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Decomposition of \( S_{12}^{\mathrm{new}}(10) \) into irreducible Hecke orbits

magma: S := CuspForms(10,12);
magma: N := Newforms(S);
sage: N = Newforms(10,12,names="a")
Label Dimension Field $q$-expansion of eigenform
10.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(32q^{2} \) \(\mathstrut-\) \(12q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut+\) \(384q^{6} \) \(\mathstrut-\) \(14176q^{7} \) \(\mathstrut-\) \(32768q^{8} \) \(\mathstrut-\) \(177003q^{9} \) \(\mathstrut+O(q^{10}) \)
10.12.1.b 1 \(\Q\) \(q \) \(\mathstrut-\) \(32q^{2} \) \(\mathstrut+\) \(738q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut-\) \(23616q^{6} \) \(\mathstrut+\) \(25574q^{7} \) \(\mathstrut-\) \(32768q^{8} \) \(\mathstrut+\) \(367497q^{9} \) \(\mathstrut+O(q^{10}) \)
10.12.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(32q^{2} \) \(\mathstrut-\) \(318q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut-\) \(10176q^{6} \) \(\mathstrut-\) \(70714q^{7} \) \(\mathstrut+\) \(32768q^{8} \) \(\mathstrut-\) \(76023q^{9} \) \(\mathstrut+O(q^{10}) \)
10.12.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(32q^{2} \) \(\mathstrut+\) \(\bigl(\alpha_{4} \) \(\mathstrut- 32\bigr)q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut+\) \(\bigl(32 \alpha_{4} \) \(\mathstrut- 1024\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 177 \alpha_{4} \) \(\mathstrut+ 66164\bigr)q^{7} \) \(\mathstrut+\) \(32768q^{8} \) \(\mathstrut+\) \(\bigl(604 \alpha_{4} \) \(\mathstrut- 90779\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{1969}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 668 x \) \(\mathstrut -\mathstrut 85344\)

Decomposition of \( S_{12}^{\mathrm{old}}(10) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(10) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/5/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 } $