Properties

Label 18.42.a.c.1.2
Level $18$
Weight $42$
Character 18.1
Self dual yes
Analytic conductor $191.649$
Analytic rank $0$
Dimension $2$
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] = [18,42,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 42, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 42);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(191.649006822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1139917559892 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.06767e6\) of defining polynomial
Character \(\chi\) \(=\) 18.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.04858e6 q^{2} +1.09951e12 q^{4} +2.54286e14 q^{5} +2.96016e17 q^{7} -1.15292e18 q^{8} +O(q^{10})\) \(q-1.04858e6 q^{2} +1.09951e12 q^{4} +2.54286e14 q^{5} +2.96016e17 q^{7} -1.15292e18 q^{8} -2.66638e20 q^{10} +1.09620e21 q^{11} +8.09333e22 q^{13} -3.10395e23 q^{14} +1.20893e24 q^{16} +5.33128e24 q^{17} -3.50762e25 q^{19} +2.79590e26 q^{20} -1.14945e27 q^{22} -6.68932e27 q^{23} +1.91866e28 q^{25} -8.48648e28 q^{26} +3.25473e29 q^{28} +1.63586e30 q^{29} +2.77961e30 q^{31} -1.26765e30 q^{32} -5.59026e30 q^{34} +7.52726e31 q^{35} +4.65417e31 q^{37} +3.67800e31 q^{38} -2.93172e32 q^{40} +8.83679e32 q^{41} -3.69256e32 q^{43} +1.20528e33 q^{44} +7.01426e33 q^{46} +1.89619e34 q^{47} +4.30577e34 q^{49} -2.01186e34 q^{50} +8.89872e34 q^{52} +2.01460e35 q^{53} +2.78747e35 q^{55} -3.41283e35 q^{56} -1.71532e36 q^{58} -5.07603e34 q^{59} -6.57027e36 q^{61} -2.91463e36 q^{62} +1.32923e36 q^{64} +2.05802e37 q^{65} +5.28414e37 q^{67} +5.86181e36 q^{68} -7.89291e37 q^{70} +1.04768e38 q^{71} -2.83453e38 q^{73} -4.88025e37 q^{74} -3.85666e37 q^{76} +3.24491e38 q^{77} -6.59452e38 q^{79} +3.07413e38 q^{80} -9.26605e38 q^{82} -1.30773e39 q^{83} +1.35567e39 q^{85} +3.87193e38 q^{86} -1.26383e39 q^{88} -2.16819e39 q^{89} +2.39575e40 q^{91} -7.35498e39 q^{92} -1.98830e40 q^{94} -8.91937e39 q^{95} -8.40052e39 q^{97} -4.51493e40 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{2} + 2199023255552 q^{4} - 97599184325580 q^{5} + 21\!\cdots\!56 q^{7}+ \cdots - 23\!\cdots\!52 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2097152 q^{2} + 2199023255552 q^{4} - 97599184325580 q^{5} + 21\!\cdots\!56 q^{7}+ \cdots - 49\!\cdots\!04 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04858e6 −0.707107
\(3\) 0 0
\(4\) 1.09951e12 0.500000
\(5\) 2.54286e14 1.19244 0.596221 0.802820i \(-0.296668\pi\)
0.596221 + 0.802820i \(0.296668\pi\)
\(6\) 0 0
\(7\) 2.96016e17 1.40218 0.701092 0.713071i \(-0.252696\pi\)
0.701092 + 0.713071i \(0.252696\pi\)
\(8\) −1.15292e18 −0.353553
\(9\) 0 0
\(10\) −2.66638e20 −0.843184
\(11\) 1.09620e21 0.491291 0.245645 0.969360i \(-0.421000\pi\)
0.245645 + 0.969360i \(0.421000\pi\)
\(12\) 0 0
\(13\) 8.09333e22 1.18111 0.590553 0.806999i \(-0.298910\pi\)
0.590553 + 0.806999i \(0.298910\pi\)
\(14\) −3.10395e23 −0.991494
\(15\) 0 0
\(16\) 1.20893e24 0.250000
\(17\) 5.33128e24 0.318148 0.159074 0.987267i \(-0.449149\pi\)
0.159074 + 0.987267i \(0.449149\pi\)
\(18\) 0 0
\(19\) −3.50762e25 −0.214074 −0.107037 0.994255i \(-0.534136\pi\)
−0.107037 + 0.994255i \(0.534136\pi\)
\(20\) 2.79590e26 0.596221
\(21\) 0 0
\(22\) −1.14945e27 −0.347395
\(23\) −6.68932e27 −0.812758 −0.406379 0.913705i \(-0.633209\pi\)
−0.406379 + 0.913705i \(0.633209\pi\)
\(24\) 0 0
\(25\) 1.91866e28 0.421917
\(26\) −8.48648e28 −0.835168
\(27\) 0 0
\(28\) 3.25473e29 0.701092
\(29\) 1.63586e30 1.71628 0.858139 0.513418i \(-0.171621\pi\)
0.858139 + 0.513418i \(0.171621\pi\)
\(30\) 0 0
\(31\) 2.77961e30 0.743137 0.371568 0.928406i \(-0.378820\pi\)
0.371568 + 0.928406i \(0.378820\pi\)
\(32\) −1.26765e30 −0.176777
\(33\) 0 0
\(34\) −5.59026e30 −0.224964
\(35\) 7.52726e31 1.67202
\(36\) 0 0
\(37\) 4.65417e31 0.330908 0.165454 0.986218i \(-0.447091\pi\)
0.165454 + 0.986218i \(0.447091\pi\)
\(38\) 3.67800e31 0.151373
\(39\) 0 0
\(40\) −2.93172e32 −0.421592
\(41\) 8.83679e32 0.765995 0.382997 0.923749i \(-0.374892\pi\)
0.382997 + 0.923749i \(0.374892\pi\)
\(42\) 0 0
\(43\) −3.69256e32 −0.120566 −0.0602829 0.998181i \(-0.519200\pi\)
−0.0602829 + 0.998181i \(0.519200\pi\)
\(44\) 1.20528e33 0.245645
\(45\) 0 0
\(46\) 7.01426e33 0.574707
\(47\) 1.89619e34 0.999714 0.499857 0.866108i \(-0.333386\pi\)
0.499857 + 0.866108i \(0.333386\pi\)
\(48\) 0 0
\(49\) 4.30577e34 0.966120
\(50\) −2.01186e34 −0.298341
\(51\) 0 0
\(52\) 8.89872e34 0.590553
\(53\) 2.01460e35 0.904761 0.452380 0.891825i \(-0.350575\pi\)
0.452380 + 0.891825i \(0.350575\pi\)
\(54\) 0 0
\(55\) 2.78747e35 0.585835
\(56\) −3.41283e35 −0.495747
\(57\) 0 0
\(58\) −1.71532e36 −1.21359
\(59\) −5.07603e34 −0.0252964 −0.0126482 0.999920i \(-0.504026\pi\)
−0.0126482 + 0.999920i \(0.504026\pi\)
\(60\) 0 0
\(61\) −6.57027e36 −1.65319 −0.826593 0.562801i \(-0.809724\pi\)
−0.826593 + 0.562801i \(0.809724\pi\)
\(62\) −2.91463e36 −0.525477
\(63\) 0 0
\(64\) 1.32923e36 0.125000
\(65\) 2.05802e37 1.40840
\(66\) 0 0
\(67\) 5.28414e37 1.94286 0.971429 0.237332i \(-0.0762729\pi\)
0.971429 + 0.237332i \(0.0762729\pi\)
\(68\) 5.86181e36 0.159074
\(69\) 0 0
\(70\) −7.89291e37 −1.18230
\(71\) 1.04768e38 1.17337 0.586684 0.809816i \(-0.300433\pi\)
0.586684 + 0.809816i \(0.300433\pi\)
\(72\) 0 0
\(73\) −2.83453e38 −1.79623 −0.898117 0.439756i \(-0.855065\pi\)
−0.898117 + 0.439756i \(0.855065\pi\)
\(74\) −4.88025e37 −0.233987
\(75\) 0 0
\(76\) −3.85666e37 −0.107037
\(77\) 3.24491e38 0.688880
\(78\) 0 0
\(79\) −6.59452e38 −0.827614 −0.413807 0.910365i \(-0.635801\pi\)
−0.413807 + 0.910365i \(0.635801\pi\)
\(80\) 3.07413e38 0.298110
\(81\) 0 0
\(82\) −9.26605e38 −0.541640
\(83\) −1.30773e39 −0.596236 −0.298118 0.954529i \(-0.596359\pi\)
−0.298118 + 0.954529i \(0.596359\pi\)
\(84\) 0 0
\(85\) 1.35567e39 0.379373
\(86\) 3.87193e38 0.0852529
\(87\) 0 0
\(88\) −1.26383e39 −0.173697
\(89\) −2.16819e39 −0.236376 −0.118188 0.992991i \(-0.537708\pi\)
−0.118188 + 0.992991i \(0.537708\pi\)
\(90\) 0 0
\(91\) 2.39575e40 1.65613
\(92\) −7.35498e39 −0.406379
\(93\) 0 0
\(94\) −1.98830e40 −0.706905
\(95\) −8.91937e39 −0.255270
\(96\) 0 0
\(97\) −8.40052e39 −0.156850 −0.0784252 0.996920i \(-0.524989\pi\)
−0.0784252 + 0.996920i \(0.524989\pi\)
\(98\) −4.51493e40 −0.683150
\(99\) 0 0
\(100\) 2.10959e40 0.210959
\(101\) −4.81454e40 −0.392615 −0.196308 0.980542i \(-0.562895\pi\)
−0.196308 + 0.980542i \(0.562895\pi\)
\(102\) 0 0
\(103\) 8.87160e40 0.483992 0.241996 0.970277i \(-0.422198\pi\)
0.241996 + 0.970277i \(0.422198\pi\)
\(104\) −9.33098e40 −0.417584
\(105\) 0 0
\(106\) −2.11246e41 −0.639763
\(107\) −5.55682e41 −1.38822 −0.694111 0.719868i \(-0.744203\pi\)
−0.694111 + 0.719868i \(0.744203\pi\)
\(108\) 0 0
\(109\) −4.77382e41 −0.815873 −0.407936 0.913010i \(-0.633752\pi\)
−0.407936 + 0.913010i \(0.633752\pi\)
\(110\) −2.92288e41 −0.414248
\(111\) 0 0
\(112\) 3.57861e41 0.350546
\(113\) −9.67742e41 −0.790045 −0.395022 0.918672i \(-0.629263\pi\)
−0.395022 + 0.918672i \(0.629263\pi\)
\(114\) 0 0
\(115\) −1.70100e42 −0.969167
\(116\) 1.79865e42 0.858139
\(117\) 0 0
\(118\) 5.32261e40 0.0178872
\(119\) 1.57814e42 0.446102
\(120\) 0 0
\(121\) −3.77687e42 −0.758634
\(122\) 6.88943e42 1.16898
\(123\) 0 0
\(124\) 3.05621e42 0.371568
\(125\) −6.68471e42 −0.689330
\(126\) 0 0
\(127\) 1.15659e43 0.861396 0.430698 0.902496i \(-0.358267\pi\)
0.430698 + 0.902496i \(0.358267\pi\)
\(128\) −1.39380e42 −0.0883883
\(129\) 0 0
\(130\) −2.15799e43 −0.995889
\(131\) −2.42843e43 −0.957776 −0.478888 0.877876i \(-0.658960\pi\)
−0.478888 + 0.877876i \(0.658960\pi\)
\(132\) 0 0
\(133\) −1.03831e43 −0.300171
\(134\) −5.54082e43 −1.37381
\(135\) 0 0
\(136\) −6.14655e42 −0.112482
\(137\) 1.54500e43 0.243308 0.121654 0.992573i \(-0.461180\pi\)
0.121654 + 0.992573i \(0.461180\pi\)
\(138\) 0 0
\(139\) 1.40766e44 1.64700 0.823501 0.567315i \(-0.192018\pi\)
0.823501 + 0.567315i \(0.192018\pi\)
\(140\) 8.27631e43 0.836011
\(141\) 0 0
\(142\) −1.09858e44 −0.829696
\(143\) 8.87189e43 0.580266
\(144\) 0 0
\(145\) 4.15976e44 2.04656
\(146\) 2.97222e44 1.27013
\(147\) 0 0
\(148\) 5.11732e43 0.165454
\(149\) −1.57473e44 −0.443493 −0.221746 0.975104i \(-0.571176\pi\)
−0.221746 + 0.975104i \(0.571176\pi\)
\(150\) 0 0
\(151\) −2.82811e44 −0.605994 −0.302997 0.952992i \(-0.597987\pi\)
−0.302997 + 0.952992i \(0.597987\pi\)
\(152\) 4.04401e43 0.0756865
\(153\) 0 0
\(154\) −3.40254e44 −0.487112
\(155\) 7.06815e44 0.886147
\(156\) 0 0
\(157\) 2.60906e44 0.251502 0.125751 0.992062i \(-0.459866\pi\)
0.125751 + 0.992062i \(0.459866\pi\)
\(158\) 6.91486e44 0.585212
\(159\) 0 0
\(160\) −3.22346e44 −0.210796
\(161\) −1.98014e45 −1.13964
\(162\) 0 0
\(163\) −1.61270e45 −0.720621 −0.360310 0.932832i \(-0.617329\pi\)
−0.360310 + 0.932832i \(0.617329\pi\)
\(164\) 9.71616e44 0.382997
\(165\) 0 0
\(166\) 1.37126e45 0.421602
\(167\) 1.22668e43 0.00333459 0.00166729 0.999999i \(-0.499469\pi\)
0.00166729 + 0.999999i \(0.499469\pi\)
\(168\) 0 0
\(169\) 1.85475e45 0.395011
\(170\) −1.42152e45 −0.268257
\(171\) 0 0
\(172\) −4.06001e44 −0.0602829
\(173\) 6.97298e45 0.919334 0.459667 0.888091i \(-0.347969\pi\)
0.459667 + 0.888091i \(0.347969\pi\)
\(174\) 0 0
\(175\) 5.67953e45 0.591606
\(176\) 1.32522e45 0.122823
\(177\) 0 0
\(178\) 2.27351e45 0.167143
\(179\) −2.57092e46 −1.68501 −0.842504 0.538690i \(-0.818920\pi\)
−0.842504 + 0.538690i \(0.818920\pi\)
\(180\) 0 0
\(181\) 1.71977e46 0.897553 0.448776 0.893644i \(-0.351860\pi\)
0.448776 + 0.893644i \(0.351860\pi\)
\(182\) −2.51213e46 −1.17106
\(183\) 0 0
\(184\) 7.71226e45 0.287353
\(185\) 1.18349e46 0.394588
\(186\) 0 0
\(187\) 5.84413e45 0.156303
\(188\) 2.08488e46 0.499857
\(189\) 0 0
\(190\) 9.35264e45 0.180503
\(191\) 2.15620e46 0.373684 0.186842 0.982390i \(-0.440175\pi\)
0.186842 + 0.982390i \(0.440175\pi\)
\(192\) 0 0
\(193\) 2.03359e46 0.284668 0.142334 0.989819i \(-0.454539\pi\)
0.142334 + 0.989819i \(0.454539\pi\)
\(194\) 8.80859e45 0.110910
\(195\) 0 0
\(196\) 4.73424e46 0.483060
\(197\) 1.48334e47 1.36359 0.681793 0.731545i \(-0.261200\pi\)
0.681793 + 0.731545i \(0.261200\pi\)
\(198\) 0 0
\(199\) −7.47977e46 −0.558985 −0.279493 0.960148i \(-0.590166\pi\)
−0.279493 + 0.960148i \(0.590166\pi\)
\(200\) −2.21206e46 −0.149170
\(201\) 0 0
\(202\) 5.04842e46 0.277621
\(203\) 4.84241e47 2.40654
\(204\) 0 0
\(205\) 2.24707e47 0.913404
\(206\) −9.30254e46 −0.342234
\(207\) 0 0
\(208\) 9.78424e46 0.295276
\(209\) −3.84504e46 −0.105172
\(210\) 0 0
\(211\) 8.14889e47 1.83362 0.916808 0.399329i \(-0.130757\pi\)
0.916808 + 0.399329i \(0.130757\pi\)
\(212\) 2.21508e47 0.452380
\(213\) 0 0
\(214\) 5.82675e47 0.981622
\(215\) −9.38966e46 −0.143768
\(216\) 0 0
\(217\) 8.22808e47 1.04201
\(218\) 5.00571e47 0.576909
\(219\) 0 0
\(220\) 3.06486e47 0.292918
\(221\) 4.31479e47 0.375766
\(222\) 0 0
\(223\) −2.94601e45 −0.00213297 −0.00106648 0.999999i \(-0.500339\pi\)
−0.00106648 + 0.999999i \(0.500339\pi\)
\(224\) −3.75245e47 −0.247873
\(225\) 0 0
\(226\) 1.01475e48 0.558646
\(227\) −1.23961e48 −0.623383 −0.311691 0.950183i \(-0.600895\pi\)
−0.311691 + 0.950183i \(0.600895\pi\)
\(228\) 0 0
\(229\) 2.51277e48 1.05567 0.527833 0.849348i \(-0.323005\pi\)
0.527833 + 0.849348i \(0.323005\pi\)
\(230\) 1.78363e48 0.685304
\(231\) 0 0
\(232\) −1.88602e48 −0.606796
\(233\) 3.44195e47 0.101393 0.0506966 0.998714i \(-0.483856\pi\)
0.0506966 + 0.998714i \(0.483856\pi\)
\(234\) 0 0
\(235\) 4.82174e48 1.19210
\(236\) −5.58116e46 −0.0126482
\(237\) 0 0
\(238\) −1.65480e48 −0.315442
\(239\) 1.24157e48 0.217177 0.108588 0.994087i \(-0.465367\pi\)
0.108588 + 0.994087i \(0.465367\pi\)
\(240\) 0 0
\(241\) 8.53681e48 1.25877 0.629385 0.777094i \(-0.283307\pi\)
0.629385 + 0.777094i \(0.283307\pi\)
\(242\) 3.96034e48 0.536435
\(243\) 0 0
\(244\) −7.22409e48 −0.826593
\(245\) 1.09490e49 1.15204
\(246\) 0 0
\(247\) −2.83883e48 −0.252844
\(248\) −3.20467e48 −0.262739
\(249\) 0 0
\(250\) 7.00942e48 0.487430
\(251\) −1.82523e49 −1.16952 −0.584759 0.811207i \(-0.698811\pi\)
−0.584759 + 0.811207i \(0.698811\pi\)
\(252\) 0 0
\(253\) −7.33281e48 −0.399300
\(254\) −1.21277e49 −0.609099
\(255\) 0 0
\(256\) 1.46150e48 0.0625000
\(257\) 3.84389e49 1.51755 0.758774 0.651354i \(-0.225799\pi\)
0.758774 + 0.651354i \(0.225799\pi\)
\(258\) 0 0
\(259\) 1.37771e49 0.463994
\(260\) 2.26282e49 0.704200
\(261\) 0 0
\(262\) 2.54639e49 0.677250
\(263\) −5.88587e49 −1.44783 −0.723915 0.689889i \(-0.757659\pi\)
−0.723915 + 0.689889i \(0.757659\pi\)
\(264\) 0 0
\(265\) 5.12285e49 1.07887
\(266\) 1.08875e49 0.212253
\(267\) 0 0
\(268\) 5.80997e49 0.971429
\(269\) 6.22517e49 0.964338 0.482169 0.876078i \(-0.339849\pi\)
0.482169 + 0.876078i \(0.339849\pi\)
\(270\) 0 0
\(271\) −8.85718e49 −1.17876 −0.589380 0.807856i \(-0.700628\pi\)
−0.589380 + 0.807856i \(0.700628\pi\)
\(272\) 6.44513e48 0.0795369
\(273\) 0 0
\(274\) −1.62005e49 −0.172045
\(275\) 2.10323e49 0.207284
\(276\) 0 0
\(277\) −1.94293e50 −1.65053 −0.825264 0.564747i \(-0.808974\pi\)
−0.825264 + 0.564747i \(0.808974\pi\)
\(278\) −1.47604e50 −1.16461
\(279\) 0 0
\(280\) −8.67834e49 −0.591149
\(281\) −2.14728e50 −1.35960 −0.679798 0.733400i \(-0.737932\pi\)
−0.679798 + 0.733400i \(0.737932\pi\)
\(282\) 0 0
\(283\) −6.68644e49 −0.366078 −0.183039 0.983106i \(-0.558593\pi\)
−0.183039 + 0.983106i \(0.558593\pi\)
\(284\) 1.15194e50 0.586684
\(285\) 0 0
\(286\) −9.30285e49 −0.410310
\(287\) 2.61583e50 1.07407
\(288\) 0 0
\(289\) −2.52383e50 −0.898782
\(290\) −4.36183e50 −1.44714
\(291\) 0 0
\(292\) −3.11660e50 −0.898117
\(293\) 2.93540e50 0.788644 0.394322 0.918972i \(-0.370979\pi\)
0.394322 + 0.918972i \(0.370979\pi\)
\(294\) 0 0
\(295\) −1.29076e49 −0.0301645
\(296\) −5.36590e49 −0.116994
\(297\) 0 0
\(298\) 1.65122e50 0.313597
\(299\) −5.41389e50 −0.959953
\(300\) 0 0
\(301\) −1.09306e50 −0.169055
\(302\) 2.96548e50 0.428502
\(303\) 0 0
\(304\) −4.24045e49 −0.0535184
\(305\) −1.67073e51 −1.97133
\(306\) 0 0
\(307\) −4.35932e50 −0.449864 −0.224932 0.974374i \(-0.572216\pi\)
−0.224932 + 0.974374i \(0.572216\pi\)
\(308\) 3.56782e50 0.344440
\(309\) 0 0
\(310\) −7.41149e50 −0.626601
\(311\) 2.38798e50 0.188992 0.0944958 0.995525i \(-0.469876\pi\)
0.0944958 + 0.995525i \(0.469876\pi\)
\(312\) 0 0
\(313\) −2.03212e51 −1.41023 −0.705117 0.709091i \(-0.749106\pi\)
−0.705117 + 0.709091i \(0.749106\pi\)
\(314\) −2.73580e50 −0.177839
\(315\) 0 0
\(316\) −7.25075e50 −0.413807
\(317\) −2.68276e51 −1.43505 −0.717525 0.696532i \(-0.754725\pi\)
−0.717525 + 0.696532i \(0.754725\pi\)
\(318\) 0 0
\(319\) 1.79323e51 0.843191
\(320\) 3.38004e50 0.149055
\(321\) 0 0
\(322\) 2.07633e51 0.805845
\(323\) −1.87001e50 −0.0681071
\(324\) 0 0
\(325\) 1.55283e51 0.498329
\(326\) 1.69103e51 0.509556
\(327\) 0 0
\(328\) −1.01881e51 −0.270820
\(329\) 5.61302e51 1.40178
\(330\) 0 0
\(331\) 7.54395e51 1.66389 0.831945 0.554858i \(-0.187227\pi\)
0.831945 + 0.554858i \(0.187227\pi\)
\(332\) −1.43787e51 −0.298118
\(333\) 0 0
\(334\) −1.28626e49 −0.00235791
\(335\) 1.34368e52 2.31674
\(336\) 0 0
\(337\) −5.36454e51 −0.818689 −0.409345 0.912380i \(-0.634243\pi\)
−0.409345 + 0.912380i \(0.634243\pi\)
\(338\) −1.94485e51 −0.279315
\(339\) 0 0
\(340\) 1.49058e51 0.189686
\(341\) 3.04700e51 0.365096
\(342\) 0 0
\(343\) −4.46968e50 −0.0475059
\(344\) 4.25723e50 0.0426264
\(345\) 0 0
\(346\) −7.31170e51 −0.650068
\(347\) 1.60783e51 0.134737 0.0673684 0.997728i \(-0.478540\pi\)
0.0673684 + 0.997728i \(0.478540\pi\)
\(348\) 0 0
\(349\) 1.51169e52 1.12601 0.563004 0.826454i \(-0.309646\pi\)
0.563004 + 0.826454i \(0.309646\pi\)
\(350\) −5.95542e51 −0.418329
\(351\) 0 0
\(352\) −1.38959e51 −0.0868487
\(353\) −8.96540e51 −0.528675 −0.264337 0.964430i \(-0.585153\pi\)
−0.264337 + 0.964430i \(0.585153\pi\)
\(354\) 0 0
\(355\) 2.66411e52 1.39917
\(356\) −2.38395e51 −0.118188
\(357\) 0 0
\(358\) 2.69581e52 1.19148
\(359\) 2.57092e52 1.07313 0.536565 0.843859i \(-0.319722\pi\)
0.536565 + 0.843859i \(0.319722\pi\)
\(360\) 0 0
\(361\) −2.56168e52 −0.954172
\(362\) −1.80331e52 −0.634666
\(363\) 0 0
\(364\) 2.63416e52 0.828064
\(365\) −7.20782e52 −2.14191
\(366\) 0 0
\(367\) 4.37801e52 1.16311 0.581557 0.813506i \(-0.302444\pi\)
0.581557 + 0.813506i \(0.302444\pi\)
\(368\) −8.08689e51 −0.203189
\(369\) 0 0
\(370\) −1.24098e52 −0.279016
\(371\) 5.96354e52 1.26864
\(372\) 0 0
\(373\) −3.46418e52 −0.660039 −0.330020 0.943974i \(-0.607055\pi\)
−0.330020 + 0.943974i \(0.607055\pi\)
\(374\) −6.12802e51 −0.110523
\(375\) 0 0
\(376\) −2.18616e52 −0.353452
\(377\) 1.32396e53 2.02710
\(378\) 0 0
\(379\) 3.98009e52 0.546750 0.273375 0.961908i \(-0.411860\pi\)
0.273375 + 0.961908i \(0.411860\pi\)
\(380\) −9.80695e51 −0.127635
\(381\) 0 0
\(382\) −2.26094e52 −0.264235
\(383\) −4.24383e52 −0.470092 −0.235046 0.971984i \(-0.575524\pi\)
−0.235046 + 0.971984i \(0.575524\pi\)
\(384\) 0 0
\(385\) 8.25136e52 0.821449
\(386\) −2.13237e52 −0.201290
\(387\) 0 0
\(388\) −9.23647e51 −0.0784252
\(389\) 1.66821e53 1.34364 0.671821 0.740713i \(-0.265512\pi\)
0.671821 + 0.740713i \(0.265512\pi\)
\(390\) 0 0
\(391\) −3.56627e52 −0.258577
\(392\) −4.96421e52 −0.341575
\(393\) 0 0
\(394\) −1.55539e53 −0.964201
\(395\) −1.67689e53 −0.986882
\(396\) 0 0
\(397\) −2.90239e53 −1.54010 −0.770052 0.637981i \(-0.779770\pi\)
−0.770052 + 0.637981i \(0.779770\pi\)
\(398\) 7.84311e52 0.395262
\(399\) 0 0
\(400\) 2.31952e52 0.105479
\(401\) −1.04436e53 −0.451222 −0.225611 0.974217i \(-0.572438\pi\)
−0.225611 + 0.974217i \(0.572438\pi\)
\(402\) 0 0
\(403\) 2.24963e53 0.877723
\(404\) −5.29365e52 −0.196308
\(405\) 0 0
\(406\) −5.07763e53 −1.70168
\(407\) 5.10189e52 0.162572
\(408\) 0 0
\(409\) 2.12912e53 0.613581 0.306790 0.951777i \(-0.400745\pi\)
0.306790 + 0.951777i \(0.400745\pi\)
\(410\) −2.35623e53 −0.645874
\(411\) 0 0
\(412\) 9.75442e52 0.241996
\(413\) −1.50259e52 −0.0354702
\(414\) 0 0
\(415\) −3.32538e53 −0.710976
\(416\) −1.02595e53 −0.208792
\(417\) 0 0
\(418\) 4.03181e52 0.0743681
\(419\) 8.36673e53 1.46950 0.734748 0.678340i \(-0.237300\pi\)
0.734748 + 0.678340i \(0.237300\pi\)
\(420\) 0 0
\(421\) 1.14983e54 1.83169 0.915845 0.401533i \(-0.131522\pi\)
0.915845 + 0.401533i \(0.131522\pi\)
\(422\) −8.54473e53 −1.29656
\(423\) 0 0
\(424\) −2.32268e53 −0.319881
\(425\) 1.02289e53 0.134232
\(426\) 0 0
\(427\) −1.94490e54 −2.31807
\(428\) −6.10979e53 −0.694111
\(429\) 0 0
\(430\) 9.84577e52 0.101659
\(431\) −1.32392e54 −1.30340 −0.651701 0.758476i \(-0.725944\pi\)
−0.651701 + 0.758476i \(0.725944\pi\)
\(432\) 0 0
\(433\) −1.08006e54 −0.967046 −0.483523 0.875332i \(-0.660643\pi\)
−0.483523 + 0.875332i \(0.660643\pi\)
\(434\) −8.62776e53 −0.736816
\(435\) 0 0
\(436\) −5.24887e53 −0.407936
\(437\) 2.34636e53 0.173990
\(438\) 0 0
\(439\) 1.14481e53 0.0773059 0.0386529 0.999253i \(-0.487693\pi\)
0.0386529 + 0.999253i \(0.487693\pi\)
\(440\) −3.21374e53 −0.207124
\(441\) 0 0
\(442\) −4.52438e53 −0.265707
\(443\) 9.71784e53 0.544870 0.272435 0.962174i \(-0.412171\pi\)
0.272435 + 0.962174i \(0.412171\pi\)
\(444\) 0 0
\(445\) −5.51341e53 −0.281864
\(446\) 3.08911e51 0.00150823
\(447\) 0 0
\(448\) 3.93472e53 0.175273
\(449\) 3.40657e54 1.44966 0.724832 0.688926i \(-0.241917\pi\)
0.724832 + 0.688926i \(0.241917\pi\)
\(450\) 0 0
\(451\) 9.68686e53 0.376326
\(452\) −1.06404e54 −0.395022
\(453\) 0 0
\(454\) 1.29982e54 0.440798
\(455\) 6.09207e54 1.97484
\(456\) 0 0
\(457\) 1.58378e54 0.469258 0.234629 0.972085i \(-0.424612\pi\)
0.234629 + 0.972085i \(0.424612\pi\)
\(458\) −2.63484e54 −0.746468
\(459\) 0 0
\(460\) −1.87027e54 −0.484583
\(461\) 2.93741e53 0.0727939 0.0363970 0.999337i \(-0.488412\pi\)
0.0363970 + 0.999337i \(0.488412\pi\)
\(462\) 0 0
\(463\) 7.57194e54 1.71711 0.858553 0.512725i \(-0.171364\pi\)
0.858553 + 0.512725i \(0.171364\pi\)
\(464\) 1.97763e54 0.429069
\(465\) 0 0
\(466\) −3.60915e53 −0.0716958
\(467\) 2.70909e54 0.515024 0.257512 0.966275i \(-0.417097\pi\)
0.257512 + 0.966275i \(0.417097\pi\)
\(468\) 0 0
\(469\) 1.56419e55 2.72424
\(470\) −5.05597e54 −0.842943
\(471\) 0 0
\(472\) 5.85227e52 0.00894362
\(473\) −4.04777e53 −0.0592328
\(474\) 0 0
\(475\) −6.72992e53 −0.0903214
\(476\) 1.73519e54 0.223051
\(477\) 0 0
\(478\) −1.30188e54 −0.153567
\(479\) 7.24145e53 0.0818366 0.0409183 0.999162i \(-0.486972\pi\)
0.0409183 + 0.999162i \(0.486972\pi\)
\(480\) 0 0
\(481\) 3.76678e54 0.390837
\(482\) −8.95149e54 −0.890084
\(483\) 0 0
\(484\) −4.15271e54 −0.379317
\(485\) −2.13613e54 −0.187035
\(486\) 0 0
\(487\) 3.33837e54 0.268653 0.134326 0.990937i \(-0.457113\pi\)
0.134326 + 0.990937i \(0.457113\pi\)
\(488\) 7.57500e54 0.584489
\(489\) 0 0
\(490\) −1.14808e55 −0.814617
\(491\) 1.88223e55 1.28086 0.640430 0.768016i \(-0.278756\pi\)
0.640430 + 0.768016i \(0.278756\pi\)
\(492\) 0 0
\(493\) 8.72124e54 0.546030
\(494\) 2.97673e54 0.178787
\(495\) 0 0
\(496\) 3.36034e54 0.185784
\(497\) 3.10131e55 1.64528
\(498\) 0 0
\(499\) −6.93745e54 −0.338953 −0.169476 0.985534i \(-0.554208\pi\)
−0.169476 + 0.985534i \(0.554208\pi\)
\(500\) −7.34991e54 −0.344665
\(501\) 0 0
\(502\) 1.91390e55 0.826974
\(503\) −2.98925e55 −1.23999 −0.619995 0.784606i \(-0.712865\pi\)
−0.619995 + 0.784606i \(0.712865\pi\)
\(504\) 0 0
\(505\) −1.22427e55 −0.468171
\(506\) 7.68901e54 0.282348
\(507\) 0 0
\(508\) 1.27168e55 0.430698
\(509\) −5.65842e55 −1.84069 −0.920345 0.391108i \(-0.872092\pi\)
−0.920345 + 0.391108i \(0.872092\pi\)
\(510\) 0 0
\(511\) −8.39067e55 −2.51865
\(512\) −1.53250e54 −0.0441942
\(513\) 0 0
\(514\) −4.03061e55 −1.07307
\(515\) 2.25592e55 0.577133
\(516\) 0 0
\(517\) 2.07860e55 0.491150
\(518\) −1.44463e55 −0.328093
\(519\) 0 0
\(520\) −2.37274e55 −0.497945
\(521\) −3.31572e55 −0.668967 −0.334483 0.942402i \(-0.608562\pi\)
−0.334483 + 0.942402i \(0.608562\pi\)
\(522\) 0 0
\(523\) 7.77683e55 1.45050 0.725250 0.688486i \(-0.241724\pi\)
0.725250 + 0.688486i \(0.241724\pi\)
\(524\) −2.67009e55 −0.478888
\(525\) 0 0
\(526\) 6.17178e55 1.02377
\(527\) 1.48189e55 0.236427
\(528\) 0 0
\(529\) −2.29924e55 −0.339424
\(530\) −5.37169e55 −0.762880
\(531\) 0 0
\(532\) −1.14163e55 −0.150085
\(533\) 7.15191e55 0.904721
\(534\) 0 0
\(535\) −1.41302e56 −1.65538
\(536\) −6.09220e55 −0.686904
\(537\) 0 0
\(538\) −6.52756e55 −0.681890
\(539\) 4.71997e55 0.474646
\(540\) 0 0
\(541\) 8.91733e55 0.831172 0.415586 0.909554i \(-0.363577\pi\)
0.415586 + 0.909554i \(0.363577\pi\)
\(542\) 9.28743e55 0.833509
\(543\) 0 0
\(544\) −6.75820e54 −0.0562411
\(545\) −1.21391e56 −0.972881
\(546\) 0 0
\(547\) −8.97461e55 −0.667230 −0.333615 0.942709i \(-0.608268\pi\)
−0.333615 + 0.942709i \(0.608268\pi\)
\(548\) 1.69875e55 0.121654
\(549\) 0 0
\(550\) −2.20539e55 −0.146572
\(551\) −5.73797e55 −0.367410
\(552\) 0 0
\(553\) −1.95208e56 −1.16047
\(554\) 2.03731e56 1.16710
\(555\) 0 0
\(556\) 1.54774e56 0.823501
\(557\) −3.47843e55 −0.178382 −0.0891911 0.996015i \(-0.528428\pi\)
−0.0891911 + 0.996015i \(0.528428\pi\)
\(558\) 0 0
\(559\) −2.98851e55 −0.142401
\(560\) 9.09990e55 0.418006
\(561\) 0 0
\(562\) 2.25159e56 0.961379
\(563\) −2.29948e55 −0.0946688 −0.0473344 0.998879i \(-0.515073\pi\)
−0.0473344 + 0.998879i \(0.515073\pi\)
\(564\) 0 0
\(565\) −2.46083e56 −0.942082
\(566\) 7.01124e55 0.258856
\(567\) 0 0
\(568\) −1.20790e56 −0.414848
\(569\) −3.18790e56 −1.05610 −0.528048 0.849214i \(-0.677076\pi\)
−0.528048 + 0.849214i \(0.677076\pi\)
\(570\) 0 0
\(571\) 1.16154e54 0.00358091 0.00179046 0.999998i \(-0.499430\pi\)
0.00179046 + 0.999998i \(0.499430\pi\)
\(572\) 9.75474e55 0.290133
\(573\) 0 0
\(574\) −2.74290e56 −0.759479
\(575\) −1.28345e56 −0.342917
\(576\) 0 0
\(577\) 6.45847e56 1.60703 0.803517 0.595281i \(-0.202959\pi\)
0.803517 + 0.595281i \(0.202959\pi\)
\(578\) 2.64643e56 0.635535
\(579\) 0 0
\(580\) 4.57371e56 1.02328
\(581\) −3.87109e56 −0.836032
\(582\) 0 0
\(583\) 2.20840e56 0.444500
\(584\) 3.26800e56 0.635065
\(585\) 0 0
\(586\) −3.07799e56 −0.557656
\(587\) 1.53676e56 0.268859 0.134430 0.990923i \(-0.457080\pi\)
0.134430 + 0.990923i \(0.457080\pi\)
\(588\) 0 0
\(589\) −9.74980e55 −0.159086
\(590\) 1.35346e55 0.0213295
\(591\) 0 0
\(592\) 5.62655e55 0.0827270
\(593\) −7.79761e56 −1.10749 −0.553746 0.832686i \(-0.686802\pi\)
−0.553746 + 0.832686i \(0.686802\pi\)
\(594\) 0 0
\(595\) 4.01300e56 0.531950
\(596\) −1.73143e56 −0.221746
\(597\) 0 0
\(598\) 5.67687e56 0.678789
\(599\) 9.18440e56 1.06121 0.530605 0.847619i \(-0.321965\pi\)
0.530605 + 0.847619i \(0.321965\pi\)
\(600\) 0 0
\(601\) 1.34315e57 1.44943 0.724714 0.689050i \(-0.241972\pi\)
0.724714 + 0.689050i \(0.241972\pi\)
\(602\) 1.14615e56 0.119540
\(603\) 0 0
\(604\) −3.10954e56 −0.302997
\(605\) −9.60405e56 −0.904626
\(606\) 0 0
\(607\) −1.37250e56 −0.120821 −0.0604105 0.998174i \(-0.519241\pi\)
−0.0604105 + 0.998174i \(0.519241\pi\)
\(608\) 4.44643e55 0.0378432
\(609\) 0 0
\(610\) 1.75188e57 1.39394
\(611\) 1.53465e57 1.18077
\(612\) 0 0
\(613\) −4.58982e56 −0.330260 −0.165130 0.986272i \(-0.552804\pi\)
−0.165130 + 0.986272i \(0.552804\pi\)
\(614\) 4.57107e56 0.318102
\(615\) 0 0
\(616\) −3.74113e56 −0.243556
\(617\) 4.69500e56 0.295658 0.147829 0.989013i \(-0.452771\pi\)
0.147829 + 0.989013i \(0.452771\pi\)
\(618\) 0 0
\(619\) −8.01009e56 −0.472040 −0.236020 0.971748i \(-0.575843\pi\)
−0.236020 + 0.971748i \(0.575843\pi\)
\(620\) 7.77151e56 0.443074
\(621\) 0 0
\(622\) −2.50398e56 −0.133637
\(623\) −6.41819e56 −0.331442
\(624\) 0 0
\(625\) −2.57233e57 −1.24390
\(626\) 2.13084e57 0.997187
\(627\) 0 0
\(628\) 2.86870e56 0.125751
\(629\) 2.48127e56 0.105278
\(630\) 0 0
\(631\) −1.26000e57 −0.500920 −0.250460 0.968127i \(-0.580582\pi\)
−0.250460 + 0.968127i \(0.580582\pi\)
\(632\) 7.60297e56 0.292606
\(633\) 0 0
\(634\) 2.81308e57 1.01473
\(635\) 2.94104e57 1.02716
\(636\) 0 0
\(637\) 3.48480e57 1.14109
\(638\) −1.88033e57 −0.596226
\(639\) 0 0
\(640\) −3.54423e56 −0.105398
\(641\) −4.29334e57 −1.23653 −0.618267 0.785968i \(-0.712165\pi\)
−0.618267 + 0.785968i \(0.712165\pi\)
\(642\) 0 0
\(643\) −3.66649e57 −0.990662 −0.495331 0.868704i \(-0.664953\pi\)
−0.495331 + 0.868704i \(0.664953\pi\)
\(644\) −2.17719e57 −0.569818
\(645\) 0 0
\(646\) 1.96085e56 0.0481590
\(647\) −7.19666e56 −0.171235 −0.0856177 0.996328i \(-0.527286\pi\)
−0.0856177 + 0.996328i \(0.527286\pi\)
\(648\) 0 0
\(649\) −5.56433e55 −0.0124279
\(650\) −1.62827e57 −0.352372
\(651\) 0 0
\(652\) −1.77318e57 −0.360310
\(653\) −1.77671e56 −0.0349862 −0.0174931 0.999847i \(-0.505569\pi\)
−0.0174931 + 0.999847i \(0.505569\pi\)
\(654\) 0 0
\(655\) −6.17515e57 −1.14209
\(656\) 1.06830e57 0.191499
\(657\) 0 0
\(658\) −5.88568e57 −0.991210
\(659\) −5.94098e57 −0.969855 −0.484928 0.874554i \(-0.661154\pi\)
−0.484928 + 0.874554i \(0.661154\pi\)
\(660\) 0 0
\(661\) 8.22733e57 1.26220 0.631101 0.775700i \(-0.282603\pi\)
0.631101 + 0.775700i \(0.282603\pi\)
\(662\) −7.91040e57 −1.17655
\(663\) 0 0
\(664\) 1.50771e57 0.210801
\(665\) −2.64027e57 −0.357936
\(666\) 0 0
\(667\) −1.09428e58 −1.39492
\(668\) 1.34874e55 0.00166729
\(669\) 0 0
\(670\) −1.40895e58 −1.63819
\(671\) −7.20231e57 −0.812194
\(672\) 0 0
\(673\) 3.40645e57 0.361404 0.180702 0.983538i \(-0.442163\pi\)
0.180702 + 0.983538i \(0.442163\pi\)
\(674\) 5.62512e57 0.578901
\(675\) 0 0
\(676\) 2.03932e57 0.197505
\(677\) 3.42769e57 0.322058 0.161029 0.986950i \(-0.448519\pi\)
0.161029 + 0.986950i \(0.448519\pi\)
\(678\) 0 0
\(679\) −2.48669e57 −0.219933
\(680\) −1.56298e57 −0.134129
\(681\) 0 0
\(682\) −3.19501e57 −0.258162
\(683\) −8.22982e57 −0.645306 −0.322653 0.946517i \(-0.604575\pi\)
−0.322653 + 0.946517i \(0.604575\pi\)
\(684\) 0 0
\(685\) 3.92872e57 0.290131
\(686\) 4.68680e56 0.0335917
\(687\) 0 0
\(688\) −4.46403e56 −0.0301414
\(689\) 1.63048e58 1.06862
\(690\) 0 0
\(691\) 1.60618e58 0.991960 0.495980 0.868334i \(-0.334809\pi\)
0.495980 + 0.868334i \(0.334809\pi\)
\(692\) 7.66688e57 0.459667
\(693\) 0 0
\(694\) −1.68593e57 −0.0952733
\(695\) 3.57948e58 1.96395
\(696\) 0 0
\(697\) 4.71115e57 0.243699
\(698\) −1.58512e58 −0.796208
\(699\) 0 0
\(700\) 6.24471e57 0.295803
\(701\) 3.61228e58 1.66174 0.830869 0.556468i \(-0.187844\pi\)
0.830869 + 0.556468i \(0.187844\pi\)
\(702\) 0 0
\(703\) −1.63251e57 −0.0708387
\(704\) 1.45710e57 0.0614113
\(705\) 0 0
\(706\) 9.40091e57 0.373829
\(707\) −1.42518e58 −0.550519
\(708\) 0 0
\(709\) −2.71836e58 −0.990965 −0.495483 0.868618i \(-0.665009\pi\)
−0.495483 + 0.868618i \(0.665009\pi\)
\(710\) −2.79352e58 −0.989364
\(711\) 0 0
\(712\) 2.49976e57 0.0835714
\(713\) −1.85937e58 −0.603990
\(714\) 0 0
\(715\) 2.25600e58 0.691934
\(716\) −2.82676e58 −0.842504
\(717\) 0 0
\(718\) −2.69580e58 −0.758817
\(719\) 1.95054e58 0.533595 0.266797 0.963753i \(-0.414035\pi\)
0.266797 + 0.963753i \(0.414035\pi\)
\(720\) 0 0
\(721\) 2.62613e58 0.678647
\(722\) 2.68611e58 0.674702
\(723\) 0 0
\(724\) 1.89091e58 0.448776
\(725\) 3.13866e58 0.724127
\(726\) 0 0
\(727\) 6.41681e58 1.39915 0.699575 0.714560i \(-0.253373\pi\)
0.699575 + 0.714560i \(0.253373\pi\)
\(728\) −2.76212e58 −0.585530
\(729\) 0 0
\(730\) 7.55795e58 1.51456
\(731\) −1.96861e57 −0.0383577
\(732\) 0 0
\(733\) 1.98495e58 0.365695 0.182847 0.983141i \(-0.441469\pi\)
0.182847 + 0.983141i \(0.441469\pi\)
\(734\) −4.59067e58 −0.822445
\(735\) 0 0
\(736\) 8.47972e57 0.143677
\(737\) 5.79246e58 0.954507
\(738\) 0 0
\(739\) 9.09593e58 1.41787 0.708934 0.705275i \(-0.249176\pi\)
0.708934 + 0.705275i \(0.249176\pi\)
\(740\) 1.30126e58 0.197294
\(741\) 0 0
\(742\) −6.25322e58 −0.897065
\(743\) 1.22229e59 1.70570 0.852849 0.522158i \(-0.174873\pi\)
0.852849 + 0.522158i \(0.174873\pi\)
\(744\) 0 0
\(745\) −4.00431e58 −0.528839
\(746\) 3.63246e58 0.466718
\(747\) 0 0
\(748\) 6.42569e57 0.0781515
\(749\) −1.64491e59 −1.94654
\(750\) 0 0
\(751\) −3.17979e57 −0.0356270 −0.0178135 0.999841i \(-0.505671\pi\)
−0.0178135 + 0.999841i \(0.505671\pi\)
\(752\) 2.29235e58 0.249929
\(753\) 0 0
\(754\) −1.38827e59 −1.43338
\(755\) −7.19147e58 −0.722612
\(756\) 0 0
\(757\) −5.74234e58 −0.546542 −0.273271 0.961937i \(-0.588106\pi\)
−0.273271 + 0.961937i \(0.588106\pi\)
\(758\) −4.17343e58 −0.386611
\(759\) 0 0
\(760\) 1.02833e58 0.0902517
\(761\) 1.47303e59 1.25842 0.629209 0.777236i \(-0.283379\pi\)
0.629209 + 0.777236i \(0.283379\pi\)
\(762\) 0 0
\(763\) −1.41313e59 −1.14400
\(764\) 2.37077e58 0.186842
\(765\) 0 0
\(766\) 4.44998e58 0.332405
\(767\) −4.10820e57 −0.0298777
\(768\) 0 0
\(769\) −1.33564e59 −0.920873 −0.460437 0.887693i \(-0.652307\pi\)
−0.460437 + 0.887693i \(0.652307\pi\)
\(770\) −8.65218e58 −0.580852
\(771\) 0 0
\(772\) 2.23595e58 0.142334
\(773\) −2.25801e59 −1.39974 −0.699868 0.714272i \(-0.746758\pi\)
−0.699868 + 0.714272i \(0.746758\pi\)
\(774\) 0 0
\(775\) 5.33312e58 0.313542
\(776\) 9.68514e57 0.0554550
\(777\) 0 0
\(778\) −1.74924e59 −0.950099
\(779\) −3.09961e58 −0.163979
\(780\) 0 0
\(781\) 1.14847e59 0.576464
\(782\) 3.73950e58 0.182842
\(783\) 0 0
\(784\) 5.20536e58 0.241530
\(785\) 6.63448e58 0.299902
\(786\) 0 0
\(787\) 3.77733e59 1.62070 0.810351 0.585945i \(-0.199277\pi\)
0.810351 + 0.585945i \(0.199277\pi\)
\(788\) 1.63095e59 0.681793
\(789\) 0 0
\(790\) 1.75835e59 0.697831
\(791\) −2.86467e59 −1.10779
\(792\) 0 0
\(793\) −5.31754e59 −1.95259
\(794\) 3.04337e59 1.08902
\(795\) 0 0
\(796\) −8.22410e58 −0.279493
\(797\) 1.32373e59 0.438433 0.219217 0.975676i \(-0.429650\pi\)
0.219217 + 0.975676i \(0.429650\pi\)
\(798\) 0 0
\(799\) 1.01091e59 0.318057
\(800\) −2.43219e58 −0.0745852
\(801\) 0 0
\(802\) 1.09509e59 0.319062
\(803\) −3.10721e59 −0.882473
\(804\) 0 0
\(805\) −5.03523e59 −1.35895
\(806\) −2.35891e59 −0.620644
\(807\) 0 0
\(808\) 5.55079e58 0.138810
\(809\) −8.51729e58 −0.207662 −0.103831 0.994595i \(-0.533110\pi\)
−0.103831 + 0.994595i \(0.533110\pi\)
\(810\) 0 0
\(811\) −1.20156e59 −0.278496 −0.139248 0.990258i \(-0.544469\pi\)
−0.139248 + 0.990258i \(0.544469\pi\)
\(812\) 5.32428e59 1.20327
\(813\) 0 0
\(814\) −5.34972e58 −0.114956
\(815\) −4.10086e59 −0.859298
\(816\) 0 0
\(817\) 1.29521e58 0.0258100
\(818\) −2.23254e59 −0.433867
\(819\) 0 0
\(820\) 2.47068e59 0.456702
\(821\) 1.35794e59 0.244820 0.122410 0.992480i \(-0.460938\pi\)
0.122410 + 0.992480i \(0.460938\pi\)
\(822\) 0 0
\(823\) −8.83563e59 −1.51545 −0.757724 0.652575i \(-0.773689\pi\)
−0.757724 + 0.652575i \(0.773689\pi\)
\(824\) −1.02283e59 −0.171117
\(825\) 0 0
\(826\) 1.57558e58 0.0250812
\(827\) 2.24391e59 0.348451 0.174226 0.984706i \(-0.444258\pi\)
0.174226 + 0.984706i \(0.444258\pi\)
\(828\) 0 0
\(829\) −8.39406e58 −0.124052 −0.0620260 0.998075i \(-0.519756\pi\)
−0.0620260 + 0.998075i \(0.519756\pi\)
\(830\) 3.48691e59 0.502736
\(831\) 0 0
\(832\) 1.07579e59 0.147638
\(833\) 2.29553e59 0.307369
\(834\) 0 0
\(835\) 3.11926e57 0.00397630
\(836\) −4.22766e58 −0.0525862
\(837\) 0 0
\(838\) −8.77315e59 −1.03909
\(839\) −7.27294e59 −0.840600 −0.420300 0.907385i \(-0.638075\pi\)
−0.420300 + 0.907385i \(0.638075\pi\)
\(840\) 0 0
\(841\) 1.76756e60 1.94561
\(842\) −1.20569e60 −1.29520
\(843\) 0 0
\(844\) 8.95980e59 0.916808
\(845\) 4.71638e59 0.471027
\(846\) 0 0
\(847\) −1.11801e60 −1.06374
\(848\) 2.43550e59 0.226190
\(849\) 0 0
\(850\) −1.07258e59 −0.0949164
\(851\) −3.11332e59 −0.268948
\(852\) 0 0
\(853\) 9.33512e58 0.0768537 0.0384269 0.999261i \(-0.487765\pi\)
0.0384269 + 0.999261i \(0.487765\pi\)
\(854\) 2.03938e60 1.63912
\(855\) 0 0
\(856\) 6.40658e59 0.490811
\(857\) 1.10562e60 0.826990 0.413495 0.910506i \(-0.364308\pi\)
0.413495 + 0.910506i \(0.364308\pi\)
\(858\) 0 0
\(859\) −6.75499e59 −0.481687 −0.240844 0.970564i \(-0.577424\pi\)
−0.240844 + 0.970564i \(0.577424\pi\)
\(860\) −1.03240e59 −0.0718838
\(861\) 0 0
\(862\) 1.38823e60 0.921644
\(863\) −5.90285e59 −0.382684 −0.191342 0.981523i \(-0.561284\pi\)
−0.191342 + 0.981523i \(0.561284\pi\)
\(864\) 0 0
\(865\) 1.77313e60 1.09625
\(866\) 1.13253e60 0.683805
\(867\) 0 0
\(868\) 9.04687e59 0.521007
\(869\) −7.22889e59 −0.406599
\(870\) 0 0
\(871\) 4.27663e60 2.29472
\(872\) 5.50384e59 0.288455
\(873\) 0 0
\(874\) −2.46033e59 −0.123030
\(875\) −1.97878e60 −0.966567
\(876\) 0 0
\(877\) −2.63402e60 −1.22780 −0.613900 0.789384i \(-0.710400\pi\)
−0.613900 + 0.789384i \(0.710400\pi\)
\(878\) −1.20042e59 −0.0546635
\(879\) 0 0
\(880\) 3.36985e59 0.146459
\(881\) −1.68258e60 −0.714445 −0.357223 0.934019i \(-0.616276\pi\)
−0.357223 + 0.934019i \(0.616276\pi\)
\(882\) 0 0
\(883\) 2.43606e60 0.987400 0.493700 0.869632i \(-0.335644\pi\)
0.493700 + 0.869632i \(0.335644\pi\)
\(884\) 4.74416e59 0.187883
\(885\) 0 0
\(886\) −1.01899e60 −0.385282
\(887\) 2.73192e60 1.00933 0.504666 0.863315i \(-0.331616\pi\)
0.504666 + 0.863315i \(0.331616\pi\)
\(888\) 0 0
\(889\) 3.42369e60 1.20784
\(890\) 5.78123e59 0.199308
\(891\) 0 0
\(892\) −3.23917e57 −0.00106648
\(893\) −6.65111e59 −0.214012
\(894\) 0 0
\(895\) −6.53750e60 −2.00927
\(896\) −4.12586e59 −0.123937
\(897\) 0 0
\(898\) −3.57205e60 −1.02507
\(899\) 4.54705e60 1.27543
\(900\) 0 0
\(901\) 1.07404e60 0.287848
\(902\) −1.01574e60 −0.266103
\(903\) 0 0
\(904\) 1.11573e60 0.279323
\(905\) 4.37314e60 1.07028
\(906\) 0 0
\(907\) −9.66132e59 −0.225989 −0.112994 0.993596i \(-0.536044\pi\)
−0.112994 + 0.993596i \(0.536044\pi\)
\(908\) −1.36296e60 −0.311691
\(909\) 0 0
\(910\) −6.38799e60 −1.39642
\(911\) 3.27643e60 0.700284 0.350142 0.936697i \(-0.386133\pi\)
0.350142 + 0.936697i \(0.386133\pi\)
\(912\) 0 0
\(913\) −1.43353e60 −0.292925
\(914\) −1.66071e60 −0.331815
\(915\) 0 0
\(916\) 2.76283e60 0.527833
\(917\) −7.18853e60 −1.34298
\(918\) 0 0
\(919\) 2.29076e60 0.409271 0.204636 0.978838i \(-0.434399\pi\)
0.204636 + 0.978838i \(0.434399\pi\)
\(920\) 1.96112e60 0.342652
\(921\) 0 0
\(922\) −3.08009e59 −0.0514731
\(923\) 8.47926e60 1.38587
\(924\) 0 0
\(925\) 8.92977e59 0.139616
\(926\) −7.93976e60 −1.21418
\(927\) 0 0
\(928\) −2.07370e60 −0.303398
\(929\) −6.62994e60 −0.948828 −0.474414 0.880302i \(-0.657340\pi\)
−0.474414 + 0.880302i \(0.657340\pi\)
\(930\) 0 0
\(931\) −1.51030e60 −0.206821
\(932\) 3.78447e59 0.0506966
\(933\) 0 0
\(934\) −2.84069e60 −0.364177
\(935\) 1.48608e60 0.186382
\(936\) 0 0
\(937\) −2.63986e60 −0.316898 −0.158449 0.987367i \(-0.550649\pi\)
−0.158449 + 0.987367i \(0.550649\pi\)
\(938\) −1.64017e61 −1.92633
\(939\) 0 0
\(940\) 5.30156e60 0.596050
\(941\) −1.40617e61 −1.54685 −0.773427 0.633885i \(-0.781459\pi\)
−0.773427 + 0.633885i \(0.781459\pi\)
\(942\) 0 0
\(943\) −5.91121e60 −0.622568
\(944\) −6.13655e58 −0.00632409
\(945\) 0 0
\(946\) 4.24439e59 0.0418839
\(947\) −1.18800e61 −1.14721 −0.573606 0.819131i \(-0.694456\pi\)
−0.573606 + 0.819131i \(0.694456\pi\)
\(948\) 0 0
\(949\) −2.29408e61 −2.12154
\(950\) 7.05683e59 0.0638669
\(951\) 0 0
\(952\) −1.81948e60 −0.157721
\(953\) 1.12712e61 0.956235 0.478117 0.878296i \(-0.341319\pi\)
0.478117 + 0.878296i \(0.341319\pi\)
\(954\) 0 0
\(955\) 5.48291e60 0.445597
\(956\) 1.36512e60 0.108588
\(957\) 0 0
\(958\) −7.59321e59 −0.0578672
\(959\) 4.57345e60 0.341163
\(960\) 0 0
\(961\) −6.26416e60 −0.447748
\(962\) −3.94975e60 −0.276364
\(963\) 0 0
\(964\) 9.38632e60 0.629385
\(965\) 5.17113e60 0.339450
\(966\) 0 0
\(967\) −6.94733e60 −0.437095 −0.218547 0.975826i \(-0.570132\pi\)
−0.218547 + 0.975826i \(0.570132\pi\)
\(968\) 4.35444e60 0.268217
\(969\) 0 0
\(970\) 2.23990e60 0.132254
\(971\) 2.12424e61 1.22803 0.614016 0.789294i \(-0.289553\pi\)
0.614016 + 0.789294i \(0.289553\pi\)
\(972\) 0 0
\(973\) 4.16689e61 2.30940
\(974\) −3.50054e60 −0.189966
\(975\) 0 0
\(976\) −7.94297e60 −0.413296
\(977\) 2.72440e61 1.38814 0.694068 0.719909i \(-0.255817\pi\)
0.694068 + 0.719909i \(0.255817\pi\)
\(978\) 0 0
\(979\) −2.37676e60 −0.116129
\(980\) 1.20385e61 0.576021
\(981\) 0 0
\(982\) −1.97366e61 −0.905705
\(983\) 6.88350e60 0.309359 0.154679 0.987965i \(-0.450566\pi\)
0.154679 + 0.987965i \(0.450566\pi\)
\(984\) 0 0
\(985\) 3.77192e61 1.62600
\(986\) −9.14488e60 −0.386101
\(987\) 0 0
\(988\) −3.12133e60 −0.126422
\(989\) 2.47007e60 0.0979908
\(990\) 0 0
\(991\) −1.65448e60 −0.0629727 −0.0314864 0.999504i \(-0.510024\pi\)
−0.0314864 + 0.999504i \(0.510024\pi\)
\(992\) −3.52357e60 −0.131369
\(993\) 0 0
\(994\) −3.25196e61 −1.16339
\(995\) −1.90200e61 −0.666557
\(996\) 0 0
\(997\) 3.62868e60 0.122039 0.0610193 0.998137i \(-0.480565\pi\)
0.0610193 + 0.998137i \(0.480565\pi\)
\(998\) 7.27445e60 0.239676
\(999\) 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 18.42.a.c.1.2 2
3.2 odd 2 2.42.a.b.1.2 2
12.11 even 2 16.42.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.42.a.b.1.2 2 3.2 odd 2
16.42.a.b.1.1 2 12.11 even 2
18.42.a.c.1.2 2 1.1 even 1 trivial