Properties

Label 26.10.a.c.1.1
Level $26$
Weight $10$
Character 26.1
Self dual yes
Analytic conductor $13.391$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,10,Mod(1,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, 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("26.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(13.3909317403\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +75.0000 q^{3} +256.000 q^{4} -1979.00 q^{5} +1200.00 q^{6} -10115.0 q^{7} +4096.00 q^{8} -14058.0 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} +75.0000 q^{3} +256.000 q^{4} -1979.00 q^{5} +1200.00 q^{6} -10115.0 q^{7} +4096.00 q^{8} -14058.0 q^{9} -31664.0 q^{10} +18850.0 q^{11} +19200.0 q^{12} +28561.0 q^{13} -161840. q^{14} -148425. q^{15} +65536.0 q^{16} -142403. q^{17} -224928. q^{18} +83302.0 q^{19} -506624. q^{20} -758625. q^{21} +301600. q^{22} -536544. q^{23} +307200. q^{24} +1.96332e6 q^{25} +456976. q^{26} -2.53058e6 q^{27} -2.58944e6 q^{28} -2.60044e6 q^{29} -2.37480e6 q^{30} -2.21400e6 q^{31} +1.04858e6 q^{32} +1.41375e6 q^{33} -2.27845e6 q^{34} +2.00176e7 q^{35} -3.59885e6 q^{36} +1.80992e7 q^{37} +1.33283e6 q^{38} +2.14208e6 q^{39} -8.10598e6 q^{40} +2.68122e7 q^{41} -1.21380e7 q^{42} -4.22535e7 q^{43} +4.82560e6 q^{44} +2.78208e7 q^{45} -8.58470e6 q^{46} +3.59150e7 q^{47} +4.91520e6 q^{48} +6.19596e7 q^{49} +3.14131e7 q^{50} -1.06802e7 q^{51} +7.31162e6 q^{52} -6.65141e7 q^{53} -4.04892e7 q^{54} -3.73042e7 q^{55} -4.14310e7 q^{56} +6.24765e6 q^{57} -4.16071e7 q^{58} -1.08164e8 q^{59} -3.79968e7 q^{60} -2.07450e8 q^{61} -3.54241e7 q^{62} +1.42197e8 q^{63} +1.67772e7 q^{64} -5.65222e7 q^{65} +2.26200e7 q^{66} +1.93016e8 q^{67} -3.64552e7 q^{68} -4.02408e7 q^{69} +3.20281e8 q^{70} -2.01833e8 q^{71} -5.75816e7 q^{72} -1.21628e8 q^{73} +2.89588e8 q^{74} +1.47249e8 q^{75} +2.13253e7 q^{76} -1.90668e8 q^{77} +3.42732e7 q^{78} +1.12872e8 q^{79} -1.29696e8 q^{80} +8.69105e7 q^{81} +4.28996e8 q^{82} +3.08254e8 q^{83} -1.94208e8 q^{84} +2.81816e8 q^{85} -6.76056e8 q^{86} -1.95033e8 q^{87} +7.72096e7 q^{88} -6.37487e6 q^{89} +4.45133e8 q^{90} -2.88895e8 q^{91} -1.37355e8 q^{92} -1.66050e8 q^{93} +5.74640e8 q^{94} -1.64855e8 q^{95} +7.86432e7 q^{96} +8.71267e8 q^{97} +9.91354e8 q^{98} -2.64993e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 75.0000 0.534584 0.267292 0.963616i \(-0.413871\pi\)
0.267292 + 0.963616i \(0.413871\pi\)
\(4\) 256.000 0.500000
\(5\) −1979.00 −1.41606 −0.708029 0.706184i \(-0.750415\pi\)
−0.708029 + 0.706184i \(0.750415\pi\)
\(6\) 1200.00 0.378008
\(7\) −10115.0 −1.59230 −0.796150 0.605100i \(-0.793133\pi\)
−0.796150 + 0.605100i \(0.793133\pi\)
\(8\) 4096.00 0.353553
\(9\) −14058.0 −0.714220
\(10\) −31664.0 −1.00130
\(11\) 18850.0 0.388190 0.194095 0.980983i \(-0.437823\pi\)
0.194095 + 0.980983i \(0.437823\pi\)
\(12\) 19200.0 0.267292
\(13\) 28561.0 0.277350
\(14\) −161840. −1.12593
\(15\) −148425. −0.757001
\(16\) 65536.0 0.250000
\(17\) −142403. −0.413522 −0.206761 0.978391i \(-0.566292\pi\)
−0.206761 + 0.978391i \(0.566292\pi\)
\(18\) −224928. −0.505030
\(19\) 83302.0 0.146644 0.0733220 0.997308i \(-0.476640\pi\)
0.0733220 + 0.997308i \(0.476640\pi\)
\(20\) −506624. −0.708029
\(21\) −758625. −0.851217
\(22\) 301600. 0.274492
\(23\) −536544. −0.399788 −0.199894 0.979817i \(-0.564060\pi\)
−0.199894 + 0.979817i \(0.564060\pi\)
\(24\) 307200. 0.189004
\(25\) 1.96332e6 1.00522
\(26\) 456976. 0.196116
\(27\) −2.53058e6 −0.916394
\(28\) −2.58944e6 −0.796150
\(29\) −2.60044e6 −0.682741 −0.341371 0.939929i \(-0.610891\pi\)
−0.341371 + 0.939929i \(0.610891\pi\)
\(30\) −2.37480e6 −0.535280
\(31\) −2.21400e6 −0.430577 −0.215288 0.976550i \(-0.569069\pi\)
−0.215288 + 0.976550i \(0.569069\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 1.41375e6 0.207520
\(34\) −2.27845e6 −0.292404
\(35\) 2.00176e7 2.25479
\(36\) −3.59885e6 −0.357110
\(37\) 1.80992e7 1.58764 0.793821 0.608151i \(-0.208089\pi\)
0.793821 + 0.608151i \(0.208089\pi\)
\(38\) 1.33283e6 0.103693
\(39\) 2.14208e6 0.148267
\(40\) −8.10598e6 −0.500652
\(41\) 2.68122e7 1.48186 0.740928 0.671585i \(-0.234386\pi\)
0.740928 + 0.671585i \(0.234386\pi\)
\(42\) −1.21380e7 −0.601901
\(43\) −4.22535e7 −1.88475 −0.942376 0.334555i \(-0.891414\pi\)
−0.942376 + 0.334555i \(0.891414\pi\)
\(44\) 4.82560e6 0.194095
\(45\) 2.78208e7 1.01138
\(46\) −8.58470e6 −0.282693
\(47\) 3.59150e7 1.07358 0.536791 0.843715i \(-0.319636\pi\)
0.536791 + 0.843715i \(0.319636\pi\)
\(48\) 4.91520e6 0.133646
\(49\) 6.19596e7 1.53542
\(50\) 3.14131e7 0.710796
\(51\) −1.06802e7 −0.221062
\(52\) 7.31162e6 0.138675
\(53\) −6.65141e7 −1.15790 −0.578951 0.815362i \(-0.696538\pi\)
−0.578951 + 0.815362i \(0.696538\pi\)
\(54\) −4.04892e7 −0.647988
\(55\) −3.73042e7 −0.549699
\(56\) −4.14310e7 −0.562963
\(57\) 6.24765e6 0.0783935
\(58\) −4.16071e7 −0.482771
\(59\) −1.08164e8 −1.16211 −0.581057 0.813863i \(-0.697361\pi\)
−0.581057 + 0.813863i \(0.697361\pi\)
\(60\) −3.79968e7 −0.378500
\(61\) −2.07450e8 −1.91836 −0.959178 0.282805i \(-0.908735\pi\)
−0.959178 + 0.282805i \(0.908735\pi\)
\(62\) −3.54241e7 −0.304464
\(63\) 1.42197e8 1.13725
\(64\) 1.67772e7 0.125000
\(65\) −5.65222e7 −0.392744
\(66\) 2.26200e7 0.146739
\(67\) 1.93016e8 1.17019 0.585094 0.810966i \(-0.301058\pi\)
0.585094 + 0.810966i \(0.301058\pi\)
\(68\) −3.64552e7 −0.206761
\(69\) −4.02408e7 −0.213720
\(70\) 3.20281e8 1.59438
\(71\) −2.01833e8 −0.942607 −0.471304 0.881971i \(-0.656216\pi\)
−0.471304 + 0.881971i \(0.656216\pi\)
\(72\) −5.75816e7 −0.252515
\(73\) −1.21628e8 −0.501281 −0.250640 0.968080i \(-0.580641\pi\)
−0.250640 + 0.968080i \(0.580641\pi\)
\(74\) 2.89588e8 1.12263
\(75\) 1.47249e8 0.537373
\(76\) 2.13253e7 0.0733220
\(77\) −1.90668e8 −0.618115
\(78\) 3.42732e7 0.104840
\(79\) 1.12872e8 0.326035 0.163017 0.986623i \(-0.447877\pi\)
0.163017 + 0.986623i \(0.447877\pi\)
\(80\) −1.29696e8 −0.354014
\(81\) 8.69105e7 0.224331
\(82\) 4.28996e8 1.04783
\(83\) 3.08254e8 0.712948 0.356474 0.934305i \(-0.383979\pi\)
0.356474 + 0.934305i \(0.383979\pi\)
\(84\) −1.94208e8 −0.425609
\(85\) 2.81816e8 0.585571
\(86\) −6.76056e8 −1.33272
\(87\) −1.95033e8 −0.364982
\(88\) 7.72096e7 0.137246
\(89\) −6.37487e6 −0.0107700 −0.00538501 0.999986i \(-0.501714\pi\)
−0.00538501 + 0.999986i \(0.501714\pi\)
\(90\) 4.45133e8 0.715151
\(91\) −2.88895e8 −0.441624
\(92\) −1.37355e8 −0.199894
\(93\) −1.66050e8 −0.230179
\(94\) 5.74640e8 0.759137
\(95\) −1.64855e8 −0.207656
\(96\) 7.86432e7 0.0945019
\(97\) 8.71267e8 0.999260 0.499630 0.866239i \(-0.333469\pi\)
0.499630 + 0.866239i \(0.333469\pi\)
\(98\) 9.91354e8 1.08570
\(99\) −2.64993e8 −0.277253
\(100\) 5.02609e8 0.502609
\(101\) −8.24412e8 −0.788312 −0.394156 0.919044i \(-0.628963\pi\)
−0.394156 + 0.919044i \(0.628963\pi\)
\(102\) −1.70884e8 −0.156315
\(103\) −1.65896e9 −1.45234 −0.726168 0.687517i \(-0.758701\pi\)
−0.726168 + 0.687517i \(0.758701\pi\)
\(104\) 1.16986e8 0.0980581
\(105\) 1.50132e9 1.20537
\(106\) −1.06423e9 −0.818761
\(107\) 1.15165e9 0.849366 0.424683 0.905342i \(-0.360386\pi\)
0.424683 + 0.905342i \(0.360386\pi\)
\(108\) −6.47827e8 −0.458197
\(109\) −2.78480e9 −1.88962 −0.944810 0.327620i \(-0.893753\pi\)
−0.944810 + 0.327620i \(0.893753\pi\)
\(110\) −5.96866e8 −0.388696
\(111\) 1.35744e9 0.848727
\(112\) −6.62897e8 −0.398075
\(113\) 6.78547e8 0.391496 0.195748 0.980654i \(-0.437287\pi\)
0.195748 + 0.980654i \(0.437287\pi\)
\(114\) 9.99624e7 0.0554325
\(115\) 1.06182e9 0.566123
\(116\) −6.65713e8 −0.341371
\(117\) −4.01511e8 −0.198089
\(118\) −1.73062e9 −0.821739
\(119\) 1.44041e9 0.658451
\(120\) −6.07949e8 −0.267640
\(121\) −2.00263e9 −0.849309
\(122\) −3.31920e9 −1.35648
\(123\) 2.01092e9 0.792175
\(124\) −5.66785e8 −0.215288
\(125\) −2.01680e7 −0.00738869
\(126\) 2.27515e9 0.804159
\(127\) −3.48292e9 −1.18803 −0.594014 0.804455i \(-0.702458\pi\)
−0.594014 + 0.804455i \(0.702458\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) −3.16901e9 −1.00756
\(130\) −9.04356e8 −0.277712
\(131\) 5.02701e9 1.49138 0.745691 0.666292i \(-0.232119\pi\)
0.745691 + 0.666292i \(0.232119\pi\)
\(132\) 3.61920e8 0.103760
\(133\) −8.42600e8 −0.233501
\(134\) 3.08825e9 0.827448
\(135\) 5.00801e9 1.29767
\(136\) −5.83283e8 −0.146202
\(137\) −6.38904e9 −1.54950 −0.774752 0.632265i \(-0.782125\pi\)
−0.774752 + 0.632265i \(0.782125\pi\)
\(138\) −6.43853e8 −0.151123
\(139\) 7.62665e9 1.73287 0.866437 0.499286i \(-0.166404\pi\)
0.866437 + 0.499286i \(0.166404\pi\)
\(140\) 5.12450e9 1.12739
\(141\) 2.69362e9 0.573920
\(142\) −3.22934e9 −0.666524
\(143\) 5.38375e8 0.107665
\(144\) −9.21305e8 −0.178555
\(145\) 5.14627e9 0.966801
\(146\) −1.94605e9 −0.354459
\(147\) 4.64697e9 0.820809
\(148\) 4.63341e9 0.793821
\(149\) −9.23455e9 −1.53489 −0.767445 0.641114i \(-0.778472\pi\)
−0.767445 + 0.641114i \(0.778472\pi\)
\(150\) 2.35598e9 0.379980
\(151\) −3.25451e9 −0.509436 −0.254718 0.967015i \(-0.581983\pi\)
−0.254718 + 0.967015i \(0.581983\pi\)
\(152\) 3.41205e8 0.0518465
\(153\) 2.00190e9 0.295346
\(154\) −3.05068e9 −0.437073
\(155\) 4.38151e9 0.609722
\(156\) 5.48371e8 0.0741334
\(157\) 1.62825e9 0.213881 0.106940 0.994265i \(-0.465895\pi\)
0.106940 + 0.994265i \(0.465895\pi\)
\(158\) 1.80595e9 0.230541
\(159\) −4.98855e9 −0.618996
\(160\) −2.07513e9 −0.250326
\(161\) 5.42714e9 0.636583
\(162\) 1.39057e9 0.158626
\(163\) −1.13187e10 −1.25590 −0.627948 0.778255i \(-0.716105\pi\)
−0.627948 + 0.778255i \(0.716105\pi\)
\(164\) 6.86393e9 0.740928
\(165\) −2.79781e9 −0.293860
\(166\) 4.93207e9 0.504130
\(167\) 1.72306e9 0.171426 0.0857131 0.996320i \(-0.472683\pi\)
0.0857131 + 0.996320i \(0.472683\pi\)
\(168\) −3.10733e9 −0.300951
\(169\) 8.15731e8 0.0769231
\(170\) 4.50905e9 0.414061
\(171\) −1.17106e9 −0.104736
\(172\) −1.08169e10 −0.942376
\(173\) 2.76347e9 0.234557 0.117278 0.993099i \(-0.462583\pi\)
0.117278 + 0.993099i \(0.462583\pi\)
\(174\) −3.12053e9 −0.258081
\(175\) −1.98589e10 −1.60061
\(176\) 1.23535e9 0.0970475
\(177\) −8.11230e9 −0.621247
\(178\) −1.01998e8 −0.00761555
\(179\) 6.86682e9 0.499939 0.249969 0.968254i \(-0.419579\pi\)
0.249969 + 0.968254i \(0.419579\pi\)
\(180\) 7.12212e9 0.505688
\(181\) −2.41534e10 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(182\) −4.62231e9 −0.312276
\(183\) −1.55587e10 −1.02552
\(184\) −2.19768e9 −0.141347
\(185\) −3.58184e10 −2.24819
\(186\) −2.65680e9 −0.162761
\(187\) −2.68430e9 −0.160525
\(188\) 9.19424e9 0.536791
\(189\) 2.55968e10 1.45917
\(190\) −2.63767e9 −0.146835
\(191\) 3.59983e10 1.95719 0.978593 0.205805i \(-0.0659813\pi\)
0.978593 + 0.205805i \(0.0659813\pi\)
\(192\) 1.25829e9 0.0668229
\(193\) −1.70031e10 −0.882107 −0.441054 0.897481i \(-0.645395\pi\)
−0.441054 + 0.897481i \(0.645395\pi\)
\(194\) 1.39403e10 0.706583
\(195\) −4.23917e9 −0.209954
\(196\) 1.58617e10 0.767709
\(197\) 3.98292e10 1.88410 0.942049 0.335477i \(-0.108897\pi\)
0.942049 + 0.335477i \(0.108897\pi\)
\(198\) −4.23989e9 −0.196048
\(199\) 1.31081e9 0.0592518 0.0296259 0.999561i \(-0.490568\pi\)
0.0296259 + 0.999561i \(0.490568\pi\)
\(200\) 8.04174e9 0.355398
\(201\) 1.44762e10 0.625563
\(202\) −1.31906e10 −0.557421
\(203\) 2.63035e10 1.08713
\(204\) −2.73414e9 −0.110531
\(205\) −5.30614e10 −2.09839
\(206\) −2.65433e10 −1.02696
\(207\) 7.54274e9 0.285537
\(208\) 1.87177e9 0.0693375
\(209\) 1.57024e9 0.0569257
\(210\) 2.40211e10 0.852327
\(211\) 2.35777e10 0.818898 0.409449 0.912333i \(-0.365721\pi\)
0.409449 + 0.912333i \(0.365721\pi\)
\(212\) −1.70276e10 −0.578951
\(213\) −1.51375e10 −0.503902
\(214\) 1.84265e10 0.600593
\(215\) 8.36196e10 2.66892
\(216\) −1.03652e10 −0.323994
\(217\) 2.23947e10 0.685607
\(218\) −4.45568e10 −1.33616
\(219\) −9.12211e9 −0.267976
\(220\) −9.54986e9 −0.274850
\(221\) −4.06717e9 −0.114690
\(222\) 2.17191e10 0.600141
\(223\) 2.38326e9 0.0645356 0.0322678 0.999479i \(-0.489727\pi\)
0.0322678 + 0.999479i \(0.489727\pi\)
\(224\) −1.06063e10 −0.281481
\(225\) −2.76003e10 −0.717947
\(226\) 1.08568e10 0.276829
\(227\) −7.46548e9 −0.186613 −0.0933064 0.995637i \(-0.529744\pi\)
−0.0933064 + 0.995637i \(0.529744\pi\)
\(228\) 1.59940e9 0.0391967
\(229\) 2.63966e10 0.634292 0.317146 0.948377i \(-0.397276\pi\)
0.317146 + 0.948377i \(0.397276\pi\)
\(230\) 1.69891e10 0.400309
\(231\) −1.43001e10 −0.330434
\(232\) −1.06514e10 −0.241386
\(233\) 2.40457e10 0.534485 0.267242 0.963629i \(-0.413888\pi\)
0.267242 + 0.963629i \(0.413888\pi\)
\(234\) −6.42417e9 −0.140070
\(235\) −7.10758e10 −1.52025
\(236\) −2.76900e10 −0.581057
\(237\) 8.46539e9 0.174293
\(238\) 2.30465e10 0.465595
\(239\) 5.96318e10 1.18219 0.591095 0.806602i \(-0.298696\pi\)
0.591095 + 0.806602i \(0.298696\pi\)
\(240\) −9.72718e9 −0.189250
\(241\) 1.25639e10 0.239910 0.119955 0.992779i \(-0.461725\pi\)
0.119955 + 0.992779i \(0.461725\pi\)
\(242\) −3.20420e10 −0.600552
\(243\) 5.63276e10 1.03632
\(244\) −5.31072e10 −0.959178
\(245\) −1.22618e11 −2.17424
\(246\) 3.21747e10 0.560153
\(247\) 2.37919e9 0.0406717
\(248\) −9.06856e9 −0.152232
\(249\) 2.31191e10 0.381130
\(250\) −3.22688e8 −0.00522459
\(251\) 2.41771e10 0.384479 0.192240 0.981348i \(-0.438425\pi\)
0.192240 + 0.981348i \(0.438425\pi\)
\(252\) 3.64023e10 0.568626
\(253\) −1.01139e10 −0.155194
\(254\) −5.57267e10 −0.840062
\(255\) 2.11362e10 0.313037
\(256\) 4.29497e9 0.0625000
\(257\) 2.96868e10 0.424488 0.212244 0.977217i \(-0.431923\pi\)
0.212244 + 0.977217i \(0.431923\pi\)
\(258\) −5.07042e10 −0.712451
\(259\) −1.83074e11 −2.52800
\(260\) −1.44697e10 −0.196372
\(261\) 3.65570e10 0.487628
\(262\) 8.04322e10 1.05457
\(263\) −7.59146e10 −0.978418 −0.489209 0.872167i \(-0.662714\pi\)
−0.489209 + 0.872167i \(0.662714\pi\)
\(264\) 5.79072e9 0.0733694
\(265\) 1.31631e11 1.63966
\(266\) −1.34816e10 −0.165110
\(267\) −4.78115e8 −0.00575747
\(268\) 4.94120e10 0.585094
\(269\) −2.57149e10 −0.299433 −0.149716 0.988729i \(-0.547836\pi\)
−0.149716 + 0.988729i \(0.547836\pi\)
\(270\) 8.01281e10 0.917589
\(271\) −8.08890e10 −0.911020 −0.455510 0.890231i \(-0.650543\pi\)
−0.455510 + 0.890231i \(0.650543\pi\)
\(272\) −9.33252e9 −0.103381
\(273\) −2.16671e10 −0.236085
\(274\) −1.02225e11 −1.09567
\(275\) 3.70085e10 0.390215
\(276\) −1.03016e10 −0.106860
\(277\) −3.40035e10 −0.347028 −0.173514 0.984831i \(-0.555512\pi\)
−0.173514 + 0.984831i \(0.555512\pi\)
\(278\) 1.22026e11 1.22533
\(279\) 3.11245e10 0.307527
\(280\) 8.19920e10 0.797188
\(281\) −7.14831e9 −0.0683951 −0.0341976 0.999415i \(-0.510888\pi\)
−0.0341976 + 0.999415i \(0.510888\pi\)
\(282\) 4.30980e10 0.405822
\(283\) −7.80508e10 −0.723333 −0.361667 0.932308i \(-0.617792\pi\)
−0.361667 + 0.932308i \(0.617792\pi\)
\(284\) −5.16694e10 −0.471304
\(285\) −1.23641e10 −0.111010
\(286\) 8.61400e9 0.0761303
\(287\) −2.71206e11 −2.35956
\(288\) −1.47409e10 −0.126258
\(289\) −9.83093e10 −0.828999
\(290\) 8.23404e10 0.683631
\(291\) 6.53450e10 0.534188
\(292\) −3.11368e10 −0.250640
\(293\) −1.26662e11 −1.00402 −0.502009 0.864862i \(-0.667406\pi\)
−0.502009 + 0.864862i \(0.667406\pi\)
\(294\) 7.43515e10 0.580399
\(295\) 2.14057e11 1.64562
\(296\) 7.41345e10 0.561316
\(297\) −4.77013e10 −0.355735
\(298\) −1.47753e11 −1.08533
\(299\) −1.53242e10 −0.110881
\(300\) 3.76957e10 0.268686
\(301\) 4.27394e11 3.00109
\(302\) −5.20722e10 −0.360226
\(303\) −6.18309e10 −0.421419
\(304\) 5.45928e9 0.0366610
\(305\) 4.10543e11 2.71650
\(306\) 3.20304e10 0.208841
\(307\) 6.15064e10 0.395182 0.197591 0.980285i \(-0.436688\pi\)
0.197591 + 0.980285i \(0.436688\pi\)
\(308\) −4.88109e10 −0.309057
\(309\) −1.24422e11 −0.776395
\(310\) 7.01042e10 0.431138
\(311\) 2.16398e11 1.31169 0.655846 0.754894i \(-0.272312\pi\)
0.655846 + 0.754894i \(0.272312\pi\)
\(312\) 8.77394e9 0.0524202
\(313\) 2.44634e11 1.44068 0.720340 0.693621i \(-0.243986\pi\)
0.720340 + 0.693621i \(0.243986\pi\)
\(314\) 2.60519e10 0.151237
\(315\) −2.81407e11 −1.61041
\(316\) 2.88952e10 0.163017
\(317\) 7.77399e10 0.432392 0.216196 0.976350i \(-0.430635\pi\)
0.216196 + 0.976350i \(0.430635\pi\)
\(318\) −7.98169e10 −0.437696
\(319\) −4.90183e10 −0.265033
\(320\) −3.32021e10 −0.177007
\(321\) 8.63740e10 0.454057
\(322\) 8.68343e10 0.450132
\(323\) −1.18625e10 −0.0606406
\(324\) 2.22491e10 0.112166
\(325\) 5.60743e10 0.278797
\(326\) −1.81100e11 −0.888053
\(327\) −2.08860e11 −1.01016
\(328\) 1.09823e11 0.523915
\(329\) −3.63280e11 −1.70946
\(330\) −4.47650e10 −0.207790
\(331\) −1.68625e11 −0.772139 −0.386070 0.922470i \(-0.626168\pi\)
−0.386070 + 0.922470i \(0.626168\pi\)
\(332\) 7.89131e10 0.356474
\(333\) −2.54439e11 −1.13393
\(334\) 2.75690e10 0.121217
\(335\) −3.81978e11 −1.65705
\(336\) −4.97172e10 −0.212804
\(337\) −7.70797e10 −0.325541 −0.162770 0.986664i \(-0.552043\pi\)
−0.162770 + 0.986664i \(0.552043\pi\)
\(338\) 1.30517e10 0.0543928
\(339\) 5.08910e10 0.209287
\(340\) 7.21448e10 0.292786
\(341\) −4.17340e10 −0.167146
\(342\) −1.87370e10 −0.0740596
\(343\) −2.18545e11 −0.852544
\(344\) −1.73070e11 −0.666361
\(345\) 7.96365e10 0.302640
\(346\) 4.42156e10 0.165857
\(347\) 3.54609e11 1.31301 0.656504 0.754322i \(-0.272034\pi\)
0.656504 + 0.754322i \(0.272034\pi\)
\(348\) −4.99285e10 −0.182491
\(349\) 1.70460e11 0.615048 0.307524 0.951540i \(-0.400500\pi\)
0.307524 + 0.951540i \(0.400500\pi\)
\(350\) −3.17743e11 −1.13180
\(351\) −7.22758e10 −0.254162
\(352\) 1.97657e10 0.0686229
\(353\) 2.96506e11 1.01636 0.508180 0.861251i \(-0.330318\pi\)
0.508180 + 0.861251i \(0.330318\pi\)
\(354\) −1.29797e11 −0.439288
\(355\) 3.99428e11 1.33479
\(356\) −1.63197e9 −0.00538501
\(357\) 1.08030e11 0.351997
\(358\) 1.09869e11 0.353510
\(359\) 7.20144e10 0.228820 0.114410 0.993434i \(-0.463502\pi\)
0.114410 + 0.993434i \(0.463502\pi\)
\(360\) 1.13954e11 0.357576
\(361\) −3.15748e11 −0.978496
\(362\) −3.86455e11 −1.18280
\(363\) −1.50197e11 −0.454026
\(364\) −7.39570e10 −0.220812
\(365\) 2.40702e11 0.709842
\(366\) −2.48940e11 −0.725153
\(367\) −1.05092e11 −0.302394 −0.151197 0.988504i \(-0.548313\pi\)
−0.151197 + 0.988504i \(0.548313\pi\)
\(368\) −3.51629e10 −0.0999471
\(369\) −3.76926e11 −1.05837
\(370\) −5.73094e11 −1.58971
\(371\) 6.72790e11 1.84373
\(372\) −4.25089e10 −0.115090
\(373\) 2.19888e11 0.588182 0.294091 0.955777i \(-0.404983\pi\)
0.294091 + 0.955777i \(0.404983\pi\)
\(374\) −4.29487e10 −0.113508
\(375\) −1.51260e9 −0.00394987
\(376\) 1.47108e11 0.379569
\(377\) −7.42712e10 −0.189358
\(378\) 4.09548e11 1.03179
\(379\) −3.14748e11 −0.783586 −0.391793 0.920053i \(-0.628145\pi\)
−0.391793 + 0.920053i \(0.628145\pi\)
\(380\) −4.22028e10 −0.103828
\(381\) −2.61219e11 −0.635100
\(382\) 5.75973e11 1.38394
\(383\) −3.41027e10 −0.0809831 −0.0404915 0.999180i \(-0.512892\pi\)
−0.0404915 + 0.999180i \(0.512892\pi\)
\(384\) 2.01327e10 0.0472510
\(385\) 3.77331e11 0.875286
\(386\) −2.72050e11 −0.623744
\(387\) 5.93999e11 1.34613
\(388\) 2.23044e11 0.499630
\(389\) 4.49612e11 0.995554 0.497777 0.867305i \(-0.334150\pi\)
0.497777 + 0.867305i \(0.334150\pi\)
\(390\) −6.78267e10 −0.148460
\(391\) 7.64055e10 0.165321
\(392\) 2.53787e11 0.542852
\(393\) 3.77026e11 0.797269
\(394\) 6.37267e11 1.33226
\(395\) −2.23374e11 −0.461684
\(396\) −6.78383e10 −0.138627
\(397\) 2.29976e11 0.464649 0.232324 0.972638i \(-0.425367\pi\)
0.232324 + 0.972638i \(0.425367\pi\)
\(398\) 2.09730e10 0.0418974
\(399\) −6.31950e10 −0.124826
\(400\) 1.28668e11 0.251304
\(401\) −6.69163e11 −1.29236 −0.646178 0.763187i \(-0.723634\pi\)
−0.646178 + 0.763187i \(0.723634\pi\)
\(402\) 2.31619e11 0.442340
\(403\) −6.32342e10 −0.119421
\(404\) −2.11049e11 −0.394156
\(405\) −1.71996e11 −0.317666
\(406\) 4.20856e11 0.768716
\(407\) 3.41171e11 0.616307
\(408\) −4.37462e10 −0.0781573
\(409\) −6.73923e11 −1.19085 −0.595423 0.803413i \(-0.703015\pi\)
−0.595423 + 0.803413i \(0.703015\pi\)
\(410\) −8.48983e11 −1.48379
\(411\) −4.79178e11 −0.828340
\(412\) −4.24693e11 −0.726168
\(413\) 1.09408e12 1.85043
\(414\) 1.20684e11 0.201905
\(415\) −6.10035e11 −1.00957
\(416\) 2.99484e10 0.0490290
\(417\) 5.71999e11 0.926366
\(418\) 2.51239e10 0.0402526
\(419\) −3.45053e11 −0.546919 −0.273460 0.961883i \(-0.588168\pi\)
−0.273460 + 0.961883i \(0.588168\pi\)
\(420\) 3.84338e11 0.602686
\(421\) −5.57817e11 −0.865411 −0.432706 0.901535i \(-0.642441\pi\)
−0.432706 + 0.901535i \(0.642441\pi\)
\(422\) 3.77243e11 0.579049
\(423\) −5.04893e11 −0.766775
\(424\) −2.72442e11 −0.409380
\(425\) −2.79582e11 −0.415680
\(426\) −2.42200e11 −0.356313
\(427\) 2.09836e12 3.05460
\(428\) 2.94823e11 0.424683
\(429\) 4.03781e10 0.0575557
\(430\) 1.33791e12 1.88721
\(431\) −6.39243e11 −0.892315 −0.446158 0.894954i \(-0.647208\pi\)
−0.446158 + 0.894954i \(0.647208\pi\)
\(432\) −1.65844e11 −0.229099
\(433\) −1.23759e12 −1.69193 −0.845965 0.533238i \(-0.820975\pi\)
−0.845965 + 0.533238i \(0.820975\pi\)
\(434\) 3.58314e11 0.484798
\(435\) 3.85971e11 0.516836
\(436\) −7.12908e11 −0.944810
\(437\) −4.46952e10 −0.0586265
\(438\) −1.45954e11 −0.189488
\(439\) −9.14852e11 −1.17560 −0.587801 0.809006i \(-0.700006\pi\)
−0.587801 + 0.809006i \(0.700006\pi\)
\(440\) −1.52798e11 −0.194348
\(441\) −8.71028e11 −1.09663
\(442\) −6.50748e10 −0.0810984
\(443\) −1.18651e12 −1.46370 −0.731852 0.681464i \(-0.761344\pi\)
−0.731852 + 0.681464i \(0.761344\pi\)
\(444\) 3.47505e11 0.424364
\(445\) 1.26159e10 0.0152510
\(446\) 3.81322e10 0.0456336
\(447\) −6.92591e11 −0.820527
\(448\) −1.69702e11 −0.199037
\(449\) −2.70808e10 −0.0314452 −0.0157226 0.999876i \(-0.505005\pi\)
−0.0157226 + 0.999876i \(0.505005\pi\)
\(450\) −4.41605e11 −0.507665
\(451\) 5.05411e11 0.575241
\(452\) 1.73708e11 0.195748
\(453\) −2.44088e11 −0.272336
\(454\) −1.19448e11 −0.131955
\(455\) 5.71722e11 0.625365
\(456\) 2.55904e10 0.0277163
\(457\) 2.02586e11 0.217263 0.108632 0.994082i \(-0.465353\pi\)
0.108632 + 0.994082i \(0.465353\pi\)
\(458\) 4.22346e11 0.448512
\(459\) 3.60361e11 0.378949
\(460\) 2.71826e11 0.283062
\(461\) 8.96346e11 0.924319 0.462159 0.886797i \(-0.347075\pi\)
0.462159 + 0.886797i \(0.347075\pi\)
\(462\) −2.28801e11 −0.233652
\(463\) −5.17740e11 −0.523597 −0.261799 0.965123i \(-0.584316\pi\)
−0.261799 + 0.965123i \(0.584316\pi\)
\(464\) −1.70423e11 −0.170685
\(465\) 3.28614e11 0.325947
\(466\) 3.84731e11 0.377938
\(467\) 9.20785e11 0.895843 0.447922 0.894073i \(-0.352164\pi\)
0.447922 + 0.894073i \(0.352164\pi\)
\(468\) −1.02787e11 −0.0990445
\(469\) −1.95235e12 −1.86329
\(470\) −1.13721e12 −1.07498
\(471\) 1.22119e11 0.114337
\(472\) −4.43040e11 −0.410869
\(473\) −7.96478e11 −0.731642
\(474\) 1.35446e11 0.123244
\(475\) 1.63548e11 0.147409
\(476\) 3.68744e11 0.329226
\(477\) 9.35055e11 0.826998
\(478\) 9.54108e11 0.835934
\(479\) 1.73253e12 1.50373 0.751867 0.659315i \(-0.229154\pi\)
0.751867 + 0.659315i \(0.229154\pi\)
\(480\) −1.55635e11 −0.133820
\(481\) 5.16932e11 0.440333
\(482\) 2.01023e11 0.169642
\(483\) 4.07036e11 0.340307
\(484\) −5.12672e11 −0.424654
\(485\) −1.72424e12 −1.41501
\(486\) 9.01242e11 0.732787
\(487\) 1.49591e12 1.20511 0.602554 0.798078i \(-0.294150\pi\)
0.602554 + 0.798078i \(0.294150\pi\)
\(488\) −8.49715e11 −0.678241
\(489\) −8.48906e11 −0.671382
\(490\) −1.96189e12 −1.53742
\(491\) −4.28954e11 −0.333076 −0.166538 0.986035i \(-0.553259\pi\)
−0.166538 + 0.986035i \(0.553259\pi\)
\(492\) 5.14795e11 0.396088
\(493\) 3.70311e11 0.282329
\(494\) 3.80670e10 0.0287592
\(495\) 5.24422e11 0.392606
\(496\) −1.45097e11 −0.107644
\(497\) 2.04155e12 1.50091
\(498\) 3.69905e11 0.269500
\(499\) −9.12174e11 −0.658606 −0.329303 0.944224i \(-0.606814\pi\)
−0.329303 + 0.944224i \(0.606814\pi\)
\(500\) −5.16301e9 −0.00369435
\(501\) 1.29230e11 0.0916416
\(502\) 3.86834e11 0.271868
\(503\) −1.26835e12 −0.883456 −0.441728 0.897149i \(-0.645634\pi\)
−0.441728 + 0.897149i \(0.645634\pi\)
\(504\) 5.82438e11 0.402080
\(505\) 1.63151e12 1.11629
\(506\) −1.61822e11 −0.109739
\(507\) 6.11798e10 0.0411218
\(508\) −8.91627e11 −0.594014
\(509\) −1.54192e12 −1.01820 −0.509098 0.860708i \(-0.670021\pi\)
−0.509098 + 0.860708i \(0.670021\pi\)
\(510\) 3.38179e11 0.221350
\(511\) 1.23027e12 0.798189
\(512\) 6.87195e10 0.0441942
\(513\) −2.10802e11 −0.134384
\(514\) 4.74990e11 0.300158
\(515\) 3.28307e12 2.05659
\(516\) −8.11267e11 −0.503779
\(517\) 6.76998e11 0.416754
\(518\) −2.92918e12 −1.78757
\(519\) 2.07261e11 0.125390
\(520\) −2.31515e11 −0.138856
\(521\) −1.48896e12 −0.885345 −0.442672 0.896683i \(-0.645969\pi\)
−0.442672 + 0.896683i \(0.645969\pi\)
\(522\) 5.84912e11 0.344805
\(523\) 2.55715e12 1.49451 0.747256 0.664536i \(-0.231371\pi\)
0.747256 + 0.664536i \(0.231371\pi\)
\(524\) 1.28691e12 0.745691
\(525\) −1.48942e12 −0.855659
\(526\) −1.21463e12 −0.691846
\(527\) 3.15281e11 0.178053
\(528\) 9.26515e10 0.0518800
\(529\) −1.51327e12 −0.840169
\(530\) 2.10610e12 1.15941
\(531\) 1.52057e12 0.830005
\(532\) −2.15706e11 −0.116751
\(533\) 7.65784e11 0.410993
\(534\) −7.64984e9 −0.00407115
\(535\) −2.27912e12 −1.20275
\(536\) 7.90592e11 0.413724
\(537\) 5.15011e11 0.267259
\(538\) −4.11438e11 −0.211731
\(539\) 1.16794e12 0.596033
\(540\) 1.28205e12 0.648833
\(541\) −2.64921e12 −1.32962 −0.664811 0.747011i \(-0.731488\pi\)
−0.664811 + 0.747011i \(0.731488\pi\)
\(542\) −1.29422e12 −0.644188
\(543\) −1.81151e12 −0.894213
\(544\) −1.49320e11 −0.0731011
\(545\) 5.51111e12 2.67581
\(546\) −3.46673e11 −0.166937
\(547\) 2.16400e12 1.03351 0.516754 0.856134i \(-0.327140\pi\)
0.516754 + 0.856134i \(0.327140\pi\)
\(548\) −1.63559e12 −0.774752
\(549\) 2.91633e12 1.37013
\(550\) 5.92136e11 0.275924
\(551\) −2.16622e11 −0.100120
\(552\) −1.64826e11 −0.0755615
\(553\) −1.14170e12 −0.519145
\(554\) −5.44056e11 −0.245386
\(555\) −2.68638e12 −1.20185
\(556\) 1.95242e12 0.866437
\(557\) −2.96364e12 −1.30460 −0.652300 0.757961i \(-0.726196\pi\)
−0.652300 + 0.757961i \(0.726196\pi\)
\(558\) 4.97991e11 0.217454
\(559\) −1.20680e12 −0.522736
\(560\) 1.31187e12 0.563697
\(561\) −2.01322e11 −0.0858141
\(562\) −1.14373e11 −0.0483627
\(563\) −3.46859e12 −1.45501 −0.727504 0.686104i \(-0.759320\pi\)
−0.727504 + 0.686104i \(0.759320\pi\)
\(564\) 6.89568e11 0.286960
\(565\) −1.34284e12 −0.554380
\(566\) −1.24881e12 −0.511474
\(567\) −8.79100e11 −0.357202
\(568\) −8.26710e11 −0.333262
\(569\) −3.83703e12 −1.53458 −0.767290 0.641300i \(-0.778395\pi\)
−0.767290 + 0.641300i \(0.778395\pi\)
\(570\) −1.97826e11 −0.0784957
\(571\) 3.35374e11 0.132028 0.0660142 0.997819i \(-0.478972\pi\)
0.0660142 + 0.997819i \(0.478972\pi\)
\(572\) 1.37824e11 0.0538323
\(573\) 2.69987e12 1.04628
\(574\) −4.33929e12 −1.66846
\(575\) −1.05341e12 −0.401874
\(576\) −2.35854e11 −0.0892775
\(577\) −1.74089e11 −0.0653852 −0.0326926 0.999465i \(-0.510408\pi\)
−0.0326926 + 0.999465i \(0.510408\pi\)
\(578\) −1.57295e12 −0.586191
\(579\) −1.27524e12 −0.471560
\(580\) 1.31745e12 0.483400
\(581\) −3.11799e12 −1.13523
\(582\) 1.04552e12 0.377728
\(583\) −1.25379e12 −0.449486
\(584\) −4.98189e11 −0.177230
\(585\) 7.94589e11 0.280505
\(586\) −2.02659e12 −0.709948
\(587\) 1.87089e12 0.650395 0.325198 0.945646i \(-0.394569\pi\)
0.325198 + 0.945646i \(0.394569\pi\)
\(588\) 1.18962e12 0.410404
\(589\) −1.84431e11 −0.0631415
\(590\) 3.42490e12 1.16363
\(591\) 2.98719e12 1.00721
\(592\) 1.18615e12 0.396911
\(593\) 4.16793e12 1.38412 0.692061 0.721839i \(-0.256703\pi\)
0.692061 + 0.721839i \(0.256703\pi\)
\(594\) −7.63221e11 −0.251543
\(595\) −2.85056e12 −0.932405
\(596\) −2.36404e12 −0.767445
\(597\) 9.83110e10 0.0316751
\(598\) −2.45188e11 −0.0784049
\(599\) 8.16635e10 0.0259183 0.0129592 0.999916i \(-0.495875\pi\)
0.0129592 + 0.999916i \(0.495875\pi\)
\(600\) 6.03131e11 0.189990
\(601\) −4.00769e12 −1.25302 −0.626511 0.779412i \(-0.715518\pi\)
−0.626511 + 0.779412i \(0.715518\pi\)
\(602\) 6.83830e12 2.12209
\(603\) −2.71341e12 −0.835772
\(604\) −8.33155e11 −0.254718
\(605\) 3.96320e12 1.20267
\(606\) −9.89294e11 −0.297988
\(607\) 1.45542e12 0.435149 0.217575 0.976044i \(-0.430185\pi\)
0.217575 + 0.976044i \(0.430185\pi\)
\(608\) 8.73485e10 0.0259232
\(609\) 1.97276e12 0.581161
\(610\) 6.56869e12 1.92086
\(611\) 1.02577e12 0.297758
\(612\) 5.12487e11 0.147673
\(613\) −2.55645e12 −0.731248 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(614\) 9.84102e11 0.279436
\(615\) −3.97961e12 −1.12177
\(616\) −7.80975e11 −0.218537
\(617\) 2.69727e11 0.0749276 0.0374638 0.999298i \(-0.488072\pi\)
0.0374638 + 0.999298i \(0.488072\pi\)
\(618\) −1.99075e12 −0.548994
\(619\) 4.28111e11 0.117206 0.0586028 0.998281i \(-0.481335\pi\)
0.0586028 + 0.998281i \(0.481335\pi\)
\(620\) 1.12167e12 0.304861
\(621\) 1.35776e12 0.366364
\(622\) 3.46237e12 0.927507
\(623\) 6.44818e10 0.0171491
\(624\) 1.40383e11 0.0370667
\(625\) −3.79469e12 −0.994755
\(626\) 3.91414e12 1.01871
\(627\) 1.17768e11 0.0304316
\(628\) 4.16831e11 0.106940
\(629\) −2.57739e12 −0.656525
\(630\) −4.50252e12 −1.13874
\(631\) −2.98421e11 −0.0749372 −0.0374686 0.999298i \(-0.511929\pi\)
−0.0374686 + 0.999298i \(0.511929\pi\)
\(632\) 4.62323e11 0.115271
\(633\) 1.76833e12 0.437770
\(634\) 1.24384e12 0.305747
\(635\) 6.89269e12 1.68231
\(636\) −1.27707e12 −0.309498
\(637\) 1.76963e12 0.425848
\(638\) −7.84293e11 −0.187407
\(639\) 2.83738e12 0.673229
\(640\) −5.31234e11 −0.125163
\(641\) −2.34144e12 −0.547799 −0.273899 0.961758i \(-0.588314\pi\)
−0.273899 + 0.961758i \(0.588314\pi\)
\(642\) 1.38198e12 0.321067
\(643\) 7.75186e12 1.78837 0.894183 0.447701i \(-0.147757\pi\)
0.894183 + 0.447701i \(0.147757\pi\)
\(644\) 1.38935e12 0.318291
\(645\) 6.27147e12 1.42676
\(646\) −1.89799e11 −0.0428794
\(647\) 3.74980e12 0.841278 0.420639 0.907228i \(-0.361806\pi\)
0.420639 + 0.907228i \(0.361806\pi\)
\(648\) 3.55985e11 0.0793130
\(649\) −2.03889e12 −0.451121
\(650\) 8.97188e11 0.197139
\(651\) 1.67960e12 0.366514
\(652\) −2.89760e12 −0.627948
\(653\) 2.82022e12 0.606979 0.303490 0.952835i \(-0.401848\pi\)
0.303490 + 0.952835i \(0.401848\pi\)
\(654\) −3.34176e12 −0.714291
\(655\) −9.94846e12 −2.11188
\(656\) 1.75717e12 0.370464
\(657\) 1.70985e12 0.358025
\(658\) −5.81248e12 −1.20877
\(659\) −1.16074e12 −0.239746 −0.119873 0.992789i \(-0.538249\pi\)
−0.119873 + 0.992789i \(0.538249\pi\)
\(660\) −7.16240e11 −0.146930
\(661\) −3.18976e11 −0.0649907 −0.0324954 0.999472i \(-0.510345\pi\)
−0.0324954 + 0.999472i \(0.510345\pi\)
\(662\) −2.69800e12 −0.545985
\(663\) −3.05038e11 −0.0613116
\(664\) 1.26261e12 0.252065
\(665\) 1.66750e12 0.330651
\(666\) −4.07103e12 −0.801807
\(667\) 1.39525e12 0.272952
\(668\) 4.41104e11 0.0857131
\(669\) 1.78745e11 0.0344997
\(670\) −6.11164e12 −1.17171
\(671\) −3.91043e12 −0.744686
\(672\) −7.95476e11 −0.150475
\(673\) 4.82897e12 0.907375 0.453687 0.891161i \(-0.350108\pi\)
0.453687 + 0.891161i \(0.350108\pi\)
\(674\) −1.23328e12 −0.230192
\(675\) −4.96832e12 −0.921176
\(676\) 2.08827e11 0.0384615
\(677\) −7.47095e12 −1.36687 −0.683434 0.730012i \(-0.739514\pi\)
−0.683434 + 0.730012i \(0.739514\pi\)
\(678\) 8.14256e11 0.147988
\(679\) −8.81286e12 −1.59112
\(680\) 1.15432e12 0.207031
\(681\) −5.59911e11 −0.0997601
\(682\) −6.67744e11 −0.118190
\(683\) −4.13060e12 −0.726306 −0.363153 0.931730i \(-0.618300\pi\)
−0.363153 + 0.931730i \(0.618300\pi\)
\(684\) −2.99791e11 −0.0523681
\(685\) 1.26439e13 2.19419
\(686\) −3.49672e12 −0.602840
\(687\) 1.97975e12 0.339082
\(688\) −2.76912e12 −0.471188
\(689\) −1.89971e12 −0.321144
\(690\) 1.27418e12 0.213999
\(691\) 4.50776e12 0.752159 0.376079 0.926587i \(-0.377272\pi\)
0.376079 + 0.926587i \(0.377272\pi\)
\(692\) 7.07449e11 0.117278
\(693\) 2.68041e12 0.441470
\(694\) 5.67375e12 0.928437
\(695\) −1.50931e13 −2.45385
\(696\) −7.98856e11 −0.129041
\(697\) −3.81814e12 −0.612780
\(698\) 2.72737e12 0.434905
\(699\) 1.80343e12 0.285727
\(700\) −5.08389e12 −0.800304
\(701\) −1.38907e12 −0.217267 −0.108633 0.994082i \(-0.534647\pi\)
−0.108633 + 0.994082i \(0.534647\pi\)
\(702\) −1.15641e12 −0.179720
\(703\) 1.50770e12 0.232818
\(704\) 3.16251e11 0.0485237
\(705\) −5.33068e12 −0.812703
\(706\) 4.74410e12 0.718675
\(707\) 8.33893e12 1.25523
\(708\) −2.07675e12 −0.310624
\(709\) 5.59251e12 0.831187 0.415593 0.909551i \(-0.363574\pi\)
0.415593 + 0.909551i \(0.363574\pi\)
\(710\) 6.39086e12 0.943836
\(711\) −1.58675e12 −0.232861
\(712\) −2.61115e10 −0.00380778
\(713\) 1.18791e12 0.172140
\(714\) 1.72849e12 0.248900
\(715\) −1.06544e12 −0.152459
\(716\) 1.75791e12 0.249969
\(717\) 4.47238e12 0.631979
\(718\) 1.15223e12 0.161800
\(719\) −8.33742e12 −1.16346 −0.581730 0.813382i \(-0.697624\pi\)
−0.581730 + 0.813382i \(0.697624\pi\)
\(720\) 1.82326e12 0.252844
\(721\) 1.67803e13 2.31255
\(722\) −5.05198e12 −0.691901
\(723\) 9.42295e11 0.128252
\(724\) −6.18328e12 −0.836364
\(725\) −5.10549e12 −0.686304
\(726\) −2.40315e12 −0.321045
\(727\) −5.13697e12 −0.682028 −0.341014 0.940058i \(-0.610770\pi\)
−0.341014 + 0.940058i \(0.610770\pi\)
\(728\) −1.18331e12 −0.156138
\(729\) 2.51391e12 0.329667
\(730\) 3.85123e12 0.501934
\(731\) 6.01702e12 0.779387
\(732\) −3.98304e12 −0.512761
\(733\) 9.72563e12 1.24437 0.622185 0.782870i \(-0.286245\pi\)
0.622185 + 0.782870i \(0.286245\pi\)
\(734\) −1.68147e12 −0.213825
\(735\) −9.19636e12 −1.16231
\(736\) −5.62607e11 −0.0706733
\(737\) 3.63834e12 0.454255
\(738\) −6.03082e12 −0.748381
\(739\) −1.15533e13 −1.42498 −0.712488 0.701684i \(-0.752432\pi\)
−0.712488 + 0.701684i \(0.752432\pi\)
\(740\) −9.16951e12 −1.12410
\(741\) 1.78439e11 0.0217424
\(742\) 1.07646e13 1.30371
\(743\) −1.04495e13 −1.25790 −0.628948 0.777447i \(-0.716514\pi\)
−0.628948 + 0.777447i \(0.716514\pi\)
\(744\) −6.80142e11 −0.0813807
\(745\) 1.82752e13 2.17349
\(746\) 3.51821e12 0.415907
\(747\) −4.33344e12 −0.509202
\(748\) −6.87180e11 −0.0802626
\(749\) −1.16490e13 −1.35245
\(750\) −2.42016e10 −0.00279298
\(751\) 1.92230e12 0.220516 0.110258 0.993903i \(-0.464832\pi\)
0.110258 + 0.993903i \(0.464832\pi\)
\(752\) 2.35372e12 0.268396
\(753\) 1.81328e12 0.205536
\(754\) −1.18834e12 −0.133897
\(755\) 6.44068e12 0.721390
\(756\) 6.55277e12 0.729587
\(757\) −1.27412e12 −0.141019 −0.0705096 0.997511i \(-0.522463\pi\)
−0.0705096 + 0.997511i \(0.522463\pi\)
\(758\) −5.03597e12 −0.554079
\(759\) −7.58539e11 −0.0829641
\(760\) −6.75245e11 −0.0734176
\(761\) 1.21619e13 1.31453 0.657267 0.753658i \(-0.271712\pi\)
0.657267 + 0.753658i \(0.271712\pi\)
\(762\) −4.17950e12 −0.449083
\(763\) 2.81682e13 3.00884
\(764\) 9.21557e12 0.978593
\(765\) −3.96176e12 −0.418227
\(766\) −5.45643e11 −0.0572637
\(767\) −3.08927e12 −0.322312
\(768\) 3.22123e11 0.0334115
\(769\) −5.98877e12 −0.617546 −0.308773 0.951136i \(-0.599918\pi\)
−0.308773 + 0.951136i \(0.599918\pi\)
\(770\) 6.03730e12 0.618920
\(771\) 2.22651e12 0.226924
\(772\) −4.35281e12 −0.441054
\(773\) −9.64838e12 −0.971956 −0.485978 0.873971i \(-0.661536\pi\)
−0.485978 + 0.873971i \(0.661536\pi\)
\(774\) 9.50399e12 0.951857
\(775\) −4.34679e12 −0.432824
\(776\) 3.56871e12 0.353292
\(777\) −1.37305e13 −1.35143
\(778\) 7.19379e12 0.703963
\(779\) 2.23351e12 0.217305
\(780\) −1.08523e12 −0.104977
\(781\) −3.80456e12 −0.365911
\(782\) 1.22249e12 0.116900
\(783\) 6.58061e12 0.625660
\(784\) 4.06059e12 0.383854
\(785\) −3.22230e12 −0.302867
\(786\) 6.03241e12 0.563754
\(787\) 1.40829e13 1.30860 0.654299 0.756236i \(-0.272964\pi\)
0.654299 + 0.756236i \(0.272964\pi\)
\(788\) 1.01963e13 0.942049
\(789\) −5.69360e12 −0.523046
\(790\) −3.57398e12 −0.326460
\(791\) −6.86350e12 −0.623378
\(792\) −1.08541e12 −0.0980238
\(793\) −5.92498e12 −0.532056
\(794\) 3.67961e12 0.328556
\(795\) 9.87235e12 0.876533
\(796\) 3.35568e11 0.0296259
\(797\) −8.45863e12 −0.742570 −0.371285 0.928519i \(-0.621083\pi\)
−0.371285 + 0.928519i \(0.621083\pi\)
\(798\) −1.01112e12 −0.0882652
\(799\) −5.11440e12 −0.443950
\(800\) 2.05869e12 0.177699
\(801\) 8.96179e10 0.00769216
\(802\) −1.07066e13 −0.913834
\(803\) −2.29269e12 −0.194592
\(804\) 3.70590e12 0.312782
\(805\) −1.07403e13 −0.901437
\(806\) −1.01175e12 −0.0844431
\(807\) −1.92862e12 −0.160072
\(808\) −3.37679e12 −0.278710
\(809\) −9.95988e12 −0.817496 −0.408748 0.912647i \(-0.634035\pi\)
−0.408748 + 0.912647i \(0.634035\pi\)
\(810\) −2.75193e12 −0.224624
\(811\) 9.83262e12 0.798133 0.399067 0.916922i \(-0.369334\pi\)
0.399067 + 0.916922i \(0.369334\pi\)
\(812\) 6.73369e12 0.543564
\(813\) −6.06668e12 −0.487016
\(814\) 5.45873e12 0.435795
\(815\) 2.23998e13 1.77842
\(816\) −6.99939e11 −0.0552656
\(817\) −3.51980e12 −0.276388
\(818\) −1.07828e13 −0.842055
\(819\) 4.06128e12 0.315417
\(820\) −1.35837e13 −1.04920
\(821\) 8.67049e12 0.666039 0.333019 0.942920i \(-0.391933\pi\)
0.333019 + 0.942920i \(0.391933\pi\)
\(822\) −7.66684e12 −0.585725
\(823\) −1.26458e13 −0.960834 −0.480417 0.877040i \(-0.659515\pi\)
−0.480417 + 0.877040i \(0.659515\pi\)
\(824\) −6.79508e12 −0.513479
\(825\) 2.77564e12 0.208603
\(826\) 1.75053e13 1.30845
\(827\) −2.04741e13 −1.52205 −0.761027 0.648720i \(-0.775305\pi\)
−0.761027 + 0.648720i \(0.775305\pi\)
\(828\) 1.93094e12 0.142768
\(829\) −1.00940e13 −0.742283 −0.371141 0.928576i \(-0.621033\pi\)
−0.371141 + 0.928576i \(0.621033\pi\)
\(830\) −9.76056e12 −0.713877
\(831\) −2.55026e12 −0.185516
\(832\) 4.79174e11 0.0346688
\(833\) −8.82324e12 −0.634929
\(834\) 9.15198e12 0.655040
\(835\) −3.40994e12 −0.242749
\(836\) 4.01982e11 0.0284629
\(837\) 5.60270e12 0.394578
\(838\) −5.52085e12 −0.386730
\(839\) 1.54505e13 1.07650 0.538249 0.842786i \(-0.319086\pi\)
0.538249 + 0.842786i \(0.319086\pi\)
\(840\) 6.14940e12 0.426163
\(841\) −7.74485e12 −0.533864
\(842\) −8.92508e12 −0.611938
\(843\) −5.36123e11 −0.0365629
\(844\) 6.03589e12 0.409449
\(845\) −1.61433e12 −0.108927
\(846\) −8.07829e12 −0.542191
\(847\) 2.02566e13 1.35235
\(848\) −4.35907e12 −0.289476
\(849\) −5.85381e12 −0.386682
\(850\) −4.47331e12 −0.293930
\(851\) −9.71104e12 −0.634721
\(852\) −3.87520e12 −0.251951
\(853\) 2.56415e13 1.65834 0.829170 0.558997i \(-0.188814\pi\)
0.829170 + 0.558997i \(0.188814\pi\)
\(854\) 3.35737e13 2.15993
\(855\) 2.31753e12 0.148312
\(856\) 4.71717e12 0.300296
\(857\) −1.44941e11 −0.00917862 −0.00458931 0.999989i \(-0.501461\pi\)
−0.00458931 + 0.999989i \(0.501461\pi\)
\(858\) 6.46050e11 0.0406980
\(859\) −9.47728e12 −0.593902 −0.296951 0.954893i \(-0.595970\pi\)
−0.296951 + 0.954893i \(0.595970\pi\)
\(860\) 2.14066e13 1.33446
\(861\) −2.03404e13 −1.26138
\(862\) −1.02279e13 −0.630962
\(863\) −4.25726e12 −0.261265 −0.130633 0.991431i \(-0.541701\pi\)
−0.130633 + 0.991431i \(0.541701\pi\)
\(864\) −2.65350e12 −0.161997
\(865\) −5.46892e12 −0.332146
\(866\) −1.98015e13 −1.19638
\(867\) −7.37319e12 −0.443169
\(868\) 5.73303e12 0.342804
\(869\) 2.12764e12 0.126563
\(870\) 6.17553e12 0.365458
\(871\) 5.51272e12 0.324552
\(872\) −1.14065e13 −0.668081
\(873\) −1.22483e13 −0.713692
\(874\) −7.15123e11 −0.0414552
\(875\) 2.03999e11 0.0117650
\(876\) −2.33526e12 −0.133988
\(877\) 1.42035e13 0.810768 0.405384 0.914147i \(-0.367138\pi\)
0.405384 + 0.914147i \(0.367138\pi\)
\(878\) −1.46376e13 −0.831276
\(879\) −9.49964e12 −0.536732
\(880\) −2.44476e12 −0.137425
\(881\) −1.12878e13 −0.631275 −0.315638 0.948880i \(-0.602218\pi\)
−0.315638 + 0.948880i \(0.602218\pi\)
\(882\) −1.39365e13 −0.775432
\(883\) 2.25433e12 0.124794 0.0623972 0.998051i \(-0.480125\pi\)
0.0623972 + 0.998051i \(0.480125\pi\)
\(884\) −1.04120e12 −0.0573452
\(885\) 1.60542e13 0.879721
\(886\) −1.89841e13 −1.03499
\(887\) −3.13450e13 −1.70024 −0.850122 0.526585i \(-0.823472\pi\)
−0.850122 + 0.526585i \(0.823472\pi\)
\(888\) 5.56009e12 0.300070
\(889\) 3.52297e13 1.89170
\(890\) 2.01854e11 0.0107841
\(891\) 1.63826e12 0.0870831
\(892\) 6.10115e11 0.0322678
\(893\) 2.99179e12 0.157434
\(894\) −1.10815e13 −0.580200
\(895\) −1.35894e13 −0.707942
\(896\) −2.71522e12 −0.140741
\(897\) −1.14932e12 −0.0592753
\(898\) −4.33294e11 −0.0222351
\(899\) 5.75739e12 0.293973
\(900\) −7.06568e12 −0.358974
\(901\) 9.47180e12 0.478819
\(902\) 8.08657e12 0.406757
\(903\) 3.20545e13 1.60433
\(904\) 2.77933e12 0.138415
\(905\) 4.77997e13 2.36868
\(906\) −3.90541e12 −0.192571
\(907\) 1.77171e13 0.869280 0.434640 0.900604i \(-0.356876\pi\)
0.434640 + 0.900604i \(0.356876\pi\)
\(908\) −1.91116e12 −0.0933064
\(909\) 1.15896e13 0.563028
\(910\) 9.14756e12 0.442200
\(911\) 2.93419e13 1.41142 0.705710 0.708501i \(-0.250628\pi\)
0.705710 + 0.708501i \(0.250628\pi\)
\(912\) 4.09446e11 0.0195984
\(913\) 5.81059e12 0.276759
\(914\) 3.24138e12 0.153628
\(915\) 3.07908e13 1.45220
\(916\) 6.75754e12 0.317146
\(917\) −5.08482e13 −2.37473
\(918\) 5.76578e12 0.267958
\(919\) −1.66013e13 −0.767754 −0.383877 0.923384i \(-0.625411\pi\)
−0.383877 + 0.923384i \(0.625411\pi\)
\(920\) 4.34922e12 0.200155
\(921\) 4.61298e12 0.211258
\(922\) 1.43415e13 0.653592
\(923\) −5.76457e12 −0.261432
\(924\) −3.66082e12 −0.165217
\(925\) 3.55345e13 1.59593
\(926\) −8.28384e12 −0.370239
\(927\) 2.33216e13 1.03729
\(928\) −2.72676e12 −0.120693
\(929\) −1.45567e13 −0.641197 −0.320598 0.947215i \(-0.603884\pi\)
−0.320598 + 0.947215i \(0.603884\pi\)
\(930\) 5.25782e12 0.230479
\(931\) 5.16136e12 0.225160
\(932\) 6.15569e12 0.267242
\(933\) 1.62299e13 0.701210
\(934\) 1.47326e13 0.633457
\(935\) 5.31222e12 0.227313
\(936\) −1.64459e12 −0.0700351
\(937\) −1.38682e13 −0.587751 −0.293875 0.955844i \(-0.594945\pi\)
−0.293875 + 0.955844i \(0.594945\pi\)
\(938\) −3.12376e13 −1.31754
\(939\) 1.83476e13 0.770164
\(940\) −1.81954e13 −0.760127
\(941\) −2.15767e13 −0.897081 −0.448540 0.893763i \(-0.648056\pi\)
−0.448540 + 0.893763i \(0.648056\pi\)
\(942\) 1.95390e12 0.0808486
\(943\) −1.43859e13 −0.592428
\(944\) −7.08864e12 −0.290528
\(945\) −5.06560e13 −2.06627
\(946\) −1.27436e13 −0.517349
\(947\) 7.20814e12 0.291238 0.145619 0.989341i \(-0.453483\pi\)
0.145619 + 0.989341i \(0.453483\pi\)
\(948\) 2.16714e12 0.0871464
\(949\) −3.47382e12 −0.139030
\(950\) 2.61677e12 0.104234
\(951\) 5.83049e12 0.231149
\(952\) 5.89990e12 0.232798
\(953\) 2.71758e13 1.06724 0.533622 0.845723i \(-0.320830\pi\)
0.533622 + 0.845723i \(0.320830\pi\)
\(954\) 1.49609e13 0.584776
\(955\) −7.12407e13 −2.77149
\(956\) 1.52657e13 0.591095
\(957\) −3.67637e12 −0.141682
\(958\) 2.77205e13 1.06330
\(959\) 6.46251e13 2.46727
\(960\) −2.49016e12 −0.0946251
\(961\) −2.15378e13 −0.814603
\(962\) 8.27092e12 0.311362
\(963\) −1.61899e13 −0.606635
\(964\) 3.21637e12 0.119955
\(965\) 3.36492e13 1.24911
\(966\) 6.51257e12 0.240633
\(967\) 1.20281e13 0.442364 0.221182 0.975233i \(-0.429009\pi\)
0.221182 + 0.975233i \(0.429009\pi\)
\(968\) −8.20275e12 −0.300276
\(969\) −8.89684e11 −0.0324174
\(970\) −2.75878e13 −1.00056
\(971\) −1.10321e12 −0.0398265 −0.0199132 0.999802i \(-0.506339\pi\)
−0.0199132 + 0.999802i \(0.506339\pi\)
\(972\) 1.44199e13 0.518159
\(973\) −7.71436e13 −2.75926
\(974\) 2.39346e13 0.852140
\(975\) 4.20557e12 0.149040
\(976\) −1.35954e13 −0.479589
\(977\) −4.02945e13 −1.41488 −0.707441 0.706773i \(-0.750150\pi\)
−0.707441 + 0.706773i \(0.750150\pi\)
\(978\) −1.35825e13 −0.474739
\(979\) −1.20166e11 −0.00418081
\(980\) −3.13902e13 −1.08712
\(981\) 3.91487e13 1.34960
\(982\) −6.86326e12 −0.235520
\(983\) 5.04270e13 1.72255 0.861276 0.508137i \(-0.169666\pi\)
0.861276 + 0.508137i \(0.169666\pi\)
\(984\) 8.23672e12 0.280076
\(985\) −7.88219e13 −2.66799
\(986\) 5.92497e12 0.199637
\(987\) −2.72460e13 −0.913852
\(988\) 6.09072e11 0.0203359
\(989\) 2.26708e13 0.753502
\(990\) 8.39075e12 0.277615
\(991\) −2.37023e13 −0.780653 −0.390327 0.920676i \(-0.627638\pi\)
−0.390327 + 0.920676i \(0.627638\pi\)
\(992\) −2.32155e12 −0.0761160
\(993\) −1.26469e13 −0.412773
\(994\) 3.26647e13 1.06131
\(995\) −2.59410e12 −0.0839040
\(996\) 5.91848e12 0.190565
\(997\) −3.80094e13 −1.21832 −0.609162 0.793045i \(-0.708494\pi\)
−0.609162 + 0.793045i \(0.708494\pi\)
\(998\) −1.45948e13 −0.465704
\(999\) −4.58015e13 −1.45491
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 26.10.a.c.1.1 1
3.2 odd 2 234.10.a.a.1.1 1
4.3 odd 2 208.10.a.b.1.1 1
13.12 even 2 338.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.c.1.1 1 1.1 even 1 trivial
208.10.a.b.1.1 1 4.3 odd 2
234.10.a.a.1.1 1 3.2 odd 2
338.10.a.b.1.1 1 13.12 even 2