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## Decomposition of $S_{12}^{\mathrm{new}}(14)$ into irreducible Hecke orbits

magma: S := CuspForms(14,12);
magma: N := Newforms(S);
sage: N = Newforms(14,12,names="a")
Label Dimension Field $q$-expansion of eigenform
14.12.1.a 1 $\Q$ $q$ $\mathstrut-$ $32q^{2}$ $\mathstrut-$ $396q^{3}$ $\mathstrut+$ $1024q^{4}$ $\mathstrut+$ $7350q^{5}$ $\mathstrut+$ $12672q^{6}$ $\mathstrut+$ $16807q^{7}$ $\mathstrut-$ $32768q^{8}$ $\mathstrut-$ $20331q^{9}$ $\mathstrut+O(q^{10})$
14.12.1.b 1 $\Q$ $q$ $\mathstrut+$ $32q^{2}$ $\mathstrut-$ $90q^{3}$ $\mathstrut+$ $1024q^{4}$ $\mathstrut-$ $7480q^{5}$ $\mathstrut-$ $2880q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut+$ $32768q^{8}$ $\mathstrut-$ $169047q^{9}$ $\mathstrut+O(q^{10})$
14.12.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut-$ $32q^{2}$ $\mathstrut+$ $\bigl(- \alpha_{3}$ $\mathstrut- 32\bigr)q^{3}$ $\mathstrut+$ $1024q^{4}$ $\mathstrut+$ $\bigl(- 21 \alpha_{3}$ $\mathstrut- 4214\bigr)q^{5}$ $\mathstrut+$ $\bigl(32 \alpha_{3}$ $\mathstrut+ 1024\bigr)q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut-$ $32768q^{8}$ $\mathstrut+$ $\bigl(- 350 \alpha_{3}$ $\mathstrut- 65803\bigr)q^{9}$ $\mathstrut+O(q^{10})$
14.12.1.d 2 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut+$ $32q^{2}$ $\mathstrut+$ $\bigl(\alpha_{4}$ $\mathstrut- 32\bigr)q^{3}$ $\mathstrut+$ $1024q^{4}$ $\mathstrut+$ $\bigl(3 \alpha_{4}$ $\mathstrut+ 2298\bigr)q^{5}$ $\mathstrut+$ $\bigl(32 \alpha_{4}$ $\mathstrut- 1024\bigr)q^{6}$ $\mathstrut+$ $16807q^{7}$ $\mathstrut+$ $32768q^{8}$ $\mathstrut+$ $\bigl(- 350 \alpha_{4}$ $\mathstrut+ 156397\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })$ $x ^{2}$ $\mathstrut +\mathstrut 414 x$ $\mathstrut -\mathstrut 110320$
$\Q(\alpha_{ 4 })$ $x ^{2}$ $\mathstrut +\mathstrut 286 x$ $\mathstrut -\mathstrut 332520$

## Decomposition of $S_{12}^{\mathrm{old}}(14)$ into lower level spaces

$S_{12}^{\mathrm{old}}(14)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/7/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 }$