# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{12}^{\mathrm{new}}(8)$ into irreducible Hecke orbits

magma: S := CuspForms(8,12);
magma: N := Newforms(S);
sage: N = Newforms(8,12,names="a")
Label Dimension Field $q$-expansion of eigenform
8.12.1.a 1 $\Q$ $q$ $\mathstrut-$ $36q^{3}$ $\mathstrut-$ $3490q^{5}$ $\mathstrut-$ $55464q^{7}$ $\mathstrut-$ $175851q^{9}$ $\mathstrut+O(q^{10})$
8.12.1.b 2 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut-$ $\frac{1}{2} \alpha_{2} q^{3}$ $\mathstrut+$ $\bigl(6 \alpha_{2}$ $\mathstrut+ 4270\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 21 \alpha_{2}$ $\mathstrut+ 44352\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 28 \alpha_{2}$ $\mathstrut+ 268533\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{109})$ $x ^{2}$ $\mathstrut +\mathstrut 112 x$ $\mathstrut -\mathstrut 1782720$

## Decomposition of $S_{12}^{\mathrm{old}}(8)$ into lower level spaces

$S_{12}^{\mathrm{old}}(8)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/4/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(4)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 }$