# Properties

 Label 180.4.i Level $180$ Weight $4$ Character orbit 180.i Rep. character $\chi_{180}(61,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $3$ Sturm bound $144$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 180.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$3$$ Sturm bound: $$144$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(180, [\chi])$$.

Total New Old
Modular forms 228 24 204
Cusp forms 204 24 180
Eisenstein series 24 0 24

## Trace form

 $$24 q + 2 q^{3} - 10 q^{5} + 12 q^{7} - 28 q^{9} + O(q^{10})$$ $$24 q + 2 q^{3} - 10 q^{5} + 12 q^{7} - 28 q^{9} - 54 q^{11} - 24 q^{13} - 20 q^{15} + 348 q^{17} - 60 q^{19} + 196 q^{21} - 84 q^{23} - 300 q^{25} - 592 q^{27} - 342 q^{29} - 60 q^{31} + 738 q^{33} + 280 q^{35} + 336 q^{37} - 308 q^{39} - 804 q^{41} - 258 q^{43} - 140 q^{45} - 276 q^{47} - 630 q^{49} + 1518 q^{51} + 3432 q^{53} + 178 q^{57} - 630 q^{59} + 822 q^{61} - 1976 q^{63} - 200 q^{65} - 186 q^{67} - 522 q^{69} - 216 q^{71} - 1500 q^{73} + 50 q^{75} - 96 q^{77} - 888 q^{79} - 412 q^{81} + 228 q^{83} - 360 q^{85} + 2700 q^{87} + 828 q^{89} + 24 q^{91} - 1376 q^{93} + 520 q^{95} - 546 q^{97} + 1512 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(180, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
180.4.i.a $2$ $10.620$ $$\Q(\sqrt{-3})$$ None $$0$$ $$9$$ $$5$$ $$7$$ $$q+(6-3\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(7-7\zeta_{6})q^{7}+\cdots$$
180.4.i.b $8$ $10.620$ 8.0.$$\cdots$$.1 None $$0$$ $$-12$$ $$20$$ $$13$$ $$q+(-2-\beta _{5})q^{3}+5\beta _{2}q^{5}+(5-3\beta _{2}+\cdots)q^{7}+\cdots$$
180.4.i.c $14$ $10.620$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$5$$ $$-35$$ $$-8$$ $$q+(-\beta _{3}+\beta _{5})q^{3}+(-5+5\beta _{2})q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(180, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(180, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$