# Properties

 Label 784.2.i Level $784$ Weight $2$ Character orbit 784.i Rep. character $\chi_{784}(177,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $36$ Newform subspaces $14$ Sturm bound $224$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$224$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(784, [\chi])$$.

Total New Old
Modular forms 272 44 228
Cusp forms 176 36 140
Eisenstein series 96 8 88

## Trace form

 $$36 q - 3 q^{3} + q^{5} - 13 q^{9} + O(q^{10})$$ $$36 q - 3 q^{3} + q^{5} - 13 q^{9} + 3 q^{11} + 4 q^{13} + 14 q^{15} + q^{17} - 5 q^{19} + 9 q^{23} - 13 q^{25} + 18 q^{27} + 16 q^{29} + 13 q^{31} - 9 q^{33} - 7 q^{37} - 26 q^{39} + 12 q^{41} - 16 q^{43} - 10 q^{45} + 3 q^{47} - 47 q^{51} - 11 q^{53} - 22 q^{55} - 54 q^{57} - 27 q^{59} - 3 q^{61} + 18 q^{65} + 15 q^{67} - 2 q^{69} + 64 q^{71} + 13 q^{73} + 4 q^{75} + 33 q^{79} + 22 q^{81} + 40 q^{83} - 50 q^{85} + 26 q^{87} + 13 q^{89} + 21 q^{93} - 33 q^{95} + 12 q^{97} + 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(784, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
784.2.i.a $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-1$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+\cdots$$
784.2.i.b $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$-4$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$$
784.2.i.c $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}-4q^{13}+\cdots$$
784.2.i.d $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$3$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots$$
784.2.i.e $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$$
784.2.i.f $2$ $6.260$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+8\zeta_{6}q^{23}+\cdots$$
784.2.i.g $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$$
784.2.i.h $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+(3+\cdots)q^{11}+\cdots$$
784.2.i.i $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+4q^{13}+(6+\cdots)q^{17}+\cdots$$
784.2.i.j $2$ $6.260$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$4$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+8q^{15}+\cdots$$
784.2.i.k $4$ $6.260$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+5\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots$$
784.2.i.l $4$ $6.260$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots$$
784.2.i.m $4$ $6.260$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots$$
784.2.i.n $4$ $6.260$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(2\beta _{1}+2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(784, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(784, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(392, [\chi])$$$$^{\oplus 2}$$