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

• Label: 756.1.h
• Level: $756$
• Weight: $1$
• Character orbit: 756.h
• Rep. character: $\chi_{756}(755,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	756.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 84 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	24	4	20
Cusp forms	12	4	8
Eisenstein series	12	0	12

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	4	0	0	0

Trace form

\( 4 q + 4 q^{4} + 4 q^{16} - 4 q^{22} - 4 q^{37} - 4 q^{46} + 4 q^{49} + 4 q^{64} - 4 q^{70} - 8 q^{85} - 4 q^{88}+O(q^{100}) \)

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.h.a	$756$	$1$	756.h	84.h	$2$	$1$	$1$	$0.377$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-21}) \)	None	✓	✓			756.1.h.a	$2$	$0$	\(-1\)	\(0\)	\(-1\)	\(-1\)	$1$		\(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
756.1.h.b	$756$	$1$	756.h	84.h	$2$	$1$	$1$	$0.377$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-21}) \)	None	✓	✓			756.1.h.a	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(1\)	$1$		\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
756.1.h.c	$756$	$1$	756.h	84.h	$2$	$1$	$1$	$0.377$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-21}) \)	None	✓	✓			756.1.h.a	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(1\)	$1$		\(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
756.1.h.d	$756$	$1$	756.h	84.h	$2$	$1$	$1$	$0.377$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-21}) \)	None	✓	✓			756.1.h.a	$2$	$0$	\(1\)	\(0\)	\(1\)	\(-1\)	$1$		\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)

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

\( S_{1}^{\mathrm{old}}(756, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)