# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{6}^{\mathrm{new}}(13)$ into irreducible Hecke orbits

magma: S := CuspForms(13,6);
magma: N := Newforms(S);
sage: N = Newforms(13,6,names="a")
Label Dimension Field $q$-expansion of eigenform
13.6.1.a 2 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(- 6 \alpha_{1}$ $\mathstrut- 29\bigr)q^{3}$ $\mathstrut+$ $\bigl(- 5 \alpha_{1}$ $\mathstrut- 34\bigr)q^{4}$ $\mathstrut+$ $\bigl(40 \alpha_{1}$ $\mathstrut+ 79\bigr)q^{5}$ $\mathstrut+$ $\bigl(\alpha_{1}$ $\mathstrut+ 12\bigr)q^{6}$ $\mathstrut+$ $\bigl(- 70 \alpha_{1}$ $\mathstrut- 193\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 41 \alpha_{1}$ $\mathstrut+ 10\bigr)q^{8}$ $\mathstrut+$ $\bigl(168 \alpha_{1}$ $\mathstrut+ 526\bigr)q^{9}$ $\mathstrut+O(q^{10})$
13.6.1.b 3 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{4} \alpha_{2} ^{2}$ $\mathstrut- \frac{3}{4} \alpha_{2}$ $\mathstrut+ \frac{45}{2}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 32\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{1}{4} \alpha_{2} ^{2}$ $\mathstrut- \frac{27}{4} \alpha_{2}$ $\mathstrut+ \frac{105}{2}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{5}{2} \alpha_{2} ^{2}$ $\mathstrut+ \frac{3}{2} \alpha_{2}$ $\mathstrut+ 111\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{3}{4} \alpha_{2} ^{2}$ $\mathstrut- \frac{9}{4} \alpha_{2}$ $\mathstrut+ \frac{79}{2}\bigr)q^{7}$ $\mathstrut+$ $\bigl(7 \alpha_{2} ^{2}$ $\mathstrut+ 20 \alpha_{2}$ $\mathstrut- 444\bigr)q^{8}$ $\mathstrut+$ $\bigl(\frac{1}{4} \alpha_{2} ^{2}$ $\mathstrut+ \frac{27}{4} \alpha_{2}$ $\mathstrut- \frac{195}{2}\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ $\Q(\sqrt{17})$ $x ^{2}$ $\mathstrut +\mathstrut 5 x$ $\mathstrut +\mathstrut 2$
$\Q(\alpha_{ 2 })\cong$ 3.3.168897.1 $x ^{3}$ $\mathstrut -\mathstrut 7 x ^{2}$ $\mathstrut -\mathstrut 84 x$ $\mathstrut +\mathstrut 444$