# Properties

 Label 1200.2.h Level $1200$ Weight $2$ Character orbit 1200.h Rep. character $\chi_{1200}(1151,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $14$ Sturm bound $480$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$480$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1200, [\chi])$$.

Total New Old
Modular forms 276 38 238
Cusp forms 204 38 166
Eisenstein series 72 0 72

## Trace form

 $$38 q - 6 q^{9} + O(q^{10})$$ $$38 q - 6 q^{9} - 4 q^{13} - 12 q^{21} + 28 q^{37} - 22 q^{49} + 36 q^{57} - 20 q^{61} + 48 q^{69} + 44 q^{73} - 6 q^{81} - 12 q^{93} - 52 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1200, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.2.h.a $2$ $9.582$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$0$$ $$0$$ $$q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}-3q^{11}+\cdots$$
1200.2.h.b $2$ $9.582$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$0$$ $$0$$ $$q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}+3q^{11}+\cdots$$
1200.2.h.c $2$ $9.582$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}-7q^{13}+\cdots$$
1200.2.h.d $2$ $9.582$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}-5q^{13}-5\zeta_{6}q^{19}+\cdots$$
1200.2.h.e $2$ $9.582$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{6}q^{3}-2\zeta_{6}q^{7}-3q^{9}+2q^{13}+\cdots$$
1200.2.h.f $2$ $9.582$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}+5q^{13}+5\zeta_{6}q^{19}+\cdots$$
1200.2.h.g $2$ $9.582$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}+7q^{13}+\cdots$$
1200.2.h.h $2$ $9.582$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$0$$ $$q+(1+\zeta_{6})q^{3}+3\zeta_{6}q^{9}-3q^{11}-2q^{13}+\cdots$$
1200.2.h.i $2$ $9.582$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$0$$ $$q+(2-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{9}+3q^{11}+\cdots$$
1200.2.h.j $4$ $9.582$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\beta _{1})q^{3}+(-1-2\beta _{1})q^{9}+\beta _{3}q^{11}+\cdots$$
1200.2.h.k $4$ $9.582$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+\beta _{2}q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots$$
1200.2.h.l $4$ $9.582$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(2+\beta _{2})q^{9}+\cdots$$
1200.2.h.m $4$ $9.582$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{3}-\zeta_{12}^{2}q^{7}+3q^{9}-2\zeta_{12}q^{11}+\cdots$$
1200.2.h.n $4$ $9.582$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1-\beta _{1})q^{3}+(-1-2\beta _{1})q^{9}-\beta _{3}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1200, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1200, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$