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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 $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{15}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 8 x \) \(\mathstrut -\mathstrut 44\)
$\Q(\alpha_{ 2 })$ \(x ^{4} \) \(\mathstrut -\mathstrut 558 x ^{2} \) \(\mathstrut +\mathstrut 140 x \) \(\mathstrut +\mathstrut 51744\)