# Properties

 Label 51.2.h Level $51$ Weight $2$ Character orbit 51.h Rep. character $\chi_{51}(19,\cdot)$ Character field $\Q(\zeta_{8})$ 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.h (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$17$$ Character field: $$\Q(\zeta_{8})$$ 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 32 8 24
Cusp forms 16 8 8
Eisenstein series 16 0 16

## Trace form

 $$8 q - 8 q^{5} - 8 q^{6} + O(q^{10})$$ $$8 q - 8 q^{5} - 8 q^{6} + 8 q^{11} - 16 q^{14} + 16 q^{16} - 8 q^{17} - 8 q^{19} + 16 q^{20} - 8 q^{22} + 8 q^{23} + 8 q^{24} - 16 q^{25} + 16 q^{26} - 8 q^{28} + 8 q^{31} + 8 q^{33} - 8 q^{34} + 32 q^{35} + 8 q^{36} - 8 q^{37} + 16 q^{39} - 8 q^{40} - 24 q^{41} + 8 q^{42} - 8 q^{43} - 8 q^{45} - 8 q^{49} - 32 q^{50} - 16 q^{52} - 32 q^{53} - 8 q^{54} - 16 q^{56} - 16 q^{57} + 24 q^{58} + 16 q^{59} - 8 q^{60} + 16 q^{61} + 16 q^{62} + 24 q^{65} - 16 q^{66} - 16 q^{67} - 24 q^{69} + 40 q^{70} - 16 q^{71} + 48 q^{73} + 64 q^{74} + 16 q^{75} + 24 q^{76} + 8 q^{78} - 16 q^{80} - 8 q^{82} + 32 q^{83} + 40 q^{85} - 32 q^{86} + 24 q^{87} - 8 q^{88} - 16 q^{91} - 32 q^{92} - 32 q^{93} - 40 q^{95} - 16 q^{96} + 8 q^{97} + 8 q^{99} + O(q^{100})$$

## 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.h.a $8$ $0.407$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+(\zeta_{16}+\zeta_{16}^{3})q^{2}+\zeta_{16}^{7}q^{3}+(\zeta_{16}^{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(51, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(51, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(17, [\chi])$$$$^{\oplus 2}$$