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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

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