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Decomposition of \( S_{10}^{\mathrm{new}}(15) \) into irreducible Hecke orbits

magma: S := CuspForms(15,10);
magma: N := Newforms(S);
sage: N = Newforms(15,10,names="a")
Label Dimension Field $q$-expansion of eigenform
15.10.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(4q^{2} \) \(\mathstrut+\) \(81q^{3} \) \(\mathstrut-\) \(496q^{4} \) \(\mathstrut+\) \(625q^{5} \) \(\mathstrut-\) \(324q^{6} \) \(\mathstrut-\) \(7680q^{7} \) \(\mathstrut+\) \(4032q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
15.10.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(22q^{2} \) \(\mathstrut-\) \(81q^{3} \) \(\mathstrut-\) \(28q^{4} \) \(\mathstrut-\) \(625q^{5} \) \(\mathstrut-\) \(1782q^{6} \) \(\mathstrut-\) \(5988q^{7} \) \(\mathstrut-\) \(11880q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
15.10.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(81q^{3} \) \(\mathstrut+\) \(\bigl(19 \alpha_{3} \) \(\mathstrut+ 580\bigr)q^{4} \) \(\mathstrut-\) \(625q^{5} \) \(\mathstrut+\) \(81 \alpha_{3} q^{6} \) \(\mathstrut+\) \(\bigl(- 56 \alpha_{3} \) \(\mathstrut- 5404\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(429 \alpha_{3} \) \(\mathstrut+ 20748\bigr)q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
15.10.1.d 2 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut-\) \(81q^{3} \) \(\mathstrut+\) \(\bigl(31 \alpha_{4} \) \(\mathstrut- 210\bigr)q^{4} \) \(\mathstrut+\) \(625q^{5} \) \(\mathstrut-\) \(81 \alpha_{4} q^{6} \) \(\mathstrut+\) \(\bigl(224 \alpha_{4} \) \(\mathstrut+ 3584\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(239 \alpha_{4} \) \(\mathstrut+ 9362\bigr)q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{4729}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 19 x \) \(\mathstrut -\mathstrut 1092\)
$\Q(\alpha_{ 4 })\cong$ \(\Q(\sqrt{241}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 31 x \) \(\mathstrut -\mathstrut 302\)

Decomposition of \( S_{10}^{\mathrm{old}}(15) \) into lower level spaces

\( S_{10}^{\mathrm{old}}(15) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/5/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(3)) }^{\oplus 2 } $