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Decomposition of \( S_{4}^{\mathrm{new}}(11) \) into irreducible Hecke orbits

magma: S := CuspForms(11,4);
magma: N := Newforms(S);
sage: N = Newforms(11,4,names="a")
Label Dimension Field $q$-expansion of eigenform
11.4.1.a 2 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- 4 \alpha_{1} \) \(\mathstrut+ 3\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(2 \alpha_{1} \) \(\mathstrut- 6\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(8 \alpha_{1} \) \(\mathstrut- 7\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 5 \alpha_{1} \) \(\mathstrut- 8\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 4 \alpha_{1} \) \(\mathstrut+ 14\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 10 \alpha_{1} \) \(\mathstrut+ 4\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(8 \alpha_{1} \) \(\mathstrut+ 14\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{3}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut -\mathstrut 2\)