# Properties

 Label 370.2.g Level $370$ Weight $2$ Character orbit 370.g Rep. character $\chi_{370}(43,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $38$ Newform subspaces $5$ 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.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$185$$ Character field: $$\Q(i)$$ Newform subspaces: $$5$$ Sturm bound: $$114$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 122 38 84
Cusp forms 106 38 68
Eisenstein series 16 0 16

## Trace form

 $$38q - 8q^{3} - 38q^{4} + 4q^{5} + O(q^{10})$$ $$38q - 8q^{3} - 38q^{4} + 4q^{5} + 2q^{10} + 8q^{12} - 4q^{14} + 38q^{16} + 4q^{17} + 10q^{18} - 12q^{19} - 4q^{20} - 16q^{22} - 10q^{25} - 8q^{26} + 40q^{27} - 18q^{29} - 16q^{31} + 8q^{35} - 12q^{37} + 8q^{39} - 2q^{40} - 56q^{42} - 6q^{45} - 8q^{47} - 8q^{48} - 8q^{50} - 6q^{53} - 12q^{55} + 4q^{56} + 18q^{58} + 20q^{59} + 22q^{61} + 40q^{62} - 38q^{64} - 24q^{65} - 8q^{66} - 16q^{67} - 4q^{68} + 88q^{69} + 16q^{70} - 16q^{71} - 10q^{72} - 2q^{73} - 26q^{74} - 72q^{75} + 12q^{76} + 40q^{77} + 20q^{78} + 32q^{79} + 4q^{80} + 10q^{81} + 48q^{82} + 20q^{83} - 16q^{86} + 24q^{87} + 16q^{88} + 46q^{89} - 40q^{90} - 40q^{91} - 36q^{94} - 40q^{95} - 112q^{97} + 34q^{98} + O(q^{100})$$

## 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.g.a $$2$$ $$2.954$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$4$$ $$2$$ $$q+iq^{2}+(-1-i)q^{3}-q^{4}+(2+i)q^{5}+\cdots$$
370.2.g.b $$2$$ $$2.954$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$-4$$ $$q+iq^{2}-q^{4}+(-2+i)q^{5}+(-2-2i)q^{7}+\cdots$$
370.2.g.c $$4$$ $$2.954$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\zeta_{8}^{2}q^{2}+2\zeta_{8}q^{3}-q^{4}+(2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots$$
370.2.g.d $$10$$ $$2.954$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$-2$$ $$2$$ $$-4$$ $$q-\beta _{6}q^{2}+(-\beta _{2}-\beta _{3}-\beta _{8})q^{3}-q^{4}+\cdots$$
370.2.g.e $$20$$ $$2.954$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$-4$$ $$2$$ $$-2$$ $$q-\beta _{10}q^{2}-\beta _{1}q^{3}-q^{4}-\beta _{8}q^{5}+\beta _{4}q^{6}+\cdots$$

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

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