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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

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