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Decomposition of \( S_{8}^{\mathrm{new}}(22) \) into irreducible Hecke orbits

magma: S := CuspForms(22,8);
magma: N := Newforms(S);
sage: N = Newforms(22,8,names="a")
Label Dimension Field $q$-expansion of eigenform
22.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(8q^{2} \) \(\mathstrut-\) \(19q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut+\) \(317q^{5} \) \(\mathstrut+\) \(152q^{6} \) \(\mathstrut-\) \(1030q^{7} \) \(\mathstrut-\) \(512q^{8} \) \(\mathstrut-\) \(1826q^{9} \) \(\mathstrut+O(q^{10}) \)
22.8.1.b 1 \(\Q\) \(q \) \(\mathstrut-\) \(8q^{2} \) \(\mathstrut+\) \(91q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut+\) \(185q^{5} \) \(\mathstrut-\) \(728q^{6} \) \(\mathstrut-\) \(722q^{7} \) \(\mathstrut-\) \(512q^{8} \) \(\mathstrut+\) \(6094q^{9} \) \(\mathstrut+O(q^{10}) \)
22.8.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(8q^{2} \) \(\mathstrut-\) \(21q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut-\) \(551q^{5} \) \(\mathstrut-\) \(168q^{6} \) \(\mathstrut+\) \(62q^{7} \) \(\mathstrut+\) \(512q^{8} \) \(\mathstrut-\) \(1746q^{9} \) \(\mathstrut+O(q^{10}) \)
22.8.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(8q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{2} \alpha_{4} \) \(\mathstrut+ 4\bigr)q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut+\) \(\bigl(\frac{1}{2} \alpha_{4} \) \(\mathstrut+ 150\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 4 \alpha_{4} \) \(\mathstrut+ 32\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(7 \alpha_{4} \) \(\mathstrut+ 680\bigr)q^{7} \) \(\mathstrut+\) \(512q^{8} \) \(\mathstrut+\) \(\bigl(\frac{23}{2} \alpha_{4} \) \(\mathstrut+ 1309\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{14881}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 62 x \) \(\mathstrut -\mathstrut 13920\)

Decomposition of \( S_{8}^{\mathrm{old}}(22) \) into lower level spaces

\( S_{8}^{\mathrm{old}}(22) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/11/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(11)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 2 } $