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Decomposition of \( S_{4}^{\mathrm{new}}(23) \) into irreducible Hecke orbits

magma: S := CuspForms(23,4);
magma: N := Newforms(S);
sage: N = Newforms(23,4,names="a")
Label Dimension Field $q$-expansion of eigenform
23.4.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(2q^{2} \) \(\mathstrut-\) \(5q^{3} \) \(\mathstrut-\) \(4q^{4} \) \(\mathstrut-\) \(6q^{5} \) \(\mathstrut+\) \(10q^{6} \) \(\mathstrut-\) \(8q^{7} \) \(\mathstrut+\) \(24q^{8} \) \(\mathstrut-\) \(2q^{9} \) \(\mathstrut+O(q^{10}) \)
23.4.1.b 4 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{11} \alpha_{2} ^{3} \) \(\mathstrut- \frac{5}{11} \alpha_{2} ^{2} \) \(\mathstrut+ \alpha_{2} \) \(\mathstrut+ \frac{71}{11}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 8\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{10}{11} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{6}{11} \alpha_{2} ^{2} \) \(\mathstrut- 20 \alpha_{2} \) \(\mathstrut+ \frac{148}{11}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{7}{11} \alpha_{2} ^{3} \) \(\mathstrut- \frac{13}{11} \alpha_{2} ^{2} \) \(\mathstrut+ 12 \alpha_{2} \) \(\mathstrut+ \frac{2}{11}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{20}{11} \alpha_{2} ^{3} \) \(\mathstrut- \frac{12}{11} \alpha_{2} ^{2} \) \(\mathstrut+ 34 \alpha_{2} \) \(\mathstrut- \frac{142}{11}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 16 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{7}{11} \alpha_{2} ^{3} \) \(\mathstrut- \frac{9}{11} \alpha_{2} ^{2} \) \(\mathstrut- 13 \alpha_{2} \) \(\mathstrut+ \frac{152}{11}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 4.4.334189.1 \(x ^{4} \) \(\mathstrut -\mathstrut 2 x ^{3} \) \(\mathstrut -\mathstrut 24 x ^{2} \) \(\mathstrut +\mathstrut 61 x \) \(\mathstrut +\mathstrut 2\)