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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.
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 $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{7} \) \(\mathstrut -\mathstrut 2560 x ^{5} \) \(\mathstrut -\mathstrut 11640 x ^{4} \) \(\mathstrut +\mathstrut 1832832 x ^{3} \) \(\mathstrut +\mathstrut 10289408 x ^{2} \) \(\mathstrut -\mathstrut 367442944 x \) \(\mathstrut -\mathstrut 1712515072\)
$\Q(\alpha_{ 2 })$ \(x ^{10} \) \(\mathstrut -\mathstrut 32 x ^{9} \) \(\mathstrut -\mathstrut 3584 x ^{8} \) \(\mathstrut +\mathstrut 116861 x ^{7} \) \(\mathstrut +\mathstrut 3703708 x ^{6} \) \(\mathstrut -\mathstrut 122455688 x ^{5} \) \(\mathstrut -\mathstrut 940607976 x ^{4} \) \(\mathstrut +\mathstrut 31631851328 x ^{3} \) \(\mathstrut +\mathstrut 49798012928 x ^{2} \) \(\mathstrut -\mathstrut 1931922774016 x \) \(\mathstrut -\mathstrut 2136806320128\)