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Decomposition of \( S_{6}^{\mathrm{new}}(21) \) into irreducible Hecke orbits

magma: S := CuspForms(21,6);
magma: N := Newforms(S);
sage: N = Newforms(21,6,names="a")
Label Dimension Field $q$-expansion of eigenform
21.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(6q^{2} \) \(\mathstrut-\) \(9q^{3} \) \(\mathstrut+\) \(4q^{4} \) \(\mathstrut+\) \(78q^{5} \) \(\mathstrut+\) \(54q^{6} \) \(\mathstrut+\) \(49q^{7} \) \(\mathstrut+\) \(168q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)
21.6.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(q^{2} \) \(\mathstrut-\) \(9q^{3} \) \(\mathstrut-\) \(31q^{4} \) \(\mathstrut-\) \(34q^{5} \) \(\mathstrut-\) \(9q^{6} \) \(\mathstrut-\) \(49q^{7} \) \(\mathstrut-\) \(63q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)
21.6.1.c 1 \(\Q\) \(q \) \(\mathstrut+\) \(5q^{2} \) \(\mathstrut+\) \(9q^{3} \) \(\mathstrut-\) \(7q^{4} \) \(\mathstrut+\) \(94q^{5} \) \(\mathstrut+\) \(45q^{6} \) \(\mathstrut-\) \(49q^{7} \) \(\mathstrut-\) \(195q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)
21.6.1.d 1 \(\Q\) \(q \) \(\mathstrut+\) \(10q^{2} \) \(\mathstrut+\) \(9q^{3} \) \(\mathstrut+\) \(68q^{4} \) \(\mathstrut-\) \(106q^{5} \) \(\mathstrut+\) \(90q^{6} \) \(\mathstrut-\) \(49q^{7} \) \(\mathstrut+\) \(360q^{8} \) \(\mathstrut+\) \(81q^{9} \) \(\mathstrut+O(q^{10}) \)

Decomposition of \( S_{6}^{\mathrm{old}}(21) \) into lower level spaces

\( S_{6}^{\mathrm{old}}(21) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/7/6/1/ }{ S^{ new }_{ 6 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/6/1/ }{ S^{ new }_{ 6 }(\Gamma_0(3)) }^{\oplus 2 } $