# Related objects

Show commands for: Magma / SageMath

## 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 $\Q$
$\Q(\alpha_{ 1 })$ $x ^{4}$ $\mathstrut +\mathstrut 9 x ^{3}$ $\mathstrut -\mathstrut 234 x ^{2}$ $\mathstrut -\mathstrut 396 x$ $\mathstrut +\mathstrut 3240$
$\Q(\alpha_{ 2 })$ $x ^{6}$ $\mathstrut -\mathstrut 15 x ^{5}$ $\mathstrut -\mathstrut 450 x ^{4}$ $\mathstrut +\mathstrut 4650 x ^{3}$ $\mathstrut +\mathstrut 64272 x ^{2}$ $\mathstrut -\mathstrut 289800 x$ $\mathstrut -\mathstrut 1974784$