# Properties

 Label 3800.2.d Level $3800$ Weight $2$ Character orbit 3800.d Rep. character $\chi_{3800}(3649,\cdot)$ Character field $\Q$ Dimension $82$ Newform subspaces $17$ Sturm bound $1200$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$17$$ Sturm bound: $$1200$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3800, [\chi])$$.

Total New Old
Modular forms 624 82 542
Cusp forms 576 82 494
Eisenstein series 48 0 48

## Trace form

 $$82 q - 78 q^{9} + O(q^{10})$$ $$82 q - 78 q^{9} - 12 q^{11} + 6 q^{19} + 16 q^{21} - 28 q^{29} - 8 q^{31} - 32 q^{39} - 4 q^{41} - 42 q^{49} + 72 q^{51} - 32 q^{61} + 8 q^{69} + 48 q^{71} - 16 q^{79} + 98 q^{81} - 12 q^{89} + 32 q^{91} - 52 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3800, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3800.2.d.a $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+iq^{7}-6q^{9}+4q^{11}+iq^{13}+\cdots$$
3800.2.d.b $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}-4q^{11}-2iq^{13}-iq^{17}+\cdots$$
3800.2.d.c $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}-q^{9}-4q^{11}-3iq^{17}+\cdots$$
3800.2.d.d $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{3}-3iq^{7}-q^{9}-3q^{11}+4iq^{13}+\cdots$$
3800.2.d.e $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}-q^{9}+4q^{11}+2iq^{13}+\cdots$$
3800.2.d.f $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-3iq^{7}+2q^{9}+2q^{11}+iq^{13}+\cdots$$
3800.2.d.g $2$ $30.343$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}-4q^{11}-3iq^{13}+3iq^{17}+\cdots$$
3800.2.d.h $4$ $30.343$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{3}+(-1+\zeta_{12}^{3})q^{9}+2q^{11}+\cdots$$
3800.2.d.i $4$ $30.343$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{3}+2\zeta_{8}^{2}q^{7}+q^{9}+(2-\zeta_{8}^{3})q^{11}+\cdots$$
3800.2.d.j $6$ $30.343$ 6.0.59105344.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(\beta _{1}+\beta _{2}-\beta _{4})q^{7}+(-3+\cdots)q^{9}+\cdots$$
3800.2.d.k $6$ $30.343$ 6.0.5161984.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}+(\beta _{3}-\beta _{5})q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots$$
3800.2.d.l $6$ $30.343$ 6.0.3356224.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+\beta _{5}q^{7}+(-1+\beta _{1})q^{9}+(\beta _{2}+\cdots)q^{13}+\cdots$$
3800.2.d.m $6$ $30.343$ 6.0.4227136.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+\beta _{3}q^{7}+(-\beta _{2}-\beta _{5})q^{9}+\cdots$$
3800.2.d.n $6$ $30.343$ 6.0.399424.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}+(\beta _{4}-\beta _{5})q^{7}-\beta _{1}q^{9}+2\beta _{1}q^{11}+\cdots$$
3800.2.d.o $6$ $30.343$ 6.0.419904.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+2\beta _{5})q^{7}+(1+\beta _{2})q^{9}+\cdots$$
3800.2.d.p $12$ $30.343$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{9}q^{7}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots$$
3800.2.d.q $12$ $30.343$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{8})q^{7}+(-1-\beta _{3}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3800, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3800, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$