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Decomposition of \( S_{4}^{\mathrm{new}}(19) \) into irreducible Hecke orbits

magma: S := CuspForms(19,4);
magma: N := Newforms(S);
sage: N = Newforms(19,4,names="a")
Label Dimension Field $q$-expansion of eigenform
19.4.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(3q^{2} \) \(\mathstrut-\) \(5q^{3} \) \(\mathstrut+\) \(q^{4} \) \(\mathstrut-\) \(12q^{5} \) \(\mathstrut+\) \(15q^{6} \) \(\mathstrut+\) \(11q^{7} \) \(\mathstrut+\) \(21q^{8} \) \(\mathstrut-\) \(2q^{9} \) \(\mathstrut+O(q^{10}) \)
19.4.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{3} \alpha_{2} ^{2} \) \(\mathstrut- \frac{4}{3} \alpha_{2} \) \(\mathstrut+ \frac{20}{3}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 8\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{1}{3} \alpha_{2} ^{2} \) \(\mathstrut- \frac{8}{3} \alpha_{2} \) \(\mathstrut+ \frac{7}{3}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{7}{3} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{2}{3} \alpha_{2} \) \(\mathstrut+ \frac{38}{3}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{4}{3} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{8}{3} \alpha_{2} \) \(\mathstrut+ \frac{17}{3}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(3 \alpha_{2} ^{2} \) \(\mathstrut+ 2 \alpha_{2} \) \(\mathstrut- 38\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(3 \alpha_{2} ^{2} \) \(\mathstrut- 29\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 3.3.3144.1 \(x ^{3} \) \(\mathstrut -\mathstrut 3 x ^{2} \) \(\mathstrut -\mathstrut 18 x \) \(\mathstrut +\mathstrut 38\)