Properties

Label 245.10.a.b.1.1
Level $245$
Weight $10$
Character 245.1
Self dual yes
Analytic conductor $126.184$
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] = [245,10,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(126.183779860\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+28.0000 q^{2} +116.000 q^{3} +272.000 q^{4} -625.000 q^{5} +3248.00 q^{6} -6720.00 q^{8} -6227.00 q^{9} +O(q^{10})\) \(q+28.0000 q^{2} +116.000 q^{3} +272.000 q^{4} -625.000 q^{5} +3248.00 q^{6} -6720.00 q^{8} -6227.00 q^{9} -17500.0 q^{10} -25548.0 q^{11} +31552.0 q^{12} +42306.0 q^{13} -72500.0 q^{15} -327424. q^{16} +526342. q^{17} -174356. q^{18} +350060. q^{19} -170000. q^{20} -715344. q^{22} -621976. q^{23} -779520. q^{24} +390625. q^{25} +1.18457e6 q^{26} -3.00556e6 q^{27} +6.72043e6 q^{29} -2.03000e6 q^{30} +6.41221e6 q^{31} -5.72723e6 q^{32} -2.96357e6 q^{33} +1.47376e7 q^{34} -1.69374e6 q^{36} -2.31768e6 q^{37} +9.80168e6 q^{38} +4.90750e6 q^{39} +4.20000e6 q^{40} +1.02247e7 q^{41} +3.01140e7 q^{43} -6.94906e6 q^{44} +3.89188e6 q^{45} -1.74153e7 q^{46} +2.36449e7 q^{47} -3.79812e7 q^{48} +1.09375e7 q^{50} +6.10557e7 q^{51} +1.15072e7 q^{52} +5.72927e7 q^{53} -8.41557e7 q^{54} +1.59675e7 q^{55} +4.06070e7 q^{57} +1.88172e8 q^{58} -8.49348e7 q^{59} -1.97200e7 q^{60} -1.46778e7 q^{61} +1.79542e8 q^{62} +7.27859e6 q^{64} -2.64412e7 q^{65} -8.29799e7 q^{66} -2.44558e8 q^{67} +1.43165e8 q^{68} -7.21492e7 q^{69} +6.19020e7 q^{71} +4.18454e7 q^{72} +2.83764e8 q^{73} -6.48951e7 q^{74} +4.53125e7 q^{75} +9.52163e7 q^{76} +1.37410e8 q^{78} +2.76107e8 q^{79} +2.04640e8 q^{80} -2.26079e8 q^{81} +2.86291e8 q^{82} +7.29960e7 q^{83} -3.28964e8 q^{85} +8.43192e8 q^{86} +7.79570e8 q^{87} +1.71683e8 q^{88} +8.96368e8 q^{89} +1.08973e8 q^{90} -1.69177e8 q^{92} +7.43816e8 q^{93} +6.62058e8 q^{94} -2.18788e8 q^{95} -6.64359e8 q^{96} -1.20581e9 q^{97} +1.59087e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 28.0000 1.23744 0.618718 0.785613i \(-0.287652\pi\)
0.618718 + 0.785613i \(0.287652\pi\)
\(3\) 116.000 0.826823 0.413411 0.910544i \(-0.364337\pi\)
0.413411 + 0.910544i \(0.364337\pi\)
\(4\) 272.000 0.531250
\(5\) −625.000 −0.447214
\(6\) 3248.00 1.02314
\(7\) 0 0
\(8\) −6720.00 −0.580049
\(9\) −6227.00 −0.316364
\(10\) −17500.0 −0.553399
\(11\) −25548.0 −0.526126 −0.263063 0.964779i \(-0.584733\pi\)
−0.263063 + 0.964779i \(0.584733\pi\)
\(12\) 31552.0 0.439250
\(13\) 42306.0 0.410825 0.205413 0.978675i \(-0.434146\pi\)
0.205413 + 0.978675i \(0.434146\pi\)
\(14\) 0 0
\(15\) −72500.0 −0.369766
\(16\) −327424. −1.24902
\(17\) 526342. 1.52844 0.764219 0.644957i \(-0.223125\pi\)
0.764219 + 0.644957i \(0.223125\pi\)
\(18\) −174356. −0.391481
\(19\) 350060. 0.616242 0.308121 0.951347i \(-0.400300\pi\)
0.308121 + 0.951347i \(0.400300\pi\)
\(20\) −170000. −0.237582
\(21\) 0 0
\(22\) −715344. −0.651048
\(23\) −621976. −0.463445 −0.231723 0.972782i \(-0.574436\pi\)
−0.231723 + 0.972782i \(0.574436\pi\)
\(24\) −779520. −0.479597
\(25\) 390625. 0.200000
\(26\) 1.18457e6 0.508370
\(27\) −3.00556e6 −1.08840
\(28\) 0 0
\(29\) 6.72043e6 1.76444 0.882218 0.470841i \(-0.156049\pi\)
0.882218 + 0.470841i \(0.156049\pi\)
\(30\) −2.03000e6 −0.457562
\(31\) 6.41221e6 1.24704 0.623519 0.781808i \(-0.285702\pi\)
0.623519 + 0.781808i \(0.285702\pi\)
\(32\) −5.72723e6 −0.965539
\(33\) −2.96357e6 −0.435013
\(34\) 1.47376e7 1.89135
\(35\) 0 0
\(36\) −1.69374e6 −0.168069
\(37\) −2.31768e6 −0.203304 −0.101652 0.994820i \(-0.532413\pi\)
−0.101652 + 0.994820i \(0.532413\pi\)
\(38\) 9.80168e6 0.762561
\(39\) 4.90750e6 0.339679
\(40\) 4.20000e6 0.259406
\(41\) 1.02247e7 0.565096 0.282548 0.959253i \(-0.408820\pi\)
0.282548 + 0.959253i \(0.408820\pi\)
\(42\) 0 0
\(43\) 3.01140e7 1.34326 0.671631 0.740886i \(-0.265594\pi\)
0.671631 + 0.740886i \(0.265594\pi\)
\(44\) −6.94906e6 −0.279504
\(45\) 3.89188e6 0.141482
\(46\) −1.74153e7 −0.573484
\(47\) 2.36449e7 0.706801 0.353401 0.935472i \(-0.385025\pi\)
0.353401 + 0.935472i \(0.385025\pi\)
\(48\) −3.79812e7 −1.03272
\(49\) 0 0
\(50\) 1.09375e7 0.247487
\(51\) 6.10557e7 1.26375
\(52\) 1.15072e7 0.218251
\(53\) 5.72927e7 0.997373 0.498686 0.866782i \(-0.333816\pi\)
0.498686 + 0.866782i \(0.333816\pi\)
\(54\) −8.41557e7 −1.34683
\(55\) 1.59675e7 0.235291
\(56\) 0 0
\(57\) 4.06070e7 0.509523
\(58\) 1.88172e8 2.18338
\(59\) −8.49348e7 −0.912539 −0.456270 0.889842i \(-0.650815\pi\)
−0.456270 + 0.889842i \(0.650815\pi\)
\(60\) −1.97200e7 −0.196438
\(61\) −1.46778e7 −0.135730 −0.0678652 0.997694i \(-0.521619\pi\)
−0.0678652 + 0.997694i \(0.521619\pi\)
\(62\) 1.79542e8 1.54313
\(63\) 0 0
\(64\) 7.27859e6 0.0542297
\(65\) −2.64412e7 −0.183727
\(66\) −8.29799e7 −0.538301
\(67\) −2.44558e8 −1.48267 −0.741336 0.671134i \(-0.765807\pi\)
−0.741336 + 0.671134i \(0.765807\pi\)
\(68\) 1.43165e8 0.811983
\(69\) −7.21492e7 −0.383187
\(70\) 0 0
\(71\) 6.19020e7 0.289096 0.144548 0.989498i \(-0.453827\pi\)
0.144548 + 0.989498i \(0.453827\pi\)
\(72\) 4.18454e7 0.183507
\(73\) 2.83764e8 1.16951 0.584755 0.811210i \(-0.301191\pi\)
0.584755 + 0.811210i \(0.301191\pi\)
\(74\) −6.48951e7 −0.251576
\(75\) 4.53125e7 0.165365
\(76\) 9.52163e7 0.327379
\(77\) 0 0
\(78\) 1.37410e8 0.420332
\(79\) 2.76107e8 0.797547 0.398773 0.917049i \(-0.369436\pi\)
0.398773 + 0.917049i \(0.369436\pi\)
\(80\) 2.04640e8 0.558580
\(81\) −2.26079e8 −0.583549
\(82\) 2.86291e8 0.699271
\(83\) 7.29960e7 0.168829 0.0844146 0.996431i \(-0.473098\pi\)
0.0844146 + 0.996431i \(0.473098\pi\)
\(84\) 0 0
\(85\) −3.28964e8 −0.683538
\(86\) 8.43192e8 1.66220
\(87\) 7.79570e8 1.45888
\(88\) 1.71683e8 0.305179
\(89\) 8.96368e8 1.51437 0.757184 0.653201i \(-0.226575\pi\)
0.757184 + 0.653201i \(0.226575\pi\)
\(90\) 1.08973e8 0.175076
\(91\) 0 0
\(92\) −1.69177e8 −0.246205
\(93\) 7.43816e8 1.03108
\(94\) 6.62058e8 0.874622
\(95\) −2.18788e8 −0.275592
\(96\) −6.64359e8 −0.798330
\(97\) −1.20581e9 −1.38295 −0.691474 0.722401i \(-0.743038\pi\)
−0.691474 + 0.722401i \(0.743038\pi\)
\(98\) 0 0
\(99\) 1.59087e8 0.166448
\(100\) 1.06250e8 0.106250
\(101\) 1.46021e9 1.39627 0.698136 0.715965i \(-0.254013\pi\)
0.698136 + 0.715965i \(0.254013\pi\)
\(102\) 1.70956e9 1.56381
\(103\) −1.08009e9 −0.945563 −0.472782 0.881180i \(-0.656750\pi\)
−0.472782 + 0.881180i \(0.656750\pi\)
\(104\) −2.84296e8 −0.238298
\(105\) 0 0
\(106\) 1.60419e9 1.23419
\(107\) 3.35949e8 0.247769 0.123884 0.992297i \(-0.460465\pi\)
0.123884 + 0.992297i \(0.460465\pi\)
\(108\) −8.17512e8 −0.578212
\(109\) −1.42521e9 −0.967072 −0.483536 0.875324i \(-0.660648\pi\)
−0.483536 + 0.875324i \(0.660648\pi\)
\(110\) 4.47090e8 0.291157
\(111\) −2.68851e8 −0.168096
\(112\) 0 0
\(113\) −2.84178e9 −1.63960 −0.819799 0.572651i \(-0.805915\pi\)
−0.819799 + 0.572651i \(0.805915\pi\)
\(114\) 1.13699e9 0.630502
\(115\) 3.88735e8 0.207259
\(116\) 1.82796e9 0.937357
\(117\) −2.63439e8 −0.129970
\(118\) −2.37817e9 −1.12921
\(119\) 0 0
\(120\) 4.87200e8 0.214482
\(121\) −1.70525e9 −0.723191
\(122\) −4.10979e8 −0.167958
\(123\) 1.18606e9 0.467234
\(124\) 1.74412e9 0.662489
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) 3.49339e9 1.19160 0.595800 0.803133i \(-0.296835\pi\)
0.595800 + 0.803133i \(0.296835\pi\)
\(128\) 3.13614e9 1.03264
\(129\) 3.49322e9 1.11064
\(130\) −7.40355e8 −0.227350
\(131\) 1.84697e9 0.547946 0.273973 0.961737i \(-0.411662\pi\)
0.273973 + 0.961737i \(0.411662\pi\)
\(132\) −8.06090e8 −0.231101
\(133\) 0 0
\(134\) −6.84762e9 −1.83471
\(135\) 1.87848e9 0.486747
\(136\) −3.53702e9 −0.886568
\(137\) 1.17238e9 0.284331 0.142166 0.989843i \(-0.454593\pi\)
0.142166 + 0.989843i \(0.454593\pi\)
\(138\) −2.02018e9 −0.474170
\(139\) 4.89001e9 1.11108 0.555538 0.831491i \(-0.312512\pi\)
0.555538 + 0.831491i \(0.312512\pi\)
\(140\) 0 0
\(141\) 2.74281e9 0.584399
\(142\) 1.73325e9 0.357738
\(143\) −1.08083e9 −0.216146
\(144\) 2.03887e9 0.395147
\(145\) −4.20027e9 −0.789080
\(146\) 7.94538e9 1.44720
\(147\) 0 0
\(148\) −6.30410e8 −0.108005
\(149\) −8.61488e9 −1.43189 −0.715947 0.698155i \(-0.754005\pi\)
−0.715947 + 0.698155i \(0.754005\pi\)
\(150\) 1.26875e9 0.204628
\(151\) 5.48905e9 0.859213 0.429607 0.903016i \(-0.358652\pi\)
0.429607 + 0.903016i \(0.358652\pi\)
\(152\) −2.35240e9 −0.357450
\(153\) −3.27753e9 −0.483543
\(154\) 0 0
\(155\) −4.00763e9 −0.557693
\(156\) 1.33484e9 0.180455
\(157\) 1.33110e9 0.174849 0.0874246 0.996171i \(-0.472136\pi\)
0.0874246 + 0.996171i \(0.472136\pi\)
\(158\) 7.73101e9 0.986914
\(159\) 6.64595e9 0.824650
\(160\) 3.57952e9 0.431802
\(161\) 0 0
\(162\) −6.33021e9 −0.722105
\(163\) 1.41097e9 0.156557 0.0782786 0.996932i \(-0.475058\pi\)
0.0782786 + 0.996932i \(0.475058\pi\)
\(164\) 2.78111e9 0.300207
\(165\) 1.85223e9 0.194544
\(166\) 2.04389e9 0.208915
\(167\) −2.48555e8 −0.0247285 −0.0123642 0.999924i \(-0.503936\pi\)
−0.0123642 + 0.999924i \(0.503936\pi\)
\(168\) 0 0
\(169\) −8.81470e9 −0.831223
\(170\) −9.21098e9 −0.845836
\(171\) −2.17982e9 −0.194957
\(172\) 8.19101e9 0.713607
\(173\) −1.66522e10 −1.41340 −0.706699 0.707515i \(-0.749816\pi\)
−0.706699 + 0.707515i \(0.749816\pi\)
\(174\) 2.18280e10 1.80527
\(175\) 0 0
\(176\) 8.36503e9 0.657144
\(177\) −9.85243e9 −0.754508
\(178\) 2.50983e10 1.87394
\(179\) 2.02956e10 1.47762 0.738811 0.673913i \(-0.235388\pi\)
0.738811 + 0.673913i \(0.235388\pi\)
\(180\) 1.05859e9 0.0751626
\(181\) 1.36159e10 0.942960 0.471480 0.881877i \(-0.343720\pi\)
0.471480 + 0.881877i \(0.343720\pi\)
\(182\) 0 0
\(183\) −1.70263e9 −0.112225
\(184\) 4.17968e9 0.268821
\(185\) 1.44855e9 0.0909203
\(186\) 2.08269e10 1.27590
\(187\) −1.34470e10 −0.804151
\(188\) 6.43142e9 0.375488
\(189\) 0 0
\(190\) −6.12605e9 −0.341027
\(191\) 1.82357e9 0.0991453 0.0495726 0.998771i \(-0.484214\pi\)
0.0495726 + 0.998771i \(0.484214\pi\)
\(192\) 8.44317e8 0.0448384
\(193\) 1.23747e10 0.641989 0.320995 0.947081i \(-0.395983\pi\)
0.320995 + 0.947081i \(0.395983\pi\)
\(194\) −3.37627e10 −1.71131
\(195\) −3.06718e9 −0.151909
\(196\) 0 0
\(197\) 1.93137e10 0.913626 0.456813 0.889563i \(-0.348991\pi\)
0.456813 + 0.889563i \(0.348991\pi\)
\(198\) 4.45445e9 0.205968
\(199\) −1.52145e10 −0.687733 −0.343867 0.939019i \(-0.611737\pi\)
−0.343867 + 0.939019i \(0.611737\pi\)
\(200\) −2.62500e9 −0.116010
\(201\) −2.83687e10 −1.22591
\(202\) 4.08860e10 1.72780
\(203\) 0 0
\(204\) 1.66071e10 0.671366
\(205\) −6.39042e9 −0.252719
\(206\) −3.02424e10 −1.17007
\(207\) 3.87304e9 0.146618
\(208\) −1.38520e10 −0.513130
\(209\) −8.94333e9 −0.324221
\(210\) 0 0
\(211\) −3.89626e10 −1.35325 −0.676624 0.736329i \(-0.736558\pi\)
−0.676624 + 0.736329i \(0.736558\pi\)
\(212\) 1.55836e10 0.529854
\(213\) 7.18063e9 0.239031
\(214\) 9.40658e9 0.306598
\(215\) −1.88213e10 −0.600725
\(216\) 2.01974e10 0.631325
\(217\) 0 0
\(218\) −3.99058e10 −1.19669
\(219\) 3.29166e10 0.966977
\(220\) 4.34316e9 0.124998
\(221\) 2.22674e10 0.627921
\(222\) −7.52783e9 −0.208009
\(223\) −1.08324e9 −0.0293328 −0.0146664 0.999892i \(-0.504669\pi\)
−0.0146664 + 0.999892i \(0.504669\pi\)
\(224\) 0 0
\(225\) −2.43242e9 −0.0632729
\(226\) −7.95698e10 −2.02890
\(227\) 4.94618e10 1.23639 0.618193 0.786027i \(-0.287865\pi\)
0.618193 + 0.786027i \(0.287865\pi\)
\(228\) 1.10451e10 0.270684
\(229\) 4.32776e10 1.03993 0.519965 0.854188i \(-0.325945\pi\)
0.519965 + 0.854188i \(0.325945\pi\)
\(230\) 1.08846e10 0.256470
\(231\) 0 0
\(232\) −4.51613e10 −1.02346
\(233\) −7.55367e9 −0.167902 −0.0839511 0.996470i \(-0.526754\pi\)
−0.0839511 + 0.996470i \(0.526754\pi\)
\(234\) −7.37630e9 −0.160830
\(235\) −1.47781e10 −0.316091
\(236\) −2.31023e10 −0.484786
\(237\) 3.20285e10 0.659430
\(238\) 0 0
\(239\) −2.76516e10 −0.548188 −0.274094 0.961703i \(-0.588378\pi\)
−0.274094 + 0.961703i \(0.588378\pi\)
\(240\) 2.37382e10 0.461847
\(241\) 8.26006e10 1.57727 0.788635 0.614861i \(-0.210788\pi\)
0.788635 + 0.614861i \(0.210788\pi\)
\(242\) −4.77469e10 −0.894904
\(243\) 3.29333e10 0.605908
\(244\) −3.99237e9 −0.0721068
\(245\) 0 0
\(246\) 3.32098e10 0.578173
\(247\) 1.48096e10 0.253168
\(248\) −4.30900e10 −0.723343
\(249\) 8.46753e9 0.139592
\(250\) −6.83594e9 −0.110680
\(251\) −2.01817e10 −0.320942 −0.160471 0.987041i \(-0.551301\pi\)
−0.160471 + 0.987041i \(0.551301\pi\)
\(252\) 0 0
\(253\) 1.58902e10 0.243831
\(254\) 9.78150e10 1.47453
\(255\) −3.81598e10 −0.565165
\(256\) 8.40854e10 1.22360
\(257\) −2.82781e10 −0.404344 −0.202172 0.979350i \(-0.564800\pi\)
−0.202172 + 0.979350i \(0.564800\pi\)
\(258\) 9.78103e10 1.37435
\(259\) 0 0
\(260\) −7.19202e9 −0.0976047
\(261\) −4.18481e10 −0.558205
\(262\) 5.17150e10 0.678049
\(263\) 1.39139e11 1.79328 0.896642 0.442756i \(-0.145999\pi\)
0.896642 + 0.442756i \(0.145999\pi\)
\(264\) 1.99152e10 0.252329
\(265\) −3.58079e10 −0.446039
\(266\) 0 0
\(267\) 1.03979e11 1.25211
\(268\) −6.65197e10 −0.787669
\(269\) −5.78883e9 −0.0674071 −0.0337035 0.999432i \(-0.510730\pi\)
−0.0337035 + 0.999432i \(0.510730\pi\)
\(270\) 5.25973e10 0.602319
\(271\) 7.46910e10 0.841214 0.420607 0.907243i \(-0.361817\pi\)
0.420607 + 0.907243i \(0.361817\pi\)
\(272\) −1.72337e11 −1.90906
\(273\) 0 0
\(274\) 3.28265e10 0.351842
\(275\) −9.97969e9 −0.105225
\(276\) −1.96246e10 −0.203568
\(277\) 2.22355e10 0.226928 0.113464 0.993542i \(-0.463805\pi\)
0.113464 + 0.993542i \(0.463805\pi\)
\(278\) 1.36920e11 1.37489
\(279\) −3.99288e10 −0.394519
\(280\) 0 0
\(281\) −1.36058e11 −1.30180 −0.650901 0.759162i \(-0.725609\pi\)
−0.650901 + 0.759162i \(0.725609\pi\)
\(282\) 7.67987e10 0.723157
\(283\) 1.71084e11 1.58551 0.792757 0.609538i \(-0.208645\pi\)
0.792757 + 0.609538i \(0.208645\pi\)
\(284\) 1.68373e10 0.153582
\(285\) −2.53793e10 −0.227866
\(286\) −3.02633e10 −0.267467
\(287\) 0 0
\(288\) 3.56635e10 0.305462
\(289\) 1.58448e11 1.33612
\(290\) −1.17608e11 −0.976437
\(291\) −1.39874e11 −1.14345
\(292\) 7.71837e10 0.621302
\(293\) −1.05732e11 −0.838115 −0.419058 0.907960i \(-0.637639\pi\)
−0.419058 + 0.907960i \(0.637639\pi\)
\(294\) 0 0
\(295\) 5.30842e10 0.408100
\(296\) 1.55748e10 0.117926
\(297\) 7.67860e10 0.572636
\(298\) −2.41217e11 −1.77188
\(299\) −2.63133e10 −0.190395
\(300\) 1.23250e10 0.0878499
\(301\) 0 0
\(302\) 1.53693e11 1.06322
\(303\) 1.69385e11 1.15447
\(304\) −1.14618e11 −0.769701
\(305\) 9.17364e9 0.0607005
\(306\) −9.17709e10 −0.598354
\(307\) −1.74144e11 −1.11889 −0.559443 0.828869i \(-0.688985\pi\)
−0.559443 + 0.828869i \(0.688985\pi\)
\(308\) 0 0
\(309\) −1.25290e11 −0.781813
\(310\) −1.12214e11 −0.690110
\(311\) 1.05907e11 0.641954 0.320977 0.947087i \(-0.395989\pi\)
0.320977 + 0.947087i \(0.395989\pi\)
\(312\) −3.29784e10 −0.197031
\(313\) 2.43558e11 1.43434 0.717171 0.696897i \(-0.245437\pi\)
0.717171 + 0.696897i \(0.245437\pi\)
\(314\) 3.72709e10 0.216365
\(315\) 0 0
\(316\) 7.51012e10 0.423697
\(317\) 1.83776e11 1.02217 0.511083 0.859531i \(-0.329245\pi\)
0.511083 + 0.859531i \(0.329245\pi\)
\(318\) 1.86087e11 1.02045
\(319\) −1.71694e11 −0.928316
\(320\) −4.54912e9 −0.0242523
\(321\) 3.89701e10 0.204861
\(322\) 0 0
\(323\) 1.84251e11 0.941888
\(324\) −6.14935e10 −0.310011
\(325\) 1.65258e10 0.0821650
\(326\) 3.95071e10 0.193730
\(327\) −1.65324e11 −0.799597
\(328\) −6.87098e10 −0.327783
\(329\) 0 0
\(330\) 5.18624e10 0.240736
\(331\) −5.81760e10 −0.266390 −0.133195 0.991090i \(-0.542524\pi\)
−0.133195 + 0.991090i \(0.542524\pi\)
\(332\) 1.98549e10 0.0896905
\(333\) 1.44322e10 0.0643182
\(334\) −6.95953e9 −0.0305999
\(335\) 1.52849e11 0.663071
\(336\) 0 0
\(337\) −3.40267e11 −1.43709 −0.718547 0.695478i \(-0.755193\pi\)
−0.718547 + 0.695478i \(0.755193\pi\)
\(338\) −2.46812e11 −1.02859
\(339\) −3.29646e11 −1.35566
\(340\) −8.94781e10 −0.363130
\(341\) −1.63819e11 −0.656100
\(342\) −6.10351e10 −0.241247
\(343\) 0 0
\(344\) −2.02366e11 −0.779157
\(345\) 4.50933e10 0.171366
\(346\) −4.66262e11 −1.74899
\(347\) −5.02625e11 −1.86107 −0.930533 0.366208i \(-0.880656\pi\)
−0.930533 + 0.366208i \(0.880656\pi\)
\(348\) 2.12043e11 0.775028
\(349\) −7.14710e10 −0.257879 −0.128939 0.991652i \(-0.541157\pi\)
−0.128939 + 0.991652i \(0.541157\pi\)
\(350\) 0 0
\(351\) −1.27153e11 −0.447142
\(352\) 1.46319e11 0.507995
\(353\) 2.55096e10 0.0874414 0.0437207 0.999044i \(-0.486079\pi\)
0.0437207 + 0.999044i \(0.486079\pi\)
\(354\) −2.75868e11 −0.933656
\(355\) −3.86887e10 −0.129288
\(356\) 2.43812e11 0.804508
\(357\) 0 0
\(358\) 5.68277e11 1.82846
\(359\) 1.49816e11 0.476029 0.238014 0.971262i \(-0.423503\pi\)
0.238014 + 0.971262i \(0.423503\pi\)
\(360\) −2.61534e10 −0.0820667
\(361\) −2.00146e11 −0.620246
\(362\) 3.81246e11 1.16685
\(363\) −1.97809e11 −0.597951
\(364\) 0 0
\(365\) −1.77352e11 −0.523021
\(366\) −4.76736e10 −0.138871
\(367\) 4.59514e11 1.32221 0.661107 0.750291i \(-0.270087\pi\)
0.661107 + 0.750291i \(0.270087\pi\)
\(368\) 2.03650e11 0.578854
\(369\) −6.36691e10 −0.178776
\(370\) 4.05594e10 0.112508
\(371\) 0 0
\(372\) 2.02318e11 0.547761
\(373\) −5.04230e11 −1.34877 −0.674386 0.738379i \(-0.735592\pi\)
−0.674386 + 0.738379i \(0.735592\pi\)
\(374\) −3.76516e11 −0.995086
\(375\) −2.83203e10 −0.0739533
\(376\) −1.58894e11 −0.409979
\(377\) 2.84315e11 0.724875
\(378\) 0 0
\(379\) −9.63136e10 −0.239779 −0.119889 0.992787i \(-0.538254\pi\)
−0.119889 + 0.992787i \(0.538254\pi\)
\(380\) −5.95102e10 −0.146408
\(381\) 4.05234e11 0.985242
\(382\) 5.10599e10 0.122686
\(383\) −6.10835e11 −1.45054 −0.725269 0.688465i \(-0.758285\pi\)
−0.725269 + 0.688465i \(0.758285\pi\)
\(384\) 3.63793e11 0.853814
\(385\) 0 0
\(386\) 3.46492e11 0.794421
\(387\) −1.87520e11 −0.424960
\(388\) −3.27980e11 −0.734691
\(389\) −6.49908e11 −1.43906 −0.719530 0.694461i \(-0.755643\pi\)
−0.719530 + 0.694461i \(0.755643\pi\)
\(390\) −8.58812e10 −0.187978
\(391\) −3.27372e11 −0.708347
\(392\) 0 0
\(393\) 2.14248e11 0.453054
\(394\) 5.40785e11 1.13055
\(395\) −1.72567e11 −0.356674
\(396\) 4.32718e10 0.0884253
\(397\) 3.67168e11 0.741836 0.370918 0.928666i \(-0.379043\pi\)
0.370918 + 0.928666i \(0.379043\pi\)
\(398\) −4.26007e11 −0.851026
\(399\) 0 0
\(400\) −1.27900e11 −0.249805
\(401\) −9.73985e11 −1.88106 −0.940530 0.339712i \(-0.889670\pi\)
−0.940530 + 0.339712i \(0.889670\pi\)
\(402\) −7.94324e11 −1.51698
\(403\) 2.71275e11 0.512315
\(404\) 3.97178e11 0.741770
\(405\) 1.41299e11 0.260971
\(406\) 0 0
\(407\) 5.92121e10 0.106964
\(408\) −4.10294e11 −0.733035
\(409\) −3.58196e11 −0.632944 −0.316472 0.948602i \(-0.602498\pi\)
−0.316472 + 0.948602i \(0.602498\pi\)
\(410\) −1.78932e11 −0.312723
\(411\) 1.35996e11 0.235091
\(412\) −2.93783e11 −0.502330
\(413\) 0 0
\(414\) 1.08445e11 0.181430
\(415\) −4.56225e10 −0.0755027
\(416\) −2.42296e11 −0.396668
\(417\) 5.67242e11 0.918662
\(418\) −2.50413e11 −0.401203
\(419\) −4.84712e11 −0.768283 −0.384141 0.923274i \(-0.625502\pi\)
−0.384141 + 0.923274i \(0.625502\pi\)
\(420\) 0 0
\(421\) 7.18298e11 1.11438 0.557192 0.830384i \(-0.311879\pi\)
0.557192 + 0.830384i \(0.311879\pi\)
\(422\) −1.09095e12 −1.67456
\(423\) −1.47237e11 −0.223607
\(424\) −3.85007e11 −0.578525
\(425\) 2.05602e11 0.305688
\(426\) 2.01058e11 0.295786
\(427\) 0 0
\(428\) 9.13782e10 0.131627
\(429\) −1.25377e11 −0.178714
\(430\) −5.26995e11 −0.743359
\(431\) 8.27297e11 1.15482 0.577409 0.816455i \(-0.304064\pi\)
0.577409 + 0.816455i \(0.304064\pi\)
\(432\) 9.84092e11 1.35944
\(433\) −8.73032e11 −1.19353 −0.596767 0.802415i \(-0.703548\pi\)
−0.596767 + 0.802415i \(0.703548\pi\)
\(434\) 0 0
\(435\) −4.87231e11 −0.652429
\(436\) −3.87657e11 −0.513757
\(437\) −2.17729e11 −0.285594
\(438\) 9.21665e11 1.19657
\(439\) −7.15061e11 −0.918867 −0.459433 0.888212i \(-0.651947\pi\)
−0.459433 + 0.888212i \(0.651947\pi\)
\(440\) −1.07302e11 −0.136480
\(441\) 0 0
\(442\) 6.23488e11 0.777012
\(443\) 5.89691e11 0.727457 0.363729 0.931505i \(-0.381504\pi\)
0.363729 + 0.931505i \(0.381504\pi\)
\(444\) −7.31275e10 −0.0893012
\(445\) −5.60230e11 −0.677246
\(446\) −3.03308e10 −0.0362975
\(447\) −9.99326e11 −1.18392
\(448\) 0 0
\(449\) −1.06477e12 −1.23636 −0.618181 0.786036i \(-0.712130\pi\)
−0.618181 + 0.786036i \(0.712130\pi\)
\(450\) −6.81078e10 −0.0782962
\(451\) −2.61220e11 −0.297312
\(452\) −7.72964e11 −0.871037
\(453\) 6.36730e11 0.710417
\(454\) 1.38493e12 1.52995
\(455\) 0 0
\(456\) −2.72879e11 −0.295548
\(457\) 1.54296e12 1.65475 0.827374 0.561651i \(-0.189834\pi\)
0.827374 + 0.561651i \(0.189834\pi\)
\(458\) 1.21177e12 1.28685
\(459\) −1.58195e12 −1.66355
\(460\) 1.05736e11 0.110106
\(461\) 8.38680e11 0.864852 0.432426 0.901669i \(-0.357658\pi\)
0.432426 + 0.901669i \(0.357658\pi\)
\(462\) 0 0
\(463\) −8.61819e11 −0.871569 −0.435784 0.900051i \(-0.643529\pi\)
−0.435784 + 0.900051i \(0.643529\pi\)
\(464\) −2.20043e12 −2.20382
\(465\) −4.64885e11 −0.461113
\(466\) −2.11503e11 −0.207768
\(467\) −1.46986e12 −1.43005 −0.715023 0.699101i \(-0.753584\pi\)
−0.715023 + 0.699101i \(0.753584\pi\)
\(468\) −7.16555e10 −0.0690468
\(469\) 0 0
\(470\) −4.13786e11 −0.391143
\(471\) 1.54408e11 0.144569
\(472\) 5.70762e11 0.529317
\(473\) −7.69353e11 −0.706725
\(474\) 8.96797e11 0.816003
\(475\) 1.36742e11 0.123248
\(476\) 0 0
\(477\) −3.56761e11 −0.315533
\(478\) −7.74245e11 −0.678348
\(479\) −2.20227e12 −1.91144 −0.955721 0.294275i \(-0.904922\pi\)
−0.955721 + 0.294275i \(0.904922\pi\)
\(480\) 4.15224e11 0.357024
\(481\) −9.80519e10 −0.0835224
\(482\) 2.31282e12 1.95177
\(483\) 0 0
\(484\) −4.63827e11 −0.384195
\(485\) 7.53631e11 0.618473
\(486\) 9.22132e11 0.749773
\(487\) 1.30073e12 1.04786 0.523932 0.851760i \(-0.324464\pi\)
0.523932 + 0.851760i \(0.324464\pi\)
\(488\) 9.86350e10 0.0787303
\(489\) 1.63672e11 0.129445
\(490\) 0 0
\(491\) 6.88947e11 0.534957 0.267479 0.963564i \(-0.413810\pi\)
0.267479 + 0.963564i \(0.413810\pi\)
\(492\) 3.22609e11 0.248218
\(493\) 3.53724e12 2.69683
\(494\) 4.14670e11 0.313279
\(495\) −9.94296e10 −0.0744376
\(496\) −2.09951e12 −1.55758
\(497\) 0 0
\(498\) 2.37091e11 0.172736
\(499\) 1.76710e12 1.27588 0.637939 0.770087i \(-0.279787\pi\)
0.637939 + 0.770087i \(0.279787\pi\)
\(500\) −6.64063e10 −0.0475164
\(501\) −2.88323e10 −0.0204461
\(502\) −5.65088e11 −0.397145
\(503\) −2.63527e12 −1.83556 −0.917780 0.397089i \(-0.870020\pi\)
−0.917780 + 0.397089i \(0.870020\pi\)
\(504\) 0 0
\(505\) −9.12633e11 −0.624432
\(506\) 4.44927e11 0.301725
\(507\) −1.02251e12 −0.687274
\(508\) 9.50203e11 0.633038
\(509\) −2.22151e12 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(510\) −1.06847e12 −0.699356
\(511\) 0 0
\(512\) 7.48685e11 0.481487
\(513\) −1.05213e12 −0.670718
\(514\) −7.91786e11 −0.500350
\(515\) 6.75053e11 0.422869
\(516\) 9.50157e11 0.590027
\(517\) −6.04080e11 −0.371867
\(518\) 0 0
\(519\) −1.93166e12 −1.16863
\(520\) 1.77685e11 0.106570
\(521\) 1.87441e12 1.11454 0.557269 0.830332i \(-0.311849\pi\)
0.557269 + 0.830332i \(0.311849\pi\)
\(522\) −1.17175e12 −0.690743
\(523\) 1.58412e12 0.925828 0.462914 0.886403i \(-0.346804\pi\)
0.462914 + 0.886403i \(0.346804\pi\)
\(524\) 5.02375e11 0.291096
\(525\) 0 0
\(526\) 3.89590e12 2.21908
\(527\) 3.37501e12 1.90602
\(528\) 9.70343e11 0.543341
\(529\) −1.41430e12 −0.785219
\(530\) −1.00262e12 −0.551945
\(531\) 5.28889e11 0.288695
\(532\) 0 0
\(533\) 4.32565e11 0.232156
\(534\) 2.91140e12 1.54941
\(535\) −2.09968e11 −0.110806
\(536\) 1.64343e12 0.860021
\(537\) 2.35429e12 1.22173
\(538\) −1.62087e11 −0.0834120
\(539\) 0 0
\(540\) 5.10945e11 0.258584
\(541\) 2.79888e12 1.40474 0.702370 0.711812i \(-0.252125\pi\)
0.702370 + 0.711812i \(0.252125\pi\)
\(542\) 2.09135e12 1.04095
\(543\) 1.57945e12 0.779660
\(544\) −3.01448e12 −1.47577
\(545\) 8.90755e11 0.432488
\(546\) 0 0
\(547\) 1.16606e12 0.556901 0.278451 0.960451i \(-0.410179\pi\)
0.278451 + 0.960451i \(0.410179\pi\)
\(548\) 3.18886e11 0.151051
\(549\) 9.13988e10 0.0429403
\(550\) −2.79431e11 −0.130210
\(551\) 2.35255e12 1.08732
\(552\) 4.84843e11 0.222267
\(553\) 0 0
\(554\) 6.22594e11 0.280809
\(555\) 1.68032e11 0.0751750
\(556\) 1.33008e12 0.590259
\(557\) 3.29033e12 1.44841 0.724204 0.689586i \(-0.242207\pi\)
0.724204 + 0.689586i \(0.242207\pi\)
\(558\) −1.11801e12 −0.488192
\(559\) 1.27400e12 0.551845
\(560\) 0 0
\(561\) −1.55985e12 −0.664890
\(562\) −3.80962e12 −1.61090
\(563\) 3.63871e12 1.52637 0.763186 0.646179i \(-0.223634\pi\)
0.763186 + 0.646179i \(0.223634\pi\)
\(564\) 7.46044e11 0.310462
\(565\) 1.77611e12 0.733251
\(566\) 4.79035e12 1.96197
\(567\) 0 0
\(568\) −4.15981e11 −0.167690
\(569\) 4.35009e12 1.73978 0.869888 0.493250i \(-0.164191\pi\)
0.869888 + 0.493250i \(0.164191\pi\)
\(570\) −7.10622e11 −0.281969
\(571\) −4.91136e12 −1.93348 −0.966739 0.255767i \(-0.917672\pi\)
−0.966739 + 0.255767i \(0.917672\pi\)
\(572\) −2.93987e11 −0.114827
\(573\) 2.11534e11 0.0819756
\(574\) 0 0
\(575\) −2.42959e11 −0.0926890
\(576\) −4.53238e10 −0.0171564
\(577\) 1.68758e12 0.633830 0.316915 0.948454i \(-0.397353\pi\)
0.316915 + 0.948454i \(0.397353\pi\)
\(578\) 4.43654e12 1.65337
\(579\) 1.43547e12 0.530811
\(580\) −1.14247e12 −0.419199
\(581\) 0 0
\(582\) −3.91647e12 −1.41495
\(583\) −1.46371e12 −0.524744
\(584\) −1.90689e12 −0.678373
\(585\) 1.64650e11 0.0581245
\(586\) −2.96051e12 −1.03711
\(587\) 4.82739e12 1.67819 0.839094 0.543987i \(-0.183086\pi\)
0.839094 + 0.543987i \(0.183086\pi\)
\(588\) 0 0
\(589\) 2.24466e12 0.768478
\(590\) 1.48636e12 0.504998
\(591\) 2.24039e12 0.755407
\(592\) 7.58865e11 0.253932
\(593\) 1.12076e12 0.372193 0.186096 0.982532i \(-0.440416\pi\)
0.186096 + 0.982532i \(0.440416\pi\)
\(594\) 2.15001e12 0.708600
\(595\) 0 0
\(596\) −2.34325e12 −0.760694
\(597\) −1.76489e12 −0.568633
\(598\) −7.36773e11 −0.235602
\(599\) 1.35944e12 0.431460 0.215730 0.976453i \(-0.430787\pi\)
0.215730 + 0.976453i \(0.430787\pi\)
\(600\) −3.04500e11 −0.0959194
\(601\) −9.50421e11 −0.297153 −0.148577 0.988901i \(-0.547469\pi\)
−0.148577 + 0.988901i \(0.547469\pi\)
\(602\) 0 0
\(603\) 1.52286e12 0.469064
\(604\) 1.49302e12 0.456457
\(605\) 1.06578e12 0.323421
\(606\) 4.74277e12 1.42858
\(607\) −3.35939e12 −1.00441 −0.502205 0.864749i \(-0.667478\pi\)
−0.502205 + 0.864749i \(0.667478\pi\)
\(608\) −2.00487e12 −0.595006
\(609\) 0 0
\(610\) 2.56862e11 0.0751131
\(611\) 1.00032e12 0.290372
\(612\) −8.91489e11 −0.256882
\(613\) −2.62257e12 −0.750162 −0.375081 0.926992i \(-0.622385\pi\)
−0.375081 + 0.926992i \(0.622385\pi\)
\(614\) −4.87603e12 −1.38455
\(615\) −7.41289e11 −0.208953
\(616\) 0 0
\(617\) −3.35285e10 −0.00931388 −0.00465694 0.999989i \(-0.501482\pi\)
−0.00465694 + 0.999989i \(0.501482\pi\)
\(618\) −3.50812e12 −0.967444
\(619\) −5.51587e12 −1.51010 −0.755051 0.655666i \(-0.772388\pi\)
−0.755051 + 0.655666i \(0.772388\pi\)
\(620\) −1.09008e12 −0.296274
\(621\) 1.86939e12 0.504414
\(622\) 2.96540e12 0.794377
\(623\) 0 0
\(624\) −1.60683e12 −0.424268
\(625\) 1.52588e11 0.0400000
\(626\) 6.81962e12 1.77491
\(627\) −1.03743e12 −0.268073
\(628\) 3.62060e11 0.0928886
\(629\) −1.21989e12 −0.310738
\(630\) 0 0
\(631\) 2.72456e12 0.684170 0.342085 0.939669i \(-0.388867\pi\)
0.342085 + 0.939669i \(0.388867\pi\)
\(632\) −1.85544e12 −0.462616
\(633\) −4.51966e12 −1.11890
\(634\) 5.14572e12 1.26487
\(635\) −2.18337e12 −0.532900
\(636\) 1.80770e12 0.438096
\(637\) 0 0
\(638\) −4.80742e12 −1.14873
\(639\) −3.85463e11 −0.0914596
\(640\) −1.96009e12 −0.461813
\(641\) 3.79136e11 0.0887022 0.0443511 0.999016i \(-0.485878\pi\)
0.0443511 + 0.999016i \(0.485878\pi\)
\(642\) 1.09116e12 0.253502
\(643\) 5.15446e12 1.18914 0.594571 0.804043i \(-0.297322\pi\)
0.594571 + 0.804043i \(0.297322\pi\)
\(644\) 0 0
\(645\) −2.18327e12 −0.496693
\(646\) 5.15904e12 1.16553
\(647\) 2.68059e12 0.601397 0.300699 0.953719i \(-0.402780\pi\)
0.300699 + 0.953719i \(0.402780\pi\)
\(648\) 1.51925e12 0.338487
\(649\) 2.16991e12 0.480111
\(650\) 4.62722e11 0.101674
\(651\) 0 0
\(652\) 3.83783e11 0.0831710
\(653\) −6.44986e12 −1.38816 −0.694082 0.719896i \(-0.744190\pi\)
−0.694082 + 0.719896i \(0.744190\pi\)
\(654\) −4.62908e12 −0.989451
\(655\) −1.15435e12 −0.245049
\(656\) −3.34780e12 −0.705818
\(657\) −1.76700e12 −0.369991
\(658\) 0 0
\(659\) −2.79549e12 −0.577396 −0.288698 0.957420i \(-0.593222\pi\)
−0.288698 + 0.957420i \(0.593222\pi\)
\(660\) 5.03807e11 0.103351
\(661\) 4.93691e11 0.100589 0.0502943 0.998734i \(-0.483984\pi\)
0.0502943 + 0.998734i \(0.483984\pi\)
\(662\) −1.62893e12 −0.329641
\(663\) 2.58302e12 0.519179
\(664\) −4.90533e11 −0.0979291
\(665\) 0 0
\(666\) 4.04102e11 0.0795897
\(667\) −4.17995e12 −0.817720
\(668\) −6.76068e10 −0.0131370
\(669\) −1.25656e11 −0.0242530
\(670\) 4.27976e12 0.820508
\(671\) 3.74989e11 0.0714113
\(672\) 0 0
\(673\) 2.83805e12 0.533277 0.266638 0.963797i \(-0.414087\pi\)
0.266638 + 0.963797i \(0.414087\pi\)
\(674\) −9.52747e12 −1.77831
\(675\) −1.17405e12 −0.217680
\(676\) −2.39760e12 −0.441587
\(677\) −7.71236e12 −1.41104 −0.705519 0.708691i \(-0.749286\pi\)
−0.705519 + 0.708691i \(0.749286\pi\)
\(678\) −9.23010e12 −1.67754
\(679\) 0 0
\(680\) 2.21064e12 0.396485
\(681\) 5.73757e12 1.02227
\(682\) −4.58693e12 −0.811882
\(683\) 1.07677e13 1.89334 0.946670 0.322204i \(-0.104424\pi\)
0.946670 + 0.322204i \(0.104424\pi\)
\(684\) −5.92912e11 −0.103571
\(685\) −7.32735e11 −0.127157
\(686\) 0 0
\(687\) 5.02021e12 0.859837
\(688\) −9.86005e12 −1.67776
\(689\) 2.42382e12 0.409746
\(690\) 1.26261e12 0.212055
\(691\) −2.02298e12 −0.337552 −0.168776 0.985654i \(-0.553981\pi\)
−0.168776 + 0.985654i \(0.553981\pi\)
\(692\) −4.52940e12 −0.750867
\(693\) 0 0
\(694\) −1.40735e13 −2.30295
\(695\) −3.05626e12 −0.496888
\(696\) −5.23871e12 −0.846219
\(697\) 5.38168e12 0.863714
\(698\) −2.00119e12 −0.319109
\(699\) −8.76226e11 −0.138825
\(700\) 0 0
\(701\) 8.66031e11 0.135457 0.0677286 0.997704i \(-0.478425\pi\)
0.0677286 + 0.997704i \(0.478425\pi\)
\(702\) −3.56029e12 −0.553310
\(703\) −8.11328e11 −0.125285
\(704\) −1.85953e11 −0.0285317
\(705\) −1.71426e12 −0.261351
\(706\) 7.14268e11 0.108203
\(707\) 0 0
\(708\) −2.67986e12 −0.400832
\(709\) −3.78834e12 −0.563042 −0.281521 0.959555i \(-0.590839\pi\)
−0.281521 + 0.959555i \(0.590839\pi\)
\(710\) −1.08328e12 −0.159985
\(711\) −1.71932e12 −0.252315
\(712\) −6.02360e12 −0.878407
\(713\) −3.98824e12 −0.577934
\(714\) 0 0
\(715\) 6.75521e11 0.0966633
\(716\) 5.52040e12 0.784986
\(717\) −3.20758e12 −0.453254
\(718\) 4.19485e12 0.589056
\(719\) −8.16972e11 −0.114006 −0.0570029 0.998374i \(-0.518154\pi\)
−0.0570029 + 0.998374i \(0.518154\pi\)
\(720\) −1.27429e12 −0.176715
\(721\) 0 0
\(722\) −5.60408e12 −0.767515
\(723\) 9.58166e12 1.30412
\(724\) 3.70353e12 0.500947
\(725\) 2.62517e12 0.352887
\(726\) −5.53864e12 −0.739927
\(727\) −3.13227e12 −0.415867 −0.207933 0.978143i \(-0.566674\pi\)
−0.207933 + 0.978143i \(0.566674\pi\)
\(728\) 0 0
\(729\) 8.27017e12 1.08453
\(730\) −4.96587e12 −0.647205
\(731\) 1.58503e13 2.05309
\(732\) −4.63115e11 −0.0596195
\(733\) 1.33197e13 1.70422 0.852112 0.523360i \(-0.175322\pi\)
0.852112 + 0.523360i \(0.175322\pi\)
\(734\) 1.28664e13 1.63616
\(735\) 0 0
\(736\) 3.56220e12 0.447474
\(737\) 6.24796e12 0.780072
\(738\) −1.78273e12 −0.221224
\(739\) −1.56702e13 −1.93274 −0.966371 0.257154i \(-0.917215\pi\)
−0.966371 + 0.257154i \(0.917215\pi\)
\(740\) 3.94006e11 0.0483014
\(741\) 1.71792e12 0.209325
\(742\) 0 0
\(743\) 7.91108e12 0.952327 0.476164 0.879357i \(-0.342027\pi\)
0.476164 + 0.879357i \(0.342027\pi\)
\(744\) −4.99844e12 −0.598076
\(745\) 5.38430e12 0.640362
\(746\) −1.41184e13 −1.66902
\(747\) −4.54546e11 −0.0534115
\(748\) −3.65758e12 −0.427205
\(749\) 0 0
\(750\) −7.92969e11 −0.0915125
\(751\) −4.01514e12 −0.460597 −0.230299 0.973120i \(-0.573970\pi\)
−0.230299 + 0.973120i \(0.573970\pi\)
\(752\) −7.74191e12 −0.882811
\(753\) −2.34108e12 −0.265362
\(754\) 7.96081e12 0.896987
\(755\) −3.43066e12 −0.384252
\(756\) 0 0
\(757\) 2.84075e12 0.314414 0.157207 0.987566i \(-0.449751\pi\)
0.157207 + 0.987566i \(0.449751\pi\)
\(758\) −2.69678e12 −0.296711
\(759\) 1.84327e12 0.201605
\(760\) 1.47025e12 0.159857
\(761\) 7.10439e12 0.767884 0.383942 0.923357i \(-0.374566\pi\)
0.383942 + 0.923357i \(0.374566\pi\)
\(762\) 1.13465e13 1.21918
\(763\) 0 0
\(764\) 4.96011e11 0.0526709
\(765\) 2.04846e12 0.216247
\(766\) −1.71034e13 −1.79495
\(767\) −3.59325e12 −0.374894
\(768\) 9.75390e12 1.01170
\(769\) 4.97052e12 0.512546 0.256273 0.966604i \(-0.417505\pi\)
0.256273 + 0.966604i \(0.417505\pi\)
\(770\) 0 0
\(771\) −3.28026e12 −0.334321
\(772\) 3.36593e12 0.341057
\(773\) −1.11565e13 −1.12388 −0.561939 0.827179i \(-0.689944\pi\)
−0.561939 + 0.827179i \(0.689944\pi\)
\(774\) −5.25056e12 −0.525861
\(775\) 2.50477e12 0.249408
\(776\) 8.10304e12 0.802177
\(777\) 0 0
\(778\) −1.81974e13 −1.78075
\(779\) 3.57925e12 0.348236
\(780\) −8.34274e11 −0.0807018
\(781\) −1.58147e12 −0.152101
\(782\) −9.16642e12 −0.876535
\(783\) −2.01987e13 −1.92041
\(784\) 0 0
\(785\) −8.31940e11 −0.0781949
\(786\) 5.99894e12 0.560626
\(787\) 6.23763e12 0.579607 0.289803 0.957086i \(-0.406410\pi\)
0.289803 + 0.957086i \(0.406410\pi\)
\(788\) 5.25334e12 0.485364
\(789\) 1.61402e13 1.48273
\(790\) −4.83188e12 −0.441361
\(791\) 0 0
\(792\) −1.06907e12 −0.0965477
\(793\) −6.20960e11 −0.0557615
\(794\) 1.02807e13 0.917975
\(795\) −4.15372e12 −0.368795
\(796\) −4.13835e12 −0.365358
\(797\) 1.83799e12 0.161355 0.0806773 0.996740i \(-0.474292\pi\)
0.0806773 + 0.996740i \(0.474292\pi\)
\(798\) 0 0
\(799\) 1.24453e13 1.08030
\(800\) −2.23720e12 −0.193108
\(801\) −5.58169e12 −0.479092
\(802\) −2.72716e13 −2.32769
\(803\) −7.24960e12 −0.615310
\(804\) −7.71629e12 −0.651263
\(805\) 0 0
\(806\) 7.59570e12 0.633957
\(807\) −6.71504e11 −0.0557337
\(808\) −9.81263e12 −0.809906
\(809\) 1.92205e13 1.57760 0.788801 0.614649i \(-0.210702\pi\)
0.788801 + 0.614649i \(0.210702\pi\)
\(810\) 3.95638e12 0.322935
\(811\) 1.70316e13 1.38249 0.691246 0.722620i \(-0.257062\pi\)
0.691246 + 0.722620i \(0.257062\pi\)
\(812\) 0 0
\(813\) 8.66416e12 0.695535
\(814\) 1.65794e12 0.132361
\(815\) −8.81855e11 −0.0700145
\(816\) −1.99911e13 −1.57845
\(817\) 1.05417e13 0.827774
\(818\) −1.00295e13 −0.783228
\(819\) 0 0
\(820\) −1.73820e12 −0.134257
\(821\) 1.33485e13 1.02539 0.512694 0.858572i \(-0.328648\pi\)
0.512694 + 0.858572i \(0.328648\pi\)
\(822\) 3.80788e12 0.290911
\(823\) −1.72858e13 −1.31338 −0.656689 0.754161i \(-0.728044\pi\)
−0.656689 + 0.754161i \(0.728044\pi\)
\(824\) 7.25817e12 0.548473
\(825\) −1.15764e12 −0.0870026
\(826\) 0 0
\(827\) 1.64651e13 1.22402 0.612012 0.790849i \(-0.290361\pi\)
0.612012 + 0.790849i \(0.290361\pi\)
\(828\) 1.05347e12 0.0778906
\(829\) 5.91786e12 0.435181 0.217590 0.976040i \(-0.430180\pi\)
0.217590 + 0.976040i \(0.430180\pi\)
\(830\) −1.27743e12 −0.0934298
\(831\) 2.57932e12 0.187629
\(832\) 3.07928e11 0.0222789
\(833\) 0 0
\(834\) 1.58828e13 1.13679
\(835\) 1.55347e11 0.0110589
\(836\) −2.43259e12 −0.172242
\(837\) −1.92723e13 −1.35728
\(838\) −1.35719e13 −0.950701
\(839\) −2.40380e13 −1.67483 −0.837414 0.546569i \(-0.815934\pi\)
−0.837414 + 0.546569i \(0.815934\pi\)
\(840\) 0 0
\(841\) 3.06570e13 2.11324
\(842\) 2.01123e13 1.37898
\(843\) −1.57827e13 −1.07636
\(844\) −1.05978e13 −0.718913
\(845\) 5.50919e12 0.371734
\(846\) −4.12263e12 −0.276699
\(847\) 0 0
\(848\) −1.87590e13 −1.24574
\(849\) 1.98457e13 1.31094
\(850\) 5.75687e12 0.378269
\(851\) 1.44154e12 0.0942203
\(852\) 1.95313e12 0.126985
\(853\) −2.26446e13 −1.46452 −0.732258 0.681027i \(-0.761534\pi\)
−0.732258 + 0.681027i \(0.761534\pi\)
\(854\) 0 0
\(855\) 1.36239e12 0.0871874
\(856\) −2.25758e12 −0.143718
\(857\) 1.20223e13 0.761334 0.380667 0.924712i \(-0.375694\pi\)
0.380667 + 0.924712i \(0.375694\pi\)
\(858\) −3.51055e12 −0.221148
\(859\) 9.44258e12 0.591727 0.295864 0.955230i \(-0.404393\pi\)
0.295864 + 0.955230i \(0.404393\pi\)
\(860\) −5.11938e12 −0.319135
\(861\) 0 0
\(862\) 2.31643e13 1.42901
\(863\) 8.80380e12 0.540283 0.270142 0.962821i \(-0.412929\pi\)
0.270142 + 0.962821i \(0.412929\pi\)
\(864\) 1.72135e13 1.05089
\(865\) 1.04076e13 0.632090
\(866\) −2.44449e13 −1.47692
\(867\) 1.83800e13 1.10474
\(868\) 0 0
\(869\) −7.05399e12 −0.419610
\(870\) −1.36425e13 −0.807340
\(871\) −1.03463e13 −0.609119
\(872\) 9.57740e12 0.560949
\(873\) 7.50858e12 0.437516
\(874\) −6.09641e12 −0.353405
\(875\) 0 0
\(876\) 8.95331e12 0.513707
\(877\) 6.73513e12 0.384457 0.192229 0.981350i \(-0.438429\pi\)
0.192229 + 0.981350i \(0.438429\pi\)
\(878\) −2.00217e13 −1.13704
\(879\) −1.22650e13 −0.692973
\(880\) −5.22814e12 −0.293884
\(881\) 7.58474e12 0.424179 0.212089 0.977250i \(-0.431973\pi\)
0.212089 + 0.977250i \(0.431973\pi\)
\(882\) 0 0
\(883\) −1.78234e13 −0.986662 −0.493331 0.869842i \(-0.664221\pi\)
−0.493331 + 0.869842i \(0.664221\pi\)
\(884\) 6.05674e12 0.333583
\(885\) 6.15777e12 0.337426
\(886\) 1.65113e13 0.900182
\(887\) 2.45177e13 1.32991 0.664956 0.746882i \(-0.268450\pi\)
0.664956 + 0.746882i \(0.268450\pi\)
\(888\) 1.80668e12 0.0975041
\(889\) 0 0
\(890\) −1.56864e13 −0.838049
\(891\) 5.77586e12 0.307020
\(892\) −2.94642e11 −0.0155830
\(893\) 8.27714e12 0.435561
\(894\) −2.79811e13 −1.46503
\(895\) −1.26847e13 −0.660812
\(896\) 0 0
\(897\) −3.05234e12 −0.157423
\(898\) −2.98134e13 −1.52992
\(899\) 4.30928e13 2.20032
\(900\) −6.61619e11 −0.0336137
\(901\) 3.01555e13 1.52442
\(902\) −7.31416e12 −0.367905
\(903\) 0 0
\(904\) 1.90968e13 0.951047
\(905\) −8.50995e12 −0.421704
\(906\) 1.78284e13 0.879096
\(907\) −8.14231e12 −0.399498 −0.199749 0.979847i \(-0.564013\pi\)
−0.199749 + 0.979847i \(0.564013\pi\)
\(908\) 1.34536e13 0.656830
\(909\) −9.09275e12 −0.441731
\(910\) 0 0
\(911\) −1.26965e13 −0.610733 −0.305367 0.952235i \(-0.598779\pi\)
−0.305367 + 0.952235i \(0.598779\pi\)
\(912\) −1.32957e13 −0.636406
\(913\) −1.86490e12 −0.0888254
\(914\) 4.32029e13 2.04765
\(915\) 1.06414e12 0.0501886
\(916\) 1.17715e13 0.552463
\(917\) 0 0
\(918\) −4.42947e13 −2.05854
\(919\) −2.53575e13 −1.17270 −0.586350 0.810058i \(-0.699436\pi\)
−0.586350 + 0.810058i \(0.699436\pi\)
\(920\) −2.61230e12 −0.120220
\(921\) −2.02007e13 −0.925120
\(922\) 2.34830e13 1.07020
\(923\) 2.61882e12 0.118768
\(924\) 0 0
\(925\) −9.05345e11 −0.0406608
\(926\) −2.41309e13 −1.07851
\(927\) 6.72569e12 0.299143
\(928\) −3.84895e13 −1.70363
\(929\) −1.89611e13 −0.835203 −0.417602 0.908630i \(-0.637129\pi\)
−0.417602 + 0.908630i \(0.637129\pi\)
\(930\) −1.30168e13 −0.570598
\(931\) 0 0
\(932\) −2.05460e12 −0.0891980
\(933\) 1.22852e13 0.530782
\(934\) −4.11561e13 −1.76959
\(935\) 8.40437e12 0.359627
\(936\) 1.77031e12 0.0753891
\(937\) −8.65559e12 −0.366833 −0.183416 0.983035i \(-0.558716\pi\)
−0.183416 + 0.983035i \(0.558716\pi\)
\(938\) 0 0
\(939\) 2.82527e13 1.18595
\(940\) −4.01964e12 −0.167923
\(941\) −3.42336e13 −1.42331 −0.711654 0.702530i \(-0.752054\pi\)
−0.711654 + 0.702530i \(0.752054\pi\)
\(942\) 4.32343e12 0.178895
\(943\) −6.35950e12 −0.261891
\(944\) 2.78097e13 1.13978
\(945\) 0 0
\(946\) −2.15419e13 −0.874527
\(947\) 2.38597e13 0.964030 0.482015 0.876163i \(-0.339905\pi\)
0.482015 + 0.876163i \(0.339905\pi\)
\(948\) 8.71174e12 0.350322
\(949\) 1.20049e13 0.480464
\(950\) 3.82878e12 0.152512
\(951\) 2.13180e13 0.845150
\(952\) 0 0
\(953\) −4.83481e13 −1.89872 −0.949360 0.314189i \(-0.898267\pi\)
−0.949360 + 0.314189i \(0.898267\pi\)
\(954\) −9.98932e12 −0.390452
\(955\) −1.13973e12 −0.0443391
\(956\) −7.52123e12 −0.291225
\(957\) −1.99165e13 −0.767553
\(958\) −6.16636e13 −2.36529
\(959\) 0 0
\(960\) −5.27698e11 −0.0200523
\(961\) 1.46768e13 0.555106
\(962\) −2.74545e12 −0.103354
\(963\) −2.09196e12 −0.0783852
\(964\) 2.24674e13 0.837925
\(965\) −7.73421e12 −0.287106
\(966\) 0 0
\(967\) −2.28025e13 −0.838617 −0.419308 0.907844i \(-0.637727\pi\)
−0.419308 + 0.907844i \(0.637727\pi\)
\(968\) 1.14593e13 0.419486
\(969\) 2.13731e13 0.778774
\(970\) 2.11017e13 0.765322
\(971\) −7.62385e12 −0.275225 −0.137612 0.990486i \(-0.543943\pi\)
−0.137612 + 0.990486i \(0.543943\pi\)
\(972\) 8.95785e12 0.321889
\(973\) 0 0
\(974\) 3.64203e13 1.29667
\(975\) 1.91699e12 0.0679359
\(976\) 4.80587e12 0.169531
\(977\) 2.97018e13 1.04293 0.521467 0.853272i \(-0.325385\pi\)
0.521467 + 0.853272i \(0.325385\pi\)
\(978\) 4.58282e12 0.160180
\(979\) −2.29004e13 −0.796749
\(980\) 0 0
\(981\) 8.87477e12 0.305947
\(982\) 1.92905e13 0.661976
\(983\) 1.08292e13 0.369917 0.184958 0.982746i \(-0.440785\pi\)
0.184958 + 0.982746i \(0.440785\pi\)
\(984\) −7.97034e12 −0.271019
\(985\) −1.20711e13 −0.408586
\(986\) 9.90428e13 3.33716
\(987\) 0 0
\(988\) 4.02822e12 0.134495
\(989\) −1.87302e13 −0.622528
\(990\) −2.78403e12 −0.0921118
\(991\) 5.29514e12 0.174400 0.0871999 0.996191i \(-0.472208\pi\)
0.0871999 + 0.996191i \(0.472208\pi\)
\(992\) −3.67242e13 −1.20406
\(993\) −6.74842e12 −0.220257
\(994\) 0 0
\(995\) 9.50909e12 0.307564
\(996\) 2.30317e12 0.0741581
\(997\) −1.64214e13 −0.526360 −0.263180 0.964747i \(-0.584771\pi\)
−0.263180 + 0.964747i \(0.584771\pi\)
\(998\) 4.94788e13 1.57882
\(999\) 6.96593e12 0.221276
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 245.10.a.b.1.1 1
7.6 odd 2 35.10.a.a.1.1 1
21.20 even 2 315.10.a.a.1.1 1
35.13 even 4 175.10.b.a.99.1 2
35.27 even 4 175.10.b.a.99.2 2
35.34 odd 2 175.10.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.a.1.1 1 7.6 odd 2
175.10.a.a.1.1 1 35.34 odd 2
175.10.b.a.99.1 2 35.13 even 4
175.10.b.a.99.2 2 35.27 even 4
245.10.a.b.1.1 1 1.1 even 1 trivial
315.10.a.a.1.1 1 21.20 even 2