# Properties

 Label 5850.2.e Level $5850$ Weight $2$ Character orbit 5850.e Rep. character $\chi_{5850}(5149,\cdot)$ Character field $\Q$ Dimension $90$ Newform subspaces $39$ Sturm bound $2520$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$39$$ Sturm bound: $$2520$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$, $$19$$, $$29$$, $$31$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(5850, [\chi])$$.

Total New Old
Modular forms 1308 90 1218
Cusp forms 1212 90 1122
Eisenstein series 96 0 96

## Trace form

 $$90 q - 90 q^{4} + O(q^{10})$$ $$90 q - 90 q^{4} - 4 q^{11} + 8 q^{14} + 90 q^{16} + 4 q^{19} + 6 q^{26} + 8 q^{34} + 24 q^{41} + 4 q^{44} + 32 q^{46} - 114 q^{49} - 8 q^{56} - 32 q^{59} + 32 q^{61} - 90 q^{64} + 80 q^{71} - 4 q^{76} - 8 q^{79} + 56 q^{86} - 48 q^{89} - 20 q^{91} - 40 q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(5850, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
5850.2.e.a $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}-6q^{11}+\cdots$$
5850.2.e.b $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-6q^{11}+\cdots$$
5850.2.e.c $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}-4q^{11}+\cdots$$
5850.2.e.d $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-4q^{11}+\cdots$$
5850.2.e.e $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{8}-4q^{11}-iq^{13}+\cdots$$
5850.2.e.f $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{8}-4q^{11}+iq^{13}+\cdots$$
5850.2.e.g $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots$$
5850.2.e.h $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots$$
5850.2.e.i $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots$$
5850.2.e.j $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}-3q^{11}+\cdots$$
5850.2.e.k $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-2q^{11}+\cdots$$
5850.2.e.l $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}-q^{11}+\cdots$$
5850.2.e.m $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+iq^{13}+\cdots$$
5850.2.e.n $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+iq^{13}+\cdots$$
5850.2.e.o $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-iq^{13}+\cdots$$
5850.2.e.p $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{8}+iq^{13}+q^{16}+\cdots$$
5850.2.e.q $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{8}-iq^{13}+q^{16}+\cdots$$
5850.2.e.r $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-iq^{13}+\cdots$$
5850.2.e.s $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+iq^{13}+\cdots$$
5850.2.e.t $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+iq^{13}+\cdots$$
5850.2.e.u $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+2q^{11}+\cdots$$
5850.2.e.v $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+2q^{11}+\cdots$$
5850.2.e.w $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+2q^{11}+\cdots$$
5850.2.e.x $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}+3q^{11}+\cdots$$
5850.2.e.y $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{8}+3q^{11}+iq^{13}+\cdots$$
5850.2.e.z $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+3iq^{7}+iq^{8}+3q^{11}+\cdots$$
5850.2.e.ba $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+5iq^{7}+iq^{8}+3q^{11}+\cdots$$
5850.2.e.bb $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+4q^{11}+\cdots$$
5850.2.e.bc $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+4q^{11}+\cdots$$
5850.2.e.bd $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+4q^{11}+\cdots$$
5850.2.e.be $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}+5q^{11}+\cdots$$
5850.2.e.bf $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+6q^{11}+\cdots$$
5850.2.e.bg $2$ $46.712$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+6q^{11}+\cdots$$
5850.2.e.bh $4$ $46.712$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}+(-2\beta _{1}-\beta _{3})q^{7}-\beta _{1}q^{8}+\cdots$$
5850.2.e.bi $4$ $46.712$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-q^{4}+(\beta _{1}-\beta _{2})q^{7}-\beta _{2}q^{8}+\cdots$$
5850.2.e.bj $4$ $46.712$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-q^{4}+(-\beta _{1}+\beta _{2})q^{7}+\beta _{1}q^{8}+\cdots$$
5850.2.e.bk $4$ $46.712$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}q^{2}-q^{4}+\zeta_{8}^{2}q^{7}+\zeta_{8}q^{8}+2\zeta_{8}^{3}q^{11}+\cdots$$
5850.2.e.bl $4$ $46.712$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}+(-\beta _{1}+\beta _{2})q^{7}-\beta _{1}q^{8}+\cdots$$
5850.2.e.bm $4$ $46.712$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-q^{4}+(-2\beta _{1}-\beta _{3})q^{7}+\beta _{1}q^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(5850, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(5850, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(390, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(585, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(650, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(975, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1170, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1950, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2925, [\chi])$$$$^{\oplus 2}$$