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Decomposition of \( S_{6}^{\mathrm{new}}(23) \) into irreducible Hecke orbits

magma: S := CuspForms(23,6);
magma: N := Newforms(S);
sage: N = Newforms(23,6,names="a")
Label Dimension Field $q$-expansion of eigenform
23.6.1.a 3 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{4} \alpha_{1} ^{2} \) \(\mathstrut- \frac{7}{2} \alpha_{1} \) \(\mathstrut- 2\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 32\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{1}{2} \alpha_{1} ^{2} \) \(\mathstrut- 38\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{5}{2} \alpha_{1} ^{2} \) \(\mathstrut- 14 \alpha_{1} \) \(\mathstrut- 2\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \alpha_{1} ^{2} \) \(\mathstrut+ 11 \alpha_{1} \) \(\mathstrut- 42\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 4 \alpha_{1} ^{2} \) \(\mathstrut- 16 \alpha_{1} \) \(\mathstrut+ 8\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{41}{4} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{173}{2} \alpha_{1} \) \(\mathstrut- 227\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
23.6.1.b 6 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{9}{412} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{71}{618} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{748}{309} \alpha_{2} ^{3} \) \(\mathstrut- \frac{4161}{412} \alpha_{2} ^{2} \) \(\mathstrut- \frac{29389}{618} \alpha_{2} \) \(\mathstrut+ \frac{49832}{309}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 32\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{15}{206} \alpha_{2} ^{5} \) \(\mathstrut- \frac{28}{103} \alpha_{2} ^{4} \) \(\mathstrut- \frac{854}{103} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{3639}{206} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{18330}{103} \alpha_{2} \) \(\mathstrut- \frac{9408}{103}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{17}{618} \alpha_{2} ^{5} \) \(\mathstrut- \frac{224}{309} \alpha_{2} ^{4} \) \(\mathstrut- \frac{183}{103} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{50639}{618} \alpha_{2} ^{2} \) \(\mathstrut- \frac{2710}{309} \alpha_{2} \) \(\mathstrut- \frac{174438}{103}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{1}{206} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{98}{103} \alpha_{2} ^{4} \) \(\mathstrut- \frac{204}{103} \alpha_{2} ^{3} \) \(\mathstrut- \frac{22573}{206} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{11962}{103} \alpha_{2} \) \(\mathstrut+ \frac{252318}{103}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 64 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{353}{1236} \alpha_{2} ^{5} \) \(\mathstrut- \frac{39}{206} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{11168}{309} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{84271}{1236} \alpha_{2} ^{2} \) \(\mathstrut- \frac{555481}{618} \alpha_{2} \) \(\mathstrut- \frac{706769}{309}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ 3.3.7925.1 \(x ^{3} \) \(\mathstrut +\mathstrut 4 x ^{2} \) \(\mathstrut -\mathstrut 48 x \) \(\mathstrut -\mathstrut 8\)
$\Q(\alpha_{ 2 })$ \(x ^{6} \) \(\mathstrut -\mathstrut 4 x ^{5} \) \(\mathstrut -\mathstrut 144 x ^{4} \) \(\mathstrut +\mathstrut 381 x ^{3} \) \(\mathstrut +\mathstrut 5928 x ^{2} \) \(\mathstrut -\mathstrut 7784 x \) \(\mathstrut -\mathstrut 77528\)