Related objects

Learn more about

Show commands for: Magma / SageMath

Decomposition of \( S_{8}^{\mathrm{new}}(7) \) into irreducible Hecke orbits

magma: S := CuspForms(7,8);
magma: N := Newforms(S);
sage: N = Newforms(7,8,names="a")
Label Dimension Field $q$-expansion of eigenform
7.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(6q^{2} \) \(\mathstrut-\) \(42q^{3} \) \(\mathstrut-\) \(92q^{4} \) \(\mathstrut-\) \(84q^{5} \) \(\mathstrut+\) \(252q^{6} \) \(\mathstrut+\) \(343q^{7} \) \(\mathstrut+\) \(1320q^{8} \) \(\mathstrut-\) \(423q^{9} \) \(\mathstrut+O(q^{10}) \)
7.8.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- 2 \alpha_{2} \) \(\mathstrut+ 44\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{2} \) \(\mathstrut+ 86\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- 10 \alpha_{2} \) \(\mathstrut+ 150\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(50 \alpha_{2} \) \(\mathstrut- 428\bigr)q^{6} \) \(\mathstrut-\) \(343q^{7} \) \(\mathstrut+\) \(\bigl(- 33 \alpha_{2} \) \(\mathstrut- 642\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 188 \alpha_{2} \) \(\mathstrut+ 605\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{865}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut -\mathstrut 214\)