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Decomposition of \( S_{8}^{\mathrm{new}}(14) \) into irreducible Hecke orbits

magma: S := CuspForms(14,8);
magma: N := Newforms(S);
sage: N = Newforms(14,8,names="a")
Label Dimension Field $q$-expansion of eigenform
14.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(8q^{2} \) \(\mathstrut-\) \(82q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut+\) \(448q^{5} \) \(\mathstrut+\) \(656q^{6} \) \(\mathstrut-\) \(343q^{7} \) \(\mathstrut-\) \(512q^{8} \) \(\mathstrut+\) \(4537q^{9} \) \(\mathstrut+O(q^{10}) \)
14.8.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(8q^{2} \) \(\mathstrut-\) \(66q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut-\) \(400q^{5} \) \(\mathstrut-\) \(528q^{6} \) \(\mathstrut-\) \(343q^{7} \) \(\mathstrut+\) \(512q^{8} \) \(\mathstrut+\) \(2169q^{9} \) \(\mathstrut+O(q^{10}) \)
14.8.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(8q^{2} \) \(\mathstrut+\) \(\bigl(\alpha_{3} \) \(\mathstrut- 8\bigr)q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut+\) \(\bigl(- 9 \alpha_{3} \) \(\mathstrut+ 450\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(8 \alpha_{3} \) \(\mathstrut- 64\bigr)q^{6} \) \(\mathstrut+\) \(343q^{7} \) \(\mathstrut+\) \(512q^{8} \) \(\mathstrut+\) \(\bigl(70 \alpha_{3} \) \(\mathstrut- 2003\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{1969}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 86 x \) \(\mathstrut -\mathstrut 120\)

Decomposition of \( S_{8}^{\mathrm{old}}(14) \) into lower level spaces

\( S_{8}^{\mathrm{old}}(14) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/7/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 2 } $