# Properties

 Label 770.2.ba Level $770$ Weight $2$ Character orbit 770.ba Rep. character $\chi_{770}(169,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $144$ Newform subspaces $3$ Sturm bound $288$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.ba (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$55$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$3$$ Sturm bound: $$288$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 608 144 464
Cusp forms 544 144 400
Eisenstein series 64 0 64

## Trace form

 $$144q + 36q^{4} - 4q^{5} + 56q^{9} + O(q^{10})$$ $$144q + 36q^{4} - 4q^{5} + 56q^{9} + 4q^{11} - 16q^{15} - 36q^{16} + 32q^{19} + 4q^{20} + 40q^{21} + 24q^{25} + 16q^{29} + 44q^{30} - 8q^{31} - 16q^{34} - 36q^{36} - 12q^{39} - 4q^{44} - 40q^{45} - 56q^{46} + 36q^{49} - 24q^{50} + 28q^{51} - 68q^{55} + 8q^{59} - 24q^{60} - 64q^{61} + 36q^{64} - 32q^{65} + 8q^{66} - 32q^{69} + 8q^{70} + 64q^{71} + 8q^{74} - 8q^{75} - 32q^{76} - 4q^{80} + 24q^{81} + 20q^{84} + 12q^{85} - 56q^{86} + 144q^{89} + 60q^{90} - 36q^{91} - 32q^{94} + 16q^{95} - 172q^{99} + O(q^{100})$$

## 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.ba.a $$8$$ $$6.148$$ $$\Q(\zeta_{20})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{20}^{7}q^{2}+(2\zeta_{20}+\zeta_{20}^{5}-\zeta_{20}^{7})q^{3}+\cdots$$
770.2.ba.b $$64$$ $$6.148$$ None $$0$$ $$0$$ $$0$$ $$0$$
770.2.ba.c $$72$$ $$6.148$$ None $$0$$ $$0$$ $$-8$$ $$0$$

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

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