# Properties

 Label 1764.1.g Level $1764$ Weight $1$ Character orbit 1764.g Rep. character $\chi_{1764}(883,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $4$ Sturm bound $336$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$336$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(1764, [\chi])$$.

Total New Old
Modular forms 42 12 30
Cusp forms 10 7 3
Eisenstein series 32 5 27

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 7 0 0 0

## Trace form

 $$7 q + q^{2} + 3 q^{4} + q^{8} + O(q^{10})$$ $$7 q + q^{2} + 3 q^{4} + q^{8} + 7 q^{16} + 4 q^{22} + q^{25} + 2 q^{29} + q^{32} + 2 q^{37} + 4 q^{46} - q^{50} - 2 q^{53} - 6 q^{58} + 3 q^{64} - 2 q^{74} - 8 q^{85} - 4 q^{88} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(1764, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1764.1.g.a $$1$$ $$0.880$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-1})$$, $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{7})$$ $$1$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+q^{4}+q^{8}+q^{16}-q^{25}+2q^{29}+\cdots$$
1764.1.g.b $$2$$ $$0.880$$ $$\Q(\sqrt{2})$$ $$D_{4}$$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q-q^{2}+q^{4}-\beta q^{5}-q^{8}+\beta q^{10}-\beta q^{13}+\cdots$$
1764.1.g.c $$2$$ $$0.880$$ $$\Q(\sqrt{-1})$$ $$D_{2}$$ $$\Q(\sqrt{-7})$$, $$\Q(\sqrt{-21})$$ $$\Q(\sqrt{3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{8}+iq^{11}+q^{16}+\cdots$$
1764.1.g.d $$2$$ $$0.880$$ $$\Q(\sqrt{2})$$ $$D_{4}$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+q^{4}-\beta q^{5}+q^{8}-\beta q^{10}+\beta q^{13}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(1764, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(1764, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 3}$$