# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(20)$ into irreducible Hecke orbits

magma: S := CuspForms(20,8);
magma: N := Newforms(S);
sage: N = Newforms(20,8,names="a")
Label Dimension Field $q$-expansion of eigenform
20.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $6q^{3}$ $\mathstrut-$ $125q^{5}$ $\mathstrut-$ $706q^{7}$ $\mathstrut-$ $2151q^{9}$ $\mathstrut+O(q^{10})$
20.8.1.b 2 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut-$ $\alpha_{2} q^{3}$ $\mathstrut+$ $125q^{5}$ $\mathstrut+$ $\bigl(9 \alpha_{2}$ $\mathstrut+ 740\bigr)q^{7}$ $\mathstrut+$ $\bigl(20 \alpha_{2}$ $\mathstrut+ 2229\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{1129})$ $x ^{2}$ $\mathstrut -\mathstrut 20 x$ $\mathstrut -\mathstrut 4416$

## Decomposition of $S_{8}^{\mathrm{old}}(20)$ into lower level spaces

$S_{8}^{\mathrm{old}}(20)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/10/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(10)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/5/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(5)) }^{\oplus 3 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 4 }$