Properties

Label 117.10.a.e.1.1
Level $117$
Weight $10$
Character 117.1
Self dual yes
Analytic conductor $60.259$
Analytic rank $1$
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] = [117,10,Mod(1,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, 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("117.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 117.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: \(60.2591928312\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(35.1685\) of defining polynomial
Character \(\chi\) \(=\) 117.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-38.1685 q^{2} +944.833 q^{4} +109.762 q^{5} +5947.44 q^{7} -16520.6 q^{8} +O(q^{10})\) \(q-38.1685 q^{2} +944.833 q^{4} +109.762 q^{5} +5947.44 q^{7} -16520.6 q^{8} -4189.45 q^{10} +25205.7 q^{11} +28561.0 q^{13} -227005. q^{14} +146811. q^{16} -109318. q^{17} -904609. q^{19} +103707. q^{20} -962062. q^{22} +435749. q^{23} -1.94108e6 q^{25} -1.09013e6 q^{26} +5.61934e6 q^{28} -6.44791e6 q^{29} +6.62308e6 q^{31} +2.85499e6 q^{32} +4.17250e6 q^{34} +652804. q^{35} +4.14357e6 q^{37} +3.45275e7 q^{38} -1.81333e6 q^{40} -1.49568e7 q^{41} +4.01789e7 q^{43} +2.38151e7 q^{44} -1.66319e7 q^{46} -6.30151e6 q^{47} -4.98153e6 q^{49} +7.40880e7 q^{50} +2.69854e7 q^{52} -1.53111e7 q^{53} +2.76663e6 q^{55} -9.82552e7 q^{56} +2.46107e8 q^{58} +1.52760e8 q^{59} +8.66321e7 q^{61} -2.52793e8 q^{62} -1.84138e8 q^{64} +3.13492e6 q^{65} -1.01034e8 q^{67} -1.03287e8 q^{68} -2.49165e7 q^{70} -4.13122e8 q^{71} -3.14453e8 q^{73} -1.58154e8 q^{74} -8.54704e8 q^{76} +1.49909e8 q^{77} -2.00580e8 q^{79} +1.61143e7 q^{80} +5.70879e8 q^{82} -6.34578e7 q^{83} -1.19990e7 q^{85} -1.53357e9 q^{86} -4.16412e8 q^{88} -3.47074e7 q^{89} +1.69865e8 q^{91} +4.11710e8 q^{92} +2.40519e8 q^{94} -9.92918e7 q^{95} -1.25403e9 q^{97} +1.90137e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8} + 84505 q^{10} - 121746 q^{11} + 142805 q^{13} - 8475 q^{14} - 322463 q^{16} + 495669 q^{17} - 840738 q^{19} + 1595607 q^{20} - 2023594 q^{22} + 592152 q^{23} + 1670362 q^{25} - 428415 q^{26} + 2587955 q^{28} - 10678182 q^{29} + 12885296 q^{31} - 3282927 q^{32} - 9934079 q^{34} - 8380731 q^{35} + 7171823 q^{37} + 25568814 q^{38} - 54359445 q^{40} - 9294012 q^{41} + 12831975 q^{43} + 41479074 q^{44} - 59319696 q^{46} - 43354215 q^{47} + 25249488 q^{49} + 16270770 q^{50} + 10310521 q^{52} - 93231780 q^{53} + 99448846 q^{55} - 199599225 q^{56} + 151020970 q^{58} - 246496182 q^{59} - 132232612 q^{61} - 158135724 q^{62} + 91019105 q^{64} - 51495483 q^{65} - 369388534 q^{67} - 238172073 q^{68} - 144857425 q^{70} - 212150457 q^{71} - 252729806 q^{73} - 192105957 q^{74} - 953775990 q^{76} - 449666118 q^{77} - 1247271728 q^{79} - 900649725 q^{80} + 169559388 q^{82} - 1696894296 q^{83} - 775363765 q^{85} - 3291621459 q^{86} - 220227222 q^{88} + 753854382 q^{89} + 288437539 q^{91} - 13876128 q^{92} + 272071215 q^{94} - 1442632962 q^{95} + 3824606 q^{97} - 1570614816 q^{98}+O(q^{100}) \) 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 \(-0.819457\pi\)
−0.843412 + 0.537267i \(0.819457\pi\)
\(3\) 0 0
\(4\) 944.833 1.84538
\(5\) 109.762 0.0785394 0.0392697 0.999229i \(-0.487497\pi\)
0.0392697 + 0.999229i \(0.487497\pi\)
\(6\) 0 0
\(7\) 5947.44 0.936244 0.468122 0.883664i \(-0.344931\pi\)
0.468122 + 0.883664i \(0.344931\pi\)
\(8\) −16520.6 −1.42600
\(9\) 0 0
\(10\) −4189.45 −0.132482
\(11\) 25205.7 0.519076 0.259538 0.965733i \(-0.416430\pi\)
0.259538 + 0.965733i \(0.416430\pi\)
\(12\) 0 0
\(13\) 28561.0 0.277350
\(14\) −227005. −1.57928
\(15\) 0 0
\(16\) 146811. 0.560039
\(17\) −109318. −0.317447 −0.158724 0.987323i \(-0.550738\pi\)
−0.158724 + 0.987323i \(0.550738\pi\)
\(18\) 0 0
\(19\) −904609. −1.59246 −0.796232 0.604992i \(-0.793176\pi\)
−0.796232 + 0.604992i \(0.793176\pi\)
\(20\) 103707. 0.144935
\(21\) 0 0
\(22\) −962062. −0.875590
\(23\) 435749. 0.324684 0.162342 0.986735i \(-0.448095\pi\)
0.162342 + 0.986735i \(0.448095\pi\)
\(24\) 0 0
\(25\) −1.94108e6 −0.993832
\(26\) −1.09013e6 −0.467841
\(27\) 0 0
\(28\) 5.61934e6 1.72772
\(29\) −6.44791e6 −1.69289 −0.846443 0.532479i \(-0.821260\pi\)
−0.846443 + 0.532479i \(0.821260\pi\)
\(30\) 0 0
\(31\) 6.62308e6 1.28805 0.644024 0.765005i \(-0.277264\pi\)
0.644024 + 0.765005i \(0.277264\pi\)
\(32\) 2.85499e6 0.481315
\(33\) 0 0
\(34\) 4.17250e6 0.535477
\(35\) 652804. 0.0735320
\(36\) 0 0
\(37\) 4.14357e6 0.363469 0.181734 0.983348i \(-0.441829\pi\)
0.181734 + 0.983348i \(0.441829\pi\)
\(38\) 3.45275e7 2.68621
\(39\) 0 0
\(40\) −1.81333e6 −0.111997
\(41\) −1.49568e7 −0.826632 −0.413316 0.910588i \(-0.635630\pi\)
−0.413316 + 0.910588i \(0.635630\pi\)
\(42\) 0 0
\(43\) 4.01789e7 1.79221 0.896107 0.443838i \(-0.146384\pi\)
0.896107 + 0.443838i \(0.146384\pi\)
\(44\) 2.38151e7 0.957891
\(45\) 0 0
\(46\) −1.66319e7 −0.547685
\(47\) −6.30151e6 −0.188367 −0.0941834 0.995555i \(-0.530024\pi\)
−0.0941834 + 0.995555i \(0.530024\pi\)
\(48\) 0 0
\(49\) −4.98153e6 −0.123447
\(50\) 7.40880e7 1.67642
\(51\) 0 0
\(52\) 2.69854e7 0.511815
\(53\) −1.53111e7 −0.266542 −0.133271 0.991080i \(-0.542548\pi\)
−0.133271 + 0.991080i \(0.542548\pi\)
\(54\) 0 0
\(55\) 2.76663e6 0.0407679
\(56\) −9.82552e7 −1.33509
\(57\) 0 0
\(58\) 2.46107e8 2.85560
\(59\) 1.52760e8 1.64126 0.820629 0.571462i \(-0.193624\pi\)
0.820629 + 0.571462i \(0.193624\pi\)
\(60\) 0 0
\(61\) 8.66321e7 0.801114 0.400557 0.916272i \(-0.368817\pi\)
0.400557 + 0.916272i \(0.368817\pi\)
\(62\) −2.52793e8 −2.17271
\(63\) 0 0
\(64\) −1.84138e8 −1.37193
\(65\) 3.13492e6 0.0217829
\(66\) 0 0
\(67\) −1.01034e8 −0.612537 −0.306268 0.951945i \(-0.599080\pi\)
−0.306268 + 0.951945i \(0.599080\pi\)
\(68\) −1.03287e8 −0.585809
\(69\) 0 0
\(70\) −2.49165e7 −0.124036
\(71\) −4.13122e8 −1.92937 −0.964685 0.263406i \(-0.915154\pi\)
−0.964685 + 0.263406i \(0.915154\pi\)
\(72\) 0 0
\(73\) −3.14453e8 −1.29599 −0.647997 0.761643i \(-0.724393\pi\)
−0.647997 + 0.761643i \(0.724393\pi\)
\(74\) −1.58154e8 −0.613108
\(75\) 0 0
\(76\) −8.54704e8 −2.93870
\(77\) 1.49909e8 0.485982
\(78\) 0 0
\(79\) −2.00580e8 −0.579383 −0.289692 0.957120i \(-0.593553\pi\)
−0.289692 + 0.957120i \(0.593553\pi\)
\(80\) 1.61143e7 0.0439851
\(81\) 0 0
\(82\) 5.70879e8 1.39438
\(83\) −6.34578e7 −0.146769 −0.0733843 0.997304i \(-0.523380\pi\)
−0.0733843 + 0.997304i \(0.523380\pi\)
\(84\) 0 0
\(85\) −1.19990e7 −0.0249321
\(86\) −1.53357e9 −3.02315
\(87\) 0 0
\(88\) −4.16412e8 −0.740204
\(89\) −3.47074e7 −0.0586364 −0.0293182 0.999570i \(-0.509334\pi\)
−0.0293182 + 0.999570i \(0.509334\pi\)
\(90\) 0 0
\(91\) 1.69865e8 0.259667
\(92\) 4.11710e8 0.599164
\(93\) 0 0
\(94\) 2.40519e8 0.317742
\(95\) −9.92918e7 −0.125071
\(96\) 0 0
\(97\) −1.25403e9 −1.43825 −0.719127 0.694879i \(-0.755458\pi\)
−0.719127 + 0.694879i \(0.755458\pi\)
\(98\) 1.90137e8 0.208233
\(99\) 0 0
\(100\) −1.83399e9 −1.83399
\(101\) 9.06459e8 0.866766 0.433383 0.901210i \(-0.357320\pi\)
0.433383 + 0.901210i \(0.357320\pi\)
\(102\) 0 0
\(103\) −4.17013e8 −0.365075 −0.182537 0.983199i \(-0.558431\pi\)
−0.182537 + 0.983199i \(0.558431\pi\)
\(104\) −4.71844e8 −0.395502
\(105\) 0 0
\(106\) 5.84402e8 0.449610
\(107\) −6.71636e8 −0.495344 −0.247672 0.968844i \(-0.579666\pi\)
−0.247672 + 0.968844i \(0.579666\pi\)
\(108\) 0 0
\(109\) 1.66748e9 1.13147 0.565734 0.824588i \(-0.308593\pi\)
0.565734 + 0.824588i \(0.308593\pi\)
\(110\) −1.05598e8 −0.0687683
\(111\) 0 0
\(112\) 8.73149e8 0.524333
\(113\) 1.84580e9 1.06496 0.532478 0.846444i \(-0.321261\pi\)
0.532478 + 0.846444i \(0.321261\pi\)
\(114\) 0 0
\(115\) 4.78287e7 0.0255005
\(116\) −6.09219e9 −3.12401
\(117\) 0 0
\(118\) −5.83063e9 −2.76851
\(119\) −6.50162e8 −0.297208
\(120\) 0 0
\(121\) −1.72262e9 −0.730560
\(122\) −3.30661e9 −1.35134
\(123\) 0 0
\(124\) 6.25770e9 2.37694
\(125\) −4.27436e8 −0.156594
\(126\) 0 0
\(127\) 1.87922e9 0.641006 0.320503 0.947248i \(-0.396148\pi\)
0.320503 + 0.947248i \(0.396148\pi\)
\(128\) 5.56650e9 1.83290
\(129\) 0 0
\(130\) −1.19655e8 −0.0367439
\(131\) −3.46045e9 −1.02663 −0.513313 0.858201i \(-0.671582\pi\)
−0.513313 + 0.858201i \(0.671582\pi\)
\(132\) 0 0
\(133\) −5.38011e9 −1.49093
\(134\) 3.85632e9 1.03324
\(135\) 0 0
\(136\) 1.80600e9 0.452680
\(137\) −5.04786e9 −1.22424 −0.612118 0.790766i \(-0.709682\pi\)
−0.612118 + 0.790766i \(0.709682\pi\)
\(138\) 0 0
\(139\) 3.59716e9 0.817322 0.408661 0.912686i \(-0.365996\pi\)
0.408661 + 0.912686i \(0.365996\pi\)
\(140\) 6.16791e8 0.135694
\(141\) 0 0
\(142\) 1.57682e10 3.25451
\(143\) 7.19899e8 0.143966
\(144\) 0 0
\(145\) −7.07736e8 −0.132958
\(146\) 1.20022e10 2.18611
\(147\) 0 0
\(148\) 3.91498e9 0.670736
\(149\) −1.27561e9 −0.212022 −0.106011 0.994365i \(-0.533808\pi\)
−0.106011 + 0.994365i \(0.533808\pi\)
\(150\) 0 0
\(151\) 4.35273e9 0.681342 0.340671 0.940183i \(-0.389346\pi\)
0.340671 + 0.940183i \(0.389346\pi\)
\(152\) 1.49447e10 2.27086
\(153\) 0 0
\(154\) −5.72181e9 −0.819766
\(155\) 7.26963e8 0.101163
\(156\) 0 0
\(157\) −1.41002e10 −1.85215 −0.926075 0.377339i \(-0.876839\pi\)
−0.926075 + 0.377339i \(0.876839\pi\)
\(158\) 7.65584e9 0.977318
\(159\) 0 0
\(160\) 3.13370e8 0.0378022
\(161\) 2.59159e9 0.303983
\(162\) 0 0
\(163\) 7.24812e8 0.0804231 0.0402116 0.999191i \(-0.487197\pi\)
0.0402116 + 0.999191i \(0.487197\pi\)
\(164\) −1.41317e10 −1.52545
\(165\) 0 0
\(166\) 2.42209e9 0.247573
\(167\) 8.33031e8 0.0828776 0.0414388 0.999141i \(-0.486806\pi\)
0.0414388 + 0.999141i \(0.486806\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) 4.57983e8 0.0420561
\(171\) 0 0
\(172\) 3.79623e10 3.30731
\(173\) −5.77955e8 −0.0490553 −0.0245277 0.999699i \(-0.507808\pi\)
−0.0245277 + 0.999699i \(0.507808\pi\)
\(174\) 0 0
\(175\) −1.15444e10 −0.930469
\(176\) 3.70046e9 0.290703
\(177\) 0 0
\(178\) 1.32473e9 0.0989094
\(179\) 1.56569e10 1.13990 0.569952 0.821678i \(-0.306962\pi\)
0.569952 + 0.821678i \(0.306962\pi\)
\(180\) 0 0
\(181\) −2.18552e10 −1.51356 −0.756781 0.653668i \(-0.773229\pi\)
−0.756781 + 0.653668i \(0.773229\pi\)
\(182\) −6.48349e9 −0.438013
\(183\) 0 0
\(184\) −7.19882e9 −0.463000
\(185\) 4.54807e8 0.0285466
\(186\) 0 0
\(187\) −2.75543e9 −0.164779
\(188\) −5.95388e9 −0.347608
\(189\) 0 0
\(190\) 3.78982e9 0.210973
\(191\) −1.67784e10 −0.912222 −0.456111 0.889923i \(-0.650758\pi\)
−0.456111 + 0.889923i \(0.650758\pi\)
\(192\) 0 0
\(193\) −2.70036e10 −1.40092 −0.700462 0.713690i \(-0.747023\pi\)
−0.700462 + 0.713690i \(0.747023\pi\)
\(194\) 4.78645e10 2.42608
\(195\) 0 0
\(196\) −4.70671e9 −0.227806
\(197\) −1.58277e10 −0.748720 −0.374360 0.927283i \(-0.622138\pi\)
−0.374360 + 0.927283i \(0.622138\pi\)
\(198\) 0 0
\(199\) 8.80397e9 0.397960 0.198980 0.980004i \(-0.436237\pi\)
0.198980 + 0.980004i \(0.436237\pi\)
\(200\) 3.20677e10 1.41721
\(201\) 0 0
\(202\) −3.45982e10 −1.46208
\(203\) −3.83485e10 −1.58495
\(204\) 0 0
\(205\) −1.64169e9 −0.0649232
\(206\) 1.59167e10 0.615817
\(207\) 0 0
\(208\) 4.19306e9 0.155327
\(209\) −2.28013e10 −0.826610
\(210\) 0 0
\(211\) −1.82054e10 −0.632308 −0.316154 0.948708i \(-0.602392\pi\)
−0.316154 + 0.948708i \(0.602392\pi\)
\(212\) −1.44665e10 −0.491870
\(213\) 0 0
\(214\) 2.56353e10 0.835558
\(215\) 4.41012e9 0.140759
\(216\) 0 0
\(217\) 3.93904e10 1.20593
\(218\) −6.36453e10 −1.90859
\(219\) 0 0
\(220\) 2.61400e9 0.0752322
\(221\) −3.12223e9 −0.0880440
\(222\) 0 0
\(223\) 1.53511e10 0.415688 0.207844 0.978162i \(-0.433355\pi\)
0.207844 + 0.978162i \(0.433355\pi\)
\(224\) 1.69799e10 0.450628
\(225\) 0 0
\(226\) −7.04514e10 −1.79639
\(227\) −4.20620e10 −1.05141 −0.525707 0.850666i \(-0.676199\pi\)
−0.525707 + 0.850666i \(0.676199\pi\)
\(228\) 0 0
\(229\) −6.68760e10 −1.60698 −0.803490 0.595318i \(-0.797026\pi\)
−0.803490 + 0.595318i \(0.797026\pi\)
\(230\) −1.82555e9 −0.0430148
\(231\) 0 0
\(232\) 1.06523e11 2.41406
\(233\) 5.19268e10 1.15422 0.577112 0.816665i \(-0.304180\pi\)
0.577112 + 0.816665i \(0.304180\pi\)
\(234\) 0 0
\(235\) −6.91668e8 −0.0147942
\(236\) 1.44333e11 3.02874
\(237\) 0 0
\(238\) 2.48157e10 0.501338
\(239\) −7.45881e10 −1.47870 −0.739348 0.673323i \(-0.764866\pi\)
−0.739348 + 0.673323i \(0.764866\pi\)
\(240\) 0 0
\(241\) 5.74852e10 1.09769 0.548845 0.835924i \(-0.315068\pi\)
0.548845 + 0.835924i \(0.315068\pi\)
\(242\) 6.57499e10 1.23233
\(243\) 0 0
\(244\) 8.18528e10 1.47836
\(245\) −5.46784e8 −0.00969545
\(246\) 0 0
\(247\) −2.58365e10 −0.441670
\(248\) −1.09417e11 −1.83676
\(249\) 0 0
\(250\) 1.63146e10 0.264147
\(251\) 1.07873e11 1.71546 0.857732 0.514097i \(-0.171873\pi\)
0.857732 + 0.514097i \(0.171873\pi\)
\(252\) 0 0
\(253\) 1.09833e10 0.168536
\(254\) −7.17272e10 −1.08126
\(255\) 0 0
\(256\) −1.18186e11 −1.71984
\(257\) 6.64074e10 0.949550 0.474775 0.880107i \(-0.342530\pi\)
0.474775 + 0.880107i \(0.342530\pi\)
\(258\) 0 0
\(259\) 2.46436e10 0.340295
\(260\) 2.96197e9 0.0401977
\(261\) 0 0
\(262\) 1.32080e11 1.73174
\(263\) 8.15356e10 1.05086 0.525432 0.850836i \(-0.323904\pi\)
0.525432 + 0.850836i \(0.323904\pi\)
\(264\) 0 0
\(265\) −1.68058e9 −0.0209340
\(266\) 2.05351e11 2.51494
\(267\) 0 0
\(268\) −9.54605e10 −1.13036
\(269\) −1.00568e11 −1.17105 −0.585523 0.810656i \(-0.699111\pi\)
−0.585523 + 0.810656i \(0.699111\pi\)
\(270\) 0 0
\(271\) −2.98757e10 −0.336477 −0.168239 0.985746i \(-0.553808\pi\)
−0.168239 + 0.985746i \(0.553808\pi\)
\(272\) −1.60491e10 −0.177783
\(273\) 0 0
\(274\) 1.92669e11 2.06507
\(275\) −4.89261e10 −0.515874
\(276\) 0 0
\(277\) −3.17195e10 −0.323718 −0.161859 0.986814i \(-0.551749\pi\)
−0.161859 + 0.986814i \(0.551749\pi\)
\(278\) −1.37298e11 −1.37868
\(279\) 0 0
\(280\) −1.07847e10 −0.104857
\(281\) −9.86953e10 −0.944318 −0.472159 0.881513i \(-0.656525\pi\)
−0.472159 + 0.881513i \(0.656525\pi\)
\(282\) 0 0
\(283\) −1.06164e11 −0.983871 −0.491935 0.870632i \(-0.663710\pi\)
−0.491935 + 0.870632i \(0.663710\pi\)
\(284\) −3.90331e11 −3.56042
\(285\) 0 0
\(286\) −2.74774e10 −0.242845
\(287\) −8.89549e10 −0.773929
\(288\) 0 0
\(289\) −1.06637e11 −0.899227
\(290\) 2.70132e10 0.224277
\(291\) 0 0
\(292\) −2.97106e11 −2.39160
\(293\) −2.26355e11 −1.79426 −0.897132 0.441763i \(-0.854353\pi\)
−0.897132 + 0.441763i \(0.854353\pi\)
\(294\) 0 0
\(295\) 1.67673e10 0.128903
\(296\) −6.84541e10 −0.518307
\(297\) 0 0
\(298\) 4.86881e10 0.357643
\(299\) 1.24454e10 0.0900511
\(300\) 0 0
\(301\) 2.38962e11 1.67795
\(302\) −1.66137e11 −1.14930
\(303\) 0 0
\(304\) −1.32806e11 −0.891841
\(305\) 9.50892e9 0.0629190
\(306\) 0 0
\(307\) −2.23009e10 −0.143285 −0.0716424 0.997430i \(-0.522824\pi\)
−0.0716424 + 0.997430i \(0.522824\pi\)
\(308\) 1.41639e11 0.896820
\(309\) 0 0
\(310\) −2.77471e10 −0.170643
\(311\) 2.71805e11 1.64754 0.823770 0.566925i \(-0.191867\pi\)
0.823770 + 0.566925i \(0.191867\pi\)
\(312\) 0 0
\(313\) −7.90775e10 −0.465697 −0.232849 0.972513i \(-0.574805\pi\)
−0.232849 + 0.972513i \(0.574805\pi\)
\(314\) 5.38183e11 3.12425
\(315\) 0 0
\(316\) −1.89515e11 −1.06918
\(317\) −2.03032e11 −1.12927 −0.564636 0.825340i \(-0.690983\pi\)
−0.564636 + 0.825340i \(0.690983\pi\)
\(318\) 0 0
\(319\) −1.62524e11 −0.878736
\(320\) −2.02114e10 −0.107751
\(321\) 0 0
\(322\) −9.89171e10 −0.512767
\(323\) 9.88899e10 0.505523
\(324\) 0 0
\(325\) −5.54391e10 −0.275639
\(326\) −2.76650e10 −0.135660
\(327\) 0 0
\(328\) 2.47095e11 1.17878
\(329\) −3.74779e10 −0.176357
\(330\) 0 0
\(331\) 3.24137e11 1.48424 0.742118 0.670270i \(-0.233822\pi\)
0.742118 + 0.670270i \(0.233822\pi\)
\(332\) −5.99570e10 −0.270844
\(333\) 0 0
\(334\) −3.17955e10 −0.139800
\(335\) −1.10897e10 −0.0481083
\(336\) 0 0
\(337\) 2.68510e11 1.13403 0.567017 0.823706i \(-0.308097\pi\)
0.567017 + 0.823706i \(0.308097\pi\)
\(338\) −3.11352e10 −0.129756
\(339\) 0 0
\(340\) −1.13370e10 −0.0460091
\(341\) 1.66939e11 0.668595
\(342\) 0 0
\(343\) −2.69628e11 −1.05182
\(344\) −6.63778e11 −2.55570
\(345\) 0 0
\(346\) 2.20597e10 0.0827478
\(347\) −2.74715e11 −1.01719 −0.508593 0.861007i \(-0.669834\pi\)
−0.508593 + 0.861007i \(0.669834\pi\)
\(348\) 0 0
\(349\) −3.54006e11 −1.27731 −0.638655 0.769494i \(-0.720509\pi\)
−0.638655 + 0.769494i \(0.720509\pi\)
\(350\) 4.40634e11 1.56954
\(351\) 0 0
\(352\) 7.19619e10 0.249839
\(353\) 2.25650e11 0.773481 0.386741 0.922189i \(-0.373601\pi\)
0.386741 + 0.922189i \(0.373601\pi\)
\(354\) 0 0
\(355\) −4.53451e10 −0.151532
\(356\) −3.27927e10 −0.108206
\(357\) 0 0
\(358\) −5.97602e11 −1.92282
\(359\) −3.39022e10 −0.107722 −0.0538608 0.998548i \(-0.517153\pi\)
−0.0538608 + 0.998548i \(0.517153\pi\)
\(360\) 0 0
\(361\) 4.95629e11 1.53594
\(362\) 8.34178e11 2.55312
\(363\) 0 0
\(364\) 1.60494e11 0.479184
\(365\) −3.45150e10 −0.101787
\(366\) 0 0
\(367\) −7.74387e10 −0.222823 −0.111412 0.993774i \(-0.535537\pi\)
−0.111412 + 0.993774i \(0.535537\pi\)
\(368\) 6.39726e10 0.181836
\(369\) 0 0
\(370\) −1.73593e10 −0.0481531
\(371\) −9.10620e10 −0.249548
\(372\) 0 0
\(373\) 6.50821e11 1.74089 0.870446 0.492264i \(-0.163831\pi\)
0.870446 + 0.492264i \(0.163831\pi\)
\(374\) 1.05171e11 0.277954
\(375\) 0 0
\(376\) 1.04105e11 0.268612
\(377\) −1.84159e11 −0.469522
\(378\) 0 0
\(379\) 1.93989e11 0.482948 0.241474 0.970407i \(-0.422369\pi\)
0.241474 + 0.970407i \(0.422369\pi\)
\(380\) −9.38141e10 −0.230803
\(381\) 0 0
\(382\) 6.40406e11 1.53876
\(383\) −4.44645e11 −1.05589 −0.527946 0.849278i \(-0.677037\pi\)
−0.527946 + 0.849278i \(0.677037\pi\)
\(384\) 0 0
\(385\) 1.64544e10 0.0381687
\(386\) 1.03069e12 2.36311
\(387\) 0 0
\(388\) −1.18485e12 −2.65412
\(389\) 1.69623e11 0.375589 0.187794 0.982208i \(-0.439866\pi\)
0.187794 + 0.982208i \(0.439866\pi\)
\(390\) 0 0
\(391\) −4.76352e10 −0.103070
\(392\) 8.22978e10 0.176036
\(393\) 0 0
\(394\) 6.04119e11 1.26296
\(395\) −2.20161e10 −0.0455044
\(396\) 0 0
\(397\) 2.94468e11 0.594951 0.297476 0.954729i \(-0.403855\pi\)
0.297476 + 0.954729i \(0.403855\pi\)
\(398\) −3.36034e11 −0.671289
\(399\) 0 0
\(400\) −2.84971e11 −0.556584
\(401\) −4.43362e11 −0.856265 −0.428133 0.903716i \(-0.640828\pi\)
−0.428133 + 0.903716i \(0.640828\pi\)
\(402\) 0 0
\(403\) 1.89162e11 0.357240
\(404\) 8.56452e11 1.59951
\(405\) 0 0
\(406\) 1.46371e12 2.67354
\(407\) 1.04441e11 0.188668
\(408\) 0 0
\(409\) −1.01253e11 −0.178918 −0.0894592 0.995990i \(-0.528514\pi\)
−0.0894592 + 0.995990i \(0.528514\pi\)
\(410\) 6.26610e10 0.109514
\(411\) 0 0
\(412\) −3.94007e11 −0.673700
\(413\) 9.08533e11 1.53662
\(414\) 0 0
\(415\) −6.96526e9 −0.0115271
\(416\) 8.15413e10 0.133493
\(417\) 0 0
\(418\) 8.70289e11 1.39435
\(419\) −7.93982e10 −0.125848 −0.0629242 0.998018i \(-0.520043\pi\)
−0.0629242 + 0.998018i \(0.520043\pi\)
\(420\) 0 0
\(421\) −6.06765e11 −0.941350 −0.470675 0.882307i \(-0.655990\pi\)
−0.470675 + 0.882307i \(0.655990\pi\)
\(422\) 6.94871e11 1.06659
\(423\) 0 0
\(424\) 2.52949e11 0.380089
\(425\) 2.12195e11 0.315489
\(426\) 0 0
\(427\) 5.15239e11 0.750038
\(428\) −6.34583e11 −0.914096
\(429\) 0 0
\(430\) −1.68328e11 −0.237436
\(431\) 2.72544e11 0.380442 0.190221 0.981741i \(-0.439080\pi\)
0.190221 + 0.981741i \(0.439080\pi\)
\(432\) 0 0
\(433\) −1.15522e12 −1.57931 −0.789655 0.613551i \(-0.789740\pi\)
−0.789655 + 0.613551i \(0.789740\pi\)
\(434\) −1.50347e12 −2.03419
\(435\) 0 0
\(436\) 1.57549e12 2.08799
\(437\) −3.94182e11 −0.517047
\(438\) 0 0
\(439\) 1.78053e11 0.228801 0.114401 0.993435i \(-0.463505\pi\)
0.114401 + 0.993435i \(0.463505\pi\)
\(440\) −4.57063e10 −0.0581351
\(441\) 0 0
\(442\) 1.19171e11 0.148515
\(443\) 6.55260e10 0.0808345 0.0404173 0.999183i \(-0.487131\pi\)
0.0404173 + 0.999183i \(0.487131\pi\)
\(444\) 0 0
\(445\) −3.80956e9 −0.00460527
\(446\) −5.85928e11 −0.701192
\(447\) 0 0
\(448\) −1.09515e12 −1.28446
\(449\) 7.98150e11 0.926779 0.463390 0.886155i \(-0.346633\pi\)
0.463390 + 0.886155i \(0.346633\pi\)
\(450\) 0 0
\(451\) −3.76997e11 −0.429085
\(452\) 1.74397e12 1.96525
\(453\) 0 0
\(454\) 1.60544e12 1.77355
\(455\) 1.86447e10 0.0203941
\(456\) 0 0
\(457\) −8.92736e11 −0.957415 −0.478708 0.877974i \(-0.658895\pi\)
−0.478708 + 0.877974i \(0.658895\pi\)
\(458\) 2.55256e12 2.71069
\(459\) 0 0
\(460\) 4.51901e10 0.0470580
\(461\) −2.32490e11 −0.239745 −0.119872 0.992789i \(-0.538249\pi\)
−0.119872 + 0.992789i \(0.538249\pi\)
\(462\) 0 0
\(463\) −1.54421e12 −1.56168 −0.780841 0.624730i \(-0.785209\pi\)
−0.780841 + 0.624730i \(0.785209\pi\)
\(464\) −9.46622e11 −0.948082
\(465\) 0 0
\(466\) −1.98197e12 −1.94697
\(467\) −1.63317e12 −1.58893 −0.794465 0.607310i \(-0.792249\pi\)
−0.794465 + 0.607310i \(0.792249\pi\)
\(468\) 0 0
\(469\) −6.00895e11 −0.573484
\(470\) 2.63999e10 0.0249553
\(471\) 0 0
\(472\) −2.52369e12 −2.34044
\(473\) 1.01274e12 0.930295
\(474\) 0 0
\(475\) 1.75592e12 1.58264
\(476\) −6.14295e11 −0.548461
\(477\) 0 0
\(478\) 2.84692e12 2.49430
\(479\) −1.54414e12 −1.34022 −0.670112 0.742260i \(-0.733754\pi\)
−0.670112 + 0.742260i \(0.733754\pi\)
\(480\) 0 0
\(481\) 1.18344e11 0.100808
\(482\) −2.19412e12 −1.85161
\(483\) 0 0
\(484\) −1.62759e12 −1.34816
\(485\) −1.37645e11 −0.112960
\(486\) 0 0
\(487\) 2.55439e11 0.205782 0.102891 0.994693i \(-0.467191\pi\)
0.102891 + 0.994693i \(0.467191\pi\)
\(488\) −1.43121e12 −1.14239
\(489\) 0 0
\(490\) 2.08699e10 0.0163545
\(491\) −1.58407e12 −1.23000 −0.615002 0.788526i \(-0.710845\pi\)
−0.615002 + 0.788526i \(0.710845\pi\)
\(492\) 0 0
\(493\) 7.04872e11 0.537402
\(494\) 9.86141e11 0.745020
\(495\) 0 0
\(496\) 9.72340e11 0.721357
\(497\) −2.45702e12 −1.80636
\(498\) 0 0
\(499\) −6.29948e11 −0.454833 −0.227417 0.973798i \(-0.573028\pi\)
−0.227417 + 0.973798i \(0.573028\pi\)
\(500\) −4.03856e11 −0.288976
\(501\) 0 0
\(502\) −4.11736e12 −2.89369
\(503\) −5.49263e11 −0.382582 −0.191291 0.981533i \(-0.561267\pi\)
−0.191291 + 0.981533i \(0.561267\pi\)
\(504\) 0 0
\(505\) 9.94949e10 0.0680753
\(506\) −4.19217e11 −0.284290
\(507\) 0 0
\(508\) 1.77555e12 1.18290
\(509\) −1.36923e11 −0.0904165 −0.0452083 0.998978i \(-0.514395\pi\)
−0.0452083 + 0.998978i \(0.514395\pi\)
\(510\) 0 0
\(511\) −1.87019e12 −1.21337
\(512\) 1.66095e12 1.06817
\(513\) 0 0
\(514\) −2.53467e12 −1.60172
\(515\) −4.57722e10 −0.0286727
\(516\) 0 0
\(517\) −1.58834e11 −0.0977767
\(518\) −9.40610e11 −0.574018
\(519\) 0 0
\(520\) −5.17906e10 −0.0310625
\(521\) 1.66627e12 0.990779 0.495389 0.868671i \(-0.335025\pi\)
0.495389 + 0.868671i \(0.335025\pi\)
\(522\) 0 0
\(523\) 1.97187e12 1.15244 0.576222 0.817293i \(-0.304526\pi\)
0.576222 + 0.817293i \(0.304526\pi\)
\(524\) −3.26955e12 −1.89451
\(525\) 0 0
\(526\) −3.11209e12 −1.77262
\(527\) −7.24021e11 −0.408887
\(528\) 0 0
\(529\) −1.61128e12 −0.894580
\(530\) 6.41452e10 0.0353121
\(531\) 0 0
\(532\) −5.08330e12 −2.75134
\(533\) −4.27182e11 −0.229266
\(534\) 0 0
\(535\) −7.37202e10 −0.0389040
\(536\) 1.66914e12 0.873479
\(537\) 0 0
\(538\) 3.83852e12 1.97535
\(539\) −1.25563e11 −0.0640784
\(540\) 0 0
\(541\) 1.54916e12 0.777513 0.388756 0.921341i \(-0.372905\pi\)
0.388756 + 0.921341i \(0.372905\pi\)
\(542\) 1.14031e12 0.567578
\(543\) 0 0
\(544\) −3.12101e11 −0.152792
\(545\) 1.83027e11 0.0888648
\(546\) 0 0
\(547\) 1.98052e12 0.945879 0.472939 0.881095i \(-0.343193\pi\)
0.472939 + 0.881095i \(0.343193\pi\)
\(548\) −4.76939e12 −2.25918
\(549\) 0 0
\(550\) 1.86744e12 0.870189
\(551\) 5.83283e12 2.69586
\(552\) 0 0
\(553\) −1.19294e12 −0.542444
\(554\) 1.21068e12 0.546056
\(555\) 0 0
\(556\) 3.39871e12 1.50827
\(557\) −8.86577e11 −0.390273 −0.195136 0.980776i \(-0.562515\pi\)
−0.195136 + 0.980776i \(0.562515\pi\)
\(558\) 0 0
\(559\) 1.14755e12 0.497071
\(560\) 9.58387e10 0.0411808
\(561\) 0 0
\(562\) 3.76705e12 1.59290
\(563\) −3.89786e12 −1.63508 −0.817539 0.575873i \(-0.804662\pi\)
−0.817539 + 0.575873i \(0.804662\pi\)
\(564\) 0 0
\(565\) 2.02599e11 0.0836410
\(566\) 4.05212e12 1.65962
\(567\) 0 0
\(568\) 6.82501e12 2.75129
\(569\) 2.26590e12 0.906223 0.453112 0.891454i \(-0.350314\pi\)
0.453112 + 0.891454i \(0.350314\pi\)
\(570\) 0 0
\(571\) −3.09546e12 −1.21860 −0.609302 0.792938i \(-0.708550\pi\)
−0.609302 + 0.792938i \(0.708550\pi\)
\(572\) 6.80184e11 0.265671
\(573\) 0 0
\(574\) 3.39527e12 1.30548
\(575\) −8.45822e11 −0.322681
\(576\) 0 0
\(577\) −3.23415e12 −1.21470 −0.607350 0.794434i \(-0.707767\pi\)
−0.607350 + 0.794434i \(0.707767\pi\)
\(578\) 4.07019e12 1.51684
\(579\) 0 0
\(580\) −6.68692e11 −0.245358
\(581\) −3.77411e11 −0.137411
\(582\) 0 0
\(583\) −3.85927e11 −0.138356
\(584\) 5.19495e12 1.84809
\(585\) 0 0
\(586\) 8.63964e12 3.02661
\(587\) 8.78491e11 0.305398 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(588\) 0 0
\(589\) −5.99129e12 −2.05117
\(590\) −6.39983e11 −0.217437
\(591\) 0 0
\(592\) 6.08321e11 0.203556
\(593\) 1.91730e12 0.636712 0.318356 0.947971i \(-0.396869\pi\)
0.318356 + 0.947971i \(0.396869\pi\)
\(594\) 0 0
\(595\) −7.13632e10 −0.0233425
\(596\) −1.20524e12 −0.391260
\(597\) 0 0
\(598\) −4.75023e11 −0.151900
\(599\) 5.10464e12 1.62011 0.810054 0.586355i \(-0.199438\pi\)
0.810054 + 0.586355i \(0.199438\pi\)
\(600\) 0 0
\(601\) 3.00419e12 0.939273 0.469636 0.882860i \(-0.344385\pi\)
0.469636 + 0.882860i \(0.344385\pi\)
\(602\) −9.12080e12 −2.83041
\(603\) 0 0
\(604\) 4.11260e12 1.25733
\(605\) −1.89079e11 −0.0573778
\(606\) 0 0
\(607\) 1.18434e12 0.354101 0.177051 0.984202i \(-0.443344\pi\)
0.177051 + 0.984202i \(0.443344\pi\)
\(608\) −2.58265e12 −0.766477
\(609\) 0 0
\(610\) −3.62941e11 −0.106133
\(611\) −1.79978e11 −0.0522436
\(612\) 0 0
\(613\) 6.42597e11 0.183809 0.0919045 0.995768i \(-0.470705\pi\)
0.0919045 + 0.995768i \(0.470705\pi\)
\(614\) 8.51193e11 0.241696
\(615\) 0 0
\(616\) −2.47659e12 −0.693011
\(617\) −5.05474e12 −1.40416 −0.702078 0.712100i \(-0.747744\pi\)
−0.702078 + 0.712100i \(0.747744\pi\)
\(618\) 0 0
\(619\) −3.09492e11 −0.0847308 −0.0423654 0.999102i \(-0.513489\pi\)
−0.0423654 + 0.999102i \(0.513489\pi\)
\(620\) 6.86859e11 0.186683
\(621\) 0 0
\(622\) −1.03744e13 −2.77911
\(623\) −2.06421e11 −0.0548980
\(624\) 0 0
\(625\) 3.74425e12 0.981533
\(626\) 3.01827e12 0.785549
\(627\) 0 0
\(628\) −1.33223e13 −3.41792
\(629\) −4.52966e11 −0.115382
\(630\) 0 0
\(631\) −7.94237e11 −0.199443 −0.0997214 0.995015i \(-0.531795\pi\)
−0.0997214 + 0.995015i \(0.531795\pi\)
\(632\) 3.31370e12 0.826202
\(633\) 0 0
\(634\) 7.74944e12 1.90488
\(635\) 2.06268e11 0.0503442
\(636\) 0 0
\(637\) −1.42278e11 −0.0342380
\(638\) 6.20328e12 1.48227
\(639\) 0 0
\(640\) 6.10991e11 0.143954
\(641\) −2.30298e12 −0.538802 −0.269401 0.963028i \(-0.586826\pi\)
−0.269401 + 0.963028i \(0.586826\pi\)
\(642\) 0 0
\(643\) −2.54593e12 −0.587350 −0.293675 0.955905i \(-0.594878\pi\)
−0.293675 + 0.955905i \(0.594878\pi\)
\(644\) 2.44862e12 0.560964
\(645\) 0 0
\(646\) −3.77448e12 −0.852728
\(647\) 2.74568e11 0.0616001 0.0308000 0.999526i \(-0.490194\pi\)
0.0308000 + 0.999526i \(0.490194\pi\)
\(648\) 0 0
\(649\) 3.85043e12 0.851937
\(650\) 2.11603e12 0.464955
\(651\) 0 0
\(652\) 6.84826e11 0.148411
\(653\) 6.05719e12 1.30365 0.651826 0.758369i \(-0.274003\pi\)
0.651826 + 0.758369i \(0.274003\pi\)
\(654\) 0 0
\(655\) −3.79827e11 −0.0806306
\(656\) −2.19582e12 −0.462946
\(657\) 0 0
\(658\) 1.43047e12 0.297484
\(659\) −4.97718e12 −1.02801 −0.514007 0.857786i \(-0.671840\pi\)
−0.514007 + 0.857786i \(0.671840\pi\)
\(660\) 0 0
\(661\) 1.79834e12 0.366409 0.183204 0.983075i \(-0.441353\pi\)
0.183204 + 0.983075i \(0.441353\pi\)
\(662\) −1.23718e13 −2.50364
\(663\) 0 0
\(664\) 1.04836e12 0.209292
\(665\) −5.90532e11 −0.117097
\(666\) 0 0
\(667\) −2.80967e12 −0.549653
\(668\) 7.87076e11 0.152940
\(669\) 0 0
\(670\) 4.23278e11 0.0811502
\(671\) 2.18362e12 0.415839
\(672\) 0 0
\(673\) 1.37693e12 0.258728 0.129364 0.991597i \(-0.458706\pi\)
0.129364 + 0.991597i \(0.458706\pi\)
\(674\) −1.02486e13 −1.91292
\(675\) 0 0
\(676\) 7.70729e11 0.141952
\(677\) −6.49642e11 −0.118857 −0.0594285 0.998233i \(-0.518928\pi\)
−0.0594285 + 0.998233i \(0.518928\pi\)
\(678\) 0 0
\(679\) −7.45828e12 −1.34656
\(680\) 1.98230e11 0.0355532
\(681\) 0 0
\(682\) −6.37181e12 −1.12780
\(683\) 1.10011e12 0.193438 0.0967190 0.995312i \(-0.469165\pi\)
0.0967190 + 0.995312i \(0.469165\pi\)
\(684\) 0 0
\(685\) −5.54064e11 −0.0961507
\(686\) 1.02913e13 1.77424
\(687\) 0 0
\(688\) 5.89869e12 1.00371
\(689\) −4.37301e11 −0.0739254
\(690\) 0 0
\(691\) 7.72289e12 1.28863 0.644315 0.764760i \(-0.277142\pi\)
0.644315 + 0.764760i \(0.277142\pi\)
\(692\) −5.46071e11 −0.0905256
\(693\) 0 0
\(694\) 1.04855e13 1.71581
\(695\) 3.94832e11 0.0641920
\(696\) 0 0
\(697\) 1.63505e12 0.262412
\(698\) 1.35119e13 2.15460
\(699\) 0 0
\(700\) −1.09076e13 −1.71707
\(701\) −1.40436e12 −0.219658 −0.109829 0.993950i \(-0.535030\pi\)
−0.109829 + 0.993950i \(0.535030\pi\)
\(702\) 0 0
\(703\) −3.74831e12 −0.578810
\(704\) −4.64131e12 −0.712137
\(705\) 0 0
\(706\) −8.61273e12 −1.30473
\(707\) 5.39111e12 0.811505
\(708\) 0 0
\(709\) 5.19764e12 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(710\) 1.73075e12 0.255607
\(711\) 0 0
\(712\) 5.73387e11 0.0836157
\(713\) 2.88600e12 0.418209
\(714\) 0 0
\(715\) 7.90176e10 0.0113070
\(716\) 1.47932e13 2.10355
\(717\) 0 0
\(718\) 1.29400e12 0.181707
\(719\) 1.27042e13 1.77284 0.886418 0.462886i \(-0.153186\pi\)
0.886418 + 0.462886i \(0.153186\pi\)
\(720\) 0 0
\(721\) −2.48016e12 −0.341799
\(722\) −1.89174e13 −2.59086
\(723\) 0 0
\(724\) −2.06495e13 −2.79309
\(725\) 1.25159e13 1.68244
\(726\) 0 0
\(727\) 9.91998e11 0.131706 0.0658531 0.997829i \(-0.479023\pi\)
0.0658531 + 0.997829i \(0.479023\pi\)
\(728\) −2.80627e12 −0.370286
\(729\) 0 0
\(730\) 1.31739e12 0.171696
\(731\) −4.39227e12 −0.568933
\(732\) 0 0
\(733\) 5.54849e12 0.709916 0.354958 0.934882i \(-0.384495\pi\)
0.354958 + 0.934882i \(0.384495\pi\)
\(734\) 2.95572e12 0.375864
\(735\) 0 0
\(736\) 1.24406e12 0.156275
\(737\) −2.54663e12 −0.317953
\(738\) 0 0
\(739\) 8.78597e12 1.08365 0.541826 0.840491i \(-0.317733\pi\)
0.541826 + 0.840491i \(0.317733\pi\)
\(740\) 4.29717e11 0.0526792
\(741\) 0 0
\(742\) 3.47570e12 0.420944
\(743\) −4.04326e12 −0.486724 −0.243362 0.969936i \(-0.578250\pi\)
−0.243362 + 0.969936i \(0.578250\pi\)
\(744\) 0 0
\(745\) −1.40014e11 −0.0166521
\(746\) −2.48408e13 −2.93658
\(747\) 0 0
\(748\) −2.60342e12 −0.304080
\(749\) −3.99451e12 −0.463763
\(750\) 0 0
\(751\) −2.52936e11 −0.0290156 −0.0145078 0.999895i \(-0.504618\pi\)
−0.0145078 + 0.999895i \(0.504618\pi\)
\(752\) −9.25130e11 −0.105493
\(753\) 0 0
\(754\) 7.02906e12 0.792001
\(755\) 4.77765e11 0.0535122
\(756\) 0 0
\(757\) 9.99638e12 1.10640 0.553199 0.833049i \(-0.313407\pi\)
0.553199 + 0.833049i \(0.313407\pi\)
\(758\) −7.40426e12 −0.814649
\(759\) 0 0
\(760\) 1.64036e12 0.178352
\(761\) 2.98848e12 0.323013 0.161507 0.986872i \(-0.448365\pi\)
0.161507 + 0.986872i \(0.448365\pi\)
\(762\) 0 0
\(763\) 9.91726e12 1.05933
\(764\) −1.58528e13 −1.68339
\(765\) 0 0
\(766\) 1.69714e13 1.78110
\(767\) 4.36299e12 0.455203
\(768\) 0 0
\(769\) 1.54887e13 1.59715 0.798576 0.601894i \(-0.205587\pi\)
0.798576 + 0.601894i \(0.205587\pi\)
\(770\) −6.28038e11 −0.0643839
\(771\) 0 0
\(772\) −2.55139e13 −2.58523
\(773\) −1.07589e13 −1.08383 −0.541915 0.840434i \(-0.682300\pi\)
−0.541915 + 0.840434i \(0.682300\pi\)
\(774\) 0 0
\(775\) −1.28559e13 −1.28010
\(776\) 2.07173e13 2.05095
\(777\) 0 0
\(778\) −6.47427e12 −0.633552
\(779\) 1.35301e13 1.31638
\(780\) 0 0
\(781\) −1.04130e13 −1.00149
\(782\) 1.81816e12 0.173861
\(783\) 0 0
\(784\) −7.31343e11 −0.0691351
\(785\) −1.54767e12 −0.145467
\(786\) 0 0
\(787\) 1.29304e13 1.20151 0.600755 0.799433i \(-0.294867\pi\)
0.600755 + 0.799433i \(0.294867\pi\)
\(788\) −1.49545e13 −1.38167
\(789\) 0 0
\(790\) 8.40321e11 0.0767579
\(791\) 1.09778e13 0.997059
\(792\) 0 0
\(793\) 2.47430e12 0.222189
\(794\) −1.12394e13 −1.00358
\(795\) 0 0
\(796\) 8.31828e12 0.734387
\(797\) −1.24713e13 −1.09484 −0.547418 0.836859i \(-0.684389\pi\)
−0.547418 + 0.836859i \(0.684389\pi\)
\(798\) 0 0
\(799\) 6.88868e11 0.0597965
\(800\) −5.54175e12 −0.478346
\(801\) 0 0
\(802\) 1.69224e13 1.44437
\(803\) −7.92600e12 −0.672719
\(804\) 0 0
\(805\) 2.84459e11 0.0238747
\(806\) −7.22002e12 −0.602602
\(807\) 0 0
\(808\) −1.49752e13 −1.23601
\(809\) 2.40763e13 1.97616 0.988079 0.153948i \(-0.0491989\pi\)
0.988079 + 0.153948i \(0.0491989\pi\)
\(810\) 0 0
\(811\) 1.14686e13 0.930928 0.465464 0.885067i \(-0.345888\pi\)
0.465464 + 0.885067i \(0.345888\pi\)
\(812\) −3.62330e13 −2.92484
\(813\) 0 0
\(814\) −3.98637e12 −0.318249
\(815\) 7.95569e10 0.00631638
\(816\) 0 0
\(817\) −3.63462e13 −2.85403
\(818\) 3.86469e12 0.301804
\(819\) 0 0
\(820\) −1.55113e12 −0.119808
\(821\) 2.50113e13 1.92129 0.960643 0.277786i \(-0.0896005\pi\)
0.960643 + 0.277786i \(0.0896005\pi\)
\(822\) 0 0
\(823\) 1.29351e13 0.982808 0.491404 0.870932i \(-0.336484\pi\)
0.491404 + 0.870932i \(0.336484\pi\)
\(824\) 6.88929e12 0.520597
\(825\) 0 0
\(826\) −3.46773e13 −2.59200
\(827\) −1.00464e13 −0.746851 −0.373425 0.927660i \(-0.621817\pi\)
−0.373425 + 0.927660i \(0.621817\pi\)
\(828\) 0 0
\(829\) 8.77304e12 0.645141 0.322570 0.946545i \(-0.395453\pi\)
0.322570 + 0.946545i \(0.395453\pi\)
\(830\) 2.65853e11 0.0194442
\(831\) 0 0
\(832\) −5.25916e12 −0.380506
\(833\) 5.44571e11 0.0391879
\(834\) 0 0
\(835\) 9.14353e10 0.00650916
\(836\) −2.15434e13 −1.52541
\(837\) 0 0
\(838\) 3.03051e12 0.212284
\(839\) 1.54259e13 1.07478 0.537392 0.843332i \(-0.319410\pi\)
0.537392 + 0.843332i \(0.319410\pi\)
\(840\) 0 0
\(841\) 2.70683e13 1.86586
\(842\) 2.31593e13 1.58789
\(843\) 0 0
\(844\) −1.72010e13 −1.16685
\(845\) 8.95364e10 0.00604149
\(846\) 0 0
\(847\) −1.02452e13 −0.683983
\(848\) −2.24784e12 −0.149274
\(849\) 0 0
\(850\) −8.09914e12 −0.532174
\(851\) 1.80556e12 0.118012
\(852\) 0 0
\(853\) −1.84928e12 −0.119600 −0.0598002 0.998210i \(-0.519046\pi\)
−0.0598002 + 0.998210i \(0.519046\pi\)
\(854\) −1.96659e13 −1.26518
\(855\) 0 0
\(856\) 1.10958e13 0.706361
\(857\) 2.34715e13 1.48637 0.743185 0.669086i \(-0.233314\pi\)
0.743185 + 0.669086i \(0.233314\pi\)
\(858\) 0 0
\(859\) 1.25121e12 0.0784079 0.0392039 0.999231i \(-0.487518\pi\)
0.0392039 + 0.999231i \(0.487518\pi\)
\(860\) 4.16683e12 0.259754
\(861\) 0 0
\(862\) −1.04026e13 −0.641739
\(863\) −2.38217e13 −1.46192 −0.730961 0.682419i \(-0.760928\pi\)
−0.730961 + 0.682419i \(0.760928\pi\)
\(864\) 0 0
\(865\) −6.34375e10 −0.00385278
\(866\) 4.40928e13 2.66402
\(867\) 0 0
\(868\) 3.72173e13 2.22539
\(869\) −5.05575e12 −0.300744
\(870\) 0 0
\(871\) −2.88564e12 −0.169887
\(872\) −2.75478e13 −1.61348
\(873\) 0 0
\(874\) 1.50453e13 0.872168
\(875\) −2.54215e12 −0.146611
\(876\) 0 0
\(877\) −1.35717e13 −0.774707 −0.387353 0.921931i \(-0.626611\pi\)
−0.387353 + 0.921931i \(0.626611\pi\)
\(878\) −6.79601e12 −0.385948
\(879\) 0 0
\(880\) 4.06171e11 0.0228316
\(881\) −2.38436e12 −0.133346 −0.0666730 0.997775i \(-0.521238\pi\)
−0.0666730 + 0.997775i \(0.521238\pi\)
\(882\) 0 0
\(883\) −5.99779e11 −0.0332023 −0.0166011 0.999862i \(-0.505285\pi\)
−0.0166011 + 0.999862i \(0.505285\pi\)
\(884\) −2.94999e12 −0.162474
\(885\) 0 0
\(886\) −2.50103e12 −0.136354
\(887\) −1.02483e13 −0.555897 −0.277949 0.960596i \(-0.589654\pi\)
−0.277949 + 0.960596i \(0.589654\pi\)
\(888\) 0 0
\(889\) 1.11766e13 0.600138
\(890\) 1.45405e11 0.00776828
\(891\) 0 0
\(892\) 1.45042e13 0.767101
\(893\) 5.70040e12 0.299967
\(894\) 0 0
\(895\) 1.71854e12 0.0895274
\(896\) 3.31065e13 1.71604
\(897\) 0 0
\(898\) −3.04642e13 −1.56331
\(899\) −4.27050e13 −2.18052
\(900\) 0 0
\(901\) 1.67378e12 0.0846130
\(902\) 1.43894e13 0.723791
\(903\) 0 0
\(904\) −3.04937e13 −1.51863
\(905\) −2.39887e12 −0.118874
\(906\) 0 0
\(907\) −2.43855e13 −1.19646 −0.598231 0.801323i \(-0.704130\pi\)
−0.598231 + 0.801323i \(0.704130\pi\)
\(908\) −3.97416e13 −1.94026
\(909\) 0 0
\(910\) −7.11641e11 −0.0344013
\(911\) −2.90386e13 −1.39683 −0.698415 0.715693i \(-0.746111\pi\)
−0.698415 + 0.715693i \(0.746111\pi\)
\(912\) 0 0
\(913\) −1.59949e12 −0.0761841
\(914\) 3.40744e13 1.61499
\(915\) 0 0
\(916\) −6.31867e13 −2.96549
\(917\) −2.05809e13 −0.961173
\(918\) 0 0
\(919\) −1.41793e13 −0.655746 −0.327873 0.944722i \(-0.606332\pi\)
−0.327873 + 0.944722i \(0.606332\pi\)
\(920\) −7.90158e11 −0.0363638
\(921\) 0 0
\(922\) 8.87378e12 0.404408
\(923\) −1.17992e13 −0.535111
\(924\) 0 0
\(925\) −8.04299e12 −0.361226
\(926\) 5.89402e13 2.63428
\(927\) 0 0
\(928\) −1.84087e13 −0.814812
\(929\) 3.56246e13 1.56920 0.784601 0.620001i \(-0.212868\pi\)
0.784601 + 0.620001i \(0.212868\pi\)
\(930\) 0 0
\(931\) 4.50634e12 0.196585
\(932\) 4.90621e13 2.12998
\(933\) 0 0
\(934\) 6.23355e13 2.68025
\(935\) −3.02442e11 −0.0129417
\(936\) 0 0
\(937\) 9.35575e12 0.396506 0.198253 0.980151i \(-0.436473\pi\)
0.198253 + 0.980151i \(0.436473\pi\)
\(938\) 2.29353e13 0.967366
\(939\) 0 0
\(940\) −6.53510e11 −0.0273009
\(941\) −1.23413e13 −0.513106 −0.256553 0.966530i \(-0.582587\pi\)
−0.256553 + 0.966530i \(0.582587\pi\)
\(942\) 0 0
\(943\) −6.51742e12 −0.268394
\(944\) 2.24269e13 0.919168
\(945\) 0 0
\(946\) −3.86546e13 −1.56924
\(947\) −2.93722e12 −0.118676 −0.0593379 0.998238i \(-0.518899\pi\)
−0.0593379 + 0.998238i \(0.518899\pi\)
\(948\) 0 0
\(949\) −8.98110e12 −0.359444
\(950\) −6.70206e13 −2.66964
\(951\) 0 0
\(952\) 1.07411e13 0.423819
\(953\) 3.90544e12 0.153374 0.0766870 0.997055i \(-0.475566\pi\)
0.0766870 + 0.997055i \(0.475566\pi\)
\(954\) 0 0
\(955\) −1.84163e12 −0.0716454
\(956\) −7.04733e13 −2.72875
\(957\) 0 0
\(958\) 5.89375e13 2.26072
\(959\) −3.00219e13 −1.14618
\(960\) 0 0
\(961\) 1.74255e13 0.659069
\(962\) −4.51703e12 −0.170045
\(963\) 0 0
\(964\) 5.43139e13 2.02565
\(965\) −2.96398e12 −0.110028
\(966\) 0 0
\(967\) −1.06697e13 −0.392405 −0.196203 0.980563i \(-0.562861\pi\)
−0.196203 + 0.980563i \(0.562861\pi\)
\(968\) 2.84587e13 1.04178
\(969\) 0 0
\(970\) 5.25371e12 0.190543
\(971\) 3.10264e12 0.112007 0.0560034 0.998431i \(-0.482164\pi\)
0.0560034 + 0.998431i \(0.482164\pi\)
\(972\) 0 0
\(973\) 2.13939e13 0.765213
\(974\) −9.74972e12 −0.347118
\(975\) 0 0
\(976\) 1.27185e13 0.448655
\(977\) 4.66167e13 1.63688 0.818438 0.574594i \(-0.194840\pi\)
0.818438 + 0.574594i \(0.194840\pi\)
\(978\) 0 0
\(979\) −8.74824e11 −0.0304368
\(980\) −5.16619e11 −0.0178918
\(981\) 0 0
\(982\) 6.04614e13 2.07480
\(983\) 3.22316e13 1.10101 0.550505 0.834832i \(-0.314435\pi\)
0.550505 + 0.834832i \(0.314435\pi\)
\(984\) 0 0
\(985\) −1.73728e12 −0.0588040
\(986\) −2.69039e13 −0.906502
\(987\) 0 0
\(988\) −2.44112e13 −0.815047
\(989\) 1.75079e13 0.581903
\(990\) 0 0
\(991\) −2.25008e13 −0.741081 −0.370540 0.928816i \(-0.620828\pi\)
−0.370540 + 0.928816i \(0.620828\pi\)
\(992\) 1.89088e13 0.619957
\(993\) 0 0
\(994\) 9.37807e13 3.04701
\(995\) 9.66342e11 0.0312556
\(996\) 0 0
\(997\) −3.13756e13 −1.00569 −0.502844 0.864377i \(-0.667713\pi\)
−0.502844 + 0.864377i \(0.667713\pi\)
\(998\) 2.40442e13 0.767224
\(999\) 0 0
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 117.10.a.e.1.1 5
3.2 odd 2 13.10.a.b.1.5 5
12.11 even 2 208.10.a.h.1.3 5
15.14 odd 2 325.10.a.b.1.1 5
39.38 odd 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 3.2 odd 2
117.10.a.e.1.1 5 1.1 even 1 trivial
169.10.a.b.1.1 5 39.38 odd 2
208.10.a.h.1.3 5 12.11 even 2
325.10.a.b.1.1 5 15.14 odd 2