Order 8 Tetramagic Series Impossibility

Theorem. There are no order 8 tetramagic series.
Proof.
S1 = 260
S2 = 11180    ≡ 2 (mod 9)
S3 = 540800   ≡ 8 (mod 9)
S4 = 27903044 ≡ 2 (mod 9)

From the Modulo 9/3 Tetramagic Series Lemma,
this tetramagic series must have
  5 entries of 1 (mod 3) and
  6 entries of 2 (mod 3).

An order 8 tetramagic series must have
at least 11 entries.