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Decomposition of \( S_{6}^{\mathrm{new}}(7) \) into irreducible Hecke orbits

magma: S := CuspForms(7,6);
magma: N := Newforms(S);
sage: N = Newforms(7,6,names="a")
Label Dimension Field $q$-expansion of eigenform
7.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(10q^{2} \) \(\mathstrut-\) \(14q^{3} \) \(\mathstrut+\) \(68q^{4} \) \(\mathstrut-\) \(56q^{5} \) \(\mathstrut+\) \(140q^{6} \) \(\mathstrut-\) \(49q^{7} \) \(\mathstrut-\) \(360q^{8} \) \(\mathstrut-\) \(47q^{9} \) \(\mathstrut+O(q^{10}) \)
7.6.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- 6 \alpha_{2} \) \(\mathstrut+ 24\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(9 \alpha_{2} \) \(\mathstrut- 38\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(10 \alpha_{2} \) \(\mathstrut- 54\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 30 \alpha_{2} \) \(\mathstrut+ 36\bigr)q^{6} \) \(\mathstrut+\) \(49q^{7} \) \(\mathstrut+\) \(\bigl(11 \alpha_{2} \) \(\mathstrut- 54\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(36 \alpha_{2} \) \(\mathstrut+ 117\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{57}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 9 x \) \(\mathstrut +\mathstrut 6\)