# Properties

 Label 52.2.f Level $52$ Weight $2$ Character orbit 52.f Rep. character $\chi_{52}(31,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $10$ Newform subspaces $2$ Sturm bound $14$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$52$$ Character field: $$\Q(i)$$ 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 18 18 0
Cusp forms 10 10 0
Eisenstein series 8 8 0

## Trace form

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

## 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.f.a $2$ $0.415$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$-6$$ $$0$$ $$q+(1+i)q^{2}+2iq^{4}+(-3-3i)q^{5}+\cdots$$
52.2.f.b $8$ $0.415$ 8.0.18939904.2 None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{2}-\beta _{5}+\beta _{7})q^{2}+(1-2\beta _{1}+\cdots)q^{3}+\cdots$$