Related objects

Learn more about

Show commands for: Magma / SageMath

Decomposition of \( S_{10}^{\mathrm{new}}(13) \) into irreducible Hecke orbits

magma: S := CuspForms(13,10);
magma: N := Newforms(S);
sage: N = Newforms(13,10,names="a")
Label Dimension Field $q$-expansion of eigenform
13.10.1.a 4 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{1}{664} \alpha_{1} ^{3} \) \(\mathstrut- \frac{129}{664} \alpha_{1} ^{2} \) \(\mathstrut- \frac{947}{332} \alpha_{1} \) \(\mathstrut+ \frac{6202}{83}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{55}{664} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{1119}{664} \alpha_{1} ^{2} \) \(\mathstrut- \frac{35231}{332} \alpha_{1} \) \(\mathstrut- \frac{48120}{83}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{12}{83} \alpha_{1} ^{3} \) \(\mathstrut- \frac{386}{83} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{3085}{83} \alpha_{1} \) \(\mathstrut+ \frac{29000}{83}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{303}{664} \alpha_{1} ^{3} \) \(\mathstrut- \frac{5887}{664} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{121419}{332} \alpha_{1} \) \(\mathstrut- \frac{94368}{83}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 1024 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{323}{664} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{19755}{664} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{85765}{332} \alpha_{1} \) \(\mathstrut- \frac{1790351}{83}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
13.10.1.b 5 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1}{1088} \alpha_{2} ^{4} \) \(\mathstrut- \frac{67}{1088} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{29}{136} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{10467}{272} \alpha_{2} \) \(\mathstrut- \frac{17547}{68}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{7}{1088} \alpha_{2} ^{4} \) \(\mathstrut- \frac{197}{1088} \alpha_{2} ^{3} \) \(\mathstrut- \frac{1055}{136} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{38589}{272} \alpha_{2} \) \(\mathstrut+ \frac{148899}{68}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{13}{272} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{395}{272} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{2085}{68} \alpha_{2} ^{2} \) \(\mathstrut- \frac{41517}{68} \alpha_{2} \) \(\mathstrut+ \frac{19521}{17}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(\frac{79}{1088} \alpha_{2} ^{4} \) \(\mathstrut- \frac{1485}{1088} \alpha_{2} ^{3} \) \(\mathstrut- \frac{9065}{136} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{114525}{272} \alpha_{2} \) \(\mathstrut+ \frac{596395}{68}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 1024 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{7}{64} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{5}{64} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{1047}{8} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{10083}{16} \alpha_{2} \) \(\mathstrut- \frac{17271}{4}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{4} \) \(\mathstrut +\mathstrut 33 x ^{3} \) \(\mathstrut -\mathstrut 1194 x ^{2} \) \(\mathstrut -\mathstrut 24936 x \) \(\mathstrut +\mathstrut 232000\)
$\Q(\alpha_{ 2 })$ \(x ^{5} \) \(\mathstrut -\mathstrut 15 x ^{4} \) \(\mathstrut -\mathstrut 1348 x ^{3} \) \(\mathstrut +\mathstrut 8508 x ^{2} \) \(\mathstrut +\mathstrut 383520 x \) \(\mathstrut -\mathstrut 1249344\)