Order 21 Hexamagic Series Impossibility

Theorem. There are no order 21 hexamagic series.
Proof.
S1 = 4641
S2 = 1366001        ≡ 8 (mod 9)
S3 = 452316501      ≡ 0 (mod 9)
S4 = 159757914953   ≡ 2 (mod 9)
S5 = 58777473899781
S6 = 22243018423773641

From the Modulo 9/3 Tetramagic Series Lemma,
this hexamagic series must have
  7 entries of 2 (mod 3),
  7 entries of 1 (mod 3), and
  7 entries of 0 (mod 3).

Let 3Aj+1, j=1..14,
 be the squares of entries that are 1 or 2 (mod 3).
Let 9Bj, j=1..7,
 be the squares of entries that are 0 (mod 3).
Let Tn = sum (Aj)n, j=1..14, n=1,2,3.
Let Un = sum (Bj)n, j=1..7,  n=1,2,3.

  S2 = 3T1 + 14 + 9U1
  S6 = 27T3 + 27T2 + 9T1 + 14 + 729U3

(S2 - 14)/3 =
  T1 + 3U1 = 455329 ≡ 1 (mod 3)
thus
  T1 ≡ 1 (mod 3).

(S6 - 14)/9 =
  3T3 + 3T2 + T1 + 81U3 = 2471446491530403 ≡ 0 (mod 3)
thus
  T1 ≡ 0 (mod 3).

T1 can't be both 0 and 1 (mod 3).