Square of the Dedekind eta function

Question

Is there a nice characterization of $\eta(\tau)^2=q^{1/12} \prod_{i=1}^\infty (1-q^i)^2$ as a unique (up to a scalar) weight one cusp form for some subgroup of $SL(2,\mathbb{Z})$?

Answer 1

OEIS sequence A002107 is the coefficients of the $q$-series for $\,\eta(\tau)^2.\,$ The sequence entry has this:

 (Magma) Basis( CuspForms( Gamma1(144), 1), 926) [1];

which means that it is the first of five forms in a basis of cusp forms of weight $1$ for $\, \Gamma_1(144).\,$ according to Magma.

Answer 2

It's known that $\eta^2$ is a weight one cusp form for the commutator subgroup $\mathrm{SL}_2(\mathbb Z)'$ of $\mathrm{SL}_2(\mathbb Z)$, which satisfies that $\mathrm{SL}_2(\mathbb Z)/\mathrm{SL}_2(\mathbb Z)'\cong\mathbb Z/12\mathbb Z$. I think it is unique up to scalar (see the arguments later), perhaps someone could compute the dimension of the relevant cusp form space using Sage or PARI/GP?

As a reference, let's look at a material https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf I found on Internet. It's said that the $\eta^2$ satisfies that for any $\gamma\in\mathrm{SL}_2(\mathbb Z)$, $\eta^2|_1\gamma=\chi(\gamma)\eta^2$ where $\chi$ is the group homomorphism $$ \chi:\mathrm{SL}_2(\mathbb Z)\to\mu_{12},\quad \begin{pmatrix} a&b\\c&d \end{pmatrix}\mapsto \zeta_{12}^{(1-c^2)(bd+3(c-1)d+c+3)+c(a+d-3)}. $$ In particular, $\eta^2$ is a weight one cusp form for $\ker(\chi)$.

It turns out that $\mathrm{SL}_2(\mathbb Z)'=\ker(\chi)$, which is generated by $\left(\begin{smallmatrix}1&-1\\-1&2\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1&1\\1&2\end{smallmatrix}\right)$. The group $$ \left\{\begin{pmatrix} a&b\\c&d \end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)~\middle|~ d\equiv 1,5\pmod{12},b\equiv c\equiv 0\pmod{12} \right\} $$ is contained in it, which is conjugated to $$ \left\{\begin{pmatrix} a&b\\c&d \end{pmatrix}\in\Gamma_0(144)~\middle|~ d\equiv 1,5\pmod{12} \right\}. $$ The $d\equiv 1,5\pmod{12}$ has index $2$ in $(\mathbb Z/144\mathbb Z)^\times$, and the above $\chi$, when $d\equiv 7,11\pmod{12}$, is $-1$. This means that $\eta^2$ corresponds to a weight one cusp form of level $144$ and central character $\left(\frac{-1}{\cdot}\right)$ (coincide with Somos' answer). From LMFDB we know that when restricted to newforms, it is unique (in fact, unique even for all weight one newforms of level $144$). I think that $\eta^2$ cannot be an oldform.