# 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

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

## 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

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