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Decomposition of \( S_{4}^{\mathrm{new}}(13) \) into irreducible Hecke orbits

magma: S := CuspForms(13,4);
magma: N := Newforms(S);
sage: N = Newforms(13,4,names="a")
Label Dimension Field $q$-expansion of eigenform
13.4.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(5q^{2} \) \(\mathstrut-\) \(7q^{3} \) \(\mathstrut+\) \(17q^{4} \) \(\mathstrut-\) \(7q^{5} \) \(\mathstrut+\) \(35q^{6} \) \(\mathstrut-\) \(13q^{7} \) \(\mathstrut-\) \(45q^{8} \) \(\mathstrut+\) \(22q^{9} \) \(\mathstrut+O(q^{10}) \)
13.4.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{2} \) \(\mathstrut+ 4\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} \) \(\mathstrut- 4\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\alpha_{2} \) \(\mathstrut- 2\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\alpha_{2} \) \(\mathstrut- 12\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(11 \alpha_{2} \) \(\mathstrut- 10\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 11 \alpha_{2} \) \(\mathstrut+ 4\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 15 \alpha_{2} \) \(\mathstrut+ 25\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{17}) \) \(x ^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut -\mathstrut 4\)