Order 20 Hexamagic Series Impossibility

Theorem. There are no order 20 hexamagic series.
Proof.
S1 = 4010              ≡ 6 (mod 7)
S2 = 1070670           ≡ 6 (mod 7)
S3 = 321602000         ≡ 6 (mod 7)
S4 = 103041066666      ≡ 6 (mod 7)
S5 = 34389866666000    ≡ 6 (mod 7)
S6 = 11805513142323810 ≡ 0 (mod 7)

From the Modulo 7 Hexamagic Series Lemma,
this hexamagic series must have
  5 entries of 1 (mod 7) and
  6 entries of 2...6 (mod 7).

An order 20 hexamagic series must contain
at least 35 entries.