# Properties

 Label 1200.3.c Level $1200$ Weight $3$ Character orbit 1200.c Rep. character $\chi_{1200}(449,\cdot)$ Character field $\Q$ Dimension $70$ Newform subspaces $13$ Sturm bound $720$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$720$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1200, [\chi])$$.

Total New Old
Modular forms 516 74 442
Cusp forms 444 70 374
Eisenstein series 72 4 68

## Trace form

 $$70 q + 2 q^{9} + O(q^{10})$$ $$70 q + 2 q^{9} - 36 q^{19} - 4 q^{21} - 60 q^{31} - 116 q^{39} - 322 q^{49} - 208 q^{51} - 132 q^{61} - 128 q^{69} - 180 q^{79} + 142 q^{81} + 104 q^{91} - 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1200, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.3.c.a $2$ $32.698$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+11iq^{7}-9q^{9}-iq^{13}+\cdots$$
1200.3.c.b $2$ $32.698$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+13iq^{7}-9q^{9}-23iq^{13}+\cdots$$
1200.3.c.c $2$ $32.698$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+2iq^{7}-9q^{9}-22iq^{13}+\cdots$$
1200.3.c.d $4$ $32.698$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-3+\beta _{3})q^{9}+(-1-2\beta _{3})q^{11}+\cdots$$
1200.3.c.e $4$ $32.698$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots$$
1200.3.c.f $4$ $32.698$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{3}-3\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots$$
1200.3.c.g $4$ $32.698$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots$$
1200.3.c.h $4$ $32.698$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+7\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots$$
1200.3.c.i $4$ $32.698$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\zeta_{8}-\zeta_{8}^{2})q^{3}-3\zeta_{8}q^{7}+(7+\zeta_{8}^{3})q^{9}+\cdots$$
1200.3.c.j $4$ $32.698$ $$\Q(i, \sqrt{35})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-8\beta _{2}q^{7}+(9+\beta _{3})q^{9}+(3+\cdots)q^{11}+\cdots$$
1200.3.c.k $8$ $32.698$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(-2\beta _{1}-2\beta _{2}+\beta _{7})q^{7}+\cdots$$
1200.3.c.l $12$ $32.698$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{6}-\beta _{10})q^{3}+(-2\beta _{6}-\beta _{7}+\beta _{8}+\cdots)q^{7}+\cdots$$
1200.3.c.m $16$ $32.698$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1200, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1200, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$