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Decomposition of \( S_{12}^{\mathrm{new}}(7) \) into irreducible Hecke orbits

magma: S := CuspForms(7,12);
magma: N := Newforms(S);
sage: N = Newforms(7,12,names="a")
Label Dimension Field $q$-expansion of eigenform
7.12.1.a 2 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(- 6 \alpha_{1} \) \(\mathstrut- 102\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(- 54 \alpha_{1} \) \(\mathstrut+ 592\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(10 \alpha_{1} \) \(\mathstrut- 6480\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(222 \alpha_{1} \) \(\mathstrut- 15840\bigr)q^{6} \) \(\mathstrut+\) \(16807q^{7} \) \(\mathstrut+\) \(\bigl(1460 \alpha_{1} \) \(\mathstrut- 142560\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- 720 \alpha_{1} \) \(\mathstrut- 71703\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
7.12.1.b 3 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(- \frac{11}{21} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{103}{7} \alpha_{2} \) \(\mathstrut+ \frac{33758}{21}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(\frac{59}{7} \alpha_{2} ^{2} \) \(\mathstrut- \frac{517}{7} \alpha_{2} \) \(\mathstrut- \frac{203864}{7}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(- \frac{538}{21} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{788}{7} \alpha_{2} \) \(\mathstrut+ \frac{2476144}{21}\bigr)q^{6} \) \(\mathstrut-\) \(16807q^{7} \) \(\mathstrut+\) \(\bigl(77 \alpha_{2} ^{2} \) \(\mathstrut- 1242 \alpha_{2} \) \(\mathstrut- 225104\bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{734}{3} \alpha_{2} ^{2} \) \(\mathstrut+ 1846 \alpha_{2} \) \(\mathstrut+ \frac{3363563}{3}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })\cong$ \(\Q(\sqrt{3369}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 54 x \) \(\mathstrut -\mathstrut 2640\)
$\Q(\alpha_{ 2 })$ \(x ^{3} \) \(\mathstrut -\mathstrut 77 x ^{2} \) \(\mathstrut -\mathstrut 2854 x \) \(\mathstrut +\mathstrut 225104\)

Decomposition of \( S_{12}^{\mathrm{old}}(7) \) into lower level spaces

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