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Decomposition of \( S_{6}^{\mathrm{new}}(22) \) into irreducible Hecke orbits

magma: S := CuspForms(22,6);
magma: N := Newforms(S);
sage: N = Newforms(22,6,names="a")
Label Dimension Field $q$-expansion of eigenform
22.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(4q^{2} \) \(\mathstrut-\) \(21q^{3} \) \(\mathstrut+\) \(16q^{4} \) \(\mathstrut+\) \(81q^{5} \) \(\mathstrut+\) \(84q^{6} \) \(\mathstrut+\) \(98q^{7} \) \(\mathstrut-\) \(64q^{8} \) \(\mathstrut+\) \(198q^{9} \) \(\mathstrut+O(q^{10}) \)
22.6.1.b 1 \(\Q\) \(q \) \(\mathstrut-\) \(4q^{2} \) \(\mathstrut+\) \(q^{3} \) \(\mathstrut+\) \(16q^{4} \) \(\mathstrut-\) \(51q^{5} \) \(\mathstrut-\) \(4q^{6} \) \(\mathstrut-\) \(166q^{7} \) \(\mathstrut-\) \(64q^{8} \) \(\mathstrut-\) \(242q^{9} \) \(\mathstrut+O(q^{10}) \)
22.6.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(4q^{2} \) \(\mathstrut-\) \(29q^{3} \) \(\mathstrut+\) \(16q^{4} \) \(\mathstrut-\) \(31q^{5} \) \(\mathstrut-\) \(116q^{6} \) \(\mathstrut-\) \(230q^{7} \) \(\mathstrut+\) \(64q^{8} \) \(\mathstrut+\) \(598q^{9} \) \(\mathstrut+O(q^{10}) \)
22.6.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(4q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1}{2} \alpha_{4} \) \(\mathstrut- 2\bigr)q^{3} \) \(\mathstrut+\) \(16q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{5}{2} \alpha_{4} \) \(\mathstrut+ 76\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(2 \alpha_{4} \) \(\mathstrut- 8\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{4} \) \(\mathstrut+ 92\bigr)q^{7} \) \(\mathstrut+\) \(64q^{8} \) \(\mathstrut+\) \(\bigl(\frac{29}{2} \alpha_{4} \) \(\mathstrut- 313\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{793}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 66 x \) \(\mathstrut +\mathstrut 296\)

Decomposition of \( S_{6}^{\mathrm{old}}(22) \) into lower level spaces

\( S_{6}^{\mathrm{old}}(22) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/11/6/1/ }{ S^{ new }_{ 6 }(\Gamma_0(11)) }^{\oplus 2 } $