Properties

Label 13.10.a.b.1.5
Level $13$
Weight $10$
Character 13.1
Self dual yes
Analytic conductor $6.695$
Analytic rank $0$
Dimension $5$
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] = [13,10,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, 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("13.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(6.69546587013\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(35.1685\) of defining polynomial
Character \(\chi\) \(=\) 13.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+38.1685 q^{2} +47.8784 q^{3} +944.833 q^{4} -109.762 q^{5} +1827.45 q^{6} +5947.44 q^{7} +16520.6 q^{8} -17390.7 q^{9} -4189.45 q^{10} -25205.7 q^{11} +45237.1 q^{12} +28561.0 q^{13} +227005. q^{14} -5255.24 q^{15} +146811. q^{16} +109318. q^{17} -663775. q^{18} -904609. q^{19} -103707. q^{20} +284754. q^{21} -962062. q^{22} -435749. q^{23} +790979. q^{24} -1.94108e6 q^{25} +1.09013e6 q^{26} -1.77503e6 q^{27} +5.61934e6 q^{28} +6.44791e6 q^{29} -200584. q^{30} +6.62308e6 q^{31} -2.85499e6 q^{32} -1.20681e6 q^{33} +4.17250e6 q^{34} -652804. q^{35} -1.64313e7 q^{36} +4.14357e6 q^{37} -3.45275e7 q^{38} +1.36745e6 q^{39} -1.81333e6 q^{40} +1.49568e7 q^{41} +1.08686e7 q^{42} +4.01789e7 q^{43} -2.38151e7 q^{44} +1.90884e6 q^{45} -1.66319e7 q^{46} +6.30151e6 q^{47} +7.02907e6 q^{48} -4.98153e6 q^{49} -7.40880e7 q^{50} +5.23397e6 q^{51} +2.69854e7 q^{52} +1.53111e7 q^{53} -6.77501e7 q^{54} +2.76663e6 q^{55} +9.82552e7 q^{56} -4.33112e7 q^{57} +2.46107e8 q^{58} -1.52760e8 q^{59} -4.96532e6 q^{60} +8.66321e7 q^{61} +2.52793e8 q^{62} -1.03430e8 q^{63} -1.84138e8 q^{64} -3.13492e6 q^{65} -4.60620e7 q^{66} -1.01034e8 q^{67} +1.03287e8 q^{68} -2.08630e7 q^{69} -2.49165e7 q^{70} +4.13122e8 q^{71} -2.87304e8 q^{72} -3.14453e8 q^{73} +1.58154e8 q^{74} -9.29357e7 q^{75} -8.54704e8 q^{76} -1.49909e8 q^{77} +5.21937e7 q^{78} -2.00580e8 q^{79} -1.61143e7 q^{80} +2.57315e8 q^{81} +5.70879e8 q^{82} +6.34578e7 q^{83} +2.69045e8 q^{84} -1.19990e7 q^{85} +1.53357e9 q^{86} +3.08715e8 q^{87} -4.16412e8 q^{88} +3.47074e7 q^{89} +7.28574e7 q^{90} +1.69865e8 q^{91} -4.11710e8 q^{92} +3.17102e8 q^{93} +2.40519e8 q^{94} +9.92918e7 q^{95} -1.36692e8 q^{96} -1.25403e9 q^{97} -1.90137e8 q^{98} +4.38343e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 15 q^{2} + 161 q^{3} + 361 q^{4} + 1803 q^{5} + 5693 q^{6} + 10099 q^{7} + 23151 q^{8} + 61060 q^{9} + 84505 q^{10} + 121746 q^{11} + 113389 q^{12} + 142805 q^{13} + 8475 q^{14} + 105973 q^{15} - 322463 q^{16}+ \cdots + 3016199848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 38.1685 1.68682 0.843412 0.537267i \(-0.180543\pi\)
0.843412 + 0.537267i \(0.180543\pi\)
\(3\) 47.8784 0.341267 0.170633 0.985335i \(-0.445419\pi\)
0.170633 + 0.985335i \(0.445419\pi\)
\(4\) 944.833 1.84538
\(5\) −109.762 −0.0785394 −0.0392697 0.999229i \(-0.512503\pi\)
−0.0392697 + 0.999229i \(0.512503\pi\)
\(6\) 1827.45 0.575657
\(7\) 5947.44 0.936244 0.468122 0.883664i \(-0.344931\pi\)
0.468122 + 0.883664i \(0.344931\pi\)
\(8\) 16520.6 1.42600
\(9\) −17390.7 −0.883537
\(10\) −4189.45 −0.132482
\(11\) −25205.7 −0.519076 −0.259538 0.965733i \(-0.583570\pi\)
−0.259538 + 0.965733i \(0.583570\pi\)
\(12\) 45237.1 0.629766
\(13\) 28561.0 0.277350
\(14\) 227005. 1.57928
\(15\) −5255.24 −0.0268029
\(16\) 146811. 0.560039
\(17\) 109318. 0.317447 0.158724 0.987323i \(-0.449262\pi\)
0.158724 + 0.987323i \(0.449262\pi\)
\(18\) −663775. −1.49037
\(19\) −904609. −1.59246 −0.796232 0.604992i \(-0.793176\pi\)
−0.796232 + 0.604992i \(0.793176\pi\)
\(20\) −103707. −0.144935
\(21\) 284754. 0.319509
\(22\) −962062. −0.875590
\(23\) −435749. −0.324684 −0.162342 0.986735i \(-0.551905\pi\)
−0.162342 + 0.986735i \(0.551905\pi\)
\(24\) 790979. 0.486647
\(25\) −1.94108e6 −0.993832
\(26\) 1.09013e6 0.467841
\(27\) −1.77503e6 −0.642789
\(28\) 5.61934e6 1.72772
\(29\) 6.44791e6 1.69289 0.846443 0.532479i \(-0.178740\pi\)
0.846443 + 0.532479i \(0.178740\pi\)
\(30\) −200584. −0.0452118
\(31\) 6.62308e6 1.28805 0.644024 0.765005i \(-0.277264\pi\)
0.644024 + 0.765005i \(0.277264\pi\)
\(32\) −2.85499e6 −0.481315
\(33\) −1.20681e6 −0.177143
\(34\) 4.17250e6 0.535477
\(35\) −652804. −0.0735320
\(36\) −1.64313e7 −1.63046
\(37\) 4.14357e6 0.363469 0.181734 0.983348i \(-0.441829\pi\)
0.181734 + 0.983348i \(0.441829\pi\)
\(38\) −3.45275e7 −2.68621
\(39\) 1.36745e6 0.0946504
\(40\) −1.81333e6 −0.111997
\(41\) 1.49568e7 0.826632 0.413316 0.910588i \(-0.364370\pi\)
0.413316 + 0.910588i \(0.364370\pi\)
\(42\) 1.08686e7 0.538956
\(43\) 4.01789e7 1.79221 0.896107 0.443838i \(-0.146384\pi\)
0.896107 + 0.443838i \(0.146384\pi\)
\(44\) −2.38151e7 −0.957891
\(45\) 1.90884e6 0.0693925
\(46\) −1.66319e7 −0.547685
\(47\) 6.30151e6 0.188367 0.0941834 0.995555i \(-0.469976\pi\)
0.0941834 + 0.995555i \(0.469976\pi\)
\(48\) 7.02907e6 0.191123
\(49\) −4.98153e6 −0.123447
\(50\) −7.40880e7 −1.67642
\(51\) 5.23397e6 0.108334
\(52\) 2.69854e7 0.511815
\(53\) 1.53111e7 0.266542 0.133271 0.991080i \(-0.457452\pi\)
0.133271 + 0.991080i \(0.457452\pi\)
\(54\) −6.77501e7 −1.08427
\(55\) 2.76663e6 0.0407679
\(56\) 9.82552e7 1.33509
\(57\) −4.33112e7 −0.543455
\(58\) 2.46107e8 2.85560
\(59\) −1.52760e8 −1.64126 −0.820629 0.571462i \(-0.806376\pi\)
−0.820629 + 0.571462i \(0.806376\pi\)
\(60\) −4.96532e6 −0.0494614
\(61\) 8.66321e7 0.801114 0.400557 0.916272i \(-0.368817\pi\)
0.400557 + 0.916272i \(0.368817\pi\)
\(62\) 2.52793e8 2.17271
\(63\) −1.03430e8 −0.827206
\(64\) −1.84138e8 −1.37193
\(65\) −3.13492e6 −0.0217829
\(66\) −4.60620e7 −0.298810
\(67\) −1.01034e8 −0.612537 −0.306268 0.951945i \(-0.599080\pi\)
−0.306268 + 0.951945i \(0.599080\pi\)
\(68\) 1.03287e8 0.585809
\(69\) −2.08630e7 −0.110804
\(70\) −2.49165e7 −0.124036
\(71\) 4.13122e8 1.92937 0.964685 0.263406i \(-0.0848458\pi\)
0.964685 + 0.263406i \(0.0848458\pi\)
\(72\) −2.87304e8 −1.25993
\(73\) −3.14453e8 −1.29599 −0.647997 0.761643i \(-0.724393\pi\)
−0.647997 + 0.761643i \(0.724393\pi\)
\(74\) 1.58154e8 0.613108
\(75\) −9.29357e7 −0.339162
\(76\) −8.54704e8 −2.93870
\(77\) −1.49909e8 −0.485982
\(78\) 5.21937e7 0.159659
\(79\) −2.00580e8 −0.579383 −0.289692 0.957120i \(-0.593553\pi\)
−0.289692 + 0.957120i \(0.593553\pi\)
\(80\) −1.61143e7 −0.0439851
\(81\) 2.57315e8 0.664175
\(82\) 5.70879e8 1.39438
\(83\) 6.34578e7 0.146769 0.0733843 0.997304i \(-0.476620\pi\)
0.0733843 + 0.997304i \(0.476620\pi\)
\(84\) 2.69045e8 0.589614
\(85\) −1.19990e7 −0.0249321
\(86\) 1.53357e9 3.02315
\(87\) 3.08715e8 0.577726
\(88\) −4.16412e8 −0.740204
\(89\) 3.47074e7 0.0586364 0.0293182 0.999570i \(-0.490666\pi\)
0.0293182 + 0.999570i \(0.490666\pi\)
\(90\) 7.28574e7 0.117053
\(91\) 1.69865e8 0.259667
\(92\) −4.11710e8 −0.599164
\(93\) 3.17102e8 0.439568
\(94\) 2.40519e8 0.317742
\(95\) 9.92918e7 0.125071
\(96\) −1.36692e8 −0.164257
\(97\) −1.25403e9 −1.43825 −0.719127 0.694879i \(-0.755458\pi\)
−0.719127 + 0.694879i \(0.755458\pi\)
\(98\) −1.90137e8 −0.208233
\(99\) 4.38343e8 0.458623
\(100\) −1.83399e9 −1.83399
\(101\) −9.06459e8 −0.866766 −0.433383 0.901210i \(-0.642680\pi\)
−0.433383 + 0.901210i \(0.642680\pi\)
\(102\) 1.99773e8 0.182741
\(103\) −4.17013e8 −0.365075 −0.182537 0.983199i \(-0.558431\pi\)
−0.182537 + 0.983199i \(0.558431\pi\)
\(104\) 4.71844e8 0.395502
\(105\) −3.12552e7 −0.0250940
\(106\) 5.84402e8 0.449610
\(107\) 6.71636e8 0.495344 0.247672 0.968844i \(-0.420334\pi\)
0.247672 + 0.968844i \(0.420334\pi\)
\(108\) −1.67710e9 −1.18619
\(109\) 1.66748e9 1.13147 0.565734 0.824588i \(-0.308593\pi\)
0.565734 + 0.824588i \(0.308593\pi\)
\(110\) 1.05598e8 0.0687683
\(111\) 1.98387e8 0.124040
\(112\) 8.73149e8 0.524333
\(113\) −1.84580e9 −1.06496 −0.532478 0.846444i \(-0.678739\pi\)
−0.532478 + 0.846444i \(0.678739\pi\)
\(114\) −1.65312e9 −0.916713
\(115\) 4.78287e7 0.0255005
\(116\) 6.09219e9 3.12401
\(117\) −4.96695e8 −0.245049
\(118\) −5.83063e9 −2.76851
\(119\) 6.50162e8 0.297208
\(120\) −8.68195e7 −0.0382210
\(121\) −1.72262e9 −0.730560
\(122\) 3.30661e9 1.35134
\(123\) 7.16109e8 0.282102
\(124\) 6.25770e9 2.37694
\(125\) 4.27436e8 0.156594
\(126\) −3.94776e9 −1.39535
\(127\) 1.87922e9 0.641006 0.320503 0.947248i \(-0.396148\pi\)
0.320503 + 0.947248i \(0.396148\pi\)
\(128\) −5.56650e9 −1.83290
\(129\) 1.92370e9 0.611623
\(130\) −1.19655e8 −0.0367439
\(131\) 3.46045e9 1.02663 0.513313 0.858201i \(-0.328418\pi\)
0.513313 + 0.858201i \(0.328418\pi\)
\(132\) −1.14023e9 −0.326896
\(133\) −5.38011e9 −1.49093
\(134\) −3.85632e9 −1.03324
\(135\) 1.94831e8 0.0504842
\(136\) 1.80600e9 0.452680
\(137\) 5.04786e9 1.22424 0.612118 0.790766i \(-0.290318\pi\)
0.612118 + 0.790766i \(0.290318\pi\)
\(138\) −7.96307e8 −0.186907
\(139\) 3.59716e9 0.817322 0.408661 0.912686i \(-0.365996\pi\)
0.408661 + 0.912686i \(0.365996\pi\)
\(140\) −6.16791e8 −0.135694
\(141\) 3.01706e8 0.0642833
\(142\) 1.57682e10 3.25451
\(143\) −7.19899e8 −0.143966
\(144\) −2.55314e9 −0.494815
\(145\) −7.07736e8 −0.132958
\(146\) −1.20022e10 −2.18611
\(147\) −2.38508e8 −0.0421283
\(148\) 3.91498e9 0.670736
\(149\) 1.27561e9 0.212022 0.106011 0.994365i \(-0.466192\pi\)
0.106011 + 0.994365i \(0.466192\pi\)
\(150\) −3.54721e9 −0.572106
\(151\) 4.35273e9 0.681342 0.340671 0.940183i \(-0.389346\pi\)
0.340671 + 0.940183i \(0.389346\pi\)
\(152\) −1.49447e10 −2.27086
\(153\) −1.90111e9 −0.280476
\(154\) −5.72181e9 −0.819766
\(155\) −7.26963e8 −0.101163
\(156\) 1.29202e9 0.174666
\(157\) −1.41002e10 −1.85215 −0.926075 0.377339i \(-0.876839\pi\)
−0.926075 + 0.377339i \(0.876839\pi\)
\(158\) −7.65584e9 −0.977318
\(159\) 7.33072e8 0.0909619
\(160\) 3.13370e8 0.0378022
\(161\) −2.59159e9 −0.303983
\(162\) 9.82132e9 1.12035
\(163\) 7.24812e8 0.0804231 0.0402116 0.999191i \(-0.487197\pi\)
0.0402116 + 0.999191i \(0.487197\pi\)
\(164\) 1.41317e10 1.52545
\(165\) 1.32462e8 0.0139127
\(166\) 2.42209e9 0.247573
\(167\) −8.33031e8 −0.0828776 −0.0414388 0.999141i \(-0.513194\pi\)
−0.0414388 + 0.999141i \(0.513194\pi\)
\(168\) 4.70430e9 0.455621
\(169\) 8.15731e8 0.0769231
\(170\) −4.57983e8 −0.0420561
\(171\) 1.57317e10 1.40700
\(172\) 3.79623e10 3.30731
\(173\) 5.77955e8 0.0490553 0.0245277 0.999699i \(-0.492192\pi\)
0.0245277 + 0.999699i \(0.492192\pi\)
\(174\) 1.17832e10 0.974522
\(175\) −1.15444e10 −0.930469
\(176\) −3.70046e9 −0.290703
\(177\) −7.31392e9 −0.560106
\(178\) 1.32473e9 0.0989094
\(179\) −1.56569e10 −1.13990 −0.569952 0.821678i \(-0.693038\pi\)
−0.569952 + 0.821678i \(0.693038\pi\)
\(180\) 1.80353e9 0.128055
\(181\) −2.18552e10 −1.51356 −0.756781 0.653668i \(-0.773229\pi\)
−0.756781 + 0.653668i \(0.773229\pi\)
\(182\) 6.48349e9 0.438013
\(183\) 4.14780e9 0.273394
\(184\) −7.19882e9 −0.463000
\(185\) −4.54807e8 −0.0285466
\(186\) 1.21033e10 0.741474
\(187\) −2.75543e9 −0.164779
\(188\) 5.95388e9 0.347608
\(189\) −1.05569e10 −0.601807
\(190\) 3.78982e9 0.210973
\(191\) 1.67784e10 0.912222 0.456111 0.889923i \(-0.349242\pi\)
0.456111 + 0.889923i \(0.349242\pi\)
\(192\) −8.81622e9 −0.468195
\(193\) −2.70036e10 −1.40092 −0.700462 0.713690i \(-0.747023\pi\)
−0.700462 + 0.713690i \(0.747023\pi\)
\(194\) −4.78645e10 −2.42608
\(195\) −1.50095e8 −0.00743378
\(196\) −4.70671e9 −0.227806
\(197\) 1.58277e10 0.748720 0.374360 0.927283i \(-0.377862\pi\)
0.374360 + 0.927283i \(0.377862\pi\)
\(198\) 1.67309e10 0.773616
\(199\) 8.80397e9 0.397960 0.198980 0.980004i \(-0.436237\pi\)
0.198980 + 0.980004i \(0.436237\pi\)
\(200\) −3.20677e10 −1.41721
\(201\) −4.83736e9 −0.209038
\(202\) −3.45982e10 −1.46208
\(203\) 3.83485e10 1.58495
\(204\) 4.94523e9 0.199917
\(205\) −1.64169e9 −0.0649232
\(206\) −1.59167e10 −0.615817
\(207\) 7.57796e9 0.286870
\(208\) 4.19306e9 0.155327
\(209\) 2.28013e10 0.826610
\(210\) −1.19296e9 −0.0423292
\(211\) −1.82054e10 −0.632308 −0.316154 0.948708i \(-0.602392\pi\)
−0.316154 + 0.948708i \(0.602392\pi\)
\(212\) 1.44665e10 0.491870
\(213\) 1.97796e10 0.658430
\(214\) 2.56353e10 0.835558
\(215\) −4.41012e9 −0.140759
\(216\) −2.93245e10 −0.916618
\(217\) 3.93904e10 1.20593
\(218\) 6.36453e10 1.90859
\(219\) −1.50555e10 −0.442280
\(220\) 2.61400e9 0.0752322
\(221\) 3.12223e9 0.0880440
\(222\) 7.57215e9 0.209233
\(223\) 1.53511e10 0.415688 0.207844 0.978162i \(-0.433355\pi\)
0.207844 + 0.978162i \(0.433355\pi\)
\(224\) −1.69799e10 −0.450628
\(225\) 3.37566e10 0.878087
\(226\) −7.04514e10 −1.79639
\(227\) 4.20620e10 1.05141 0.525707 0.850666i \(-0.323801\pi\)
0.525707 + 0.850666i \(0.323801\pi\)
\(228\) −4.09219e10 −1.00288
\(229\) −6.68760e10 −1.60698 −0.803490 0.595318i \(-0.797026\pi\)
−0.803490 + 0.595318i \(0.797026\pi\)
\(230\) 1.82555e9 0.0430148
\(231\) −7.17741e9 −0.165849
\(232\) 1.06523e11 2.41406
\(233\) −5.19268e10 −1.15422 −0.577112 0.816665i \(-0.695820\pi\)
−0.577112 + 0.816665i \(0.695820\pi\)
\(234\) −1.89581e10 −0.413355
\(235\) −6.91668e8 −0.0147942
\(236\) −1.44333e11 −3.02874
\(237\) −9.60345e9 −0.197724
\(238\) 2.48157e10 0.501338
\(239\) 7.45881e10 1.47870 0.739348 0.673323i \(-0.235134\pi\)
0.739348 + 0.673323i \(0.235134\pi\)
\(240\) −7.71525e8 −0.0150107
\(241\) 5.74852e10 1.09769 0.548845 0.835924i \(-0.315068\pi\)
0.548845 + 0.835924i \(0.315068\pi\)
\(242\) −6.57499e10 −1.23233
\(243\) 4.72577e10 0.869449
\(244\) 8.18528e10 1.47836
\(245\) 5.46784e8 0.00969545
\(246\) 2.73328e10 0.475857
\(247\) −2.58365e10 −0.441670
\(248\) 1.09417e11 1.83676
\(249\) 3.03826e9 0.0500873
\(250\) 1.63146e10 0.264147
\(251\) −1.07873e11 −1.71546 −0.857732 0.514097i \(-0.828127\pi\)
−0.857732 + 0.514097i \(0.828127\pi\)
\(252\) −9.77240e10 −1.52651
\(253\) 1.09833e10 0.168536
\(254\) 7.17272e10 1.08126
\(255\) −5.74492e8 −0.00850850
\(256\) −1.18186e11 −1.71984
\(257\) −6.64074e10 −0.949550 −0.474775 0.880107i \(-0.657470\pi\)
−0.474775 + 0.880107i \(0.657470\pi\)
\(258\) 7.34247e10 1.03170
\(259\) 2.46436e10 0.340295
\(260\) −2.96197e9 −0.0401977
\(261\) −1.12133e11 −1.49573
\(262\) 1.32080e11 1.73174
\(263\) −8.15356e10 −1.05086 −0.525432 0.850836i \(-0.676096\pi\)
−0.525432 + 0.850836i \(0.676096\pi\)
\(264\) −1.99371e10 −0.252607
\(265\) −1.68058e9 −0.0209340
\(266\) −2.05351e11 −2.51494
\(267\) 1.66174e9 0.0200107
\(268\) −9.54605e10 −1.13036
\(269\) 1.00568e11 1.17105 0.585523 0.810656i \(-0.300889\pi\)
0.585523 + 0.810656i \(0.300889\pi\)
\(270\) 7.43640e9 0.0851580
\(271\) −2.98757e10 −0.336477 −0.168239 0.985746i \(-0.553808\pi\)
−0.168239 + 0.985746i \(0.553808\pi\)
\(272\) 1.60491e10 0.177783
\(273\) 8.13286e9 0.0886158
\(274\) 1.92669e11 2.06507
\(275\) 4.89261e10 0.515874
\(276\) −1.97120e10 −0.204475
\(277\) −3.17195e10 −0.323718 −0.161859 0.986814i \(-0.551749\pi\)
−0.161859 + 0.986814i \(0.551749\pi\)
\(278\) 1.37298e11 1.37868
\(279\) −1.15180e11 −1.13804
\(280\) −1.07847e10 −0.104857
\(281\) 9.86953e10 0.944318 0.472159 0.881513i \(-0.343475\pi\)
0.472159 + 0.881513i \(0.343475\pi\)
\(282\) 1.15157e10 0.108435
\(283\) −1.06164e11 −0.983871 −0.491935 0.870632i \(-0.663710\pi\)
−0.491935 + 0.870632i \(0.663710\pi\)
\(284\) 3.90331e11 3.56042
\(285\) 4.75393e9 0.0426826
\(286\) −2.74774e10 −0.242845
\(287\) 8.89549e10 0.773929
\(288\) 4.96501e10 0.425260
\(289\) −1.06637e11 −0.899227
\(290\) −2.70132e10 −0.224277
\(291\) −6.00410e10 −0.490828
\(292\) −2.97106e11 −2.39160
\(293\) 2.26355e11 1.79426 0.897132 0.441763i \(-0.145647\pi\)
0.897132 + 0.441763i \(0.145647\pi\)
\(294\) −9.10348e9 −0.0710631
\(295\) 1.67673e10 0.128903
\(296\) 6.84541e10 0.518307
\(297\) 4.47407e10 0.333656
\(298\) 4.86881e10 0.357643
\(299\) −1.24454e10 −0.0900511
\(300\) −8.78087e10 −0.625881
\(301\) 2.38962e11 1.67795
\(302\) 1.66137e11 1.14930
\(303\) −4.33998e10 −0.295798
\(304\) −1.32806e11 −0.891841
\(305\) −9.50892e9 −0.0629190
\(306\) −7.25625e10 −0.473114
\(307\) −2.23009e10 −0.143285 −0.0716424 0.997430i \(-0.522824\pi\)
−0.0716424 + 0.997430i \(0.522824\pi\)
\(308\) −1.41639e11 −0.896820
\(309\) −1.99659e10 −0.124588
\(310\) −2.77471e10 −0.170643
\(311\) −2.71805e11 −1.64754 −0.823770 0.566925i \(-0.808133\pi\)
−0.823770 + 0.566925i \(0.808133\pi\)
\(312\) 2.25911e10 0.134972
\(313\) −7.90775e10 −0.465697 −0.232849 0.972513i \(-0.574805\pi\)
−0.232849 + 0.972513i \(0.574805\pi\)
\(314\) −5.38183e11 −3.12425
\(315\) 1.13527e10 0.0649683
\(316\) −1.89515e11 −1.06918
\(317\) 2.03032e11 1.12927 0.564636 0.825340i \(-0.309017\pi\)
0.564636 + 0.825340i \(0.309017\pi\)
\(318\) 2.79802e10 0.153437
\(319\) −1.62524e11 −0.878736
\(320\) 2.02114e10 0.107751
\(321\) 3.21568e10 0.169044
\(322\) −9.89171e10 −0.512767
\(323\) −9.88899e10 −0.505523
\(324\) 2.43120e11 1.22565
\(325\) −5.54391e10 −0.275639
\(326\) 2.76650e10 0.135660
\(327\) 7.98365e10 0.386132
\(328\) 2.47095e11 1.17878
\(329\) 3.74779e10 0.176357
\(330\) 5.05586e9 0.0234683
\(331\) 3.24137e11 1.48424 0.742118 0.670270i \(-0.233822\pi\)
0.742118 + 0.670270i \(0.233822\pi\)
\(332\) 5.99570e10 0.270844
\(333\) −7.20594e10 −0.321138
\(334\) −3.17955e10 −0.139800
\(335\) 1.10897e10 0.0481083
\(336\) 4.18050e10 0.178937
\(337\) 2.68510e11 1.13403 0.567017 0.823706i \(-0.308097\pi\)
0.567017 + 0.823706i \(0.308097\pi\)
\(338\) 3.11352e10 0.129756
\(339\) −8.83739e10 −0.363434
\(340\) −1.13370e10 −0.0460091
\(341\) −1.66939e11 −0.668595
\(342\) 6.00457e11 2.37336
\(343\) −2.69628e11 −1.05182
\(344\) 6.63778e11 2.55570
\(345\) 2.28996e9 0.00870247
\(346\) 2.20597e10 0.0827478
\(347\) 2.74715e11 1.01719 0.508593 0.861007i \(-0.330166\pi\)
0.508593 + 0.861007i \(0.330166\pi\)
\(348\) 2.91684e11 1.06612
\(349\) −3.54006e11 −1.27731 −0.638655 0.769494i \(-0.720509\pi\)
−0.638655 + 0.769494i \(0.720509\pi\)
\(350\) −4.40634e11 −1.56954
\(351\) −5.06966e10 −0.178277
\(352\) 7.19619e10 0.249839
\(353\) −2.25650e11 −0.773481 −0.386741 0.922189i \(-0.626399\pi\)
−0.386741 + 0.922189i \(0.626399\pi\)
\(354\) −2.79161e11 −0.944801
\(355\) −4.53451e10 −0.151532
\(356\) 3.27927e10 0.108206
\(357\) 3.11287e10 0.101427
\(358\) −5.97602e11 −1.92282
\(359\) 3.39022e10 0.107722 0.0538608 0.998548i \(-0.482847\pi\)
0.0538608 + 0.998548i \(0.482847\pi\)
\(360\) 3.15351e10 0.0989538
\(361\) 4.95629e11 1.53594
\(362\) −8.34178e11 −2.55312
\(363\) −8.24764e10 −0.249316
\(364\) 1.60494e11 0.479184
\(365\) 3.45150e10 0.101787
\(366\) 1.58315e11 0.461167
\(367\) −7.74387e10 −0.222823 −0.111412 0.993774i \(-0.535537\pi\)
−0.111412 + 0.993774i \(0.535537\pi\)
\(368\) −6.39726e10 −0.181836
\(369\) −2.60109e11 −0.730360
\(370\) −1.73593e10 −0.0481531
\(371\) 9.10620e10 0.249548
\(372\) 2.99609e11 0.811169
\(373\) 6.50821e11 1.74089 0.870446 0.492264i \(-0.163831\pi\)
0.870446 + 0.492264i \(0.163831\pi\)
\(374\) −1.05171e11 −0.277954
\(375\) 2.04649e10 0.0534404
\(376\) 1.04105e11 0.268612
\(377\) 1.84159e11 0.469522
\(378\) −4.02940e11 −1.01514
\(379\) 1.93989e11 0.482948 0.241474 0.970407i \(-0.422369\pi\)
0.241474 + 0.970407i \(0.422369\pi\)
\(380\) 9.38141e10 0.230803
\(381\) 8.99743e10 0.218754
\(382\) 6.40406e11 1.53876
\(383\) 4.44645e11 1.05589 0.527946 0.849278i \(-0.322963\pi\)
0.527946 + 0.849278i \(0.322963\pi\)
\(384\) −2.66515e11 −0.625506
\(385\) 1.64544e10 0.0381687
\(386\) −1.03069e12 −2.36311
\(387\) −6.98737e11 −1.58349
\(388\) −1.18485e12 −2.65412
\(389\) −1.69623e11 −0.375589 −0.187794 0.982208i \(-0.560134\pi\)
−0.187794 + 0.982208i \(0.560134\pi\)
\(390\) −5.72889e9 −0.0125395
\(391\) −4.76352e10 −0.103070
\(392\) −8.22978e10 −0.176036
\(393\) 1.65681e11 0.350353
\(394\) 6.04119e11 1.26296
\(395\) 2.20161e10 0.0455044
\(396\) 4.14161e11 0.846332
\(397\) 2.94468e11 0.594951 0.297476 0.954729i \(-0.403855\pi\)
0.297476 + 0.954729i \(0.403855\pi\)
\(398\) 3.36034e11 0.671289
\(399\) −2.57591e11 −0.508806
\(400\) −2.84971e11 −0.556584
\(401\) 4.43362e11 0.856265 0.428133 0.903716i \(-0.359172\pi\)
0.428133 + 0.903716i \(0.359172\pi\)
\(402\) −1.84635e11 −0.352611
\(403\) 1.89162e11 0.357240
\(404\) −8.56452e11 −1.59951
\(405\) −2.82434e10 −0.0521639
\(406\) 1.46371e12 2.67354
\(407\) −1.04441e11 −0.188668
\(408\) 8.64682e10 0.154485
\(409\) −1.01253e11 −0.178918 −0.0894592 0.995990i \(-0.528514\pi\)
−0.0894592 + 0.995990i \(0.528514\pi\)
\(410\) −6.26610e10 −0.109514
\(411\) 2.41684e11 0.417791
\(412\) −3.94007e11 −0.673700
\(413\) −9.08533e11 −1.53662
\(414\) 2.89239e11 0.483900
\(415\) −6.96526e9 −0.0115271
\(416\) −8.15413e10 −0.133493
\(417\) 1.72226e11 0.278925
\(418\) 8.70289e11 1.39435
\(419\) 7.93982e10 0.125848 0.0629242 0.998018i \(-0.479957\pi\)
0.0629242 + 0.998018i \(0.479957\pi\)
\(420\) −2.95310e10 −0.0463080
\(421\) −6.06765e11 −0.941350 −0.470675 0.882307i \(-0.655990\pi\)
−0.470675 + 0.882307i \(0.655990\pi\)
\(422\) −6.94871e11 −1.06659
\(423\) −1.09587e11 −0.166429
\(424\) 2.52949e11 0.380089
\(425\) −2.12195e11 −0.315489
\(426\) 7.54958e11 1.11066
\(427\) 5.15239e11 0.750038
\(428\) 6.34583e11 0.914096
\(429\) −3.44676e10 −0.0491307
\(430\) −1.68328e11 −0.237436
\(431\) −2.72544e11 −0.380442 −0.190221 0.981741i \(-0.560920\pi\)
−0.190221 + 0.981741i \(0.560920\pi\)
\(432\) −2.60593e11 −0.359987
\(433\) −1.15522e12 −1.57931 −0.789655 0.613551i \(-0.789740\pi\)
−0.789655 + 0.613551i \(0.789740\pi\)
\(434\) 1.50347e12 2.03419
\(435\) −3.38853e10 −0.0453742
\(436\) 1.57549e12 2.08799
\(437\) 3.94182e11 0.517047
\(438\) −5.74646e11 −0.746048
\(439\) 1.78053e11 0.228801 0.114401 0.993435i \(-0.463505\pi\)
0.114401 + 0.993435i \(0.463505\pi\)
\(440\) 4.57063e10 0.0581351
\(441\) 8.66321e10 0.109070
\(442\) 1.19171e11 0.148515
\(443\) −6.55260e10 −0.0808345 −0.0404173 0.999183i \(-0.512869\pi\)
−0.0404173 + 0.999183i \(0.512869\pi\)
\(444\) 1.87443e11 0.228900
\(445\) −3.80956e9 −0.00460527
\(446\) 5.85928e11 0.701192
\(447\) 6.10742e10 0.0723559
\(448\) −1.09515e12 −1.28446
\(449\) −7.98150e11 −0.926779 −0.463390 0.886155i \(-0.653367\pi\)
−0.463390 + 0.886155i \(0.653367\pi\)
\(450\) 1.28844e12 1.48118
\(451\) −3.76997e11 −0.429085
\(452\) −1.74397e12 −1.96525
\(453\) 2.08402e11 0.232519
\(454\) 1.60544e12 1.77355
\(455\) −1.86447e10 −0.0203941
\(456\) −7.15526e11 −0.774968
\(457\) −8.92736e11 −0.957415 −0.478708 0.877974i \(-0.658895\pi\)
−0.478708 + 0.877974i \(0.658895\pi\)
\(458\) −2.55256e12 −2.71069
\(459\) −1.94042e11 −0.204051
\(460\) 4.51901e10 0.0470580
\(461\) 2.32490e11 0.239745 0.119872 0.992789i \(-0.461751\pi\)
0.119872 + 0.992789i \(0.461751\pi\)
\(462\) −2.73951e11 −0.279759
\(463\) −1.54421e12 −1.56168 −0.780841 0.624730i \(-0.785209\pi\)
−0.780841 + 0.624730i \(0.785209\pi\)
\(464\) 9.46622e11 0.948082
\(465\) −3.48058e10 −0.0345234
\(466\) −1.98197e12 −1.94697
\(467\) 1.63317e12 1.58893 0.794465 0.607310i \(-0.207751\pi\)
0.794465 + 0.607310i \(0.207751\pi\)
\(468\) −4.69293e11 −0.452208
\(469\) −6.00895e11 −0.573484
\(470\) −2.63999e10 −0.0249553
\(471\) −6.75094e11 −0.632077
\(472\) −2.52369e12 −2.34044
\(473\) −1.01274e12 −0.930295
\(474\) −3.66549e11 −0.333526
\(475\) 1.75592e12 1.58264
\(476\) 6.14295e11 0.548461
\(477\) −2.66271e11 −0.235500
\(478\) 2.84692e12 2.49430
\(479\) 1.54414e12 1.34022 0.670112 0.742260i \(-0.266246\pi\)
0.670112 + 0.742260i \(0.266246\pi\)
\(480\) 1.50036e10 0.0129006
\(481\) 1.18344e11 0.100808
\(482\) 2.19412e12 1.85161
\(483\) −1.24081e11 −0.103739
\(484\) −1.62759e12 −1.34816
\(485\) 1.37645e11 0.112960
\(486\) 1.80375e12 1.46661
\(487\) 2.55439e11 0.205782 0.102891 0.994693i \(-0.467191\pi\)
0.102891 + 0.994693i \(0.467191\pi\)
\(488\) 1.43121e12 1.14239
\(489\) 3.47028e10 0.0274457
\(490\) 2.08699e10 0.0163545
\(491\) 1.58407e12 1.23000 0.615002 0.788526i \(-0.289155\pi\)
0.615002 + 0.788526i \(0.289155\pi\)
\(492\) 6.76603e11 0.520584
\(493\) 7.04872e11 0.537402
\(494\) −9.86141e11 −0.745020
\(495\) −4.81135e10 −0.0360200
\(496\) 9.72340e11 0.721357
\(497\) 2.45702e12 1.80636
\(498\) 1.15966e11 0.0844884
\(499\) −6.29948e11 −0.454833 −0.227417 0.973798i \(-0.573028\pi\)
−0.227417 + 0.973798i \(0.573028\pi\)
\(500\) 4.03856e11 0.288976
\(501\) −3.98842e10 −0.0282834
\(502\) −4.11736e12 −2.89369
\(503\) 5.49263e11 0.382582 0.191291 0.981533i \(-0.438733\pi\)
0.191291 + 0.981533i \(0.438733\pi\)
\(504\) −1.70872e12 −1.17960
\(505\) 9.94949e10 0.0680753
\(506\) 4.19217e11 0.284290
\(507\) 3.90559e10 0.0262513
\(508\) 1.77555e12 1.18290
\(509\) 1.36923e11 0.0904165 0.0452083 0.998978i \(-0.485605\pi\)
0.0452083 + 0.998978i \(0.485605\pi\)
\(510\) −2.19275e10 −0.0143523
\(511\) −1.87019e12 −1.21337
\(512\) −1.66095e12 −1.06817
\(513\) 1.60570e12 1.02362
\(514\) −2.53467e12 −1.60172
\(515\) 4.57722e10 0.0286727
\(516\) 1.81758e12 1.12867
\(517\) −1.58834e11 −0.0977767
\(518\) 9.40610e11 0.574018
\(519\) 2.76715e10 0.0167410
\(520\) −5.17906e10 −0.0310625
\(521\) −1.66627e12 −0.990779 −0.495389 0.868671i \(-0.664975\pi\)
−0.495389 + 0.868671i \(0.664975\pi\)
\(522\) −4.27996e12 −2.52303
\(523\) 1.97187e12 1.15244 0.576222 0.817293i \(-0.304526\pi\)
0.576222 + 0.817293i \(0.304526\pi\)
\(524\) 3.26955e12 1.89451
\(525\) −5.52730e11 −0.317538
\(526\) −3.11209e12 −1.77262
\(527\) 7.24021e11 0.408887
\(528\) −1.77172e11 −0.0992072
\(529\) −1.61128e12 −0.894580
\(530\) −6.41452e10 −0.0353121
\(531\) 2.65660e12 1.45011
\(532\) −5.08330e12 −2.75134
\(533\) 4.27182e11 0.229266
\(534\) 6.34260e10 0.0337545
\(535\) −7.37202e10 −0.0389040
\(536\) −1.66914e12 −0.873479
\(537\) −7.49629e11 −0.389011
\(538\) 3.83852e12 1.97535
\(539\) 1.25563e11 0.0640784
\(540\) 1.84083e11 0.0931624
\(541\) 1.54916e12 0.777513 0.388756 0.921341i \(-0.372905\pi\)
0.388756 + 0.921341i \(0.372905\pi\)
\(542\) −1.14031e12 −0.567578
\(543\) −1.04639e12 −0.516529
\(544\) −3.12101e11 −0.152792
\(545\) −1.83027e11 −0.0888648
\(546\) 3.10419e11 0.149479
\(547\) 1.98052e12 0.945879 0.472939 0.881095i \(-0.343193\pi\)
0.472939 + 0.881095i \(0.343193\pi\)
\(548\) 4.76939e12 2.25918
\(549\) −1.50659e12 −0.707814
\(550\) 1.86744e12 0.870189
\(551\) −5.83283e12 −2.69586
\(552\) −3.44668e11 −0.158007
\(553\) −1.19294e12 −0.542444
\(554\) −1.21068e12 −0.546056
\(555\) −2.17754e10 −0.00974200
\(556\) 3.39871e12 1.50827
\(557\) 8.86577e11 0.390273 0.195136 0.980776i \(-0.437485\pi\)
0.195136 + 0.980776i \(0.437485\pi\)
\(558\) −4.39623e12 −1.91967
\(559\) 1.14755e12 0.497071
\(560\) −9.58387e10 −0.0411808
\(561\) −1.31926e11 −0.0562336
\(562\) 3.76705e12 1.59290
\(563\) 3.89786e12 1.63508 0.817539 0.575873i \(-0.195338\pi\)
0.817539 + 0.575873i \(0.195338\pi\)
\(564\) 2.85062e11 0.118627
\(565\) 2.02599e11 0.0836410
\(566\) −4.05212e12 −1.65962
\(567\) 1.53037e12 0.621830
\(568\) 6.82501e12 2.75129
\(569\) −2.26590e12 −0.906223 −0.453112 0.891454i \(-0.649686\pi\)
−0.453112 + 0.891454i \(0.649686\pi\)
\(570\) 1.81450e11 0.0719981
\(571\) −3.09546e12 −1.21860 −0.609302 0.792938i \(-0.708550\pi\)
−0.609302 + 0.792938i \(0.708550\pi\)
\(572\) −6.80184e11 −0.265671
\(573\) 8.03323e11 0.311311
\(574\) 3.39527e12 1.30548
\(575\) 8.45822e11 0.322681
\(576\) 3.20228e12 1.21215
\(577\) −3.23415e12 −1.21470 −0.607350 0.794434i \(-0.707767\pi\)
−0.607350 + 0.794434i \(0.707767\pi\)
\(578\) −4.07019e12 −1.51684
\(579\) −1.29289e12 −0.478089
\(580\) −6.68692e11 −0.245358
\(581\) 3.77411e11 0.137411
\(582\) −2.29167e12 −0.827941
\(583\) −3.85927e11 −0.138356
\(584\) −5.19495e12 −1.84809
\(585\) 5.45183e10 0.0192460
\(586\) 8.63964e12 3.02661
\(587\) −8.78491e11 −0.305398 −0.152699 0.988273i \(-0.548796\pi\)
−0.152699 + 0.988273i \(0.548796\pi\)
\(588\) −2.25350e11 −0.0777427
\(589\) −5.99129e12 −2.05117
\(590\) 6.39983e11 0.217437
\(591\) 7.57805e11 0.255513
\(592\) 6.08321e11 0.203556
\(593\) −1.91730e12 −0.636712 −0.318356 0.947971i \(-0.603131\pi\)
−0.318356 + 0.947971i \(0.603131\pi\)
\(594\) 1.70769e12 0.562819
\(595\) −7.13632e10 −0.0233425
\(596\) 1.20524e12 0.391260
\(597\) 4.21520e11 0.135811
\(598\) −4.75023e11 −0.151900
\(599\) −5.10464e12 −1.62011 −0.810054 0.586355i \(-0.800562\pi\)
−0.810054 + 0.586355i \(0.800562\pi\)
\(600\) −1.53535e12 −0.483645
\(601\) 3.00419e12 0.939273 0.469636 0.882860i \(-0.344385\pi\)
0.469636 + 0.882860i \(0.344385\pi\)
\(602\) 9.12080e12 2.83041
\(603\) 1.75705e12 0.541199
\(604\) 4.11260e12 1.25733
\(605\) 1.89079e11 0.0573778
\(606\) −1.65650e12 −0.498960
\(607\) 1.18434e12 0.354101 0.177051 0.984202i \(-0.443344\pi\)
0.177051 + 0.984202i \(0.443344\pi\)
\(608\) 2.58265e12 0.766477
\(609\) 1.83607e12 0.540892
\(610\) −3.62941e11 −0.106133
\(611\) 1.79978e11 0.0522436
\(612\) −1.79623e12 −0.517584
\(613\) 6.42597e11 0.183809 0.0919045 0.995768i \(-0.470705\pi\)
0.0919045 + 0.995768i \(0.470705\pi\)
\(614\) −8.51193e11 −0.241696
\(615\) −7.86017e10 −0.0221561
\(616\) −2.47659e12 −0.693011
\(617\) 5.05474e12 1.40416 0.702078 0.712100i \(-0.252256\pi\)
0.702078 + 0.712100i \(0.252256\pi\)
\(618\) −7.62068e11 −0.210158
\(619\) −3.09492e11 −0.0847308 −0.0423654 0.999102i \(-0.513489\pi\)
−0.0423654 + 0.999102i \(0.513489\pi\)
\(620\) −6.86859e11 −0.186683
\(621\) 7.73466e11 0.208703
\(622\) −1.03744e13 −2.77911
\(623\) 2.06421e11 0.0548980
\(624\) 2.00757e11 0.0530079
\(625\) 3.74425e12 0.981533
\(626\) −3.01827e12 −0.785549
\(627\) 1.09169e12 0.282094
\(628\) −1.33223e13 −3.41792
\(629\) 4.52966e11 0.115382
\(630\) 4.33315e11 0.109590
\(631\) −7.94237e11 −0.199443 −0.0997214 0.995015i \(-0.531795\pi\)
−0.0997214 + 0.995015i \(0.531795\pi\)
\(632\) −3.31370e12 −0.826202
\(633\) −8.71644e11 −0.215786
\(634\) 7.74944e12 1.90488
\(635\) −2.06268e11 −0.0503442
\(636\) 6.92631e11 0.167859
\(637\) −1.42278e11 −0.0342380
\(638\) −6.20328e12 −1.48227
\(639\) −7.18446e12 −1.70467
\(640\) 6.10991e11 0.143954
\(641\) 2.30298e12 0.538802 0.269401 0.963028i \(-0.413174\pi\)
0.269401 + 0.963028i \(0.413174\pi\)
\(642\) 1.22738e12 0.285148
\(643\) −2.54593e12 −0.587350 −0.293675 0.955905i \(-0.594878\pi\)
−0.293675 + 0.955905i \(0.594878\pi\)
\(644\) −2.44862e12 −0.560964
\(645\) −2.11149e11 −0.0480365
\(646\) −3.77448e12 −0.852728
\(647\) −2.74568e11 −0.0616001 −0.0308000 0.999526i \(-0.509806\pi\)
−0.0308000 + 0.999526i \(0.509806\pi\)
\(648\) 4.25099e12 0.947115
\(649\) 3.85043e12 0.851937
\(650\) −2.11603e12 −0.464955
\(651\) 1.88595e12 0.411543
\(652\) 6.84826e11 0.148411
\(653\) −6.05719e12 −1.30365 −0.651826 0.758369i \(-0.725997\pi\)
−0.651826 + 0.758369i \(0.725997\pi\)
\(654\) 3.04724e12 0.651338
\(655\) −3.79827e11 −0.0806306
\(656\) 2.19582e12 0.462946
\(657\) 5.46855e12 1.14506
\(658\) 1.43047e12 0.297484
\(659\) 4.97718e12 1.02801 0.514007 0.857786i \(-0.328160\pi\)
0.514007 + 0.857786i \(0.328160\pi\)
\(660\) 1.25154e11 0.0256742
\(661\) 1.79834e12 0.366409 0.183204 0.983075i \(-0.441353\pi\)
0.183204 + 0.983075i \(0.441353\pi\)
\(662\) 1.23718e13 2.50364
\(663\) 1.49487e11 0.0300465
\(664\) 1.04836e12 0.209292
\(665\) 5.90532e11 0.117097
\(666\) −2.75040e12 −0.541703
\(667\) −2.80967e12 −0.549653
\(668\) −7.87076e11 −0.152940
\(669\) 7.34986e11 0.141860
\(670\) 4.23278e11 0.0811502
\(671\) −2.18362e12 −0.415839
\(672\) −8.12969e11 −0.153784
\(673\) 1.37693e12 0.258728 0.129364 0.991597i \(-0.458706\pi\)
0.129364 + 0.991597i \(0.458706\pi\)
\(674\) 1.02486e13 1.91292
\(675\) 3.44547e12 0.638824
\(676\) 7.70729e11 0.141952
\(677\) 6.49642e11 0.118857 0.0594285 0.998233i \(-0.481072\pi\)
0.0594285 + 0.998233i \(0.481072\pi\)
\(678\) −3.37310e12 −0.613049
\(679\) −7.45828e12 −1.34656
\(680\) −1.98230e11 −0.0355532
\(681\) 2.01386e12 0.358813
\(682\) −6.37181e12 −1.12780
\(683\) −1.10011e12 −0.193438 −0.0967190 0.995312i \(-0.530835\pi\)
−0.0967190 + 0.995312i \(0.530835\pi\)
\(684\) 1.48639e13 2.59645
\(685\) −5.54064e11 −0.0961507
\(686\) −1.02913e13 −1.77424
\(687\) −3.20192e12 −0.548409
\(688\) 5.89869e12 1.00371
\(689\) 4.37301e11 0.0739254
\(690\) 8.74044e10 0.0146795
\(691\) 7.72289e12 1.28863 0.644315 0.764760i \(-0.277142\pi\)
0.644315 + 0.764760i \(0.277142\pi\)
\(692\) 5.46071e11 0.0905256
\(693\) 2.60702e12 0.429383
\(694\) 1.04855e13 1.71581
\(695\) −3.94832e11 −0.0641920
\(696\) 5.10016e12 0.823838
\(697\) 1.63505e12 0.262412
\(698\) −1.35119e13 −2.15460
\(699\) −2.48617e12 −0.393898
\(700\) −1.09076e13 −1.71707
\(701\) 1.40436e12 0.219658 0.109829 0.993950i \(-0.464970\pi\)
0.109829 + 0.993950i \(0.464970\pi\)
\(702\) −1.93501e12 −0.300723
\(703\) −3.74831e12 −0.578810
\(704\) 4.64131e12 0.712137
\(705\) −3.31159e10 −0.00504877
\(706\) −8.61273e12 −1.30473
\(707\) −5.39111e12 −0.811505
\(708\) −6.91043e12 −1.03361
\(709\) 5.19764e12 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(710\) −1.73075e12 −0.255607
\(711\) 3.48822e12 0.511907
\(712\) 5.73387e11 0.0836157
\(713\) −2.88600e12 −0.418209
\(714\) 1.18814e12 0.171090
\(715\) 7.90176e10 0.0113070
\(716\) −1.47932e13 −2.10355
\(717\) 3.57116e12 0.504630
\(718\) 1.29400e12 0.181707
\(719\) −1.27042e13 −1.77284 −0.886418 0.462886i \(-0.846814\pi\)
−0.886418 + 0.462886i \(0.846814\pi\)
\(720\) 2.80238e11 0.0388625
\(721\) −2.48016e12 −0.341799
\(722\) 1.89174e13 2.59086
\(723\) 2.75230e12 0.374605
\(724\) −2.06495e13 −2.79309
\(725\) −1.25159e13 −1.68244
\(726\) −3.14800e12 −0.420552
\(727\) 9.91998e11 0.131706 0.0658531 0.997829i \(-0.479023\pi\)
0.0658531 + 0.997829i \(0.479023\pi\)
\(728\) 2.80627e12 0.370286
\(729\) −2.80211e12 −0.367461
\(730\) 1.31739e12 0.171696
\(731\) 4.39227e12 0.568933
\(732\) 3.91898e12 0.504514
\(733\) 5.54849e12 0.709916 0.354958 0.934882i \(-0.384495\pi\)
0.354958 + 0.934882i \(0.384495\pi\)
\(734\) −2.95572e12 −0.375864
\(735\) 2.61791e10 0.00330873
\(736\) 1.24406e12 0.156275
\(737\) 2.54663e12 0.317953
\(738\) −9.92797e12 −1.23199
\(739\) 8.78597e12 1.08365 0.541826 0.840491i \(-0.317733\pi\)
0.541826 + 0.840491i \(0.317733\pi\)
\(740\) −4.29717e11 −0.0526792
\(741\) −1.23701e12 −0.150727
\(742\) 3.47570e12 0.420944
\(743\) 4.04326e12 0.486724 0.243362 0.969936i \(-0.421750\pi\)
0.243362 + 0.969936i \(0.421750\pi\)
\(744\) 5.23871e12 0.626825
\(745\) −1.40014e11 −0.0166521
\(746\) 2.48408e13 2.93658
\(747\) −1.10357e12 −0.129676
\(748\) −2.60342e12 −0.304080
\(749\) 3.99451e12 0.463763
\(750\) 7.81116e11 0.0901446
\(751\) −2.52936e11 −0.0290156 −0.0145078 0.999895i \(-0.504618\pi\)
−0.0145078 + 0.999895i \(0.504618\pi\)
\(752\) 9.25130e11 0.105493
\(753\) −5.16479e12 −0.585431
\(754\) 7.02906e12 0.792001
\(755\) −4.77765e11 −0.0535122
\(756\) −9.97448e12 −1.11056
\(757\) 9.99638e12 1.10640 0.553199 0.833049i \(-0.313407\pi\)
0.553199 + 0.833049i \(0.313407\pi\)
\(758\) 7.40426e12 0.814649
\(759\) 5.25864e11 0.0575156
\(760\) 1.64036e12 0.178352
\(761\) −2.98848e12 −0.323013 −0.161507 0.986872i \(-0.551635\pi\)
−0.161507 + 0.986872i \(0.551635\pi\)
\(762\) 3.43418e12 0.369000
\(763\) 9.91726e12 1.05933
\(764\) 1.58528e13 1.68339
\(765\) 2.08670e11 0.0220284
\(766\) 1.69714e13 1.78110
\(767\) −4.36299e12 −0.455203
\(768\) −5.65858e12 −0.586924
\(769\) 1.54887e13 1.59715 0.798576 0.601894i \(-0.205587\pi\)
0.798576 + 0.601894i \(0.205587\pi\)
\(770\) 6.28038e11 0.0643839
\(771\) −3.17948e12 −0.324050
\(772\) −2.55139e13 −2.58523
\(773\) 1.07589e13 1.08383 0.541915 0.840434i \(-0.317700\pi\)
0.541915 + 0.840434i \(0.317700\pi\)
\(774\) −2.66697e13 −2.67106
\(775\) −1.28559e13 −1.28010
\(776\) −2.07173e13 −2.05095
\(777\) 1.17990e12 0.116131
\(778\) −6.47427e12 −0.633552
\(779\) −1.35301e13 −1.31638
\(780\) −1.41814e11 −0.0137181
\(781\) −1.04130e13 −1.00149
\(782\) −1.81816e12 −0.173861
\(783\) −1.14452e13 −1.08817
\(784\) −7.31343e11 −0.0691351
\(785\) 1.54767e12 0.145467
\(786\) 6.32379e12 0.590985
\(787\) 1.29304e13 1.20151 0.600755 0.799433i \(-0.294867\pi\)
0.600755 + 0.799433i \(0.294867\pi\)
\(788\) 1.49545e13 1.38167
\(789\) −3.90379e12 −0.358625
\(790\) 8.40321e11 0.0767579
\(791\) −1.09778e13 −0.997059
\(792\) 7.24168e12 0.653997
\(793\) 2.47430e12 0.222189
\(794\) 1.12394e13 1.00358
\(795\) −8.04635e10 −0.00714409
\(796\) 8.31828e12 0.734387
\(797\) 1.24713e13 1.09484 0.547418 0.836859i \(-0.315611\pi\)
0.547418 + 0.836859i \(0.315611\pi\)
\(798\) −9.83185e12 −0.858267
\(799\) 6.88868e11 0.0597965
\(800\) 5.54175e12 0.478346
\(801\) −6.03585e11 −0.0518075
\(802\) 1.69224e13 1.44437
\(803\) 7.92600e12 0.672719
\(804\) −4.57049e12 −0.385755
\(805\) 2.84459e11 0.0238747
\(806\) 7.22002e12 0.602602
\(807\) 4.81502e12 0.399639
\(808\) −1.49752e13 −1.23601
\(809\) −2.40763e13 −1.97616 −0.988079 0.153948i \(-0.950801\pi\)
−0.988079 + 0.153948i \(0.950801\pi\)
\(810\) −1.07801e12 −0.0879913
\(811\) 1.14686e13 0.930928 0.465464 0.885067i \(-0.345888\pi\)
0.465464 + 0.885067i \(0.345888\pi\)
\(812\) 3.62330e13 2.92484
\(813\) −1.43040e12 −0.114829
\(814\) −3.98637e12 −0.318249
\(815\) −7.95569e10 −0.00631638
\(816\) 7.68403e11 0.0606713
\(817\) −3.63462e13 −2.85403
\(818\) −3.86469e12 −0.301804
\(819\) −2.95406e12 −0.229426
\(820\) −1.55113e12 −0.119808
\(821\) −2.50113e13 −1.92129 −0.960643 0.277786i \(-0.910399\pi\)
−0.960643 + 0.277786i \(0.910399\pi\)
\(822\) 9.22470e12 0.704740
\(823\) 1.29351e13 0.982808 0.491404 0.870932i \(-0.336484\pi\)
0.491404 + 0.870932i \(0.336484\pi\)
\(824\) −6.88929e12 −0.520597
\(825\) 2.34250e12 0.176051
\(826\) −3.46773e13 −2.59200
\(827\) 1.00464e13 0.746851 0.373425 0.927660i \(-0.378183\pi\)
0.373425 + 0.927660i \(0.378183\pi\)
\(828\) 7.15990e12 0.529384
\(829\) 8.77304e12 0.645141 0.322570 0.946545i \(-0.395453\pi\)
0.322570 + 0.946545i \(0.395453\pi\)
\(830\) −2.65853e11 −0.0194442
\(831\) −1.51868e12 −0.110474
\(832\) −5.25916e12 −0.380506
\(833\) −5.44571e11 −0.0391879
\(834\) 6.57361e12 0.470497
\(835\) 9.14353e10 0.00650916
\(836\) 2.15434e13 1.52541
\(837\) −1.17561e13 −0.827943
\(838\) 3.03051e12 0.212284
\(839\) −1.54259e13 −1.07478 −0.537392 0.843332i \(-0.680590\pi\)
−0.537392 + 0.843332i \(0.680590\pi\)
\(840\) −5.16354e11 −0.0357842
\(841\) 2.70683e13 1.86586
\(842\) −2.31593e13 −1.58789
\(843\) 4.72537e12 0.322264
\(844\) −1.72010e13 −1.16685
\(845\) −8.95364e10 −0.00604149
\(846\) −4.18279e12 −0.280737
\(847\) −1.02452e13 −0.683983
\(848\) 2.24784e12 0.149274
\(849\) −5.08296e12 −0.335762
\(850\) −8.09914e12 −0.532174
\(851\) −1.80556e12 −0.118012
\(852\) 1.86884e13 1.21505
\(853\) −1.84928e12 −0.119600 −0.0598002 0.998210i \(-0.519046\pi\)
−0.0598002 + 0.998210i \(0.519046\pi\)
\(854\) 1.96659e13 1.26518
\(855\) −1.72675e12 −0.110505
\(856\) 1.10958e13 0.706361
\(857\) −2.34715e13 −1.48637 −0.743185 0.669086i \(-0.766686\pi\)
−0.743185 + 0.669086i \(0.766686\pi\)
\(858\) −1.31558e12 −0.0828749
\(859\) 1.25121e12 0.0784079 0.0392039 0.999231i \(-0.487518\pi\)
0.0392039 + 0.999231i \(0.487518\pi\)
\(860\) −4.16683e12 −0.259754
\(861\) 4.25902e12 0.264116
\(862\) −1.04026e13 −0.641739
\(863\) 2.38217e13 1.46192 0.730961 0.682419i \(-0.239072\pi\)
0.730961 + 0.682419i \(0.239072\pi\)
\(864\) 5.06768e12 0.309384
\(865\) −6.34375e10 −0.00385278
\(866\) −4.40928e13 −2.66402
\(867\) −5.10563e12 −0.306876
\(868\) 3.72173e13 2.22539
\(869\) 5.05575e12 0.300744
\(870\) −1.29335e12 −0.0765383
\(871\) −2.88564e12 −0.169887
\(872\) 2.75478e13 1.61348
\(873\) 2.18084e13 1.27075
\(874\) 1.50453e13 0.872168
\(875\) 2.54215e12 0.146611
\(876\) −1.42249e13 −0.816173
\(877\) −1.35717e13 −0.774707 −0.387353 0.921931i \(-0.626611\pi\)
−0.387353 + 0.921931i \(0.626611\pi\)
\(878\) 6.79601e12 0.385948
\(879\) 1.08375e13 0.612323
\(880\) 4.06171e11 0.0228316
\(881\) 2.38436e12 0.133346 0.0666730 0.997775i \(-0.478762\pi\)
0.0666730 + 0.997775i \(0.478762\pi\)
\(882\) 3.30662e12 0.183982
\(883\) −5.99779e11 −0.0332023 −0.0166011 0.999862i \(-0.505285\pi\)
−0.0166011 + 0.999862i \(0.505285\pi\)
\(884\) 2.94999e12 0.162474
\(885\) 8.02792e11 0.0439904
\(886\) −2.50103e12 −0.136354
\(887\) 1.02483e13 0.555897 0.277949 0.960596i \(-0.410346\pi\)
0.277949 + 0.960596i \(0.410346\pi\)
\(888\) 3.27747e12 0.176881
\(889\) 1.11766e13 0.600138
\(890\) −1.45405e11 −0.00776828
\(891\) −6.48579e12 −0.344757
\(892\) 1.45042e13 0.767101
\(893\) −5.70040e12 −0.299967
\(894\) 2.33111e12 0.122052
\(895\) 1.71854e12 0.0895274
\(896\) −3.31065e13 −1.71604
\(897\) −5.95867e11 −0.0307315
\(898\) −3.04642e13 −1.56331
\(899\) 4.27050e13 2.18052
\(900\) 3.18944e13 1.62040
\(901\) 1.67378e12 0.0846130
\(902\) −1.43894e13 −0.723791
\(903\) 1.14411e13 0.572628
\(904\) −3.04937e13 −1.51863
\(905\) 2.39887e12 0.118874
\(906\) 7.95437e12 0.392219
\(907\) −2.43855e13 −1.19646 −0.598231 0.801323i \(-0.704130\pi\)
−0.598231 + 0.801323i \(0.704130\pi\)
\(908\) 3.97416e13 1.94026
\(909\) 1.57639e13 0.765820
\(910\) −7.11641e11 −0.0344013
\(911\) 2.90386e13 1.39683 0.698415 0.715693i \(-0.253889\pi\)
0.698415 + 0.715693i \(0.253889\pi\)
\(912\) −6.35855e12 −0.304356
\(913\) −1.59949e12 −0.0761841
\(914\) −3.40744e13 −1.61499
\(915\) −4.55272e11 −0.0214722
\(916\) −6.31867e13 −2.96549
\(917\) 2.05809e13 0.961173
\(918\) −7.40630e12 −0.344199
\(919\) −1.41793e13 −0.655746 −0.327873 0.944722i \(-0.606332\pi\)
−0.327873 + 0.944722i \(0.606332\pi\)
\(920\) 7.90158e11 0.0363638
\(921\) −1.06773e12 −0.0488984
\(922\) 8.87378e12 0.404408
\(923\) 1.17992e13 0.535111
\(924\) −6.78146e12 −0.306055
\(925\) −8.04299e12 −0.361226
\(926\) −5.89402e13 −2.63428
\(927\) 7.25212e12 0.322557
\(928\) −1.84087e13 −0.814812
\(929\) −3.56246e13 −1.56920 −0.784601 0.620001i \(-0.787132\pi\)
−0.784601 + 0.620001i \(0.787132\pi\)
\(930\) −1.32849e12 −0.0582349
\(931\) 4.50634e12 0.196585
\(932\) −4.90621e13 −2.12998
\(933\) −1.30136e13 −0.562250
\(934\) 6.23355e13 2.68025
\(935\) 3.02442e11 0.0129417
\(936\) −8.20568e12 −0.349441
\(937\) 9.35575e12 0.396506 0.198253 0.980151i \(-0.436473\pi\)
0.198253 + 0.980151i \(0.436473\pi\)
\(938\) −2.29353e13 −0.967366
\(939\) −3.78610e12 −0.158927
\(940\) −6.53510e11 −0.0273009
\(941\) 1.23413e13 0.513106 0.256553 0.966530i \(-0.417413\pi\)
0.256553 + 0.966530i \(0.417413\pi\)
\(942\) −2.57673e13 −1.06620
\(943\) −6.51742e12 −0.268394
\(944\) −2.24269e13 −0.919168
\(945\) 1.15875e12 0.0472656
\(946\) −3.86546e13 −1.56924
\(947\) 2.93722e12 0.118676 0.0593379 0.998238i \(-0.481101\pi\)
0.0593379 + 0.998238i \(0.481101\pi\)
\(948\) −9.07366e12 −0.364876
\(949\) −8.98110e12 −0.359444
\(950\) 6.70206e13 2.66964
\(951\) 9.72086e12 0.385383
\(952\) 1.07411e13 0.423819
\(953\) −3.90544e12 −0.153374 −0.0766870 0.997055i \(-0.524434\pi\)
−0.0766870 + 0.997055i \(0.524434\pi\)
\(954\) −1.01631e13 −0.397247
\(955\) −1.84163e12 −0.0716454
\(956\) 7.04733e13 2.72875
\(957\) −7.78137e12 −0.299883
\(958\) 5.89375e13 2.26072
\(959\) 3.00219e13 1.14618
\(960\) 9.67687e11 0.0367718
\(961\) 1.74255e13 0.659069
\(962\) 4.51703e12 0.170045
\(963\) −1.16802e13 −0.437655
\(964\) 5.43139e13 2.02565
\(965\) 2.96398e12 0.110028
\(966\) −4.73599e12 −0.174990
\(967\) −1.06697e13 −0.392405 −0.196203 0.980563i \(-0.562861\pi\)
−0.196203 + 0.980563i \(0.562861\pi\)
\(968\) −2.84587e13 −1.04178
\(969\) −4.73469e12 −0.172518
\(970\) 5.25371e12 0.190543
\(971\) −3.10264e12 −0.112007 −0.0560034 0.998431i \(-0.517836\pi\)
−0.0560034 + 0.998431i \(0.517836\pi\)
\(972\) 4.46506e13 1.60446
\(973\) 2.13939e13 0.765213
\(974\) 9.74972e12 0.347118
\(975\) −2.65434e12 −0.0940665
\(976\) 1.27185e13 0.448655
\(977\) −4.66167e13 −1.63688 −0.818438 0.574594i \(-0.805160\pi\)
−0.818438 + 0.574594i \(0.805160\pi\)
\(978\) 1.32455e12 0.0462961
\(979\) −8.74824e11 −0.0304368
\(980\) 5.16619e11 0.0178918
\(981\) −2.89986e13 −0.999694
\(982\) 6.04614e13 2.07480
\(983\) −3.22316e13 −1.10101 −0.550505 0.834832i \(-0.685565\pi\)
−0.550505 + 0.834832i \(0.685565\pi\)
\(984\) 1.18305e13 0.402278
\(985\) −1.73728e12 −0.0588040
\(986\) 2.69039e13 0.906502
\(987\) 1.79438e12 0.0601849
\(988\) −2.44112e13 −0.815047
\(989\) −1.75079e13 −0.581903
\(990\) −1.83642e12 −0.0607594
\(991\) −2.25008e13 −0.741081 −0.370540 0.928816i \(-0.620828\pi\)
−0.370540 + 0.928816i \(0.620828\pi\)
\(992\) −1.89088e13 −0.619957
\(993\) 1.55192e13 0.506520
\(994\) 9.37807e13 3.04701
\(995\) −9.66342e11 −0.0312556
\(996\) 2.87064e12 0.0924299
\(997\) −3.13756e13 −1.00569 −0.502844 0.864377i \(-0.667713\pi\)
−0.502844 + 0.864377i \(0.667713\pi\)
\(998\) −2.40442e13 −0.767224
\(999\) −7.35495e12 −0.233633
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 13.10.a.b.1.5 5
3.2 odd 2 117.10.a.e.1.1 5
4.3 odd 2 208.10.a.h.1.3 5
5.4 even 2 325.10.a.b.1.1 5
13.12 even 2 169.10.a.b.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.b.1.5 5 1.1 even 1 trivial
117.10.a.e.1.1 5 3.2 odd 2
169.10.a.b.1.1 5 13.12 even 2
208.10.a.h.1.3 5 4.3 odd 2
325.10.a.b.1.1 5 5.4 even 2