# Related objects

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.
Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ $x ^{5}$ $\mathstrut +\mathstrut 16 x ^{4}$ $\mathstrut -\mathstrut 320 x ^{3}$ $\mathstrut -\mathstrut 3136 x ^{2}$ $\mathstrut +\mathstrut 25680 x$ $\mathstrut +\mathstrut 10816$
$\Q(\alpha_{ 2 })$ $x ^{8}$ $\mathstrut -\mathstrut 832 x ^{6}$ $\mathstrut -\mathstrut 1059 x ^{5}$ $\mathstrut +\mathstrut 203052 x ^{4}$ $\mathstrut +\mathstrut 678328 x ^{3}$ $\mathstrut -\mathstrut 13424272 x ^{2}$ $\mathstrut -\mathstrut 73308944 x$ $\mathstrut -\mathstrut 37372224$