The answer is in Rick Miranda and David R. Morrison, Embeddings of Integral QuadraticForms https://web.math.ucsb.edu/~drm/manuscripts/eiqf.pdf Chapter VIII Theorem 7.5 (2) and the following Lemmas

Namely, for any $k$ the quadratic form is

• 2-regular (Lemma 7.7 (1))
• 3-semiregular (Lemma 7.6 (2))
• p-regular for all $p\neq 2,3$ (Lemma 7.6 (1))

If you have a finite number of examples in mind you can use sage. I implemented spinor genera for the recent version of sageMath following Conway Sloane's description in SPLAG.

for k in range(1,1): 
`    D = matrix(ZZ,3,3,[2,1,0,1,2,0,0,0,-3*2*k]) 
    rep = Genus(D).representatives() 
    len(rep)

The answer is in Rick Miranda and David R. Morrison, Embeddings of Integral QuadraticForms https://web.math.ucsb.edu/~drm/manuscripts/eiqf.pdf Chapter VIII Theorem 7.5 (2) and the following Lemmas

Namely, for any $k$ the quadratic form is

• 2-regular (Lemma 7.7 (1))
• 3-semiregular (Lemma 7.6 (2))
• p-regular for all $p\neq 2,3$ (Lemma 7.6 (1))

If you have a finite number of examples in mind you can use sage. I implemented spinor genera for the recent version of sageMath following Conway Sloane's description in SPLAG.

for k in range(1,100): 
    D = matrix(ZZ,3,3,[2,1,0,1,2,0,0,0,-3*2*k]) 
    rep = Genus(D).representatives() 
    len(rep)