Space of modular forms of level 3700, weight 1, and Character 3700.b

• Label: 3700.1.b
• Level: $3700$
• Weight: $1$
• Character orbit: 3700.b
• Rep. character: $\chi_{3700}(3551,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $570$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3700.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 148 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(570\)
Trace bound:			\(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3700, [\chi])\).

	Total	New	Old
Modular forms	22	16	6
Cusp forms	10	10	0
Eisenstein series	12	6	6

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	10	0	0	0

Trace form

\( 10 q - 2 q^{4} + 2 q^{9} + 10 q^{16} - 8 q^{21} - 4 q^{26} + 4 q^{34} + 6 q^{36} - 8 q^{41} - 4 q^{46} + 2 q^{49} - 2 q^{64} + 2 q^{74} + 2 q^{81} + 8 q^{84} - 4 q^{86}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3700, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3700.1.b.a	$3700$	$1$	3700.b	148.b	$2$	$1$	$1$	$1.847$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-37}) \)	None	✓	✓			3700.1.b.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+q^{4}-q^{8}+q^{9}+q^{16}-q^{18}+\cdots\)
3700.1.b.b	$3700$	$1$	3700.b	148.b	$2$	$1$	$1$	$1.847$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-37}) \)	None	✓	✓			3700.1.b.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+q^{4}-q^{8}+q^{9}+q^{16}-q^{18}+\cdots\)
3700.1.b.c	$3700$	$1$	3700.b	148.b	$2$	$1$	$1$	$1.847$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-37}) \)	None	✓	✓			3700.1.b.a	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}+q^{8}+q^{9}+q^{16}+q^{18}+\cdots\)
3700.1.b.d	$3700$	$1$	3700.b	148.b	$2$	$1$	$1$	$1.847$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-37}) \)	None	✓	✓			3700.1.b.a	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}+q^{8}+q^{9}+q^{16}+q^{18}+\cdots\)
3700.1.b.e	$3700$	$1$	3700.b	148.b	$2$	$2$	$2$	$1.847$	\(\Q(\sqrt{-1}) \)	$D_{2}$	\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-185}) \)	\(\Q(\sqrt{185}) \)					740.1.g.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-i q^{2}-q^{4}+i q^{8}+q^{9}-2 i q^{13}+\cdots\)
3700.1.b.f	$3700$	$1$	3700.b	148.b	$2$	$4$	$4$	$1.847$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-185}) \)	None			✓		740.1.g.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$		\(q+\zeta_{8}^{2}q^{2}+(-\zeta_{8}-\zeta_{8}^{3})q^{3}-q^{4}+\cdots\)