# Properties

 Label 770.2.i Level $770$ Weight $2$ Character orbit 770.i Rep. character $\chi_{770}(221,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $48$ Newform subspaces $13$ Sturm bound $288$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$13$$ Sturm bound: $$288$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$13$$

## Dimensions

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

Total New Old
Modular forms 304 48 256
Cusp forms 272 48 224
Eisenstein series 32 0 32

## Trace form

 $$48 q - 24 q^{4} - 4 q^{5} - 8 q^{6} + 16 q^{7} - 12 q^{9} + O(q^{10})$$ $$48 q - 24 q^{4} - 4 q^{5} - 8 q^{6} + 16 q^{7} - 12 q^{9} - 16 q^{13} + 8 q^{14} - 24 q^{16} - 16 q^{19} + 8 q^{20} + 4 q^{21} + 4 q^{24} - 24 q^{25} - 8 q^{28} - 24 q^{29} + 4 q^{30} + 16 q^{31} - 8 q^{33} - 16 q^{34} - 8 q^{35} + 24 q^{36} - 16 q^{37} - 24 q^{38} - 8 q^{39} + 24 q^{41} - 8 q^{42} + 80 q^{43} - 8 q^{45} - 4 q^{46} + 32 q^{47} + 4 q^{49} - 48 q^{51} + 8 q^{52} - 24 q^{53} + 28 q^{54} - 4 q^{56} + 128 q^{57} - 8 q^{58} - 8 q^{59} + 44 q^{61} - 48 q^{63} + 48 q^{64} - 8 q^{67} - 8 q^{69} - 12 q^{70} + 64 q^{71} - 64 q^{73} + 16 q^{74} + 32 q^{76} - 8 q^{77} + 64 q^{78} - 32 q^{79} - 4 q^{80} - 24 q^{81} - 32 q^{82} - 32 q^{83} - 8 q^{84} - 16 q^{85} - 4 q^{86} - 48 q^{87} + 28 q^{89} - 24 q^{91} - 80 q^{93} - 24 q^{94} + 8 q^{95} + 4 q^{96} + 16 q^{98} + O(q^{100})$$

## 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.i.a $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-1$$ $$1$$ $$5$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.b $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-1$$ $$-1$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.c $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-1$$ $$-1$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.d $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-3$$ $$-1$$ $$1$$ $$q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.e $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$-1$$ $$-5$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.f $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$-1$$ $$5$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.g $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$1$$ $$-1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.h $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$1$$ $$5$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.i $$2$$ $$6.148$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$-1$$ $$5$$ $$q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
770.2.i.j $$6$$ $$6.148$$ $$\Q(\zeta_{18})$$ None $$3$$ $$0$$ $$-3$$ $$0$$ $$q+(1-\zeta_{18}^{3})q^{2}+(-\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{4}+\cdots)q^{3}+\cdots$$
770.2.i.k $$6$$ $$6.148$$ 6.0.64827.1 None $$3$$ $$0$$ $$3$$ $$0$$ $$q+(1-\beta _{5})q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots$$
770.2.i.l $$8$$ $$6.148$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-4$$ $$1$$ $$4$$ $$3$$ $$q+(-1-\beta _{5})q^{2}+\beta _{1}q^{3}+\beta _{5}q^{4}+(1+\cdots)q^{5}+\cdots$$
770.2.i.m $$10$$ $$6.148$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$-5$$ $$0$$ $$-5$$ $$0$$ $$q+(-1+\beta _{6})q^{2}+(\beta _{1}-\beta _{2})q^{3}-\beta _{6}q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(770, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(154, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(385, [\chi])$$$$^{\oplus 2}$$