Series expansion for theta function of even lattice

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Please let me answer using an epic style, since it reflects my experience with these objects.

Let $A$ be the quadratic form from the above post...

sage: A = QuadraticForm( QQ, 2, [2, -1, 2] )
sage: A
Quadratic form in 2 variables over Rational Field with coefficients: 
[ 2 -1 ]
[ * 2 ]

A longer time i thought this is the quadratic form corresponding to the symmetrically extended matrix with those entries, i.e. to...

sage: MA = matrix( QQ, 2, 2, [ 2, -1, -1, 2] )
sage: MA
[ 2 -1]
[-1  2]

since what else should be the star?! ...till i got an error, then searching for it was typing something like...

sage: A
Quadratic form in 2 variables over Rational Field with coefficients: 
[ 2 -1 ]
[ * 2 ]
sage: A.matrix()    # oh, there is a matrix method!
[ 4 -1]
[-1  4]
sage: MA = matrix( QQ, 2, 2, [ 2, -1, -1, 2] )
sage: MA
[ 2 -1]
[-1  2]

sage: v = vector( [100, 10] )
sage: A(v)
19200
sage: 2 * 100^2 - 2*100*10 + 10^2
18100
sage: 2 * 100^2 -   100*10 + 10^2
19100
sage: ( v * A.matrix() * v.column() ) / 2
(19200)
sage: ( v * MA         * v.column() ) / 2
(9100)
sage: ( v * MA         * v.column() )
(18200)

And the quadratic form A is not quite reflected by the symmetic matrix MA. What is going on here? I think, the answer to the question is already here, despite of the fact that there is only one more question...

In the given posted situation with the Q, i suspect the quadratic form written for the above matrix MA instead is needed / intended, the one which maps the vector $x$ with components $s,t$ to $$q_A(x)=\frac 12\langle x,x\rangle_A =\frac 12(2s^2-2st+t^2) = x^2-xy+y^2\ .$$

Since the expected theta series is an other one, let us consider "the other quadratic form", and the theta series associated to it...

sage: B = QuadraticForm( QQ, 2, [1, -1, 1] )
sage: B
Quadratic form in 2 variables over Rational Field with coefficients: 
[ 1 -1 ]
[ * 1 ]
sage: B.theta_series( 20 )
1 + 6*q + 6*q^3 + 6*q^4 + 12*q^7 + 6*q^9 + 6*q^12 + 12*q^13 + 6*q^16 + 12*q^19 + O(q^20)

...is in agreement with

sage: ModularForms( Gamma1(3), 1, prec=20 ).0
1 + 6*q + 6*q^3 + 6*q^4 + 12*q^7 + 6*q^9 + 6*q^12 + 12*q^13 + 6*q^16 + 12*q^19 + O(q^20)

and in fact the first $1000$ coefficients do coincide... For this we may try successively to get them (right):

sage: b = B.theta_series( 1000 )
sage: m = ModularForms( Gamma1(3), 1, prec=20 ).0

sage: b.coefficients()[:20]    # !only non zero coefficients shown
[1, 6, 6, 6, 12, 6, 6, 12, 6, 12, 12, 6, 6, 12, 12, 6, 12, 12, 12, 6]
sage: m.coefficients( 1000 )[:20]   # !first coeff is not shown
[6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6, 0, 0, 12, 0]

sage: b.coefficients()[:20]    # !only non zero coefficients shown
[1, 6, 6, 6, 12, 6, 6, 12, 6, 12, 12, 6, 6, 12, 12, 6, 12, 12, 12, 6]
sage: b.polynomial().coeffs()[:20]
[1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6, 0, 0, 12]
sage: m.coefficients(1000)[:20]    # !first coefficient, the starting one, not shown
[6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6, 0, 0, 12, 0]

sage: b.polynomial().coeffs()[1:20] == m.coefficients(1000)[:19]
True
sage: b.polynomial().coeffs()[1:1000] == m.coefficients(1000)[:999]
True

I hope we are both at home now, know which maths object is which sage object.