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Decomposition of \( S_{8}^{\mathrm{new}}(21) \) into irreducible Hecke orbits

magma: S := CuspForms(21,8);
magma: N := Newforms(S);
sage: N = Newforms(21,8,names="a")
Label Dimension Field $q$-expansion of eigenform
21.8.1.a 1 \(\Q\) \(q \) \(\mathstrut+\) \(2q^{2} \) \(\mathstrut+\) \(27q^{3} \) \(\mathstrut-\) \(124q^{4} \) \(\mathstrut-\) \(278q^{5} \) \(\mathstrut+\) \(54q^{6} \) \(\mathstrut-\) \(343q^{7} \) \(\mathstrut-\) \(504q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)
21.8.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut-\) \(27q^{3} \) \(\mathstrut+\) \(\bigl(- 9 \alpha_{2} \) \(\mathstrut+ 118\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- 20 \alpha_{2} \) \(\mathstrut- 270\bigr)q^{5} \) \(\mathstrut-\) \(27 \alpha_{2} q^{6} \) \(\mathstrut+\) \(343q^{7} \) \(\mathstrut+\) \(\bigl(71 \alpha_{2} \) \(\mathstrut- 2214\bigr)q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)
21.8.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut-\) \(27q^{3} \) \(\mathstrut+\) \(\bigl(12 \alpha_{3} \) \(\mathstrut+ 104\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(8 \alpha_{3} \) \(\mathstrut- 60\bigr)q^{5} \) \(\mathstrut-\) \(27 \alpha_{3} q^{6} \) \(\mathstrut-\) \(343q^{7} \) \(\mathstrut+\) \(\bigl(120 \alpha_{3} \) \(\mathstrut+ 2784\bigr)q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)
21.8.1.d 3 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut+\) \(27q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{2} \) \(\mathstrut- 128\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- 2 \alpha_{4} ^{2} \) \(\mathstrut+ 16 \alpha_{4} \) \(\mathstrut+ 384\bigr)q^{5} \) \(\mathstrut+\) \(27 \alpha_{4} q^{6} \) \(\mathstrut+\) \(343q^{7} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{4} ^{2} \) \(\mathstrut+ 44 \alpha_{4} \) \(\mathstrut+ 792\bigr)q^{8} \) \(\mathstrut+\) \(729q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{1065}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 9 x \) \(\mathstrut -\mathstrut 246\)
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{67}) \) \(x ^{2} \) \(\mathstrut -\mathstrut 12 x \) \(\mathstrut -\mathstrut 232\)
$\Q(\alpha_{ 4 })\cong$ 3.3.2910828.1 \(x ^{3} \) \(\mathstrut +\mathstrut 3 x ^{2} \) \(\mathstrut -\mathstrut 300 x \) \(\mathstrut -\mathstrut 792\)