# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{4}^{\mathrm{new}}(21)$ into irreducible Hecke orbits

magma: S := CuspForms(21,4);
magma: N := Newforms(S);
sage: N = Newforms(21,4,names="a")
Label Dimension Field $q$-expansion of eigenform
21.4.1.a 1 $\Q$ $q$ $\mathstrut-$ $3q^{2}$ $\mathstrut-$ $3q^{3}$ $\mathstrut+$ $q^{4}$ $\mathstrut-$ $18q^{5}$ $\mathstrut+$ $9q^{6}$ $\mathstrut+$ $7q^{7}$ $\mathstrut+$ $21q^{8}$ $\mathstrut+$ $9q^{9}$ $\mathstrut+O(q^{10})$
21.4.1.b 1 $\Q$ $q$ $\mathstrut+$ $4q^{2}$ $\mathstrut-$ $3q^{3}$ $\mathstrut+$ $8q^{4}$ $\mathstrut-$ $4q^{5}$ $\mathstrut-$ $12q^{6}$ $\mathstrut-$ $7q^{7}$ $\mathstrut+$ $9q^{9}$ $\mathstrut+O(q^{10})$
21.4.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $3q^{3}$ $\mathstrut+$ $\bigl(- 3 \alpha_{3}$ $\mathstrut+ 4\bigr)q^{4}$ $\mathstrut-$ $2 \alpha_{3} q^{5}$ $\mathstrut+$ $3 \alpha_{3} q^{6}$ $\mathstrut+$ $7q^{7}$ $\mathstrut+$ $\bigl(5 \alpha_{3}$ $\mathstrut- 36\bigr)q^{8}$ $\mathstrut+$ $9q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{57})$ $x ^{2}$ $\mathstrut +\mathstrut 3 x$ $\mathstrut -\mathstrut 12$

## Decomposition of $S_{4}^{\mathrm{old}}(21)$ into lower level spaces

$S_{4}^{\mathrm{old}}(21)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/7/4/1/ }{ S^{ new }_{ 4 }(\Gamma_0(7)) }^{\oplus 2 }$