# Properties

 Label 1600.2.l Level $1600$ Weight $2$ Character orbit 1600.l Rep. character $\chi_{1600}(401,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $70$ Newform subspaces $9$ Sturm bound $480$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.l (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$9$$ Sturm bound: $$480$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 528 82 446
Cusp forms 432 70 362
Eisenstein series 96 12 84

## Trace form

 $$70 q - 2 q^{3} + O(q^{10})$$ $$70 q - 2 q^{3} + 2 q^{11} + 2 q^{13} + 4 q^{17} + 6 q^{19} - 12 q^{21} + 16 q^{27} + 10 q^{29} + 4 q^{33} + 10 q^{37} + 18 q^{43} - 24 q^{47} - 30 q^{49} - 28 q^{51} - 6 q^{53} - 30 q^{59} + 10 q^{61} + 44 q^{63} + 30 q^{67} + 36 q^{69} - 20 q^{77} - 34 q^{81} + 38 q^{83} + 60 q^{91} + 32 q^{93} + 4 q^{97} - 18 q^{99} + O(q^{100})$$

## 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.l.a $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+i)q^{3}-2iq^{7}+iq^{9}+(-1+\cdots)q^{11}+\cdots$$
1600.2.l.b $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+i)q^{3}+iq^{9}+(3+3i)q^{11}+\cdots$$
1600.2.l.c $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(1-i)q^{3}+iq^{9}+(3+3i)q^{11}+(-3+\cdots)q^{13}+\cdots$$
1600.2.l.d $4$ $12.776$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+\beta _{3})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots$$
1600.2.l.e $4$ $12.776$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots$$
1600.2.l.f $12$ $12.776$ 12.0.$$\cdots$$.1 None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}+(\beta _{2}+\beta _{3})q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots$$
1600.2.l.g $12$ $12.776$ 12.0.$$\cdots$$.1 None $$0$$ $$2$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+(-\beta _{2}-\beta _{3})q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots$$
1600.2.l.h $16$ $12.776$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{14}q^{3}+(-\beta _{2}+\beta _{3}+\beta _{5}-\beta _{8}+\cdots)q^{7}+\cdots$$
1600.2.l.i $16$ $12.776$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{3}+(-\beta _{3}-\beta _{6}-\beta _{8}+\beta _{9}-\beta _{10}+\cdots)q^{7}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 3}$$