# Properties

 Label 1300.2.bb Level $1300$ Weight $2$ Character orbit 1300.bb Rep. character $\chi_{1300}(549,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $44$ Newform subspaces $7$ Sturm bound $420$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.bb (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$420$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1300, [\chi])$$.

Total New Old
Modular forms 456 44 412
Cusp forms 384 44 340
Eisenstein series 72 0 72

## Trace form

 $$44 q + 22 q^{9} + O(q^{10})$$ $$44 q + 22 q^{9} - 12 q^{11} - 2 q^{19} - 4 q^{21} - 2 q^{29} + 8 q^{31} - 2 q^{39} + 16 q^{41} + 44 q^{49} + 92 q^{51} - 2 q^{59} - 2 q^{61} + 18 q^{69} + 12 q^{71} + 16 q^{79} + 2 q^{81} + 14 q^{89} - 32 q^{91} - 56 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1300, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1300.2.bb.a $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+(-\zeta_{12}+\zeta_{12}^{3})q^{7}-2\zeta_{12}^{2}q^{9}+\cdots$$
1300.2.bb.b $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+(-\zeta_{12}+\zeta_{12}^{3})q^{7}-2\zeta_{12}^{2}q^{9}+\cdots$$
1300.2.bb.c $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+(-5\zeta_{12}+5\zeta_{12}^{3})q^{7}+\cdots$$
1300.2.bb.d $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\zeta_{12}q^{3}+(4\zeta_{12}-4\zeta_{12}^{3})q^{7}+\zeta_{12}^{2}q^{9}+\cdots$$
1300.2.bb.e $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\zeta_{12}q^{3}+(-3\zeta_{12}+3\zeta_{12}^{3})q^{7}+\cdots$$
1300.2.bb.f $4$ $10.381$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\zeta_{12}q^{3}+(\zeta_{12}-\zeta_{12}^{3})q^{7}+6\zeta_{12}^{2}q^{9}+\cdots$$
1300.2.bb.g $20$ $10.381$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{9}+\beta _{15})q^{7}+(1+\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1300, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1300, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(260, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(650, [\chi])$$$$^{\oplus 2}$$