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Decomposition of \( S_{12}^{\mathrm{new}}(9) \) into irreducible Hecke orbits

magma: S := CuspForms(9,12);
magma: N := Newforms(S);
sage: N = Newforms(9,12,names="a")
Label Dimension Field $q$-expansion of eigenform
9.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(78q^{2} \) \(\mathstrut+\) \(4036q^{4} \) \(\mathstrut+\) \(5370q^{5} \) \(\mathstrut-\) \(27760q^{7} \) \(\mathstrut-\) \(155064q^{8} \) \(\mathstrut+O(q^{10}) \)
9.12.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(24q^{2} \) \(\mathstrut-\) \(1472q^{4} \) \(\mathstrut-\) \(4830q^{5} \) \(\mathstrut-\) \(16744q^{7} \) \(\mathstrut-\) \(84480q^{8} \) \(\mathstrut+O(q^{10}) \)
9.12.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(472q^{4} \) \(\mathstrut+\) \(224 \alpha_{3} q^{5} \) \(\mathstrut+\) \(58100q^{7} \) \(\mathstrut-\) \(1576 \alpha_{3} q^{8} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{70}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 2520\)

Decomposition of \( S_{12}^{\mathrm{old}}(9) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(9) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 3 } $