# Properties

 Label 770.2.bg Level $770$ Weight $2$ Character orbit 770.bg Rep. character $\chi_{770}(81,\cdot)$ Character field $\Q(\zeta_{15})$ Dimension $256$ Newform subspaces $6$ Sturm bound $288$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.bg (of order $$15$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$77$$ Character field: $$\Q(\zeta_{15})$$ Newform subspaces: $$6$$ Sturm bound: $$288$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 1216 256 960
Cusp forms 1088 256 832
Eisenstein series 128 0 128

## Trace form

 $$256q + 32q^{4} + 44q^{9} + O(q^{10})$$ $$256q + 32q^{4} + 44q^{9} + 16q^{10} - 2q^{11} + 16q^{13} - 2q^{14} - 24q^{15} + 32q^{16} - 28q^{17} + 16q^{18} + 4q^{19} + 32q^{21} - 8q^{22} + 24q^{23} + 32q^{25} - 20q^{26} + 72q^{27} + 40q^{29} - 16q^{33} + 64q^{34} - 4q^{35} - 48q^{36} + 8q^{37} + 8q^{38} + 4q^{39} - 4q^{40} + 84q^{41} + 40q^{42} - 2q^{44} + 4q^{46} + 8q^{47} + 20q^{49} + 4q^{51} - 8q^{52} - 8q^{53} - 16q^{55} - 80q^{57} + 8q^{58} + 16q^{59} - 8q^{60} + 56q^{61} + 16q^{62} - 64q^{63} - 64q^{64} - 4q^{65} - 32q^{67} - 28q^{68} - 320q^{69} + 64q^{71} - 24q^{72} - 52q^{73} - 20q^{74} - 8q^{76} - 84q^{77} - 192q^{78} - 44q^{79} - 20q^{81} - 40q^{82} - 144q^{83} - 72q^{84} + 48q^{85} - 16q^{86} - 104q^{87} - 16q^{88} - 72q^{89} + 40q^{90} + 58q^{91} + 72q^{92} - 4q^{93} - 42q^{94} - 128q^{97} - 64q^{98} + 148q^{99} + O(q^{100})$$

## 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.bg.a $$8$$ $$6.148$$ $$\Q(\zeta_{15})$$ None $$-1$$ $$1$$ $$-1$$ $$-4$$ $$q-\zeta_{15}q^{2}+(2-\zeta_{15}+2\zeta_{15}^{3}-2\zeta_{15}^{4}+\cdots)q^{3}+\cdots$$
770.2.bg.b $$8$$ $$6.148$$ $$\Q(\zeta_{15})$$ None $$1$$ $$-1$$ $$1$$ $$-4$$ $$q+\zeta_{15}q^{2}+(-2+\zeta_{15}-2\zeta_{15}^{3}+2\zeta_{15}^{4}+\cdots)q^{3}+\cdots$$
770.2.bg.c $$48$$ $$6.148$$ None $$-6$$ $$-4$$ $$-6$$ $$12$$
770.2.bg.d $$48$$ $$6.148$$ None $$6$$ $$4$$ $$6$$ $$-4$$
770.2.bg.e $$72$$ $$6.148$$ None $$-9$$ $$3$$ $$9$$ $$2$$
770.2.bg.f $$72$$ $$6.148$$ None $$9$$ $$-3$$ $$-9$$ $$-2$$

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

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