# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{12}^{\mathrm{new}}(7)$ into irreducible Hecke orbits

magma: S := CuspForms(7,12);
magma: N := Newforms(S);
sage: N = Newforms(7,12,names="a")
Label Dimension Field $q$-expansion of eigenform
7.12.1.a 2 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(- 6 \alpha_{1}$ $\mathstrut- 102\bigr)q^{3}$ $\mathstrut+$ $\bigl(- 54 \alpha_{1}$ $\mathstrut+ 592\bigr)q^{4}$ $\mathstrut+$ $\bigl(10 \alpha_{1}$ $\mathstrut- 6480\bigr)q^{5}$ $\mathstrut+$ $\bigl(222 \alpha_{1}$ $\mathstrut- 15840\bigr)q^{6}$ $\mathstrut+$ $16807q^{7}$ $\mathstrut+$ $\bigl(1460 \alpha_{1}$ $\mathstrut- 142560\bigr)q^{8}$ $\mathstrut+$ $\bigl(- 720 \alpha_{1}$ $\mathstrut- 71703\bigr)q^{9}$ $\mathstrut+O(q^{10})$
7.12.1.b 3 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $\bigl(- \frac{11}{21} \alpha_{2} ^{2}$ $\mathstrut+ \frac{103}{7} \alpha_{2}$ $\mathstrut+ \frac{33758}{21}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{2} ^{2}$ $\mathstrut- 2048\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{59}{7} \alpha_{2} ^{2}$ $\mathstrut- \frac{517}{7} \alpha_{2}$ $\mathstrut- \frac{203864}{7}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{538}{21} \alpha_{2} ^{2}$ $\mathstrut+ \frac{788}{7} \alpha_{2}$ $\mathstrut+ \frac{2476144}{21}\bigr)q^{6}$ $\mathstrut-$ $16807q^{7}$ $\mathstrut+$ $\bigl(77 \alpha_{2} ^{2}$ $\mathstrut- 1242 \alpha_{2}$ $\mathstrut- 225104\bigr)q^{8}$ $\mathstrut+$ $\bigl(- \frac{734}{3} \alpha_{2} ^{2}$ $\mathstrut+ 1846 \alpha_{2}$ $\mathstrut+ \frac{3363563}{3}\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ $\Q(\sqrt{3369})$ $x ^{2}$ $\mathstrut +\mathstrut 54 x$ $\mathstrut -\mathstrut 2640$
$\Q(\alpha_{ 2 })$ $x ^{3}$ $\mathstrut -\mathstrut 77 x ^{2}$ $\mathstrut -\mathstrut 2854 x$ $\mathstrut +\mathstrut 225104$

## Decomposition of $S_{12}^{\mathrm{old}}(7)$ into lower level spaces

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