# Properties

 Label 1225.2.b Level $1225$ Weight $2$ Character orbit 1225.b Rep. character $\chi_{1225}(99,\cdot)$ Character field $\Q$ Dimension $56$ Newform subspaces $14$ Sturm bound $280$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$280$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$, $$19$$, $$31$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1225, [\chi])$$.

Total New Old
Modular forms 164 66 98
Cusp forms 116 56 60
Eisenstein series 48 10 38

## Trace form

 $$56q - 56q^{4} + 8q^{6} - 36q^{9} + O(q^{10})$$ $$56q - 56q^{4} + 8q^{6} - 36q^{9} - 8q^{11} + 40q^{16} - 8q^{19} - 28q^{24} + 40q^{29} + 20q^{31} - 36q^{34} + 48q^{36} - 16q^{39} - 8q^{41} + 64q^{44} + 24q^{46} - 52q^{51} + 24q^{54} + 4q^{59} - 16q^{61} - 36q^{64} - 8q^{66} - 12q^{69} - 68q^{71} + 60q^{74} + 52q^{76} - 4q^{79} + 8q^{81} - 24q^{86} + 48q^{89} + 52q^{94} - 20q^{96} + 44q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1225, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1225.2.b.a $$2$$ $$9.782$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+3iq^{3}-2q^{4}-6q^{6}-6q^{9}+\cdots$$
1225.2.b.b $$2$$ $$9.782$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}-3iq^{3}-2q^{4}+6q^{6}-6q^{9}+\cdots$$
1225.2.b.c $$2$$ $$9.782$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+q^{4}+3iq^{8}+3q^{9}+4q^{11}+\cdots$$
1225.2.b.d $$2$$ $$9.782$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2q^{4}+2q^{9}-3q^{11}-2iq^{12}+\cdots$$
1225.2.b.e $$4$$ $$9.782$$ $$\Q(i, \sqrt{21})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-4+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{8}+\cdots$$
1225.2.b.f $$4$$ $$9.782$$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots$$
1225.2.b.g $$4$$ $$9.782$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{3}+(-1-\zeta_{8}^{3})q^{4}+\cdots$$
1225.2.b.h $$4$$ $$9.782$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{3}+(-1-\zeta_{8}^{3})q^{4}+\cdots$$
1225.2.b.i $$4$$ $$9.782$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}+(\zeta_{8}+\zeta_{8}^{2})q^{3}+(-2-\zeta_{8}^{3})q^{6}+\cdots$$
1225.2.b.j $$4$$ $$9.782$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}+(-\zeta_{8}-\zeta_{8}^{2})q^{3}+(2+\zeta_{8}^{3})q^{6}+\cdots$$
1225.2.b.k $$4$$ $$9.782$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(2\beta _{1}+2\beta _{3})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots$$
1225.2.b.l $$6$$ $$9.782$$ 6.0.4227136.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-\beta _{3}+\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$
1225.2.b.m $$6$$ $$9.782$$ 6.0.4227136.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{3}-\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$
1225.2.b.n $$8$$ $$9.782$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{2}-\beta _{3})q^{2}+(-\beta _{5}+\beta _{7})q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1225, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1225, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(245, [\chi])$$$$^{\oplus 2}$$