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Decomposition of $S_{8}^{\mathrm{new}}(16)$ into irreducible Hecke orbits

magma: S := CuspForms(16,8);
magma: N := Newforms(S);
sage: N = Newforms(16,8,names="a")
Label Dimension Field $q$-expansion of eigenform
16.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $44q^{3}$ $\mathstrut+$ $430q^{5}$ $\mathstrut+$ $1224q^{7}$ $\mathstrut-$ $251q^{9}$ $\mathstrut+O(q^{10})$
16.8.1.b 1 $\Q$ $q$ $\mathstrut-$ $12q^{3}$ $\mathstrut-$ $210q^{5}$ $\mathstrut-$ $1016q^{7}$ $\mathstrut-$ $2043q^{9}$ $\mathstrut+O(q^{10})$
16.8.1.c 1 $\Q$ $q$ $\mathstrut+$ $84q^{3}$ $\mathstrut-$ $82q^{5}$ $\mathstrut+$ $456q^{7}$ $\mathstrut+$ $4869q^{9}$ $\mathstrut+O(q^{10})$

Decomposition of $S_{8}^{\mathrm{old}}(16)$ into lower level spaces

$S_{8}^{\mathrm{old}}(16)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/8/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(8)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/8/1/ }{ S^{ new }_{ 8 }(\Gamma_0(2)) }^{\oplus 4 }$