# Properties

 Label 240.3.c Level $240$ Weight $3$ Character orbit 240.c Rep. character $\chi_{240}(209,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $5$ Sturm bound $144$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 240.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$144$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$7$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(240, [\chi])$$.

Total New Old
Modular forms 108 26 82
Cusp forms 84 22 62
Eisenstein series 24 4 20

## Trace form

 $$22 q - 2 q^{9} + O(q^{10})$$ $$22 q - 2 q^{9} - 14 q^{15} + 20 q^{19} + 8 q^{21} - 10 q^{25} + 36 q^{31} + 32 q^{39} + 24 q^{45} - 154 q^{49} + 116 q^{51} - 64 q^{55} + 140 q^{61} + 36 q^{69} - 192 q^{75} - 28 q^{79} - 154 q^{81} + 44 q^{85} - 384 q^{91} - 32 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(240, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
240.3.c.a $1$ $6.540$ $$\Q$$ $$\Q(\sqrt{-15})$$ $$0$$ $$-3$$ $$-5$$ $$0$$ $$q-3q^{3}-5q^{5}+9q^{9}+15q^{15}+14q^{17}+\cdots$$
240.3.c.b $1$ $6.540$ $$\Q$$ $$\Q(\sqrt{-15})$$ $$0$$ $$3$$ $$5$$ $$0$$ $$q+3q^{3}+5q^{5}+9q^{9}+15q^{15}-14q^{17}+\cdots$$
240.3.c.c $4$ $6.540$ $$\Q(\sqrt{2}, \sqrt{-17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(2\beta _{2}+\beta _{3})q^{5}+(-2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
240.3.c.d $4$ $6.540$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(-2\beta _{2}+2\beta _{3})q^{7}+\cdots$$
240.3.c.e $12$ $6.540$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{7}q^{5}-\beta _{9}q^{7}+(1-\beta _{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(240, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(240, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$