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Decomposition of \( S_{12}^{\mathrm{new}}(16) \) into irreducible Hecke orbits

magma: S := CuspForms(16,12);
magma: N := Newforms(S);
sage: N = Newforms(16,12,names="a")
Label Dimension Field $q$-expansion of eigenform
16.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(252q^{3} \) \(\mathstrut+\) \(4830q^{5} \) \(\mathstrut+\) \(16744q^{7} \) \(\mathstrut-\) \(113643q^{9} \) \(\mathstrut+O(q^{10}) \)
16.12.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(36q^{3} \) \(\mathstrut-\) \(3490q^{5} \) \(\mathstrut+\) \(55464q^{7} \) \(\mathstrut-\) \(175851q^{9} \) \(\mathstrut+O(q^{10}) \)
16.12.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(516q^{3} \) \(\mathstrut-\) \(10530q^{5} \) \(\mathstrut-\) \(49304q^{7} \) \(\mathstrut+\) \(89109q^{9} \) \(\mathstrut+O(q^{10}) \)
16.12.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\frac{1}{2} \alpha_{4} q^{3} \) \(\mathstrut+\) \(\bigl(6 \alpha_{4} \) \(\mathstrut+ 4270\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(21 \alpha_{4} \) \(\mathstrut- 44352\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 28 \alpha_{4} \) \(\mathstrut+ 268533\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{109}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 112 x \) \(\mathstrut -\mathstrut 1782720\)

Decomposition of \( S_{12}^{\mathrm{old}}(16) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(16) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/8/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(8)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/4/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(4)) }^{\oplus 3 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 5 } $