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Decomposition of \( S_{12}^{\mathrm{new}}(21) \) into irreducible Hecke orbits

magma: S := CuspForms(21,12);
magma: N := Newforms(S);
sage: N = Newforms(21,12,names="a")
Label Dimension Field $q$-expansion of eigenform
21.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(62q^{2} \) \(\mathstrut+\) \(243q^{3} \) \(\mathstrut+\) \(1796q^{4} \) \(\mathstrut-\) \(3310q^{5} \) \(\mathstrut-\) \(15066q^{6} \) \(\mathstrut-\) \(16807q^{7} \) \(\mathstrut+\) \(15624q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
21.12.1.b 1 \(\Q\) \(q \) \(\mathstrut+\) \(8q^{2} \) \(\mathstrut+\) \(243q^{3} \) \(\mathstrut-\) \(1984q^{4} \) \(\mathstrut+\) \(4390q^{5} \) \(\mathstrut+\) \(1944q^{6} \) \(\mathstrut-\) \(16807q^{7} \) \(\mathstrut-\) \(32256q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
21.12.1.c 3 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut-\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{3} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{37}{4} \alpha_{3} ^{2} \) \(\mathstrut+ 551 \alpha_{3} \) \(\mathstrut- 11266\bigr)q^{5} \) \(\mathstrut-\) \(243 \alpha_{3} q^{6} \) \(\mathstrut-\) \(16807q^{7} \) \(\mathstrut+\) \(\bigl(- 68 \alpha_{3} ^{2} \) \(\mathstrut- 2376 \alpha_{3} \) \(\mathstrut+ 7232\bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
21.12.1.d 3 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut-\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{46}{13} \alpha_{4} ^{2} \) \(\mathstrut+ \frac{310}{13} \alpha_{4} \) \(\mathstrut+ \frac{122790}{13}\bigr)q^{5} \) \(\mathstrut-\) \(243 \alpha_{4} q^{6} \) \(\mathstrut+\) \(16807q^{7} \) \(\mathstrut+\) \(\bigl(- 33 \alpha_{4} ^{2} \) \(\mathstrut- 1186 \alpha_{4} \) \(\mathstrut+ 69840\bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
21.12.1.e 4 $\Q(\alpha_{ 5 })$ \(q \) \(\mathstrut+\) \(\alpha_{5} q^{2} \) \(\mathstrut+\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{5} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{1}{12} \alpha_{5} ^{3} \) \(\mathstrut+ \frac{15}{4} \alpha_{5} ^{2} \) \(\mathstrut+ 311 \alpha_{5} \) \(\mathstrut- 4698\bigr)q^{5} \) \(\mathstrut+\) \(243 \alpha_{5} q^{6} \) \(\mathstrut+\) \(16807q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{5} ^{3} \) \(\mathstrut- 4096 \alpha_{5} \bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })$ \(x ^{3} \) \(\mathstrut +\mathstrut 68 x ^{2} \) \(\mathstrut -\mathstrut 1720 x \) \(\mathstrut -\mathstrut 7232\)
$\Q(\alpha_{ 4 })$ \(x ^{3} \) \(\mathstrut +\mathstrut 33 x ^{2} \) \(\mathstrut -\mathstrut 2910 x \) \(\mathstrut -\mathstrut 69840\)
$\Q(\alpha_{ 5 })$ \(x ^{4} \) \(\mathstrut -\mathstrut 45 x ^{3} \) \(\mathstrut -\mathstrut 5484 x ^{2} \) \(\mathstrut +\mathstrut 154872 x \) \(\mathstrut +\mathstrut 3433536\)

Decomposition of \( S_{12}^{\mathrm{old}}(21) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(21) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/7/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 } $