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Decomposition of \( S_{6}^{\mathrm{new}}(19) \) into irreducible Hecke orbits

magma: S := CuspForms(19,6);
magma: N := Newforms(S);
sage: N = Newforms(19,6,names="a")
Label Dimension Field $q$-expansion of eigenform
19.6.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(6q^{2} \) \(\mathstrut+\) \(4q^{3} \) \(\mathstrut+\) \(4q^{4} \) \(\mathstrut+\) \(54q^{5} \) \(\mathstrut-\) \(24q^{6} \) \(\mathstrut+\) \(248q^{7} \) \(\mathstrut+\) \(168q^{8} \) \(\mathstrut-\) \(227q^{9} \) \(\mathstrut+O(q^{10}) \)
19.6.1.b 1 \(\Q\) \(q \) \(\mathstrut-\) \(2q^{2} \) \(\mathstrut-\) \(q^{3} \) \(\mathstrut-\) \(28q^{4} \) \(\mathstrut-\) \(24q^{5} \) \(\mathstrut+\) \(2q^{6} \) \(\mathstrut-\) \(167q^{7} \) \(\mathstrut+\) \(120q^{8} \) \(\mathstrut-\) \(242q^{9} \) \(\mathstrut+O(q^{10}) \)
19.6.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{3} \) \(\mathstrut- 14\bigr)q^{3} \) \(\mathstrut-\) \(7 \alpha_{3} q^{4} \) \(\mathstrut+\) \(\bigl(5 \alpha_{3} \) \(\mathstrut- 49\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(7 \alpha_{3} \) \(\mathstrut- 96\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(14 \alpha_{3} \) \(\mathstrut+ 85\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(17 \alpha_{3} \) \(\mathstrut- 224\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(21 \alpha_{3} \) \(\mathstrut+ 241\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
19.6.1.d 4 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{3}{22} \alpha_{4} ^{3} \) \(\mathstrut- \frac{15}{22} \alpha_{4} ^{2} \) \(\mathstrut- \frac{105}{11} \alpha_{4} \) \(\mathstrut+ \frac{543}{11}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{2} \) \(\mathstrut- 32\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{23}{22} \alpha_{4} ^{3} \) \(\mathstrut+ \frac{27}{22} \alpha_{4} ^{2} \) \(\mathstrut+ \frac{893}{11} \alpha_{4} \) \(\mathstrut- \frac{1512}{11}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{6}{11} \alpha_{4} ^{3} \) \(\mathstrut+ \frac{3}{11} \alpha_{4} ^{2} \) \(\mathstrut- \frac{618}{11} \alpha_{4} \) \(\mathstrut+ \frac{1710}{11}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{9}{11} \alpha_{4} ^{3} \) \(\mathstrut- \frac{23}{11} \alpha_{4} ^{2} \) \(\mathstrut- \frac{630}{11} \alpha_{4} \) \(\mathstrut+ \frac{1399}{11}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{3} \) \(\mathstrut- 64 \alpha_{4} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{45}{22} \alpha_{4} ^{3} \) \(\mathstrut- \frac{225}{22} \alpha_{4} ^{2} \) \(\mathstrut- \frac{2367}{11} \alpha_{4} \) \(\mathstrut+ \frac{12006}{11}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{177}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 7 x \) \(\mathstrut -\mathstrut 32\)
$\Q(\alpha_{ 4 })$ \(x ^{4} \) \(\mathstrut -\mathstrut 9 x ^{3} \) \(\mathstrut -\mathstrut 72 x ^{2} \) \(\mathstrut +\mathstrut 774 x \) \(\mathstrut -\mathstrut 1140\)