# Related objects

Show commands for: Magma / SageMath

## Decomposition of $S_{10}^{\mathrm{new}}(23)$ into irreducible Hecke orbits

magma: S := CuspForms(23,10);
magma: N := Newforms(S);
sage: N = Newforms(23,10,names="a")
Label Dimension Field $q$-expansion of eigenform
23.10.1.a 7 $\Q(\alpha_{ 1 })$ $q$ $\mathstrut+$ $\alpha_{1} q^{2}$ $\mathstrut+$ $\bigl(\frac{7713}{730833920} \alpha_{1} ^{6}$ $\mathstrut- \frac{52833}{182708480} \alpha_{1} ^{5}$ $\mathstrut- \frac{832207}{45677120} \alpha_{1} ^{4}$ $\mathstrut+ \frac{32964761}{91354240} \alpha_{1} ^{3}$ $\mathstrut+ \frac{180306507}{22838560} \alpha_{1} ^{2}$ $\mathstrut- \frac{577718629}{5709640} \alpha_{1}$ $\mathstrut- \frac{864914887}{1427410}\bigr)q^{3}$ $\mathstrut+$ $\bigl(\alpha_{1} ^{2}$ $\mathstrut- 512\bigr)q^{4}$ $\mathstrut+$ $\bigl(- \frac{25921}{365416960} \alpha_{1} ^{6}$ $\mathstrut+ \frac{165341}{91354240} \alpha_{1} ^{5}$ $\mathstrut+ \frac{2837689}{22838560} \alpha_{1} ^{4}$ $\mathstrut- \frac{98355337}{45677120} \alpha_{1} ^{3}$ $\mathstrut- \frac{651066159}{11419280} \alpha_{1} ^{2}$ $\mathstrut+ \frac{1561294403}{2854820} \alpha_{1}$ $\mathstrut+ \frac{3970847059}{713705}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- \frac{52833}{182708480} \alpha_{1} ^{6}$ $\mathstrut+ \frac{401873}{45677120} \alpha_{1} ^{5}$ $\mathstrut+ \frac{5523397}{11419280} \alpha_{1} ^{4}$ $\mathstrut- \frac{261463281}{22838560} \alpha_{1} ^{3}$ $\mathstrut- \frac{1197735847}{5709640} \alpha_{1} ^{2}$ $\mathstrut+ \frac{4670412119}{1427410} \alpha_{1}$ $\mathstrut+ \frac{12899051514}{713705}\bigr)q^{6}$ $\mathstrut+$ $\bigl(\frac{3093}{73083392} \alpha_{1} ^{6}$ $\mathstrut- \frac{50501}{18270848} \alpha_{1} ^{5}$ $\mathstrut- \frac{140405}{4567712} \alpha_{1} ^{4}$ $\mathstrut+ \frac{30868269}{9135424} \alpha_{1} ^{3}$ $\mathstrut- \frac{563921}{2283856} \alpha_{1} ^{2}$ $\mathstrut- \frac{452140963}{570964} \alpha_{1}$ $\mathstrut- \frac{379284861}{142741}\bigr)q^{7}$ $\mathstrut+$ $\bigl(\alpha_{1} ^{3}$ $\mathstrut- 1024 \alpha_{1} \bigr)q^{8}$ $\mathstrut+$ $\bigl(- \frac{352883}{730833920} \alpha_{1} ^{6}$ $\mathstrut+ \frac{3272203}{182708480} \alpha_{1} ^{5}$ $\mathstrut+ \frac{29502297}{45677120} \alpha_{1} ^{4}$ $\mathstrut- \frac{1898614491}{91354240} \alpha_{1} ^{3}$ $\mathstrut- \frac{3823957257}{22838560} \alpha_{1} ^{2}$ $\mathstrut+ \frac{23814969739}{5709640} \alpha_{1}$ $\mathstrut+ \frac{1576380587}{1427410}\bigr)q^{9}$ $\mathstrut+O(q^{10})$
23.10.1.b 10 $\Q(\alpha_{ 2 })$ $q + \ldots^\ast$

${}^\ast$: The Fourier coefficients of this newform are large. They are available for download.
Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ $x ^{7}$ $\mathstrut -\mathstrut 2560 x ^{5}$ $\mathstrut -\mathstrut 11640 x ^{4}$ $\mathstrut +\mathstrut 1832832 x ^{3}$ $\mathstrut +\mathstrut 10289408 x ^{2}$ $\mathstrut -\mathstrut 367442944 x$ $\mathstrut -\mathstrut 1712515072$
$\Q(\alpha_{ 2 })$ $x ^{10}$ $\mathstrut -\mathstrut 32 x ^{9}$ $\mathstrut -\mathstrut 3584 x ^{8}$ $\mathstrut +\mathstrut 116861 x ^{7}$ $\mathstrut +\mathstrut 3703708 x ^{6}$ $\mathstrut -\mathstrut 122455688 x ^{5}$ $\mathstrut -\mathstrut 940607976 x ^{4}$ $\mathstrut +\mathstrut 31631851328 x ^{3}$ $\mathstrut +\mathstrut 49798012928 x ^{2}$ $\mathstrut -\mathstrut 1931922774016 x$ $\mathstrut -\mathstrut 2136806320128$