# Properties

 Label 370.2.b Level $370$ Weight $2$ Character orbit 370.b Rep. character $\chi_{370}(149,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $4$ Sturm bound $114$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$114$$ Trace bound: $$5$$ 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 62 18 44
Cusp forms 54 18 36
Eisenstein series 8 0 8

## Trace form

 $$18 q - 18 q^{4} - 4 q^{5} - 6 q^{9} + O(q^{10})$$ $$18 q - 18 q^{4} - 4 q^{5} - 6 q^{9} - 4 q^{10} - 4 q^{14} + 18 q^{16} - 20 q^{19} + 4 q^{20} + 8 q^{21} + 16 q^{25} - 16 q^{29} + 20 q^{30} + 8 q^{31} + 16 q^{35} + 6 q^{36} - 40 q^{39} + 4 q^{40} - 4 q^{41} + 20 q^{45} - 12 q^{46} - 54 q^{49} + 8 q^{50} + 8 q^{51} + 36 q^{55} + 4 q^{56} + 20 q^{59} - 18 q^{64} - 8 q^{66} - 80 q^{69} + 24 q^{71} - 10 q^{74} + 50 q^{75} + 20 q^{76} - 48 q^{79} - 4 q^{80} - 14 q^{81} - 8 q^{84} + 36 q^{85} - 4 q^{86} - 60 q^{89} + 50 q^{90} + 48 q^{91} - 36 q^{94} + 36 q^{95} + 4 q^{99} + O(q^{100})$$

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

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

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