$(20.108)$ in Iwaniec and Kowalski

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c} e\left(\dfrac{d}{c}(Q(h)+h^tm\right) .$$

Lemma $20.13$ claim that if $(c,2|A|D)=1$,then $$G_m\left(\dfrac{d}{c}\right)=\left(\dfrac{|A|}{c}\right) \left(\varepsilon_c\left(\dfrac{2d}{c}\right)\sqrt{c}\right)^re\left(-\dfrac{\bar{d}}{c} Q^*(m)\right),$$ where $\bar{d} d\equiv 1\,\text{mod}\,c.$ But my calculation is $e\left(-\dfrac{d}{c} Q^*(m)\right)$. It's a crucial difference because $(20.114)$ will be $$K^{(c_0)}(l,m,n;c_1)\ll (l+Q^*(m),n,c_1)^{1/2}c_1^{(r+1)/2}\tau(c_1).$$ And the following calculation will be different. Which equation is correct?