# Properties

 Label 370.2.n Level $370$ Weight $2$ Character orbit 370.n Rep. character $\chi_{370}(269,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $36$ Newform subspaces $6$ Sturm bound $114$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.n (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$185$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$114$$ Trace bound: $$7$$ 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 124 36 88
Cusp forms 108 36 72
Eisenstein series 16 0 16

## Trace form

 $$36q + 18q^{4} + 4q^{5} + 18q^{9} + O(q^{10})$$ $$36q + 18q^{4} + 4q^{5} + 18q^{9} + 4q^{10} - 12q^{11} - 8q^{14} - 18q^{16} + 14q^{19} - 4q^{20} + 16q^{21} - 10q^{25} - 12q^{26} + 16q^{29} - 8q^{30} - 32q^{31} - 12q^{34} + 8q^{35} + 36q^{36} - 20q^{39} + 2q^{40} + 4q^{41} - 6q^{44} + 64q^{45} - 6q^{46} - 6q^{49} - 32q^{50} - 8q^{51} - 12q^{55} - 4q^{56} + 46q^{59} - 24q^{61} - 36q^{64} - 30q^{65} + 8q^{66} - 52q^{69} + 12q^{70} + 36q^{71} - 2q^{74} + 16q^{75} - 14q^{76} + 12q^{79} - 8q^{80} - 34q^{81} + 32q^{84} - 24q^{85} + 28q^{86} - 30q^{89} - 14q^{90} - 6q^{94} + 24q^{95} - 94q^{99} + O(q^{100})$$

## 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.n.a $$4$$ $$2.954$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-8$$ $$-6$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-2+\zeta_{12}^{3})q^{5}+\cdots$$
370.2.n.b $$4$$ $$2.954$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
370.2.n.c $$4$$ $$2.954$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$6$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(2-\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
370.2.n.d $$4$$ $$2.954$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+(3\zeta_{12}-3\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
370.2.n.e $$8$$ $$2.954$$ 8.0.303595776.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-2\beta _{5}q^{3}-\beta _{4}q^{4}+(\beta _{1}+\beta _{5}+\cdots)q^{5}+\cdots$$
370.2.n.f $$12$$ $$2.954$$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}+(\beta _{5}-\beta _{6}-\beta _{7}+\beta _{9})q^{3}+\cdots$$

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

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