Space of modular forms of level 56, weight 4, and Character 56.i

• Label: 56.4.i
• Level: $56$
• Weight: $4$
• Character orbit: 56.i
• Rep. character: $\chi_{56}(9,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	56.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(56, [\chi])\).

	Total	New	Old
Modular forms	56	12	44
Cusp forms	40	12	28
Eisenstein series	16	0	16

Trace form

\( 12 q + 6 q^{3} - 10 q^{5} - 24 q^{7} - 68 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(56, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
56.4.i.a	$56$	$4$	56.i	7.c	$3$	$6$	$3$	$3.304$	6.0.11163123.4	None		$2$	$0$	\(0\)	\(-1\)	\(-13\)	\(-20\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}+(-5+5\beta _{1}-\beta _{3}-\beta _{4}-2\beta _{5})q^{5}+\cdots\)
56.4.i.b	$56$	$4$	56.i	7.c	$3$	$6$	$3$	$3.304$	6.0.11163123.4	None		$2$	$0$	\(0\)	\(7\)	\(3\)	\(-4\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(56, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(56, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)