# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{10}^{\mathrm{new}}(19)$ into irreducible Hecke orbits

magma: S := CuspForms(19,10);
magma: N := Newforms(S);
sage: N = Newforms(19,10,names="a")
Label Dimension Field $q$-expansion of eigenform
19.10.1.a 6 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(- \frac{61}{3181056} \alpha_{1} ^{5}$ $\mathstrut+ \frac{983}{3181056} \alpha_{1} ^{4}$ $\mathstrut+ \frac{72547}{1590528} \alpha_{1} ^{3}$ $\mathstrut- \frac{11431}{12426} \alpha_{1} ^{2}$ $\mathstrut- \frac{4544977}{198816} \alpha_{1}$ $\mathstrut+ \frac{19099889}{49704}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{1} ^{2}$ $\mathstrut- 512\bigr)q^{4}$ $\mathstrut+$ $\bigl(\frac{121}{530176} \alpha_{1} ^{5}$ $\mathstrut- \frac{11825}{1590528} \alpha_{1} ^{4}$ $\mathstrut- \frac{156127}{265088} \alpha_{1} ^{3}$ $\mathstrut+ \frac{672287}{49704} \alpha_{1} ^{2}$ $\mathstrut+ \frac{9016261}{33136} \alpha_{1}$ $\mathstrut- \frac{112773743}{24852}\bigr)q^{5}$ $\mathstrut+$ $\bigl(\frac{749}{795264} \alpha_{1} ^{5}$ $\mathstrut+ \frac{10745}{795264} \alpha_{1} ^{4}$ $\mathstrut- \frac{732707}{397632} \alpha_{1} ^{3}$ $\mathstrut- \frac{68333}{6213} \alpha_{1} ^{2}$ $\mathstrut+ \frac{30791393}{49704} \alpha_{1}$ $\mathstrut- \frac{22000321}{12426}\bigr)q^{6}$ $\mathstrut+$ $\bigl(- \frac{6089}{3181056} \alpha_{1} ^{5}$ $\mathstrut- \frac{29943}{1060352} \alpha_{1} ^{4}$ $\mathstrut+ \frac{5437607}{1590528} \alpha_{1} ^{3}$ $\mathstrut+ \frac{533579}{16568} \alpha_{1} ^{2}$ $\mathstrut- \frac{239711357}{198816} \alpha_{1}$ $\mathstrut- \frac{53955825}{16568}\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{1} ^{3}$ $\mathstrut- 1024 \alpha_{1} \bigr)q^{8}$ $\mathstrut+$ $\bigl(\frac{105}{55808} \alpha_{1} ^{5}$ $\mathstrut+ \frac{6885}{55808} \alpha_{1} ^{4}$ $\mathstrut- \frac{54487}{27904} \alpha_{1} ^{3}$ $\mathstrut- \frac{75939}{436} \alpha_{1} ^{2}$ $\mathstrut- \frac{1462707}{3488} \alpha_{1}$ $\mathstrut+ \frac{28825819}{872}\bigr)q^{9}$ $\mathstrut+O(q^{10})$
19.10.1.b 8 $\Q(\alpha_{ 2 })$ $q + \ldots^\ast$

${}^\ast$: The Fourier coefficients of this newform are large. They are available for download.

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ $x ^{6}$ $\mathstrut +\mathstrut 33 x ^{5}$ $\mathstrut -\mathstrut 1674 x ^{4}$ $\mathstrut -\mathstrut 48120 x ^{3}$ $\mathstrut +\mathstrut 618576 x ^{2}$ $\mathstrut +\mathstrut 12266496 x$ $\mathstrut -\mathstrut 92329216$
$\Q(\alpha_{ 2 })$ $x ^{8}$ $\mathstrut -\mathstrut 15 x ^{7}$ $\mathstrut -\mathstrut 3258 x ^{6}$ $\mathstrut +\mathstrut 41238 x ^{5}$ $\mathstrut +\mathstrut 2972568 x ^{4}$ $\mathstrut -\mathstrut 23100984 x ^{3}$ $\mathstrut -\mathstrut 693287712 x ^{2}$ $\mathstrut -\mathstrut 3693191040 x$ $\mathstrut -\mathstrut 5784998400$