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Decomposition of \( S_{10}^{\mathrm{new}}(20) \) into irreducible Hecke orbits

magma: S := CuspForms(20,10);
magma: N := Newforms(S);
sage: N = Newforms(20,10,names="a")
Label Dimension Field $q$-expansion of eigenform
20.10.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(48q^{3} \) \(\mathstrut+\) \(625q^{5} \) \(\mathstrut-\) \(532q^{7} \) \(\mathstrut-\) \(17379q^{9} \) \(\mathstrut+O(q^{10}) \)
20.10.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\frac{1}{2} \alpha_{2} q^{3} \) \(\mathstrut-\) \(625q^{5} \) \(\mathstrut+\) \(\bigl(\frac{69}{2} \alpha_{2} \) \(\mathstrut+ 8780\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 130 \alpha_{2} \) \(\mathstrut- 16359\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{79}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 520 x \) \(\mathstrut -\mathstrut 13296\)

Decomposition of \( S_{10}^{\mathrm{old}}(20) \) into lower level spaces

\( S_{10}^{\mathrm{old}}(20) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/10/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(10)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/5/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(5)) }^{\oplus 3 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/4/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(4)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(2)) }^{\oplus 4 } $