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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 $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{19}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 20 x \) \(\mathstrut +\mathstrut 24\)