The $\mathrm{mod}\:2$ reduction of the Eisenstein series of level $\Gamma_0(3)$ and weight $2$ is $1$

Question

A SageMath computation, if I am interpreting correctly, for the commandEisensteinForms(Gamma0(3),2).basis() is telling me that there is a unique modular form of level $\Gamma_0(3)$ and weight $2$ with Fourier coefficients in $\mathbb{Q}$ up to rescaling. I suspect that this form can be rescaled so that the constant term in the Fourier expansion is $1$ and all other terms are integers.

1. Is it true that the form can be so rescaled?
2. If it is true, then is it true that the non-constant terms are all even integers (i.e. is there a theorem directly stating so or can I carry out a finite computation which supplied with some theorem will prove this)? I am aware of the Swinnerton-Dyer congruence for Eisenstein series (with trivial characters) of even weight$\geq 4$ but it does not apply here. One could try to apply the Sturm bound but the issue is that the constant modular form is not of weight $2$ (I suppose there are some versions which apply to modular form of equal level and distinct weights but I am not confident that I am parsing the literature correctly).

Answer 1

Let $E_2(z)= 1-24 \sum_{n=1}^\infty e^{2i \pi nz} \sigma(n)$ then the only weight $2$ level $3$ form is $$\frac{3 E_2(3z) - E_2(z)}{2} = 1-\sum_{n=1}^\infty e^{2i \pi nz} (36 \sigma(n/3) - 12 \sigma(n)) \in M_2(\Gamma_0(3))$$