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Decomposition of \( S_{12}^{\mathrm{new}}(17) \) into irreducible Hecke orbits

magma: S := CuspForms(17,12);
magma: N := Newforms(S);
sage: N = Newforms(17,12,names="a")
Label Dimension Field $q$-expansion of eigenform
17.12.1.a 6 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{223}{437669760} \alpha_{1} ^{5} \) \(\mathstrut- \frac{6481}{13262720} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{303473}{54708720} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{333520697}{109417440} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{132001597}{27354360} \alpha_{1} \) \(\mathstrut- \frac{334470111}{103615}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{669}{9118120} \alpha_{1} ^{5} \) \(\mathstrut+ \frac{13043}{1657840} \alpha_{1} ^{4} \) \(\mathstrut- \frac{8867879}{18236240} \alpha_{1} ^{3} \) \(\mathstrut- \frac{258423497}{4559060} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{1207519507}{4559060} \alpha_{1} \) \(\mathstrut+ \frac{6562687944}{103615}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{35311}{72944960} \alpha_{1} ^{5} \) \(\mathstrut+ \frac{8369}{6631360} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{54763727}{18236240} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{261805109}{18236240} \alpha_{1} ^{2} \) \(\mathstrut- \frac{3621202594}{1139765} \alpha_{1} \) \(\mathstrut- \frac{645684012}{103615}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{139129}{145889920} \alpha_{1} ^{5} \) \(\mathstrut+ \frac{677949}{13262720} \alpha_{1} ^{4} \) \(\mathstrut- \frac{15010773}{2279530} \alpha_{1} ^{3} \) \(\mathstrut- \frac{13190551511}{36472480} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{55888473039}{9118120} \alpha_{1} \) \(\mathstrut+ \frac{37346012609}{103615}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 4096 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{27871}{12872640} \alpha_{1} ^{5} \) \(\mathstrut- \frac{49637}{390080} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{50076907}{3218160} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{2964044189}{3218160} \alpha_{1} ^{2} \) \(\mathstrut- \frac{3676227704}{201135} \alpha_{1} \) \(\mathstrut- \frac{5767294089}{6095}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
17.12.1.b 8 $\Q(\alpha_{ 2 })$ $q + \ldots^\ast$

${}^\ast$: 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 $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{6} \) \(\mathstrut +\mathstrut 9 x ^{5} \) \(\mathstrut -\mathstrut 8410 x ^{4} \) \(\mathstrut -\mathstrut 88580 x ^{3} \) \(\mathstrut +\mathstrut 18705368 x ^{2} \) \(\mathstrut +\mathstrut 99820416 x \) \(\mathstrut -\mathstrut 12230355456\)
$\Q(\alpha_{ 2 })$ \(x ^{8} \) \(\mathstrut -\mathstrut 55 x ^{7} \) \(\mathstrut -\mathstrut 12058 x ^{6} \) \(\mathstrut +\mathstrut 726176 x ^{5} \) \(\mathstrut +\mathstrut 33040416 x ^{4} \) \(\mathstrut -\mathstrut 2383120192 x ^{3} \) \(\mathstrut +\mathstrut 15657421696 x ^{2} \) \(\mathstrut +\mathstrut 351034071040 x \) \(\mathstrut +\mathstrut 166084534272\)

Decomposition of \( S_{12}^{\mathrm{old}}(17) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(17) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 2 } $