# Properties

 Label 810.4.e Level $810$ Weight $4$ Character orbit 810.e Rep. character $\chi_{810}(271,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $96$ Newform subspaces $34$ Sturm bound $648$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 810.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$34$$ Sturm bound: $$648$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(810, [\chi])$$.

Total New Old
Modular forms 1020 96 924
Cusp forms 924 96 828
Eisenstein series 96 0 96

## Trace form

 $$96 q - 192 q^{4} + 120 q^{7} + O(q^{10})$$ $$96 q - 192 q^{4} + 120 q^{7} - 240 q^{13} - 768 q^{16} + 1560 q^{19} - 72 q^{22} - 1200 q^{25} - 960 q^{28} + 300 q^{31} + 360 q^{34} + 3360 q^{37} + 264 q^{43} + 1008 q^{46} - 5148 q^{49} - 960 q^{52} - 204 q^{61} + 6144 q^{64} + 1380 q^{67} + 360 q^{70} + 1848 q^{73} - 3120 q^{76} - 6072 q^{79} - 4176 q^{82} - 1440 q^{85} - 288 q^{88} + 8376 q^{91} + 1224 q^{94} - 5244 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(810, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
810.4.e.a $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-5$$ $$-14$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.b $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-5$$ $$-8$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.c $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-5$$ $$4$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.d $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-5$$ $$34$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.e $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$-32$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.f $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$-14$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.g $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$-2$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.h $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$4$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.i $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$4$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.j $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$13$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.k $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$22$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.l $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$28$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.m $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$-32$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.n $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$-14$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.o $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$-2$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.p $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$4$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.q $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$4$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.r $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$13$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.s $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$22$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.t $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$28$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
810.4.e.u $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$5$$ $$-14$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.v $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$5$$ $$-8$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.w $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$5$$ $$4$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.x $2$ $47.792$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$5$$ $$34$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
810.4.e.y $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{-163})$$ None $$-4$$ $$0$$ $$-10$$ $$-7$$ $$q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots$$
810.4.e.z $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{401})$$ None $$-4$$ $$0$$ $$-10$$ $$-1$$ $$q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-5+5\beta _{1}+\cdots)q^{5}+\cdots$$
810.4.e.ba $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{-1027})$$ None $$-4$$ $$0$$ $$-10$$ $$5$$ $$q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots$$
810.4.e.bb $4$ $47.792$ $$\Q(\zeta_{12})$$ None $$-4$$ $$0$$ $$-10$$ $$26$$ $$q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}-5\zeta_{12}q^{5}+\cdots$$
810.4.e.bc $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{-163})$$ None $$4$$ $$0$$ $$10$$ $$-7$$ $$q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots$$
810.4.e.bd $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{401})$$ None $$4$$ $$0$$ $$10$$ $$-1$$ $$q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(5-5\beta _{1}+\cdots)q^{5}+\cdots$$
810.4.e.be $4$ $47.792$ $$\Q(\sqrt{-3}, \sqrt{-1027})$$ None $$4$$ $$0$$ $$10$$ $$5$$ $$q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots$$
810.4.e.bf $4$ $47.792$ $$\Q(\zeta_{12})$$ None $$4$$ $$0$$ $$10$$ $$26$$ $$q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+5\zeta_{12}q^{5}+\cdots$$
810.4.e.bg $8$ $47.792$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$-8$$ $$0$$ $$20$$ $$-2$$ $$q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots$$
810.4.e.bh $8$ $47.792$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$8$$ $$0$$ $$-20$$ $$-2$$ $$q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(810, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(810, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(405, [\chi])$$$$^{\oplus 2}$$