# Properties

 Label 51.2.e Level $51$ Weight $2$ Character orbit 51.e Rep. character $\chi_{51}(4,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $8$ Newform subspaces $1$ Sturm bound $12$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$51 = 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 51.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$17$$ Character field: $$\Q(i)$$ Newform subspaces: $$1$$ Sturm bound: $$12$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(51, [\chi])$$.

Total New Old
Modular forms 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

## Trace form

 $$8 q - 12 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + O(q^{10})$$ $$8 q - 12 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{13} + 24 q^{14} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 12 q^{20} - 8 q^{21} + 12 q^{22} - 16 q^{23} - 16 q^{24} - 8 q^{28} + 4 q^{29} + 24 q^{30} - 8 q^{31} + 12 q^{33} - 12 q^{34} + 8 q^{35} + 8 q^{37} + 24 q^{38} + 8 q^{39} + 36 q^{40} + 28 q^{41} - 32 q^{44} + 4 q^{45} - 20 q^{46} - 8 q^{47} - 32 q^{48} - 60 q^{50} - 4 q^{51} - 16 q^{52} + 4 q^{54} - 4 q^{55} - 24 q^{56} - 4 q^{58} + 16 q^{61} + 16 q^{62} - 4 q^{63} + 4 q^{64} + 16 q^{65} + 8 q^{67} + 60 q^{68} - 12 q^{69} + 24 q^{71} + 12 q^{72} + 20 q^{73} - 28 q^{74} - 16 q^{75} + 20 q^{78} + 20 q^{79} + 60 q^{80} - 8 q^{81} - 40 q^{82} + 48 q^{84} - 32 q^{85} - 8 q^{86} - 8 q^{88} - 8 q^{89} - 36 q^{91} + 56 q^{92} + 8 q^{95} + 8 q^{96} - 12 q^{97} - 76 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(51, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
51.2.e.a $8$ $0.407$ 8.0.836829184.2 None $$0$$ $$0$$ $$-4$$ $$-4$$ $$q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-1-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots$$