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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

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