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Decomposition of \( S_{10}^{\mathrm{new}}(22) \) into irreducible Hecke orbits

magma: S := CuspForms(22,10);
magma: N := Newforms(S);
sage: N = Newforms(22,10,names="a")
Label Dimension Field $q$-expansion of eigenform
22.10.1.a 1 \(\Q\) \(q \) \(\mathstrut+\) \(16q^{2} \) \(\mathstrut-\) \(41q^{3} \) \(\mathstrut+\) \(256q^{4} \) \(\mathstrut-\) \(1039q^{5} \) \(\mathstrut-\) \(656q^{6} \) \(\mathstrut-\) \(3482q^{7} \) \(\mathstrut+\) \(4096q^{8} \) \(\mathstrut-\) \(18002q^{9} \) \(\mathstrut+O(q^{10}) \)
22.10.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(16q^{2} \) \(\mathstrut+\) \(137q^{3} \) \(\mathstrut+\) \(256q^{4} \) \(\mathstrut-\) \(595q^{5} \) \(\mathstrut+\) \(2192q^{6} \) \(\mathstrut+\) \(11354q^{7} \) \(\mathstrut+\) \(4096q^{8} \) \(\mathstrut-\) \(914q^{9} \) \(\mathstrut+O(q^{10}) \)
22.10.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(16q^{2} \) \(\mathstrut+\) \(201q^{3} \) \(\mathstrut+\) \(256q^{4} \) \(\mathstrut+\) \(2349q^{5} \) \(\mathstrut+\) \(3216q^{6} \) \(\mathstrut-\) \(8806q^{7} \) \(\mathstrut+\) \(4096q^{8} \) \(\mathstrut+\) \(20718q^{9} \) \(\mathstrut+O(q^{10}) \)
22.10.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut-\) \(16q^{2} \) \(\mathstrut+\) \(\bigl(\alpha_{4} \) \(\mathstrut+ 16\bigr)q^{3} \) \(\mathstrut+\) \(256q^{4} \) \(\mathstrut+\) \(\bigl(\frac{97}{9} \alpha_{4} \) \(\mathstrut+ \frac{226}{9}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{4} \) \(\mathstrut- 256\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{110}{3} \alpha_{4} \) \(\mathstrut- \frac{8320}{3}\bigr)q^{7} \) \(\mathstrut-\) \(4096q^{8} \) \(\mathstrut+\) \(\bigl(- 21 \alpha_{4} \) \(\mathstrut- 2127\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
22.10.1.e 2 $\Q(\alpha_{ 5 })$ \(q \) \(\mathstrut-\) \(16q^{2} \) \(\mathstrut+\) \(\bigl(\alpha_{5} \) \(\mathstrut+ 16\bigr)q^{3} \) \(\mathstrut+\) \(256q^{4} \) \(\mathstrut+\) \(\bigl(- 10 \alpha_{5} \) \(\mathstrut- 729\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{5} \) \(\mathstrut- 256\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 11 \alpha_{5} \) \(\mathstrut+ 4109\bigr)q^{7} \) \(\mathstrut-\) \(4096q^{8} \) \(\mathstrut+\) \(\bigl(34 \alpha_{5} \) \(\mathstrut+ 10204\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{889}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 53 x \) \(\mathstrut -\mathstrut 17300\)
$\Q(\alpha_{ 5 })\cong$ \(\Q(\sqrt{463}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut -\mathstrut 29631\)

Decomposition of \( S_{10}^{\mathrm{old}}(22) \) into lower level spaces

\( S_{10}^{\mathrm{old}}(22) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/11/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(11)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(2)) }^{\oplus 2 } $