# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{8}^{\mathrm{new}}(13)$ into irreducible Hecke orbits

magma: S := CuspForms(13,8);
magma: N := Newforms(S);
sage: N = Newforms(13,8,names="a")
Label Dimension Field $q$-expansion of eigenform
13.8.1.a 1 $\Q$ $q$ $\mathstrut+$ $10q^{2}$ $\mathstrut-$ $73q^{3}$ $\mathstrut-$ $28q^{4}$ $\mathstrut-$ $295q^{5}$ $\mathstrut-$ $730q^{6}$ $\mathstrut+$ $1373q^{7}$ $\mathstrut-$ $1560q^{8}$ $\mathstrut+$ $3142q^{9}$ $\mathstrut+O(q^{10})$
13.8.1.b 2 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- 3 \alpha_{2}$ $\mathstrut- 6\bigr)q^{3}$ $\mathstrut+$ $\bigl(- 19 \alpha_{2}$ $\mathstrut- 134\bigr)q^{4}$ $\mathstrut+$ $\bigl(11 \alpha_{2}$ $\mathstrut- 72\bigr)q^{5}$ $\mathstrut+$ $\bigl(51 \alpha_{2}$ $\mathstrut+ 18\bigr)q^{6}$ $\mathstrut+$ $\bigl(- 9 \alpha_{2}$ $\mathstrut- 1090\bigr)q^{7}$ $\mathstrut+$ $\bigl(99 \alpha_{2}$ $\mathstrut+ 114\bigr)q^{8}$ $\mathstrut+$ $\bigl(- 135 \alpha_{2}$ $\mathstrut- 2205\bigr)q^{9}$ $\mathstrut+O(q^{10})$
13.8.1.c 4 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{4} \alpha_{3} ^{3}$ $\mathstrut- \frac{3}{4} \alpha_{3} ^{2}$ $\mathstrut+ 65 \alpha_{3}$ $\mathstrut+ 278\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{3} ^{2}$ $\mathstrut- 128\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{13}{4} \alpha_{3} ^{3}$ $\mathstrut+ \frac{15}{4} \alpha_{3} ^{2}$ $\mathstrut- 805 \alpha_{3}$ $\mathstrut- 2292\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{9}{2} \alpha_{3} ^{3}$ $\mathstrut- \frac{5}{2} \alpha_{3} ^{2}$ $\mathstrut+ 1094 \alpha_{3}$ $\mathstrut+ 3220\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{3}{4} \alpha_{3} ^{3}$ $\mathstrut- \frac{25}{4} \alpha_{3} ^{2}$ $\mathstrut+ 171 \alpha_{3}$ $\mathstrut+ 2052\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{3} ^{3}$ $\mathstrut- 256 \alpha_{3} \bigr)q^{8}$ $\mathstrut+$ $\bigl(- \frac{67}{4} \alpha_{3} ^{3}$ $\mathstrut- \frac{129}{4} \alpha_{3} ^{2}$ $\mathstrut+ 4139 \alpha_{3}$ $\mathstrut+ 15527\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{337})$ $x ^{2}$ $\mathstrut +\mathstrut 19 x$ $\mathstrut +\mathstrut 6$
$\Q(\alpha_{ 3 })$ $x ^{4}$ $\mathstrut -\mathstrut 15 x ^{3}$ $\mathstrut -\mathstrut 270 x ^{2}$ $\mathstrut +\mathstrut 3264 x$ $\mathstrut +\mathstrut 12880$