# Properties

 Label 370.2.h Level $370$ Weight $2$ Character orbit 370.h Rep. character $\chi_{370}(117,\cdot)$ 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.h (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

 $$38q - 2q^{2} + 8q^{3} + 38q^{4} - 6q^{5} - 2q^{8} + O(q^{10})$$ $$38q - 2q^{2} + 8q^{3} + 38q^{4} - 6q^{5} - 2q^{8} + 2q^{10} + 8q^{12} - 8q^{13} + 4q^{14} - 8q^{15} + 38q^{16} + 12q^{19} - 6q^{20} - 8q^{23} + 10q^{25} - 8q^{26} - 40q^{27} + 18q^{29} - 16q^{31} - 2q^{32} - 8q^{35} - 38q^{37} - 8q^{39} + 2q^{40} + 16q^{43} + 4q^{45} - 8q^{47} + 8q^{48} - 10q^{50} - 8q^{52} - 6q^{53} + 12q^{55} + 4q^{56} + 48q^{57} - 18q^{58} - 20q^{59} - 8q^{60} + 22q^{61} - 40q^{62} + 38q^{64} + 24q^{65} - 8q^{66} + 16q^{67} - 88q^{69} + 16q^{70} - 16q^{71} + 2q^{73} + 26q^{74} - 72q^{75} + 12q^{76} - 40q^{77} - 20q^{78} - 32q^{79} - 6q^{80} + 10q^{81} + 20q^{83} - 16q^{86} - 46q^{89} - 40q^{90} - 40q^{91} - 8q^{92} - 48q^{93} + 36q^{94} + 40q^{95} + O(q^{100})$$

## 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.h.a $$2$$ $$2.954$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$-2$$ $$-4$$ $$q+q^{2}+q^{4}+(-1+2i)q^{5}+(-2+2i)q^{7}+\cdots$$
370.2.h.b $$2$$ $$2.954$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$2$$ $$-2$$ $$2$$ $$q+q^{2}+(1-i)q^{3}+q^{4}+(-1-2i)q^{5}+\cdots$$
370.2.h.c $$4$$ $$2.954$$ $$\Q(\zeta_{8})$$ None $$4$$ $$0$$ $$0$$ $$8$$ $$q+q^{2}+2\zeta_{8}q^{3}+q^{4}+(-\zeta_{8}-2\zeta_{8}^{3})q^{5}+\cdots$$
370.2.h.d $$10$$ $$2.954$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$10$$ $$2$$ $$2$$ $$-4$$ $$q+q^{2}+(\beta _{1}-\beta _{4})q^{3}+q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$
370.2.h.e $$20$$ $$2.954$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-20$$ $$4$$ $$-4$$ $$-2$$ $$q-q^{2}-\beta _{4}q^{3}+q^{4}+\beta _{13}q^{5}+\beta _{4}q^{6}+\cdots$$

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

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