# Properties

 Label 1216.3.e Level $1216$ Weight $3$ Character orbit 1216.e Rep. character $\chi_{1216}(1025,\cdot)$ Character field $\Q$ Dimension $78$ Newform subspaces $16$ Sturm bound $480$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q$$ Newform subspaces: $$16$$ Sturm bound: $$480$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1216, [\chi])$$.

Total New Old
Modular forms 332 82 250
Cusp forms 308 78 230
Eisenstein series 24 4 20

## Trace form

 $$78 q + 4 q^{5} - 226 q^{9} + O(q^{10})$$ $$78 q + 4 q^{5} - 226 q^{9} - 4 q^{17} + 346 q^{25} + 68 q^{45} + 458 q^{49} - 64 q^{57} + 292 q^{61} - 4 q^{73} + 152 q^{77} + 366 q^{81} + 568 q^{85} - 320 q^{93} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1216, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1216.3.e.a $1$ $33.134$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$9$$ $$-5$$ $$q+9q^{5}-5q^{7}+9q^{9}-3q^{11}+15q^{17}+\cdots$$
1216.3.e.b $1$ $33.134$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$9$$ $$5$$ $$q+9q^{5}+5q^{7}+9q^{9}+3q^{11}+15q^{17}+\cdots$$
1216.3.e.c $2$ $33.134$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$-14$$ $$-22$$ $$q+\beta q^{3}-7q^{5}-11q^{7}-23q^{9}+3q^{11}+\cdots$$
1216.3.e.d $2$ $33.134$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$-14$$ $$22$$ $$q+\beta q^{3}-7q^{5}+11q^{7}-23q^{9}-3q^{11}+\cdots$$
1216.3.e.e $2$ $33.134$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-9$$ $$-5$$ $$q+(-4-\beta )q^{5}+(-4+3\beta )q^{7}+9q^{9}+\cdots$$
1216.3.e.f $2$ $33.134$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-9$$ $$5$$ $$q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+9q^{9}+\cdots$$
1216.3.e.g $2$ $33.134$ $$\Q(\sqrt{-13})$$ None $$0$$ $$0$$ $$-8$$ $$-10$$ $$q+\beta q^{3}-4q^{5}-5q^{7}-4q^{9}+10q^{11}+\cdots$$
1216.3.e.h $2$ $33.134$ $$\Q(\sqrt{-13})$$ None $$0$$ $$0$$ $$-8$$ $$10$$ $$q+\beta q^{3}-4q^{5}+5q^{7}-4q^{9}-10q^{11}+\cdots$$
1216.3.e.i $2$ $33.134$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$2$$ $$-10$$ $$q+\beta q^{3}+q^{5}-5q^{7}+q^{9}+5q^{11}+\cdots$$
1216.3.e.j $2$ $33.134$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$2$$ $$10$$ $$q+\beta q^{3}+q^{5}+5q^{7}+q^{9}-5q^{11}+\cdots$$
1216.3.e.k $2$ $33.134$ $$\Q(\sqrt{-29})$$ None $$0$$ $$0$$ $$8$$ $$-2$$ $$q+\beta q^{3}+4q^{5}-q^{7}-20q^{9}-14q^{11}+\cdots$$
1216.3.e.l $2$ $33.134$ $$\Q(\sqrt{-29})$$ None $$0$$ $$0$$ $$8$$ $$2$$ $$q+\beta q^{3}+4q^{5}+q^{7}-20q^{9}+14q^{11}+\cdots$$
1216.3.e.m $8$ $33.134$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$14$$ $$-6$$ $$q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(-1+\beta _{3})q^{7}+\cdots$$
1216.3.e.n $8$ $33.134$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$14$$ $$6$$ $$q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(1-\beta _{3})q^{7}+\cdots$$
1216.3.e.o $20$ $33.134$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{11}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+(-3+\beta _{5}+\cdots)q^{9}+\cdots$$
1216.3.e.p $20$ $33.134$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{7}q^{3}-\beta _{6}q^{5}+\beta _{1}q^{7}+(-3+\beta _{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1216, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1216, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(152, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(304, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(608, [\chi])$$$$^{\oplus 2}$$