1) X0(50) Eta Products Final.sagews

R = QQ[['q']]
q = R.gens()[0]

#Q denotes q^(1/24). etaQ denotes eta * Q
etaQ = qexp_eta(R,100)

F = q^(-1) * (etaQ(q^2)*etaQ(q^(25)))/(etaQ(q) * etaQ(q^(50)))
G = q^(-3) * (etaQ(q^2)^2*etaQ(q^(25)))/(etaQ(q) * etaQ(q^(50))^2)
H = 2*G - F^3 + 2*F^2 + 2*F - 1

H^2 + O(q^(10))

print "\n"

F^6 - 4*F^5 - 10*F^3 - 4 * F + 1 + O(q^(10))

q^-6 + 2*q^-5 + q^-4 - 8*q^-3 - 28*q^-2 - 60*q^-1 - 102 - 138*q - 120*q^2 + 72*q^3 + 702*q^4 + 2282*q^5 + 5781*q^6 + 12952*q^7 + 26880*q^8 + 52842*q^9 + O(q^10) q^-6 + 2*q^-5 + q^-4 - 8*q^-3 - 28*q^-2 - 60*q^-1 - 102 - 138*q - 120*q^2 + 72*q^3 + 702*q^4 + 2282*q^5 + 5781*q^6 + 12952*q^7 + 26880*q^8 + 52842*q^9 + O(q^10)

X =  (2*F^2 - F + 2)/(F+1)^2
Y = -(4*F^3 - 3*F^2 - 3*F - 5*G + 4)/(F + 1)^3
Y^2 + X*Y + Y + O(q^(10))
print "\n"
X^3 - X - 2  + O(q^(10))

4 - 55*q + 315*q^2 - 1410*q^3 + 5455*q^4 - 19025*q^5 + 61555*q^6 - 187880*q^7 + 547510*q^8 - 1536060*q^9 + O(q^10) 4 - 55*q + 315*q^2 - 1410*q^3 + 5455*q^4 - 19025*q^5 + 61555*q^6 - 187880*q^7 + 547510*q^8 - 1536060*q^9 + O(q^10)

E = EllipticCurve([1,0,1,-1,-2])
E
f = -1*q*derivative(X)/(2*Y + X + 1) + O(q^(100))
coeffs = f.list()
P = Primes()
p = P.first()
myTable = [["p", "E(Fp)", "a_p"]]
while p < 100:
    if E.has_good_reduction(p):
        Emodp = E.reduction(p).cardinality()
        ap = coeffs[p-1]
        myRow = [p, Emodp, ap]
        myTable.append(myRow)
    p = P.next(p)
table(myTable)

Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field p E(Fp) a_p 3 3 1 7 6 2 11 15 -3 13 18 -4 17 21 -3 19 15 5 23 18 6 29 30 0 31 30 2 37 36 2 41 45 -3 43 48 -4 47 36 12 53 48 6 59 60 0 61 60 2 67 81 -13 71 60 12 73 63 11 79 90 -10 83 93 -9 89 75 15 97 96 2

nf = ModularForms(Gamma0(50),2,prec=50).newforms()[0]
nf

q - q^2 + q^3 + q^4 - q^6 + 2*q^7 - q^8 - 2*q^9 - 3*q^11 + q^12 - 4*q^13 - 2*q^14 + q^16 - 3*q^17 + 2*q^18 + 5*q^19 + 2*q^21 + 3*q^22 + 6*q^23 - q^24 + 4*q^26 - 5*q^27 + 2*q^28 + 2*q^31 - q^32 - 3*q^33 + 3*q^34 - 2*q^36 + 2*q^37 - 5*q^38 - 4*q^39 - 3*q^41 - 2*q^42 - 4*q^43 - 3*q^44 - 6*q^46 + 12*q^47 + q^48 - 3*q^49 + O(q^50)

E = EllipticCurve([1,0,1,-1,-2])
E.reduction(29).cardinality()

30

E.conductor()

50

ModularForms(Gamma0(2),2,prec=50).newforms()

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