# Properties

 Label 370.2.e Level $370$ Weight $2$ Character orbit 370.e Rep. character $\chi_{370}(121,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $6$ Sturm bound $114$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$37$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$6$$ Sturm bound: $$114$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(370, [\chi])$$.

Total New Old
Modular forms 120 20 100
Cusp forms 104 20 84
Eisenstein series 16 0 16

## Trace form

 $$20q + 4q^{3} - 10q^{4} - 8q^{7} - 2q^{9} + O(q^{10})$$ $$20q + 4q^{3} - 10q^{4} - 8q^{7} - 2q^{9} - 4q^{10} + 4q^{11} + 4q^{12} - 8q^{13} - 8q^{14} - 10q^{16} + 8q^{17} + 10q^{19} - 16q^{21} - 40q^{23} - 10q^{25} + 4q^{26} + 16q^{27} - 8q^{28} + 16q^{29} + 4q^{30} + 12q^{33} - 12q^{34} + 4q^{36} - 12q^{37} - 32q^{38} - 12q^{39} + 2q^{40} - 4q^{41} - 20q^{42} - 2q^{44} - 32q^{45} + 10q^{46} + 32q^{47} - 8q^{48} - 18q^{49} + 72q^{51} - 8q^{52} + 16q^{53} - 24q^{54} + 12q^{55} + 4q^{56} + 20q^{58} + 26q^{59} + 24q^{61} + 4q^{62} - 32q^{63} + 20q^{64} + 6q^{65} + 56q^{66} - 28q^{67} - 16q^{68} - 20q^{69} + 4q^{70} - 28q^{71} + 40q^{73} + 2q^{74} - 8q^{75} + 10q^{76} + 20q^{77} + 12q^{78} + 20q^{79} - 2q^{81} + 24q^{83} + 32q^{84} + 28q^{86} + 36q^{87} + 30q^{89} + 2q^{90} + 8q^{91} + 20q^{92} + 24q^{93} - 18q^{94} + 40q^{97} - 24q^{98} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(370, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
370.2.e.a $$2$$ $$2.954$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-2$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{2}-2\zeta_{6}q^{3}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots$$
370.2.e.b $$2$$ $$2.954$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$1$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+\cdots$$
370.2.e.c $$2$$ $$2.954$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$1$$ $$0$$ $$q+(1-\zeta_{6})q^{2}+3\zeta_{6}q^{3}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+\cdots$$
370.2.e.d $$4$$ $$2.954$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$2$$ $$-2$$ $$0$$ $$q+(-1+\zeta_{12})q^{2}+\zeta_{12}q^{3}-\zeta_{12}q^{4}+\cdots$$
370.2.e.e $$4$$ $$2.954$$ $$\Q(\sqrt{-3}, \sqrt{10})$$ None $$2$$ $$2$$ $$-2$$ $$-4$$ $$q-\beta _{2}q^{2}+(1+\beta _{2})q^{3}+(-1-\beta _{2})q^{4}+\cdots$$
370.2.e.f $$6$$ $$2.954$$ 6.0.2696112.1 None $$-3$$ $$0$$ $$3$$ $$-2$$ $$q-\beta _{4}q^{2}+\beta _{5}q^{3}+(-1+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(370, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(370, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(37, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(74, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(185, [\chi])$$$$^{\oplus 2}$$