# Construction of Bases for Modular Functions for $\Gamma_0(121)$ related to $p(11n+6)$

Supplement material to the article by Ralf Hemmecke, Peter Paule, and Silviu Radu

$t = \frac{\eta(\tau)}{\eta(121\tau)} = q^{-5}-{q^{-4}}-{q^{-3}}+O(q^{-2})$

$u = \frac{\eta(11\tau)^{12}}{\eta(\tau)\eta(121\tau)^{11}} = q^{-50}+q^{-49}+2\,q^{-48}+O(q^{-47})$

$z = \frac{1}{11}(u - t^{10}) = q^{-49}-3\,q^{-48}+3\,q^{-47}+O(q^{-46})$

$f = \frac{\eta(\tau)^{11}\eta(11\tau)}{\eta(121\tau)^{11}} q^{\frac{13}{24}} \sum_{n=1}^\infty p(11n+6)q^n = 11\,q^{-54}+176\,q^{-53}+935\,q^{-52}+O(q^{-51})$g

dc.input
The polynomials $d_1, \ldots, d_5$ with $d=d_1\,d_2^2\,d_3^2\,d_4\,d_5$ and the polynomials $c_0,\ldots,c_4$ such that $d(t)f = \sum_{k=0}^4 c_k(t) z^k$.
bf.input
The order-complete basis $B^{(f)} = \{1,b^{(f)}_1,\ldots,b^{(f)}_4\}$ such that $\mathbb{Q}[t, u, f]=\langle B^{(f)} \rangle_{\mathbb{Q}[t]}$.
f2.input
Relation showing that $f_2 \in \mathbb{Q}[t, u, f]$.
bj.input
The integral basis $B^{(j)} = \{1,b^{(j)}_1,\ldots,b^{(j)}_4\}$ such that $\mathbb{Q}[t,u,j^\infty_0,j^\infty_2]=\langle B^{(j)} \rangle_{\mathbb{Q}[t]}=M^\infty(121)$ together with a representations of the basis elements in terms of $t$, $u$, $j(\tau)$ (variable $J0$) and $j_2(\tau)=j(11^2\tau)$ (variable $J2$). Note that $j^\infty_0 := t^{25}\,u\,j$ and $j^\infty_2 := t\, u\,j_2$.
fj.input
Relation showing that $f$ can be expressed as a polynomial in terms of $t$, $u$, $j(\tau)$ (variable $J0$) and $j_2(\tau)=j(11^2\tau)$ (variable $J2$). Note that $j^\infty_0 := t^{25}\,u\,j$ and $j^\infty_2 := t\, u\,j_2$.
bg.input
The order-complete basis $B^{(g)} = \{1,b^{(g)}_1,\ldots,b^{(g)}_4\}$ such that $\mathbb{Q}[t, u, g]=\langle B^{(g)} \rangle_{\mathbb{Q}[t]}$ together with a representations of the basis elements in terms of $t$, $u$, and $g$.
bfg.input
The integral basis $B^{(fg)} = \{1,b^{(fg)}_1,\ldots,b^{(fg)}_4\}$ such that $\mathbb{Q}[t,u,f,g]=\langle B^{(fg)} \rangle_{\mathbb{Q}[t]}=M^\infty(121)$ together with a representations of the basis elements in terms of $t$, $u$, $f$, and $g$.
bgh.input
The integral basis $B^{(gh)} = \{1,b^{(gh)}_1,\ldots,b^{(gh)}_4\}$ such that $\mathbb{Q}[t, u, g, h]=\langle B^{(gh)} \rangle_{\mathbb{Q}[t]=M^\infty(121)}$ together with a representations of the basis elements in terms of $t$, $u$, $g$, and $h$.
bh.input
The integral basis $B^{(h)} = \{1,b^{(h)}_1,\ldots,b^{(h)}_4\}$ such that $\mathbb{Q}[t, u, h]=\langle B^{(h)} \rangle_{\mathbb{Q}[t]=M^\infty(121)}$ together with a representations of the basis elements in terms of $t$, $u$, and $h$.
ds.input
The polynomials $d_1, \ldots, d_6$ with $d^*=d_1\,{d_2}^2\,d_3^2\,d_4\,d_5\,d_6$ such that $d^*(t)h \in \mathbb{Q}[t,u]$.
dz.input
The factors of the discriminant $D(T)$ (with respect to $Z$) of the polynomial $p(T,Z)$ such that $p(t,z)=0$ together with their exponents. $D(T) = dz_1^4\,dz_2^2\,dz_3^4\,dz_4^4\,dz_5\,dz_6^2\,dz_7^2\,dz_8^2 = T^4\,dz_5 \, d^*$.
p.maple
Minimal polynomial p(T, U) and Maple commands to compute an integral basis for $M^\infty(121)$.
v.input
The polynomial $v(T,U)$ from the basis $\{1,U,U^2,U^3,v(T,U)\}$ returned by Maple.
bv.input
The integral basis $B^{(v)} = \{1,b^{(v)}_1,\ldots,b^{(v)}_4\}$ such that $\mathbb{Q}[t, u, v(t,u)]=\langle B^{(v)} \rangle_{\mathbb{Q}[t]}=M^\infty(121)$ together with a representations of the basis elements in terms of $t$, $u$, and $v(t,u)$ given by the polynomials $p_{i,j}(T)$, $i=1,\ldots,4$, $j=0,\ldots,4$.

The Computation has been mostly done by means of the QEta package. The scripts to compute the above data are available upon request.

Ralf Hemmecke