# Properties

 Label 338.2.i Level $338$ Weight $2$ Character orbit 338.i Rep. character $\chi_{338}(3,\cdot)$ Character field $\Q(\zeta_{39})$ Dimension $360$ Newform subspaces $2$ Sturm bound $91$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.i (of order $$39$$ and degree $$24$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$169$$ Character field: $$\Q(\zeta_{39})$$ Newform subspaces: $$2$$ Sturm bound: $$91$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(338, [\chi])$$.

Total New Old
Modular forms 1128 360 768
Cusp forms 1032 360 672
Eisenstein series 96 0 96

## Trace form

 $$360q + q^{2} + 15q^{4} + 2q^{5} + 4q^{7} - 2q^{8} + 11q^{9} + O(q^{10})$$ $$360q + q^{2} + 15q^{4} + 2q^{5} + 4q^{7} - 2q^{8} + 11q^{9} - q^{10} + 4q^{11} + 45q^{13} - 4q^{14} - 52q^{15} + 15q^{16} + q^{17} + 6q^{18} - q^{20} - 24q^{22} - 8q^{23} - 12q^{25} + 5q^{26} + 12q^{27} + 4q^{28} - 7q^{29} + 98q^{30} - 60q^{31} + q^{32} - 6q^{34} - 14q^{35} + 11q^{36} + 3q^{37} + 4q^{38} - 52q^{39} - 37q^{40} - 9q^{41} - 6q^{42} - 12q^{43} - 8q^{44} - 127q^{45} - 4q^{46} + 42q^{47} - 13q^{49} - 4q^{50} - 102q^{51} + 2q^{52} - 35q^{53} + 156q^{54} - 116q^{55} + 2q^{56} - 78q^{57} - 53q^{58} - 4q^{59} - 52q^{60} - 3q^{61} - 4q^{62} - 142q^{63} - 30q^{64} + 7q^{65} + 16q^{66} - 152q^{67} + q^{68} - 16q^{69} - 200q^{70} + 148q^{71} - 3q^{72} - 22q^{73} - 96q^{74} - 334q^{75} - 26q^{76} + 8q^{77} + 40q^{79} - q^{80} - q^{81} - 13q^{82} + 260q^{83} - 94q^{85} - 62q^{86} - 72q^{87} + 2q^{88} - 266q^{89} + 22q^{90} - 44q^{91} + 16q^{92} - 286q^{93} - 66q^{94} + 184q^{95} - 76q^{97} + 9q^{98} + 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(338, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
338.2.i.a $$168$$ $$2.699$$ None $$-7$$ $$0$$ $$0$$ $$14$$
338.2.i.b $$192$$ $$2.699$$ None $$8$$ $$0$$ $$2$$ $$-10$$

## Decomposition of $$S_{2}^{\mathrm{old}}(338, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(338, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 2}$$