# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{10}^{\mathrm{new}}(15)$ into irreducible Hecke orbits

magma: S := CuspForms(15,10);
magma: N := Newforms(S);
sage: N = Newforms(15,10,names="a")
Label Dimension Field $q$-expansion of eigenform
15.10.1.a 1 $\Q$ $q$ $\mathstrut-$ $4q^{2}$ $\mathstrut+$ $81q^{3}$ $\mathstrut-$ $496q^{4}$ $\mathstrut+$ $625q^{5}$ $\mathstrut-$ $324q^{6}$ $\mathstrut-$ $7680q^{7}$ $\mathstrut+$ $4032q^{8}$ $\mathstrut+$ $6561q^{9}$ $\mathstrut+O(q^{10})$
15.10.1.b 1 $\Q$ $q$ $\mathstrut+$ $22q^{2}$ $\mathstrut-$ $81q^{3}$ $\mathstrut-$ $28q^{4}$ $\mathstrut-$ $625q^{5}$ $\mathstrut-$ $1782q^{6}$ $\mathstrut-$ $5988q^{7}$ $\mathstrut-$ $11880q^{8}$ $\mathstrut+$ $6561q^{9}$ $\mathstrut+O(q^{10})$
15.10.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut+$ $81q^{3}$ $\mathstrut+$ $\bigl(19 \alpha_{3}$ $\mathstrut+ 580\bigr)q^{4}$ $\mathstrut-$ $625q^{5}$ $\mathstrut+$ $81 \alpha_{3} q^{6}$ $\mathstrut+$ $\bigl(- 56 \alpha_{3}$ $\mathstrut- 5404\bigr)q^{7}$ $\mathstrut+$ $\bigl(429 \alpha_{3}$ $\mathstrut+ 20748\bigr)q^{8}$ $\mathstrut+$ $6561q^{9}$ $\mathstrut+O(q^{10})$
15.10.1.d 2 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut+$ $\alpha_{4} q^{2}$ $\mathstrut-$ $81q^{3}$ $\mathstrut+$ $\bigl(31 \alpha_{4}$ $\mathstrut- 210\bigr)q^{4}$ $\mathstrut+$ $625q^{5}$ $\mathstrut-$ $81 \alpha_{4} q^{6}$ $\mathstrut+$ $\bigl(224 \alpha_{4}$ $\mathstrut+ 3584\bigr)q^{7}$ $\mathstrut+$ $\bigl(239 \alpha_{4}$ $\mathstrut+ 9362\bigr)q^{8}$ $\mathstrut+$ $6561q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{4729})$ $x ^{2}$ $\mathstrut -\mathstrut 19 x$ $\mathstrut -\mathstrut 1092$
$\Q(\alpha_{ 4 })\cong$ $\Q(\sqrt{241})$ $x ^{2}$ $\mathstrut -\mathstrut 31 x$ $\mathstrut -\mathstrut 302$

## Decomposition of $S_{10}^{\mathrm{old}}(15)$ into lower level spaces

$S_{10}^{\mathrm{old}}(15)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/5/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(3)) }^{\oplus 2 }$