# Properties

 Label 51.2.d Level $51$ Weight $2$ Character orbit 51.d Rep. character $\chi_{51}(16,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $2$ Sturm bound $12$ Trace bound $2$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 8 4 4
Cusp forms 4 4 0
Eisenstein series 4 0 4

## Trace form

 $$4q - 2q^{2} + 2q^{4} - 6q^{8} - 4q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{4} - 6q^{8} - 4q^{9} + 2q^{13} - 6q^{15} - 10q^{16} + 10q^{17} + 2q^{18} + 2q^{19} + 4q^{21} + 2q^{25} + 8q^{26} + 12q^{30} + 26q^{32} + 2q^{33} - 14q^{34} - 12q^{35} - 2q^{36} - 28q^{38} + 16q^{42} - 10q^{43} - 20q^{47} - 12q^{49} + 26q^{50} - 10q^{51} - 8q^{52} + 30q^{55} - 24q^{59} - 12q^{60} - 2q^{64} - 28q^{66} + 14q^{68} + 10q^{69} + 24q^{70} + 6q^{72} + 28q^{76} + 52q^{77} + 4q^{81} + 12q^{83} - 16q^{84} - 6q^{85} - 4q^{86} - 12q^{87} - 40q^{89} + 28q^{93} - 8q^{94} - 30q^{98} + O(q^{100})$$

## 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.d.a $$2$$ $$0.407$$ $$\Q(\sqrt{-1})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q-2q^{2}+iq^{3}+2q^{4}+3iq^{5}-2iq^{6}+\cdots$$
51.2.d.b $$2$$ $$0.407$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+iq^{3}-q^{4}+iq^{6}-4iq^{7}+\cdots$$