# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{10}^{\mathrm{new}}(7)$ into irreducible Hecke orbits

magma: S := CuspForms(7,10);
magma: N := Newforms(S);
sage: N = Newforms(7,10,names="a")
Label Dimension Field $q$-expansion of eigenform
7.10.1.a 2 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(- 11 \alpha_{1}$ $\mathstrut- 76\bigr)q^{3}$ $\mathstrut+$ $\bigl(- 6 \alpha_{1}$ $\mathstrut- 328\bigr)q^{4}$ $\mathstrut+$ $\bigl(95 \alpha_{1}$ $\mathstrut- 834\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 10 \alpha_{1}$ $\mathstrut- 2024\bigr)q^{6}$ $\mathstrut-$ $2401q^{7}$ $\mathstrut+$ $\bigl(- 804 \alpha_{1}$ $\mathstrut- 1104\bigr)q^{8}$ $\mathstrut+$ $\bigl(946 \alpha_{1}$ $\mathstrut+ 8357\bigr)q^{9}$ $\mathstrut+O(q^{10})$
7.10.1.b 3 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{1}{7} \alpha_{2} ^{2}$ $\mathstrut+ \frac{15}{7} \alpha_{2}$ $\mathstrut+ \frac{1122}{7}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 512\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{13}{7} \alpha_{2} ^{2}$ $\mathstrut- \frac{197}{7} \alpha_{2}$ $\mathstrut+ \frac{18408}{7}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{6}{7} \alpha_{2} ^{2}$ $\mathstrut- \frac{204}{7} \alpha_{2}$ $\mathstrut+ \frac{19080}{7}\bigr)q^{6}$ $\mathstrut+$ $2401q^{7}$ $\mathstrut+$ $\bigl(21 \alpha_{2} ^{2}$ $\mathstrut+ 302 \alpha_{2}$ $\mathstrut- 19080\bigr)q^{8}$ $\mathstrut+$ $\bigl(- 18 \alpha_{2} ^{2}$ $\mathstrut+ 54 \alpha_{2}$ $\mathstrut+ 9513\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ $\Q(\sqrt{193})$ $x ^{2}$ $\mathstrut +\mathstrut 6 x$ $\mathstrut -\mathstrut 184$
$\Q(\alpha_{ 2 })$ $x ^{3}$ $\mathstrut -\mathstrut 21 x ^{2}$ $\mathstrut -\mathstrut 1326 x$ $\mathstrut +\mathstrut 19080$