Trouble verifying modularity properties

asked 2014-09-19 18:52:37 -0500

updated 2014-09-19 19:12:25 -0500

Hey all,

I'm having a bit of trouble doing some numerical things with modular forms, and I simply can't figure out where I'm going wrong.

The $j$ function should satisfy $j(\gamma \tau) = j(\tau)$ for every $\tau$ in the upper half plane and every $\gamma\in SL_2(\mathbb{Z})$. I wrote some code to numerically compute some values of the $j$ function (there may be a better way to do it for the $j$ function, but I hope to eventually migrate this code to work for other modular forms which I define). Here it is below.

## Starts Laurent series in q
R.<q> = LaurentSeriesRing(QQ)

I = CC.0 # imaginary unit

precision = 75

##evaluates a function using its q-expansion
def evaluate(f,z):
    result = 0
    coeffs = f.coefficients()
    exps   = f.exponents()
    for i in range(0,len(coeffs)):
        result = result + coeffs[i]*z^(exps[i])
    return result

## computes the action of a member of the modular group on tau in the upper half plane
def action(gamma,tau):
    return ((gamma[0]*tau + gamma[1])/(gamma[2]*tau + gamma[3]))

## Produce Eisenstein series with specified weight and precision of q-expansion
def eis(weight,precision):
    t = EisensteinForms(1,weight)
    t.set_precision(precision)
    t = t.eisenstein_series()
    e = t[0].q_expansion()
    return e*(1/e.list()[0])

## gives you q which corresponds to tau 
def qt(tau):
    return exp(2*pi*I*tau)

## Defining delta cusp form
delta = CuspForms(1,12).0
delta = delta.q_expansion(precision)

# Computes j function
g2 = 60*eis(4,precision)/240
j = 1728*g2^3/(27*delta)

Now when I run the following code:

tau = 1+I
gamma = [3,-1,4,-1]
print(evaluate(j,qt(tau)).n()) #j(tau)
print(evaluate(j,qt(action(gamma,tau))).n()) #j(gamma tau)

the values $j(\tau)$ and $j(\gamma\tau)$ are not equal! I would appreciate any help.

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