# Properties

 Label 825.2.n Level $825$ Weight $2$ Character orbit 825.n Rep. character $\chi_{825}(301,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $152$ Newform subspaces $16$ Sturm bound $240$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.n (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$11$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$16$$ Sturm bound: $$240$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(825, [\chi])$$.

Total New Old
Modular forms 528 152 376
Cusp forms 432 152 280
Eisenstein series 96 0 96

## Trace form

 $$152 q - 4 q^{2} - 42 q^{4} + 2 q^{6} - 6 q^{7} + 8 q^{8} - 38 q^{9} + O(q^{10})$$ $$152 q - 4 q^{2} - 42 q^{4} + 2 q^{6} - 6 q^{7} + 8 q^{8} - 38 q^{9} + 10 q^{11} + 8 q^{12} + 6 q^{13} - 6 q^{14} - 34 q^{16} + 2 q^{17} + 6 q^{18} + 20 q^{19} - 16 q^{21} + 36 q^{22} + 20 q^{23} + 24 q^{24} - 18 q^{26} - 16 q^{28} - 4 q^{29} - 8 q^{32} + 6 q^{33} - 40 q^{34} - 42 q^{36} + 46 q^{37} + 46 q^{38} + 16 q^{39} - 28 q^{41} - 22 q^{42} + 24 q^{43} + 70 q^{44} + 70 q^{46} + 18 q^{47} - 40 q^{48} - 76 q^{49} - 8 q^{51} - 62 q^{52} - 26 q^{53} - 8 q^{54} - 156 q^{56} - 8 q^{57} + 44 q^{58} - 14 q^{59} - 14 q^{61} + 12 q^{62} - 6 q^{63} - 118 q^{64} - 12 q^{66} + 12 q^{67} + 68 q^{68} - 2 q^{69} + 4 q^{71} - 2 q^{72} + 16 q^{73} + 50 q^{74} + 48 q^{76} + 94 q^{77} - 80 q^{78} + 34 q^{79} - 38 q^{81} + 50 q^{82} - 30 q^{83} + 72 q^{84} + 64 q^{87} + 42 q^{88} + 76 q^{89} + 26 q^{91} + 70 q^{92} - 52 q^{93} + 54 q^{94} + 20 q^{96} - 30 q^{97} - 136 q^{98} + 20 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(825, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
825.2.n.a $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$-3$$ $$-1$$ $$0$$ $$-3$$ $$q+(-1+\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$$
825.2.n.b $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$-2$$ $$1$$ $$0$$ $$8$$ $$q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}-\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots$$
825.2.n.c $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$1$$ $$-1$$ $$0$$ $$-1$$ $$q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
825.2.n.d $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$2$$ $$-1$$ $$0$$ $$-8$$ $$q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots$$
825.2.n.e $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$3$$ $$1$$ $$0$$ $$3$$ $$q+(1-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$$
825.2.n.f $$4$$ $$6.588$$ $$\Q(\zeta_{10})$$ None $$3$$ $$1$$ $$0$$ $$3$$ $$q+(1-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$$
825.2.n.g $$8$$ $$6.588$$ 8.0.13140625.1 None $$-4$$ $$2$$ $$0$$ $$-3$$ $$q+(-1-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\beta _{5}+\cdots)q^{2}+\cdots$$
825.2.n.h $$8$$ $$6.588$$ 8.0.819390625.1 None $$-3$$ $$2$$ $$0$$ $$-7$$ $$q-\beta _{1}q^{2}+(1+\beta _{2}+\beta _{3}-\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots$$
825.2.n.i $$8$$ $$6.588$$ 8.0.819390625.1 None $$-2$$ $$-2$$ $$0$$ $$-9$$ $$q-\beta _{4}q^{2}-\beta _{6}q^{3}+(1-\beta _{1}+2\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots$$
825.2.n.j $$8$$ $$6.588$$ $$\Q(\zeta_{15})$$ None $$-2$$ $$-2$$ $$0$$ $$5$$ $$q+(1-2\zeta_{15}+\zeta_{15}^{3}-\zeta_{15}^{4}+\zeta_{15}^{5}+\cdots)q^{2}+\cdots$$
825.2.n.k $$8$$ $$6.588$$ 8.0.13140625.1 None $$0$$ $$2$$ $$0$$ $$-1$$ $$q+(-1+\beta _{3}-\beta _{4}+\beta _{5}+\beta _{6})q^{2}+(1+\cdots)q^{3}+\cdots$$
825.2.n.l $$8$$ $$6.588$$ 8.0.819390625.1 None $$3$$ $$-2$$ $$0$$ $$7$$ $$q+\beta _{1}q^{2}+(-1-\beta _{2}-\beta _{3}+\beta _{6})q^{3}+\cdots$$
825.2.n.m $$16$$ $$6.588$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-2$$ $$-4$$ $$0$$ $$4$$ $$q+(-\beta _{1}+\beta _{2}-\beta _{4}-\beta _{6})q^{2}-\beta _{11}q^{3}+\cdots$$
825.2.n.n $$16$$ $$6.588$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$2$$ $$4$$ $$0$$ $$-4$$ $$q+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{6})q^{2}+\beta _{11}q^{3}+\cdots$$
825.2.n.o $$24$$ $$6.588$$ None $$-2$$ $$6$$ $$0$$ $$-4$$
825.2.n.p $$24$$ $$6.588$$ None $$2$$ $$-6$$ $$0$$ $$4$$

## Decomposition of $$S_{2}^{\mathrm{old}}(825, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(825, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(33, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(165, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(275, [\chi])$$$$^{\oplus 2}$$