# Properties

 Label 770.2.m Level $770$ Weight $2$ Character orbit 770.m Rep. character $\chi_{770}(43,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $72$ Newform subspaces $6$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.m (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$55$$ Character field: $$\Q(i)$$ Newform subspaces: $$6$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(770, [\chi])$$.

Total New Old
Modular forms 304 72 232
Cusp forms 272 72 200
Eisenstein series 32 0 32

## Trace form

 $$72 q - 8 q^{3} + O(q^{10})$$ $$72 q - 8 q^{3} + 8 q^{11} + 8 q^{12} - 8 q^{15} - 72 q^{16} - 8 q^{20} - 8 q^{22} + 8 q^{23} + 40 q^{25} + 16 q^{27} + 32 q^{31} - 40 q^{33} - 88 q^{36} + 8 q^{37} + 48 q^{38} - 24 q^{45} + 24 q^{47} + 8 q^{48} - 40 q^{53} - 40 q^{60} - 32 q^{66} + 40 q^{67} - 16 q^{70} - 96 q^{71} + 72 q^{75} - 16 q^{77} + 32 q^{78} - 120 q^{81} + 32 q^{82} + 128 q^{86} - 8 q^{88} + 48 q^{91} - 8 q^{92} + 32 q^{93} + 88 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(770, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
770.2.m.a $$4$$ $$6.148$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(-2-\zeta_{8}^{2})q^{5}+\cdots$$
770.2.m.b $$4$$ $$6.148$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(1+2\zeta_{8}^{2})q^{5}+\cdots$$
770.2.m.c $$4$$ $$6.148$$ $$\Q(\zeta_{8})$$ None $$0$$ $$8$$ $$-4$$ $$0$$ $$q+\zeta_{8}q^{2}+(2+2\zeta_{8}^{2})q^{3}+\zeta_{8}^{2}q^{4}+\cdots$$
770.2.m.d $$8$$ $$6.148$$ 8.0.40960000.1 None $$0$$ $$-4$$ $$16$$ $$0$$ $$q-\beta _{7}q^{2}+\beta _{5}q^{3}-\beta _{2}q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots$$
770.2.m.e $$16$$ $$6.148$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$-8$$ $$-8$$ $$0$$ $$q-\beta _{10}q^{2}+(-1-\beta _{3}+\beta _{7}-\beta _{15})q^{3}+\cdots$$
770.2.m.f $$36$$ $$6.148$$ None $$0$$ $$-4$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(770, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(770, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(110, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(385, [\chi])$$$$^{\oplus 2}$$