# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{12}^{\mathrm{new}}(9)$ into irreducible Hecke orbits

magma: S := CuspForms(9,12);
magma: N := Newforms(S);
sage: N = Newforms(9,12,names="a")
Label Dimension Field $q$-expansion of eigenform
9.12.1.a 1 $\Q$ $q$ $\mathstrut-$ $78q^{2}$ $\mathstrut+$ $4036q^{4}$ $\mathstrut+$ $5370q^{5}$ $\mathstrut-$ $27760q^{7}$ $\mathstrut-$ $155064q^{8}$ $\mathstrut+O(q^{10})$
9.12.1.b 1 $\Q$ $q$ $\mathstrut+$ $24q^{2}$ $\mathstrut-$ $1472q^{4}$ $\mathstrut-$ $4830q^{5}$ $\mathstrut-$ $16744q^{7}$ $\mathstrut-$ $84480q^{8}$ $\mathstrut+O(q^{10})$
9.12.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $472q^{4}$ $\mathstrut+$ $224 \alpha_{3} q^{5}$ $\mathstrut+$ $58100q^{7}$ $\mathstrut-$ $1576 \alpha_{3} q^{8}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{70})$ $x ^{2}$ $\mathstrut -\mathstrut 2520$

## Decomposition of $S_{12}^{\mathrm{old}}(9)$ into lower level spaces

$S_{12}^{\mathrm{old}}(9)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 3 }$