Related objects

Learn more about

Show commands for: Magma / SageMath

Decomposition of \( S_{12}^{\mathrm{new}}(11) \) into irreducible Hecke orbits

magma: S := CuspForms(11,12);
magma: N := Newforms(S);
sage: N = Newforms(11,12,names="a")
Label Dimension Field $q$-expansion of eigenform
11.12.1.a 3 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{4}{17} \alpha_{1} ^{2} \) \(\mathstrut- \frac{110}{17} \alpha_{1} \) \(\mathstrut+ \frac{9773}{17}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{90}{17} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{1540}{17} \alpha_{1} \) \(\mathstrut- \frac{311395}{17}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{110}{17} \alpha_{1} ^{2} \) \(\mathstrut- \frac{8227}{17} \alpha_{1} \) \(\mathstrut+ \frac{368896}{17}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{440}{17} \alpha_{1} ^{2} \) \(\mathstrut- \frac{21110}{17} \alpha_{1} \) \(\mathstrut+ \frac{1291202}{17}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(404 \alpha_{1} \) \(\mathstrut- 92224\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{348}{17} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{19668}{17} \alpha_{1} \) \(\mathstrut- \frac{2167122}{17}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
11.12.1.b 5 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{36792} \alpha_{2} ^{4} \) \(\mathstrut- \frac{1}{252} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{2503}{6132} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{86591}{4599} \alpha_{2} \) \(\mathstrut- \frac{3160763}{4599}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{17}{36792} \alpha_{2} ^{4} \) \(\mathstrut- \frac{13}{252} \alpha_{2} ^{3} \) \(\mathstrut- \frac{6689}{2044} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{1427732}{4599} \alpha_{2} \) \(\mathstrut+ \frac{12880711}{4599}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{89}{18396} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{25}{126} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{69467}{3066} \alpha_{2} ^{2} \) \(\mathstrut- \frac{1661575}{4599} \alpha_{2} \) \(\mathstrut- \frac{10945616}{4599}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{8}{657} \alpha_{2} ^{4} \) \(\mathstrut- \frac{7}{18} \alpha_{2} ^{3} \) \(\mathstrut- \frac{5441}{73} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{975835}{657} \alpha_{2} \) \(\mathstrut+ \frac{46021382}{657}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 4096 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{43}{5256} \alpha_{2} ^{4} \) \(\mathstrut- \frac{25}{36} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{48353}{876} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{1505600}{657} \alpha_{2} \) \(\mathstrut+ \frac{61715638}{657}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ 3.3.202533.1 \(x ^{3} \) \(\mathstrut -\mathstrut 4500 x \) \(\mathstrut +\mathstrut 92224\)
$\Q(\alpha_{ 2 })$ \(x ^{5} \) \(\mathstrut -\mathstrut 32 x ^{4} \) \(\mathstrut -\mathstrut 7718 x ^{3} \) \(\mathstrut +\mathstrut 140876 x ^{2} \) \(\mathstrut +\mathstrut 11993504 x \) \(\mathstrut -\mathstrut 87564928\)

Decomposition of \( S_{12}^{\mathrm{old}}(11) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(11) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 2 } $