# Properties

 Label 225.4.b Level $225$ Weight $4$ Character orbit 225.b Rep. character $\chi_{225}(199,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $9$ Sturm bound $120$ Trace bound $11$

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## Defining parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$120$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(225, [\chi])$$.

Total New Old
Modular forms 102 24 78
Cusp forms 78 22 56
Eisenstein series 24 2 22

## Trace form

 $$22 q - 98 q^{4} + O(q^{10})$$ $$22 q - 98 q^{4} - 90 q^{11} + 288 q^{14} + 314 q^{16} - 154 q^{19} + 516 q^{26} + 708 q^{29} + 396 q^{31} + 194 q^{34} - 1326 q^{41} + 402 q^{44} - 324 q^{46} - 1742 q^{49} + 660 q^{56} + 936 q^{59} + 268 q^{61} - 1066 q^{64} - 2064 q^{71} - 852 q^{74} - 286 q^{76} + 4860 q^{79} + 4860 q^{86} - 1206 q^{89} - 1584 q^{91} + 4160 q^{94} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.4.b.a $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-17q^{4}+6iq^{7}-9iq^{8}-50q^{11}+\cdots$$
225.4.b.b $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-17q^{4}-6iq^{7}-9iq^{8}+50q^{11}+\cdots$$
225.4.b.c $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}-8q^{4}+3iq^{7}-2^{5}q^{11}+\cdots$$
225.4.b.d $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-q^{4}-20iq^{7}+21iq^{8}+\cdots$$
225.4.b.e $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+7q^{4}+24iq^{7}+15iq^{8}+\cdots$$
225.4.b.f $2$ $13.275$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+7q^{4}-6iq^{7}+15iq^{8}+43q^{11}+\cdots$$
225.4.b.g $2$ $13.275$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+8q^{4}+2iq^{7}+7iq^{13}+2^{6}q^{16}+\cdots$$
225.4.b.h $4$ $13.275$ $$\Q(i, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{3})q^{2}+(-12+2\beta _{2})q^{4}+\cdots$$
225.4.b.i $4$ $13.275$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-2q^{4}+3\beta _{1}q^{7}-6\beta _{2}q^{8}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

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