# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(5)$ into irreducible Hecke orbits

magma: S := CuspForms(5,8);
magma: N := Newforms(S);
sage: N = Newforms(5,8,names="a")
Label Dimension Field $q$-expansion of eigenform
5.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $14q^{2}$ $\mathstrut-$ $48q^{3}$ $\mathstrut+$ $68q^{4}$ $\mathstrut+$ $125q^{5}$ $\mathstrut+$ $672q^{6}$ $\mathstrut-$ $1644q^{7}$ $\mathstrut+$ $840q^{8}$ $\mathstrut+$ $117q^{9}$ $\mathstrut+O(q^{10})$
5.8.1.b 2 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- 8 \alpha_{2}$ $\mathstrut+ 90\bigr)q^{3}$ $\mathstrut+$ $\bigl(20 \alpha_{2}$ $\mathstrut- 152\bigr)q^{4}$ $\mathstrut-$ $125q^{5}$ $\mathstrut+$ $\bigl(- 70 \alpha_{2}$ $\mathstrut+ 192\bigr)q^{6}$ $\mathstrut+$ $\bigl(56 \alpha_{2}$ $\mathstrut- 610\bigr)q^{7}$ $\mathstrut+$ $\bigl(120 \alpha_{2}$ $\mathstrut- 480\bigr)q^{8}$ $\mathstrut+$ $\bigl(- 160 \alpha_{2}$ $\mathstrut+ 4377\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{19})$ $x ^{2}$ $\mathstrut -\mathstrut 20 x$ $\mathstrut +\mathstrut 24$