Space of modular forms of level 784, weight 2, and Character 784.f

• Label: 784.2.f
• Level: $784$
• Weight: $2$
• Character orbit: 784.f
• Rep. character: $\chi_{784}(783,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $5$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	136	20	116
Cusp forms	88	20	68
Eisenstein series	48	0	48

Trace form

\( 20 q + 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(784, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.2.f.a	$784$	$2$	784.f	28.d	$2$	$2$	$2$	$6.260$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+\zeta_{6}q^{5}-2q^{9}-\zeta_{6}q^{11}-\zeta_{6}q^{15}+\cdots\)
784.2.f.b	$784$	$2$	784.f	28.d	$2$	$2$	$2$	$6.260$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\zeta_{6}q^{5}-2q^{9}+\zeta_{6}q^{11}+\zeta_{6}q^{15}+\cdots\)
784.2.f.c	$784$	$2$	784.f	28.d	$2$	$4$	$4$	$6.260$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{5}-3q^{9}+(\beta _{1}+\beta _{2})q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)
784.2.f.d	$784$	$2$	784.f	28.d	$2$	$4$	$4$	$6.260$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+4q^{9}+\beta _{3}q^{11}+\cdots\)
784.2.f.e	$784$	$2$	784.f	28.d	$2$	$8$	$8$	$6.260$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{2}-\beta _{4})q^{5}+(3-3\beta _{1})q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(784, [\chi]) \cong \)