# Properties

 Label 1300.2.ba Level $1300$ Weight $2$ Character orbit 1300.ba Rep. character $\chi_{1300}(49,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $40$ Newform subspaces $4$ Sturm bound $420$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.ba (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$420$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 456 40 416
Cusp forms 384 40 344
Eisenstein series 72 0 72

## Trace form

 $$40 q + 20 q^{9} + O(q^{10})$$ $$40 q + 20 q^{9} + 12 q^{11} + 6 q^{19} - 24 q^{29} - 34 q^{39} + 30 q^{41} - 22 q^{49} - 60 q^{51} + 42 q^{59} + 4 q^{61} + 42 q^{69} + 36 q^{71} - 16 q^{79} + 28 q^{81} - 30 q^{89} - 12 q^{91} + O(q^{100})$$

## 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.ba.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.ba.b $8$ $10.381$ 8.0.22581504.2 None $$0$$ $$-6$$ $$0$$ $$6$$ $$q+(\beta _{2}+\beta _{6})q^{3}+(1-\beta _{4}-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots$$
1300.2.ba.c $8$ $10.381$ 8.0.22581504.2 None $$0$$ $$6$$ $$0$$ $$-6$$ $$q+(-\beta _{2}-\beta _{6})q^{3}+(-1+\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots$$
1300.2.ba.d $20$ $10.381$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{18}q^{3}+(-\beta _{10}-\beta _{16}-\beta _{19})q^{7}+\cdots$$

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

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