Space of modular forms of level 1216, weight 3, and Character 1216.g

• Label: 1216.3.g
• Level: $1216$
• Weight: $3$
• Character orbit: 1216.g
• Rep. character: $\chi_{1216}(417,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 152 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

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

	Total	New	Old
Modular forms	332	80	252
Cusp forms	308	80	228
Eisenstein series	24	0	24

Trace form

\( 80 q + 240 q^{9} + O(q^{10}) \)

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1216.3.g.a	$1216$	$3$	1216.g	152.g	$2$	$4$	$4$	$33.134$	\(\Q(i, \sqrt{19})\)	\(\Q(\sqrt{-19}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{5}-3\beta _{2}q^{7}-9q^{9}+3\beta _{1}q^{11}+\cdots\)
1216.3.g.b	$1216$	$3$	1216.g	152.g	$2$	$4$	$4$	$33.134$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+(\zeta_{8}+2\zeta_{8}^{2})q^{3}+23q^{9}+7\zeta_{8}q^{11}+\cdots\)
1216.3.g.c	$1216$	$3$	1216.g	152.g	$2$	$8$	$8$	$33.134$	8.0.2702336256.1	\(\Q(\sqrt{-19}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-\beta _{3}-\beta _{6})q^{5}+(\beta _{1}-\beta _{7})q^{7}-9q^{9}+\cdots\)
1216.3.g.d	$1216$	$3$	1216.g	152.g	$2$	$16$	$16$	$33.134$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{4}q^{5}+(\beta _{5}+\beta _{6})q^{7}+(-1+\cdots)q^{9}+\cdots\)
1216.3.g.e	$1216$	$3$	1216.g	152.g	$2$	$48$	$48$	$33.134$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

\( S_{3}^{\mathrm{old}}(1216, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(304, [\chi])\)\(^{\oplus 3}\)