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Decomposition of \( S_{10}^{\mathrm{new}}(17) \) into irreducible Hecke orbits

magma: S := CuspForms(17,10);
magma: N := Newforms(S);
sage: N = Newforms(17,10,names="a")
Label Dimension Field $q$-expansion of eigenform
17.10.1.a 5 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{7}{16192} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{311}{16192} \alpha_{1} ^{3} \) \(\mathstrut- \frac{4603}{8096} \alpha_{1} ^{2} \) \(\mathstrut- \frac{2915}{184} \alpha_{1} \) \(\mathstrut+ \frac{40198}{253}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{89}{8096} \alpha_{1} ^{4} \) \(\mathstrut- \frac{3665}{8096} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{35537}{4048} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{26515}{92} \alpha_{1} \) \(\mathstrut- \frac{323970}{253}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{5}{1012} \alpha_{1} ^{4} \) \(\mathstrut- \frac{67}{1012} \alpha_{1} ^{3} \) \(\mathstrut- \frac{2565}{506} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{232}{23} \alpha_{1} \) \(\mathstrut- \frac{50064}{253}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{213}{16192} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{16981}{16192} \alpha_{1} ^{3} \) \(\mathstrut- \frac{9081}{8096} \alpha_{1} ^{2} \) \(\mathstrut- \frac{160157}{184} \alpha_{1} \) \(\mathstrut- \frac{733462}{253}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 1024 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{841}{8096} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{16257}{8096} \alpha_{1} ^{3} \) \(\mathstrut- \frac{445601}{4048} \alpha_{1} ^{2} \) \(\mathstrut- \frac{120131}{92} \alpha_{1} \) \(\mathstrut+ \frac{2623039}{253}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
17.10.1.b 7 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{41053}{6260544000} \alpha_{2} ^{6} \) \(\mathstrut+ \frac{2167}{46374400} \alpha_{2} ^{5} \) \(\mathstrut- \frac{62116439}{3130272000} \alpha_{2} ^{4} \) \(\mathstrut- \frac{271417537}{1565136000} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{11699176237}{782568000} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{1647981773}{10869000} \alpha_{2} \) \(\mathstrut- \frac{6071804633}{5434500}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{44783}{2608560000} \alpha_{2} ^{6} \) \(\mathstrut+ \frac{16111}{57968000} \alpha_{2} ^{5} \) \(\mathstrut- \frac{69407929}{1304280000} \alpha_{2} ^{4} \) \(\mathstrut- \frac{519815207}{652140000} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{13363936907}{326070000} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{1271785039}{2264375} \alpha_{2} \) \(\mathstrut- \frac{6330304663}{2264375}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{62873}{1565136000} \alpha_{2} ^{6} \) \(\mathstrut- \frac{3053}{11593600} \alpha_{2} ^{5} \) \(\mathstrut- \frac{93362599}{782568000} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{71234683}{391284000} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{16335856517}{195642000} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{1303871893}{2717250} \alpha_{2} \) \(\mathstrut- \frac{6022516153}{1358625}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{5047}{417369600} \alpha_{2} ^{6} \) \(\mathstrut+ \frac{401}{9274880} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{9765461}{208684800} \alpha_{2} ^{4} \) \(\mathstrut- \frac{25349837}{104342400} \alpha_{2} ^{3} \) \(\mathstrut- \frac{2523525463}{52171200} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{217681173}{724600} \alpha_{2} \) \(\mathstrut+ \frac{3259164867}{362300}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 1024 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{79391}{69561600} \alpha_{2} ^{6} \) \(\mathstrut- \frac{19381}{4637440} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{116030533}{34780800} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{341174339}{17390400} \alpha_{2} ^{3} \) \(\mathstrut- \frac{20670650039}{8695200} \alpha_{2} ^{2} \) \(\mathstrut- \frac{7433802543}{362300} \alpha_{2} \) \(\mathstrut+ \frac{30397636803}{181150}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{5} \) \(\mathstrut +\mathstrut 33 x ^{4} \) \(\mathstrut -\mathstrut 1162 x ^{3} \) \(\mathstrut -\mathstrut 24920 x ^{2} \) \(\mathstrut +\mathstrut 344192 x \) \(\mathstrut +\mathstrut 457728\)
$\Q(\alpha_{ 2 })$ \(x ^{7} \) \(\mathstrut +\mathstrut x ^{6} \) \(\mathstrut -\mathstrut 2986 x ^{5} \) \(\mathstrut -\mathstrut 8252 x ^{4} \) \(\mathstrut +\mathstrut 2252056 x ^{3} \) \(\mathstrut +\mathstrut 10388768 x ^{2} \) \(\mathstrut -\mathstrut 243559296 x \) \(\mathstrut +\mathstrut 675998208\)