Representations by some quinary quadratic forms of level 8

Let rQ(n) be the representation number of a nonnegative integer n by a certain quinary quadratic form Q of level 8. Then the associated theta function is a modular form of weight 5/2 for Γ0(8) associated with the character (8⋅). We express this theta function as a linear combination of Hecke eigenforms and find the general formula of the representation number rQ(n). As a consequence, we show that rQ(n) satisfies some partially multiplicative relations by applying Fricke involution and Hecke operators on the associated theta functions.