Properties

Label 234.10.a.a.1.1
Level $234$
Weight $10$
Character 234.1
Self dual yes
Analytic conductor $120.518$
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] = [234,10,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, 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("234.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(120.518385662\)
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 \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{2} +256.000 q^{4} +1979.00 q^{5} -10115.0 q^{7} -4096.00 q^{8} +O(q^{10})\) \(q-16.0000 q^{2} +256.000 q^{4} +1979.00 q^{5} -10115.0 q^{7} -4096.00 q^{8} -31664.0 q^{10} -18850.0 q^{11} +28561.0 q^{13} +161840. q^{14} +65536.0 q^{16} +142403. q^{17} +83302.0 q^{19} +506624. q^{20} +301600. q^{22} +536544. q^{23} +1.96332e6 q^{25} -456976. q^{26} -2.58944e6 q^{28} +2.60044e6 q^{29} -2.21400e6 q^{31} -1.04858e6 q^{32} -2.27845e6 q^{34} -2.00176e7 q^{35} +1.80992e7 q^{37} -1.33283e6 q^{38} -8.10598e6 q^{40} -2.68122e7 q^{41} -4.22535e7 q^{43} -4.82560e6 q^{44} -8.58470e6 q^{46} -3.59150e7 q^{47} +6.19596e7 q^{49} -3.14131e7 q^{50} +7.31162e6 q^{52} +6.65141e7 q^{53} -3.73042e7 q^{55} +4.14310e7 q^{56} -4.16071e7 q^{58} +1.08164e8 q^{59} -2.07450e8 q^{61} +3.54241e7 q^{62} +1.67772e7 q^{64} +5.65222e7 q^{65} +1.93016e8 q^{67} +3.64552e7 q^{68} +3.20281e8 q^{70} +2.01833e8 q^{71} -1.21628e8 q^{73} -2.89588e8 q^{74} +2.13253e7 q^{76} +1.90668e8 q^{77} +1.12872e8 q^{79} +1.29696e8 q^{80} +4.28996e8 q^{82} -3.08254e8 q^{83} +2.81816e8 q^{85} +6.76056e8 q^{86} +7.72096e7 q^{88} +6.37487e6 q^{89} -2.88895e8 q^{91} +1.37355e8 q^{92} +5.74640e8 q^{94} +1.64855e8 q^{95} +8.71267e8 q^{97} -9.91354e8 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 256.000 0.500000
\(5\) 1979.00 1.41606 0.708029 0.706184i \(-0.249585\pi\)
0.708029 + 0.706184i \(0.249585\pi\)
\(6\) 0 0
\(7\) −10115.0 −1.59230 −0.796150 0.605100i \(-0.793133\pi\)
−0.796150 + 0.605100i \(0.793133\pi\)
\(8\) −4096.00 −0.353553
\(9\) 0 0
\(10\) −31664.0 −1.00130
\(11\) −18850.0 −0.388190 −0.194095 0.980983i \(-0.562177\pi\)
−0.194095 + 0.980983i \(0.562177\pi\)
\(12\) 0 0
\(13\) 28561.0 0.277350
\(14\) 161840. 1.12593
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 142403. 0.413522 0.206761 0.978391i \(-0.433708\pi\)
0.206761 + 0.978391i \(0.433708\pi\)
\(18\) 0 0
\(19\) 83302.0 0.146644 0.0733220 0.997308i \(-0.476640\pi\)
0.0733220 + 0.997308i \(0.476640\pi\)
\(20\) 506624. 0.708029
\(21\) 0 0
\(22\) 301600. 0.274492
\(23\) 536544. 0.399788 0.199894 0.979817i \(-0.435940\pi\)
0.199894 + 0.979817i \(0.435940\pi\)
\(24\) 0 0
\(25\) 1.96332e6 1.00522
\(26\) −456976. −0.196116
\(27\) 0 0
\(28\) −2.58944e6 −0.796150
\(29\) 2.60044e6 0.682741 0.341371 0.939929i \(-0.389109\pi\)
0.341371 + 0.939929i \(0.389109\pi\)
\(30\) 0 0
\(31\) −2.21400e6 −0.430577 −0.215288 0.976550i \(-0.569069\pi\)
−0.215288 + 0.976550i \(0.569069\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) 0 0
\(34\) −2.27845e6 −0.292404
\(35\) −2.00176e7 −2.25479
\(36\) 0 0
\(37\) 1.80992e7 1.58764 0.793821 0.608151i \(-0.208089\pi\)
0.793821 + 0.608151i \(0.208089\pi\)
\(38\) −1.33283e6 −0.103693
\(39\) 0 0
\(40\) −8.10598e6 −0.500652
\(41\) −2.68122e7 −1.48186 −0.740928 0.671585i \(-0.765614\pi\)
−0.740928 + 0.671585i \(0.765614\pi\)
\(42\) 0 0
\(43\) −4.22535e7 −1.88475 −0.942376 0.334555i \(-0.891414\pi\)
−0.942376 + 0.334555i \(0.891414\pi\)
\(44\) −4.82560e6 −0.194095
\(45\) 0 0
\(46\) −8.58470e6 −0.282693
\(47\) −3.59150e7 −1.07358 −0.536791 0.843715i \(-0.680364\pi\)
−0.536791 + 0.843715i \(0.680364\pi\)
\(48\) 0 0
\(49\) 6.19596e7 1.53542
\(50\) −3.14131e7 −0.710796
\(51\) 0 0
\(52\) 7.31162e6 0.138675
\(53\) 6.65141e7 1.15790 0.578951 0.815362i \(-0.303462\pi\)
0.578951 + 0.815362i \(0.303462\pi\)
\(54\) 0 0
\(55\) −3.73042e7 −0.549699
\(56\) 4.14310e7 0.562963
\(57\) 0 0
\(58\) −4.16071e7 −0.482771
\(59\) 1.08164e8 1.16211 0.581057 0.813863i \(-0.302639\pi\)
0.581057 + 0.813863i \(0.302639\pi\)
\(60\) 0 0
\(61\) −2.07450e8 −1.91836 −0.959178 0.282805i \(-0.908735\pi\)
−0.959178 + 0.282805i \(0.908735\pi\)
\(62\) 3.54241e7 0.304464
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) 5.65222e7 0.392744
\(66\) 0 0
\(67\) 1.93016e8 1.17019 0.585094 0.810966i \(-0.301058\pi\)
0.585094 + 0.810966i \(0.301058\pi\)
\(68\) 3.64552e7 0.206761
\(69\) 0 0
\(70\) 3.20281e8 1.59438
\(71\) 2.01833e8 0.942607 0.471304 0.881971i \(-0.343784\pi\)
0.471304 + 0.881971i \(0.343784\pi\)
\(72\) 0 0
\(73\) −1.21628e8 −0.501281 −0.250640 0.968080i \(-0.580641\pi\)
−0.250640 + 0.968080i \(0.580641\pi\)
\(74\) −2.89588e8 −1.12263
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 4.28996e8 1.04783
\(83\) −3.08254e8 −0.712948 −0.356474 0.934305i \(-0.616021\pi\)
−0.356474 + 0.934305i \(0.616021\pi\)
\(84\) 0 0
\(85\) 2.81816e8 0.585571
\(86\) 6.76056e8 1.33272
\(87\) 0 0
\(88\) 7.72096e7 0.137246
\(89\) 6.37487e6 0.0107700 0.00538501 0.999986i \(-0.498286\pi\)
0.00538501 + 0.999986i \(0.498286\pi\)
\(90\) 0 0
\(91\) −2.88895e8 −0.441624
\(92\) 1.37355e8 0.199894
\(93\) 0 0
\(94\) 5.74640e8 0.759137
\(95\) 1.64855e8 0.207656
\(96\) 0 0
\(97\) 8.71267e8 0.999260 0.499630 0.866239i \(-0.333469\pi\)
0.499630 + 0.866239i \(0.333469\pi\)
\(98\) −9.91354e8 −1.08570
\(99\) 0 0
\(100\) 5.02609e8 0.502609
\(101\) 8.24412e8 0.788312 0.394156 0.919044i \(-0.371037\pi\)
0.394156 + 0.919044i \(0.371037\pi\)
\(102\) 0 0
\(103\) −1.65896e9 −1.45234 −0.726168 0.687517i \(-0.758701\pi\)
−0.726168 + 0.687517i \(0.758701\pi\)
\(104\) −1.16986e8 −0.0980581
\(105\) 0 0
\(106\) −1.06423e9 −0.818761
\(107\) −1.15165e9 −0.849366 −0.424683 0.905342i \(-0.639614\pi\)
−0.424683 + 0.905342i \(0.639614\pi\)
\(108\) 0 0
\(109\) −2.78480e9 −1.88962 −0.944810 0.327620i \(-0.893753\pi\)
−0.944810 + 0.327620i \(0.893753\pi\)
\(110\) 5.96866e8 0.388696
\(111\) 0 0
\(112\) −6.62897e8 −0.398075
\(113\) −6.78547e8 −0.391496 −0.195748 0.980654i \(-0.562713\pi\)
−0.195748 + 0.980654i \(0.562713\pi\)
\(114\) 0 0
\(115\) 1.06182e9 0.566123
\(116\) 6.65713e8 0.341371
\(117\) 0 0
\(118\) −1.73062e9 −0.821739
\(119\) −1.44041e9 −0.658451
\(120\) 0 0
\(121\) −2.00263e9 −0.849309
\(122\) 3.31920e9 1.35648
\(123\) 0 0
\(124\) −5.66785e8 −0.215288
\(125\) 2.01680e7 0.00738869
\(126\) 0 0
\(127\) −3.48292e9 −1.18803 −0.594014 0.804455i \(-0.702458\pi\)
−0.594014 + 0.804455i \(0.702458\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) 0 0
\(130\) −9.04356e8 −0.277712
\(131\) −5.02701e9 −1.49138 −0.745691 0.666292i \(-0.767881\pi\)
−0.745691 + 0.666292i \(0.767881\pi\)
\(132\) 0 0
\(133\) −8.42600e8 −0.233501
\(134\) −3.08825e9 −0.827448
\(135\) 0 0
\(136\) −5.83283e8 −0.146202
\(137\) 6.38904e9 1.54950 0.774752 0.632265i \(-0.217875\pi\)
0.774752 + 0.632265i \(0.217875\pi\)
\(138\) 0 0
\(139\) 7.62665e9 1.73287 0.866437 0.499286i \(-0.166404\pi\)
0.866437 + 0.499286i \(0.166404\pi\)
\(140\) −5.12450e9 −1.12739
\(141\) 0 0
\(142\) −3.22934e9 −0.666524
\(143\) −5.38375e8 −0.107665
\(144\) 0 0
\(145\) 5.14627e9 0.966801
\(146\) 1.94605e9 0.354459
\(147\) 0 0
\(148\) 4.63341e9 0.793821
\(149\) 9.23455e9 1.53489 0.767445 0.641114i \(-0.221528\pi\)
0.767445 + 0.641114i \(0.221528\pi\)
\(150\) 0 0
\(151\) −3.25451e9 −0.509436 −0.254718 0.967015i \(-0.581983\pi\)
−0.254718 + 0.967015i \(0.581983\pi\)
\(152\) −3.41205e8 −0.0518465
\(153\) 0 0
\(154\) −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
\(159\) 0 0
\(160\) −2.07513e9 −0.250326
\(161\) −5.42714e9 −0.636583
\(162\) 0 0
\(163\) −1.13187e10 −1.25590 −0.627948 0.778255i \(-0.716105\pi\)
−0.627948 + 0.778255i \(0.716105\pi\)
\(164\) −6.86393e9 −0.740928
\(165\) 0 0
\(166\) 4.93207e9 0.504130
\(167\) −1.72306e9 −0.171426 −0.0857131 0.996320i \(-0.527317\pi\)
−0.0857131 + 0.996320i \(0.527317\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) −4.50905e9 −0.414061
\(171\) 0 0
\(172\) −1.08169e10 −0.942376
\(173\) −2.76347e9 −0.234557 −0.117278 0.993099i \(-0.537417\pi\)
−0.117278 + 0.993099i \(0.537417\pi\)
\(174\) 0 0
\(175\) −1.98589e10 −1.60061
\(176\) −1.23535e9 −0.0970475
\(177\) 0 0
\(178\) −1.01998e8 −0.00761555
\(179\) −6.86682e9 −0.499939 −0.249969 0.968254i \(-0.580421\pi\)
−0.249969 + 0.968254i \(0.580421\pi\)
\(180\) 0 0
\(181\) −2.41534e10 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(182\) 4.62231e9 0.312276
\(183\) 0 0
\(184\) −2.19768e9 −0.141347
\(185\) 3.58184e10 2.24819
\(186\) 0 0
\(187\) −2.68430e9 −0.160525
\(188\) −9.19424e9 −0.536791
\(189\) 0 0
\(190\) −2.63767e9 −0.146835
\(191\) −3.59983e10 −1.95719 −0.978593 0.205805i \(-0.934019\pi\)
−0.978593 + 0.205805i \(0.934019\pi\)
\(192\) 0 0
\(193\) −1.70031e10 −0.882107 −0.441054 0.897481i \(-0.645395\pi\)
−0.441054 + 0.897481i \(0.645395\pi\)
\(194\) −1.39403e10 −0.706583
\(195\) 0 0
\(196\) 1.58617e10 0.767709
\(197\) −3.98292e10 −1.88410 −0.942049 0.335477i \(-0.891103\pi\)
−0.942049 + 0.335477i \(0.891103\pi\)
\(198\) 0 0
\(199\) 1.31081e9 0.0592518 0.0296259 0.999561i \(-0.490568\pi\)
0.0296259 + 0.999561i \(0.490568\pi\)
\(200\) −8.04174e9 −0.355398
\(201\) 0 0
\(202\) −1.31906e10 −0.557421
\(203\) −2.63035e10 −1.08713
\(204\) 0 0
\(205\) −5.30614e10 −2.09839
\(206\) 2.65433e10 1.02696
\(207\) 0 0
\(208\) 1.87177e9 0.0693375
\(209\) −1.57024e9 −0.0569257
\(210\) 0 0
\(211\) 2.35777e10 0.818898 0.409449 0.912333i \(-0.365721\pi\)
0.409449 + 0.912333i \(0.365721\pi\)
\(212\) 1.70276e10 0.578951
\(213\) 0 0
\(214\) 1.84265e10 0.600593
\(215\) −8.36196e10 −2.66892
\(216\) 0 0
\(217\) 2.23947e10 0.685607
\(218\) 4.45568e10 1.33616
\(219\) 0 0
\(220\) −9.54986e9 −0.274850
\(221\) 4.06717e9 0.114690
\(222\) 0 0
\(223\) 2.38326e9 0.0645356 0.0322678 0.999479i \(-0.489727\pi\)
0.0322678 + 0.999479i \(0.489727\pi\)
\(224\) 1.06063e10 0.281481
\(225\) 0 0
\(226\) 1.08568e10 0.276829
\(227\) 7.46548e9 0.186613 0.0933064 0.995637i \(-0.470256\pi\)
0.0933064 + 0.995637i \(0.470256\pi\)
\(228\) 0 0
\(229\) 2.63966e10 0.634292 0.317146 0.948377i \(-0.397276\pi\)
0.317146 + 0.948377i \(0.397276\pi\)
\(230\) −1.69891e10 −0.400309
\(231\) 0 0
\(232\) −1.06514e10 −0.241386
\(233\) −2.40457e10 −0.534485 −0.267242 0.963629i \(-0.586112\pi\)
−0.267242 + 0.963629i \(0.586112\pi\)
\(234\) 0 0
\(235\) −7.10758e10 −1.52025
\(236\) 2.76900e10 0.581057
\(237\) 0 0
\(238\) 2.30465e10 0.465595
\(239\) −5.96318e10 −1.18219 −0.591095 0.806602i \(-0.701304\pi\)
−0.591095 + 0.806602i \(0.701304\pi\)
\(240\) 0 0
\(241\) 1.25639e10 0.239910 0.119955 0.992779i \(-0.461725\pi\)
0.119955 + 0.992779i \(0.461725\pi\)
\(242\) 3.20420e10 0.600552
\(243\) 0 0
\(244\) −5.31072e10 −0.959178
\(245\) 1.22618e11 2.17424
\(246\) 0 0
\(247\) 2.37919e9 0.0406717
\(248\) 9.06856e9 0.152232
\(249\) 0 0
\(250\) −3.22688e8 −0.00522459
\(251\) −2.41771e10 −0.384479 −0.192240 0.981348i \(-0.561575\pi\)
−0.192240 + 0.981348i \(0.561575\pi\)
\(252\) 0 0
\(253\) −1.01139e10 −0.155194
\(254\) 5.57267e10 0.840062
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) −2.96868e10 −0.424488 −0.212244 0.977217i \(-0.568077\pi\)
−0.212244 + 0.977217i \(0.568077\pi\)
\(258\) 0 0
\(259\) −1.83074e11 −2.52800
\(260\) 1.44697e10 0.196372
\(261\) 0 0
\(262\) 8.04322e10 1.05457
\(263\) 7.59146e10 0.978418 0.489209 0.872167i \(-0.337286\pi\)
0.489209 + 0.872167i \(0.337286\pi\)
\(264\) 0 0
\(265\) 1.31631e11 1.63966
\(266\) 1.34816e10 0.165110
\(267\) 0 0
\(268\) 4.94120e10 0.585094
\(269\) 2.57149e10 0.299433 0.149716 0.988729i \(-0.452164\pi\)
0.149716 + 0.988729i \(0.452164\pi\)
\(270\) 0 0
\(271\) −8.08890e10 −0.911020 −0.455510 0.890231i \(-0.650543\pi\)
−0.455510 + 0.890231i \(0.650543\pi\)
\(272\) 9.33252e9 0.103381
\(273\) 0 0
\(274\) −1.02225e11 −1.09567
\(275\) −3.70085e10 −0.390215
\(276\) 0 0
\(277\) −3.40035e10 −0.347028 −0.173514 0.984831i \(-0.555512\pi\)
−0.173514 + 0.984831i \(0.555512\pi\)
\(278\) −1.22026e11 −1.22533
\(279\) 0 0
\(280\) 8.19920e10 0.797188
\(281\) 7.14831e9 0.0683951 0.0341976 0.999415i \(-0.489112\pi\)
0.0341976 + 0.999415i \(0.489112\pi\)
\(282\) 0 0
\(283\) −7.80508e10 −0.723333 −0.361667 0.932308i \(-0.617792\pi\)
−0.361667 + 0.932308i \(0.617792\pi\)
\(284\) 5.16694e10 0.471304
\(285\) 0 0
\(286\) 8.61400e9 0.0761303
\(287\) 2.71206e11 2.35956
\(288\) 0 0
\(289\) −9.83093e10 −0.828999
\(290\) −8.23404e10 −0.683631
\(291\) 0 0
\(292\) −3.11368e10 −0.250640
\(293\) 1.26662e11 1.00402 0.502009 0.864862i \(-0.332594\pi\)
0.502009 + 0.864862i \(0.332594\pi\)
\(294\) 0 0
\(295\) 2.14057e11 1.64562
\(296\) −7.41345e10 −0.561316
\(297\) 0 0
\(298\) −1.47753e11 −1.08533
\(299\) 1.53242e10 0.110881
\(300\) 0 0
\(301\) 4.27394e11 3.00109
\(302\) 5.20722e10 0.360226
\(303\) 0 0
\(304\) 5.45928e9 0.0366610
\(305\) −4.10543e11 −2.71650
\(306\) 0 0
\(307\) 6.15064e10 0.395182 0.197591 0.980285i \(-0.436688\pi\)
0.197591 + 0.980285i \(0.436688\pi\)
\(308\) 4.88109e10 0.309057
\(309\) 0 0
\(310\) 7.01042e10 0.431138
\(311\) −2.16398e11 −1.31169 −0.655846 0.754894i \(-0.727688\pi\)
−0.655846 + 0.754894i \(0.727688\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 2.88952e10 0.163017
\(317\) −7.77399e10 −0.432392 −0.216196 0.976350i \(-0.569365\pi\)
−0.216196 + 0.976350i \(0.569365\pi\)
\(318\) 0 0
\(319\) −4.90183e10 −0.265033
\(320\) 3.32021e10 0.177007
\(321\) 0 0
\(322\) 8.68343e10 0.450132
\(323\) 1.18625e10 0.0606406
\(324\) 0 0
\(325\) 5.60743e10 0.278797
\(326\) 1.81100e11 0.888053
\(327\) 0 0
\(328\) 1.09823e11 0.523915
\(329\) 3.63280e11 1.70946
\(330\) 0 0
\(331\) −1.68625e11 −0.772139 −0.386070 0.922470i \(-0.626168\pi\)
−0.386070 + 0.922470i \(0.626168\pi\)
\(332\) −7.89131e10 −0.356474
\(333\) 0 0
\(334\) 2.75690e10 0.121217
\(335\) 3.81978e11 1.65705
\(336\) 0 0
\(337\) −7.70797e10 −0.325541 −0.162770 0.986664i \(-0.552043\pi\)
−0.162770 + 0.986664i \(0.552043\pi\)
\(338\) −1.30517e10 −0.0543928
\(339\) 0 0
\(340\) 7.21448e10 0.292786
\(341\) 4.17340e10 0.167146
\(342\) 0 0
\(343\) −2.18545e11 −0.852544
\(344\) 1.73070e11 0.666361
\(345\) 0 0
\(346\) 4.42156e10 0.165857
\(347\) −3.54609e11 −1.31301 −0.656504 0.754322i \(-0.727966\pi\)
−0.656504 + 0.754322i \(0.727966\pi\)
\(348\) 0 0
\(349\) 1.70460e11 0.615048 0.307524 0.951540i \(-0.400500\pi\)
0.307524 + 0.951540i \(0.400500\pi\)
\(350\) 3.17743e11 1.13180
\(351\) 0 0
\(352\) 1.97657e10 0.0686229
\(353\) −2.96506e11 −1.01636 −0.508180 0.861251i \(-0.669682\pi\)
−0.508180 + 0.861251i \(0.669682\pi\)
\(354\) 0 0
\(355\) 3.99428e11 1.33479
\(356\) 1.63197e9 0.00538501
\(357\) 0 0
\(358\) 1.09869e11 0.353510
\(359\) −7.20144e10 −0.228820 −0.114410 0.993434i \(-0.536498\pi\)
−0.114410 + 0.993434i \(0.536498\pi\)
\(360\) 0 0
\(361\) −3.15748e11 −0.978496
\(362\) 3.86455e11 1.18280
\(363\) 0 0
\(364\) −7.39570e10 −0.220812
\(365\) −2.40702e11 −0.709842
\(366\) 0 0
\(367\) −1.05092e11 −0.302394 −0.151197 0.988504i \(-0.548313\pi\)
−0.151197 + 0.988504i \(0.548313\pi\)
\(368\) 3.51629e10 0.0999471
\(369\) 0 0
\(370\) −5.73094e11 −1.58971
\(371\) −6.72790e11 −1.84373
\(372\) 0 0
\(373\) 2.19888e11 0.588182 0.294091 0.955777i \(-0.404983\pi\)
0.294091 + 0.955777i \(0.404983\pi\)
\(374\) 4.29487e10 0.113508
\(375\) 0 0
\(376\) 1.47108e11 0.379569
\(377\) 7.42712e10 0.189358
\(378\) 0 0
\(379\) −3.14748e11 −0.783586 −0.391793 0.920053i \(-0.628145\pi\)
−0.391793 + 0.920053i \(0.628145\pi\)
\(380\) 4.22028e10 0.103828
\(381\) 0 0
\(382\) 5.75973e11 1.38394
\(383\) 3.41027e10 0.0809831 0.0404915 0.999180i \(-0.487108\pi\)
0.0404915 + 0.999180i \(0.487108\pi\)
\(384\) 0 0
\(385\) 3.77331e11 0.875286
\(386\) 2.72050e11 0.623744
\(387\) 0 0
\(388\) 2.23044e11 0.499630
\(389\) −4.49612e11 −0.995554 −0.497777 0.867305i \(-0.665850\pi\)
−0.497777 + 0.867305i \(0.665850\pi\)
\(390\) 0 0
\(391\) 7.64055e10 0.165321
\(392\) −2.53787e11 −0.542852
\(393\) 0 0
\(394\) 6.37267e11 1.33226
\(395\) 2.23374e11 0.461684
\(396\) 0 0
\(397\) 2.29976e11 0.464649 0.232324 0.972638i \(-0.425367\pi\)
0.232324 + 0.972638i \(0.425367\pi\)
\(398\) −2.09730e10 −0.0418974
\(399\) 0 0
\(400\) 1.28668e11 0.251304
\(401\) 6.69163e11 1.29236 0.646178 0.763187i \(-0.276366\pi\)
0.646178 + 0.763187i \(0.276366\pi\)
\(402\) 0 0
\(403\) −6.32342e10 −0.119421
\(404\) 2.11049e11 0.394156
\(405\) 0 0
\(406\) 4.20856e11 0.768716
\(407\) −3.41171e11 −0.616307
\(408\) 0 0
\(409\) −6.73923e11 −1.19085 −0.595423 0.803413i \(-0.703015\pi\)
−0.595423 + 0.803413i \(0.703015\pi\)
\(410\) 8.48983e11 1.48379
\(411\) 0 0
\(412\) −4.24693e11 −0.726168
\(413\) −1.09408e12 −1.85043
\(414\) 0 0
\(415\) −6.10035e11 −1.00957
\(416\) −2.99484e10 −0.0490290
\(417\) 0 0
\(418\) 2.51239e10 0.0402526
\(419\) 3.45053e11 0.546919 0.273460 0.961883i \(-0.411832\pi\)
0.273460 + 0.961883i \(0.411832\pi\)
\(420\) 0 0
\(421\) −5.57817e11 −0.865411 −0.432706 0.901535i \(-0.642441\pi\)
−0.432706 + 0.901535i \(0.642441\pi\)
\(422\) −3.77243e11 −0.579049
\(423\) 0 0
\(424\) −2.72442e11 −0.409380
\(425\) 2.79582e11 0.415680
\(426\) 0 0
\(427\) 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 \(0.352792\pi\)
\(432\) 0 0
\(433\) −1.23759e12 −1.69193 −0.845965 0.533238i \(-0.820975\pi\)
−0.845965 + 0.533238i \(0.820975\pi\)
\(434\) −3.58314e11 −0.484798
\(435\) 0 0
\(436\) −7.12908e11 −0.944810
\(437\) 4.46952e10 0.0586265
\(438\) 0 0
\(439\) −9.14852e11 −1.17560 −0.587801 0.809006i \(-0.700006\pi\)
−0.587801 + 0.809006i \(0.700006\pi\)
\(440\) 1.52798e11 0.194348
\(441\) 0 0
\(442\) −6.50748e10 −0.0810984
\(443\) 1.18651e12 1.46370 0.731852 0.681464i \(-0.238656\pi\)
0.731852 + 0.681464i \(0.238656\pi\)
\(444\) 0 0
\(445\) 1.26159e10 0.0152510
\(446\) −3.81322e10 −0.0456336
\(447\) 0 0
\(448\) −1.69702e11 −0.199037
\(449\) 2.70808e10 0.0314452 0.0157226 0.999876i \(-0.494995\pi\)
0.0157226 + 0.999876i \(0.494995\pi\)
\(450\) 0 0
\(451\) 5.05411e11 0.575241
\(452\) −1.73708e11 −0.195748
\(453\) 0 0
\(454\) −1.19448e11 −0.131955
\(455\) −5.71722e11 −0.625365
\(456\) 0 0
\(457\) 2.02586e11 0.217263 0.108632 0.994082i \(-0.465353\pi\)
0.108632 + 0.994082i \(0.465353\pi\)
\(458\) −4.22346e11 −0.448512
\(459\) 0 0
\(460\) 2.71826e11 0.283062
\(461\) −8.96346e11 −0.924319 −0.462159 0.886797i \(-0.652925\pi\)
−0.462159 + 0.886797i \(0.652925\pi\)
\(462\) 0 0
\(463\) −5.17740e11 −0.523597 −0.261799 0.965123i \(-0.584316\pi\)
−0.261799 + 0.965123i \(0.584316\pi\)
\(464\) 1.70423e11 0.170685
\(465\) 0 0
\(466\) 3.84731e11 0.377938
\(467\) −9.20785e11 −0.895843 −0.447922 0.894073i \(-0.647836\pi\)
−0.447922 + 0.894073i \(0.647836\pi\)
\(468\) 0 0
\(469\) −1.95235e12 −1.86329
\(470\) 1.13721e12 1.07498
\(471\) 0 0
\(472\) −4.43040e11 −0.410869
\(473\) 7.96478e11 0.731642
\(474\) 0 0
\(475\) 1.63548e11 0.147409
\(476\) −3.68744e11 −0.329226
\(477\) 0 0
\(478\) 9.54108e11 0.835934
\(479\) −1.73253e12 −1.50373 −0.751867 0.659315i \(-0.770846\pi\)
−0.751867 + 0.659315i \(0.770846\pi\)
\(480\) 0 0
\(481\) 5.16932e11 0.440333
\(482\) −2.01023e11 −0.169642
\(483\) 0 0
\(484\) −5.12672e11 −0.424654
\(485\) 1.72424e12 1.41501
\(486\) 0 0
\(487\) 1.49591e12 1.20511 0.602554 0.798078i \(-0.294150\pi\)
0.602554 + 0.798078i \(0.294150\pi\)
\(488\) 8.49715e11 0.678241
\(489\) 0 0
\(490\) −1.96189e12 −1.53742
\(491\) 4.28954e11 0.333076 0.166538 0.986035i \(-0.446741\pi\)
0.166538 + 0.986035i \(0.446741\pi\)
\(492\) 0 0
\(493\) 3.70311e11 0.282329
\(494\) −3.80670e10 −0.0287592
\(495\) 0 0
\(496\) −1.45097e11 −0.107644
\(497\) −2.04155e12 −1.50091
\(498\) 0 0
\(499\) −9.12174e11 −0.658606 −0.329303 0.944224i \(-0.606814\pi\)
−0.329303 + 0.944224i \(0.606814\pi\)
\(500\) 5.16301e9 0.00369435
\(501\) 0 0
\(502\) 3.86834e11 0.271868
\(503\) 1.26835e12 0.883456 0.441728 0.897149i \(-0.354366\pi\)
0.441728 + 0.897149i \(0.354366\pi\)
\(504\) 0 0
\(505\) 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 \(0.329979\pi\)
\(510\) 0 0
\(511\) 1.23027e12 0.798189
\(512\) −6.87195e10 −0.0441942
\(513\) 0 0
\(514\) 4.74990e11 0.300158
\(515\) −3.28307e12 −2.05659
\(516\) 0 0
\(517\) 6.76998e11 0.416754
\(518\) 2.92918e12 1.78757
\(519\) 0 0
\(520\) −2.31515e11 −0.138856
\(521\) 1.48896e12 0.885345 0.442672 0.896683i \(-0.354031\pi\)
0.442672 + 0.896683i \(0.354031\pi\)
\(522\) 0 0
\(523\) 2.55715e12 1.49451 0.747256 0.664536i \(-0.231371\pi\)
0.747256 + 0.664536i \(0.231371\pi\)
\(524\) −1.28691e12 −0.745691
\(525\) 0 0
\(526\) −1.21463e12 −0.691846
\(527\) −3.15281e11 −0.178053
\(528\) 0 0
\(529\) −1.51327e12 −0.840169
\(530\) −2.10610e12 −1.15941
\(531\) 0 0
\(532\) −2.15706e11 −0.116751
\(533\) −7.65784e11 −0.410993
\(534\) 0 0
\(535\) −2.27912e12 −1.20275
\(536\) −7.90592e11 −0.413724
\(537\) 0 0
\(538\) −4.11438e11 −0.211731
\(539\) −1.16794e12 −0.596033
\(540\) 0 0
\(541\) −2.64921e12 −1.32962 −0.664811 0.747011i \(-0.731488\pi\)
−0.664811 + 0.747011i \(0.731488\pi\)
\(542\) 1.29422e12 0.644188
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 5.92136e11 0.275924
\(551\) 2.16622e11 0.100120
\(552\) 0 0
\(553\) −1.14170e12 −0.519145
\(554\) 5.44056e11 0.245386
\(555\) 0 0
\(556\) 1.95242e12 0.866437
\(557\) 2.96364e12 1.30460 0.652300 0.757961i \(-0.273804\pi\)
0.652300 + 0.757961i \(0.273804\pi\)
\(558\) 0 0
\(559\) −1.20680e12 −0.522736
\(560\) −1.31187e12 −0.563697
\(561\) 0 0
\(562\) −1.14373e11 −0.0483627
\(563\) 3.46859e12 1.45501 0.727504 0.686104i \(-0.240680\pi\)
0.727504 + 0.686104i \(0.240680\pi\)
\(564\) 0 0
\(565\) −1.34284e12 −0.554380
\(566\) 1.24881e12 0.511474
\(567\) 0 0
\(568\) −8.26710e11 −0.333262
\(569\) 3.83703e12 1.53458 0.767290 0.641300i \(-0.221605\pi\)
0.767290 + 0.641300i \(0.221605\pi\)
\(570\) 0 0
\(571\) 3.35374e11 0.132028 0.0660142 0.997819i \(-0.478972\pi\)
0.0660142 + 0.997819i \(0.478972\pi\)
\(572\) −1.37824e11 −0.0538323
\(573\) 0 0
\(574\) −4.33929e12 −1.66846
\(575\) 1.05341e12 0.401874
\(576\) 0 0
\(577\) −1.74089e11 −0.0653852 −0.0326926 0.999465i \(-0.510408\pi\)
−0.0326926 + 0.999465i \(0.510408\pi\)
\(578\) 1.57295e12 0.586191
\(579\) 0 0
\(580\) 1.31745e12 0.483400
\(581\) 3.11799e12 1.13523
\(582\) 0 0
\(583\) −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 \(0.605431\pi\)
\(588\) 0 0
\(589\) −1.84431e11 −0.0631415
\(590\) −3.42490e12 −1.16363
\(591\) 0 0
\(592\) 1.18615e12 0.396911
\(593\) −4.16793e12 −1.38412 −0.692061 0.721839i \(-0.743297\pi\)
−0.692061 + 0.721839i \(0.743297\pi\)
\(594\) 0 0
\(595\) −2.85056e12 −0.932405
\(596\) 2.36404e12 0.767445
\(597\) 0 0
\(598\) −2.45188e11 −0.0784049
\(599\) −8.16635e10 −0.0259183 −0.0129592 0.999916i \(-0.504125\pi\)
−0.0129592 + 0.999916i \(0.504125\pi\)
\(600\) 0 0
\(601\) −4.00769e12 −1.25302 −0.626511 0.779412i \(-0.715518\pi\)
−0.626511 + 0.779412i \(0.715518\pi\)
\(602\) −6.83830e12 −2.12209
\(603\) 0 0
\(604\) −8.33155e11 −0.254718
\(605\) −3.96320e12 −1.20267
\(606\) 0 0
\(607\) 1.45542e12 0.435149 0.217575 0.976044i \(-0.430185\pi\)
0.217575 + 0.976044i \(0.430185\pi\)
\(608\) −8.73485e10 −0.0259232
\(609\) 0 0
\(610\) 6.56869e12 1.92086
\(611\) −1.02577e12 −0.297758
\(612\) 0 0
\(613\) −2.55645e12 −0.731248 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(614\) −9.84102e11 −0.279436
\(615\) 0 0
\(616\) −7.80975e11 −0.218537
\(617\) −2.69727e11 −0.0749276 −0.0374638 0.999298i \(-0.511928\pi\)
−0.0374638 + 0.999298i \(0.511928\pi\)
\(618\) 0 0
\(619\) 4.28111e11 0.117206 0.0586028 0.998281i \(-0.481335\pi\)
0.0586028 + 0.998281i \(0.481335\pi\)
\(620\) −1.12167e12 −0.304861
\(621\) 0 0
\(622\) 3.46237e12 0.927507
\(623\) −6.44818e10 −0.0171491
\(624\) 0 0
\(625\) −3.79469e12 −0.994755
\(626\) −3.91414e12 −1.01871
\(627\) 0 0
\(628\) 4.16831e11 0.106940
\(629\) 2.57739e12 0.656525
\(630\) 0 0
\(631\) −2.98421e11 −0.0749372 −0.0374686 0.999298i \(-0.511929\pi\)
−0.0374686 + 0.999298i \(0.511929\pi\)
\(632\) −4.62323e11 −0.115271
\(633\) 0 0
\(634\) 1.24384e12 0.305747
\(635\) −6.89269e12 −1.68231
\(636\) 0 0
\(637\) 1.76963e12 0.425848
\(638\) 7.84293e11 0.187407
\(639\) 0 0
\(640\) −5.31234e11 −0.125163
\(641\) 2.34144e12 0.547799 0.273899 0.961758i \(-0.411686\pi\)
0.273899 + 0.961758i \(0.411686\pi\)
\(642\) 0 0
\(643\) 7.75186e12 1.78837 0.894183 0.447701i \(-0.147757\pi\)
0.894183 + 0.447701i \(0.147757\pi\)
\(644\) −1.38935e12 −0.318291
\(645\) 0 0
\(646\) −1.89799e11 −0.0428794
\(647\) −3.74980e12 −0.841278 −0.420639 0.907228i \(-0.638194\pi\)
−0.420639 + 0.907228i \(0.638194\pi\)
\(648\) 0 0
\(649\) −2.03889e12 −0.451121
\(650\) −8.97188e11 −0.197139
\(651\) 0 0
\(652\) −2.89760e12 −0.627948
\(653\) −2.82022e12 −0.606979 −0.303490 0.952835i \(-0.598152\pi\)
−0.303490 + 0.952835i \(0.598152\pi\)
\(654\) 0 0
\(655\) −9.94846e12 −2.11188
\(656\) −1.75717e12 −0.370464
\(657\) 0 0
\(658\) −5.81248e12 −1.20877
\(659\) 1.16074e12 0.239746 0.119873 0.992789i \(-0.461751\pi\)
0.119873 + 0.992789i \(0.461751\pi\)
\(660\) 0 0
\(661\) −3.18976e11 −0.0649907 −0.0324954 0.999472i \(-0.510345\pi\)
−0.0324954 + 0.999472i \(0.510345\pi\)
\(662\) 2.69800e12 0.545985
\(663\) 0 0
\(664\) 1.26261e12 0.252065
\(665\) −1.66750e12 −0.330651
\(666\) 0 0
\(667\) 1.39525e12 0.272952
\(668\) −4.41104e11 −0.0857131
\(669\) 0 0
\(670\) −6.11164e12 −1.17171
\(671\) 3.91043e12 0.744686
\(672\) 0 0
\(673\) 4.82897e12 0.907375 0.453687 0.891161i \(-0.350108\pi\)
0.453687 + 0.891161i \(0.350108\pi\)
\(674\) 1.23328e12 0.230192
\(675\) 0 0
\(676\) 2.08827e11 0.0384615
\(677\) 7.47095e12 1.36687 0.683434 0.730012i \(-0.260486\pi\)
0.683434 + 0.730012i \(0.260486\pi\)
\(678\) 0 0
\(679\) −8.81286e12 −1.59112
\(680\) −1.15432e12 −0.207031
\(681\) 0 0
\(682\) −6.67744e11 −0.118190
\(683\) 4.13060e12 0.726306 0.363153 0.931730i \(-0.381700\pi\)
0.363153 + 0.931730i \(0.381700\pi\)
\(684\) 0 0
\(685\) 1.26439e13 2.19419
\(686\) 3.49672e12 0.602840
\(687\) 0 0
\(688\) −2.76912e12 −0.471188
\(689\) 1.89971e12 0.321144
\(690\) 0 0
\(691\) 4.50776e12 0.752159 0.376079 0.926587i \(-0.377272\pi\)
0.376079 + 0.926587i \(0.377272\pi\)
\(692\) −7.07449e11 −0.117278
\(693\) 0 0
\(694\) 5.67375e12 0.928437
\(695\) 1.50931e13 2.45385
\(696\) 0 0
\(697\) −3.81814e12 −0.612780
\(698\) −2.72737e12 −0.434905
\(699\) 0 0
\(700\) −5.08389e12 −0.800304
\(701\) 1.38907e12 0.217267 0.108633 0.994082i \(-0.465353\pi\)
0.108633 + 0.994082i \(0.465353\pi\)
\(702\) 0 0
\(703\) 1.50770e12 0.232818
\(704\) −3.16251e11 −0.0485237
\(705\) 0 0
\(706\) 4.74410e12 0.718675
\(707\) −8.33893e12 −1.25523
\(708\) 0 0
\(709\) 5.59251e12 0.831187 0.415593 0.909551i \(-0.363574\pi\)
0.415593 + 0.909551i \(0.363574\pi\)
\(710\) −6.39086e12 −0.943836
\(711\) 0 0
\(712\) −2.61115e10 −0.00380778
\(713\) −1.18791e12 −0.172140
\(714\) 0 0
\(715\) −1.06544e12 −0.152459
\(716\) −1.75791e12 −0.249969
\(717\) 0 0
\(718\) 1.15223e12 0.161800
\(719\) 8.33742e12 1.16346 0.581730 0.813382i \(-0.302376\pi\)
0.581730 + 0.813382i \(0.302376\pi\)
\(720\) 0 0
\(721\) 1.67803e13 2.31255
\(722\) 5.05198e12 0.691901
\(723\) 0 0
\(724\) −6.18328e12 −0.836364
\(725\) 5.10549e12 0.686304
\(726\) 0 0
\(727\) −5.13697e12 −0.682028 −0.341014 0.940058i \(-0.610770\pi\)
−0.341014 + 0.940058i \(0.610770\pi\)
\(728\) 1.18331e12 0.156138
\(729\) 0 0
\(730\) 3.85123e12 0.501934
\(731\) −6.01702e12 −0.779387
\(732\) 0 0
\(733\) 9.72563e12 1.24437 0.622185 0.782870i \(-0.286245\pi\)
0.622185 + 0.782870i \(0.286245\pi\)
\(734\) 1.68147e12 0.213825
\(735\) 0 0
\(736\) −5.62607e11 −0.0706733
\(737\) −3.63834e12 −0.454255
\(738\) 0 0
\(739\) −1.15533e13 −1.42498 −0.712488 0.701684i \(-0.752432\pi\)
−0.712488 + 0.701684i \(0.752432\pi\)
\(740\) 9.16951e12 1.12410
\(741\) 0 0
\(742\) 1.07646e13 1.30371
\(743\) 1.04495e13 1.25790 0.628948 0.777447i \(-0.283486\pi\)
0.628948 + 0.777447i \(0.283486\pi\)
\(744\) 0 0
\(745\) 1.82752e13 2.17349
\(746\) −3.51821e12 −0.415907
\(747\) 0 0
\(748\) −6.87180e11 −0.0802626
\(749\) 1.16490e13 1.35245
\(750\) 0 0
\(751\) 1.92230e12 0.220516 0.110258 0.993903i \(-0.464832\pi\)
0.110258 + 0.993903i \(0.464832\pi\)
\(752\) −2.35372e12 −0.268396
\(753\) 0 0
\(754\) −1.18834e12 −0.133897
\(755\) −6.44068e12 −0.721390
\(756\) 0 0
\(757\) −1.27412e12 −0.141019 −0.0705096 0.997511i \(-0.522463\pi\)
−0.0705096 + 0.997511i \(0.522463\pi\)
\(758\) 5.03597e12 0.554079
\(759\) 0 0
\(760\) −6.75245e11 −0.0734176
\(761\) −1.21619e13 −1.31453 −0.657267 0.753658i \(-0.728288\pi\)
−0.657267 + 0.753658i \(0.728288\pi\)
\(762\) 0 0
\(763\) 2.81682e13 3.00884
\(764\) −9.21557e12 −0.978593
\(765\) 0 0
\(766\) −5.45643e11 −0.0572637
\(767\) 3.08927e12 0.322312
\(768\) 0 0
\(769\) −5.98877e12 −0.617546 −0.308773 0.951136i \(-0.599918\pi\)
−0.308773 + 0.951136i \(0.599918\pi\)
\(770\) −6.03730e12 −0.618920
\(771\) 0 0
\(772\) −4.35281e12 −0.441054
\(773\) 9.64838e12 0.971956 0.485978 0.873971i \(-0.338464\pi\)
0.485978 + 0.873971i \(0.338464\pi\)
\(774\) 0 0
\(775\) −4.34679e12 −0.432824
\(776\) −3.56871e12 −0.353292
\(777\) 0 0
\(778\) 7.19379e12 0.703963
\(779\) −2.23351e12 −0.217305
\(780\) 0 0
\(781\) −3.80456e12 −0.365911
\(782\) −1.22249e12 −0.116900
\(783\) 0 0
\(784\) 4.06059e12 0.383854
\(785\) 3.22230e12 0.302867
\(786\) 0 0
\(787\) 1.40829e13 1.30860 0.654299 0.756236i \(-0.272964\pi\)
0.654299 + 0.756236i \(0.272964\pi\)
\(788\) −1.01963e13 −0.942049
\(789\) 0 0
\(790\) −3.57398e12 −0.326460
\(791\) 6.86350e12 0.623378
\(792\) 0 0
\(793\) −5.92498e12 −0.532056
\(794\) −3.67961e12 −0.328556
\(795\) 0 0
\(796\) 3.35568e11 0.0296259
\(797\) 8.45863e12 0.742570 0.371285 0.928519i \(-0.378917\pi\)
0.371285 + 0.928519i \(0.378917\pi\)
\(798\) 0 0
\(799\) −5.11440e12 −0.443950
\(800\) −2.05869e12 −0.177699
\(801\) 0 0
\(802\) −1.07066e13 −0.913834
\(803\) 2.29269e12 0.194592
\(804\) 0 0
\(805\) −1.07403e13 −0.901437
\(806\) 1.01175e12 0.0844431
\(807\) 0 0
\(808\) −3.37679e12 −0.278710
\(809\) 9.95988e12 0.817496 0.408748 0.912647i \(-0.365965\pi\)
0.408748 + 0.912647i \(0.365965\pi\)
\(810\) 0 0
\(811\) 9.83262e12 0.798133 0.399067 0.916922i \(-0.369334\pi\)
0.399067 + 0.916922i \(0.369334\pi\)
\(812\) −6.73369e12 −0.543564
\(813\) 0 0
\(814\) 5.45873e12 0.435795
\(815\) −2.23998e13 −1.77842
\(816\) 0 0
\(817\) −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 \(0.608067\pi\)
\(822\) 0 0
\(823\) −1.26458e13 −0.960834 −0.480417 0.877040i \(-0.659515\pi\)
−0.480417 + 0.877040i \(0.659515\pi\)
\(824\) 6.79508e12 0.513479
\(825\) 0 0
\(826\) 1.75053e13 1.30845
\(827\) 2.04741e13 1.52205 0.761027 0.648720i \(-0.224695\pi\)
0.761027 + 0.648720i \(0.224695\pi\)
\(828\) 0 0
\(829\) −1.00940e13 −0.742283 −0.371141 0.928576i \(-0.621033\pi\)
−0.371141 + 0.928576i \(0.621033\pi\)
\(830\) 9.76056e12 0.713877
\(831\) 0 0
\(832\) 4.79174e11 0.0346688
\(833\) 8.82324e12 0.634929
\(834\) 0 0
\(835\) −3.40994e12 −0.242749
\(836\) −4.01982e11 −0.0284629
\(837\) 0 0
\(838\) −5.52085e12 −0.386730
\(839\) −1.54505e13 −1.07650 −0.538249 0.842786i \(-0.680914\pi\)
−0.538249 + 0.842786i \(0.680914\pi\)
\(840\) 0 0
\(841\) −7.74485e12 −0.533864
\(842\) 8.92508e12 0.611938
\(843\) 0 0
\(844\) 6.03589e12 0.409449
\(845\) 1.61433e12 0.108927
\(846\) 0 0
\(847\) 2.02566e13 1.35235
\(848\) 4.35907e12 0.289476
\(849\) 0 0
\(850\) −4.47331e12 −0.293930
\(851\) 9.71104e12 0.634721
\(852\) 0 0
\(853\) 2.56415e13 1.65834 0.829170 0.558997i \(-0.188814\pi\)
0.829170 + 0.558997i \(0.188814\pi\)
\(854\) −3.35737e13 −2.15993
\(855\) 0 0
\(856\) 4.71717e12 0.300296
\(857\) 1.44941e11 0.00917862 0.00458931 0.999989i \(-0.498539\pi\)
0.00458931 + 0.999989i \(0.498539\pi\)
\(858\) 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
\(861\) 0 0
\(862\) −1.02279e13 −0.630962
\(863\) 4.25726e12 0.261265 0.130633 0.991431i \(-0.458299\pi\)
0.130633 + 0.991431i \(0.458299\pi\)
\(864\) 0 0
\(865\) −5.46892e12 −0.332146
\(866\) 1.98015e13 1.19638
\(867\) 0 0
\(868\) 5.73303e12 0.342804
\(869\) −2.12764e12 −0.126563
\(870\) 0 0
\(871\) 5.51272e12 0.324552
\(872\) 1.14065e13 0.668081
\(873\) 0 0
\(874\) −7.15123e11 −0.0414552
\(875\) −2.03999e11 −0.0117650
\(876\) 0 0
\(877\) 1.42035e13 0.810768 0.405384 0.914147i \(-0.367138\pi\)
0.405384 + 0.914147i \(0.367138\pi\)
\(878\) 1.46376e13 0.831276
\(879\) 0 0
\(880\) −2.44476e12 −0.137425
\(881\) 1.12878e13 0.631275 0.315638 0.948880i \(-0.397782\pi\)
0.315638 + 0.948880i \(0.397782\pi\)
\(882\) 0 0
\(883\) 2.25433e12 0.124794 0.0623972 0.998051i \(-0.480125\pi\)
0.0623972 + 0.998051i \(0.480125\pi\)
\(884\) 1.04120e12 0.0573452
\(885\) 0 0
\(886\) −1.89841e13 −1.03499
\(887\) 3.13450e13 1.70024 0.850122 0.526585i \(-0.176528\pi\)
0.850122 + 0.526585i \(0.176528\pi\)
\(888\) 0 0
\(889\) 3.52297e13 1.89170
\(890\) −2.01854e11 −0.0107841
\(891\) 0 0
\(892\) 6.10115e11 0.0322678
\(893\) −2.99179e12 −0.157434
\(894\) 0 0
\(895\) −1.35894e13 −0.707942
\(896\) 2.71522e12 0.140741
\(897\) 0 0
\(898\) −4.33294e11 −0.0222351
\(899\) −5.75739e12 −0.293973
\(900\) 0 0
\(901\) 9.47180e12 0.478819
\(902\) −8.08657e12 −0.406757
\(903\) 0 0
\(904\) 2.77933e12 0.138415
\(905\) −4.77997e13 −2.36868
\(906\) 0 0
\(907\) 1.77171e13 0.869280 0.434640 0.900604i \(-0.356876\pi\)
0.434640 + 0.900604i \(0.356876\pi\)
\(908\) 1.91116e12 0.0933064
\(909\) 0 0
\(910\) 9.14756e12 0.442200
\(911\) −2.93419e13 −1.41142 −0.705710 0.708501i \(-0.749372\pi\)
−0.705710 + 0.708501i \(0.749372\pi\)
\(912\) 0 0
\(913\) 5.81059e12 0.276759
\(914\) −3.24138e12 −0.153628
\(915\) 0 0
\(916\) 6.75754e12 0.317146
\(917\) 5.08482e13 2.37473
\(918\) 0 0
\(919\) −1.66013e13 −0.767754 −0.383877 0.923384i \(-0.625411\pi\)
−0.383877 + 0.923384i \(0.625411\pi\)
\(920\) −4.34922e12 −0.200155
\(921\) 0 0
\(922\) 1.43415e13 0.653592
\(923\) 5.76457e12 0.261432
\(924\) 0 0
\(925\) 3.55345e13 1.59593
\(926\) 8.28384e12 0.370239
\(927\) 0 0
\(928\) −2.72676e12 −0.120693
\(929\) 1.45567e13 0.641197 0.320598 0.947215i \(-0.396116\pi\)
0.320598 + 0.947215i \(0.396116\pi\)
\(930\) 0 0
\(931\) 5.16136e12 0.225160
\(932\) −6.15569e12 −0.267242
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) −1.81954e13 −0.760127
\(941\) 2.15767e13 0.897081 0.448540 0.893763i \(-0.351944\pi\)
0.448540 + 0.893763i \(0.351944\pi\)
\(942\) 0 0
\(943\) −1.43859e13 −0.592428
\(944\) 7.08864e12 0.290528
\(945\) 0 0
\(946\) −1.27436e13 −0.517349
\(947\) −7.20814e12 −0.291238 −0.145619 0.989341i \(-0.546517\pi\)
−0.145619 + 0.989341i \(0.546517\pi\)
\(948\) 0 0
\(949\) −3.47382e12 −0.139030
\(950\) −2.61677e12 −0.104234
\(951\) 0 0
\(952\) 5.89990e12 0.232798
\(953\) −2.71758e13 −1.06724 −0.533622 0.845723i \(-0.679170\pi\)
−0.533622 + 0.845723i \(0.679170\pi\)
\(954\) 0 0
\(955\) −7.12407e13 −2.77149
\(956\) −1.52657e13 −0.591095
\(957\) 0 0
\(958\) 2.77205e13 1.06330
\(959\) −6.46251e13 −2.46727
\(960\) 0 0
\(961\) −2.15378e13 −0.814603
\(962\) −8.27092e12 −0.311362
\(963\) 0 0
\(964\) 3.21637e12 0.119955
\(965\) −3.36492e13 −1.24911
\(966\) 0 0
\(967\) 1.20281e13 0.442364 0.221182 0.975233i \(-0.429009\pi\)
0.221182 + 0.975233i \(0.429009\pi\)
\(968\) 8.20275e12 0.300276
\(969\) 0 0
\(970\) −2.75878e13 −1.00056
\(971\) 1.10321e12 0.0398265 0.0199132 0.999802i \(-0.493661\pi\)
0.0199132 + 0.999802i \(0.493661\pi\)
\(972\) 0 0
\(973\) −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 \(0.249850\pi\)
\(978\) 0 0
\(979\) −1.20166e11 −0.00418081
\(980\) 3.13902e13 1.08712
\(981\) 0 0
\(982\) −6.86326e12 −0.235520
\(983\) −5.04270e13 −1.72255 −0.861276 0.508137i \(-0.830334\pi\)
−0.861276 + 0.508137i \(0.830334\pi\)
\(984\) 0 0
\(985\) −7.88219e13 −2.66799
\(986\) −5.92497e12 −0.199637
\(987\) 0 0
\(988\) 6.09072e11 0.0203359
\(989\) −2.26708e13 −0.753502
\(990\) 0 0
\(991\) −2.37023e13 −0.780653 −0.390327 0.920676i \(-0.627638\pi\)
−0.390327 + 0.920676i \(0.627638\pi\)
\(992\) 2.32155e12 0.0761160
\(993\) 0 0
\(994\) 3.26647e13 1.06131
\(995\) 2.59410e12 0.0839040
\(996\) 0 0
\(997\) −3.80094e13 −1.21832 −0.609162 0.793045i \(-0.708494\pi\)
−0.609162 + 0.793045i \(0.708494\pi\)
\(998\) 1.45948e13 0.465704
\(999\) 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 234.10.a.a.1.1 1
3.2 odd 2 26.10.a.c.1.1 1
12.11 even 2 208.10.a.b.1.1 1
39.38 odd 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 3.2 odd 2
208.10.a.b.1.1 1 12.11 even 2
234.10.a.a.1.1 1 1.1 even 1 trivial
338.10.a.b.1.1 1 39.38 odd 2