Related objects

Learn more about

Show commands for: Magma / SageMath

Decomposition of \( S_{8}^{\mathrm{new}}(23) \) into irreducible Hecke orbits

magma: S := CuspForms(23,8);
magma: N := Newforms(S);
sage: N = Newforms(23,8,names="a")
Label Dimension Field $q$-expansion of eigenform
23.8.1.a 5 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{5}{3104} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{53}{1552} \alpha_{1} ^{3} \) \(\mathstrut- \frac{495}{776} \alpha_{1} ^{2} \) \(\mathstrut- \frac{3247}{388} \alpha_{1} \) \(\mathstrut+ \frac{4622}{97}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{43}{1552} \alpha_{1} ^{4} \) \(\mathstrut- \frac{417}{776} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{2511}{388} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{16129}{194} \alpha_{1} \) \(\mathstrut- \frac{34180}{97}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{13}{1552} \alpha_{1} ^{4} \) \(\mathstrut- \frac{95}{776} \alpha_{1} ^{3} \) \(\mathstrut- \frac{1287}{388} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{1219}{194} \alpha_{1} \) \(\mathstrut- \frac{1690}{97}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{15}{388} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{415}{388} \alpha_{1} ^{3} \) \(\mathstrut- \frac{933}{194} \alpha_{1} ^{2} \) \(\mathstrut- \frac{20452}{97} \alpha_{1} \) \(\mathstrut- \frac{16724}{97}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 256 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{371}{3104} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{2303}{1552} \alpha_{1} ^{3} \) \(\mathstrut- \frac{23925}{776} \alpha_{1} ^{2} \) \(\mathstrut- \frac{85417}{388} \alpha_{1} \) \(\mathstrut+ \frac{31175}{97}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
23.8.1.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 $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{5} \) \(\mathstrut +\mathstrut 16 x ^{4} \) \(\mathstrut -\mathstrut 320 x ^{3} \) \(\mathstrut -\mathstrut 3136 x ^{2} \) \(\mathstrut +\mathstrut 25680 x \) \(\mathstrut +\mathstrut 10816\)
$\Q(\alpha_{ 2 })$ \(x ^{8} \) \(\mathstrut -\mathstrut 832 x ^{6} \) \(\mathstrut -\mathstrut 1059 x ^{5} \) \(\mathstrut +\mathstrut 203052 x ^{4} \) \(\mathstrut +\mathstrut 678328 x ^{3} \) \(\mathstrut -\mathstrut 13424272 x ^{2} \) \(\mathstrut -\mathstrut 73308944 x \) \(\mathstrut -\mathstrut 37372224\)