# Properties

 Label 135.4.b Level $135$ Weight $4$ Character orbit 135.b Rep. character $\chi_{135}(109,\cdot)$ Character field $\Q$ Dimension $24$ Newform subspaces $3$ Sturm bound $72$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 135.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$72$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(135, [\chi])$$.

Total New Old
Modular forms 60 24 36
Cusp forms 48 24 24
Eisenstein series 12 0 12

## Trace form

 $$24 q - 102 q^{4} + O(q^{10})$$ $$24 q - 102 q^{4} + 72 q^{10} + 438 q^{16} - 108 q^{19} - 354 q^{25} + 180 q^{31} + 1830 q^{34} - 582 q^{40} - 258 q^{46} - 2460 q^{49} + 1458 q^{55} - 2088 q^{61} - 2700 q^{64} - 4158 q^{70} + 7914 q^{76} + 1440 q^{79} - 432 q^{85} - 4104 q^{91} - 2556 q^{94} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(135, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
135.4.b.a $4$ $7.965$ $$\Q(i, \sqrt{5})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+(-13-\beta _{3})q^{4}-5\beta _{1}q^{5}+\cdots$$
135.4.b.b $8$ $7.965$ 8.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}-\beta _{4}-2\beta _{5}+\cdots)q^{5}+\cdots$$
135.4.b.c $12$ $7.965$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}+(-5-\beta _{1})q^{4}-\beta _{3}q^{5}+\beta _{8}q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(135, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(135, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$