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Decomposition of \( S_{8}^{\mathrm{new}}(19) \) into irreducible Hecke orbits

magma: S := CuspForms(19,8);
magma: N := Newforms(S);
sage: N = Newforms(19,8,names="a")
Label Dimension Field $q$-expansion of eigenform
19.8.1.a 4 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{72} \alpha_{1} ^{3} \) \(\mathstrut- \frac{3}{8} \alpha_{1} ^{2} \) \(\mathstrut- \frac{1}{2} \alpha_{1} \) \(\mathstrut+ \frac{53}{2}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{3}{8} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{33}{8} \alpha_{1} ^{2} \) \(\mathstrut- \frac{169}{2} \alpha_{1} \) \(\mathstrut- \frac{525}{2}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{1} ^{3} \) \(\mathstrut- \frac{15}{4} \alpha_{1} ^{2} \) \(\mathstrut+ 21 \alpha_{1} \) \(\mathstrut+ 45\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{3}{2} \alpha_{1} ^{3} \) \(\mathstrut- \frac{29}{2} \alpha_{1} ^{2} \) \(\mathstrut+ 294 \alpha_{1} \) \(\mathstrut+ 143\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 256 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{47}{72} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{93}{8} \alpha_{1} ^{2} \) \(\mathstrut- \frac{13}{2} \alpha_{1} \) \(\mathstrut- \frac{3757}{2}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
19.8.1.b 6 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{43}{35136} \alpha_{2} ^{5} \) \(\mathstrut- \frac{1235}{35136} \alpha_{2} ^{4} \) \(\mathstrut- \frac{607}{4392} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{137375}{17568} \alpha_{2} ^{2} \) \(\mathstrut- \frac{48479}{8784} \alpha_{2} \) \(\mathstrut- \frac{128750}{549}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{49}{17568} \alpha_{2} ^{5} \) \(\mathstrut- \frac{1169}{17568} \alpha_{2} ^{4} \) \(\mathstrut- \frac{365}{1098} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{94613}{8784} \alpha_{2} ^{2} \) \(\mathstrut- \frac{16073}{4392} \alpha_{2} \) \(\mathstrut+ \frac{44375}{549}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{295}{17568} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{7247}{17568} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{4675}{2196} \alpha_{2} ^{3} \) \(\mathstrut- \frac{739403}{8784} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{527675}{4392} \alpha_{2} \) \(\mathstrut+ \frac{1326808}{549}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{211}{35136} \alpha_{2} ^{5} \) \(\mathstrut- \frac{2315}{35136} \alpha_{2} ^{4} \) \(\mathstrut- \frac{7345}{4392} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{102455}{17568} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{819361}{8784} \alpha_{2} \) \(\mathstrut+ \frac{442075}{549}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 256 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{75}{3904} \alpha_{2} ^{5} \) \(\mathstrut- \frac{2835}{3904} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{445}{488} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{309471}{1952} \alpha_{2} ^{2} \) \(\mathstrut- \frac{639855}{976} \alpha_{2} \) \(\mathstrut- \frac{220999}{61}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{4} \) \(\mathstrut +\mathstrut 9 x ^{3} \) \(\mathstrut -\mathstrut 234 x ^{2} \) \(\mathstrut -\mathstrut 396 x \) \(\mathstrut +\mathstrut 3240\)
$\Q(\alpha_{ 2 })$ \(x ^{6} \) \(\mathstrut -\mathstrut 15 x ^{5} \) \(\mathstrut -\mathstrut 450 x ^{4} \) \(\mathstrut +\mathstrut 4650 x ^{3} \) \(\mathstrut +\mathstrut 64272 x ^{2} \) \(\mathstrut -\mathstrut 289800 x \) \(\mathstrut -\mathstrut 1974784\)