# Properties

 Label 1078.2.e Level $1078$ Weight $2$ Character orbit 1078.e Rep. character $\chi_{1078}(67,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $64$ Newform subspaces $22$ Sturm bound $336$ Trace bound $23$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$22$$ Sturm bound: $$336$$ Trace bound: $$23$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1078, [\chi])$$.

Total New Old
Modular forms 368 64 304
Cusp forms 304 64 240
Eisenstein series 64 0 64

## Trace form

 $$64 q - 32 q^{4} - 4 q^{5} - 8 q^{6} - 24 q^{9} + O(q^{10})$$ $$64 q - 32 q^{4} - 4 q^{5} - 8 q^{6} - 24 q^{9} - 8 q^{13} + 8 q^{15} - 32 q^{16} + 12 q^{17} - 16 q^{18} + 8 q^{20} - 32 q^{23} + 4 q^{24} - 40 q^{25} - 12 q^{26} + 24 q^{27} + 8 q^{29} - 4 q^{31} - 4 q^{33} + 8 q^{34} + 48 q^{36} + 48 q^{37} - 4 q^{38} + 44 q^{39} + 24 q^{41} - 40 q^{43} + 4 q^{45} - 4 q^{46} + 16 q^{47} - 32 q^{50} - 36 q^{51} + 4 q^{52} - 20 q^{53} + 16 q^{54} - 8 q^{57} + 12 q^{58} - 4 q^{59} - 4 q^{60} + 36 q^{61} - 24 q^{62} + 64 q^{64} + 20 q^{65} - 8 q^{66} + 52 q^{67} + 12 q^{68} - 24 q^{69} + 56 q^{71} - 16 q^{72} - 16 q^{73} + 20 q^{74} - 60 q^{75} + 64 q^{78} + 20 q^{79} - 4 q^{80} - 64 q^{81} - 8 q^{82} - 40 q^{83} - 144 q^{85} - 20 q^{86} + 12 q^{87} - 72 q^{90} + 64 q^{92} + 44 q^{93} - 8 q^{94} - 36 q^{95} + 4 q^{96} + 96 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1078, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1078.2.e.a $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-2$$ $$-2$$ $$0$$ $$q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.b $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$0$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$$
1078.2.e.c $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$2$$ $$0$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots$$
1078.2.e.d $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$0$$ $$0$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.e $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$2$$ $$2$$ $$0$$ $$q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.f $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$3$$ $$2$$ $$0$$ $$q-\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.g $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-3$$ $$-4$$ $$0$$ $$q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.h $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-2$$ $$-2$$ $$0$$ $$q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.i $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-4$$ $$0$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots$$
1078.2.e.j $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+4\zeta_{6}q^{5}+\cdots$$
1078.2.e.k $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$0$$ $$0$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.l $2$ $8.608$ $$\Q(\sqrt{-3})$$ None $$1$$ $$2$$ $$2$$ $$0$$ $$q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
1078.2.e.m $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$-2$$ $$-4$$ $$0$$ $$q+\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(-1+\cdots)q^{4}+\cdots$$
1078.2.e.n $4$ $8.608$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$-2$$ $$-2$$ $$2$$ $$0$$ $$q+\beta _{1}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(-1+\cdots)q^{4}+\cdots$$
1078.2.e.o $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+2\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+\cdots$$
1078.2.e.p $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(-3\beta _{1}+\cdots)q^{5}+\cdots$$
1078.2.e.q $4$ $8.608$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$-2$$ $$2$$ $$-2$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots$$
1078.2.e.r $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(-1-\beta _{2})q^{4}-\beta _{3}q^{6}+\cdots$$
1078.2.e.s $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
1078.2.e.t $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}-2\beta _{1}q^{5}-q^{8}+\cdots$$
1078.2.e.u $4$ $8.608$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}-\beta _{1}q^{5}-q^{8}+\cdots$$
1078.2.e.v $4$ $8.608$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$2$$ $$0$$ $$2$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1078, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1078, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(154, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(539, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 4}$$