# Properties

 Label 52.2.l Level $52$ Weight $2$ Character orbit 52.l Rep. character $\chi_{52}(7,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $20$ Newform subspaces $2$ Sturm bound $14$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.l (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$52$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$2$$ Sturm bound: $$14$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(52, [\chi])$$.

Total New Old
Modular forms 36 36 0
Cusp forms 20 20 0
Eisenstein series 16 16 0

## Trace form

 $$20 q - 4 q^{2} - 6 q^{4} - 6 q^{5} - 14 q^{6} + 2 q^{8} - 2 q^{9} + O(q^{10})$$ $$20 q - 4 q^{2} - 6 q^{4} - 6 q^{5} - 14 q^{6} + 2 q^{8} - 2 q^{9} - 6 q^{10} - 8 q^{13} + 8 q^{14} + 6 q^{16} - 12 q^{17} + 6 q^{18} + 14 q^{20} - 28 q^{21} + 10 q^{24} + 26 q^{26} + 12 q^{28} - 4 q^{29} + 42 q^{30} + 36 q^{32} - 20 q^{33} + 10 q^{34} - 6 q^{36} + 10 q^{37} - 64 q^{40} + 20 q^{41} - 28 q^{42} - 8 q^{44} + 20 q^{45} - 46 q^{46} - 10 q^{48} + 60 q^{49} - 6 q^{50} - 56 q^{52} - 4 q^{53} - 16 q^{54} - 60 q^{56} + 12 q^{57} - 74 q^{58} - 24 q^{60} + 14 q^{61} - 18 q^{62} + 4 q^{65} + 56 q^{66} + 20 q^{68} - 12 q^{69} + 28 q^{70} + 68 q^{72} + 42 q^{73} + 14 q^{74} + 22 q^{76} + 68 q^{78} + 68 q^{80} + 30 q^{81} + 54 q^{82} + 84 q^{84} - 22 q^{85} + 16 q^{86} + 36 q^{88} - 58 q^{89} - 12 q^{92} - 92 q^{93} - 38 q^{94} - 72 q^{96} - 38 q^{97} - 16 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(52, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
52.2.l.a $4$ $0.415$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$-2$$ $$0$$ $$6$$ $$0$$ $$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
52.2.l.b $16$ $0.415$ 16.0.$$\cdots$$.1 None $$-2$$ $$0$$ $$-12$$ $$0$$ $$q-\beta _{12}q^{2}+(\beta _{3}+\beta _{12}-\beta _{13}+\beta _{14}+\cdots)q^{3}+\cdots$$