# Properties

 Label 370.2.q Level $370$ Weight $2$ Character orbit 370.q Rep. character $\chi_{370}(97,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $76$ 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.q (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$185$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$6$$ 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 244 76 168
Cusp forms 212 76 136
Eisenstein series 32 0 32

## Trace form

 $$76 q + 2 q^{2} + 4 q^{3} - 38 q^{4} + 6 q^{5} - 4 q^{8} + O(q^{10})$$ $$76 q + 2 q^{2} + 4 q^{3} - 38 q^{4} + 6 q^{5} - 4 q^{8} - 2 q^{10} + 4 q^{12} + 8 q^{13} + 8 q^{14} + 8 q^{15} - 38 q^{16} - 12 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{26} + 64 q^{27} - 18 q^{29} - 24 q^{30} + 16 q^{31} + 2 q^{32} + 12 q^{33} - 16 q^{35} + 14 q^{37} - 64 q^{39} - 14 q^{40} - 18 q^{41} - 84 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 8 q^{47} - 8 q^{48} + 72 q^{49} - 20 q^{50} + 8 q^{52} - 42 q^{53} - 12 q^{55} - 4 q^{56} + 24 q^{57} + 6 q^{58} + 20 q^{59} + 8 q^{60} - 4 q^{61} - 20 q^{62} + 76 q^{64} + 12 q^{65} + 8 q^{66} + 68 q^{67} - 80 q^{69} + 8 q^{70} - 8 q^{71} + 4 q^{73} + 52 q^{74} - 120 q^{75} - 24 q^{76} - 44 q^{77} + 20 q^{78} - 16 q^{79} - 6 q^{80} + 62 q^{81} - 8 q^{83} - 8 q^{86} - 36 q^{87} + 58 q^{89} + 40 q^{90} - 20 q^{91} - 4 q^{92} + 72 q^{93} - 36 q^{94} + 68 q^{95} - 24 q^{98} + O(q^{100})$$

## 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.q.a $4$ $2.954$ $$\Q(\zeta_{12})$$ None $$-2$$ $$4$$ $$-4$$ $$4$$ $$q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots$$
370.2.q.b $4$ $2.954$ $$\Q(\zeta_{12})$$ None $$-2$$ $$6$$ $$2$$ $$-2$$ $$q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}+\zeta_{12}^{2}-2\zeta_{12}^{3})q^{3}+\cdots$$
370.2.q.c $8$ $2.954$ $$\Q(\zeta_{24})$$ None $$4$$ $$4$$ $$4$$ $$12$$ $$q+\zeta_{24}^{4}q^{2}+(\zeta_{24}^{4}+\zeta_{24}^{5}+\zeta_{24}^{6}+\cdots)q^{3}+\cdots$$
370.2.q.d $12$ $2.954$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-6$$ $$0$$ $$2$$ $$-4$$ $$q+(-1-\beta _{7})q^{2}+(1-\beta _{1}-\beta _{3}+2\beta _{4}+\cdots)q^{3}+\cdots$$
370.2.q.e $16$ $2.954$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-8$$ $$-8$$ $$-4$$ $$0$$ $$q+(-1-\beta _{14})q^{2}+(-1+\beta _{2}+\beta _{13}+\cdots)q^{3}+\cdots$$
370.2.q.f $32$ $2.954$ None $$16$$ $$-2$$ $$6$$ $$-10$$

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

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