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Decomposition of $S_{10}^{\mathrm{new}}(14)$ into irreducible Hecke orbits

magma: S := CuspForms(14,10);
magma: N := Newforms(S);
sage: N = Newforms(14,10,names="a")
Label Dimension Field $q$-expansion of eigenform
14.10.1.a 1 $\Q$ $q$ $\mathstrut-$ $16q^{2}$ $\mathstrut-$ $6q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $560q^{5}$ $\mathstrut+$ $96q^{6}$ $\mathstrut-$ $2401q^{7}$ $\mathstrut-$ $4096q^{8}$ $\mathstrut-$ $19647q^{9}$ $\mathstrut+O(q^{10})$
14.10.1.b 1 $\Q$ $q$ $\mathstrut+$ $16q^{2}$ $\mathstrut+$ $170q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $544q^{5}$ $\mathstrut+$ $2720q^{6}$ $\mathstrut-$ $2401q^{7}$ $\mathstrut+$ $4096q^{8}$ $\mathstrut+$ $9217q^{9}$ $\mathstrut+O(q^{10})$
14.10.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut-$ $16q^{2}$ $\mathstrut+$ $\bigl(\alpha_{3}$ $\mathstrut+ 16\bigr)q^{3}$ $\mathstrut+$ $256q^{4}$ $\mathstrut+$ $\bigl(\frac{21}{5} \alpha_{3}$ $\mathstrut- \frac{6342}{5}\bigr)q^{5}$ $\mathstrut+$ $\bigl(- 16 \alpha_{3}$ $\mathstrut- 256\bigr)q^{6}$ $\mathstrut+$ $2401q^{7}$ $\mathstrut-$ $4096q^{8}$ $\mathstrut+$ $\bigl(- 14 \alpha_{3}$ $\mathstrut+ 37669\bigr)q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{2305})$ $x ^{2}$ $\mathstrut +\mathstrut 46 x$ $\mathstrut -\mathstrut 57096$

Decomposition of $S_{10}^{\mathrm{old}}(14)$ into lower level spaces

$S_{10}^{\mathrm{old}}(14)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/7/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(7)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/2/10/1/ }{ S^{ new }_{ 10 }(\Gamma_0(2)) }^{\oplus 2 }$