# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(11)$ into irreducible Hecke orbits

magma: S := CuspForms(11,8);
magma: N := Newforms(S);
sage: N = Newforms(11,8,names="a")
Label Dimension Field $q$-expansion of eigenform
11.8.1.a 2 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(- 6 \alpha_{1}$ $\mathstrut- 27\bigr)q^{3}$ $\mathstrut+$ $\bigl(- 8 \alpha_{1}$ $\mathstrut- 84\bigr)q^{4}$ $\mathstrut+$ $\bigl(20 \alpha_{1}$ $\mathstrut- 155\bigr)q^{5}$ $\mathstrut+$ $\bigl(21 \alpha_{1}$ $\mathstrut- 264\bigr)q^{6}$ $\mathstrut+$ $\bigl(82 \alpha_{1}$ $\mathstrut- 286\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 148 \alpha_{1}$ $\mathstrut- 352\bigr)q^{8}$ $\mathstrut+$ $\bigl(36 \alpha_{1}$ $\mathstrut+ 126\bigr)q^{9}$ $\mathstrut+O(q^{10})$
11.8.1.b 4 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{252} \alpha_{2} ^{3}$ $\mathstrut- \frac{5}{18} \alpha_{2} ^{2}$ $\mathstrut+ \frac{349}{126} \alpha_{2}$ $\mathstrut+ \frac{205}{3}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 128\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{3}{28} \alpha_{2} ^{3}$ $\mathstrut- \frac{1}{2} \alpha_{2} ^{2}$ $\mathstrut- \frac{543}{14} \alpha_{2}$ $\mathstrut+ 285\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{5}{18} \alpha_{2} ^{3}$ $\mathstrut+ \frac{5}{9} \alpha_{2} ^{2}$ $\mathstrut+ \frac{620}{9} \alpha_{2}$ $\mathstrut+ \frac{616}{3}\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{1}{14} \alpha_{2} ^{3}$ $\mathstrut- 5 \alpha_{2} ^{2}$ $\mathstrut- \frac{15}{7} \alpha_{2}$ $\mathstrut+ 1430\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{3}$ $\mathstrut- 256 \alpha_{2} \bigr)q^{8}$ $\mathstrut+$ $\bigl(- \frac{215}{252} \alpha_{2} ^{3}$ $\mathstrut+ \frac{77}{18} \alpha_{2} ^{2}$ $\mathstrut+ \frac{32195}{126} \alpha_{2}$ $\mathstrut- \frac{2482}{3}\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ $\Q(\sqrt{15})$ $x ^{2}$ $\mathstrut +\mathstrut 8 x$ $\mathstrut -\mathstrut 44$
$\Q(\alpha_{ 2 })$ $x ^{4}$ $\mathstrut -\mathstrut 558 x ^{2}$ $\mathstrut +\mathstrut 140 x$ $\mathstrut +\mathstrut 51744$