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Decomposition of \( S_{6}^{\mathrm{new}}(15) \) into irreducible Hecke orbits

magma: S := CuspForms(15,6);
magma: N := Newforms(S);
sage: N = Newforms(15,6,names="a")
Label Dimension Field $q$-expansion of eigenform
15.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(2q^{2} \) \(\mathstrut-\) \(9q^{3} \) \(\mathstrut-\) \(28q^{4} \) \(\mathstrut-\) \(25q^{5} \) \(\mathstrut+\) \(18q^{6} \) \(\mathstrut-\) \(132q^{7} \) \(\mathstrut+\) \(120q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)
15.6.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(7q^{2} \) \(\mathstrut+\) \(9q^{3} \) \(\mathstrut+\) \(17q^{4} \) \(\mathstrut-\) \(25q^{5} \) \(\mathstrut+\) \(63q^{6} \) \(\mathstrut+\) \(12q^{7} \) \(\mathstrut-\) \(105q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)
15.6.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut-\) \(9q^{3} \) \(\mathstrut+\) \(\bigl(- \alpha_{3} \) \(\mathstrut+ 70\bigr)q^{4} \) \(\mathstrut+\) \(25q^{5} \) \(\mathstrut-\) \(9 \alpha_{3} q^{6} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{3} \) \(\mathstrut- 64\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(39 \alpha_{3} \) \(\mathstrut- 102\bigr)q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{409}) \) \(x ^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut -\mathstrut 102\)

Decomposition of \( S_{6}^{\mathrm{old}}(15) \) into lower level spaces

\( S_{6}^{\mathrm{old}}(15) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/5/6/1/ }{ S^{ new }_{ 6 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/6/1/ }{ S^{ new }_{ 6 }(\Gamma_0(3)) }^{\oplus 2 } $