# Properties

 Label 50.3.f Level $50$ Weight $3$ Character orbit 50.f Rep. character $\chi_{50}(3,\cdot)$ Character field $\Q(\zeta_{20})$ Dimension $40$ Newform subspaces $2$ Sturm bound $22$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 50.f (of order $$20$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{20})$$ Newform subspaces: $$2$$ Sturm bound: $$22$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(50, [\chi])$$.

Total New Old
Modular forms 136 40 96
Cusp forms 104 40 64
Eisenstein series 32 0 32

## Trace form

 $$40 q + 2 q^{2} + 4 q^{3} - 4 q^{7} - 4 q^{8} + O(q^{10})$$ $$40 q + 2 q^{2} + 4 q^{3} - 4 q^{7} - 4 q^{8} + 10 q^{10} + 8 q^{12} - 6 q^{13} - 20 q^{15} + 40 q^{16} - 154 q^{17} - 152 q^{18} - 200 q^{19} - 40 q^{20} - 96 q^{22} - 36 q^{23} + 110 q^{25} - 20 q^{26} + 220 q^{27} + 88 q^{28} + 200 q^{29} + 240 q^{30} + 32 q^{32} + 328 q^{33} + 250 q^{34} + 200 q^{35} - 60 q^{36} - 54 q^{37} + 40 q^{38} - 400 q^{39} + 20 q^{40} + 80 q^{41} - 16 q^{42} - 156 q^{43} - 330 q^{45} - 44 q^{47} - 16 q^{48} - 50 q^{50} - 12 q^{52} + 74 q^{53} + 240 q^{55} + 560 q^{57} - 80 q^{58} + 100 q^{59} - 240 q^{60} - 120 q^{61} - 616 q^{62} - 536 q^{63} - 790 q^{65} - 684 q^{67} - 332 q^{68} - 700 q^{69} - 280 q^{70} + 120 q^{71} - 4 q^{72} + 14 q^{73} + 60 q^{75} + 512 q^{77} + 376 q^{78} + 400 q^{79} - 190 q^{81} + 624 q^{82} + 904 q^{83} + 600 q^{84} + 460 q^{85} + 980 q^{87} + 32 q^{88} + 1450 q^{89} + 970 q^{90} + 368 q^{92} + 348 q^{93} + 40 q^{95} + 86 q^{97} + 82 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(50, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.3.f.a $16$ $1.362$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-4$$ $$2$$ $$0$$ $$-2$$ $$q+(\beta _{1}+\beta _{9})q^{2}+(-1+\beta _{2}-\beta _{3}+\beta _{7}+\cdots)q^{3}+\cdots$$
50.3.f.b $24$ $1.362$ None $$6$$ $$2$$ $$0$$ $$-2$$

## Decomposition of $$S_{3}^{\mathrm{old}}(50, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(50, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 2}$$