# Properties

 Label 315.3.h Level $315$ Weight $3$ Character orbit 315.h Rep. character $\chi_{315}(181,\cdot)$ Character field $\Q$ Dimension $28$ Newform subspaces $4$ Sturm bound $144$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 315.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$144$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(315, [\chi])$$.

Total New Old
Modular forms 104 28 76
Cusp forms 88 28 60
Eisenstein series 16 0 16

## Trace form

 $$28 q + 2 q^{2} + 58 q^{4} - 6 q^{7} - 2 q^{8} + O(q^{10})$$ $$28 q + 2 q^{2} + 58 q^{4} - 6 q^{7} - 2 q^{8} + 14 q^{11} + 62 q^{14} + 162 q^{16} + 80 q^{22} - 4 q^{23} - 140 q^{25} + 14 q^{28} - 142 q^{29} + 294 q^{32} - 30 q^{35} - 156 q^{37} - 120 q^{43} - 180 q^{44} - 172 q^{46} - 176 q^{49} - 10 q^{50} - 256 q^{53} + 238 q^{56} - 136 q^{58} + 186 q^{64} + 90 q^{65} + 80 q^{67} + 60 q^{70} - 184 q^{71} - 44 q^{74} - 40 q^{77} + 326 q^{79} - 90 q^{85} - 140 q^{86} + 52 q^{88} + 54 q^{91} - 36 q^{92} - 622 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.3.h.a $2$ $8.583$ $$\Q(\sqrt{-5})$$ None $$-4$$ $$0$$ $$0$$ $$-4$$ $$q-2q^{2}+\beta q^{5}+(-2-3\beta )q^{7}+8q^{8}+\cdots$$
315.3.h.b $2$ $8.583$ $$\Q(\sqrt{-5})$$ None $$2$$ $$0$$ $$0$$ $$14$$ $$q+q^{2}-3q^{4}+\beta q^{5}+7q^{7}-7q^{8}+\cdots$$
315.3.h.c $12$ $8.583$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+(2-\beta _{2})q^{4}+\beta _{7}q^{5}+(-1+\cdots)q^{7}+\cdots$$
315.3.h.d $12$ $8.583$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$4$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{2}q^{2}+(4-\beta _{1})q^{4}-\beta _{8}q^{5}+(-1+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(315, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$