# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{12}^{\mathrm{new}}(21)$ into irreducible Hecke orbits

magma: S := CuspForms(21,12);
magma: N := Newforms(S);
sage: N = Newforms(21,12,names="a")
Label Dimension Field $q$-expansion of eigenform
21.12.1.a 1 $\Q$ $q$ $\mathstrut-$ $62q^{2}$ $\mathstrut+$ $243q^{3}$ $\mathstrut+$ $1796q^{4}$ $\mathstrut-$ $3310q^{5}$ $\mathstrut-$ $15066q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut+$ $15624q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
21.12.1.b 1 $\Q$ $q$ $\mathstrut+$ $8q^{2}$ $\mathstrut+$ $243q^{3}$ $\mathstrut-$ $1984q^{4}$ $\mathstrut+$ $4390q^{5}$ $\mathstrut+$ $1944q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut-$ $32256q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
21.12.1.c 3 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut-$ $243q^{3}$ $\mathstrut+$ $\bigl(\alpha_{3} ^{2}$ $\mathstrut- 2048\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{37}{4} \alpha_{3} ^{2}$ $\mathstrut+ 551 \alpha_{3}$ $\mathstrut- 11266\bigr)q^{5}$ $\mathstrut-$ $243 \alpha_{3} q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut+$ $\bigl(- 68 \alpha_{3} ^{2}$ $\mathstrut- 2376 \alpha_{3}$ $\mathstrut+ 7232\bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
21.12.1.d 3 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut+$ $\alpha_{4} q^{2}$ $\mathstrut-$ $243q^{3}$ $\mathstrut+$ $\bigl(\alpha_{4} ^{2}$ $\mathstrut- 2048\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{46}{13} \alpha_{4} ^{2}$ $\mathstrut+ \frac{310}{13} \alpha_{4}$ $\mathstrut+ \frac{122790}{13}\bigr)q^{5}$ $\mathstrut-$ $243 \alpha_{4} q^{6}$ $\mathstrut+$ $16807q^{7}$ $\mathstrut+$ $\bigl(- 33 \alpha_{4} ^{2}$ $\mathstrut- 1186 \alpha_{4}$ $\mathstrut+ 69840\bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
21.12.1.e 4 $\Q(\alpha_{ 5 })$ $q$ $\mathstrut+$ $\alpha_{5} q^{2}$ $\mathstrut+$ $243q^{3}$ $\mathstrut+$ $\bigl(\alpha_{5} ^{2}$ $\mathstrut- 2048\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{1}{12} \alpha_{5} ^{3}$ $\mathstrut+ \frac{15}{4} \alpha_{5} ^{2}$ $\mathstrut+ 311 \alpha_{5}$ $\mathstrut- 4698\bigr)q^{5}$ $\mathstrut+$ $243 \alpha_{5} q^{6}$ $\mathstrut+$ $16807q^{7}$ $\mathstrut+$ $\bigl(\alpha_{5} ^{3}$ $\mathstrut- 4096 \alpha_{5} \bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })$ $x ^{3}$ $\mathstrut +\mathstrut 68 x ^{2}$ $\mathstrut -\mathstrut 1720 x$ $\mathstrut -\mathstrut 7232$
$\Q(\alpha_{ 4 })$ $x ^{3}$ $\mathstrut +\mathstrut 33 x ^{2}$ $\mathstrut -\mathstrut 2910 x$ $\mathstrut -\mathstrut 69840$
$\Q(\alpha_{ 5 })$ $x ^{4}$ $\mathstrut -\mathstrut 45 x ^{3}$ $\mathstrut -\mathstrut 5484 x ^{2}$ $\mathstrut +\mathstrut 154872 x$ $\mathstrut +\mathstrut 3433536$

## Decomposition of $S_{12}^{\mathrm{old}}(21)$ into lower level spaces

$S_{12}^{\mathrm{old}}(21)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/7/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 }$