# Properties

 Label 840.2.f Level $840$ Weight $2$ Character orbit 840.f Rep. character $\chi_{840}(41,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $2$ Sturm bound $384$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 840.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$384$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(840, [\chi])$$.

Total New Old
Modular forms 208 32 176
Cusp forms 176 32 144
Eisenstein series 32 0 32

## Trace form

 $$32q + 4q^{7} - 4q^{9} + O(q^{10})$$ $$32q + 4q^{7} - 4q^{9} + 12q^{21} + 32q^{25} + 24q^{37} + 12q^{39} + 64q^{43} - 8q^{49} + 12q^{51} - 28q^{63} - 8q^{79} - 12q^{81} + 40q^{91} - 64q^{93} - 116q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(840, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
840.2.f.a $$16$$ $$6.707$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-16$$ $$2$$ $$q+\beta _{2}q^{3}-q^{5}+\beta _{4}q^{7}-\beta _{1}q^{9}+\beta _{8}q^{11}+\cdots$$
840.2.f.b $$16$$ $$6.707$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$16$$ $$2$$ $$q-\beta _{2}q^{3}+q^{5}+\beta _{9}q^{7}-\beta _{1}q^{9}+\beta _{8}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(840, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(840, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 2}$$