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Decomposition of \( S_{10}^{\mathrm{new}}(11) \) into irreducible Hecke orbits

magma: S := CuspForms(11,10);
magma: N := Newforms(S);
sage: N = Newforms(11,10,names="a")
Label Dimension Field $q$-expansion of eigenform
11.10.1.a 3 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{8} \alpha_{1} ^{2} \) \(\mathstrut- \frac{11}{2} \alpha_{1} \) \(\mathstrut+ 40\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{3}{8} \alpha_{1} ^{2} \) \(\mathstrut- \frac{77}{2} \alpha_{1} \) \(\mathstrut- 302\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{11}{2} \alpha_{1} ^{2} \) \(\mathstrut- 113 \alpha_{1} \) \(\mathstrut+ 836\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{11}{4} \alpha_{1} ^{2} \) \(\mathstrut+ 155 \alpha_{1} \) \(\mathstrut- 4664\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(200 \alpha_{1} \) \(\mathstrut- 6688\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{315}{8} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{2277}{2} \alpha_{1} \) \(\mathstrut- 27279\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
11.10.1.b 5 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1}{312} \alpha_{2} ^{4} \) \(\mathstrut- \frac{1}{12} \alpha_{2} ^{3} \) \(\mathstrut- \frac{571}{156} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{787}{13} \alpha_{2} \) \(\mathstrut+ \frac{14127}{13}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{29}{1560} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{9}{20} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{837}{52} \alpha_{2} ^{2} \) \(\mathstrut- \frac{44354}{195} \alpha_{2} \) \(\mathstrut- \frac{163411}{65}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{5}{156} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{7}{6} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{3115}{78} \alpha_{2} ^{2} \) \(\mathstrut- \frac{35063}{39} \alpha_{2} \) \(\mathstrut- \frac{176968}{13}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{7}{78} \alpha_{2} ^{4} \) \(\mathstrut- \frac{7}{2} \alpha_{2} ^{3} \) \(\mathstrut- \frac{1120}{13} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{92953}{39} \alpha_{2} \) \(\mathstrut+ \frac{415030}{13}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 1024 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{5}{312} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{37}{12} \alpha_{2} ^{3} \) \(\mathstrut- \frac{15491}{156} \alpha_{2} ^{2} \) \(\mathstrut- \frac{28734}{13} \alpha_{2} \) \(\mathstrut+ \frac{536802}{13}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ 3.3.2659452.1 \(x ^{3} \) \(\mathstrut -\mathstrut 1224 x \) \(\mathstrut +\mathstrut 6688\)
$\Q(\alpha_{ 2 })$ \(x ^{5} \) \(\mathstrut -\mathstrut 16 x ^{4} \) \(\mathstrut -\mathstrut 1506 x ^{3} \) \(\mathstrut +\mathstrut 6428 x ^{2} \) \(\mathstrut +\mathstrut 619552 x \) \(\mathstrut +\mathstrut 4247232\)