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Decomposition of \( S_{12}^{\mathrm{new}}(22) \) into irreducible Hecke orbits

magma: S := CuspForms(22,12);
magma: N := Newforms(S);
sage: N = Newforms(22,12,names="a")
Label Dimension Field $q$-expansion of eigenform
22.12.1.a 2 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(32q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{2} \alpha_{1} \) \(\mathstrut+ 16\bigr)q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(\bigl(\frac{25}{3} \alpha_{1} \) \(\mathstrut- \frac{8015}{3}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{1} \) \(\mathstrut+ 512\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{55}{2} \alpha_{1} \) \(\mathstrut- 55757\bigr)q^{7} \) \(\mathstrut+\) \(32768q^{8} \) \(\mathstrut+\) \(\bigl(213 \alpha_{1} \) \(\mathstrut- 38676\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
22.12.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut-\) \(32q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1}{2} \alpha_{2} \) \(\mathstrut+ 16\bigr)q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(\bigl(\frac{1}{420} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{9}{7} \alpha_{2} \) \(\mathstrut- \frac{542314}{105}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{2} \) \(\mathstrut- 512\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{1}{20} \alpha_{2} ^{2} \) \(\mathstrut- \frac{129}{2} \alpha_{2} \) \(\mathstrut+ \frac{279764}{5}\bigr)q^{7} \) \(\mathstrut-\) \(32768q^{8} \) \(\mathstrut+\) \(\bigl(\frac{1}{4} \alpha_{2} ^{2} \) \(\mathstrut+ 16 \alpha_{2} \) \(\mathstrut- 176891\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
22.12.1.c 3 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut-\) \(32q^{2} \) \(\mathstrut+\) \(\bigl(\alpha_{3} \) \(\mathstrut+ 32\bigr)q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(\bigl(\frac{19}{81} \alpha_{3} ^{2} \) \(\mathstrut- \frac{1424}{27} \alpha_{3} \) \(\mathstrut- \frac{2236258}{81}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 32 \alpha_{3} \) \(\mathstrut- 1024\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{47}{81} \alpha_{3} ^{2} \) \(\mathstrut+ \frac{7969}{27} \alpha_{3} \) \(\mathstrut+ \frac{7002236}{81}\bigr)q^{7} \) \(\mathstrut-\) \(32768q^{8} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{2} \) \(\mathstrut+ 64 \alpha_{3} \) \(\mathstrut- 176123\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
22.12.1.d 3 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(32q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{2} \alpha_{4} \) \(\mathstrut+ 16\bigr)q^{3} \) \(\mathstrut+\) \(1024q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{1}{108} \alpha_{4} ^{2} \) \(\mathstrut- \frac{151}{27} \alpha_{4} \) \(\mathstrut+ \frac{81478}{9}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 16 \alpha_{4} \) \(\mathstrut+ 512\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{1}{108} \alpha_{4} ^{2} \) \(\mathstrut+ \frac{59}{54} \alpha_{4} \) \(\mathstrut+ \frac{79460}{9}\bigr)q^{7} \) \(\mathstrut+\) \(32768q^{8} \) \(\mathstrut+\) \(\bigl(\frac{1}{4} \alpha_{4} ^{2} \) \(\mathstrut- 16 \alpha_{4} \) \(\mathstrut- 176891\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{331}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 916 x \) \(\mathstrut -\mathstrut 552860\)
$\Q(\alpha_{ 2 })$ \(x ^{3} \) \(\mathstrut +\mathstrut 236 x ^{2} \) \(\mathstrut -\mathstrut 2108716 x \) \(\mathstrut +\mathstrut 575269024\)
$\Q(\alpha_{ 3 })$ \(x ^{3} \) \(\mathstrut -\mathstrut 10 x ^{2} \) \(\mathstrut -\mathstrut 214591 x \) \(\mathstrut -\mathstrut 35229956\)
$\Q(\alpha_{ 4 })$ \(x ^{3} \) \(\mathstrut +\mathstrut 644 x ^{2} \) \(\mathstrut -\mathstrut 1557836 x \) \(\mathstrut -\mathstrut 458689440\)

Decomposition of \( S_{12}^{\mathrm{old}}(22) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(22) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/11/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(11)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 } $