Space of modular forms of level 784, weight 6, and Character 784.f

• Label: 784.6.f
• Level: $784$
• Weight: $6$
• Character orbit: 784.f
• Rep. character: $\chi_{784}(783,\cdot)$
• Character field: $\Q$
• Dimension: $100$
• Newform subspaces: $6$
• Sturm bound: $672$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(672\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(784, [\chi])\).

	Total	New	Old
Modular forms	584	100	484
Cusp forms	536	100	436
Eisenstein series	48	0	48

Trace form

\( 100 q + 8100 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(784, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.6.f.a	$784$	$6$	784.f	28.d	$2$	$4$	$4$	$125.741$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(5\beta _{1}-23\beta _{2})q^{5}-3^{5}q^{9}+(188\beta _{1}+\cdots)q^{13}+\cdots\)
784.6.f.b	$784$	$6$	784.f	28.d	$2$	$12$	$12$	$125.741$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{2}\cdot 7^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(91+\beta _{10}+\cdots)q^{9}+\cdots\)
784.6.f.c	$784$	$6$	784.f	28.d	$2$	$14$	$14$	$125.741$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(-18\)	\(0\)	\(0\)	$2^{27}\cdot 3^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{3}+(2\beta _{1}-\beta _{5})q^{5}+(75+\cdots)q^{9}+\cdots\)
784.6.f.d	$784$	$6$	784.f	28.d	$2$	$14$	$14$	$125.741$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(18\)	\(0\)	\(0\)	$2^{27}\cdot 3^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{3}+(-2\beta _{1}+\beta _{5})q^{5}+(75+\cdots)q^{9}+\cdots\)
784.6.f.e	$784$	$6$	784.f	28.d	$2$	$16$	$16$	$125.741$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{34}\cdot 7^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{9}q^{3}+(\beta _{10}-\beta _{13})q^{5}+(107+\beta _{1}+\cdots)q^{9}+\cdots\)
784.6.f.f	$784$	$6$	784.f	28.d	$2$	$40$	$40$	$125.741$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(784, [\chi]) \cong \)