# Properties

 Label 225.2.h Level $225$ Weight $2$ Character orbit 225.h Rep. character $\chi_{225}(46,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $44$ Newform subspaces $5$ Sturm bound $60$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.h (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$5$$ Sturm bound: $$60$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(225, [\chi])$$.

Total New Old
Modular forms 136 52 84
Cusp forms 104 44 60
Eisenstein series 32 8 24

## Trace form

 $$44 q + 4 q^{2} - 12 q^{4} + 11 q^{5} - 10 q^{7} - 7 q^{8} + O(q^{10})$$ $$44 q + 4 q^{2} - 12 q^{4} + 11 q^{5} - 10 q^{7} - 7 q^{8} - 4 q^{10} - 4 q^{11} - 3 q^{13} + 11 q^{14} + 8 q^{16} + 2 q^{17} + 7 q^{19} - 21 q^{20} - 14 q^{22} - 25 q^{23} - 11 q^{25} + 44 q^{26} + 5 q^{28} - q^{29} + 15 q^{31} - 30 q^{32} - 20 q^{34} + 25 q^{35} - 5 q^{37} + 21 q^{38} + 7 q^{40} - 2 q^{41} - 54 q^{43} - 32 q^{44} - 3 q^{46} + 18 q^{47} - 14 q^{49} - 76 q^{50} - 12 q^{52} - 45 q^{53} - 36 q^{55} + 18 q^{58} - 12 q^{59} + 25 q^{61} - 20 q^{62} - 7 q^{64} - 12 q^{65} - 42 q^{67} + 128 q^{68} - 20 q^{70} - 16 q^{71} + 45 q^{73} + 66 q^{74} - 36 q^{76} + 60 q^{77} + 31 q^{79} + 169 q^{80} + 120 q^{82} + 17 q^{83} - 22 q^{85} - 39 q^{86} + 82 q^{88} - 28 q^{89} - 8 q^{91} - 10 q^{92} - 27 q^{94} - 91 q^{95} + 48 q^{97} - 53 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.2.h.a $4$ $1.797$ $$\Q(\zeta_{10})$$ None $$1$$ $$0$$ $$-5$$ $$0$$ $$q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{4}+\cdots$$
225.2.h.b $4$ $1.797$ $$\Q(\zeta_{10})$$ None $$2$$ $$0$$ $$5$$ $$-2$$ $$q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\zeta_{10}^{3})q^{4}+\cdots$$
225.2.h.c $8$ $1.797$ 8.0.26265625.1 None $$1$$ $$0$$ $$5$$ $$4$$ $$q+(\beta _{3}+\beta _{7})q^{2}+(-1+2\beta _{1}+2\beta _{3}+\cdots)q^{4}+\cdots$$
225.2.h.d $12$ $1.797$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$6$$ $$-12$$ $$q+\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{5}-2\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots$$
225.2.h.e $16$ $1.797$ 16.0.$$\cdots$$.3 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{3}-\beta _{10}+\beta _{12}-\beta _{14})q^{2}+(-\beta _{8}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

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