# Properties

 Label 1200.3.l Level $1200$ Weight $3$ Character orbit 1200.l Rep. character $\chi_{1200}(401,\cdot)$ Character field $\Q$ Dimension $73$ Newform subspaces $25$ Sturm bound $720$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$25$$ Sturm bound: $$720$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$

## Dimensions

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

Total New Old
Modular forms 516 79 437
Cusp forms 444 73 371
Eisenstein series 72 6 66

## Trace form

 $$73 q - q^{3} + 6 q^{7} - 3 q^{9} + O(q^{10})$$ $$73 q - q^{3} + 6 q^{7} - 3 q^{9} + 2 q^{13} - 2 q^{19} + 14 q^{21} - 25 q^{27} + 30 q^{31} + 16 q^{33} - 14 q^{37} + 26 q^{39} - 82 q^{43} + 419 q^{49} - 44 q^{51} - 22 q^{57} + 90 q^{61} - 58 q^{63} + 62 q^{67} + 68 q^{69} + 10 q^{73} + 294 q^{79} - 63 q^{81} + 144 q^{87} - 124 q^{91} + 114 q^{93} - 166 q^{97} + 352 q^{99} + O(q^{100})$$

## 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.l.a $1$ $32.698$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$-13$$ $$q-3q^{3}-13q^{7}+9q^{9}-23q^{13}-11q^{19}+\cdots$$
1200.3.l.b $1$ $32.698$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$2$$ $$q-3q^{3}+2q^{7}+9q^{9}+22q^{13}-26q^{19}+\cdots$$
1200.3.l.c $1$ $32.698$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$11$$ $$q-3q^{3}+11q^{7}+9q^{9}+q^{13}+37q^{19}+\cdots$$
1200.3.l.d $1$ $32.698$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$-11$$ $$q+3q^{3}-11q^{7}+9q^{9}-q^{13}+37q^{19}+\cdots$$
1200.3.l.e $1$ $32.698$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$13$$ $$q+3q^{3}+13q^{7}+9q^{9}+23q^{13}-11q^{19}+\cdots$$
1200.3.l.f $2$ $32.698$ $$\Q(\sqrt{-11})$$ None $$0$$ $$-5$$ $$0$$ $$0$$ $$q+(-3+\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots$$
1200.3.l.g $2$ $32.698$ $$\Q(\sqrt{-5})$$ None $$0$$ $$-4$$ $$0$$ $$-12$$ $$q+(-2+\beta )q^{3}-6q^{7}+(-1-4\beta )q^{9}+\cdots$$
1200.3.l.h $2$ $32.698$ $$\Q(\sqrt{-5})$$ None $$0$$ $$-4$$ $$0$$ $$16$$ $$q+(-2+\beta )q^{3}+8q^{7}+(-1-4\beta )q^{9}+\cdots$$
1200.3.l.i $2$ $32.698$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-2$$ $$0$$ $$-14$$ $$q+(-1+\beta )q^{3}-7q^{7}+(-7-2\beta )q^{9}+\cdots$$
1200.3.l.j $2$ $32.698$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-2$$ $$0$$ $$2$$ $$q+(-1-\beta )q^{3}+q^{7}+(-7+2\beta )q^{9}+\cdots$$
1200.3.l.k $2$ $32.698$ $$\Q(\sqrt{-35})$$ None $$0$$ $$-1$$ $$0$$ $$-16$$ $$q-\beta q^{3}-8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots$$
1200.3.l.l $2$ $32.698$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-9q^{9}+14iq^{17}-22q^{19}+\cdots$$
1200.3.l.m $2$ $32.698$ $$\Q(\sqrt{-35})$$ None $$0$$ $$1$$ $$0$$ $$16$$ $$q+\beta q^{3}+8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots$$
1200.3.l.n $2$ $32.698$ $$\Q(\sqrt{-2})$$ None $$0$$ $$2$$ $$0$$ $$-12$$ $$q+(1+\beta )q^{3}-6q^{7}+(-7+2\beta )q^{9}+\cdots$$
1200.3.l.o $2$ $32.698$ $$\Q(\sqrt{-2})$$ None $$0$$ $$2$$ $$0$$ $$-2$$ $$q+(1-\beta )q^{3}-q^{7}+(-7-2\beta )q^{9}+3\beta q^{11}+\cdots$$
1200.3.l.p $2$ $32.698$ $$\Q(\sqrt{-2})$$ None $$0$$ $$2$$ $$0$$ $$14$$ $$q+(1-\beta )q^{3}+7q^{7}+(-7-2\beta )q^{9}+\cdots$$
1200.3.l.q $2$ $32.698$ $$\Q(\sqrt{-5})$$ None $$0$$ $$4$$ $$0$$ $$-16$$ $$q+(2-\beta )q^{3}-8q^{7}+(-1-4\beta )q^{9}+\cdots$$
1200.3.l.r $2$ $32.698$ $$\Q(\sqrt{-5})$$ None $$0$$ $$4$$ $$0$$ $$4$$ $$q+(2+\beta )q^{3}+2q^{7}+(-1+4\beta )q^{9}+\cdots$$
1200.3.l.s $2$ $32.698$ $$\Q(\sqrt{-11})$$ None $$0$$ $$5$$ $$0$$ $$0$$ $$q+(3-\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots$$
1200.3.l.t $4$ $32.698$ $$\Q(\sqrt{-2}, \sqrt{-17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}+\beta _{2})q^{7}+\cdots$$
1200.3.l.u $4$ $32.698$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(1+\beta _{3})q^{3}+(2+\beta _{1}-2\beta _{2}-2\beta _{3})q^{7}+\cdots$$
1200.3.l.v $6$ $32.698$ 6.0.574198272.1 None $$0$$ $$-4$$ $$0$$ $$10$$ $$q+(-1-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots$$
1200.3.l.w $6$ $32.698$ 6.0.574198272.1 None $$0$$ $$4$$ $$0$$ $$-10$$ $$q+(1+\beta _{2})q^{3}+(-2+\beta _{1})q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots$$
1200.3.l.x $8$ $32.698$ 8.0.$$\cdots$$.5 None $$0$$ $$-4$$ $$0$$ $$16$$ $$q+\beta _{2}q^{3}+(2+\beta _{3}-\beta _{4}-\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots$$
1200.3.l.y $12$ $32.698$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}-\beta _{5}q^{7}+(-1-\beta _{9})q^{9}+(\beta _{7}+\cdots)q^{11}+\cdots$$

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

$$S_{3}^{\mathrm{old}}(1200, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$