# Properties

 Label 1216.2.i Level $1216$ Weight $2$ Character orbit 1216.i Rep. character $\chi_{1216}(577,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $76$ Newform subspaces $18$ Sturm bound $320$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$18$$ Sturm bound: $$320$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1216, [\chi])$$.

Total New Old
Modular forms 344 84 260
Cusp forms 296 76 220
Eisenstein series 48 8 40

## Trace form

 $$76 q + 2 q^{5} - 36 q^{9} + O(q^{10})$$ $$76 q + 2 q^{5} - 36 q^{9} - 6 q^{13} - 2 q^{17} - 32 q^{25} + 2 q^{29} - 24 q^{33} + 24 q^{37} + 6 q^{41} - 24 q^{45} + 44 q^{49} + 2 q^{53} - 6 q^{57} - 38 q^{61} + 12 q^{65} + 60 q^{69} - 2 q^{73} + 16 q^{77} - 22 q^{81} + 22 q^{85} - 2 q^{89} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1216, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1216.2.i.a $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$2$$ $$4$$ $$q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+2q^{7}+\cdots$$
1216.2.i.b $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-4$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.c $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.d $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$8$$ $$q+(-1+\zeta_{6})q^{3}+4q^{7}+2\zeta_{6}q^{9}+3q^{11}+\cdots$$
1216.2.i.e $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$3$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.f $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-4$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.g $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.h $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$-8$$ $$q+(1-\zeta_{6})q^{3}-4q^{7}+2\zeta_{6}q^{9}-3q^{11}+\cdots$$
1216.2.i.i $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$3$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
1216.2.i.j $2$ $9.710$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$2$$ $$-4$$ $$q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}-2q^{7}+\cdots$$
1216.2.i.k $4$ $9.710$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$2$$ $$-4$$ $$q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+\cdots$$
1216.2.i.l $4$ $9.710$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$2$$ $$4$$ $$q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{3})q^{7}+\cdots$$
1216.2.i.m $6$ $9.710$ 6.0.2696112.1 None $$0$$ $$-1$$ $$1$$ $$4$$ $$q+(-\beta _{1}+\beta _{5})q^{3}+\beta _{1}q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots$$
1216.2.i.n $6$ $9.710$ 6.0.2696112.1 None $$0$$ $$1$$ $$1$$ $$-4$$ $$q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
1216.2.i.o $8$ $9.710$ 8.0.$$\cdots$$.5 None $$0$$ $$-4$$ $$-2$$ $$-8$$ $$q+(-1-\beta _{2}+\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{1}+\cdots)q^{5}+\cdots$$
1216.2.i.p $8$ $9.710$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(\beta _{2}-\beta _{4})q^{3}-\beta _{1}q^{5}+(-2+2\beta _{1}+\cdots)q^{9}+\cdots$$
1216.2.i.q $8$ $9.710$ 8.0.$$\cdots$$.5 None $$0$$ $$4$$ $$-2$$ $$8$$ $$q+(-\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+\cdots$$
1216.2.i.r $12$ $9.710$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+(\beta _{5}-\beta _{8})q^{3}+\beta _{10}q^{5}+(\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1216, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1216, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(152, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(304, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(608, [\chi])$$$$^{\oplus 2}$$