# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{4}^{\mathrm{new}}(19)$ into irreducible Hecke orbits

magma: S := CuspForms(19,4);
magma: N := Newforms(S);
sage: N = Newforms(19,4,names="a")
Label Dimension Field $q$-expansion of eigenform
19.4.1.a 1 $\Q$ $q$ $\mathstrut-$ $3q^{2}$ $\mathstrut-$ $5q^{3}$ $\mathstrut+$ $q^{4}$ $\mathstrut-$ $12q^{5}$ $\mathstrut+$ $15q^{6}$ $\mathstrut+$ $11q^{7}$ $\mathstrut+$ $21q^{8}$ $\mathstrut-$ $2q^{9}$ $\mathstrut+O(q^{10})$
19.4.1.b 3 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{3} \alpha_{2} ^{2}$ $\mathstrut- \frac{4}{3} \alpha_{2}$ $\mathstrut+ \frac{20}{3}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 8\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{1}{3} \alpha_{2} ^{2}$ $\mathstrut- \frac{8}{3} \alpha_{2}$ $\mathstrut+ \frac{7}{3}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{7}{3} \alpha_{2} ^{2}$ $\mathstrut+ \frac{2}{3} \alpha_{2}$ $\mathstrut+ \frac{38}{3}\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{4}{3} \alpha_{2} ^{2}$ $\mathstrut+ \frac{8}{3} \alpha_{2}$ $\mathstrut+ \frac{17}{3}\bigr)q^{7}$ $\mathstrut+$ $\bigl(3 \alpha_{2} ^{2}$ $\mathstrut+ 2 \alpha_{2}$ $\mathstrut- 38\bigr)q^{8}$ $\mathstrut+$ $\bigl(3 \alpha_{2} ^{2}$ $\mathstrut- 29\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 3.3.3144.1 $x ^{3}$ $\mathstrut -\mathstrut 3 x ^{2}$ $\mathstrut -\mathstrut 18 x$ $\mathstrut +\mathstrut 38$