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Decomposition of \( S_{6}^{\mathrm{new}}(13) \) into irreducible Hecke orbits

magma: S := CuspForms(13,6);
magma: N := Newforms(S);
sage: N = Newforms(13,6,names="a")
Label Dimension Field $q$-expansion of eigenform
13.6.1.a 2 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- 6 \alpha_{1} \) \(\mathstrut- 29\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 5 \alpha_{1} \) \(\mathstrut- 34\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(40 \alpha_{1} \) \(\mathstrut+ 79\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\alpha_{1} \) \(\mathstrut+ 12\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 70 \alpha_{1} \) \(\mathstrut- 193\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 41 \alpha_{1} \) \(\mathstrut+ 10\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(168 \alpha_{1} \) \(\mathstrut+ 526\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
13.6.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{2} ^{2} \) \(\mathstrut- \frac{3}{4} \alpha_{2} \) \(\mathstrut+ \frac{45}{2}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 32\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{2} ^{2} \) \(\mathstrut- \frac{27}{4} \alpha_{2} \) \(\mathstrut+ \frac{105}{2}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{5}{2} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{3}{2} \alpha_{2} \) \(\mathstrut+ 111\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{3}{4} \alpha_{2} ^{2} \) \(\mathstrut- \frac{9}{4} \alpha_{2} \) \(\mathstrut+ \frac{79}{2}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(7 \alpha_{2} ^{2} \) \(\mathstrut+ 20 \alpha_{2} \) \(\mathstrut- 444\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{1}{4} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{27}{4} \alpha_{2} \) \(\mathstrut- \frac{195}{2}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{17}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 5 x \) \(\mathstrut +\mathstrut 2\)
$\Q(\alpha_{ 2 })\cong$ 3.3.168897.1 \(x ^{3} \) \(\mathstrut -\mathstrut 7 x ^{2} \) \(\mathstrut -\mathstrut 84 x \) \(\mathstrut +\mathstrut 444\)