# Related objects

Show commands for: Magma / SageMath

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