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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.
Click on the label in the table above for more information about each newform.

The coefficient fields are:

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