A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$sumlimits_{x in mathbb{Z}^4 } {f(c x), c in mathbb{R}, c ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$varepsilon = prodlimits_{i = 1}^k {left( {frac{{det Q}}{{p_i }}} right)} ne 0$$ for the Legendre symbol $$left( {frac{ cdot }{p}} right)$$ ; define $$B_i (Q,x) = left{ {begin{array}{*{20}c} {1 for Q(x) equiv 0bmod p_i } , {0 for Q(x)not equiv 0bmod p_i } end{array} } right.$$ and forr∈?,r>0, $$F(Q,f,r) = sumlimits_{x in mathbb{Z}^4 } {left( {prodlimits_{i = 1}^k {left( {B_i (Q,x) - frac{1}{{p_i }}} right)} } right)f(r^{ - {1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = varepsilon F(Q^{ - 1} ,hat f,m)$$