −0.282877
\(397\) −1.47288e11 −0.297584 −0.148792 0.988869i \(-0.547538\pi\)
−0.148792 + 0.988869i \(0.547538\pi\)
\(398\) −6.45389e10 −0.128928
\(399\) 4.87815e11 0.963555
\(400\) 6.56000e10 0.128125
\(401\) −3.22101e11 −0.622075 −0.311037 0.950398i \(-0.600676\pi\)
−0.311037 + 0.950398i \(0.600676\pi\)
\(402\) −4.13010e10 −0.0788757
\(403\) 3.90121e11 0.736761
\(404\) 9.70690e10 0.181286
\(405\) 1.31928e11 0.243662
\(406\) −1.42409e11 −0.260117
\(407\) 6.89948e11 1.24635
\(408\) −4.33244e11 −0.774037
\(409\) −3.96423e11 −0.700493 −0.350247 0.936658i \(-0.613902\pi\)
−0.350247 + 0.936658i \(0.613902\pi\)
\(410\) 5.52813e10 0.0966164
\(411\) 8.15455e11 1.40965
\(412\) 3.35209e11 0.573163
\(413\) −3.61357e11 −0.611170
\(414\) −2.84895e10 −0.0476632
\(415\) −1.63566e11 −0.270693
\(416\) 6.11627e11 1.00131
\(417\) −2.87648e11 −0.465853
\(418\) −3.72895e11 −0.597438
\(419\) −1.23162e12 −1.95215 −0.976074 0.217441i \(-0.930229\pi\)
−0.976074 + 0.217441i \(0.930229\pi\)
\(420\) −1.35405e11 −0.212330
\(421\) −1.17500e12 −1.82293 −0.911465 0.411379i \(-0.865047\pi\)
−0.911465 + 0.411379i \(0.865047\pi\)
\(422\) 6.43671e10 0.0988002
\(423\) 2.37784e11 0.361120
\(424\) 3.05192e11 0.458593
\(425\) 1.93298e11 0.287393
\(426\) −1.73250e11 −0.254877
\(427\) 1.94066e11 0.282503
\(428\) −4.73585e11 −0.682183
\(429\) −6.10707e11 −0.870513
\(430\) −3.19840e10 −0.0451153
\(431\) −2.32327e11 −0.324304 −0.162152 0.986766i \(-0.551843\pi\)
−0.162152 + 0.986766i \(0.551843\pi\)
\(432\) 5.04846e11 0.697400
\(433\) −1.51554e11 −0.207191 −0.103596 0.994620i \(-0.533035\pi\)
−0.103596 + 0.994620i \(0.533035\pi\)
\(434\) 1.14196e11 0.154506
\(435\) 2.98993e11 0.400368
\(436\) 1.11847e12 1.48230
\(437\) 5.37209e11 0.704655
\(438\) 3.75900e11 0.488021
\(439\) −5.21385e11 −0.669990 −0.334995 0.942220i \(-0.608735\pi\)
−0.334995 + 0.942220i \(0.608735\pi\)
\(440\) 2.21798e11 0.282112
\(441\) 1.49515e11 0.188240
\(442\) 4.58952e11 0.571962
\(443\) 1.10169e12 1.35908 0.679539 0.733639i \(-0.262180\pi\)
0.679539 + 0.733639i \(0.262180\pi\)
\(444\) −7.62574e11 −0.931234
\(445\) 1.24616e10 0.0150645
\(446\) −2.54894e11 −0.305038
\(447\) −1.67045e11 −0.197901
\(448\) −1.85706e11 −0.217809
\(449\) 5.85672e11 0.680058 0.340029 0.940415i \(-0.389563\pi\)
0.340029 + 0.940415i \(0.389563\pi\)
\(450\) 2.08969e10 0.0240229
\(451\) −5.10888e11 −0.581475
\(452\) −4.99817e11 −0.563233
\(453\) −4.16687e11 −0.464909
\(454\) −2.42097e11 −0.267447
\(455\) 3.07370e11 0.336210
\(456\) 8.83172e11 0.956541
\(457\) −5.70804e11 −0.612159 −0.306079 0.952006i \(-0.599017\pi\)
−0.306079 + 0.952006i \(0.599017\pi\)
\(458\) −1.64881e11 −0.175096
\(459\) 1.48758e12 1.56432
\(460\) −1.49115e11 −0.155279
\(461\) −7.60279e11 −0.784005 −0.392002 0.919964i \(-0.628218\pi\)
−0.392002 + 0.919964i \(0.628218\pi\)
\(462\) −1.78765e11 −0.182555
\(463\) −7.14289e11 −0.722370 −0.361185 0.932494i \(-0.617628\pi\)
−0.361185 + 0.932494i \(0.617628\pi\)
\(464\) 7.04725e11 0.705812
\(465\) −2.39758e11 −0.237813
\(466\) −2.89388e10 −0.0284279
\(467\) 7.82521e11 0.761325 0.380662 0.924714i \(-0.375696\pi\)
0.380662 + 0.924714i \(0.375696\pi\)
\(468\) −3.47312e11 −0.334668
\(469\) −1.92104e11 −0.183340
\(470\) −1.77796e11 −0.168066
\(471\) −6.32917e11 −0.592588
\(472\) −6.54226e11 −0.606720
\(473\) 2.95583e11 0.271522
\(474\) 8.66772e10 0.0788683
\(475\) −3.94039e11 −0.355155
\(476\) −9.40406e11 −0.839622
\(477\) −2.65732e11 −0.235023
\(478\) −1.77533e10 −0.0155544
\(479\) −6.95738e11 −0.603860 −0.301930 0.953330i \(-0.597631\pi\)
−0.301930 + 0.953330i \(0.597631\pi\)
\(480\) −3.75890e11 −0.323203
\(481\) 1.73105e12 1.47454
\(482\) −2.06382e11 −0.174164
\(483\) 2.57537e11 0.215316
\(484\) 9.98003e10 0.0826661
\(485\) 1.21896e10 0.0100035
\(486\) 2.80856e11 0.228360
\(487\) −1.65196e12 −1.33082 −0.665408 0.746480i \(-0.731742\pi\)
−0.665408 + 0.746480i \(0.731742\pi\)
\(488\) 3.51350e11 0.280447
\(489\) 1.91107e12 1.51142
\(490\) −1.11795e11 −0.0876074
\(491\) 1.19989e12 0.931694 0.465847 0.884865i \(-0.345750\pi\)
0.465847 + 0.884865i \(0.345750\pi\)
\(492\) 5.64665e11 0.434458
\(493\) 2.07655e12 1.58318
\(494\) −9.35578e11 −0.706820
\(495\) −1.93121e11 −0.144579
\(496\) −5.65109e11 −0.419242
\(497\) −8.05842e11 −0.592442
\(498\) 2.38676e11 0.173891
\(499\) 1.43146e12 1.03354 0.516768 0.856125i \(-0.327135\pi\)
0.516768 + 0.856125i \(0.327135\pi\)
\(500\) 1.09375e11 0.0782624
\(501\) −1.59086e12 −1.12814
\(502\) 7.13478e11 0.501434
\(503\) 1.41833e12 0.987919 0.493959 0.869485i \(-0.335549\pi\)
0.493959 + 0.869485i \(0.335549\pi\)
\(504\) −2.17853e11 −0.150392
\(505\) 1.35420e11 0.0926555
\(506\) −1.96866e11 −0.133504
\(507\) −3.23326e11 −0.217323
\(508\) −1.39800e11 −0.0931367
\(509\) −1.16463e12 −0.769059 −0.384529 0.923113i \(-0.625636\pi\)
−0.384529 + 0.923113i \(0.625636\pi\)
\(510\) −2.82060e11 −0.184619
\(511\) 1.74843e12 1.13437
\(512\) −1.54632e12 −0.994455
\(513\) −3.03245e12 −1.93315
\(514\) 8.80466e11 0.556389
\(515\) 4.67646e11 0.292944
\(516\) −3.26697e11 −0.202872
\(517\) 1.64312e12 1.01149
\(518\) 5.06711e11 0.309226
\(519\) −1.78980e12 −1.08281
\(520\) 5.56483e11 0.333762
\(521\) −8.36946e11 −0.497654 −0.248827 0.968548i \(-0.580045\pi\)
−0.248827 + 0.968548i \(0.580045\pi\)
\(522\) 2.24490e11 0.132337
\(523\) −2.60288e12 −1.52124 −0.760618 0.649199i \(-0.775104\pi\)
−0.760618 + 0.649199i \(0.775104\pi\)
\(524\) 1.26202e12 0.731267
\(525\) −1.88902e11 −0.108522
\(526\) −1.65099e11 −0.0940394
\(527\) −1.66516e12 −0.940389
\(528\) 8.84638e11 0.495351
\(529\) −1.51754e12 −0.842538
\(530\) 1.98693e11 0.109381
\(531\) 5.69636e11 0.310937
\(532\) 1.91703e12 1.03759
\(533\) −1.28180e12 −0.687934
\(534\) −1.81840e10 −0.00967731
\(535\) −6.60693e11 −0.348664
\(536\) −3.47798e11 −0.182006
\(537\) 2.87168e12 1.49022
\(538\) 2.11912e11 0.109052
\(539\) 1.03317e12 0.527255
\(540\) 8.41730e11 0.425992
\(541\) 3.07406e12 1.54286 0.771428 0.636317i \(-0.219543\pi\)
0.771428 + 0.636317i \(0.219543\pi\)
\(542\) −2.75203e11 −0.136980
\(543\) −1.19109e12 −0.587956
\(544\) −2.61061e12 −1.27805
\(545\) 1.56037e12 0.757606
\(546\) −4.48514e11 −0.215978
\(547\) 1.32972e12 0.635062 0.317531 0.948248i \(-0.397146\pi\)
0.317531 + 0.948248i \(0.397146\pi\)
\(548\) 3.20460e12 1.51796
\(549\) −3.05921e11 −0.143726
\(550\) 1.44400e11 0.0672876
\(551\) −4.23307e12 −1.95647
\(552\) 4.66262e11 0.213749
\(553\) 4.03163e11 0.183323
\(554\) 8.48183e11 0.382557
\(555\) −1.06386e12 −0.475955
\(556\) −1.13041e12 −0.501647
\(557\) 2.49418e12 1.09794 0.548971 0.835841i \(-0.315020\pi\)
0.548971 + 0.835841i \(0.315020\pi\)
\(558\) −1.80016e11 −0.0786061
\(559\) 7.41606e11 0.321233
\(560\) −4.45240e11 −0.191315
\(561\) 2.60669e12 1.11111
\(562\) −8.02392e11 −0.339292
\(563\) 1.06689e12 0.447541 0.223770 0.974642i \(-0.428163\pi\)
0.223770 + 0.974642i \(0.428163\pi\)
\(564\) −1.81608e12 −0.755750
\(565\) −6.97290e11 −0.287869
\(566\) 1.99437e10 0.00816831
\(567\) −8.95420e11 −0.363834
\(568\) −1.45895e12 −0.588129
\(569\) −4.57278e12 −1.82884 −0.914420 0.404768i \(-0.867352\pi\)
−0.914420 + 0.404768i \(0.867352\pi\)
\(570\) 5.74982e11 0.228148
\(571\) −2.32305e12 −0.914525 −0.457263 0.889332i \(-0.651170\pi\)
−0.457263 + 0.889332i \(0.651170\pi\)
\(572\) −2.39997e12 −0.937398
\(573\) 1.10423e12 0.427922
\(574\) −3.75205e11 −0.144266
\(575\) −2.08029e11 −0.0793631
\(576\) 2.92744e11 0.110812
\(577\) 1.01144e12 0.379881 0.189940 0.981796i \(-0.439171\pi\)
0.189940 + 0.981796i \(0.439171\pi\)
\(578\) −1.01025e12 −0.376489
\(579\) 2.33051e12 0.861784
\(580\) 1.17499e12 0.431130
\(581\) 1.11016e12 0.404196
\(582\) −1.77871e10 −0.00642615
\(583\) −1.83624e12 −0.658296
\(584\) 3.16547e12 1.12611
\(585\) −4.84532e11 −0.171049
\(586\) 6.77654e11 0.237394
\(587\) −1.10143e12 −0.382899 −0.191449 0.981502i \(-0.561319\pi\)
−0.191449 + 0.981502i \(0.561319\pi\)
\(588\) −1.14192e12 −0.393947
\(589\) 3.39444e12 1.16211
\(590\) −4.25928e11 −0.144711
\(591\) 2.87772e12 0.970296
\(592\) −2.50751e12 −0.839065
\(593\) 9.84366e11 0.326897 0.163448 0.986552i \(-0.447738\pi\)
0.163448 + 0.986552i \(0.447738\pi\)
\(594\) 1.11128e12 0.366255
\(595\) −1.31195e12 −0.429132
\(596\) −6.56457e11 −0.213107
\(597\) −9.19679e11 −0.296314
\(598\) −4.93929e11 −0.157946
\(599\) 4.51397e12 1.43264 0.716321 0.697771i \(-0.245825\pi\)
0.716321 + 0.697771i \(0.245825\pi\)
\(600\) −3.42000e11 −0.107732
\(601\) −1.44212e12 −0.450885 −0.225443 0.974256i \(-0.572383\pi\)
−0.225443 + 0.974256i \(0.572383\pi\)
\(602\) 2.17082e11 0.0673657
\(603\) 3.02829e11 0.0932758
\(604\) −1.63751e12 −0.500630
\(605\) 1.39230e11 0.0422508
\(606\) −1.97605e11 −0.0595211
\(607\) −4.12105e12 −1.23214 −0.616069 0.787692i \(-0.711276\pi\)
−0.616069 + 0.787692i \(0.711276\pi\)
\(608\) 5.32176e12 1.57939
\(609\) −2.02932e12 −0.597824
\(610\) 2.28743e11 0.0668904
\(611\) 4.12252e12 1.19668
\(612\) 1.48244e12 0.427164
\(613\) 1.38470e12 0.396080 0.198040 0.980194i \(-0.436542\pi\)
0.198040 + 0.980194i \(0.436542\pi\)
\(614\) −5.08406e10 −0.0144362
\(615\) 7.87759e11 0.222052
\(616\) −1.50539e12 −0.421246
\(617\) 6.22714e11 0.172984 0.0864918 0.996253i \(-0.472434\pi\)
0.0864918 + 0.996253i \(0.472434\pi\)
\(618\) −6.82390e11 −0.188185
\(619\) 1.89438e12 0.518631 0.259315 0.965793i \(-0.416503\pi\)
0.259315 + 0.965793i \(0.416503\pi\)
\(620\) −9.42208e11 −0.256085
\(621\) −1.60095e12 −0.431983
\(622\) −2.17002e12 −0.581309
\(623\) −8.45797e10 −0.0224942
\(624\) 2.21952e12 0.586042
\(625\) 1.52588e11 0.0400000
\(626\) −1.30799e12 −0.340424
\(627\) −5.31375e12 −1.37308
\(628\) −2.48725e12 −0.638119
\(629\) −7.38866e12 −1.88208
\(630\) −1.41831e11 −0.0358707
\(631\) −1.10160e12 −0.276624 −0.138312 0.990389i \(-0.544168\pi\)
−0.138312 + 0.990389i \(0.544168\pi\)
\(632\) 7.29914e11 0.181989
\(633\) 9.17231e11 0.227071
\(634\) −2.02885e12 −0.498710
\(635\) −1.95034e11 −0.0476023
\(636\) 2.02953e12 0.491856
\(637\) 2.59217e12 0.623787
\(638\) 1.55125e12 0.370672
\(639\) 1.27031e12 0.301409
\(640\) −1.90710e12 −0.449328
\(641\) −4.85252e12 −1.13529 −0.567644 0.823274i \(-0.692145\pi\)
−0.567644 + 0.823274i \(0.692145\pi\)
\(642\) 9.64083e11 0.223979
\(643\) 1.57487e12 0.363325 0.181662 0.983361i \(-0.441852\pi\)
0.181662 + 0.983361i \(0.441852\pi\)
\(644\) 1.01207e12 0.231860
\(645\) −4.55772e11 −0.103688
\(646\) 3.99334e12 0.902172
\(647\) 6.93025e12 1.55482 0.777410 0.628995i \(-0.216533\pi\)
0.777410 + 0.628995i \(0.216533\pi\)
\(648\) −1.62113e12 −0.361185
\(649\) 3.93626e12 0.870928
\(650\) 3.62294e11 0.0796069
\(651\) 1.62729e12 0.355099
\(652\) 7.51015e12 1.62755
\(653\) −8.15499e12 −1.75515 −0.877575 0.479440i \(-0.840840\pi\)
−0.877575 + 0.479440i \(0.840840\pi\)
\(654\) −2.27689e12 −0.486679
\(655\) 1.76063e12 0.373751
\(656\) 1.85674e12 0.391458
\(657\) −2.75619e12 −0.577118
\(658\) 1.20674e12 0.250955
\(659\) −2.92553e12 −0.604255 −0.302127 0.953268i \(-0.597697\pi\)
−0.302127 + 0.953268i \(0.597697\pi\)
\(660\) 1.47496e12 0.302575
\(661\) −1.62921e12 −0.331947 −0.165974 0.986130i \(-0.553077\pi\)
−0.165974 + 0.986130i \(0.553077\pi\)
\(662\) 1.92051e12 0.388648
\(663\) 6.54007e12 1.31453
\(664\) 2.00990e12 0.401254
\(665\) 2.67442e12 0.530313
\(666\) −7.98768e11 −0.157321
\(667\) −2.23480e12 −0.437193
\(668\) −6.25181e12 −1.21482
\(669\) −3.63225e12 −0.701064
\(670\) −2.26431e11 −0.0434109
\(671\) −2.11395e12 −0.402572
\(672\) 2.55124e12 0.482603
\(673\) −2.87521e12 −0.540258 −0.270129 0.962824i \(-0.587066\pi\)
−0.270129 + 0.962824i \(0.587066\pi\)
\(674\) 4.11049e11 0.0767228
\(675\) 1.17429e12 0.217725
\(676\) −1.27061e12 −0.234020
\(677\) 3.19620e12 0.584769 0.292385 0.956301i \(-0.405551\pi\)
0.292385 + 0.956301i \(0.405551\pi\)
\(678\) 1.01749e12 0.184925
\(679\) −8.27332e10 −0.0149371
\(680\) −2.37524e12 −0.426008
\(681\) −3.44988e12 −0.614670
\(682\) −1.24393e12 −0.220174
\(683\) −5.47901e11 −0.0963406 −0.0481703 0.998839i \(-0.515339\pi\)
−0.0481703 + 0.998839i \(0.515339\pi\)
\(684\) −3.02196e12 −0.527881
\(685\) 4.47070e12 0.775832
\(686\) 2.12822e12 0.366908
\(687\) −2.34955e12 −0.402420
\(688\) −1.07425e12 −0.182792
\(689\) −4.60705e12 −0.778819
\(690\) 3.03556e11 0.0509821
\(691\) 5.90996e11 0.0986128 0.0493064 0.998784i \(-0.484299\pi\)
0.0493064 + 0.998784i \(0.484299\pi\)
\(692\) −7.03361e12 −1.16601
\(693\) 1.31075e12 0.215884
\(694\) −2.21248e12 −0.362044
\(695\) −1.57702e12 −0.256392
\(696\) −3.67402e12 −0.593472
\(697\) 5.47110e12 0.878066
\(698\) −3.73274e12 −0.595221
\(699\) −4.12378e11 −0.0653354
\(700\) −7.42350e11 −0.116860
\(701\) 7.12738e12 1.11481 0.557403 0.830242i \(-0.311798\pi\)
0.557403 + 0.830242i \(0.311798\pi\)
\(702\) 2.78815e12 0.433310
\(703\) 1.50619e13 2.32584
\(704\) 2.02290e12 0.310382
\(705\) −2.53359e12 −0.386265
\(706\) 8.07981e11 0.122400
\(707\) −9.19122e11 −0.138352
\(708\) −4.35060e12 −0.650728
\(709\) 9.09269e11 0.135140 0.0675701 0.997715i \(-0.478475\pi\)
0.0675701 + 0.997715i \(0.478475\pi\)
\(710\) −9.49837e11 −0.140277
\(711\) −6.35538e11 −0.0932671
\(712\) −1.53129e11 −0.0223304
\(713\) 1.79206e12 0.259687
\(714\) 1.91440e12 0.275670
\(715\) −3.34817e12 −0.479105
\(716\) 1.12852e13 1.60472
\(717\) −2.52985e11 −0.0357485
\(718\) 1.33158e12 0.186985
\(719\) 1.08017e13 1.50734 0.753669 0.657254i \(-0.228282\pi\)
0.753669 + 0.657254i \(0.228282\pi\)
\(720\) 7.01868e11 0.0973327
\(721\) −3.17401e12 −0.437421
\(722\) −5.55895e12 −0.761334
\(723\) −2.94094e12 −0.400280
\(724\) −4.68077e12 −0.633132
\(725\) 1.63921e12 0.220351
\(726\) −2.03165e11 −0.0271415
\(727\) −9.77691e12 −1.29807 −0.649033 0.760760i \(-0.724826\pi\)
−0.649033 + 0.760760i \(0.724826\pi\)
\(728\) −3.77696e12 −0.498370
\(729\) 8.15697e12 1.06968
\(730\) 2.06085e12 0.268593
\(731\) −3.16540e12 −0.410016
\(732\) 2.33647e12 0.300788
\(733\) 3.52295e11 0.0450753 0.0225377 0.999746i \(-0.492825\pi\)
0.0225377 + 0.999746i \(0.492825\pi\)
\(734\) 2.37905e12 0.302531
\(735\) −1.59308e12 −0.201347
\(736\) 2.80957e12 0.352931
\(737\) 2.09258e12 0.261264
\(738\) 5.91466e11 0.0733966
\(739\) −2.72124e12 −0.335635 −0.167818 0.985818i \(-0.553672\pi\)
−0.167818 + 0.985818i \(0.553672\pi\)
\(740\) −4.18078e12 −0.512524
\(741\) −1.33320e13 −1.62447
\(742\) −1.34857e12 −0.163326
\(743\) −5.85311e12 −0.704590 −0.352295 0.935889i \(-0.614599\pi\)
−0.352295 + 0.935889i \(0.614599\pi\)
\(744\) 2.94615e12 0.352514
\(745\) −9.15816e11 −0.108919
\(746\) −1.56571e12 −0.185091
\(747\) −1.75003e12 −0.205638
\(748\) 1.02438e13 1.19648
\(749\) 4.48426e12 0.520622
\(750\) −2.22656e11 −0.0256956
\(751\) −5.13723e12 −0.589317 −0.294659 0.955603i \(-0.595206\pi\)
−0.294659 + 0.955603i \(0.595206\pi\)
\(752\) −5.97166e12 −0.680950
\(753\) 1.01671e13 1.15244
\(754\) 3.89203e12 0.438536
\(755\) −2.28447e12 −0.255873
\(756\) −5.71299e12 −0.636086
\(757\) −1.45713e13 −1.61275 −0.806373 0.591407i \(-0.798573\pi\)
−0.806373 + 0.591407i \(0.798573\pi\)
\(758\) 1.33607e12 0.147001
\(759\) −2.80534e12 −0.306830
\(760\) 4.84195e12 0.526453
\(761\) 1.03936e13 1.12340 0.561700 0.827341i \(-0.310148\pi\)
0.561700 + 0.827341i \(0.310148\pi\)
\(762\) 2.84593e11 0.0305793
\(763\) −1.05905e13 −1.13125
\(764\) 4.33944e12 0.460801
\(765\) 2.06813e12 0.218324
\(766\) −5.76626e11 −0.0605152
\(767\) 9.87591e12 1.03038
\(768\) 2.27600e11 0.0236073
\(769\) −3.91664e12 −0.403873 −0.201936 0.979399i \(-0.564723\pi\)
−0.201936 + 0.979399i \(0.564723\pi\)
\(770\) −9.80072e11 −0.100473
\(771\) 1.25466e13 1.27874
\(772\) 9.15851e12 0.927998
\(773\) −6.36894e12 −0.641592 −0.320796 0.947148i \(-0.603950\pi\)
−0.320796 + 0.947148i \(0.603950\pi\)
\(774\) −3.42203e11 −0.0342728
\(775\) −1.31446e12 −0.130885
\(776\) −1.49786e11 −0.0148284
\(777\) 7.22063e12 0.710690
\(778\) −6.33800e12 −0.620217
\(779\) −1.11529e13 −1.08510
\(780\) 3.70061e12 0.357971
\(781\) 8.77802e12 0.844242
\(782\) 2.10824e12 0.201600
\(783\) 1.26151e13 1.19940
\(784\) −3.75489e12 −0.354956
\(785\) −3.46994e12 −0.326143
\(786\) −2.56911e12 −0.240094
\(787\) 6.55343e11 0.0608951 0.0304475 0.999536i \(-0.490307\pi\)
0.0304475 + 0.999536i \(0.490307\pi\)
\(788\) 1.13089e13 1.04485
\(789\) −2.35267e12 −0.216129
\(790\) 4.75204e11 0.0434069
\(791\) 4.73265e12 0.429843
\(792\) 2.37307e12 0.214312
\(793\) −5.30382e12 −0.476277
\(794\) 1.17830e12 0.105212
\(795\) 2.83137e12 0.251388
\(796\) −3.61418e12 −0.319081
\(797\) 3.83799e12 0.336931 0.168466 0.985708i \(-0.446119\pi\)
0.168466 + 0.985708i \(0.446119\pi\)
\(798\) −3.90252e12 −0.340668
\(799\) −1.75962e13 −1.52742
\(800\) −2.06080e12 −0.177882
\(801\) 1.33330e11 0.0114441
\(802\) 2.57681e12 0.219937
\(803\) −1.90456e13 −1.61650
\(804\) −2.31285e12 −0.195207
\(805\) 1.41193e12 0.118504
\(806\) −3.12097e12 −0.260484
\(807\) 3.01974e12 0.250633
\(808\) −1.66404e12 −0.137345
\(809\) 2.67581e12 0.219627 0.109814 0.993952i \(-0.464975\pi\)
0.109814 + 0.993952i \(0.464975\pi\)
\(810\) −1.05542e12 −0.0861476
\(811\) −2.32586e13 −1.88795 −0.943973 0.330023i \(-0.892943\pi\)
−0.943973 + 0.330023i \(0.892943\pi\)
\(812\) −7.97489e12 −0.643758
\(813\) −3.92165e12 −0.314819
\(814\) −5.51959e12 −0.440653
\(815\) 1.04773e13 0.831843
\(816\) −9.47360e12 −0.748014
\(817\) 6.45270e12 0.506690
\(818\) 3.17138e12 0.247662
\(819\) 3.28861e12 0.255408
\(820\) 3.09575e12 0.239113
\(821\) 1.65740e13 1.27316 0.636580 0.771211i \(-0.280349\pi\)
0.636580 + 0.771211i \(0.280349\pi\)
\(822\) −6.52364e12 −0.498387
\(823\) 1.37332e13 1.04345 0.521726 0.853113i \(-0.325288\pi\)
0.521726 + 0.853113i \(0.325288\pi\)
\(824\) −5.74644e12 −0.434236
\(825\) 2.05770e12 0.154646
\(826\) 2.89086e12 0.216081
\(827\) 1.80036e13 1.33840 0.669198 0.743085i \(-0.266638\pi\)
0.669198 + 0.743085i \(0.266638\pi\)
\(828\) −1.59541e12 −0.117960
\(829\) 2.11116e13 1.55248 0.776240 0.630438i \(-0.217125\pi\)
0.776240 + 0.630438i \(0.217125\pi\)
\(830\) 1.30853e12 0.0957046
\(831\) 1.20866e13 0.879225
\(832\) 5.07536e12 0.367208
\(833\) −1.10642e13 −0.796191
\(834\) 2.30118e12 0.164704
\(835\) −8.72183e12 −0.620896
\(836\) −2.08821e13 −1.47858
\(837\) −1.01159e13 −0.712425
\(838\) 9.85294e12 0.690188
\(839\) −3.47988e12 −0.242458 −0.121229 0.992625i \(-0.538683\pi\)
−0.121229 + 0.992625i \(0.538683\pi\)
\(840\) 2.32122e12 0.160864
\(841\) 3.10254e12 0.213863
\(842\) 9.40003e12 0.644503
\(843\) −1.14341e13 −0.779789
\(844\) 3.60456e12 0.244518
\(845\) −1.77262e12 −0.119608
\(846\) −1.90227e12 −0.127675
\(847\) −9.44984e11 −0.0630883
\(848\) 6.67354e12 0.443175
\(849\) 2.84198e11 0.0187731
\(850\) −1.54638e12 −0.101609
\(851\) 7.95175e12 0.519733
\(852\) −9.70202e12 −0.630789
\(853\) 1.12686e13 0.728784 0.364392 0.931246i \(-0.381277\pi\)
0.364392 + 0.931246i \(0.381277\pi\)
\(854\) −1.55253e12 −0.0998800
\(855\) −4.21590e12 −0.269801
\(856\) 8.11860e12 0.516832
\(857\) 3.75227e12 0.237618 0.118809 0.992917i \(-0.462092\pi\)
0.118809 + 0.992917i \(0.462092\pi\)
\(858\) 4.88566e12 0.307773
\(859\) 1.57014e13 0.983944 0.491972 0.870611i \(-0.336276\pi\)
0.491972 + 0.870611i \(0.336276\pi\)
\(860\) −1.79110e12 −0.111655
\(861\) −5.34668e12 −0.331566
\(862\) 1.85862e12 0.114659
\(863\) −1.39316e13 −0.854975 −0.427487 0.904021i \(-0.640601\pi\)
−0.427487 + 0.904021i \(0.640601\pi\)
\(864\) −1.58595e13 −0.968231
\(865\) −9.81251e12 −0.595947
\(866\) 1.21243e12 0.0732531
\(867\) −1.43960e13 −0.865279
\(868\) 6.39495e12 0.382383
\(869\) −4.39165e12 −0.261239
\(870\) −2.39194e12 −0.141551
\(871\) 5.25021e12 0.309097
\(872\) −1.91738e13 −1.12301
\(873\) 1.30419e11 0.00759935
\(874\) −4.29767e12 −0.249133
\(875\) −1.03564e12 −0.0597275
\(876\) 2.10504e13 1.20779
\(877\) −1.59832e13 −0.912357 −0.456179 0.889888i \(-0.650782\pi\)
−0.456179 + 0.889888i \(0.650782\pi\)
\(878\) 4.17108e12 0.236877
\(879\) 9.65658e12 0.545599
\(880\) 4.84999e12 0.272627
\(881\) −3.16227e13 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(882\) −1.19612e12 −0.0665527
\(883\) −1.48526e13 −0.822202 −0.411101 0.911590i \(-0.634856\pi\)
−0.411101 + 0.911590i \(0.634856\pi\)
\(884\) 2.57013e13 1.41553
\(885\) −6.06948e12 −0.332588
\(886\) −8.81356e12 −0.480507
\(887\) −9.39325e12 −0.509518 −0.254759 0.967005i \(-0.581996\pi\)
−0.254759 + 0.967005i \(0.581996\pi\)
\(888\) 1.30727e13 0.705516
\(889\) 1.32373e12 0.0710792
\(890\) −9.96932e10 −0.00532611
\(891\) 9.75378e12 0.518470
\(892\) −1.42741e13 −0.754930
\(893\) 3.58699e13 1.88755
\(894\) 1.33636e12 0.0699687
\(895\) 1.57438e13 0.820176
\(896\) 1.29439e13 0.670930
\(897\) −7.03849e12 −0.363006
\(898\) −4.68538e12 −0.240437
\(899\) −1.41210e13 −0.721018
\(900\) 1.17022e12 0.0594536
\(901\) 1.96643e13 0.994071
\(902\) 4.08710e12 0.205582
\(903\) 3.09341e12 0.154826
\(904\) 8.56830e12 0.426714
\(905\) −6.53010e12 −0.323594
\(906\) 3.33350e12 0.164370
\(907\) 2.64429e13 1.29741 0.648704 0.761041i \(-0.275311\pi\)
0.648704 + 0.761041i \(0.275311\pi\)
\(908\) −1.35574e13 −0.661898
\(909\) 1.44888e12 0.0703876
\(910\) −2.45896e12 −0.118868
\(911\) −3.41645e13 −1.64339 −0.821697 0.569924i \(-0.806972\pi\)
−0.821697 + 0.569924i \(0.806972\pi\)
\(912\) 1.93120e13 0.924381
\(913\) −1.20929e13 −0.575987
\(914\) 4.56643e12 0.216431
\(915\) 3.25959e12 0.153733
\(916\) −9.23333e12 −0.433340
\(917\) −1.19498e13 −0.558081
\(918\) −1.19007e13 −0.553069
\(919\) 3.14366e13 1.45384 0.726920 0.686723i \(-0.240951\pi\)
0.726920 + 0.686723i \(0.240951\pi\)
\(920\) 2.55626e12 0.117641
\(921\) −7.24478e11 −0.0331785
\(922\) 6.08223e12 0.277188
\(923\) 2.20237e13 0.998809
\(924\) −1.00108e13 −0.451801
\(925\) −5.83256e12 −0.261952
\(926\) 5.71431e12 0.255396
\(927\) 5.00344e12 0.222541
\(928\) −2.21387e13 −0.979909
\(929\) −8.77007e12 −0.386307 −0.193153 0.981169i \(-0.561872\pi\)
−0.193153 + 0.981169i \(0.561872\pi\)
\(930\) 1.91807e12 0.0840795
\(931\) 2.25545e13 0.983918
\(932\) −1.62057e12 −0.0703553
\(933\) −3.09228e13 −1.33602
\(934\) −6.26017e12 −0.269169
\(935\) 1.42910e13 0.611521
\(936\) 5.95393e12 0.253549
\(937\) −1.66294e13 −0.704770 −0.352385 0.935855i \(-0.614629\pi\)
−0.352385 + 0.935855i \(0.614629\pi\)
\(938\) 1.53683e12 0.0648206
\(939\) −1.86389e13 −0.782392
\(940\) −9.95656e12 −0.415943
\(941\) −7.56052e12 −0.314339 −0.157169 0.987572i \(-0.550237\pi\)
−0.157169 + 0.987572i \(0.550237\pi\)
\(942\) 5.06334e12 0.209512
\(943\) −5.88806e12 −0.242476
\(944\) −1.43057e13 −0.586322
\(945\) −7.97013e12 −0.325104
\(946\) −2.36466e12 −0.0959974
\(947\) 3.40925e13 1.37747 0.688737 0.725011i \(-0.258165\pi\)
0.688737 + 0.725011i \(0.258165\pi\)
\(948\) 4.85393e12 0.195189
\(949\) −4.77846e13 −1.91245
\(950\) 3.15231e12 0.125566
\(951\) −2.89111e13 −1.14618
\(952\) 1.61212e13 0.636110
\(953\) 3.45885e13 1.35836 0.679178 0.733973i \(-0.262336\pi\)
0.679178 + 0.733973i \(0.262336\pi\)
\(954\) 2.12586e12 0.0830933
\(955\) 6.05391e12 0.235516
\(956\) −9.94185e11 −0.0384952
\(957\) 2.21054e13 0.851911
\(958\) 5.56591e12 0.213497
\(959\) −3.03435e13 −1.15846
\(960\) −3.11919e12 −0.118528
\(961\) −1.51162e13 −0.571726
\(962\) −1.38484e13 −0.521329
\(963\) −7.06889e12 −0.264870
\(964\) −1.15574e13 −0.431035
\(965\) 1.27769e13 0.474301
\(966\) −2.06029e12 −0.0761258
\(967\) 4.59891e13 1.69136 0.845680 0.533691i \(-0.179195\pi\)
0.845680 + 0.533691i \(0.179195\pi\)
\(968\) −1.71086e12 −0.0626290
\(969\) 5.69050e13 2.07345
\(970\) −9.75168e10 −0.00353677
\(971\) 2.72663e13 0.984327 0.492164 0.870503i \(-0.336206\pi\)
0.492164 + 0.870503i \(0.336206\pi\)
\(972\) 1.57279e13 0.565163
\(973\) 1.07035e13 0.382842
\(974\) 1.32156e13 0.470515
\(975\) 5.16269e12 0.182959
\(976\) 7.68284e12 0.271018
\(977\) −4.92228e13 −1.72839 −0.864193 0.503160i \(-0.832171\pi\)
−0.864193 + 0.503160i \(0.832171\pi\)
\(978\) −1.52885e13 −0.534368
\(979\) 9.21324e11 0.0320546
\(980\) −6.26053e12 −0.216817
\(981\) 1.66947e13 0.575530
\(982\) −9.59909e12 −0.329404
\(983\) 3.98337e13 1.36069 0.680346 0.732891i \(-0.261830\pi\)
0.680346 + 0.732891i \(0.261830\pi\)
\(984\) −9.67998e12 −0.329152
\(985\) 1.57770e13 0.534023
\(986\) −1.66124e13 −0.559740
\(987\) 1.71960e13 0.576766
\(988\) −5.23924e13 −1.74929
\(989\) 3.40664e12 0.113225
\(990\) 1.54496e12 0.0511164
\(991\) −2.56388e13 −0.844434 −0.422217 0.906495i \(-0.638748\pi\)
−0.422217 + 0.906495i \(0.638748\pi\)
\(992\) 1.77527e13 0.582052
\(993\) 2.73673e13 0.893224
\(994\) 6.44674e12 0.209460
\(995\) −5.04210e12 −0.163083
\(996\) 1.33659e13 0.430358
\(997\) −4.97241e13 −1.59382 −0.796909 0.604099i \(-0.793533\pi\)
−0.796909 + 0.604099i \(0.793533\pi\)
\(998\) −1.14517e13 −0.365410
\(999\) −4.48863e13 −1.42584
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 5.10.a.a.1.1 1
3.2 odd 2 45.10.a.c.1.1 1
4.3 odd 2 80.10.a.d.1.1 1
5.2 odd 4 25.10.b.a.24.1 2
5.3 odd 4 25.10.b.a.24.2 2
5.4 even 2 25.10.a.a.1.1 1
7.6 odd 2 245.10.a.a.1.1 1
8.3 odd 2 320.10.a.c.1.1 1
8.5 even 2 320.10.a.h.1.1 1
15.2 even 4 225.10.b.d.199.2 2
15.8 even 4 225.10.b.d.199.1 2
15.14 odd 2 225.10.a.b.1.1 1
20.3 even 4 400.10.c.e.49.2 2
20.7 even 4 400.10.c.e.49.1 2
20.19 odd 2 400.10.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.10.a.a.1.1 1 1.1 even 1 trivial
25.10.a.a.1.1 1 5.4 even 2
25.10.b.a.24.1 2 5.2 odd 4
25.10.b.a.24.2 2 5.3 odd 4
45.10.a.c.1.1 1 3.2 odd 2
80.10.a.d.1.1 1 4.3 odd 2
225.10.a.b.1.1 1 15.14 odd 2
225.10.b.d.199.1 2 15.8 even 4
225.10.b.d.199.2 2 15.2 even 4
245.10.a.a.1.1 1 7.6 odd 2
320.10.a.c.1.1 1 8.3 odd 2
320.10.a.h.1.1 1 8.5 even 2
400.10.a.c.1.1 1 20.19 odd 2
400.10.c.e.49.1 2 20.7 even 4
400.10.c.e.49.2 2 20.3 even 4