# Properties

 Label 225.3.g Level $225$ Weight $3$ Character orbit 225.g Rep. character $\chi_{225}(82,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $28$ Newform subspaces $7$ Sturm bound $90$ Trace bound $16$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 225.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$7$$ Sturm bound: $$90$$ Trace bound: $$16$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(225, [\chi])$$.

Total New Old
Modular forms 144 32 112
Cusp forms 96 28 68
Eisenstein series 48 4 44

## Trace form

 $$28 q - 4 q^{2} + 16 q^{7} + 12 q^{8} + O(q^{10})$$ $$28 q - 4 q^{2} + 16 q^{7} + 12 q^{8} + 44 q^{11} - 8 q^{13} - 228 q^{16} - 40 q^{17} + 80 q^{22} + 56 q^{23} + 128 q^{26} - 64 q^{28} + 96 q^{31} - 76 q^{32} - 104 q^{37} - 96 q^{38} + 140 q^{41} - 32 q^{43} + 520 q^{46} + 128 q^{47} + 40 q^{52} + 56 q^{53} - 600 q^{56} + 312 q^{58} - 568 q^{61} + 88 q^{62} - 80 q^{67} - 104 q^{68} - 712 q^{71} - 296 q^{73} - 140 q^{76} + 88 q^{77} - 328 q^{82} - 16 q^{83} + 1064 q^{86} + 288 q^{88} + 520 q^{91} + 104 q^{92} + 40 q^{97} - 188 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.3.g.a $4$ $6.131$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
225.3.g.b $4$ $6.131$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\beta _{1}q^{2}+8\beta _{2}q^{4}+5\beta _{1}q^{7}+8\beta _{3}q^{8}+\cdots$$
225.3.g.c $4$ $6.131$ $$\Q(i, \sqrt{30})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+11\beta _{2}q^{4}+7\beta _{3}q^{8}-61q^{16}+\cdots$$
225.3.g.d $4$ $6.131$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{2}q^{4}+\beta _{1}q^{7}-3\beta _{3}q^{13}-2^{4}q^{16}+\cdots$$
225.3.g.e $4$ $6.131$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{4}+4\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots$$
225.3.g.f $4$ $6.131$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{4}-6\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots$$
225.3.g.g $4$ $6.131$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$20$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(5+5\beta _{2})q^{7}-3\beta _{3}q^{8}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(225, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$