Are Weight 12 Modular Forms the Only Ones Without Zeros on the Upper Half Plane?

[MathematicalPhysicist]

I am asked to find all the modular forms with weight k which don't have zeros on the upper half plane.

I know that a modular form with weight k is composed of an Eisenstein series with index k and a cusp form with weight k, and I have at my disposal the zeros formula for modular forms.

So I know that for k<12 (and k is obviously even cause Eisenstein series vanish for odd indices), they have zeros on the upper half plane, so I should be looking at modular form with weight 12m where m is positive integer.

Now if I have this equation for k=12m:
$$ord_{\infty} f + \sum_{p\neq i ,exp(2\pi i/3); p \in H/SL_2(Z)} ord_p f=m$$
where H is the upper half plane, so the only f which it zero isn't in H is for $$ord_{\infty} f =m$$.

Is that enough?

Thanks.

 

[MathematicalPhysicist]

Any tips?

 

[MathematicalPhysicist]

Anyone?

 

[Petek]

I looked at this, but don't have any useful information. You might get a better response at Math Stack Exchange.

 

[Stephen Tashi]

Also worth a try is the SOSCybermathboard http://www.sosmath.com/CBB/