Space of modular forms of level 1400, weight 1, and Character 1400.m

• Label: 1400.1.m
• Level: $1400$
• Weight: $1$
• Character orbit: 1400.m
• Rep. character: $\chi_{1400}(1301,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1400.m (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1400, [\chi])\).

	Total	New	Old
Modular forms	24	15	9
Cusp forms	12	9	3
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	9	0	0	0

Trace form

\( 9 q + q^{2} + 3 q^{4} + q^{7} + q^{8} - q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1400, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1400.1.m.a	$1400$	$1$	1400.m	56.h	$2$	$1$	$1$	$0.699$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \)	\(\Q(\sqrt{2}) \)	✓	$4$	$0$	\(1\)	\(0\)	\(0\)	\(1\)	$1$		\(q+q^{2}+q^{4}+q^{7}+q^{8}-q^{9}+q^{14}+\cdots\)
1400.1.m.b	$1400$	$1$	1400.m	56.h	$2$	$2$	$2$	$0.699$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-14}) \)	None	✓	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)	$1$		\(q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+q^{7}-q^{8}+\cdots\)
1400.1.m.c	$1400$	$1$	1400.m	56.h	$2$	$2$	$2$	$0.699$	\(\Q(\sqrt{-3}) \)	$D_{6}$	\(\Q(\sqrt{-7}) \)	None		$4$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)	$1$		\(q+\zeta_{6}^{2}q^{2}-\zeta_{6}q^{4}+q^{7}+q^{8}-q^{9}+\cdots\)
1400.1.m.d	$1400$	$1$	1400.m	56.h	$2$	$2$	$2$	$0.699$	\(\Q(\sqrt{-3}) \)	$D_{6}$	\(\Q(\sqrt{-7}) \)	None		$4$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)	$1$		\(q-\zeta_{6}^{2}q^{2}-\zeta_{6}q^{4}-q^{7}-q^{8}-q^{9}+\cdots\)
1400.1.m.e	$1400$	$1$	1400.m	56.h	$2$	$2$	$2$	$0.699$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-14}) \)	None	✓	$4$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)	$1$		\(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}-q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1400, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1400, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)