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Decomposition of \( S_{10}^{\mathrm{new}}(21) \) into irreducible Hecke orbits

magma: S := CuspForms(21,10);
magma: N := Newforms(S);
sage: N = Newforms(21,10,names="a")
Label Dimension Field $q$-expansion of eigenform
21.10.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(24q^{2} \) \(\mathstrut+\) \(81q^{3} \) \(\mathstrut+\) \(64q^{4} \) \(\mathstrut-\) \(144q^{5} \) \(\mathstrut-\) \(1944q^{6} \) \(\mathstrut+\) \(2401q^{7} \) \(\mathstrut+\) \(10752q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
21.10.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut-\) \(81q^{3} \) \(\mathstrut+\) \(\bigl(9 \alpha_{2} \) \(\mathstrut+ 56\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- 70 \alpha_{2} \) \(\mathstrut+ 900\bigr)q^{5} \) \(\mathstrut-\) \(81 \alpha_{2} q^{6} \) \(\mathstrut-\) \(2401q^{7} \) \(\mathstrut+\) \(\bigl(- 375 \alpha_{2} \) \(\mathstrut+ 5112\bigr)q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
21.10.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut-\) \(81q^{3} \) \(\mathstrut+\) \(\bigl(30 \alpha_{3} \) \(\mathstrut- 392\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(70 \alpha_{3} \) \(\mathstrut- 486\bigr)q^{5} \) \(\mathstrut-\) \(81 \alpha_{3} q^{6} \) \(\mathstrut+\) \(2401q^{7} \) \(\mathstrut+\) \(\bigl(- 4 \alpha_{3} \) \(\mathstrut+ 3600\bigr)q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)
21.10.1.d 3 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut+\) \(81q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{2} \) \(\mathstrut- 512\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{2}{5} \alpha_{4} ^{2} \) \(\mathstrut+ \frac{46}{5} \alpha_{4} \) \(\mathstrut+ \frac{6338}{5}\bigr)q^{5} \) \(\mathstrut+\) \(81 \alpha_{4} q^{6} \) \(\mathstrut-\) \(2401q^{7} \) \(\mathstrut+\) \(\bigl(- 13 \alpha_{4} ^{2} \) \(\mathstrut+ 498 \alpha_{4} \) \(\mathstrut+ 10984\bigr)q^{8} \) \(\mathstrut+\) \(6561q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{2353}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 9 x \) \(\mathstrut -\mathstrut 568\)
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{345}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 30 x \) \(\mathstrut -\mathstrut 120\)
$\Q(\alpha_{ 4 })$ \(x ^{3} \) \(\mathstrut +\mathstrut 13 x ^{2} \) \(\mathstrut -\mathstrut 1522 x \) \(\mathstrut -\mathstrut 10984\)

Decomposition of \( S_{10}^{\mathrm{old}}(21) \) into lower level spaces

\( S_{10}^{\mathrm{old}}(21) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/7/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(3)) }^{\oplus 2 } $