Connection between the SU(n) Hierarchy and Modular Forms: Modular Properties of the Exact Polynomial n(k)

We investigate the connection between the exact polynomial n(k) = (103k⁴ − 370k³ + 101k² + 478k)/12, which generates the hierarchy of groups SU(n(k)), and modular forms and Ramanujan congruences. Key results:— The ratios n(k)/n(1) for k = -1, 0, 1, 2, 3 have denominators dividing n(1) = 26.— Periodicity n(k) mod 12 corresponds to the weight of the modular form Δ(τ).— n(2) = 4 is the unique perfect square among n(k) for k > 0.— The prime 691 appears in both n(-4) and ζ(12), connecting to the Ramanujan congruence.— All r₄(n(k)) are multiples of 12.— τ(n(k))/n(k) is an integer for k = -1.— n = 4 is the unique index for which σ₃(n) is prime. Languages: Russian, English. License: UAL v1.0.