# Properties

 Label 1600.2.n Level $1600$ Weight $2$ Character orbit 1600.n Rep. character $\chi_{1600}(1343,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $68$ Newform subspaces $22$ Sturm bound $480$ Trace bound $21$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.n (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$22$$ Sturm bound: $$480$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1600, [\chi])$$.

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

## Trace form

 $$68 q + O(q^{10})$$ $$68 q - 4 q^{13} + 12 q^{17} + 8 q^{21} - 8 q^{33} - 4 q^{37} - 8 q^{41} - 52 q^{53} + 16 q^{57} + 72 q^{61} + 12 q^{73} + 72 q^{77} - 108 q^{81} + 56 q^{93} + 28 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1600, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1600.2.n.a $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$4$$ $$q+(-2-2i)q^{3}+(2-2i)q^{7}+5iq^{9}+\cdots$$
1600.2.n.b $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$-2$$ $$0$$ $$-6$$ $$q+(-1-i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots$$
1600.2.n.c $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$-6$$ $$q+(-1-i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots$$
1600.2.n.d $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$2$$ $$q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}+4iq^{11}+\cdots$$
1600.2.n.e $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$2$$ $$q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}+6iq^{11}+\cdots$$
1600.2.n.f $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$2$$ $$q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}-4iq^{11}+\cdots$$
1600.2.n.g $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{9}+(-5+5i)q^{13}+(5+5i)q^{17}+\cdots$$
1600.2.n.h $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{9}+(-1+i)q^{13}+(-3-3i)q^{17}+\cdots$$
1600.2.n.i $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-2$$ $$q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}-4iq^{11}+\cdots$$
1600.2.n.j $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-2$$ $$q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}-6iq^{11}+\cdots$$
1600.2.n.k $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-2$$ $$q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}+4iq^{11}+\cdots$$
1600.2.n.l $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$2$$ $$0$$ $$6$$ $$q+(1+i)q^{3}+(3-3i)q^{7}-iq^{9}+6q^{21}+\cdots$$
1600.2.n.m $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$6$$ $$q+(1+i)q^{3}+(3-3i)q^{7}-iq^{9}+2iq^{11}+\cdots$$
1600.2.n.n $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$4$$ $$0$$ $$-4$$ $$q+(2+2i)q^{3}+(-2+2i)q^{7}+5iq^{9}+\cdots$$
1600.2.n.o $4$ $12.776$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots$$
1600.2.n.p $4$ $12.776$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots$$
1600.2.n.q $4$ $12.776$ $$\Q(i, \sqrt{5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}-\beta _{2}q^{7}-7\beta _{1}q^{9}-10q^{21}+\cdots$$
1600.2.n.r $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{3}-\zeta_{12}^{3}q^{7}+3\zeta_{12}q^{9}+\cdots$$
1600.2.n.s $4$ $12.776$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots$$
1600.2.n.t $4$ $12.776$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots$$
1600.2.n.u $8$ $12.776$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{5}q^{3}-4\zeta_{24}q^{7}+2\zeta_{24}^{3}q^{9}+\cdots$$
1600.2.n.v $8$ $12.776$ 8.0.3317760000.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+2\beta _{3}q^{9}+\beta _{7}q^{11}+2\beta _{4}q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1600, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$