The Monstrous Moonshine relationships between the Monster group (the largest sporadic finite simple group) and modular functions were investigated after observations of several numerical coincidences:

Several of the lowest coefficients of the Fourier expansion of the $j$ modular function are small sums of the smallest dimensions of irreducible representations of the Monster group. For example, the linear term is $196884q$, and the two smallest irreducible representations of the Monster group have dimensions $1$ and $196883$. The quadratic coefficient is the sum of the dimensions of the three smallest irreducible representations. This led to the conjecture of the existence and then the construction of an infinite graded vertex algebra on which the Monster acts whose pieces have dimensions equal to the Fourier coefficients of $j$.

I'm not sure whether computers were involved historically, but this was a shocking pattern of numerical coincidences of a type which is much easier to find now with tools like Sloane's online encyclopedia of integer sequences.

Also, it was noted that the primes $p$ such that $\mathbb H^2/\Gamma_0(p)^*$ has genus zero are precisely the primes dividing the order of the Monster group.

The Monstrous Moonshine relationships between the Monster group (the largest sporadic finite simple group) and modular functions were investigated after observations of several numerical coincidences:

Several of the lowest coefficients of the Fourier expansion of the $j$ modular function are small sums of the smallest dimensions of irreducible representations of the Monster group. For example, the linear term is $196884q$, and the two smallest irreducible representations of the Monster group have dimensions $1$ and $196883$. The quadratic coefficient is the sum of the dimensions of the three smallest irreducible representations. This led to the conjecture of the existence and then the construction of an infinite graded vertex algebra on which the Monster acts whose pieces have dimensions equal to the Fourier coefficients of $j$.

I'm not sure whether computers were involved historically, but this was a shocking pattern of numerical coincidences of a type which is much easier to find now with tools like Sloane's online encyclopedia of integer sequences.

Also, it was noted that the primes $p$ such that $\mathbb H^2/\Gamma_0(p)^*$ has genus zero are precisely the primes dividing the order of the Monster group.

The Monstrous Moonshine relationships between the Monster group (the largest sporadic finite simple group) and modular functions were investigated after observations of several numerical coincidences:

Several of the lowest coefficients of the Fourier expansion of the $j$ modular function are small sums of the smallest dimensions of irreducible representations of the Monster group. For example, the linear term is $196884q$, and the two smallest irreducible representations of the Monster group have dimensions $1$ and $196883$. The quadratic coefficient is the sum of the dimensions of the three smallest irreducible representations. This led to the conjecture of the existence and then the construction of an infinite graded algebra on which the Monster acts whose pieces have dimensions equal to the Fourier coefficients of $j$.

I'm not sure whether computers were involved historically, but this was a shocking pattern of numerical coincidences of a type which is much easier to find now with tools like Sloane's online encyclopedia of integer sequences.

Also, it was noted that the primes $p$ such that $\mathbb H^2/\Gamma_0(p)^*$ has genus zero are precisely the primes dividing the order of the Monster group.

The Monstrous Moonshine relationships between the Monster group (the largest sporadic finite simple group) and modular functions were investigated after observations of several numerical coincidences:

Several of the lowest coefficients of the Fourier expansion of the $j$ modular function are small sums of the smallest dimensions of irreducible representations of the Monster group. For example, the linear term is $196884q$, and the two smallest irreducible representations of the Monster group have dimensions $1$ and $196883$. The quadratic coefficient is the sum of the dimensions of the three smallest irreducible representations. This led to the conjecture of the existence and then the construction of an infinite graded vertex algebra on which the Monster acts whose pieces have dimensions equal to the Fourier coefficients of $j$.

I'm not sure whether computers were involved historically, but this was a shocking pattern of numerical coincidences of a type which is much easier to find now with tools like Sloane's online encyclopedia of integer sequences.

Also, it was noted that the primes $p$ such that $\mathbb H^2/\Gamma_0(p)^*$ has genus zero are precisely the primes dividing the order of the Monster group.