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Decomposition of \( S_{8}^{\mathrm{new}}(20) \) into irreducible Hecke orbits

magma: S := CuspForms(20,8);
magma: N := Newforms(S);
sage: N = Newforms(20,8,names="a")
Label Dimension Field $q$-expansion of eigenform
20.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(6q^{3} \) \(\mathstrut-\) \(125q^{5} \) \(\mathstrut-\) \(706q^{7} \) \(\mathstrut-\) \(2151q^{9} \) \(\mathstrut+O(q^{10}) \)
20.8.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut-\) \(\alpha_{2} q^{3} \) \(\mathstrut+\) \(125q^{5} \) \(\mathstrut+\) \(\bigl(9 \alpha_{2} \) \(\mathstrut+ 740\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(20 \alpha_{2} \) \(\mathstrut+ 2229\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{1129}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 20 x \) \(\mathstrut -\mathstrut 4416\)

Decomposition of \( S_{8}^{\mathrm{old}}(20) \) into lower level spaces

\( S_{8}^{\mathrm{old}}(20) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/10/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(10)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/5/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(5)) }^{\oplus 3 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 4 } $