# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{10}^{\mathrm{new}}(22)$ into irreducible Hecke orbits

magma: S := CuspForms(22,10);
magma: N := Newforms(S);
sage: N = Newforms(22,10,names="a")
Label Dimension Field $q$-expansion of eigenform
22.10.1.a 1 $\Q$ $q$ $\mathstrut+$ $16q^{2}$ $\mathstrut-$ $41q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut-$ $1039q^{5}$ $\mathstrut-$ $656q^{6}$ $\mathstrut-$ $3482q^{7}$ $\mathstrut+$ $4096q^{8}$ $\mathstrut-$ $18002q^{9}$ $\mathstrut+O(q^{10})$
22.10.1.b 1 $\Q$ $q$ $\mathstrut+$ $16q^{2}$ $\mathstrut+$ $137q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut-$ $595q^{5}$ $\mathstrut+$ $2192q^{6}$ $\mathstrut+$ $11354q^{7}$ $\mathstrut+$ $4096q^{8}$ $\mathstrut-$ $914q^{9}$ $\mathstrut+O(q^{10})$
22.10.1.c 1 $\Q$ $q$ $\mathstrut+$ $16q^{2}$ $\mathstrut+$ $201q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $2349q^{5}$ $\mathstrut+$ $3216q^{6}$ $\mathstrut-$ $8806q^{7}$ $\mathstrut+$ $4096q^{8}$ $\mathstrut+$ $20718q^{9}$ $\mathstrut+O(q^{10})$
22.10.1.d 2 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut-$ $16q^{2}$ $\mathstrut+$ $\bigl(\alpha_{4}$ $\mathstrut+ 16\bigr)q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $\bigl(\frac{97}{9} \alpha_{4}$ $\mathstrut+ \frac{226}{9}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 16 \alpha_{4}$ $\mathstrut- 256\bigr)q^{6}$ $\mathstrut+$ $\bigl(\frac{110}{3} \alpha_{4}$ $\mathstrut- \frac{8320}{3}\bigr)q^{7}$ $\mathstrut-$ $4096q^{8}$ $\mathstrut+$ $\bigl(- 21 \alpha_{4}$ $\mathstrut- 2127\bigr)q^{9}$ $\mathstrut+O(q^{10})$
22.10.1.e 2 $\Q(\alpha_{ 5 })$ $q$ $\mathstrut-$ $16q^{2}$ $\mathstrut+$ $\bigl(\alpha_{5}$ $\mathstrut+ 16\bigr)q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $\bigl(- 10 \alpha_{5}$ $\mathstrut- 729\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 16 \alpha_{5}$ $\mathstrut- 256\bigr)q^{6}$ $\mathstrut+$ $\bigl(- 11 \alpha_{5}$ $\mathstrut+ 4109\bigr)q^{7}$ $\mathstrut-$ $4096q^{8}$ $\mathstrut+$ $\bigl(34 \alpha_{5}$ $\mathstrut+ 10204\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 4 })\cong$ $\Q(\sqrt{889})$ $x ^{2}$ $\mathstrut +\mathstrut 53 x$ $\mathstrut -\mathstrut 17300$
$\Q(\alpha_{ 5 })\cong$ $\Q(\sqrt{463})$ $x ^{2}$ $\mathstrut -\mathstrut 2 x$ $\mathstrut -\mathstrut 29631$

## Decomposition of $S_{10}^{\mathrm{old}}(22)$ into lower level spaces

$S_{10}^{\mathrm{old}}(22)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/11/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(11)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(2)) }^{\oplus 2 }$