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Decomposition of \( S_{12}^{\mathrm{new}}(5) \) into irreducible Hecke orbits

magma: S := CuspForms(5,12);
magma: N := Newforms(S);
sage: N = Newforms(5,12,names="a")
Label Dimension Field $q$-expansion of eigenform
5.12.1.a 1 \(\Q\) \(q \) \(\mathstrut+\) \(34q^{2} \) \(\mathstrut-\) \(792q^{3} \) \(\mathstrut-\) \(892q^{4} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut-\) \(26928q^{6} \) \(\mathstrut-\) \(17556q^{7} \) \(\mathstrut-\) \(99960q^{8} \) \(\mathstrut+\) \(450117q^{9} \) \(\mathstrut+O(q^{10}) \)
5.12.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{16}{3} \alpha_{2} \) \(\mathstrut- \frac{170}{3}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 20 \alpha_{2} \) \(\mathstrut+ 3288\bigr)q^{4} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{490}{3} \alpha_{2} \) \(\mathstrut+ \frac{85376}{3}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(176 \alpha_{2} \) \(\mathstrut+ 30710\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(1640 \alpha_{2} \) \(\mathstrut- 106720\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{3520}{3} \alpha_{2} \) \(\mathstrut- \frac{66469}{3}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{151}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 20 x \) \(\mathstrut -\mathstrut 5336\)

Decomposition of \( S_{12}^{\mathrm{old}}(5) \) into lower level spaces

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