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Decomposition of \( S_{6}^{\mathrm{new}}(11) \) into irreducible Hecke orbits

magma: S := CuspForms(11,6);
magma: N := Newforms(S);
sage: N = Newforms(11,6,names="a")
Label Dimension Field $q$-expansion of eigenform
11.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(4q^{2} \) \(\mathstrut-\) \(15q^{3} \) \(\mathstrut-\) \(16q^{4} \) \(\mathstrut-\) \(19q^{5} \) \(\mathstrut+\) \(60q^{6} \) \(\mathstrut+\) \(10q^{7} \) \(\mathstrut+\) \(192q^{8} \) \(\mathstrut-\) \(18q^{9} \) \(\mathstrut+O(q^{10}) \)
11.6.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{6} \alpha_{2} ^{2} \) \(\mathstrut- \frac{5}{3} \alpha_{2} \) \(\mathstrut+ \frac{64}{3}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 32\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{3}{2} \alpha_{2} ^{2} \) \(\mathstrut- 7 \alpha_{2} \) \(\mathstrut+ 98\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{5}{3} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{19}{3} \alpha_{2} \) \(\mathstrut+ \frac{94}{3}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(5 \alpha_{2} ^{2} \) \(\mathstrut+ 10 \alpha_{2} \) \(\mathstrut- 272\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(26 \alpha_{2} \) \(\mathstrut- 188\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{11}{6} \alpha_{2} ^{2} \) \(\mathstrut- \frac{79}{3} \alpha_{2} \) \(\mathstrut+ \frac{323}{3}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })$ \(x ^{3} \) \(\mathstrut -\mathstrut 90 x \) \(\mathstrut +\mathstrut 188\)