The number of representations of arithmetic progressions by integral quadratic forms

Let f be a positive definite integral quadratic forms and let r(n, f) be the number of representations of an integer n by f. In this article, we prove that if f(z) is a modular form of weight $$\frac{k}{2}$$ k 2 and level N, then $$f_{(m,r)}(z)$$ f ( m , r ) ( z ) is a modular form of weight $$\frac{k}{2}$$ k 2 and level $$Nm^2$$ N m 2 (see Definition 2.3 for the definition of $$f_{(m,r)}(z)$$ f ( m , r ) ( z ) ). As applications, we prove that if $$n\equiv 3 \ (\textrm{mod} \ 8)$$ n ≡ 3 ( mod 8 ) , then $$\begin{aligned} r(n,x^2+7y^2+7z^2)=r(n,2x^2+4y^2+2xy+7z^2), \end{aligned}$$ r ( n , x 2 + 7 y 2 + 7 z 2 ) = r ( n , 2 x 2 + 4 y 2 + 2 x y + 7 z 2 ) , and if $$n\equiv 1 \ (\textrm{mod}\ 3)$$ n ≡ 1 ( mod 3 ) , then $$\begin{aligned} r(n,x^2+y^2+2z^2+3t^2+3w^2)=r(n,x^2+y^2+2z^2+2t^2+2zt+6w^2). \end{aligned}$$ r ( n , x 2 + y 2 + 2 z 2 + 3 t 2 + 3 w 2 ) = r ( n , x 2 + y 2 + 2 z 2 + 2 t 2 + 2 z t + 6 w 2 ) .