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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.
Click on the label in the table above for more information about each newform.

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 } $