Density of modular forms with transcendental zeros

For an even positive integer k, let M_k,Z(SL₂(Z)) be the set of modular forms of weight k on SL₂(Z) with integral Fourier coefficients. Let M_k,Z^tran(SL₂(Z)) be the subset of M_k,Z(SL₂(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL₂(Z). For a modular form f(z)=∑_n=0^∞a_f(n)e^2πinz of weight k(f), let ϖ(f):=∑n=0r_k(f)|a_f(n)|, where r_k(f)=dim_C⁡M_k(f),Z(SL₂(Z))⊗C−1. In this paper, we prove that if k=12 or k≥16, then [Formula presented] as X→∞, where α_k denotes the sum of the volumes of certain polytopes. Moreover, if we let M_Z=∪_k=0^∞M_k,Z(SL₂(Z)) (resp. M_Z^tran=∪_k=0^∞M_k,Z^tran(SL₂(Z))) and φ is a monotone increasing function on R⁺ such that φ(x+1)−φ(x)≥Cx² for some positive number C, then we prove [Formula presented]