Eccoci.

Euler discovered the remarkable quadratic formula:

$n^2 + n + 41$

It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + 41 + 41$ is clearly divisible by $41$.

The incredible formula $n^2 - 79n + 1601$ was discovered, which produces $80$ primes for the consecutive values $0 \le n \le 79$. The product of the coefficients, $-79$ and $1601$, is $-126479$.

Considering quadratics of the form:

$n^2 + an + b$, where $|a| < 1000$ and $|b| \le 1000$

where $|n|$ is the modulus/absolute value of $n$
e.g. $|11| = 11$ and $|-4| = 4$

Find the product of the coefficients, $a$ and $b$, for the quadratic expression that produces the maximum number of primes for consecutive values of $n$, starting with $n = 0$.