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Decomposition of \( S_{12}^{\mathrm{new}}(8) \) into irreducible Hecke orbits

magma: S := CuspForms(8,12);
magma: N := Newforms(S);
sage: N = Newforms(8,12,names="a")
Label Dimension Field $q$-expansion of eigenform
8.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(36q^{3} \) \(\mathstrut-\) \(3490q^{5} \) \(\mathstrut-\) \(55464q^{7} \) \(\mathstrut-\) \(175851q^{9} \) \(\mathstrut+O(q^{10}) \)
8.12.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut-\) \(\frac{1}{2} \alpha_{2} q^{3} \) \(\mathstrut+\) \(\bigl(6 \alpha_{2} \) \(\mathstrut+ 4270\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 21 \alpha_{2} \) \(\mathstrut+ 44352\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 28 \alpha_{2} \) \(\mathstrut+ 268533\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{109}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 112 x \) \(\mathstrut -\mathstrut 1782720\)

Decomposition of \( S_{12}^{\mathrm{old}}(8) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(8) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/4/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(4)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 } $