Properties

Label 338.10.a.b.1.1
Level $338$
Weight $10$
Character 338.1
Self dual yes
Analytic conductor $174.082$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,10,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, 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("338.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(174.082112623\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

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

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 \(-0.249585\pi\)
0.708029 + 0.706184i \(0.249585\pi\)
\(6\) −1200.00 −0.378008
\(7\) 10115.0 1.59230 0.796150 0.605100i \(-0.206867\pi\)
0.796150 + 0.605100i \(0.206867\pi\)
\(8\) −4096.00 −0.353553
\(9\) −14058.0 −0.714220
\(10\) −31664.0 −1.00130
\(11\) −18850.0 −0.388190 −0.194095 0.980983i \(-0.562177\pi\)
−0.194095 + 0.980983i \(0.562177\pi\)
\(12\) 19200.0 0.267292
\(13\) 0 0
\(14\) −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 \(-0.523360\pi\)
−0.0733220 + 0.997308i \(0.523360\pi\)
\(20\) 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
\(26\) 0 0
\(27\) −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 \(-0.430931\pi\)
0.215288 + 0.976550i \(0.430931\pi\)
\(32\) −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 \(-0.791911\pi\)
−0.793821 + 0.608151i \(0.791911\pi\)
\(38\) 1.33283e6 0.103693
\(39\) 0 0
\(40\) −8.10598e6 −0.500652
\(41\) −2.68122e7 −1.48186 −0.740928 0.671585i \(-0.765614\pi\)
−0.740928 + 0.671585i \(0.765614\pi\)
\(42\) −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 \(-0.680364\pi\)
−0.536791 + 0.843715i \(0.680364\pi\)
\(48\) 4.91520e6 0.133646
\(49\) 6.19596e7 1.53542
\(50\) −3.14131e7 −0.710796
\(51\) −1.06802e7 −0.221062
\(52\) 0 0
\(53\) −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 \(-0.302639\pi\)
0.581057 + 0.813863i \(0.302639\pi\)
\(60\) 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
\(65\) 0 0
\(66\) 2.26200e7 0.146739
\(67\) −1.93016e8 −1.17019 −0.585094 0.810966i \(-0.698942\pi\)
−0.585094 + 0.810966i \(0.698942\pi\)
\(68\) −3.64552e7 −0.206761
\(69\) −4.02408e7 −0.213720
\(70\) −3.20281e8 −1.59438
\(71\) 2.01833e8 0.942607 0.471304 0.881971i \(-0.343784\pi\)
0.471304 + 0.881971i \(0.343784\pi\)
\(72\) 5.75816e7 0.252515
\(73\) 1.21628e8 0.501281 0.250640 0.968080i \(-0.419359\pi\)
0.250640 + 0.968080i \(0.419359\pi\)
\(74\) 2.89588e8 1.12263
\(75\) 1.47249e8 0.537373
\(76\) −2.13253e7 −0.0733220
\(77\) −1.90668e8 −0.618115
\(78\) 0 0
\(79\) 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 \(-0.616021\pi\)
−0.356474 + 0.934305i \(0.616021\pi\)
\(84\) 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 \(-0.498286\pi\)
0.00538501 + 0.999986i \(0.498286\pi\)
\(90\) 4.45133e8 0.715151
\(91\) 0 0
\(92\) −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 \(-0.666531\pi\)
−0.499630 + 0.866239i \(0.666531\pi\)
\(98\) −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 \(0.758701\pi\)
\(104\) 0 0
\(105\) 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 \(-0.106247\pi\)
0.944810 + 0.327620i \(0.106247\pi\)
\(110\) 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
\(117\) 0 0
\(118\) −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
\(130\) 0 0
\(131\) 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 \(-0.217875\pi\)
0.774752 + 0.632265i \(0.217875\pi\)
\(138\) 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
\(143\) 0 0
\(144\) −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 \(-0.221528\pi\)
0.767445 + 0.641114i \(0.221528\pi\)
\(150\) −2.35598e9 −0.379980
\(151\) 3.25451e9 0.509436 0.254718 0.967015i \(-0.418017\pi\)
0.254718 + 0.967015i \(0.418017\pi\)
\(152\) 3.41205e8 0.0518465
\(153\) 2.00190e9 0.295346
\(154\) 3.05068e9 0.437073
\(155\) 4.38151e9 0.609722
\(156\) 0 0
\(157\) 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 \(-0.283895\pi\)
0.627948 + 0.778255i \(0.283895\pi\)
\(164\) −6.86393e9 −0.740928
\(165\) −2.79781e9 −0.293860
\(166\) 4.93207e9 0.504130
\(167\) −1.72306e9 −0.171426 −0.0857131 0.996320i \(-0.527317\pi\)
−0.0857131 + 0.996320i \(0.527317\pi\)
\(168\) −3.10733e9 −0.300951
\(169\) 0 0
\(170\) 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 \(0.815323\pi\)
\(182\) 0 0
\(183\) −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 \(-0.354605\pi\)
0.441054 + 0.897481i \(0.354605\pi\)
\(194\) 1.39403e10 0.706583
\(195\) 0 0
\(196\) 1.58617e10 0.767709
\(197\) −3.98292e10 −1.88410 −0.942049 0.335477i \(-0.891103\pi\)
−0.942049 + 0.335477i \(0.891103\pi\)
\(198\) −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
\(208\) 0 0
\(209\) 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
\(221\) 0 0
\(222\) 2.17191e10 0.600141
\(223\) −2.38326e9 −0.0645356 −0.0322678 0.999479i \(-0.510273\pi\)
−0.0322678 + 0.999479i \(0.510273\pi\)
\(224\) −1.06063e10 −0.281481
\(225\) −2.76003e10 −0.717947
\(226\) −1.08568e10 −0.276829
\(227\) 7.46548e9 0.186613 0.0933064 0.995637i \(-0.470256\pi\)
0.0933064 + 0.995637i \(0.470256\pi\)
\(228\) −1.59940e9 −0.0391967
\(229\) −2.63966e10 −0.634292 −0.317146 0.948377i \(-0.602724\pi\)
−0.317146 + 0.948377i \(0.602724\pi\)
\(230\) 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 \(0.413888\pi\)
\(234\) 0 0
\(235\) −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 \(-0.701304\pi\)
−0.591095 + 0.806602i \(0.701304\pi\)
\(240\) 9.72718e9 0.189250
\(241\) −1.25639e10 −0.239910 −0.119955 0.992779i \(-0.538275\pi\)
−0.119955 + 0.992779i \(0.538275\pi\)
\(242\) 3.20420e10 0.600552
\(243\) 5.63276e10 1.03632
\(244\) −5.31072e10 −0.959178
\(245\) 1.22618e11 2.17424
\(246\) 3.21747e10 0.560153
\(247\) 0 0
\(248\) −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
\(260\) 0 0
\(261\) 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 \(-0.349457\pi\)
0.455510 + 0.890231i \(0.349457\pi\)
\(272\) −9.33252e9 −0.103381
\(273\) 0 0
\(274\) −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 \(-0.489112\pi\)
0.0341976 + 0.999415i \(0.489112\pi\)
\(282\) 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
\(286\) 0 0
\(287\) −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 \(-0.332594\pi\)
0.502009 + 0.864862i \(0.332594\pi\)
\(294\) −7.43515e10 −0.580399
\(295\) 2.14057e11 1.64562
\(296\) 7.41345e10 0.561316
\(297\) 4.77013e10 0.355735
\(298\) −1.47753e11 −1.08533
\(299\) 0 0
\(300\) 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 \(-0.563312\pi\)
−0.197591 + 0.980285i \(0.563312\pi\)
\(308\) −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 \(0.272312\pi\)
\(312\) 0 0
\(313\) 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 \(-0.569365\pi\)
−0.216196 + 0.976350i \(0.569365\pi\)
\(318\) 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
\(325\) 0 0
\(326\) −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 \(-0.373832\pi\)
0.386070 + 0.922470i \(0.373832\pi\)
\(332\) −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 \(0.552043\pi\)
\(338\) 0 0
\(339\) 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 \(-0.599500\pi\)
−0.307524 + 0.951540i \(0.599500\pi\)
\(350\) −3.17743e11 −1.13180
\(351\) 0 0
\(352\) 1.97657e10 0.0686229
\(353\) −2.96506e11 −1.01636 −0.508180 0.861251i \(-0.669682\pi\)
−0.508180 + 0.861251i \(0.669682\pi\)
\(354\) −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 \(-0.536498\pi\)
−0.114410 + 0.993434i \(0.536498\pi\)
\(360\) 1.13954e11 0.357576
\(361\) −3.15748e11 −0.978496
\(362\) 3.86455e11 1.18280
\(363\) −1.50197e11 −0.454026
\(364\) 0 0
\(365\) 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
\(377\) 0 0
\(378\) 4.09548e11 1.03179
\(379\) 3.14748e11 0.783586 0.391793 0.920053i \(-0.371855\pi\)
0.391793 + 0.920053i \(0.371855\pi\)
\(380\) −4.22028e10 −0.103828
\(381\) −2.61219e11 −0.635100
\(382\) −5.75973e11 −1.38394
\(383\) 3.41027e10 0.0809831 0.0404915 0.999180i \(-0.487108\pi\)
0.0404915 + 0.999180i \(0.487108\pi\)
\(384\) −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 \(0.334150\pi\)
\(390\) 0 0
\(391\) 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 \(-0.574633\pi\)
−0.232324 + 0.972638i \(0.574633\pi\)
\(398\) −2.09730e10 −0.0418974
\(399\) −6.31950e10 −0.124826
\(400\) 1.28668e11 0.251304
\(401\) 6.69163e11 1.29236 0.646178 0.763187i \(-0.276366\pi\)
0.646178 + 0.763187i \(0.276366\pi\)
\(402\) 2.31619e11 0.442340
\(403\) 0 0
\(404\) −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 \(-0.296985\pi\)
0.595423 + 0.803413i \(0.296985\pi\)
\(410\) 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
\(416\) 0 0
\(417\) 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 \(-0.357559\pi\)
0.432706 + 0.901535i \(0.357559\pi\)
\(422\) −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
\(429\) 0 0
\(430\) 1.33791e12 1.88721
\(431\) 6.39243e11 0.892315 0.446158 0.894954i \(-0.352792\pi\)
0.446158 + 0.894954i \(0.352792\pi\)
\(432\) −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
\(442\) 0 0
\(443\) −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 \(-0.494995\pi\)
0.0157226 + 0.999876i \(0.494995\pi\)
\(450\) 4.41605e11 0.507665
\(451\) 5.05411e11 0.575241
\(452\) 1.73708e11 0.195748
\(453\) 2.44088e11 0.272336
\(454\) −1.19448e11 −0.131955
\(455\) 0 0
\(456\) 2.55904e10 0.0277163
\(457\) −2.02586e11 −0.217263 −0.108632 0.994082i \(-0.534647\pi\)
−0.108632 + 0.994082i \(0.534647\pi\)
\(458\) 4.22346e11 0.448512
\(459\) 3.60361e11 0.378949
\(460\) −2.71826e11 −0.283062
\(461\) −8.96346e11 −0.924319 −0.462159 0.886797i \(-0.652925\pi\)
−0.462159 + 0.886797i \(0.652925\pi\)
\(462\) 2.28801e11 0.233652
\(463\) 5.17740e11 0.523597 0.261799 0.965123i \(-0.415684\pi\)
0.261799 + 0.965123i \(0.415684\pi\)
\(464\) −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 \(0.352164\pi\)
\(468\) 0 0
\(469\) −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 \(-0.770846\pi\)
−0.751867 + 0.659315i \(0.770846\pi\)
\(480\) −1.55635e11 −0.133820
\(481\) 0 0
\(482\) 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 \(-0.705850\pi\)
−0.602554 + 0.798078i \(0.705850\pi\)
\(488\) 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
\(494\) 0 0
\(495\) 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 \(-0.393186\pi\)
0.329303 + 0.944224i \(0.393186\pi\)
\(500\) 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
\(507\) 0 0
\(508\) −8.91627e11 −0.594014
\(509\) 1.54192e12 1.01820 0.509098 0.860708i \(-0.329979\pi\)
0.509098 + 0.860708i \(0.329979\pi\)
\(510\) 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
\(520\) 0 0
\(521\) −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
\(533\) 0 0
\(534\) −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 \(-0.268512\pi\)
0.664811 + 0.747011i \(0.268512\pi\)
\(542\) −1.29422e12 −0.644188
\(543\) −1.81151e12 −0.894213
\(544\) 1.49320e11 0.0731011
\(545\) 5.51111e12 2.67581
\(546\) 0 0
\(547\) 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 \(-0.273804\pi\)
0.652300 + 0.757961i \(0.273804\pi\)
\(558\) 4.97991e11 0.217454
\(559\) 0 0
\(560\) 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 \(0.478972\pi\)
\(572\) 0 0
\(573\) 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 \(-0.489592\pi\)
0.0326926 + 0.999465i \(0.489592\pi\)
\(578\) 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
\(585\) 0 0
\(586\) −2.02659e12 −0.709948
\(587\) −1.87089e12 −0.650395 −0.325198 0.945646i \(-0.605431\pi\)
−0.325198 + 0.945646i \(0.605431\pi\)
\(588\) 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 \(-0.743297\pi\)
−0.692061 + 0.721839i \(0.743297\pi\)
\(594\) −7.63221e11 −0.251543
\(595\) −2.85056e12 −0.932405
\(596\) 2.36404e12 0.767445
\(597\) 9.83110e10 0.0316751
\(598\) 0 0
\(599\) 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
\(611\) 0 0
\(612\) 5.12487e11 0.147673
\(613\) 2.55645e12 0.731248 0.365624 0.930763i \(-0.380856\pi\)
0.365624 + 0.930763i \(0.380856\pi\)
\(614\) 9.84102e11 0.279436
\(615\) −3.97961e12 −1.12177
\(616\) 7.80975e11 0.218537
\(617\) −2.69727e11 −0.0749276 −0.0374638 0.999298i \(-0.511928\pi\)
−0.0374638 + 0.999298i \(0.511928\pi\)
\(618\) 1.99075e12 0.548994
\(619\) −4.28111e11 −0.117206 −0.0586028 0.998281i \(-0.518665\pi\)
−0.0586028 + 0.998281i \(0.518665\pi\)
\(620\) 1.12167e12 0.304861
\(621\) 1.35776e12 0.366364
\(622\) −3.46237e12 −0.927507
\(623\) 6.44818e10 0.0171491
\(624\) 0 0
\(625\) −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 \(-0.488071\pi\)
0.0374686 + 0.999298i \(0.488071\pi\)
\(632\) −4.62323e11 −0.115271
\(633\) 1.76833e12 0.437770
\(634\) 1.24384e12 0.305747
\(635\) −6.89269e12 −1.68231
\(636\) −1.27707e12 −0.309498
\(637\) 0 0
\(638\) −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 \(-0.852243\pi\)
−0.894183 + 0.447701i \(0.852243\pi\)
\(644\) −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
\(650\) 0 0
\(651\) 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 \(-0.489655\pi\)
0.0324954 + 0.999472i \(0.489655\pi\)
\(662\) −2.69800e12 −0.545985
\(663\) 0 0
\(664\) 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
\(676\) 0 0
\(677\) −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 \(-0.381700\pi\)
0.363153 + 0.931730i \(0.381700\pi\)
\(684\) 2.99791e11 0.0523681
\(685\) 1.26439e13 2.19419
\(686\) −3.49672e12 −0.602840
\(687\) −1.97975e12 −0.339082
\(688\) −2.76912e12 −0.471188
\(689\) 0 0
\(690\) 1.27418e12 0.213999
\(691\) −4.50776e12 −0.752159 −0.376079 0.926587i \(-0.622728\pi\)
−0.376079 + 0.926587i \(0.622728\pi\)
\(692\) 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 \(0.534647\pi\)
\(702\) 0 0
\(703\) 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 \(-0.636426\pi\)
−0.415593 + 0.909551i \(0.636426\pi\)
\(710\) −6.39086e12 −0.943836
\(711\) −1.58675e12 −0.232861
\(712\) −2.61115e10 −0.00380778
\(713\) −1.18791e12 −0.172140
\(714\) 1.72849e12 0.248900
\(715\) 0 0
\(716\) 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 \(0.610770\pi\)
\(728\) 0 0
\(729\) 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 \(-0.713755\pi\)
−0.622185 + 0.782870i \(0.713755\pi\)
\(734\) 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 \(-0.247568\pi\)
0.712488 + 0.701684i \(0.247568\pi\)
\(740\) −9.16951e12 −1.12410
\(741\) 0 0
\(742\) 1.07646e13 1.30371
\(743\) 1.04495e13 1.25790 0.628948 0.777447i \(-0.283486\pi\)
0.628948 + 0.777447i \(0.283486\pi\)
\(744\) −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
\(754\) 0 0
\(755\) 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 \(-0.728288\pi\)
−0.657267 + 0.753658i \(0.728288\pi\)
\(762\) 4.17950e12 0.449083
\(763\) 2.81682e13 3.00884
\(764\) 9.21557e12 0.978593
\(765\) 3.96176e12 0.418227
\(766\) −5.45643e11 −0.0572637
\(767\) 0 0
\(768\) 3.22123e11 0.0334115
\(769\) 5.98877e12 0.617546 0.308773 0.951136i \(-0.400082\pi\)
0.308773 + 0.951136i \(0.400082\pi\)
\(770\) 6.03730e12 0.618920
\(771\) 2.22651e12 0.226924
\(772\) 4.35281e12 0.441054
\(773\) 9.64838e12 0.971956 0.485978 0.873971i \(-0.338464\pi\)
0.485978 + 0.873971i \(0.338464\pi\)
\(774\) −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
\(780\) 0 0
\(781\) −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 \(-0.727036\pi\)
−0.654299 + 0.756236i \(0.727036\pi\)
\(788\) −1.01963e13 −0.942049
\(789\) −5.69360e12 −0.523046
\(790\) −3.57398e12 −0.326460
\(791\) 6.86350e12 0.623378
\(792\) −1.08541e12 −0.0980238
\(793\) 0 0
\(794\) 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
\(806\) 0 0
\(807\) −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 \(-0.630666\pi\)
−0.399067 + 0.916922i \(0.630666\pi\)
\(812\) −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
\(819\) 0 0
\(820\) −1.35837e13 −1.04920
\(821\) −8.67049e12 −0.666039 −0.333019 0.942920i \(-0.608067\pi\)
−0.333019 + 0.942920i \(0.608067\pi\)
\(822\) −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 \(-0.224695\pi\)
0.761027 + 0.648720i \(0.224695\pi\)
\(828\) 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
\(832\) 0 0
\(833\) −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 \(-0.680914\pi\)
−0.538249 + 0.842786i \(0.680914\pi\)
\(840\) −6.14940e12 −0.426163
\(841\) −7.74485e12 −0.533864
\(842\) −8.92508e12 −0.611938
\(843\) 5.36123e11 0.0365629
\(844\) 6.03589e12 0.409449
\(845\) 0 0
\(846\) −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 \(-0.811186\pi\)
−0.829170 + 0.558997i \(0.811186\pi\)
\(854\) 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 \(0.501461\pi\)
\(858\) 0 0
\(859\) −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 \(-0.458299\pi\)
0.130633 + 0.991431i \(0.458299\pi\)
\(864\) 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
\(871\) 0 0
\(872\) −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 \(-0.632862\pi\)
−0.405384 + 0.914147i \(0.632862\pi\)
\(878\) 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 \(0.480125\pi\)
\(884\) 0 0
\(885\) 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
\(897\) 0 0
\(898\) −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
\(910\) 0 0
\(911\) 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
\(923\) 0 0
\(924\) −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 \(-0.396116\pi\)
0.320598 + 0.947215i \(0.396116\pi\)
\(930\) −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
\(936\) 0 0
\(937\) −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 \(-0.351944\pi\)
0.448540 + 0.893763i \(0.351944\pi\)
\(942\) −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 \(-0.546517\pi\)
−0.145619 + 0.989341i \(0.546517\pi\)
\(948\) 2.16714e12 0.0871464
\(949\) 0 0
\(950\) 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
\(962\) 0 0
\(963\) −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 \(-0.570991\pi\)
−0.221182 + 0.975233i \(0.570991\pi\)
\(968\) 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
\(975\) 0 0
\(976\) −1.35954e13 −0.479589
\(977\) 4.02945e13 1.41488 0.707441 0.706773i \(-0.249850\pi\)
0.707441 + 0.706773i \(0.249850\pi\)
\(978\) −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 \(-0.830334\pi\)
−0.861276 + 0.508137i \(0.830334\pi\)
\(984\) 8.23672e12 0.280076
\(985\) −7.88219e13 −2.66799
\(986\) −5.92497e12 −0.199637
\(987\) −2.72460e13 −0.913852
\(988\) 0 0
\(989\) 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 338.10.a.b.1.1 1
13.12 even 2 26.10.a.c.1.1 1
39.38 odd 2 234.10.a.a.1.1 1
52.51 odd 2 208.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.c.1.1 1 13.12 even 2
208.10.a.b.1.1 1 52.51 odd 2
234.10.a.a.1.1 1 39.38 odd 2
338.10.a.b.1.1 1 1.1 even 1 trivial