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Decomposition of \( S_{8}^{\mathrm{new}}(15) \) into irreducible Hecke orbits

magma: S := CuspForms(15,8);
magma: N := Newforms(S);
sage: N = Newforms(15,8,names="a")
Label Dimension Field $q$-expansion of eigenform
15.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(22q^{2} \) \(\mathstrut+\) \(27q^{3} \) \(\mathstrut+\) \(356q^{4} \) \(\mathstrut-\) \(125q^{5} \) \(\mathstrut-\) \(594q^{6} \) \(\mathstrut-\) \(420q^{7} \) \(\mathstrut-\) \(5016q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)
15.8.1.b 1 \(\Q\) \(q \) \(\mathstrut-\) \(13q^{2} \) \(\mathstrut-\) \(27q^{3} \) \(\mathstrut+\) \(41q^{4} \) \(\mathstrut-\) \(125q^{5} \) \(\mathstrut+\) \(351q^{6} \) \(\mathstrut+\) \(1380q^{7} \) \(\mathstrut+\) \(1131q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)
15.8.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(27q^{3} \) \(\mathstrut+\) \(\bigl(7 \alpha_{3} \) \(\mathstrut+ 10\bigr)q^{4} \) \(\mathstrut+\) \(125q^{5} \) \(\mathstrut+\) \(27 \alpha_{3} q^{6} \) \(\mathstrut+\) \(\bigl(- 56 \alpha_{3} \) \(\mathstrut+ 848\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 69 \alpha_{3} \) \(\mathstrut+ 966\bigr)q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{601}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut -\mathstrut 138\)

Decomposition of \( S_{8}^{\mathrm{old}}(15) \) into lower level spaces

\( S_{8}^{\mathrm{old}}(15) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/5/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(3)) }^{\oplus 2 } $