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Decomposition of \( S_{12}^{\mathrm{new}}(20) \) into irreducible Hecke orbits

magma: S := CuspForms(20,12);
magma: N := Newforms(S);
sage: N = Newforms(20,12,names="a")
Label Dimension Field $q$-expansion of eigenform
20.12.1.a 1 \(\Q\) \(q \) \(\mathstrut+\) \(306q^{3} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut-\) \(32074q^{7} \) \(\mathstrut-\) \(83511q^{9} \) \(\mathstrut+O(q^{10}) \)
20.12.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{3} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut+\) \(\bigl(111 \alpha_{2} \) \(\mathstrut- 10540\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(220 \alpha_{2} \) \(\mathstrut- 2331\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })$ \(x ^{2} \) \(\mathstrut -\mathstrut 220 x \) \(\mathstrut -\mathstrut 174816\)

Decomposition of \( S_{12}^{\mathrm{old}}(20) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(20) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/10/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(10)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/5/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(5)) }^{\oplus 3 }\oplus \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 6 } $