# Properties

 Label 1200.2.v Level $1200$ Weight $2$ Character orbit 1200.v Rep. character $\chi_{1200}(257,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $68$ Newform subspaces $13$ Sturm bound $480$ Trace bound $21$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.v (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$13$$ Sturm bound: $$480$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

## Dimensions

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

Total New Old
Modular forms 552 76 476
Cusp forms 408 68 340
Eisenstein series 144 8 136

## Trace form

 $$68 q - 2 q^{3} - 4 q^{7} + O(q^{10})$$ $$68 q - 2 q^{3} - 4 q^{7} + 4 q^{13} + 12 q^{21} - 14 q^{27} + 24 q^{31} - 4 q^{33} + 20 q^{37} + 12 q^{43} + 52 q^{51} - 20 q^{57} + 24 q^{61} + 48 q^{63} + 20 q^{67} - 4 q^{73} - 20 q^{81} + 20 q^{87} + 32 q^{91} + 8 q^{93} - 4 q^{97} + O(q^{100})$$

## 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.v.a $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-2+2\zeta_{8}^{2}+\cdots)q^{7}+\cdots$$
1200.2.v.b $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-1+\zeta_{8}^{2}+\cdots)q^{7}+\cdots$$
1200.2.v.c $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$12$$ $$q+(-1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(3+3\zeta_{8}^{2}+\cdots)q^{7}+\cdots$$
1200.2.v.d $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+(-1-\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{7}+\cdots$$
1200.2.v.e $4$ $9.582$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+2\beta _{1}q^{7}-3\beta _{2}q^{9}-4\beta _{3}q^{13}+\cdots$$
1200.2.v.f $4$ $9.582$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+\beta _{1}q^{7}-3\beta _{2}q^{9}+3\beta _{3}q^{13}+\cdots$$
1200.2.v.g $4$ $9.582$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}-3\beta _{2}q^{9}+4\beta _{1}q^{17}-4\beta _{2}q^{19}+\cdots$$
1200.2.v.h $4$ $9.582$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+3\beta _{1}q^{7}-3\beta _{2}q^{9}-\beta _{3}q^{13}+\cdots$$
1200.2.v.i $4$ $9.582$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$2$$ $$0$$ $$-4$$ $$q+(1-\beta _{3})q^{3}+(-1+\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots$$
1200.2.v.j $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$-4$$ $$q+(1-\zeta_{8}^{2})q^{3}+(-1-\zeta_{8}+\zeta_{8}^{2}+\zeta_{8}^{3})q^{7}+\cdots$$
1200.2.v.k $4$ $9.582$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(2-2\zeta_{8}^{2})q^{7}+\cdots$$
1200.2.v.l $8$ $9.582$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{3}q^{3}-2\zeta_{24}q^{7}+(2\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{9}+\cdots$$
1200.2.v.m $16$ $9.582$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{9}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{5})q^{7}+(-\beta _{3}+\cdots)q^{9}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1200, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$