# Learn more about

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(22)$ into irreducible Hecke orbits

magma: S := CuspForms(22,8);
magma: N := Newforms(S);
sage: N = Newforms(22,8,names="a")
Label Dimension Field $q$-expansion of eigenform
22.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $8q^{2}$ $\mathstrut-$ $19q^{3}$ $\mathstrut+$ $64q^{4}$ $\mathstrut+$ $317q^{5}$ $\mathstrut+$ $152q^{6}$ $\mathstrut-$ $1030q^{7}$ $\mathstrut-$ $512q^{8}$ $\mathstrut-$ $1826q^{9}$ $\mathstrut+O(q^{10})$
22.8.1.b 1 $\Q$ $q$ $\mathstrut-$ $8q^{2}$ $\mathstrut+$ $91q^{3}$ $\mathstrut+$ $64q^{4}$ $\mathstrut+$ $185q^{5}$ $\mathstrut-$ $728q^{6}$ $\mathstrut-$ $722q^{7}$ $\mathstrut-$ $512q^{8}$ $\mathstrut+$ $6094q^{9}$ $\mathstrut+O(q^{10})$
22.8.1.c 1 $\Q$ $q$ $\mathstrut+$ $8q^{2}$ $\mathstrut-$ $21q^{3}$ $\mathstrut+$ $64q^{4}$ $\mathstrut-$ $551q^{5}$ $\mathstrut-$ $168q^{6}$ $\mathstrut+$ $62q^{7}$ $\mathstrut+$ $512q^{8}$ $\mathstrut-$ $1746q^{9}$ $\mathstrut+O(q^{10})$
22.8.1.d 2 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut+$ $8q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{2} \alpha_{4}$ $\mathstrut+ 4\bigr)q^{3}$ $\mathstrut+$ $64q^{4}$ $\mathstrut+$ $\bigl(\frac{1}{2} \alpha_{4}$ $\mathstrut+ 150\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 4 \alpha_{4}$ $\mathstrut+ 32\bigr)q^{6}$ $\mathstrut+$ $\bigl(7 \alpha_{4}$ $\mathstrut+ 680\bigr)q^{7}$ $\mathstrut+$ $512q^{8}$ $\mathstrut+$ $\bigl(\frac{23}{2} \alpha_{4}$ $\mathstrut+ 1309\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ $\Q(\sqrt{14881})$ $x ^{2}$ $\mathstrut -\mathstrut 62 x$ $\mathstrut -\mathstrut 13920$

## Decomposition of $S_{8}^{\mathrm{old}}(22)$ into lower level spaces

$S_{8}^{\mathrm{old}}(22)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/11/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(11)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 2 }$