 Solution

Write $w=2k$. For a positive-determinant real matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, extend the slash operator for modular forms byLet ${\mathcal{M}}_n$ be the integral matrices of determinant $n>0$. Each left $S{L}_2(\mathbb{Z})$ orbit has a unique representative $\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$ with $ad=n$, $a,d>0$, $0\le b<d$. To obtain it, the Euclidean algorithm on the first column produces $(a,0)$ with $a>0$; the determinant fixes $d$, and a row shear reduces $b$ modulo $d$. This also proves uniqueness.

Define the normalized Hecke operatorRight multiplication by $S{L}_2(\mathbb{Z})$ permutes these left orbits, and the slash operation is associative. Hence the sum satisfies the same modular form transformation law. It is holomorphic on $\mathbb{H}$ since each argument stays there.

For $f={\sum }_{r\ge 0}c(r)q^r$, summing the phase $e^{2\pi irb/d}$ over $b$ gives zero unless $d\mid r$, and gives $d$ when $d\mid r$. For the resulting term to contribute to $q^m$, one needs $a\mid m$ and $r=dm/a=nm/a^2$. Its scalar factor is $n^{w-1}{d}^{1-w}=a^{w-1}$. Thuswhere $\gcd (n,0)=n$. All powers are nonnegative, so the sum is holomorphic at infinity and belongs to $M_w$; level one has only this modular cusp class. Its constant coefficient is ${\sigma }_{w-1}(n)c(0)$, so modular cusp forms are preserved. This is the full Fourier coefficients of a composite-index Hecke operator formula, including the constant term.