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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 $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{1009})$ $x ^{2}$ $\mathstrut +\mathstrut 10 x$ $\mathstrut -\mathstrut 984$