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Decomposition of \( S_{10}^{\mathrm{new}}(5) \) into irreducible Hecke orbits

magma: S := CuspForms(5,10);
magma: N := Newforms(S);
sage: N = Newforms(5,10,names="a")
Label Dimension Field $q$-expansion of eigenform
5.10.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(8q^{2} \) \(\mathstrut-\) \(114q^{3} \) \(\mathstrut-\) \(448q^{4} \) \(\mathstrut-\) \(625q^{5} \) \(\mathstrut+\) \(912q^{6} \) \(\mathstrut+\) \(4242q^{7} \) \(\mathstrut+\) \(7680q^{8} \) \(\mathstrut-\) \(6687q^{9} \) \(\mathstrut+O(q^{10}) \)
5.10.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- 2 \alpha_{2} \) \(\mathstrut+ 120\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 10 \alpha_{2} \) \(\mathstrut+ 472\bigr)q^{4} \) \(\mathstrut+\) \(625q^{5} \) \(\mathstrut+\) \(\bigl(140 \alpha_{2} \) \(\mathstrut- 1968\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 214 \alpha_{2} \) \(\mathstrut- 220\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(60 \alpha_{2} \) \(\mathstrut- 9840\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 520 \alpha_{2} \) \(\mathstrut- 1347\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{1009}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 10 x \) \(\mathstrut -\mathstrut 984\)