Given

\[f(x) = x^{13} + 2x^{12} + 3x^{11} + 4x^{10} + \cdots + 13x + 14,\]

denote

\[ N = f(a) \times f\left(a^2\right) \times f\left(a^3\right) \times\cdots \times f\left(a^{14} \right),\]

where \(a = \cos\left( \frac{2\pi}{15} \right) + i \sin\left( \frac{2\pi}{15} \right) .\)

Then what is the value of \(M \) such that \( N^\frac{1}{M} = 15 ?\)

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