Properties

Label 21.20.g.a.5.1
Level $21$
Weight $20$
Character 21.5
Analytic conductor $48.052$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,20,Mod(5,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(48.0515062768\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 5.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.20.g.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(29524.5 - 17046.0i) q^{3} +(-262144. - 454047. i) q^{4} +(4.84282e7 + 9.51504e7i) q^{7} +(5.81131e8 - 1.00655e9i) q^{9} +O(q^{10})\) \(q+(29524.5 - 17046.0i) q^{3} +(-262144. - 454047. i) q^{4} +(4.84282e7 + 9.51504e7i) q^{7} +(5.81131e8 - 1.00655e9i) q^{9} +(-1.54793e10 - 8.93700e9i) q^{12} +7.60595e10i q^{13} +(-1.37439e11 + 2.38051e11i) q^{16} +(1.02386e12 + 5.91128e11i) q^{19} +(3.05175e12 + 1.98376e12i) q^{21} +(9.53674e12 + 1.65181e13i) q^{25} -3.96238e13i q^{27} +(3.05076e13 - 4.69318e13i) q^{28} +(-2.39571e14 + 1.38317e14i) q^{31} -6.09360e14 q^{36} +(7.90506e14 - 1.36920e15i) q^{37} +(1.29651e15 + 2.24562e15i) q^{39} +6.01253e15 q^{43} +9.37113e15i q^{48} +(-6.70831e15 + 9.21593e15i) q^{49} +(3.45346e16 - 1.99385e16i) q^{52} +4.03054e16 q^{57} +(1.01134e17 + 5.83897e16i) q^{61} +(1.23917e17 + 6.54950e15i) q^{63} +1.44115e17 q^{64} +(2.20587e17 + 3.82068e17i) q^{67} +(-1.43298e17 + 8.27332e16i) q^{73} +(5.63135e17 + 3.25126e17i) q^{75} -6.19843e17i q^{76} +(-1.31195e17 + 2.27236e17i) q^{79} +(-6.75426e17 - 1.16987e18i) q^{81} +(1.00722e17 - 1.90567e18i) q^{84} +(-7.23710e18 + 3.68343e18i) q^{91} +(-4.71548e18 + 8.16746e18i) q^{93} -1.15735e19i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9} - 30958682112 q^{12} - 274877906944 q^{16} + 2047727487699 q^{19} + 6103502217954 q^{21} + 19073486328125 q^{25} + 61015133782016 q^{28} - 479142790563765 q^{31} - 12\!\cdots\!92 q^{36}+ \cdots - 94\!\cdots\!95 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 29524.5 17046.0i 0.866025 0.500000i
\(4\) −262144. 454047.i −0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 4.84282e7 + 9.51504e7i 0.453594 + 0.891209i
\(8\) 0 0
\(9\) 5.81131e8 1.00655e9i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −1.54793e10 8.93700e9i −0.866025 0.500000i
\(13\) 7.60595e10i 1.98926i 0.103490 + 0.994631i \(0.466999\pi\)
−0.103490 + 0.994631i \(0.533001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.37439e11 + 2.38051e11i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 1.02386e12 + 5.91128e11i 0.727919 + 0.420264i 0.817660 0.575701i \(-0.195271\pi\)
−0.0897415 + 0.995965i \(0.528604\pi\)
\(20\) 0 0
\(21\) 3.05175e12 + 1.98376e12i 0.838428 + 0.545012i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 9.53674e12 + 1.65181e13i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 3.96238e13i 1.00000i
\(28\) 3.05076e13 4.69318e13i 0.545012 0.838428i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.39571e14 + 1.38317e14i −1.62742 + 0.939590i −0.642557 + 0.766238i \(0.722126\pi\)
−0.984860 + 0.173352i \(0.944540\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.09360e14 −1.00000
\(37\) 7.90506e14 1.36920e15i 0.999974 1.73201i 0.493715 0.869624i \(-0.335638\pi\)
0.506258 0.862382i \(-0.331028\pi\)
\(38\) 0 0
\(39\) 1.29651e15 + 2.24562e15i 0.994631 + 1.72275i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 6.01253e15 1.82434 0.912172 0.409807i \(-0.134404\pi\)
0.912172 + 0.409807i \(0.134404\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 9.37113e15i 1.00000i
\(49\) −6.70831e15 + 9.21593e15i −0.588505 + 0.808493i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.45346e16 1.99385e16i 1.72275 0.994631i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.03054e16 0.840528
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 1.01134e17 + 5.83897e16i 1.10730 + 0.639297i 0.938127 0.346290i \(-0.112559\pi\)
0.169168 + 0.985587i \(0.445892\pi\)
\(62\) 0 0
\(63\) 1.23917e17 + 6.54950e15i 0.998606 + 0.0527804i
\(64\) 1.44115e17 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.20587e17 + 3.82068e17i 0.990535 + 1.71566i 0.614139 + 0.789198i \(0.289503\pi\)
0.376396 + 0.926459i \(0.377163\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.43298e17 + 8.27332e16i −0.284887 + 0.164480i −0.635634 0.771991i \(-0.719261\pi\)
0.350747 + 0.936470i \(0.385928\pi\)
\(74\) 0 0
\(75\) 5.63135e17 + 3.25126e17i 0.866025 + 0.500000i
\(76\) 6.19843e17i 0.840528i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.31195e17 + 2.27236e17i −0.123157 + 0.213314i −0.921011 0.389537i \(-0.872635\pi\)
0.797854 + 0.602851i \(0.205968\pi\)
\(80\) 0 0
\(81\) −6.75426e17 1.16987e18i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00722e17 1.90567e18i 0.0527804 0.998606i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −7.23710e18 + 3.68343e18i −1.77285 + 0.902317i
\(92\) 0 0
\(93\) −4.71548e18 + 8.16746e18i −0.939590 + 1.62742i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.15735e19i 1.54572i −0.634575 0.772861i \(-0.718825\pi\)
0.634575 0.772861i \(-0.281175\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000e18 8.66025e18i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −2.10657e19 1.21623e19i −1.59082 0.918461i −0.993166 0.116708i \(-0.962766\pi\)
−0.597655 0.801753i \(-0.703901\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −1.79910e19 + 1.03871e19i −0.866025 + 0.500000i
\(109\) 1.58738e19 + 2.74942e19i 0.700049 + 1.21252i 0.968449 + 0.249213i \(0.0801720\pi\)
−0.268399 + 0.963308i \(0.586495\pi\)
\(110\) 0 0
\(111\) 5.38998e19i 1.99995i
\(112\) −2.93066e19 1.54897e18i −0.998606 0.0527804i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.65576e19 + 4.42005e19i 1.72275 + 0.994631i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.05795e19 + 5.29653e19i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.25604e20 + 7.25177e19i 1.62742 + 0.939590i
\(125\) 0 0
\(126\) 0 0
\(127\) 3.92316e19 0.405043 0.202521 0.979278i \(-0.435086\pi\)
0.202521 + 0.979278i \(0.435086\pi\)
\(128\) 0 0
\(129\) 1.77517e20 1.02489e20i 1.57993 0.912172i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) −6.66217e18 + 1.26048e20i −0.0443634 + 0.839357i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 3.87845e20i 1.69831i −0.528142 0.849156i \(-0.677111\pi\)
0.528142 0.849156i \(-0.322889\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.59740e20 + 2.76678e20i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −4.09649e19 + 3.86445e20i −0.105414 + 0.994428i
\(148\) −8.28905e20 −1.99995
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 1.77389e19 + 3.07246e19i 0.0353708 + 0.0612640i 0.883169 0.469055i \(-0.155405\pi\)
−0.847798 + 0.530319i \(0.822072\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 6.79744e20 1.17735e21i 0.994631 1.72275i
\(157\) −3.31927e20 + 1.91638e20i −0.457084 + 0.263898i −0.710817 0.703377i \(-0.751675\pi\)
0.253733 + 0.967274i \(0.418341\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.34847e20 + 4.06767e20i −0.226466 + 0.392250i −0.956758 0.290884i \(-0.906050\pi\)
0.730292 + 0.683135i \(0.239384\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −4.32313e21 −2.95716
\(170\) 0 0
\(171\) 1.19000e21 6.87045e20i 0.727919 0.420264i
\(172\) −1.57615e21 2.72997e21i −0.912172 1.57993i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −1.10986e21 + 1.70737e21i −0.545012 + 0.838428i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 1.59372e21i 0.568152i 0.958802 + 0.284076i \(0.0916868\pi\)
−0.958802 + 0.284076i \(0.908313\pi\)
\(182\) 0 0
\(183\) 3.98124e21 1.27859
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.77022e21 1.91891e21i 0.891209 0.453594i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 4.25493e21 2.45658e21i 0.866025 0.500000i
\(193\) 3.42509e21 + 5.93243e21i 0.663556 + 1.14931i 0.979675 + 0.200592i \(0.0642867\pi\)
−0.316119 + 0.948719i \(0.602380\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 5.94301e21 + 6.29984e20i 0.994428 + 0.105414i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −8.60187e21 + 4.96629e21i −1.24591 + 0.719329i −0.970292 0.241937i \(-0.922217\pi\)
−0.275623 + 0.961266i \(0.588884\pi\)
\(200\) 0 0
\(201\) 1.30255e22 + 7.52025e21i 1.71566 + 0.990535i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.81061e22 1.04535e22i −1.72275 0.994631i
\(209\) 0 0
\(210\) 0 0
\(211\) 2.39936e22 1.99257 0.996283 0.0861410i \(-0.0274536\pi\)
0.996283 + 0.0861410i \(0.0274536\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.47629e22 1.60969e22i −1.57556 1.02418i
\(218\) 0 0
\(219\) −2.82054e21 + 4.88531e21i −0.164480 + 0.284887i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.84214e22i 0.904537i 0.891882 + 0.452269i \(0.149385\pi\)
−0.891882 + 0.452269i \(0.850615\pi\)
\(224\) 0 0
\(225\) 2.21684e22 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) −1.05658e22 1.83005e22i −0.420264 0.727919i
\(229\) −1.24485e22 7.18715e21i −0.474985 0.274233i 0.243339 0.969941i \(-0.421757\pi\)
−0.718324 + 0.695708i \(0.755091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.94536e21i 0.246314i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −6.33683e22 + 3.65857e22i −1.48837 + 0.859310i −0.999912 0.0132790i \(-0.995773\pi\)
−0.488456 + 0.872589i \(0.662440\pi\)
\(242\) 0 0
\(243\) −3.98832e22 2.30266e22i −0.866025 0.500000i
\(244\) 6.12260e22i 1.27859i
\(245\) 0 0
\(246\) 0 0
\(247\) −4.49609e22 + 7.78746e22i −0.836015 + 1.44802i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −2.95102e22 5.79808e22i −0.453594 0.891209i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −3.77789e22 6.54350e22i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 1.68562e23 + 8.90921e21i 1.99716 + 0.105558i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.15651e23 2.00314e23i 0.990535 1.71566i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 2.00409e23 + 1.15706e23i 1.54422 + 0.891557i 0.998565 + 0.0535483i \(0.0170531\pi\)
0.545657 + 0.838009i \(0.316280\pi\)
\(272\) 0 0
\(273\) −1.50884e23 + 2.32115e23i −1.08417 + 1.66785i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.11289e23 + 1.92758e23i 0.696456 + 1.20630i 0.969688 + 0.244348i \(0.0785740\pi\)
−0.273232 + 0.961948i \(0.588093\pi\)
\(278\) 0 0
\(279\) 3.21520e23i 1.87918i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.26314e23 1.30662e23i 1.15542 0.667082i 0.205218 0.978716i \(-0.434210\pi\)
0.950202 + 0.311634i \(0.100876\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.19536e23 2.07043e23i 0.500000 0.866025i
\(290\) 0 0
\(291\) −1.97281e23 3.41701e23i −0.772861 1.33864i
\(292\) 7.51294e22 + 4.33760e22i 0.284887 + 0.164480i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 3.40920e23i 1.00000i
\(301\) 2.91176e23 + 5.72095e23i 0.827512 + 1.62587i
\(302\) 0 0
\(303\) 0 0
\(304\) −2.81438e23 + 1.62488e23i −0.727919 + 0.420264i
\(305\) 0 0
\(306\) 0 0
\(307\) 6.85262e23i 1.61451i −0.590200 0.807257i \(-0.700951\pi\)
0.590200 0.807257i \(-0.299049\pi\)
\(308\) 0 0
\(309\) −8.29271e23 −1.83692
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −2.95718e23 1.70733e23i −0.579704 0.334692i 0.181312 0.983426i \(-0.441966\pi\)
−0.761016 + 0.648734i \(0.775299\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.37567e23 0.246314
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −3.54118e23 + 6.13350e23i −0.500000 + 0.866025i
\(325\) −1.25636e24 + 7.25360e23i −1.72275 + 0.994631i
\(326\) 0 0
\(327\) 9.37331e23 + 5.41168e23i 1.21252 + 0.700049i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.32820e23 7.49666e23i 0.498817 0.863976i −0.501182 0.865342i \(-0.667101\pi\)
0.999999 + 0.00136580i \(0.000434747\pi\)
\(332\) 0 0
\(333\) −9.18774e23 1.59136e24i −0.999974 1.73201i
\(334\) 0 0
\(335\) 0 0
\(336\) −8.91667e23 + 4.53827e23i −0.891209 + 0.453594i
\(337\) 1.06890e24 1.03861 0.519305 0.854589i \(-0.326191\pi\)
0.519305 + 0.854589i \(0.326191\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.20177e24 1.91987e23i −0.987479 0.157753i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 1.51610e24i 1.05654i −0.849076 0.528270i \(-0.822841\pi\)
0.849076 0.528270i \(-0.177159\pi\)
\(350\) 0 0
\(351\) 3.01376e24 1.98926
\(352\) 0 0
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −2.90345e23 5.02893e23i −0.146756 0.254189i
\(362\) 0 0
\(363\) 2.08503e24i 1.00000i
\(364\) 3.56961e24 + 2.32039e24i 1.66785 + 1.08417i
\(365\) 0 0
\(366\) 0 0
\(367\) −3.98463e24 + 2.30053e24i −1.72211 + 0.994261i −0.807562 + 0.589782i \(0.799214\pi\)
−0.914547 + 0.404479i \(0.867453\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 4.94454e24 1.87918
\(373\) −6.38263e23 + 1.10550e24i −0.236464 + 0.409568i −0.959697 0.281036i \(-0.909322\pi\)
0.723233 + 0.690604i \(0.242655\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.06002e24 1.92931 0.964654 0.263520i \(-0.0848835\pi\)
0.964654 + 0.263520i \(0.0848835\pi\)
\(380\) 0 0
\(381\) 1.15829e24 6.68742e23i 0.350777 0.202521i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.49407e24 6.05190e24i 0.912172 1.57993i
\(388\) −5.25489e24 + 3.03391e24i −1.33864 + 0.772861i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.95150e23 2.28140e23i −0.0809567 0.0467404i 0.458975 0.888449i \(-0.348217\pi\)
−0.539932 + 0.841709i \(0.681550\pi\)
\(398\) 0 0
\(399\) 1.95192e24 + 3.83508e24i 0.381258 + 0.749086i
\(400\) −5.24288e24 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −1.05203e25 1.82217e25i −1.86909 3.23736i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.10693e25 + 6.39084e24i −1.70902 + 0.986703i −0.773242 + 0.634112i \(0.781366\pi\)
−0.935778 + 0.352591i \(0.885301\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.27531e25i 1.83692i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.61119e24 1.14509e25i −0.849156 1.47078i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 4.68847e24 0.549985 0.274993 0.961446i \(-0.411325\pi\)
0.274993 + 0.961446i \(0.411325\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.58068e23 + 1.24506e25i −0.0674848 + 1.27681i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 9.43249e24 + 5.44585e24i 0.866025 + 0.500000i
\(433\) 2.20683e25i 1.98213i −0.133365 0.991067i \(-0.542578\pi\)
0.133365 0.991067i \(-0.457422\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.32243e24 1.44149e25i 0.700049 1.21252i
\(437\) 0 0
\(438\) 0 0
\(439\) −5.08816e24 2.93765e24i −0.401003 0.231519i 0.285913 0.958255i \(-0.407703\pi\)
−0.686917 + 0.726736i \(0.741036\pi\)
\(440\) 0 0
\(441\) 5.37787e24 + 1.21079e25i 0.405923 + 0.913907i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −2.44730e25 + 1.41295e25i −1.73201 + 0.999974i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 6.97924e24 + 1.37126e25i 0.453594 + 0.891209i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.04746e24 + 6.04753e23i 0.0612640 + 0.0353708i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.26766e25 2.19566e25i 0.682024 1.18130i −0.292338 0.956315i \(-0.594433\pi\)
0.974362 0.224986i \(-0.0722335\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.04121e25 1.92084 0.960421 0.278552i \(-0.0898545\pi\)
0.960421 + 0.278552i \(0.0898545\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 4.63476e25i 1.98926i
\(469\) −2.56713e25 + 3.94919e25i −1.07971 + 1.66098i
\(470\) 0 0
\(471\) −6.53332e24 + 1.13160e25i −0.263898 + 0.457084i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.25497e25i 0.840528i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 1.04140e26 + 6.01255e25i 3.44541 + 1.98921i
\(482\) 0 0
\(483\) 0 0
\(484\) 3.20650e25 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −3.06835e25 5.31453e25i −0.902359 1.56293i −0.824424 0.565972i \(-0.808501\pi\)
−0.0779343 0.996958i \(-0.524832\pi\)
\(488\) 0 0
\(489\) 1.60128e25i 0.452932i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 7.60404e25i 1.87918i
\(497\) 0 0
\(498\) 0 0
\(499\) −2.08933e25 + 3.61882e25i −0.487586 + 0.844523i −0.999898 0.0142759i \(-0.995456\pi\)
0.512312 + 0.858799i \(0.328789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.27638e26 + 7.36920e25i −2.56098 + 1.47858i
\(508\) −1.02843e25 1.78130e25i −0.202521 0.350777i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −1.48118e25 9.62825e24i −0.275809 0.179287i
\(512\) 0 0
\(513\) 2.34227e25 4.05693e25i 0.420264 0.727919i
\(514\) 0 0
\(515\) 0 0
\(516\) −9.30700e25 5.37340e25i −1.57993 0.912172i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −7.66777e25 4.42699e25i −1.14526 0.661214i −0.197530 0.980297i \(-0.563292\pi\)
−0.947727 + 0.319083i \(0.896625\pi\)
\(524\) 0 0
\(525\) −3.66426e24 + 6.93278e25i −0.0527804 + 0.998606i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.73077e25 6.46189e25i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 5.89783e25 3.00179e25i 0.749086 0.381258i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.03584e25 3.52618e25i 0.220480 0.381883i −0.734474 0.678637i \(-0.762571\pi\)
0.954954 + 0.296754i \(0.0959042\pi\)
\(542\) 0 0
\(543\) 2.71664e25 + 4.70537e25i 0.284076 + 0.492034i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.22629e26 1.19596 0.597978 0.801513i \(-0.295971\pi\)
0.597978 + 0.801513i \(0.295971\pi\)
\(548\) 0 0
\(549\) 1.17544e26 6.78641e25i 1.10730 0.639297i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.79751e25 1.47860e24i −0.245970 0.0130005i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.76100e26 + 1.01671e26i −1.47078 + 0.849156i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 4.57310e26i 3.62910i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.86041e25 1.20922e26i 0.545012 0.838428i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −1.15237e26 1.99596e26i −0.747391 1.29452i −0.949069 0.315067i \(-0.897973\pi\)
0.201678 0.979452i \(-0.435360\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 8.37498e25 1.45059e26i 0.500000 0.866025i
\(577\) 2.89027e26 1.66870e26i 1.69734 0.979959i 0.749070 0.662491i \(-0.230501\pi\)
0.948269 0.317469i \(-0.102833\pi\)
\(578\) 0 0
\(579\) 2.02248e26 + 1.16768e26i 1.14931 + 0.663556i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.86203e26 8.27044e25i 0.913907 0.405923i
\(589\) −3.27051e26 −1.57950
\(590\) 0 0
\(591\) 0 0
\(592\) 2.17293e26 + 3.76362e26i 0.999974 + 1.73201i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.69311e26 + 2.93254e26i −0.719329 + 1.24591i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 3.11996e26i 1.24406i −0.782994 0.622030i \(-0.786308\pi\)
0.782994 0.622030i \(-0.213692\pi\)
\(602\) 0 0
\(603\) 5.12760e26 1.98107
\(604\) 9.30028e24 1.61086e25i 0.0353708 0.0612640i
\(605\) 0 0
\(606\) 0 0
\(607\) −4.32601e26 2.49762e26i −1.56962 0.906222i −0.996212 0.0869525i \(-0.972287\pi\)
−0.573409 0.819269i \(-0.694379\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.99985e26 + 3.46385e26i 0.660881 + 1.14468i 0.980384 + 0.197095i \(0.0631507\pi\)
−0.319503 + 0.947585i \(0.603516\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.04142e26 + 6.01263e25i −0.313736 + 0.181135i −0.648597 0.761132i \(-0.724644\pi\)
0.334861 + 0.942267i \(0.391311\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −7.12763e26 −1.98926
\(625\) −1.81899e26 + 3.15058e26i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 1.74025e26 + 1.00474e26i 0.457084 + 0.263898i
\(629\) 0 0
\(630\) 0 0
\(631\) −7.88321e26 −1.97890 −0.989450 0.144873i \(-0.953723\pi\)
−0.989450 + 0.144873i \(0.953723\pi\)
\(632\) 0 0
\(633\) 7.08400e26 4.08995e26i 1.72561 0.996283i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −7.00959e26 5.10231e26i −1.60830 1.17069i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 8.31292e26i 1.74482i −0.488779 0.872408i \(-0.662557\pi\)
0.488779 0.872408i \(-0.337443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.00550e27 5.31448e25i −1.87656 0.0991839i
\(652\) 2.46255e26 0.452932
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.92315e26i 0.328959i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 4.48193e26 2.58764e26i 0.723687 0.417821i −0.0924209 0.995720i \(-0.529461\pi\)
0.816108 + 0.577899i \(0.196127\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.14011e26 + 5.43883e26i 0.452269 + 0.783352i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.57491e26 −1.16704 −0.583521 0.812098i \(-0.698325\pi\)
−0.583521 + 0.812098i \(0.698325\pi\)
\(674\) 0 0
\(675\) 6.54510e26 3.77882e26i 0.866025 0.500000i
\(676\) 1.13328e27 + 1.96290e27i 1.47858 + 2.56098i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 1.10122e27 5.60482e26i 1.37756 0.701130i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) −6.23901e26 3.60210e26i −0.727919 0.420264i
\(685\) 0 0
\(686\) 0 0
\(687\) −4.90048e26 −0.548466
\(688\) −8.26356e26 + 1.43129e27i −0.912172 + 1.57993i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.12985e27 + 6.52318e26i 1.19668 + 0.690905i 0.959814 0.280637i \(-0.0905457\pi\)
0.236868 + 0.971542i \(0.423879\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.06617e27 + 5.63513e25i 0.998606 + 0.0527804i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.61874e27 9.34580e26i 1.45580 0.840506i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.17157e27 2.02922e27i 0.971916 1.68341i 0.282156 0.959368i \(-0.408950\pi\)
0.689759 0.724039i \(-0.257716\pi\)
\(710\) 0 0
\(711\) 1.52482e26 + 2.64107e26i 0.123157 + 0.213314i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 1.37072e26 2.59340e27i 0.0969536 1.83436i
\(722\) 0 0
\(723\) −1.24728e27 + 2.16035e27i −0.859310 + 1.48837i
\(724\) 7.23621e26 4.17783e26i 0.492034 0.284076i
\(725\) 0 0
\(726\) 0 0
\(727\) 2.89394e27i 1.89197i 0.324217 + 0.945983i \(0.394899\pi\)
−0.324217 + 0.945983i \(0.605101\pi\)
\(728\) 0 0
\(729\) −1.57004e27 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1.04366e27 1.80767e27i −0.639297 1.10730i
\(733\) −2.23012e27 1.28756e27i −1.34846 0.778537i −0.360433 0.932785i \(-0.617371\pi\)
−0.988032 + 0.154249i \(0.950704\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.60254e26 + 1.31680e27i 0.425438 + 0.736881i 0.996461 0.0840533i \(-0.0267866\pi\)
−0.571023 + 0.820934i \(0.693453\pi\)
\(740\) 0 0
\(741\) 3.06561e27i 1.67203i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.13011e26 1.23497e27i 0.342387 0.593031i −0.642489 0.766295i \(-0.722098\pi\)
0.984875 + 0.173264i \(0.0554313\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.85961e27 1.20882e27i −0.838428 0.545012i
\(757\) 3.34562e26 0.148959 0.0744793 0.997223i \(-0.476271\pi\)
0.0744793 + 0.997223i \(0.476271\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) −1.84735e27 + 2.84189e27i −0.763071 + 1.17388i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −2.23081e27 1.28796e27i −0.866025 0.500000i
\(769\) 8.39780e26i 0.322007i −0.986954 0.161003i \(-0.948527\pi\)
0.986954 0.161003i \(-0.0514730\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.79573e27 3.11030e27i 0.663556 1.14931i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −4.56946e27 2.63818e27i −1.62742 0.939590i
\(776\) 0 0
\(777\) 5.12859e27 2.61027e27i 1.78237 0.907164i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.27188e27 2.86355e27i −0.405923 0.913907i
\(785\) 0 0
\(786\) 0 0
\(787\) −5.18994e27 + 2.99642e27i −1.59736 + 0.922236i −0.605366 + 0.795947i \(0.706973\pi\)
−0.991994 + 0.126289i \(0.959693\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.44109e27 + 7.69220e27i −1.27173 + 2.20270i
\(794\) 0 0
\(795\) 0 0
\(796\) 4.50985e27 + 2.60377e27i 1.24591 + 0.719329i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 7.88555e27i 1.98107i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 2.92108e27i 0.675843i −0.941174 0.337922i \(-0.890276\pi\)
0.941174 0.337922i \(-0.109724\pi\)
\(812\) 0 0
\(813\) 7.88932e27 1.78311
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.15601e27 + 3.55417e27i 1.32797 + 0.766707i
\(818\) 0 0
\(819\) −4.98152e26 + 9.42504e27i −0.104994 + 1.98649i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −2.10110e27 3.63921e27i −0.422812 0.732332i 0.573401 0.819275i \(-0.305624\pi\)
−0.996213 + 0.0869425i \(0.972290\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 6.24939e27 3.60809e27i 1.17373 0.677656i 0.219178 0.975685i \(-0.429663\pi\)
0.954557 + 0.298029i \(0.0963292\pi\)
\(830\) 0 0
\(831\) 6.57149e27 + 3.79405e27i 1.20630 + 0.696456i
\(832\) 1.09613e28i 1.98926i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.48063e27 + 9.49272e27i 0.939590 + 1.62742i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 6.10326e27 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −6.28979e27 1.08942e28i −0.996283 1.72561i
\(845\) 0 0
\(846\) 0 0
\(847\) −6.52059e27 3.44640e26i −0.998606 0.0527804i
\(848\) 0 0
\(849\) 4.45454e27 7.71548e27i 0.667082 1.15542i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.30743e28i 1.87242i −0.351444 0.936209i \(-0.614309\pi\)
0.351444 0.936209i \(-0.385691\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 7.12643e27 + 4.11445e27i 0.954855 + 0.551286i 0.894586 0.446897i \(-0.147471\pi\)
0.0602690 + 0.998182i \(0.480804\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.15045e27i 1.00000i
\(868\) −8.17294e26 + 1.54632e28i −0.0991839 + 1.87656i
\(869\) 0 0
\(870\) 0 0
\(871\) −2.90599e28 + 1.67778e28i −3.41289 + 1.97043i
\(872\) 0 0
\(873\) −1.16492e28 6.72569e27i −1.33864 0.772861i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.95755e27 0.328959
\(877\) 7.38731e27 1.27952e28i 0.812812 1.40783i −0.0980766 0.995179i \(-0.531269\pi\)
0.910889 0.412653i \(-0.135398\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.93667e28 −1.99723 −0.998617 0.0525783i \(-0.983256\pi\)
−0.998617 + 0.0525783i \(0.983256\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 1.89992e27 + 3.73291e27i 0.183725 + 0.360978i
\(890\) 0 0
\(891\) 0 0
\(892\) 8.36418e27 4.82906e27i 0.783352 0.452269i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −5.81131e27 1.00655e28i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 1.83487e28 + 1.19274e28i 1.52958 + 0.994290i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.49346e27 1.64432e28i −0.758849 1.31437i −0.943438 0.331549i \(-0.892429\pi\)
0.184589 0.982816i \(-0.440905\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −5.53953e27 + 9.59476e27i −0.420264 + 0.727919i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 7.53627e27i 0.548466i
\(917\) 0 0
\(918\) 0 0
\(919\) 4.36969e27 7.56852e27i 0.308286 0.533966i −0.669702 0.742630i \(-0.733578\pi\)
0.977987 + 0.208664i \(0.0669114\pi\)
\(920\) 0 0
\(921\) −1.16810e28 2.02320e28i −0.807257 1.39821i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.01554e28 1.99995
\(926\) 0 0
\(927\) −2.44838e28 + 1.41357e28i −1.59082 + 0.918461i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) −1.23162e28 + 5.47039e27i −0.768165 + 0.341190i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.83178e28i 1.07485i 0.843313 + 0.537423i \(0.180602\pi\)
−0.843313 + 0.537423i \(0.819398\pi\)
\(938\) 0 0
\(939\) −1.16412e28 −0.669384
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 4.06161e27 2.34497e27i 0.213314 0.123157i
\(949\) −6.29264e27 1.08992e28i −0.327193 0.566715i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.74276e28 4.75061e28i 1.26566 2.19218i
\(962\) 0 0
\(963\) 0 0
\(964\) 3.32232e28 + 1.91815e28i 1.48837 + 0.859310i
\(965\) 0 0
\(966\) 0 0
\(967\) 3.83031e28 1.66603 0.833014 0.553251i \(-0.186613\pi\)
0.833014 + 0.553251i \(0.186613\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 2.41451e28i 1.00000i
\(973\) 3.69036e28 1.87826e28i 1.51355 0.770344i
\(974\) 0 0
\(975\) −2.47289e28 + 4.28318e28i −0.994631 + 1.72275i
\(976\) −2.77995e28 + 1.60500e28i −1.10730 + 0.639297i
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.68990e28 1.40010
\(982\) 0 0
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.71449e28 1.67203
\(989\) 0 0
\(990\) 0 0
\(991\) 2.57098e28 + 4.45307e28i 0.885929 + 1.53447i 0.844645 + 0.535326i \(0.179811\pi\)
0.0412835 + 0.999147i \(0.486855\pi\)
\(992\) 0 0
\(993\) 2.95113e28i 0.997633i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.08959e28 + 1.20643e28i −0.679919 + 0.392552i −0.799825 0.600234i \(-0.795074\pi\)
0.119905 + 0.992785i \(0.461741\pi\)
\(998\) 0 0
\(999\) −5.42527e28 3.13228e28i −1.73201 0.999974i
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 21.20.g.a.5.1 2
3.2 odd 2 CM 21.20.g.a.5.1 2
7.3 odd 6 inner 21.20.g.a.17.1 yes 2
21.17 even 6 inner 21.20.g.a.17.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.20.g.a.5.1 2 1.1 even 1 trivial
21.20.g.a.5.1 2 3.2 odd 2 CM
21.20.g.a.17.1 yes 2 7.3 odd 6 inner
21.20.g.a.17.1 yes 2 21.17 even 6 inner