# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{4}^{\mathrm{new}}(23)$ into irreducible Hecke orbits

magma: S := CuspForms(23,4);
magma: N := Newforms(S);
sage: N = Newforms(23,4,names="a")
Label Dimension Field $q$-expansion of eigenform
23.4.1.a 1 $\Q$ $q$ $\mathstrut-$ $2q^{2}$ $\mathstrut-$ $5q^{3}$ $\mathstrut-$ $4q^{4}$ $\mathstrut-$ $6q^{5}$ $\mathstrut+$ $10q^{6}$ $\mathstrut-$ $8q^{7}$ $\mathstrut+$ $24q^{8}$ $\mathstrut-$ $2q^{9}$ $\mathstrut+O(q^{10})$
23.4.1.b 4 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{11} \alpha_{2} ^{3}$ $\mathstrut- \frac{5}{11} \alpha_{2} ^{2}$ $\mathstrut+ \alpha_{2}$ $\mathstrut+ \frac{71}{11}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 8\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{10}{11} \alpha_{2} ^{3}$ $\mathstrut+ \frac{6}{11} \alpha_{2} ^{2}$ $\mathstrut- 20 \alpha_{2}$ $\mathstrut+ \frac{148}{11}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{7}{11} \alpha_{2} ^{3}$ $\mathstrut- \frac{13}{11} \alpha_{2} ^{2}$ $\mathstrut+ 12 \alpha_{2}$ $\mathstrut+ \frac{2}{11}\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{20}{11} \alpha_{2} ^{3}$ $\mathstrut- \frac{12}{11} \alpha_{2} ^{2}$ $\mathstrut+ 34 \alpha_{2}$ $\mathstrut- \frac{142}{11}\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{3}$ $\mathstrut- 16 \alpha_{2} \bigr)q^{8}$ $\mathstrut+$ $\bigl(\frac{7}{11} \alpha_{2} ^{3}$ $\mathstrut- \frac{9}{11} \alpha_{2} ^{2}$ $\mathstrut- 13 \alpha_{2}$ $\mathstrut+ \frac{152}{11}\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 4.4.334189.1 $x ^{4}$ $\mathstrut -\mathstrut 2 x ^{3}$ $\mathstrut -\mathstrut 24 x ^{2}$ $\mathstrut +\mathstrut 61 x$ $\mathstrut +\mathstrut 2$