求下面表达式的值，对 \(10^9 + 7\) 取模。
\[\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{m} \operatorname*{lcm}(i,j)^{\gcd(i,j)}
\]
\(1 \leq n,m \leq 5 \times 10^5\)。
\[\begin{aligned}
& \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{m} \operatorname*{lcm}(i,j)^{\gcd(i,j)} \\
= & \sum\limits_{d=1}^{\min(n,m)} \sum\limits_{i=1}^{\lfloor \frac{n}{d} \rfloor} \sum\limits_{j=1}^{\lfloor \frac{m}{d} \rfloor} (d \times i \times j)^d [\gcd(i,j)=1] \\
= & \sum\limits_{d=1}^{\min(n,m)} d^d \sum\limits_{i=1}^{\lfloor \frac{n}{d} \rfloor} \sum\limits_{j=1}^{\lfloor \frac{m}{d} \rfloor} i^d \times j^d \times \sum\limits_{e \mid \gcd(i,j)} \mu(e) \\
= & \sum\limits_{d=1}^{\min(n,m)} d^d \sum\limits_{e=1}^{\lfloor \frac{\min(n,m)}{d} \rfloor} \mu(e) \times e^{2d} \times \sum\limits_{i=1}^{\lfloor \frac{n}{de} \rfloor} \sum\limits_{j=1}^{\lfloor \frac{m}{de} \rfloor} i^d \times j^d \\
= & \sum\limits_{d=1}^{\min(n,m)} d^d \sum\limits_{e=1}^{\lfloor \frac{\min(n,m)}{d} \rfloor} \mu(e) \times e^{2d} \times (\sum\limits_{i=1}^{\lfloor \frac{n}{de} \rfloor} i^d) \times (\sum\limits_{i=1}^{\lfloor \frac{m}{de} \rfloor} i^d) \\
\end{aligned}
\]