Modulo 7 Hexamagic Series Lemma
If A,B,C,D,E,F are the number
of 1,2,3,4,5,6 (mod 7) entries, respectively,
in a hexamagic series, then
  A ≡ 6S1 + 6S2 + 6S3 + 6S4 + 6S5 + 6S6 (mod 7)
  B ≡ 3S1 + 5S2 + 6S3 + 3S4 + 5S5 + 6S6 (mod 7)
  C ≡ 2S1 + 3S2 +  S3 + 5S4 + 4S5 + 6S6 (mod 7)
  D ≡ 5S1 + 3S2 + 6S3 + 5S4 + 3S5 + 6S6 (mod 7)
  E ≡ 4S1 + 5S2 +  S3 + 3S4 + 2S5 + 6S6 (mod 7)
  F ≡  S1 + 6S2 +  S3 + 6S4 +  S5 + 6S6 (mod 7)

Proof

[1] A + 2B + 3C + 4D + 5E + 6F ≡ S1 (mod 7)
[2] A + 4B + 2C + 2D + 4E +  F ≡ S2 (mod 7)
[3] A +  B + 6C +  D + 6E + 6F ≡ S3 (mod 7)
[4] A + 2B + 4C + 4D + 2E +  F ≡ S4 (mod 7)
[5] A + 4B + 5C + 2D + 3E + 6F ≡ S5 (mod 7)
[6] A +  B +  C +  D +  E +  F ≡ S6 (mod 7)

Solve [6] for A.
[6a] A ≡ S6 + 6B + 6C + 6D + 6E + 6F  (mod 7)
[1a] S6 +  B + 2C + 3D + 4E + 5F ≡ S1 (mod 7)
[2a] S6 + 3B +  C +  D + 3E      ≡ S2 (mod 7)
[3a] S6      + 5C      + 5E + 5F ≡ S3 (mod 7)
[4a] S6 +  B + 3C + 3D +  E      ≡ S4 (mod 7)
[5a] S6 + 3B + 4C +  D + 2E + 5F ≡ S5 (mod 7)

Solve [5a] for D.
[5b] D ≡  S5 + 6S6 + 4B + 3C + 5E + 2F (mod 7)
[6b] A ≡ 6S5 + 2S6 + 2B + 3C +  E + 4F (mod 7)
[1b]     3S5 + 5S6 + 6B + 4C + 5E + 4F ≡ S1 (mod 7)
[2b]      S5            + 4C +  E + 2F ≡ S2 (mod 7)
[3b]            S6      + 5C + 5E + 5F ≡ S3 (mod 7)
[4b]     3S5 + 5S6 + 6B + 5C + 2E + 6F ≡ S4 (mod 7)

Solve [2b] for E.
[2c] E ≡  S2 + 6S5            + 3C + 5F (mod 7)
[5c] D ≡ 5S2 + 3S5 + 6S6 + 4B + 4C + 6F (mod 7)
[6c] A ≡  S2 + 5S5 + 2S6 + 2B + 6C + 2F (mod 7)
[1c]     5S2 + 5S5 + 5S6 + 6B + 5C +  F ≡ S1 (mod 7)
[3c]     5S2 + 2S5 +  S6      + 6C + 2F ≡ S3 (mod 7)
[4c]     2S2 +  S5 + 5S6 + 6B + 4C + 2F ≡ S4 (mod 7)

Solve [1c] for F.
[1d] F ≡  S1 + 2S2 + 2S5 + 2S6 +  B + 2C (mod 7)
[2d] E ≡ 5S1 + 4S2 + 2S5 + 3S6 + 5B + 6C (mod 7)
[5d] D ≡ 6S1 + 3S2 +  S5 + 4S6 + 3B + 2C (mod 7)
[6d] A ≡ 2S1 + 5S2 + 2S5 + 6S6 + 4B + 3C (mod 7)
[3d]     2S1 + 2S2 + 6S5 + 5S6 + 2B + 3C ≡ S3 (mod 7)
[4d]     2S1 + 6S2 + 5S5 + 2S6 +  B +  C ≡ S4 (mod 7)

Solve [4d] for C.
[4d] C ≡ 5S1 +  S2 +  S4 + 2S5 + 5S6 + 6B (mod 7)
[1d] F ≡ 4S1 + 4S2 + 2S4 + 6S5 + 5S6 + 6B (mod 7)
[2d] E ≡       3S2 + 6S4       + 5S6 + 6B (mod 7)
[5d] D ≡ 2S1 + 5S2 + 2S4 + 5S5        + B (mod 7)
[6d] A ≡ 3S1 +  S2 + 3S4 +  S5        + B (mod 7)
[3d]     3S1 + 5S2 + 3S4 + 5S5 + 6S6 + 6B ≡ S3 (mod 7)

Multiply [3d] by 6 and solve for B.
[3d] B ≡ 3S1 + 5S2 + 6S3 + 3S4 + 5S5 + 6S6 (mod 7)
[4d] C ≡ 2S1 + 3S2 +  S3 + 5S4 + 4S5 + 6S6 (mod 7)
[1d] F ≡  S1 + 6S2 +  S3 + 6S4 +  S5 + 6S6 (mod 7)
[2d] E ≡ 4S1 + 5S2 +  S3 + 3S4 + 2S5 + 6S6 (mod 7)
[5d] D ≡ 5S1 + 3S2 + 6S3 + 5S4 + 3S5 + 6S6 (mod 7)
[6d] A ≡ 6S1 + 6S2 + 6S3 + 6S4 + 6S5 + 6S6 (mod 7)