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Decomposition of \( S_{12}^{\mathrm{new}}(23) \) into irreducible Hecke orbits

magma: S := CuspForms(23,12);
magma: N := Newforms(S);
sage: N = Newforms(23,12,names="a")
Label Dimension Field $q$-expansion of eigenform
23.12.1.a 8 $\Q(\alpha_{ 1 })$ $q + \ldots^\ast$
23.12.1.b 11 $\Q(\alpha_{ 2 })$ $q + \ldots^\ast$

${}^\ast$: 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 $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{8} \) \(\mathstrut +\mathstrut 32 x ^{7} \) \(\mathstrut -\mathstrut 10240 x ^{6} \) \(\mathstrut -\mathstrut 243056 x ^{5} \) \(\mathstrut +\mathstrut 32897712 x ^{4} \) \(\mathstrut +\mathstrut 475545152 x ^{3} \) \(\mathstrut -\mathstrut 32355612672 x ^{2} \) \(\mathstrut -\mathstrut 118888316928 x \) \(\mathstrut +\mathstrut 6030172585984\)
$\Q(\alpha_{ 2 })$ \(x ^{11} \) \(\mathstrut -\mathstrut 32 x ^{10} \) \(\mathstrut -\mathstrut 16384 x ^{9} \) \(\mathstrut +\mathstrut 453021 x ^{8} \) \(\mathstrut +\mathstrut 91689644 x ^{7} \) \(\mathstrut -\mathstrut 2171413128 x ^{6} \) \(\mathstrut -\mathstrut 205410732416 x ^{5} \) \(\mathstrut +\mathstrut 4226023192464 x ^{4} \) \(\mathstrut +\mathstrut 148783481621696 x ^{3} \) \(\mathstrut -\mathstrut 3199980978790400 x ^{2} \) \(\mathstrut +\mathstrut 5191695182200832 x \) \(\mathstrut +\mathstrut 60024577238663168\)