# Properties

 Label 1200.4.f Level $1200$ Weight $4$ Character orbit 1200.f Rep. character $\chi_{1200}(49,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $24$ Sturm bound $960$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$24$$ Sturm bound: $$960$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(1200, [\chi])$$.

Total New Old
Modular forms 756 54 702
Cusp forms 684 54 630
Eisenstein series 72 0 72

## Trace form

 $$54 q - 486 q^{9} + O(q^{10})$$ $$54 q - 486 q^{9} + 252 q^{19} + 284 q^{29} + 636 q^{31} - 312 q^{39} - 356 q^{41} - 2030 q^{49} + 132 q^{51} + 1376 q^{59} + 756 q^{61} - 528 q^{69} - 224 q^{71} - 2240 q^{79} + 4374 q^{81} - 1932 q^{89} - 812 q^{91} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(1200, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.4.f.a $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+4iq^{7}-9q^{9}-72q^{11}+\cdots$$
1200.4.f.b $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+24iq^{7}-9q^{9}-52q^{11}+\cdots$$
1200.4.f.c $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+iq^{7}-9q^{9}-42q^{11}+67iq^{13}+\cdots$$
1200.4.f.d $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+8iq^{7}-9q^{9}-6^{2}q^{11}+\cdots$$
1200.4.f.e $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+2^{5}iq^{7}-9q^{9}-6^{2}q^{11}+\cdots$$
1200.4.f.f $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+19iq^{7}-9q^{9}-22q^{11}+\cdots$$
1200.4.f.g $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+8iq^{7}-9q^{9}-20q^{11}+\cdots$$
1200.4.f.h $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+20iq^{7}-9q^{9}-2^{4}q^{11}+\cdots$$
1200.4.f.i $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+5iq^{7}-9q^{9}-14q^{11}+\cdots$$
1200.4.f.j $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+2^{4}iq^{7}-9q^{9}-12q^{11}+\cdots$$
1200.4.f.k $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+13iq^{7}-9q^{9}-6q^{11}+\cdots$$
1200.4.f.l $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-9q^{9}-4q^{11}+54iq^{13}+\cdots$$
1200.4.f.m $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+20iq^{7}-9q^{9}+24q^{11}+\cdots$$
1200.4.f.n $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+28iq^{7}-9q^{9}+24q^{11}+\cdots$$
1200.4.f.o $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+2^{4}iq^{7}-9q^{9}+28q^{11}+\cdots$$
1200.4.f.p $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+24iq^{7}-9q^{9}+28q^{11}+\cdots$$
1200.4.f.q $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+23iq^{7}-9q^{9}+30q^{11}+\cdots$$
1200.4.f.r $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+4iq^{7}-9q^{9}+48q^{11}+\cdots$$
1200.4.f.s $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+7iq^{7}-9q^{9}+54q^{11}+\cdots$$
1200.4.f.t $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+20iq^{7}-9q^{9}+56q^{11}+\cdots$$
1200.4.f.u $2$ $70.802$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+2^{5}iq^{7}-9q^{9}+60q^{11}+\cdots$$
1200.4.f.v $4$ $70.802$ $$\Q(i, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}+(-13\beta _{1}+\beta _{3})q^{7}-9q^{9}+\cdots$$
1200.4.f.w $4$ $70.802$ $$\Q(i, \sqrt{181})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}+(-3\beta _{1}+\beta _{2})q^{7}-9q^{9}+\cdots$$
1200.4.f.x $4$ $70.802$ $$\Q(i, \sqrt{109})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{7}-9q^{9}+(8+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(1200, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(1200, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$