# Properties

 Label 50.2.d Level $50$ Weight $2$ Character orbit 50.d Rep. character $\chi_{50}(11,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $12$ Newform subspaces $2$ Sturm bound $15$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 50.d (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$2$$ Sturm bound: $$15$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(50, [\chi])$$.

Total New Old
Modular forms 36 12 24
Cusp forms 20 12 8
Eisenstein series 16 0 16

## Trace form

 $$12q - q^{2} - 4q^{3} - 3q^{4} + 5q^{5} - 2q^{6} - 8q^{7} - q^{8} - 9q^{9} + O(q^{10})$$ $$12q - q^{2} - 4q^{3} - 3q^{4} + 5q^{5} - 2q^{6} - 8q^{7} - q^{8} - 9q^{9} - 5q^{10} + 4q^{11} + 6q^{12} - 14q^{13} - 4q^{14} - 3q^{16} - 8q^{17} + 12q^{18} + 10q^{19} - 16q^{21} + 8q^{22} + 6q^{23} + 8q^{24} + 25q^{25} + 18q^{26} + 20q^{27} + 2q^{28} - 6q^{31} + 4q^{32} + 22q^{33} + 11q^{34} - 30q^{35} - 9q^{36} - 23q^{37} - 20q^{38} - 8q^{39} - 5q^{40} + 4q^{41} - 22q^{42} - 4q^{43} - 6q^{44} + 5q^{45} + 8q^{46} + 22q^{47} + 6q^{48} + 4q^{49} - 25q^{50} - 16q^{51} - 14q^{52} - 9q^{53} - 20q^{54} - 10q^{55} - 4q^{56} - 30q^{58} + 4q^{61} + 28q^{62} + 16q^{63} - 3q^{64} - 15q^{65} - 24q^{66} + 2q^{67} + 42q^{68} + 52q^{69} + 40q^{70} - 6q^{71} - 13q^{72} + 6q^{73} + 26q^{74} + 50q^{75} + 24q^{77} + 24q^{78} - 5q^{80} - 33q^{81} + 38q^{82} + 26q^{83} + 14q^{84} + 35q^{85} - 22q^{86} - 2q^{88} - 5q^{89} + 25q^{90} + 4q^{91} + 6q^{92} - 68q^{93} - 24q^{94} - 60q^{95} - 2q^{96} - 58q^{97} - 17q^{98} - 68q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(50, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
50.2.d.a $$4$$ $$0.399$$ $$\Q(\zeta_{10})$$ None $$1$$ $$-1$$ $$5$$ $$-12$$ $$q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-1+\cdots)q^{3}+\cdots$$
50.2.d.b $$8$$ $$0.399$$ 8.0.58140625.2 None $$-2$$ $$-3$$ $$0$$ $$4$$ $$q+(-1+\beta _{2}+\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(50, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(50, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 2}$$