$\DeclareMathOperator{\field}{\mathbb{F}}$ Let $n$ be a positive integer, and let $q : \field_2^{n} \rightarrow \field_2$ be a quadratic form, specified in coordinates as

$$q(x)=\sum_{i =1}^n \alpha_i x_i + \sum_{\substack{j,k=1\\ j<k}}^n \beta_{i,j} x_i x_j$$

for some $\alpha_i, \beta_{j,k} \in \field_2$. For each integer $a\in \{0,\dots, n\}$, I am interested in the quantity

$$ C_{q,a}=\sum_{\substack{x \in \field_2^{n}\\ |x|=a}} (-1)^{q(x)} \in \mathbb{Z},$$

where $|x|$ denotes the Hamming weight of $x$.

Is there a simple formula for $C_{q,a}$? Have these quantities $C_{q,a}$ been studied before?

$\DeclareMathOperator{\field}{\mathbb{F}}$ Let $n$ be a positive integer, and let $q : \field_2^{n} \rightarrow \field_2$ be a quadratic form, specified in coordinates as

$$q(x)=\sum_{i =1}^n \alpha_i x_i + \sum_{\substack{j,k=1\\ j<k}}^n \beta_{j,k} x_j x_k$$

for some $\alpha_i, \beta_{j,k} \in \field_2$. For each integer $a\in \{0,\dots, n\}$, I am interested in the quantity

$$ C_{q,a}=\sum_{\substack{x \in \field_2^{n}\\ |x|=a}} (-1)^{q(x)} \in \mathbb{Z},$$

where $|x|$ denotes the Hamming weight of $x$.

Is there a simple formula for $C_{q,a}$? Have these quantities $C_{q,a}$ been studied before?