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Decomposition of \( S_{4}^{\mathrm{new}}(21) \) into irreducible Hecke orbits

magma: S := CuspForms(21,4);
magma: N := Newforms(S);
sage: N = Newforms(21,4,names="a")
Label Dimension Field $q$-expansion of eigenform
21.4.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(3q^{2} \) \(\mathstrut-\) \(3q^{3} \) \(\mathstrut+\) \(q^{4} \) \(\mathstrut-\) \(18q^{5} \) \(\mathstrut+\) \(9q^{6} \) \(\mathstrut+\) \(7q^{7} \) \(\mathstrut+\) \(21q^{8} \) \(\mathstrut+\) \(9q^{9} \) \(\mathstrut+O(q^{10}) \)
21.4.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(4q^{2} \) \(\mathstrut-\) \(3q^{3} \) \(\mathstrut+\) \(8q^{4} \) \(\mathstrut-\) \(4q^{5} \) \(\mathstrut-\) \(12q^{6} \) \(\mathstrut-\) \(7q^{7} \) \(\mathstrut+\) \(9q^{9} \) \(\mathstrut+O(q^{10}) \)
21.4.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut+\) \(3q^{3} \) \(\mathstrut+\) \(\bigl(- 3 \alpha_{3} \) \(\mathstrut+ 4\bigr)q^{4} \) \(\mathstrut-\) \(2 \alpha_{3} q^{5} \) \(\mathstrut+\) \(3 \alpha_{3} q^{6} \) \(\mathstrut+\) \(7q^{7} \) \(\mathstrut+\) \(\bigl(5 \alpha_{3} \) \(\mathstrut- 36\bigr)q^{8} \) \(\mathstrut+\) \(9q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{57}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut -\mathstrut 12\)

Decomposition of \( S_{4}^{\mathrm{old}}(21) \) into lower level spaces

\( S_{4}^{\mathrm{old}}(21) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/7/4/1/ }{ S^{ new }_{ 4 }(\Gamma_0(7)) }^{\oplus 2 } $