# Properties

 Label 225.4.e Level $225$ Weight $4$ Character orbit 225.e Rep. character $\chi_{225}(76,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $108$ Newform subspaces $7$ Sturm bound $120$ Trace bound $2$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 192 120 72
Cusp forms 168 108 60
Eisenstein series 24 12 12

## Trace form

 $$108 q - q^{2} + 5 q^{3} - 203 q^{4} - 27 q^{6} - 5 q^{7} + 42 q^{8} - 75 q^{9} + O(q^{10})$$ $$108 q - q^{2} + 5 q^{3} - 203 q^{4} - 27 q^{6} - 5 q^{7} + 42 q^{8} - 75 q^{9} + 12 q^{11} - 158 q^{12} + 13 q^{13} + 174 q^{14} - 715 q^{16} + 302 q^{17} - 256 q^{18} - 146 q^{19} - 69 q^{21} + 3 q^{22} - 159 q^{23} + 135 q^{24} - 648 q^{26} + 308 q^{27} + 52 q^{28} + 69 q^{29} - 81 q^{31} + 61 q^{32} + 1024 q^{33} + 85 q^{34} + 2103 q^{36} - 236 q^{37} - 649 q^{38} - 711 q^{39} + 678 q^{41} - 2184 q^{42} - 86 q^{43} + 186 q^{44} + 512 q^{46} - q^{47} - 431 q^{48} - 1841 q^{49} - 75 q^{51} + 532 q^{52} + 1916 q^{53} - 1743 q^{54} + 1974 q^{56} + 1475 q^{57} + 534 q^{58} + 738 q^{59} - 405 q^{61} + 1416 q^{62} - 2331 q^{63} + 4562 q^{64} + 1116 q^{66} - 374 q^{67} - 2039 q^{68} + 1917 q^{69} - 6672 q^{71} + 3069 q^{72} - 902 q^{73} + 1764 q^{74} + 251 q^{76} - 1221 q^{77} + 6508 q^{78} - 383 q^{79} + 3489 q^{81} - 138 q^{82} - 1695 q^{83} - 4524 q^{84} - 591 q^{86} - 6127 q^{87} + 1653 q^{88} - 4488 q^{89} - 1934 q^{91} - 4998 q^{92} - 1251 q^{93} + 814 q^{94} - 7308 q^{96} + 1210 q^{97} - 1850 q^{98} - 27 q^{99} + O(q^{100})$$

## 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.e.a $4$ $13.275$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$-1$$ $$6$$ $$0$$ $$9$$ $$q+(-\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2}+2\beta _{3})q^{3}+\cdots$$
225.4.e.b $4$ $13.275$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$3$$ $$3$$ $$0$$ $$7$$ $$q+(\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots$$
225.4.e.c $6$ $13.275$ 6.0.15759792.1 None $$-1$$ $$-9$$ $$0$$ $$-43$$ $$q+(\beta _{1}-\beta _{5})q^{2}+(-1-\beta _{2}+\beta _{5})q^{3}+\cdots$$
225.4.e.d $14$ $13.275$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$-2$$ $$5$$ $$0$$ $$22$$ $$q+(\beta _{1}-\beta _{2})q^{2}+(-\beta _{7}+\beta _{10})q^{3}+(-5+\cdots)q^{4}+\cdots$$
225.4.e.e $24$ $13.275$ None $$-4$$ $$-1$$ $$0$$ $$6$$
225.4.e.f $24$ $13.275$ None $$4$$ $$1$$ $$0$$ $$-6$$
225.4.e.g $32$ $13.275$ None $$0$$ $$0$$ $$0$$ $$0$$

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

$$S_{4}^{\mathrm{old}}(225, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$