Properties

Label 3.10.a.b.1.1
Level $3$
Weight $10$
Character 3.1
Self dual yes
Analytic conductor $1.545$
Analytic rank $0$
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] = [3,10,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, 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("3.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(1.54510750849\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+18.0000 q^{2} +81.0000 q^{3} -188.000 q^{4} -1530.00 q^{5} +1458.00 q^{6} +9128.00 q^{7} -12600.0 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+18.0000 q^{2} +81.0000 q^{3} -188.000 q^{4} -1530.00 q^{5} +1458.00 q^{6} +9128.00 q^{7} -12600.0 q^{8} +6561.00 q^{9} -27540.0 q^{10} +21132.0 q^{11} -15228.0 q^{12} +31214.0 q^{13} +164304. q^{14} -123930. q^{15} -130544. q^{16} -279342. q^{17} +118098. q^{18} +144020. q^{19} +287640. q^{20} +739368. q^{21} +380376. q^{22} -1.76350e6 q^{23} -1.02060e6 q^{24} +387775. q^{25} +561852. q^{26} +531441. q^{27} -1.71606e6 q^{28} +4.69251e6 q^{29} -2.23074e6 q^{30} -369088. q^{31} +4.10141e6 q^{32} +1.71169e6 q^{33} -5.02816e6 q^{34} -1.39658e7 q^{35} -1.23347e6 q^{36} +9.34708e6 q^{37} +2.59236e6 q^{38} +2.52833e6 q^{39} +1.92780e7 q^{40} -7.22684e6 q^{41} +1.33086e7 q^{42} -2.31475e7 q^{43} -3.97282e6 q^{44} -1.00383e7 q^{45} -3.17429e7 q^{46} +2.29719e7 q^{47} -1.05741e7 q^{48} +4.29668e7 q^{49} +6.97995e6 q^{50} -2.26267e7 q^{51} -5.86823e6 q^{52} +7.84772e7 q^{53} +9.56594e6 q^{54} -3.23320e7 q^{55} -1.15013e8 q^{56} +1.16656e7 q^{57} +8.44652e7 q^{58} -2.03107e7 q^{59} +2.32988e7 q^{60} -1.79340e8 q^{61} -6.64358e6 q^{62} +5.98888e7 q^{63} +1.40664e8 q^{64} -4.77574e7 q^{65} +3.08105e7 q^{66} +2.74528e8 q^{67} +5.25163e7 q^{68} -1.42843e8 q^{69} -2.51385e8 q^{70} -3.63426e7 q^{71} -8.26686e7 q^{72} -2.47090e8 q^{73} +1.68247e8 q^{74} +3.14098e7 q^{75} -2.70758e7 q^{76} +1.92893e8 q^{77} +4.55100e7 q^{78} +1.91875e8 q^{79} +1.99732e8 q^{80} +4.30467e7 q^{81} -1.30083e8 q^{82} -2.76159e8 q^{83} -1.39001e8 q^{84} +4.27393e8 q^{85} -4.16655e8 q^{86} +3.80093e8 q^{87} -2.66263e8 q^{88} -6.78997e8 q^{89} -1.80690e8 q^{90} +2.84921e8 q^{91} +3.31537e8 q^{92} -2.98961e7 q^{93} +4.13494e8 q^{94} -2.20351e8 q^{95} +3.32214e8 q^{96} -5.67658e8 q^{97} +7.73402e8 q^{98} +1.38647e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.0000 0.795495 0.397748 0.917495i \(-0.369792\pi\)
0.397748 + 0.917495i \(0.369792\pi\)
\(3\) 81.0000 0.577350
\(4\) −188.000 −0.367188
\(5\) −1530.00 −1.09478 −0.547389 0.836878i \(-0.684378\pi\)
−0.547389 + 0.836878i \(0.684378\pi\)
\(6\) 1458.00 0.459279
\(7\) 9128.00 1.43693 0.718463 0.695565i \(-0.244846\pi\)
0.718463 + 0.695565i \(0.244846\pi\)
\(8\) −12600.0 −1.08759
\(9\) 6561.00 0.333333
\(10\) −27540.0 −0.870891
\(11\) 21132.0 0.435185 0.217592 0.976040i \(-0.430180\pi\)
0.217592 + 0.976040i \(0.430180\pi\)
\(12\) −15228.0 −0.211996
\(13\) 31214.0 0.303113 0.151556 0.988449i \(-0.451571\pi\)
0.151556 + 0.988449i \(0.451571\pi\)
\(14\) 164304. 1.14307
\(15\) −123930. −0.632071
\(16\) −130544. −0.497986
\(17\) −279342. −0.811178 −0.405589 0.914056i \(-0.632934\pi\)
−0.405589 + 0.914056i \(0.632934\pi\)
\(18\) 118098. 0.265165
\(19\) 144020. 0.253531 0.126766 0.991933i \(-0.459540\pi\)
0.126766 + 0.991933i \(0.459540\pi\)
\(20\) 287640. 0.401989
\(21\) 739368. 0.829610
\(22\) 380376. 0.346187
\(23\) −1.76350e6 −1.31401 −0.657006 0.753885i \(-0.728177\pi\)
−0.657006 + 0.753885i \(0.728177\pi\)
\(24\) −1.02060e6 −0.627921
\(25\) 387775. 0.198541
\(26\) 561852. 0.241125
\(27\) 531441. 0.192450
\(28\) −1.71606e6 −0.527621
\(29\) 4.69251e6 1.23201 0.616005 0.787742i \(-0.288750\pi\)
0.616005 + 0.787742i \(0.288750\pi\)
\(30\) −2.23074e6 −0.502809
\(31\) −369088. −0.0717798 −0.0358899 0.999356i \(-0.511427\pi\)
−0.0358899 + 0.999356i \(0.511427\pi\)
\(32\) 4.10141e6 0.691446
\(33\) 1.71169e6 0.251254
\(34\) −5.02816e6 −0.645288
\(35\) −1.39658e7 −1.57312
\(36\) −1.23347e6 −0.122396
\(37\) 9.34708e6 0.819914 0.409957 0.912105i \(-0.365544\pi\)
0.409957 + 0.912105i \(0.365544\pi\)
\(38\) 2.59236e6 0.201683
\(39\) 2.52833e6 0.175002
\(40\) 1.92780e7 1.19067
\(41\) −7.22684e6 −0.399412 −0.199706 0.979856i \(-0.563999\pi\)
−0.199706 + 0.979856i \(0.563999\pi\)
\(42\) 1.33086e7 0.659950
\(43\) −2.31475e7 −1.03251 −0.516257 0.856434i \(-0.672675\pi\)
−0.516257 + 0.856434i \(0.672675\pi\)
\(44\) −3.97282e6 −0.159794
\(45\) −1.00383e7 −0.364926
\(46\) −3.17429e7 −1.04529
\(47\) 2.29719e7 0.686683 0.343342 0.939211i \(-0.388441\pi\)
0.343342 + 0.939211i \(0.388441\pi\)
\(48\) −1.05741e7 −0.287512
\(49\) 4.29668e7 1.06476
\(50\) 6.97995e6 0.157938
\(51\) −2.26267e7 −0.468334
\(52\) −5.86823e6 −0.111299
\(53\) 7.84772e7 1.36616 0.683081 0.730343i \(-0.260640\pi\)
0.683081 + 0.730343i \(0.260640\pi\)
\(54\) 9.56594e6 0.153093
\(55\) −3.23320e7 −0.476431
\(56\) −1.15013e8 −1.56279
\(57\) 1.16656e7 0.146376
\(58\) 8.44652e7 0.980058
\(59\) −2.03107e7 −0.218218 −0.109109 0.994030i \(-0.534800\pi\)
−0.109109 + 0.994030i \(0.534800\pi\)
\(60\) 2.32988e7 0.232089
\(61\) −1.79340e8 −1.65841 −0.829207 0.558942i \(-0.811207\pi\)
−0.829207 + 0.558942i \(0.811207\pi\)
\(62\) −6.64358e6 −0.0571005
\(63\) 5.98888e7 0.478975
\(64\) 1.40664e8 1.04803
\(65\) −4.77574e7 −0.331842
\(66\) 3.08105e7 0.199871
\(67\) 2.74528e8 1.66437 0.832186 0.554496i \(-0.187089\pi\)
0.832186 + 0.554496i \(0.187089\pi\)
\(68\) 5.25163e7 0.297854
\(69\) −1.42843e8 −0.758645
\(70\) −2.51385e8 −1.25141
\(71\) −3.63426e7 −0.169728 −0.0848641 0.996393i \(-0.527046\pi\)
−0.0848641 + 0.996393i \(0.527046\pi\)
\(72\) −8.26686e7 −0.362530
\(73\) −2.47090e8 −1.01836 −0.509180 0.860660i \(-0.670051\pi\)
−0.509180 + 0.860660i \(0.670051\pi\)
\(74\) 1.68247e8 0.652237
\(75\) 3.14098e7 0.114628
\(76\) −2.70758e7 −0.0930935
\(77\) 1.92893e8 0.625328
\(78\) 4.55100e7 0.139213
\(79\) 1.91875e8 0.554238 0.277119 0.960836i \(-0.410620\pi\)
0.277119 + 0.960836i \(0.410620\pi\)
\(80\) 1.99732e8 0.545184
\(81\) 4.30467e7 0.111111
\(82\) −1.30083e8 −0.317730
\(83\) −2.76159e8 −0.638717 −0.319358 0.947634i \(-0.603467\pi\)
−0.319358 + 0.947634i \(0.603467\pi\)
\(84\) −1.39001e8 −0.304622
\(85\) 4.27393e8 0.888060
\(86\) −4.16655e8 −0.821359
\(87\) 3.80093e8 0.711301
\(88\) −2.66263e8 −0.473303
\(89\) −6.78997e8 −1.14713 −0.573566 0.819160i \(-0.694440\pi\)
−0.573566 + 0.819160i \(0.694440\pi\)
\(90\) −1.80690e8 −0.290297
\(91\) 2.84921e8 0.435551
\(92\) 3.31537e8 0.482489
\(93\) −2.98961e7 −0.0414421
\(94\) 4.13494e8 0.546253
\(95\) −2.20351e8 −0.277561
\(96\) 3.32214e8 0.399206
\(97\) −5.67658e8 −0.651049 −0.325524 0.945534i \(-0.605541\pi\)
−0.325524 + 0.945534i \(0.605541\pi\)
\(98\) 7.73402e8 0.847009
\(99\) 1.38647e8 0.145062
\(100\) −7.29017e7 −0.0729017
\(101\) 1.62282e9 1.55176 0.775881 0.630879i \(-0.217306\pi\)
0.775881 + 0.630879i \(0.217306\pi\)
\(102\) −4.07281e8 −0.372557
\(103\) −1.75103e9 −1.53294 −0.766470 0.642280i \(-0.777989\pi\)
−0.766470 + 0.642280i \(0.777989\pi\)
\(104\) −3.93296e8 −0.329663
\(105\) −1.13123e9 −0.908239
\(106\) 1.41259e9 1.08677
\(107\) −1.54296e9 −1.13796 −0.568980 0.822352i \(-0.692662\pi\)
−0.568980 + 0.822352i \(0.692662\pi\)
\(108\) −9.99109e7 −0.0706653
\(109\) 4.57665e8 0.310548 0.155274 0.987871i \(-0.450374\pi\)
0.155274 + 0.987871i \(0.450374\pi\)
\(110\) −5.81975e8 −0.378998
\(111\) 7.57113e8 0.473377
\(112\) −1.19161e9 −0.715569
\(113\) 3.26794e9 1.88548 0.942739 0.333531i \(-0.108240\pi\)
0.942739 + 0.333531i \(0.108240\pi\)
\(114\) 2.09981e8 0.116442
\(115\) 2.69815e9 1.43855
\(116\) −8.82192e8 −0.452379
\(117\) 2.04795e8 0.101038
\(118\) −3.65592e8 −0.173591
\(119\) −2.54983e9 −1.16560
\(120\) 1.56152e9 0.687435
\(121\) −1.91139e9 −0.810614
\(122\) −3.22812e9 −1.31926
\(123\) −5.85374e8 −0.230601
\(124\) 6.93885e7 0.0263566
\(125\) 2.39499e9 0.877421
\(126\) 1.07800e9 0.381023
\(127\) 9.28879e8 0.316842 0.158421 0.987372i \(-0.449360\pi\)
0.158421 + 0.987372i \(0.449360\pi\)
\(128\) 4.32029e8 0.142255
\(129\) −1.87495e9 −0.596122
\(130\) −8.59634e8 −0.263978
\(131\) −9.88659e8 −0.293309 −0.146655 0.989188i \(-0.546851\pi\)
−0.146655 + 0.989188i \(0.546851\pi\)
\(132\) −3.21798e8 −0.0922573
\(133\) 1.31461e9 0.364306
\(134\) 4.94151e9 1.32400
\(135\) −8.13105e8 −0.210690
\(136\) 3.51971e9 0.882230
\(137\) −5.73253e7 −0.0139028 −0.00695142 0.999976i \(-0.502213\pi\)
−0.00695142 + 0.999976i \(0.502213\pi\)
\(138\) −2.57118e9 −0.603498
\(139\) −4.65052e9 −1.05666 −0.528330 0.849039i \(-0.677182\pi\)
−0.528330 + 0.849039i \(0.677182\pi\)
\(140\) 2.62558e9 0.577629
\(141\) 1.86072e9 0.396457
\(142\) −6.54168e8 −0.135018
\(143\) 6.59614e8 0.131910
\(144\) −8.56499e8 −0.165995
\(145\) −7.17954e9 −1.34878
\(146\) −4.44761e9 −0.810101
\(147\) 3.48031e9 0.614738
\(148\) −1.75725e9 −0.301062
\(149\) −1.40236e9 −0.233089 −0.116545 0.993185i \(-0.537182\pi\)
−0.116545 + 0.993185i \(0.537182\pi\)
\(150\) 5.65376e8 0.0911857
\(151\) 1.01548e10 1.58955 0.794773 0.606907i \(-0.207590\pi\)
0.794773 + 0.606907i \(0.207590\pi\)
\(152\) −1.81465e9 −0.275738
\(153\) −1.83276e9 −0.270393
\(154\) 3.47207e9 0.497445
\(155\) 5.64705e8 0.0785830
\(156\) −4.75327e8 −0.0642587
\(157\) 9.36605e9 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(158\) 3.45375e9 0.440893
\(159\) 6.35665e9 0.788754
\(160\) −6.27515e9 −0.756980
\(161\) −1.60972e10 −1.88814
\(162\) 7.74841e8 0.0883883
\(163\) −7.34780e8 −0.0815292 −0.0407646 0.999169i \(-0.512979\pi\)
−0.0407646 + 0.999169i \(0.512979\pi\)
\(164\) 1.35865e9 0.146659
\(165\) −2.61889e9 −0.275068
\(166\) −4.97087e9 −0.508096
\(167\) 1.55584e9 0.154789 0.0773947 0.997001i \(-0.475340\pi\)
0.0773947 + 0.997001i \(0.475340\pi\)
\(168\) −9.31604e9 −0.902276
\(169\) −9.63019e9 −0.908123
\(170\) 7.69308e9 0.706448
\(171\) 9.44915e8 0.0845104
\(172\) 4.35173e9 0.379126
\(173\) 1.90448e10 1.61648 0.808238 0.588856i \(-0.200421\pi\)
0.808238 + 0.588856i \(0.200421\pi\)
\(174\) 6.84168e9 0.565837
\(175\) 3.53961e9 0.285288
\(176\) −2.75866e9 −0.216716
\(177\) −1.64516e9 −0.125988
\(178\) −1.22220e10 −0.912537
\(179\) 4.32852e9 0.315138 0.157569 0.987508i \(-0.449634\pi\)
0.157569 + 0.987508i \(0.449634\pi\)
\(180\) 1.88721e9 0.133996
\(181\) −9.56757e8 −0.0662595 −0.0331298 0.999451i \(-0.510547\pi\)
−0.0331298 + 0.999451i \(0.510547\pi\)
\(182\) 5.12859e9 0.346479
\(183\) −1.45265e10 −0.957485
\(184\) 2.22200e10 1.42911
\(185\) −1.43010e10 −0.897624
\(186\) −5.38130e8 −0.0329670
\(187\) −5.90306e9 −0.353012
\(188\) −4.31871e9 −0.252141
\(189\) 4.85099e9 0.276537
\(190\) −3.96631e9 −0.220798
\(191\) −1.76438e10 −0.959274 −0.479637 0.877467i \(-0.659232\pi\)
−0.479637 + 0.877467i \(0.659232\pi\)
\(192\) 1.13938e10 0.605079
\(193\) 1.30993e10 0.679581 0.339790 0.940501i \(-0.389644\pi\)
0.339790 + 0.940501i \(0.389644\pi\)
\(194\) −1.02178e10 −0.517906
\(195\) −3.86835e9 −0.191589
\(196\) −8.07775e9 −0.390965
\(197\) 4.74589e9 0.224502 0.112251 0.993680i \(-0.464194\pi\)
0.112251 + 0.993680i \(0.464194\pi\)
\(198\) 2.49565e9 0.115396
\(199\) −1.92102e10 −0.868348 −0.434174 0.900829i \(-0.642960\pi\)
−0.434174 + 0.900829i \(0.642960\pi\)
\(200\) −4.88596e9 −0.215931
\(201\) 2.22368e10 0.960926
\(202\) 2.92108e10 1.23442
\(203\) 4.28332e10 1.77031
\(204\) 4.25382e9 0.171966
\(205\) 1.10571e10 0.437268
\(206\) −3.15185e10 −1.21945
\(207\) −1.15703e10 −0.438004
\(208\) −4.07480e9 −0.150946
\(209\) 3.04343e9 0.110333
\(210\) −2.03622e10 −0.722500
\(211\) −1.81273e10 −0.629597 −0.314798 0.949159i \(-0.601937\pi\)
−0.314798 + 0.949159i \(0.601937\pi\)
\(212\) −1.47537e10 −0.501637
\(213\) −2.94375e9 −0.0979926
\(214\) −2.77732e10 −0.905241
\(215\) 3.54156e10 1.13037
\(216\) −6.69616e9 −0.209307
\(217\) −3.36904e9 −0.103142
\(218\) 8.23797e9 0.247039
\(219\) −2.00143e10 −0.587951
\(220\) 6.07841e9 0.174939
\(221\) −8.71938e9 −0.245878
\(222\) 1.36280e10 0.376569
\(223\) 3.91081e9 0.105900 0.0529499 0.998597i \(-0.483138\pi\)
0.0529499 + 0.998597i \(0.483138\pi\)
\(224\) 3.74377e10 0.993556
\(225\) 2.54419e9 0.0661803
\(226\) 5.88230e10 1.49989
\(227\) −3.58126e10 −0.895199 −0.447599 0.894234i \(-0.647721\pi\)
−0.447599 + 0.894234i \(0.647721\pi\)
\(228\) −2.19314e9 −0.0537476
\(229\) 5.30445e9 0.127462 0.0637310 0.997967i \(-0.479700\pi\)
0.0637310 + 0.997967i \(0.479700\pi\)
\(230\) 4.85667e10 1.14436
\(231\) 1.56243e10 0.361033
\(232\) −5.91256e10 −1.33992
\(233\) −2.54172e10 −0.564970 −0.282485 0.959272i \(-0.591159\pi\)
−0.282485 + 0.959272i \(0.591159\pi\)
\(234\) 3.68631e9 0.0803749
\(235\) −3.51470e10 −0.751766
\(236\) 3.81840e9 0.0801268
\(237\) 1.55419e10 0.319989
\(238\) −4.58970e10 −0.927231
\(239\) −6.33494e10 −1.25589 −0.627946 0.778257i \(-0.716104\pi\)
−0.627946 + 0.778257i \(0.716104\pi\)
\(240\) 1.61783e10 0.314762
\(241\) 1.01929e11 1.94635 0.973173 0.230073i \(-0.0738965\pi\)
0.973173 + 0.230073i \(0.0738965\pi\)
\(242\) −3.44050e10 −0.644840
\(243\) 3.48678e9 0.0641500
\(244\) 3.37159e10 0.608949
\(245\) −6.57392e10 −1.16567
\(246\) −1.05367e10 −0.183442
\(247\) 4.49544e9 0.0768486
\(248\) 4.65051e9 0.0780671
\(249\) −2.23689e10 −0.368763
\(250\) 4.31097e10 0.697984
\(251\) 9.64745e10 1.53420 0.767098 0.641530i \(-0.221700\pi\)
0.767098 + 0.641530i \(0.221700\pi\)
\(252\) −1.12591e10 −0.175874
\(253\) −3.72662e10 −0.571838
\(254\) 1.67198e10 0.252046
\(255\) 3.46189e10 0.512722
\(256\) −6.42434e10 −0.934864
\(257\) −7.99591e9 −0.114332 −0.0571661 0.998365i \(-0.518206\pi\)
−0.0571661 + 0.998365i \(0.518206\pi\)
\(258\) −3.37490e10 −0.474212
\(259\) 8.53201e10 1.17816
\(260\) 8.97839e9 0.121848
\(261\) 3.07876e10 0.410670
\(262\) −1.77959e10 −0.233326
\(263\) 1.06269e11 1.36964 0.684821 0.728712i \(-0.259881\pi\)
0.684821 + 0.728712i \(0.259881\pi\)
\(264\) −2.15673e10 −0.273262
\(265\) −1.20070e11 −1.49564
\(266\) 2.36631e10 0.289803
\(267\) −5.49988e10 −0.662296
\(268\) −5.16113e10 −0.611137
\(269\) −1.00886e11 −1.17475 −0.587377 0.809313i \(-0.699840\pi\)
−0.587377 + 0.809313i \(0.699840\pi\)
\(270\) −1.46359e10 −0.167603
\(271\) −2.86345e10 −0.322498 −0.161249 0.986914i \(-0.551552\pi\)
−0.161249 + 0.986914i \(0.551552\pi\)
\(272\) 3.64664e10 0.403955
\(273\) 2.30786e10 0.251465
\(274\) −1.03186e9 −0.0110596
\(275\) 8.19446e9 0.0864019
\(276\) 2.68545e10 0.278565
\(277\) 4.12251e10 0.420729 0.210365 0.977623i \(-0.432535\pi\)
0.210365 + 0.977623i \(0.432535\pi\)
\(278\) −8.37094e10 −0.840568
\(279\) −2.42159e9 −0.0239266
\(280\) 1.75970e11 1.71091
\(281\) 1.21800e11 1.16538 0.582692 0.812693i \(-0.301999\pi\)
0.582692 + 0.812693i \(0.301999\pi\)
\(282\) 3.34930e10 0.315379
\(283\) 2.32820e10 0.215765 0.107883 0.994164i \(-0.465593\pi\)
0.107883 + 0.994164i \(0.465593\pi\)
\(284\) 6.83242e9 0.0623221
\(285\) −1.78484e10 −0.160250
\(286\) 1.18731e10 0.104934
\(287\) −6.59666e10 −0.573925
\(288\) 2.69093e10 0.230482
\(289\) −4.05559e10 −0.341990
\(290\) −1.29232e11 −1.07295
\(291\) −4.59803e10 −0.375883
\(292\) 4.64528e10 0.373929
\(293\) −9.12801e10 −0.723555 −0.361778 0.932264i \(-0.617830\pi\)
−0.361778 + 0.932264i \(0.617830\pi\)
\(294\) 6.26456e10 0.489021
\(295\) 3.10753e10 0.238900
\(296\) −1.17773e11 −0.891731
\(297\) 1.12304e10 0.0837513
\(298\) −2.52425e10 −0.185421
\(299\) −5.50458e10 −0.398294
\(300\) −5.90504e9 −0.0420898
\(301\) −2.11290e11 −1.48365
\(302\) 1.82786e11 1.26448
\(303\) 1.31449e11 0.895910
\(304\) −1.88009e10 −0.126255
\(305\) 2.74390e11 1.81560
\(306\) −3.29897e10 −0.215096
\(307\) −4.39070e10 −0.282105 −0.141053 0.990002i \(-0.545049\pi\)
−0.141053 + 0.990002i \(0.545049\pi\)
\(308\) −3.62639e10 −0.229613
\(309\) −1.41833e11 −0.885043
\(310\) 1.01647e10 0.0625124
\(311\) −7.64835e10 −0.463603 −0.231801 0.972763i \(-0.574462\pi\)
−0.231801 + 0.972763i \(0.574462\pi\)
\(312\) −3.18570e10 −0.190331
\(313\) 2.68488e11 1.58116 0.790580 0.612359i \(-0.209779\pi\)
0.790580 + 0.612359i \(0.209779\pi\)
\(314\) 1.68589e11 0.978691
\(315\) −9.16299e10 −0.524372
\(316\) −3.60725e10 −0.203509
\(317\) 1.52056e11 0.845738 0.422869 0.906191i \(-0.361023\pi\)
0.422869 + 0.906191i \(0.361023\pi\)
\(318\) 1.14420e11 0.627450
\(319\) 9.91621e10 0.536152
\(320\) −2.15216e11 −1.14736
\(321\) −1.24980e11 −0.657001
\(322\) −2.89749e11 −1.50200
\(323\) −4.02308e10 −0.205659
\(324\) −8.09278e9 −0.0407986
\(325\) 1.21040e10 0.0601803
\(326\) −1.32260e10 −0.0648561
\(327\) 3.70709e10 0.179295
\(328\) 9.10582e10 0.434397
\(329\) 2.09687e11 0.986713
\(330\) −4.71400e10 −0.218815
\(331\) −2.63057e11 −1.20455 −0.602274 0.798289i \(-0.705739\pi\)
−0.602274 + 0.798289i \(0.705739\pi\)
\(332\) 5.19179e10 0.234529
\(333\) 6.13262e10 0.273305
\(334\) 2.80051e10 0.123134
\(335\) −4.20028e11 −1.82212
\(336\) −9.65201e10 −0.413134
\(337\) 2.97183e11 1.25513 0.627566 0.778564i \(-0.284051\pi\)
0.627566 + 0.778564i \(0.284051\pi\)
\(338\) −1.73343e11 −0.722407
\(339\) 2.64703e11 1.08858
\(340\) −8.03499e10 −0.326085
\(341\) −7.79957e9 −0.0312375
\(342\) 1.70085e10 0.0672276
\(343\) 2.38530e10 0.0930507
\(344\) 2.91658e11 1.12295
\(345\) 2.18550e11 0.830549
\(346\) 3.42807e11 1.28590
\(347\) 1.40219e11 0.519186 0.259593 0.965718i \(-0.416412\pi\)
0.259593 + 0.965718i \(0.416412\pi\)
\(348\) −7.14575e10 −0.261181
\(349\) −3.14891e11 −1.13618 −0.568088 0.822968i \(-0.692317\pi\)
−0.568088 + 0.822968i \(0.692317\pi\)
\(350\) 6.37130e10 0.226946
\(351\) 1.65884e10 0.0583341
\(352\) 8.66710e10 0.300907
\(353\) −5.31195e11 −1.82082 −0.910411 0.413704i \(-0.864235\pi\)
−0.910411 + 0.413704i \(0.864235\pi\)
\(354\) −2.96129e10 −0.100223
\(355\) 5.56043e10 0.185815
\(356\) 1.27652e11 0.421212
\(357\) −2.06537e11 −0.672961
\(358\) 7.79133e10 0.250691
\(359\) −4.97684e11 −1.58135 −0.790676 0.612235i \(-0.790271\pi\)
−0.790676 + 0.612235i \(0.790271\pi\)
\(360\) 1.26483e11 0.396891
\(361\) −3.01946e11 −0.935722
\(362\) −1.72216e10 −0.0527091
\(363\) −1.54822e11 −0.468008
\(364\) −5.35652e10 −0.159929
\(365\) 3.78047e11 1.11488
\(366\) −2.61478e11 −0.761675
\(367\) 2.77743e11 0.799183 0.399591 0.916693i \(-0.369152\pi\)
0.399591 + 0.916693i \(0.369152\pi\)
\(368\) 2.30214e11 0.654359
\(369\) −4.74153e10 −0.133137
\(370\) −2.57419e11 −0.714056
\(371\) 7.16340e11 1.96307
\(372\) 5.62047e9 0.0152170
\(373\) 2.55319e11 0.682957 0.341478 0.939890i \(-0.389072\pi\)
0.341478 + 0.939890i \(0.389072\pi\)
\(374\) −1.06255e11 −0.280819
\(375\) 1.93994e11 0.506579
\(376\) −2.89446e11 −0.746830
\(377\) 1.46472e11 0.373438
\(378\) 8.73179e10 0.219983
\(379\) 8.12103e10 0.202178 0.101089 0.994877i \(-0.467767\pi\)
0.101089 + 0.994877i \(0.467767\pi\)
\(380\) 4.14259e10 0.101917
\(381\) 7.52392e10 0.182929
\(382\) −3.17589e11 −0.763097
\(383\) −4.13123e11 −0.981036 −0.490518 0.871431i \(-0.663192\pi\)
−0.490518 + 0.871431i \(0.663192\pi\)
\(384\) 3.49943e10 0.0821310
\(385\) −2.95126e11 −0.684596
\(386\) 2.35788e11 0.540603
\(387\) −1.51871e11 −0.344171
\(388\) 1.06720e11 0.239057
\(389\) 2.53052e11 0.560321 0.280160 0.959953i \(-0.409612\pi\)
0.280160 + 0.959953i \(0.409612\pi\)
\(390\) −6.96303e10 −0.152408
\(391\) 4.92618e11 1.06590
\(392\) −5.41381e11 −1.15802
\(393\) −8.00814e10 −0.169342
\(394\) 8.54259e10 0.178590
\(395\) −2.93568e11 −0.606768
\(396\) −2.60656e10 −0.0532648
\(397\) −6.53562e10 −0.132047 −0.0660236 0.997818i \(-0.521031\pi\)
−0.0660236 + 0.997818i \(0.521031\pi\)
\(398\) −3.45784e11 −0.690766
\(399\) 1.06484e11 0.210332
\(400\) −5.06217e10 −0.0988705
\(401\) −2.40886e11 −0.465225 −0.232612 0.972570i \(-0.574727\pi\)
−0.232612 + 0.972570i \(0.574727\pi\)
\(402\) 4.00262e11 0.764412
\(403\) −1.15207e10 −0.0217574
\(404\) −3.05091e11 −0.569788
\(405\) −6.58615e10 −0.121642
\(406\) 7.70998e11 1.40827
\(407\) 1.97522e11 0.356814
\(408\) 2.85096e11 0.509356
\(409\) −3.78340e11 −0.668540 −0.334270 0.942477i \(-0.608490\pi\)
−0.334270 + 0.942477i \(0.608490\pi\)
\(410\) 1.99027e11 0.347844
\(411\) −4.64335e9 −0.00802681
\(412\) 3.29193e11 0.562876
\(413\) −1.85396e11 −0.313563
\(414\) −2.08265e11 −0.348430
\(415\) 4.22524e11 0.699253
\(416\) 1.28021e11 0.209586
\(417\) −3.76693e11 −0.610063
\(418\) 5.47818e10 0.0877693
\(419\) 9.68766e11 1.53552 0.767760 0.640737i \(-0.221371\pi\)
0.767760 + 0.640737i \(0.221371\pi\)
\(420\) 2.12672e11 0.333494
\(421\) −1.43789e11 −0.223077 −0.111539 0.993760i \(-0.535578\pi\)
−0.111539 + 0.993760i \(0.535578\pi\)
\(422\) −3.26292e11 −0.500841
\(423\) 1.50719e11 0.228894
\(424\) −9.88812e11 −1.48582
\(425\) −1.08322e11 −0.161052
\(426\) −5.29876e10 −0.0779527
\(427\) −1.63701e12 −2.38302
\(428\) 2.90076e11 0.417845
\(429\) 5.34288e10 0.0761583
\(430\) 6.37481e11 0.899207
\(431\) 2.06890e11 0.288796 0.144398 0.989520i \(-0.453875\pi\)
0.144398 + 0.989520i \(0.453875\pi\)
\(432\) −6.93764e10 −0.0958374
\(433\) −4.81883e10 −0.0658789 −0.0329394 0.999457i \(-0.510487\pi\)
−0.0329394 + 0.999457i \(0.510487\pi\)
\(434\) −6.06426e10 −0.0820492
\(435\) −5.81543e11 −0.778718
\(436\) −8.60411e10 −0.114029
\(437\) −2.53979e11 −0.333143
\(438\) −3.60257e11 −0.467712
\(439\) 4.37941e11 0.562763 0.281381 0.959596i \(-0.409207\pi\)
0.281381 + 0.959596i \(0.409207\pi\)
\(440\) 4.07383e11 0.518162
\(441\) 2.81905e11 0.354919
\(442\) −1.56949e11 −0.195595
\(443\) 7.58665e11 0.935908 0.467954 0.883753i \(-0.344991\pi\)
0.467954 + 0.883753i \(0.344991\pi\)
\(444\) −1.42337e11 −0.173818
\(445\) 1.03887e12 1.25585
\(446\) 7.03946e10 0.0842427
\(447\) −1.13591e11 −0.134574
\(448\) 1.28398e12 1.50594
\(449\) −1.37280e11 −0.159403 −0.0797017 0.996819i \(-0.525397\pi\)
−0.0797017 + 0.996819i \(0.525397\pi\)
\(450\) 4.57955e10 0.0526461
\(451\) −1.52718e11 −0.173818
\(452\) −6.14373e11 −0.692324
\(453\) 8.22535e11 0.917725
\(454\) −6.44627e11 −0.712126
\(455\) −4.35930e11 −0.476832
\(456\) −1.46987e11 −0.159198
\(457\) 1.31091e11 0.140589 0.0702944 0.997526i \(-0.477606\pi\)
0.0702944 + 0.997526i \(0.477606\pi\)
\(458\) 9.54800e10 0.101395
\(459\) −1.48454e11 −0.156111
\(460\) −5.07252e11 −0.528218
\(461\) −8.03508e11 −0.828583 −0.414291 0.910144i \(-0.635971\pi\)
−0.414291 + 0.910144i \(0.635971\pi\)
\(462\) 2.81238e11 0.287200
\(463\) −1.78582e12 −1.80602 −0.903009 0.429621i \(-0.858647\pi\)
−0.903009 + 0.429621i \(0.858647\pi\)
\(464\) −6.12579e11 −0.613524
\(465\) 4.57411e10 0.0453699
\(466\) −4.57509e11 −0.449431
\(467\) 6.44366e11 0.626912 0.313456 0.949603i \(-0.398513\pi\)
0.313456 + 0.949603i \(0.398513\pi\)
\(468\) −3.85015e10 −0.0370997
\(469\) 2.50590e12 2.39158
\(470\) −6.32646e11 −0.598026
\(471\) 7.58650e11 0.710309
\(472\) 2.55914e11 0.237332
\(473\) −4.89152e11 −0.449334
\(474\) 2.79753e11 0.254550
\(475\) 5.58474e10 0.0503363
\(476\) 4.79369e11 0.427995
\(477\) 5.14889e11 0.455387
\(478\) −1.14029e12 −0.999056
\(479\) −1.87701e12 −1.62913 −0.814565 0.580073i \(-0.803024\pi\)
−0.814565 + 0.580073i \(0.803024\pi\)
\(480\) −5.08287e11 −0.437043
\(481\) 2.91760e11 0.248526
\(482\) 1.83472e12 1.54831
\(483\) −1.30387e12 −1.09012
\(484\) 3.59341e11 0.297647
\(485\) 8.68516e11 0.712755
\(486\) 6.27621e10 0.0510310
\(487\) 1.49355e12 1.20321 0.601603 0.798795i \(-0.294529\pi\)
0.601603 + 0.798795i \(0.294529\pi\)
\(488\) 2.25968e12 1.80368
\(489\) −5.95172e10 −0.0470709
\(490\) −1.18331e12 −0.927287
\(491\) −7.47313e11 −0.580278 −0.290139 0.956985i \(-0.593702\pi\)
−0.290139 + 0.956985i \(0.593702\pi\)
\(492\) 1.10050e11 0.0846736
\(493\) −1.31082e12 −0.999379
\(494\) 8.09179e10 0.0611327
\(495\) −2.12130e11 −0.158810
\(496\) 4.81822e10 0.0357453
\(497\) −3.31736e11 −0.243887
\(498\) −4.02640e11 −0.293349
\(499\) 1.01300e12 0.731402 0.365701 0.930732i \(-0.380829\pi\)
0.365701 + 0.930732i \(0.380829\pi\)
\(500\) −4.50257e11 −0.322178
\(501\) 1.26023e11 0.0893677
\(502\) 1.73654e12 1.22044
\(503\) 2.03399e11 0.141675 0.0708373 0.997488i \(-0.477433\pi\)
0.0708373 + 0.997488i \(0.477433\pi\)
\(504\) −7.54599e11 −0.520929
\(505\) −2.48292e12 −1.69884
\(506\) −6.70792e11 −0.454894
\(507\) −7.80045e11 −0.524305
\(508\) −1.74629e11 −0.116340
\(509\) 5.93699e11 0.392045 0.196023 0.980599i \(-0.437197\pi\)
0.196023 + 0.980599i \(0.437197\pi\)
\(510\) 6.23139e11 0.407868
\(511\) −2.25543e12 −1.46331
\(512\) −1.37758e12 −0.885935
\(513\) 7.65381e10 0.0487921
\(514\) −1.43926e11 −0.0909508
\(515\) 2.67907e12 1.67823
\(516\) 3.52490e11 0.218888
\(517\) 4.85442e11 0.298834
\(518\) 1.53576e12 0.937217
\(519\) 1.54263e12 0.933273
\(520\) 6.01743e11 0.360908
\(521\) −1.74950e12 −1.04027 −0.520133 0.854085i \(-0.674118\pi\)
−0.520133 + 0.854085i \(0.674118\pi\)
\(522\) 5.54176e11 0.326686
\(523\) 3.20555e11 0.187346 0.0936731 0.995603i \(-0.470139\pi\)
0.0936731 + 0.995603i \(0.470139\pi\)
\(524\) 1.85868e11 0.107699
\(525\) 2.86708e11 0.164711
\(526\) 1.91285e12 1.08954
\(527\) 1.03102e11 0.0582262
\(528\) −2.23451e11 −0.125121
\(529\) 1.30877e12 0.726627
\(530\) −2.16126e12 −1.18978
\(531\) −1.33258e11 −0.0727392
\(532\) −2.47148e11 −0.133769
\(533\) −2.25579e11 −0.121067
\(534\) −9.89978e11 −0.526854
\(535\) 2.36072e12 1.24581
\(536\) −3.45906e12 −1.81016
\(537\) 3.50610e11 0.181945
\(538\) −1.81595e12 −0.934512
\(539\) 9.07974e11 0.463366
\(540\) 1.52864e11 0.0773628
\(541\) 3.50229e11 0.175778 0.0878889 0.996130i \(-0.471988\pi\)
0.0878889 + 0.996130i \(0.471988\pi\)
\(542\) −5.15421e11 −0.256546
\(543\) −7.74974e10 −0.0382549
\(544\) −1.14570e12 −0.560885
\(545\) −7.00228e11 −0.339981
\(546\) 4.15415e11 0.200039
\(547\) −2.46037e12 −1.17505 −0.587527 0.809205i \(-0.699898\pi\)
−0.587527 + 0.809205i \(0.699898\pi\)
\(548\) 1.07772e10 0.00510495
\(549\) −1.17665e12 −0.552804
\(550\) 1.47500e11 0.0687323
\(551\) 6.75815e11 0.312353
\(552\) 1.79982e12 0.825095
\(553\) 1.75143e12 0.796399
\(554\) 7.42051e11 0.334688
\(555\) −1.15838e12 −0.518244
\(556\) 8.74299e11 0.387992
\(557\) 1.04356e12 0.459376 0.229688 0.973264i \(-0.426229\pi\)
0.229688 + 0.973264i \(0.426229\pi\)
\(558\) −4.35886e10 −0.0190335
\(559\) −7.22525e11 −0.312968
\(560\) 1.82316e12 0.783390
\(561\) −4.78147e11 −0.203812
\(562\) 2.19240e12 0.927057
\(563\) 3.81849e12 1.60179 0.800893 0.598808i \(-0.204359\pi\)
0.800893 + 0.598808i \(0.204359\pi\)
\(564\) −3.49816e11 −0.145574
\(565\) −4.99995e12 −2.06418
\(566\) 4.19076e11 0.171640
\(567\) 3.92930e11 0.159658
\(568\) 4.57917e11 0.184595
\(569\) 4.84062e11 0.193596 0.0967979 0.995304i \(-0.469140\pi\)
0.0967979 + 0.995304i \(0.469140\pi\)
\(570\) −3.21271e11 −0.127478
\(571\) 1.77044e10 0.00696978 0.00348489 0.999994i \(-0.498891\pi\)
0.00348489 + 0.999994i \(0.498891\pi\)
\(572\) −1.24007e11 −0.0484357
\(573\) −1.42915e12 −0.553837
\(574\) −1.18740e12 −0.456555
\(575\) −6.83840e11 −0.260885
\(576\) 9.22896e11 0.349343
\(577\) −7.46985e11 −0.280557 −0.140278 0.990112i \(-0.544800\pi\)
−0.140278 + 0.990112i \(0.544800\pi\)
\(578\) −7.30007e11 −0.272052
\(579\) 1.06105e12 0.392356
\(580\) 1.34975e12 0.495255
\(581\) −2.52078e12 −0.917789
\(582\) −8.27645e11 −0.299013
\(583\) 1.65838e12 0.594532
\(584\) 3.11333e12 1.10756
\(585\) −3.13336e11 −0.110614
\(586\) −1.64304e12 −0.575585
\(587\) −4.33948e12 −1.50857 −0.754286 0.656546i \(-0.772017\pi\)
−0.754286 + 0.656546i \(0.772017\pi\)
\(588\) −6.54298e11 −0.225724
\(589\) −5.31561e10 −0.0181984
\(590\) 5.59356e11 0.190044
\(591\) 3.84417e11 0.129616
\(592\) −1.22020e12 −0.408305
\(593\) 1.90120e12 0.631366 0.315683 0.948865i \(-0.397766\pi\)
0.315683 + 0.948865i \(0.397766\pi\)
\(594\) 2.02147e11 0.0666238
\(595\) 3.90125e12 1.27608
\(596\) 2.63644e11 0.0855874
\(597\) −1.55603e12 −0.501341
\(598\) −9.90824e11 −0.316841
\(599\) −5.49649e12 −1.74447 −0.872237 0.489083i \(-0.837332\pi\)
−0.872237 + 0.489083i \(0.837332\pi\)
\(600\) −3.95763e11 −0.124668
\(601\) 4.77698e12 1.49355 0.746773 0.665079i \(-0.231602\pi\)
0.746773 + 0.665079i \(0.231602\pi\)
\(602\) −3.80322e12 −1.18023
\(603\) 1.80118e12 0.554791
\(604\) −1.90909e12 −0.583661
\(605\) 2.92442e12 0.887443
\(606\) 2.36608e12 0.712692
\(607\) 8.25000e11 0.246663 0.123332 0.992366i \(-0.460642\pi\)
0.123332 + 0.992366i \(0.460642\pi\)
\(608\) 5.90685e11 0.175303
\(609\) 3.46949e12 1.02209
\(610\) 4.93902e12 1.44430
\(611\) 7.17045e11 0.208142
\(612\) 3.44559e11 0.0992848
\(613\) −1.46985e12 −0.420438 −0.210219 0.977654i \(-0.567418\pi\)
−0.210219 + 0.977654i \(0.567418\pi\)
\(614\) −7.90326e11 −0.224413
\(615\) 8.95622e11 0.252457
\(616\) −2.43045e12 −0.680101
\(617\) 5.17373e12 1.43721 0.718606 0.695418i \(-0.244781\pi\)
0.718606 + 0.695418i \(0.244781\pi\)
\(618\) −2.55300e12 −0.704048
\(619\) −4.80274e12 −1.31487 −0.657433 0.753513i \(-0.728358\pi\)
−0.657433 + 0.753513i \(0.728358\pi\)
\(620\) −1.06164e11 −0.0288547
\(621\) −9.37194e11 −0.252882
\(622\) −1.37670e12 −0.368794
\(623\) −6.19789e12 −1.64834
\(624\) −3.30059e11 −0.0871487
\(625\) −4.42170e12 −1.15912
\(626\) 4.83279e12 1.25780
\(627\) 2.46518e11 0.0637007
\(628\) −1.76082e12 −0.451748
\(629\) −2.61103e12 −0.665096
\(630\) −1.64934e12 −0.417135
\(631\) 5.93992e12 1.49159 0.745794 0.666177i \(-0.232071\pi\)
0.745794 + 0.666177i \(0.232071\pi\)
\(632\) −2.41762e12 −0.602784
\(633\) −1.46831e12 −0.363498
\(634\) 2.73700e12 0.672780
\(635\) −1.42119e12 −0.346872
\(636\) −1.19505e12 −0.289620
\(637\) 1.34116e12 0.322741
\(638\) 1.78492e12 0.426506
\(639\) −2.38444e11 −0.0565761
\(640\) −6.61004e11 −0.155738
\(641\) 6.47738e12 1.51544 0.757720 0.652580i \(-0.226313\pi\)
0.757720 + 0.652580i \(0.226313\pi\)
\(642\) −2.24963e12 −0.522641
\(643\) −1.74519e12 −0.402619 −0.201309 0.979528i \(-0.564520\pi\)
−0.201309 + 0.979528i \(0.564520\pi\)
\(644\) 3.02627e12 0.693301
\(645\) 2.86867e12 0.652621
\(646\) −7.24155e11 −0.163601
\(647\) 4.97244e12 1.11558 0.557790 0.829982i \(-0.311650\pi\)
0.557790 + 0.829982i \(0.311650\pi\)
\(648\) −5.42389e11 −0.120843
\(649\) −4.29205e11 −0.0949650
\(650\) 2.17872e11 0.0478731
\(651\) −2.72892e11 −0.0595492
\(652\) 1.38139e11 0.0299365
\(653\) −1.45057e12 −0.312198 −0.156099 0.987741i \(-0.549892\pi\)
−0.156099 + 0.987741i \(0.549892\pi\)
\(654\) 6.67276e11 0.142628
\(655\) 1.51265e12 0.321109
\(656\) 9.43420e11 0.198901
\(657\) −1.62115e12 −0.339453
\(658\) 3.77437e12 0.784925
\(659\) −3.05175e12 −0.630325 −0.315162 0.949038i \(-0.602059\pi\)
−0.315162 + 0.949038i \(0.602059\pi\)
\(660\) 4.92351e11 0.101001
\(661\) −1.88315e12 −0.383688 −0.191844 0.981425i \(-0.561447\pi\)
−0.191844 + 0.981425i \(0.561447\pi\)
\(662\) −4.73503e12 −0.958213
\(663\) −7.06270e11 −0.141958
\(664\) 3.47961e12 0.694662
\(665\) −2.01136e12 −0.398834
\(666\) 1.10387e12 0.217412
\(667\) −8.27522e12 −1.61888
\(668\) −2.92498e11 −0.0568367
\(669\) 3.16776e11 0.0611412
\(670\) −7.56051e12 −1.44949
\(671\) −3.78981e12 −0.721716
\(672\) 3.03245e12 0.573630
\(673\) 7.84359e12 1.47383 0.736914 0.675986i \(-0.236282\pi\)
0.736914 + 0.675986i \(0.236282\pi\)
\(674\) 5.34929e12 0.998451
\(675\) 2.06080e11 0.0382092
\(676\) 1.81047e12 0.333451
\(677\) 2.32420e12 0.425230 0.212615 0.977136i \(-0.431802\pi\)
0.212615 + 0.977136i \(0.431802\pi\)
\(678\) 4.76466e12 0.865961
\(679\) −5.18158e12 −0.935509
\(680\) −5.38516e12 −0.965846
\(681\) −2.90082e12 −0.516843
\(682\) −1.40392e11 −0.0248493
\(683\) −6.89265e12 −1.21197 −0.605986 0.795475i \(-0.707221\pi\)
−0.605986 + 0.795475i \(0.707221\pi\)
\(684\) −1.77644e11 −0.0310312
\(685\) 8.77077e10 0.0152205
\(686\) 4.29354e11 0.0740214
\(687\) 4.29660e11 0.0735902
\(688\) 3.02176e12 0.514177
\(689\) 2.44959e12 0.414101
\(690\) 3.93390e12 0.660697
\(691\) 1.11430e13 1.85931 0.929654 0.368434i \(-0.120106\pi\)
0.929654 + 0.368434i \(0.120106\pi\)
\(692\) −3.58043e12 −0.593550
\(693\) 1.26557e12 0.208443
\(694\) 2.52393e12 0.413010
\(695\) 7.11530e12 1.15681
\(696\) −4.78918e12 −0.773605
\(697\) 2.01876e12 0.323994
\(698\) −5.66803e12 −0.903822
\(699\) −2.05879e12 −0.326186
\(700\) −6.65447e11 −0.104754
\(701\) −5.78015e12 −0.904082 −0.452041 0.891997i \(-0.649304\pi\)
−0.452041 + 0.891997i \(0.649304\pi\)
\(702\) 2.98591e11 0.0464045
\(703\) 1.34617e12 0.207874
\(704\) 2.97251e12 0.456085
\(705\) −2.84691e12 −0.434032
\(706\) −9.56151e12 −1.44846
\(707\) 1.48131e13 2.22977
\(708\) 3.09291e11 0.0462612
\(709\) −3.39902e12 −0.505180 −0.252590 0.967573i \(-0.581282\pi\)
−0.252590 + 0.967573i \(0.581282\pi\)
\(710\) 1.00088e12 0.147815
\(711\) 1.25889e12 0.184746
\(712\) 8.55537e12 1.24761
\(713\) 6.50885e11 0.0943195
\(714\) −3.71766e12 −0.535337
\(715\) −1.00921e12 −0.144412
\(716\) −8.13761e11 −0.115715
\(717\) −5.13130e12 −0.725089
\(718\) −8.95831e12 −1.25796
\(719\) 2.71272e11 0.0378551 0.0189276 0.999821i \(-0.493975\pi\)
0.0189276 + 0.999821i \(0.493975\pi\)
\(720\) 1.31044e12 0.181728
\(721\) −1.59834e13 −2.20272
\(722\) −5.43503e12 −0.744362
\(723\) 8.25623e12 1.12372
\(724\) 1.79870e11 0.0243297
\(725\) 1.81964e12 0.244604
\(726\) −2.78680e12 −0.372298
\(727\) −2.13863e12 −0.283943 −0.141972 0.989871i \(-0.545344\pi\)
−0.141972 + 0.989871i \(0.545344\pi\)
\(728\) −3.59001e12 −0.473701
\(729\) 2.82430e11 0.0370370
\(730\) 6.80485e12 0.886881
\(731\) 6.46606e12 0.837552
\(732\) 2.73099e12 0.351577
\(733\) 4.60033e12 0.588601 0.294301 0.955713i \(-0.404913\pi\)
0.294301 + 0.955713i \(0.404913\pi\)
\(734\) 4.99938e12 0.635746
\(735\) −5.32487e12 −0.673002
\(736\) −7.23282e12 −0.908568
\(737\) 5.80133e12 0.724309
\(738\) −8.53475e11 −0.105910
\(739\) 8.60245e12 1.06102 0.530508 0.847680i \(-0.322001\pi\)
0.530508 + 0.847680i \(0.322001\pi\)
\(740\) 2.68859e12 0.329596
\(741\) 3.64131e11 0.0443686
\(742\) 1.28941e13 1.56161
\(743\) 1.31407e12 0.158187 0.0790934 0.996867i \(-0.474797\pi\)
0.0790934 + 0.996867i \(0.474797\pi\)
\(744\) 3.76691e11 0.0450720
\(745\) 2.14561e12 0.255181
\(746\) 4.59574e12 0.543289
\(747\) −1.81188e12 −0.212906
\(748\) 1.10977e12 0.129622
\(749\) −1.40841e13 −1.63516
\(750\) 3.49189e12 0.402981
\(751\) −7.95810e12 −0.912913 −0.456457 0.889746i \(-0.650882\pi\)
−0.456457 + 0.889746i \(0.650882\pi\)
\(752\) −2.99884e12 −0.341958
\(753\) 7.81443e12 0.885768
\(754\) 2.63650e12 0.297068
\(755\) −1.55368e13 −1.74020
\(756\) −9.11987e11 −0.101541
\(757\) −6.95936e12 −0.770261 −0.385131 0.922862i \(-0.625844\pi\)
−0.385131 + 0.922862i \(0.625844\pi\)
\(758\) 1.46179e12 0.160832
\(759\) −3.01856e12 −0.330151
\(760\) 2.77642e12 0.301873
\(761\) −1.38632e13 −1.49841 −0.749207 0.662336i \(-0.769565\pi\)
−0.749207 + 0.662336i \(0.769565\pi\)
\(762\) 1.35431e12 0.145519
\(763\) 4.17757e12 0.446234
\(764\) 3.31704e12 0.352233
\(765\) 2.80413e12 0.296020
\(766\) −7.43621e12 −0.780409
\(767\) −6.33977e11 −0.0661446
\(768\) −5.20371e12 −0.539744
\(769\) 1.35604e13 1.39832 0.699158 0.714967i \(-0.253558\pi\)
0.699158 + 0.714967i \(0.253558\pi\)
\(770\) −5.31227e12 −0.544593
\(771\) −6.47669e11 −0.0660098
\(772\) −2.46267e12 −0.249534
\(773\) 4.96740e12 0.500405 0.250203 0.968194i \(-0.419503\pi\)
0.250203 + 0.968194i \(0.419503\pi\)
\(774\) −2.73367e12 −0.273786
\(775\) −1.43123e11 −0.0142512
\(776\) 7.15248e12 0.708075
\(777\) 6.91093e12 0.680208
\(778\) 4.55494e12 0.445732
\(779\) −1.04081e12 −0.101263
\(780\) 7.27250e11 0.0703490
\(781\) −7.67993e11 −0.0738631
\(782\) 8.86713e12 0.847916
\(783\) 2.49379e12 0.237100
\(784\) −5.60905e12 −0.530234
\(785\) −1.43301e13 −1.34690
\(786\) −1.44146e12 −0.134711
\(787\) −5.25809e12 −0.488587 −0.244293 0.969701i \(-0.578556\pi\)
−0.244293 + 0.969701i \(0.578556\pi\)
\(788\) −8.92227e11 −0.0824342
\(789\) 8.60781e12 0.790763
\(790\) −5.28423e12 −0.482681
\(791\) 2.98298e13 2.70929
\(792\) −1.74695e12 −0.157768
\(793\) −5.59792e12 −0.502686
\(794\) −1.17641e12 −0.105043
\(795\) −9.72568e12 −0.863511
\(796\) 3.61152e12 0.318846
\(797\) 1.61816e13 1.42056 0.710281 0.703918i \(-0.248568\pi\)
0.710281 + 0.703918i \(0.248568\pi\)
\(798\) 1.91671e12 0.167318
\(799\) −6.41701e12 −0.557022
\(800\) 1.59042e12 0.137280
\(801\) −4.45490e12 −0.382377
\(802\) −4.33596e12 −0.370084
\(803\) −5.22150e12 −0.443175
\(804\) −4.18052e12 −0.352840
\(805\) 2.46287e13 2.06709
\(806\) −2.07373e11 −0.0173079
\(807\) −8.17180e12 −0.678245
\(808\) −2.04476e13 −1.68768
\(809\) −1.39276e13 −1.14317 −0.571583 0.820544i \(-0.693671\pi\)
−0.571583 + 0.820544i \(0.693671\pi\)
\(810\) −1.18551e12 −0.0967657
\(811\) 1.38190e13 1.12171 0.560857 0.827913i \(-0.310472\pi\)
0.560857 + 0.827913i \(0.310472\pi\)
\(812\) −8.05265e12 −0.650035
\(813\) −2.31939e12 −0.186194
\(814\) 3.55540e12 0.283844
\(815\) 1.12421e12 0.0892564
\(816\) 2.95378e12 0.233224
\(817\) −3.33370e12 −0.261774
\(818\) −6.81012e12 −0.531820
\(819\) 1.86937e12 0.145184
\(820\) −2.07873e12 −0.160559
\(821\) 3.48578e12 0.267766 0.133883 0.990997i \(-0.457255\pi\)
0.133883 + 0.990997i \(0.457255\pi\)
\(822\) −8.35803e10 −0.00638529
\(823\) −1.21472e13 −0.922948 −0.461474 0.887154i \(-0.652679\pi\)
−0.461474 + 0.887154i \(0.652679\pi\)
\(824\) 2.20629e13 1.66721
\(825\) 6.63751e11 0.0498842
\(826\) −3.33712e12 −0.249438
\(827\) −1.70702e13 −1.26901 −0.634504 0.772919i \(-0.718796\pi\)
−0.634504 + 0.772919i \(0.718796\pi\)
\(828\) 2.17522e12 0.160830
\(829\) 5.58071e12 0.410387 0.205194 0.978721i \(-0.434218\pi\)
0.205194 + 0.978721i \(0.434218\pi\)
\(830\) 7.60543e12 0.556253
\(831\) 3.33923e12 0.242908
\(832\) 4.39068e12 0.317671
\(833\) −1.20024e13 −0.863707
\(834\) −6.78047e12 −0.485302
\(835\) −2.38044e12 −0.169460
\(836\) −5.72165e11 −0.0405129
\(837\) −1.96148e11 −0.0138140
\(838\) 1.74378e13 1.22150
\(839\) 1.28642e13 0.896304 0.448152 0.893957i \(-0.352082\pi\)
0.448152 + 0.893957i \(0.352082\pi\)
\(840\) 1.42535e13 0.987793
\(841\) 7.51250e12 0.517849
\(842\) −2.58819e12 −0.177457
\(843\) 9.86580e12 0.672835
\(844\) 3.40793e12 0.231180
\(845\) 1.47342e13 0.994193
\(846\) 2.71293e12 0.182084
\(847\) −1.74471e13 −1.16479
\(848\) −1.02447e13 −0.680329
\(849\) 1.88584e12 0.124572
\(850\) −1.94979e12 −0.128116
\(851\) −1.64835e13 −1.07738
\(852\) 5.53426e11 0.0359817
\(853\) 1.03465e13 0.669147 0.334574 0.942370i \(-0.391408\pi\)
0.334574 + 0.942370i \(0.391408\pi\)
\(854\) −2.94663e13 −1.89568
\(855\) −1.44572e12 −0.0925202
\(856\) 1.94413e13 1.23763
\(857\) 5.63216e12 0.356666 0.178333 0.983970i \(-0.442930\pi\)
0.178333 + 0.983970i \(0.442930\pi\)
\(858\) 9.61718e11 0.0605836
\(859\) −2.28289e13 −1.43059 −0.715295 0.698822i \(-0.753708\pi\)
−0.715295 + 0.698822i \(0.753708\pi\)
\(860\) −6.65814e12 −0.415059
\(861\) −5.34329e12 −0.331356
\(862\) 3.72401e12 0.229736
\(863\) 4.62918e12 0.284090 0.142045 0.989860i \(-0.454632\pi\)
0.142045 + 0.989860i \(0.454632\pi\)
\(864\) 2.17966e12 0.133069
\(865\) −2.91386e13 −1.76968
\(866\) −8.67389e11 −0.0524063
\(867\) −3.28503e12 −0.197448
\(868\) 6.33379e11 0.0378726
\(869\) 4.05470e12 0.241196
\(870\) −1.04678e13 −0.619466
\(871\) 8.56913e12 0.504493
\(872\) −5.76658e12 −0.337749
\(873\) −3.72440e12 −0.217016
\(874\) −4.57162e12 −0.265014
\(875\) 2.18614e13 1.26079
\(876\) 3.76268e12 0.215888
\(877\) 5.33611e12 0.304598 0.152299 0.988334i \(-0.451332\pi\)
0.152299 + 0.988334i \(0.451332\pi\)
\(878\) 7.88294e12 0.447675
\(879\) −7.39369e12 −0.417745
\(880\) 4.22074e12 0.237256
\(881\) 7.87890e12 0.440630 0.220315 0.975429i \(-0.429291\pi\)
0.220315 + 0.975429i \(0.429291\pi\)
\(882\) 5.07429e12 0.282336
\(883\) 1.38973e13 0.769320 0.384660 0.923058i \(-0.374319\pi\)
0.384660 + 0.923058i \(0.374319\pi\)
\(884\) 1.63924e12 0.0902835
\(885\) 2.51710e12 0.137929
\(886\) 1.36560e13 0.744510
\(887\) −5.86607e12 −0.318193 −0.159097 0.987263i \(-0.550858\pi\)
−0.159097 + 0.987263i \(0.550858\pi\)
\(888\) −9.53963e12 −0.514841
\(889\) 8.47881e12 0.455278
\(890\) 1.86996e13 0.999026
\(891\) 9.09663e11 0.0483538
\(892\) −7.35232e11 −0.0388851
\(893\) 3.30841e12 0.174096
\(894\) −2.04464e12 −0.107053
\(895\) −6.62263e12 −0.345006
\(896\) 3.94356e12 0.204410
\(897\) −4.45871e12 −0.229955
\(898\) −2.47103e12 −0.126805
\(899\) −1.73195e12 −0.0884334
\(900\) −4.78308e11 −0.0243006
\(901\) −2.19220e13 −1.10820
\(902\) −2.74892e12 −0.138271
\(903\) −1.71145e13 −0.856583
\(904\) −4.11761e13 −2.05063
\(905\) 1.46384e12 0.0725395
\(906\) 1.48056e13 0.730045
\(907\) 8.52250e12 0.418152 0.209076 0.977899i \(-0.432954\pi\)
0.209076 + 0.977899i \(0.432954\pi\)
\(908\) 6.73277e12 0.328706
\(909\) 1.06473e13 0.517254
\(910\) −7.84674e12 −0.379317
\(911\) 1.63369e13 0.785843 0.392921 0.919572i \(-0.371464\pi\)
0.392921 + 0.919572i \(0.371464\pi\)
\(912\) −1.52288e12 −0.0728934
\(913\) −5.83580e12 −0.277960
\(914\) 2.35964e12 0.111838
\(915\) 2.22256e13 1.04823
\(916\) −9.97236e11 −0.0468024
\(917\) −9.02448e12 −0.421464
\(918\) −2.67217e12 −0.124186
\(919\) 3.24712e13 1.50169 0.750843 0.660481i \(-0.229648\pi\)
0.750843 + 0.660481i \(0.229648\pi\)
\(920\) −3.39967e13 −1.56456
\(921\) −3.55647e12 −0.162873
\(922\) −1.44631e13 −0.659134
\(923\) −1.13440e12 −0.0514468
\(924\) −2.93737e12 −0.132567
\(925\) 3.62456e12 0.162786
\(926\) −3.21447e13 −1.43668
\(927\) −1.14885e13 −0.510980
\(928\) 1.92459e13 0.851868
\(929\) −1.24571e13 −0.548716 −0.274358 0.961628i \(-0.588465\pi\)
−0.274358 + 0.961628i \(0.588465\pi\)
\(930\) 8.23339e11 0.0360916
\(931\) 6.18808e12 0.269949
\(932\) 4.77843e12 0.207450
\(933\) −6.19516e12 −0.267661
\(934\) 1.15986e13 0.498706
\(935\) 9.03167e12 0.386470
\(936\) −2.58042e12 −0.109888
\(937\) 9.29012e12 0.393725 0.196863 0.980431i \(-0.436925\pi\)
0.196863 + 0.980431i \(0.436925\pi\)
\(938\) 4.51061e13 1.90249
\(939\) 2.17475e13 0.912883
\(940\) 6.60763e12 0.276039
\(941\) −3.24831e13 −1.35053 −0.675265 0.737575i \(-0.735971\pi\)
−0.675265 + 0.737575i \(0.735971\pi\)
\(942\) 1.36557e13 0.565048
\(943\) 1.27445e13 0.524832
\(944\) 2.65143e12 0.108669
\(945\) −7.42202e12 −0.302746
\(946\) −8.80474e12 −0.357443
\(947\) −4.16258e13 −1.68185 −0.840926 0.541151i \(-0.817989\pi\)
−0.840926 + 0.541151i \(0.817989\pi\)
\(948\) −2.92187e12 −0.117496
\(949\) −7.71265e12 −0.308678
\(950\) 1.00525e12 0.0400423
\(951\) 1.23165e13 0.488287
\(952\) 3.21279e13 1.26770
\(953\) 1.86479e13 0.732339 0.366169 0.930548i \(-0.380669\pi\)
0.366169 + 0.930548i \(0.380669\pi\)
\(954\) 9.26800e12 0.362258
\(955\) 2.69950e13 1.05019
\(956\) 1.19097e13 0.461148
\(957\) 8.03213e12 0.309547
\(958\) −3.37861e13 −1.29596
\(959\) −5.23265e11 −0.0199774
\(960\) −1.74325e13 −0.662428
\(961\) −2.63034e13 −0.994848
\(962\) 5.25167e12 0.197702
\(963\) −1.01233e13 −0.379320
\(964\) −1.91626e13 −0.714674
\(965\) −2.00420e13 −0.743991
\(966\) −2.34697e13 −0.867183
\(967\) −2.32990e13 −0.856875 −0.428437 0.903572i \(-0.640936\pi\)
−0.428437 + 0.903572i \(0.640936\pi\)
\(968\) 2.40835e13 0.881617
\(969\) −3.25870e12 −0.118737
\(970\) 1.56333e13 0.566993
\(971\) −3.63628e13 −1.31272 −0.656358 0.754450i \(-0.727904\pi\)
−0.656358 + 0.754450i \(0.727904\pi\)
\(972\) −6.55515e11 −0.0235551
\(973\) −4.24500e13 −1.51834
\(974\) 2.68839e13 0.957145
\(975\) 9.80425e11 0.0347451
\(976\) 2.34118e13 0.825866
\(977\) 5.18141e13 1.81938 0.909688 0.415291i \(-0.136320\pi\)
0.909688 + 0.415291i \(0.136320\pi\)
\(978\) −1.07131e12 −0.0374447
\(979\) −1.43486e13 −0.499214
\(980\) 1.23590e13 0.428021
\(981\) 3.00274e12 0.103516
\(982\) −1.34516e13 −0.461608
\(983\) −2.54862e13 −0.870592 −0.435296 0.900287i \(-0.643356\pi\)
−0.435296 + 0.900287i \(0.643356\pi\)
\(984\) 7.37571e12 0.250799
\(985\) −7.26121e12 −0.245780
\(986\) −2.35947e13 −0.795001
\(987\) 1.69847e13 0.569679
\(988\) −8.45143e11 −0.0282178
\(989\) 4.08205e13 1.35673
\(990\) −3.81834e12 −0.126333
\(991\) 3.92830e13 1.29382 0.646910 0.762567i \(-0.276061\pi\)
0.646910 + 0.762567i \(0.276061\pi\)
\(992\) −1.51378e12 −0.0496318
\(993\) −2.13076e13 −0.695447
\(994\) −5.97124e12 −0.194011
\(995\) 2.93917e13 0.950649
\(996\) 4.20535e12 0.135405
\(997\) 4.98389e13 1.59750 0.798748 0.601665i \(-0.205496\pi\)
0.798748 + 0.601665i \(0.205496\pi\)
\(998\) 1.82340e13 0.581827
\(999\) 4.96742e12 0.157792
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 3.10.a.b.1.1 1
3.2 odd 2 9.10.a.a.1.1 1
4.3 odd 2 48.10.a.a.1.1 1
5.2 odd 4 75.10.b.c.49.2 2
5.3 odd 4 75.10.b.c.49.1 2
5.4 even 2 75.10.a.b.1.1 1
7.6 odd 2 147.10.a.c.1.1 1
8.3 odd 2 192.10.a.n.1.1 1
8.5 even 2 192.10.a.g.1.1 1
9.2 odd 6 81.10.c.d.28.1 2
9.4 even 3 81.10.c.b.55.1 2
9.5 odd 6 81.10.c.d.55.1 2
9.7 even 3 81.10.c.b.28.1 2
11.10 odd 2 363.10.a.a.1.1 1
12.11 even 2 144.10.a.m.1.1 1
15.2 even 4 225.10.b.c.199.1 2
15.8 even 4 225.10.b.c.199.2 2
15.14 odd 2 225.10.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.10.a.b.1.1 1 1.1 even 1 trivial
9.10.a.a.1.1 1 3.2 odd 2
48.10.a.a.1.1 1 4.3 odd 2
75.10.a.b.1.1 1 5.4 even 2
75.10.b.c.49.1 2 5.3 odd 4
75.10.b.c.49.2 2 5.2 odd 4
81.10.c.b.28.1 2 9.7 even 3
81.10.c.b.55.1 2 9.4 even 3
81.10.c.d.28.1 2 9.2 odd 6
81.10.c.d.55.1 2 9.5 odd 6
144.10.a.m.1.1 1 12.11 even 2
147.10.a.c.1.1 1 7.6 odd 2
192.10.a.g.1.1 1 8.5 even 2
192.10.a.n.1.1 1 8.3 odd 2
225.10.a.e.1.1 1 15.14 odd 2
225.10.b.c.199.1 2 15.2 even 4
225.10.b.c.199.2 2 15.8 even 4
363.10.a.a.1.1 1 11.10 odd 2