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Decomposition of \( S_{8}^{\mathrm{new}}(17) \) into irreducible Hecke orbits

magma: S := CuspForms(17,8);
magma: N := Newforms(S);
sage: N = Newforms(17,8,names="a")
Label Dimension Field $q$-expansion of eigenform
17.8.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(2q^{2} \) \(\mathstrut+\) \(18q^{3} \) \(\mathstrut-\) \(124q^{4} \) \(\mathstrut-\) \(10q^{5} \) \(\mathstrut-\) \(36q^{6} \) \(\mathstrut-\) \(902q^{7} \) \(\mathstrut+\) \(504q^{8} \) \(\mathstrut-\) \(1863q^{9} \) \(\mathstrut+O(q^{10}) \)
17.8.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{2} ^{2} \) \(\mathstrut- \frac{17}{4} \alpha_{2} \) \(\mathstrut+ \frac{47}{2}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(2 \alpha_{2} ^{2} \) \(\mathstrut+ 6 \alpha_{2} \) \(\mathstrut- 474\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{9}{2} \alpha_{2} ^{2} \) \(\mathstrut- \frac{105}{2} \alpha_{2} \) \(\mathstrut+ 423\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{35}{4} \alpha_{2} ^{2} \) \(\mathstrut- \frac{259}{4} \alpha_{2} \) \(\mathstrut+ \frac{2557}{2}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut+ 48 \alpha_{2} \) \(\mathstrut- 1692\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{55}{2} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{719}{2} \alpha_{2} \) \(\mathstrut- 5336\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
17.8.1.c 6 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{119}{214848} \alpha_{3} ^{5} \) \(\mathstrut- \frac{2237}{214848} \alpha_{3} ^{4} \) \(\mathstrut- \frac{6559}{35808} \alpha_{3} ^{3} \) \(\mathstrut+ \frac{18353}{6714} \alpha_{3} ^{2} \) \(\mathstrut+ \frac{21449}{1492} \alpha_{3} \) \(\mathstrut- \frac{40069}{373}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{13}{11936} \alpha_{3} ^{5} \) \(\mathstrut+ \frac{119}{11936} \alpha_{3} ^{4} \) \(\mathstrut+ \frac{3115}{5968} \alpha_{3} ^{3} \) \(\mathstrut- \frac{1773}{373} \alpha_{3} ^{2} \) \(\mathstrut- \frac{25063}{746} \alpha_{3} \) \(\mathstrut+ \frac{176304}{373}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{113}{53712} \alpha_{3} ^{5} \) \(\mathstrut+ \frac{5453}{53712} \alpha_{3} ^{4} \) \(\mathstrut- \frac{467}{2238} \alpha_{3} ^{3} \) \(\mathstrut- \frac{428377}{13428} \alpha_{3} ^{2} \) \(\mathstrut+ \frac{47158}{373} \alpha_{3} \) \(\mathstrut+ \frac{882742}{373}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{51}{23872} \alpha_{3} ^{5} \) \(\mathstrut- \frac{1599}{23872} \alpha_{3} ^{4} \) \(\mathstrut+ \frac{23353}{11936} \alpha_{3} ^{3} \) \(\mathstrut+ \frac{22827}{746} \alpha_{3} ^{2} \) \(\mathstrut- \frac{445501}{1492} \alpha_{3} \) \(\mathstrut- \frac{917983}{373}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{3} \) \(\mathstrut- 256 \alpha_{3} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{169}{11936} \alpha_{3} ^{5} \) \(\mathstrut- \frac{1547}{11936} \alpha_{3} ^{4} \) \(\mathstrut- \frac{40495}{5968} \alpha_{3} ^{3} \) \(\mathstrut+ \frac{8129}{373} \alpha_{3} ^{2} \) \(\mathstrut+ \frac{480987}{746} \alpha_{3} \) \(\mathstrut+ \frac{844605}{373}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ 3.3.694349.1 \(x ^{3} \) \(\mathstrut -\mathstrut x ^{2} \) \(\mathstrut -\mathstrut 304 x \) \(\mathstrut +\mathstrut 1692\)
$\Q(\alpha_{ 3 })$ \(x ^{6} \) \(\mathstrut -\mathstrut 15 x ^{5} \) \(\mathstrut -\mathstrut 514 x ^{4} \) \(\mathstrut +\mathstrut 5312 x ^{3} \) \(\mathstrut +\mathstrut 83552 x ^{2} \) \(\mathstrut -\mathstrut 422208 x \) \(\mathstrut -\mathstrut 4272768\)