# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(15)$ into irreducible Hecke orbits

magma: S := CuspForms(15,8);
magma: N := Newforms(S);
sage: N = Newforms(15,8,names="a")
Label Dimension Field $q$-expansion of eigenform
15.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $22q^{2}$ $\mathstrut+$ $27q^{3}$ $\mathstrut+$ $356q^{4}$ $\mathstrut-$ $125q^{5}$ $\mathstrut-$ $594q^{6}$ $\mathstrut-$ $420q^{7}$ $\mathstrut-$ $5016q^{8}$ $\mathstrut+$ $729q^{9}$ $\mathstrut+O(q^{10})$
15.8.1.b 1 $\Q$ $q$ $\mathstrut-$ $13q^{2}$ $\mathstrut-$ $27q^{3}$ $\mathstrut+$ $41q^{4}$ $\mathstrut-$ $125q^{5}$ $\mathstrut+$ $351q^{6}$ $\mathstrut+$ $1380q^{7}$ $\mathstrut+$ $1131q^{8}$ $\mathstrut+$ $729q^{9}$ $\mathstrut+O(q^{10})$
15.8.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $27q^{3}$ $\mathstrut+$ $\bigl(7 \alpha_{3}$ $\mathstrut+ 10\bigr)q^{4}$ $\mathstrut+$ $125q^{5}$ $\mathstrut+$ $27 \alpha_{3} q^{6}$ $\mathstrut+$ $\bigl(- 56 \alpha_{3}$ $\mathstrut+ 848\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 69 \alpha_{3}$ $\mathstrut+ 966\bigr)q^{8}$ $\mathstrut+$ $729q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{601})$ $x ^{2}$ $\mathstrut -\mathstrut 7 x$ $\mathstrut -\mathstrut 138$

## Decomposition of $S_{8}^{\mathrm{old}}(15)$ into lower level spaces

$S_{8}^{\mathrm{old}}(15)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/5/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(3)) }^{\oplus 2 }$