Space of modular forms of level 756, weight 1, and Character 756.bk

• Label: 756.1.bk
• Level: $756$
• Weight: $1$
• Character orbit: 756.bk
• Rep. character: $\chi_{756}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $144$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(144\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(756, [\chi])\).

	Total	New	Old
Modular forms	50	2	48
Cusp forms	14	2	12
Eisenstein series	36	0	36

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	2	0	0	0

Trace form

\( 2 q - q^{7} + 4 q^{13} + q^{19} - q^{25} + q^{31} - 2 q^{37} - 2 q^{43} - q^{49} + q^{61} - 2 q^{67} + q^{73} - 2 q^{79} - 2 q^{91} - 2 q^{97}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
756.1.bk.a	$756$	$1$	756.bk	21.h	$6$	$2$	$1$	$0.377$	\(\Q(\sqrt{-3}) \)	$D_{3}$	\(\Q(\sqrt{-3}) \)	None		✓	✓	✓	756.1.bk.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)	$1$		\(q-\zeta_{6}q^{7}+q^{13}-\zeta_{6}^{2}q^{19}-\zeta_{6}q^{25}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(756, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)