# Properties

 Label 360.2.q Level $360$ Weight $2$ Character orbit 360.q Rep. character $\chi_{360}(121,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $5$ Sturm bound $144$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$5$$ Sturm bound: $$144$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(360, [\chi])$$.

Total New Old
Modular forms 160 24 136
Cusp forms 128 24 104
Eisenstein series 32 0 32

## Trace form

 $$24 q - 2 q^{3} - 2 q^{5} + 4 q^{9} + O(q^{10})$$ $$24 q - 2 q^{3} - 2 q^{5} + 4 q^{9} + 2 q^{11} + 28 q^{17} - 12 q^{19} + 20 q^{21} - 12 q^{25} + 4 q^{27} - 6 q^{29} + 12 q^{31} - 14 q^{33} - 24 q^{35} - 36 q^{39} - 4 q^{41} + 18 q^{43} - 12 q^{45} + 12 q^{47} - 6 q^{49} + 14 q^{51} + 8 q^{53} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 8 q^{63} - 8 q^{65} + 6 q^{67} - 58 q^{69} - 8 q^{71} + 36 q^{73} - 2 q^{75} - 40 q^{77} + 4 q^{81} + 32 q^{83} + 8 q^{87} - 36 q^{89} - 24 q^{91} + 8 q^{93} - 18 q^{97} - 80 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(360, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
360.2.q.a $2$ $2.875$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$1$$ $$0$$ $$q+(-1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+3\zeta_{6}q^{9}+\cdots$$
360.2.q.b $4$ $2.875$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$-4$$ $$q-\zeta_{12}^{2}q^{3}+\zeta_{12}q^{5}+(-2+2\zeta_{12}+\cdots)q^{7}+\cdots$$
360.2.q.c $4$ $2.875$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$2$$ $$-2$$ $$q+(\beta _{1}+\beta _{2}-\beta _{3})q^{3}+\beta _{2}q^{5}+(-1+\cdots)q^{7}+\cdots$$
360.2.q.d $6$ $2.875$ 6.0.954288.1 None $$0$$ $$-1$$ $$-3$$ $$5$$ $$q-\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
360.2.q.e $8$ $2.875$ 8.0.856615824.2 None $$0$$ $$0$$ $$-4$$ $$1$$ $$q-\beta _{4}q^{3}-\beta _{1}q^{5}+(-\beta _{2}+\beta _{4})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(360, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(360, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$