Space of modular forms of level 735, weight 1, and Character 735.f

• Label: 735.1.f
• Level: $735$
• Weight: $1$
• Character orbit: 735.f
• Rep. character: $\chi_{735}(344,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	735.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(735, [\chi])\).

	Total	New	Old
Modular forms	22	16	6
Cusp forms	6	6	0
Eisenstein series	16	10	6

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	6	0	0	0

Trace form

\( 6 q + 2 q^{4} + 6 q^{9} - 2 q^{15} - 2 q^{16} + 6 q^{25} + 2 q^{36} - 8 q^{46} - 4 q^{51} - 6 q^{60} - 6 q^{64} - 4 q^{79} + 6 q^{81} - 4 q^{85}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(735, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
735.1.f.a	$735$	$1$	735.f	15.d	$2$	$1$	$1$	$0.367$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-35}) \)	\(\Q(\sqrt{21}) \)	✓	✓			735.1.f.a	$4$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)	$1$		\(q-q^{3}-q^{4}-q^{5}+q^{9}+q^{12}+q^{15}+\cdots\)
735.1.f.b	$735$	$1$	735.f	15.d	$2$	$1$	$1$	$0.367$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-35}) \)	\(\Q(\sqrt{21}) \)	✓	✓			735.1.f.a	$4$	$0$	\(0\)	\(1\)	\(1\)	\(0\)	$1$		\(q+q^{3}-q^{4}+q^{5}+q^{9}-q^{12}+q^{15}+\cdots\)
735.1.f.c	$735$	$1$	735.f	15.d	$2$	$2$	$2$	$0.367$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None	✓	✓			735.1.f.c	$4$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)	$1$		\(q-\beta q^{2}-q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{9}+\cdots\)
735.1.f.d	$735$	$1$	735.f	15.d	$2$	$2$	$2$	$0.367$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None	✓	✓			735.1.f.c	$4$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)	$1$		\(q-\beta q^{2}+q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{9}+\cdots\)