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Decomposition of \( S_{8}^{\mathrm{new}}(13) \) into irreducible Hecke orbits

magma: S := CuspForms(13,8);
magma: N := Newforms(S);
sage: N = Newforms(13,8,names="a")
Label Dimension Field $q$-expansion of eigenform
13.8.1.a 1 \(\Q\) \(q \) \(\mathstrut+\) \(10q^{2} \) \(\mathstrut-\) \(73q^{3} \) \(\mathstrut-\) \(28q^{4} \) \(\mathstrut-\) \(295q^{5} \) \(\mathstrut-\) \(730q^{6} \) \(\mathstrut+\) \(1373q^{7} \) \(\mathstrut-\) \(1560q^{8} \) \(\mathstrut+\) \(3142q^{9} \) \(\mathstrut+O(q^{10}) \)
13.8.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{2} \) \(\mathstrut- 6\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 19 \alpha_{2} \) \(\mathstrut- 134\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(11 \alpha_{2} \) \(\mathstrut- 72\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(51 \alpha_{2} \) \(\mathstrut+ 18\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- 9 \alpha_{2} \) \(\mathstrut- 1090\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(99 \alpha_{2} \) \(\mathstrut+ 114\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 135 \alpha_{2} \) \(\mathstrut- 2205\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
13.8.1.c 4 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{3} ^{3} \) \(\mathstrut- \frac{3}{4} \alpha_{3} ^{2} \) \(\mathstrut+ 65 \alpha_{3} \) \(\mathstrut+ 278\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{13}{4} \alpha_{3} ^{3} \) \(\mathstrut+ \frac{15}{4} \alpha_{3} ^{2} \) \(\mathstrut- 805 \alpha_{3} \) \(\mathstrut- 2292\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{9}{2} \alpha_{3} ^{3} \) \(\mathstrut- \frac{5}{2} \alpha_{3} ^{2} \) \(\mathstrut+ 1094 \alpha_{3} \) \(\mathstrut+ 3220\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{3}{4} \alpha_{3} ^{3} \) \(\mathstrut- \frac{25}{4} \alpha_{3} ^{2} \) \(\mathstrut+ 171 \alpha_{3} \) \(\mathstrut+ 2052\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{3} \) \(\mathstrut- 256 \alpha_{3} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{67}{4} \alpha_{3} ^{3} \) \(\mathstrut- \frac{129}{4} \alpha_{3} ^{2} \) \(\mathstrut+ 4139 \alpha_{3} \) \(\mathstrut+ 15527\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{337}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 19 x \) \(\mathstrut +\mathstrut 6\)
$\Q(\alpha_{ 3 })$ \(x ^{4} \) \(\mathstrut -\mathstrut 15 x ^{3} \) \(\mathstrut -\mathstrut 270 x ^{2} \) \(\mathstrut +\mathstrut 3264 x \) \(\mathstrut +\mathstrut 12880\)