# Properties

 Label 117.4.b Level $117$ Weight $4$ Character orbit 117.b Rep. character $\chi_{117}(64,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $5$ Sturm bound $56$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$56$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(117, [\chi])$$.

Total New Old
Modular forms 46 18 28
Cusp forms 38 16 22
Eisenstein series 8 2 6

## Trace form

 $$16 q - 58 q^{4} + O(q^{10})$$ $$16 q - 58 q^{4} + 102 q^{10} + 38 q^{13} + 174 q^{14} + 154 q^{16} - 222 q^{17} + 432 q^{22} - 156 q^{23} - 226 q^{25} + 198 q^{26} + 720 q^{29} - 714 q^{35} + 12 q^{38} - 822 q^{40} + 350 q^{43} - 554 q^{49} + 292 q^{52} + 708 q^{53} - 2112 q^{55} - 2358 q^{56} - 1604 q^{61} - 1008 q^{62} + 2054 q^{64} - 570 q^{65} + 5694 q^{68} - 2658 q^{74} - 2040 q^{77} + 1180 q^{79} + 960 q^{82} - 1344 q^{88} + 390 q^{91} + 7884 q^{92} + 522 q^{94} - 1620 q^{95} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(117, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
117.4.b.a $2$ $6.903$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-3iq^{5}+5iq^{7}+7iq^{8}+\cdots$$
117.4.b.b $2$ $6.903$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+8q^{4}+\zeta_{6}q^{7}+(35-\zeta_{6})q^{13}+2^{6}q^{16}+\cdots$$
117.4.b.c $4$ $6.903$ 4.0.8112.1 $$\Q(\sqrt{-39})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-8-\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{5}+\cdots$$
117.4.b.d $4$ $6.903$ 4.0.5054412.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}-6\beta _{1}q^{7}+\cdots$$
117.4.b.e $4$ $6.903$ 4.0.1362828.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-4+\beta _{3})q^{4}+(2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(117, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(117, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$