# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{12}^{\mathrm{new}}(19)$ into irreducible Hecke orbits

magma: S := CuspForms(19,12);
magma: N := Newforms(S);
sage: N = Newforms(19,12,names="a")
Label Dimension Field $q$-expansion of eigenform
19.12.1.a 7 $\Q(\alpha_{ 1 })$ $q + \ldots^\ast$
19.12.1.b 9 $\Q(\alpha_{ 2 })$ $q + \ldots^\ast$

${}^\ast$: The Fourier coefficients of this newform are large. They are available for download.

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ $x ^{7}$ $\mathstrut +\mathstrut 9 x ^{6}$ $\mathstrut -\mathstrut 10458 x ^{5}$ $\mathstrut -\mathstrut 57780 x ^{4}$ $\mathstrut +\mathstrut 33079800 x ^{3}$ $\mathstrut -\mathstrut 56001024 x ^{2}$ $\mathstrut -\mathstrut 33006306816 x$ $\mathstrut +\mathstrut 384276234240$
$\Q(\alpha_{ 2 })$ $x ^{9}$ $\mathstrut -\mathstrut 87 x ^{8}$ $\mathstrut -\mathstrut 11322 x ^{7}$ $\mathstrut +\mathstrut 1111298 x ^{6}$ $\mathstrut +\mathstrut 23062032 x ^{5}$ $\mathstrut -\mathstrut 3703642344 x ^{4}$ $\mathstrut +\mathstrut 49908522752 x ^{3}$ $\mathstrut +\mathstrut 1569947506944 x ^{2}$ $\mathstrut -\mathstrut 33506908250112 x$ $\mathstrut +\mathstrut 139839674073088$

## Decomposition of $S_{12}^{\mathrm{old}}(19)$ into lower level spaces

$S_{12}^{\mathrm{old}}(19)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 2 }$