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Decomposition of \( S_{10}^{\mathrm{new}}(7) \) into irreducible Hecke orbits

magma: S := CuspForms(7,10);
magma: N := Newforms(S);
sage: N = Newforms(7,10,names="a")
Label Dimension Field $q$-expansion of eigenform
7.10.1.a 2 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- 11 \alpha_{1} \) \(\mathstrut- 76\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 6 \alpha_{1} \) \(\mathstrut- 328\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(95 \alpha_{1} \) \(\mathstrut- 834\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- 10 \alpha_{1} \) \(\mathstrut- 2024\bigr)q^{6} \) \(\mathstrut-\) \(2401q^{7} \) \(\mathstrut+\) \(\bigl(- 804 \alpha_{1} \) \(\mathstrut- 1104\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(946 \alpha_{1} \) \(\mathstrut+ 8357\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
7.10.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{7} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{15}{7} \alpha_{2} \) \(\mathstrut+ \frac{1122}{7}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{13}{7} \alpha_{2} ^{2} \) \(\mathstrut- \frac{197}{7} \alpha_{2} \) \(\mathstrut+ \frac{18408}{7}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{6}{7} \alpha_{2} ^{2} \) \(\mathstrut- \frac{204}{7} \alpha_{2} \) \(\mathstrut+ \frac{19080}{7}\bigr)q^{6} \) \(\mathstrut+\) \(2401q^{7} \) \(\mathstrut+\) \(\bigl(21 \alpha_{2} ^{2} \) \(\mathstrut+ 302 \alpha_{2} \) \(\mathstrut- 19080\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 18 \alpha_{2} ^{2} \) \(\mathstrut+ 54 \alpha_{2} \) \(\mathstrut+ 9513\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{193}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 184\)
$\Q(\alpha_{ 2 })$ \(x ^{3} \) \(\mathstrut -\mathstrut 21 x ^{2} \) \(\mathstrut -\mathstrut 1326 x \) \(\mathstrut +\mathstrut 19080\)