Exact Linear Representations of the Polynomial n(k) via Modular Forms: n(k) = A·r₂(m) + B·τ(m) + C·r₄(m)

We establish exact linear representations of the polynomial n(k) = (103k⁴ − 370k³ + 101k² + 478k)/12 via classical modular forms. For nine values of k, we prove that n(k) = A·r₂(m) + B·τ(m) + C·r₄(m) with integer coefficients. A universal constant A = −15 is discovered, connecting seven of the nine representations. Explicit formulas are provided for k = −3, −1, 1, 2, 3, 5, 6, 7, 8 with m ∈ {1, 3, 4, 5}. Connection with divisor sum functions σₐ(k) is established for k = 1, 2, 3, 4, 5. Languages: Russian, English. License: UAL v1.0.