# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(17)$ into irreducible Hecke orbits

magma: S := CuspForms(17,8);
magma: N := Newforms(S);
sage: N = Newforms(17,8,names="a")
Label Dimension Field $q$-expansion of eigenform
17.8.1.a 1 $\Q$ $q$ $\mathstrut-$ $2q^{2}$ $\mathstrut+$ $18q^{3}$ $\mathstrut-$ $124q^{4}$ $\mathstrut-$ $10q^{5}$ $\mathstrut-$ $36q^{6}$ $\mathstrut-$ $902q^{7}$ $\mathstrut+$ $504q^{8}$ $\mathstrut-$ $1863q^{9}$ $\mathstrut+O(q^{10})$
17.8.1.b 3 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{4} \alpha_{2} ^{2}$ $\mathstrut- \frac{17}{4} \alpha_{2}$ $\mathstrut+ \frac{47}{2}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 128\bigr)q^{4}$ $\mathstrut+$ $\bigl(2 \alpha_{2} ^{2}$ $\mathstrut+ 6 \alpha_{2}$ $\mathstrut- 474\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{9}{2} \alpha_{2} ^{2}$ $\mathstrut- \frac{105}{2} \alpha_{2}$ $\mathstrut+ 423\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{35}{4} \alpha_{2} ^{2}$ $\mathstrut- \frac{259}{4} \alpha_{2}$ $\mathstrut+ \frac{2557}{2}\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut+ 48 \alpha_{2}$ $\mathstrut- 1692\bigr)q^{8}$ $\mathstrut+$ $\bigl(\frac{55}{2} \alpha_{2} ^{2}$ $\mathstrut+ \frac{719}{2} \alpha_{2}$ $\mathstrut- 5336\bigr)q^{9}$ $\mathstrut+O(q^{10})$
17.8.1.c 6 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $\bigl(\frac{119}{214848} \alpha_{3} ^{5}$ $\mathstrut- \frac{2237}{214848} \alpha_{3} ^{4}$ $\mathstrut- \frac{6559}{35808} \alpha_{3} ^{3}$ $\mathstrut+ \frac{18353}{6714} \alpha_{3} ^{2}$ $\mathstrut+ \frac{21449}{1492} \alpha_{3}$ $\mathstrut- \frac{40069}{373}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{3} ^{2}$ $\mathstrut- 128\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{13}{11936} \alpha_{3} ^{5}$ $\mathstrut+ \frac{119}{11936} \alpha_{3} ^{4}$ $\mathstrut+ \frac{3115}{5968} \alpha_{3} ^{3}$ $\mathstrut- \frac{1773}{373} \alpha_{3} ^{2}$ $\mathstrut- \frac{25063}{746} \alpha_{3}$ $\mathstrut+ \frac{176304}{373}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{113}{53712} \alpha_{3} ^{5}$ $\mathstrut+ \frac{5453}{53712} \alpha_{3} ^{4}$ $\mathstrut- \frac{467}{2238} \alpha_{3} ^{3}$ $\mathstrut- \frac{428377}{13428} \alpha_{3} ^{2}$ $\mathstrut+ \frac{47158}{373} \alpha_{3}$ $\mathstrut+ \frac{882742}{373}\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{51}{23872} \alpha_{3} ^{5}$ $\mathstrut- \frac{1599}{23872} \alpha_{3} ^{4}$ $\mathstrut+ \frac{23353}{11936} \alpha_{3} ^{3}$ $\mathstrut+ \frac{22827}{746} \alpha_{3} ^{2}$ $\mathstrut- \frac{445501}{1492} \alpha_{3}$ $\mathstrut- \frac{917983}{373}\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{3} ^{3}$ $\mathstrut- 256 \alpha_{3} \bigr)q^{8}$ $\mathstrut+$ $\bigl(\frac{169}{11936} \alpha_{3} ^{5}$ $\mathstrut- \frac{1547}{11936} \alpha_{3} ^{4}$ $\mathstrut- \frac{40495}{5968} \alpha_{3} ^{3}$ $\mathstrut+ \frac{8129}{373} \alpha_{3} ^{2}$ $\mathstrut+ \frac{480987}{746} \alpha_{3}$ $\mathstrut+ \frac{844605}{373}\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 3.3.694349.1 $x ^{3}$ $\mathstrut -\mathstrut x ^{2}$ $\mathstrut -\mathstrut 304 x$ $\mathstrut +\mathstrut 1692$
$\Q(\alpha_{ 3 })$ $x ^{6}$ $\mathstrut -\mathstrut 15 x ^{5}$ $\mathstrut -\mathstrut 514 x ^{4}$ $\mathstrut +\mathstrut 5312 x ^{3}$ $\mathstrut +\mathstrut 83552 x ^{2}$ $\mathstrut -\mathstrut 422208 x$ $\mathstrut -\mathstrut 4272768$