Space of modular forms of level 416, weight 1, and Character 416.h

• Label: 416.1.h
• Level: $416$
• Weight: $1$
• Character orbit: 416.h
• Rep. character: $\chi_{416}(207,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $56$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	416.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 104 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(56\)
Trace bound:			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(416, [\chi])\).

	Total	New	Old
Modular forms	16	4	12
Cusp forms	8	2	6
Eisenstein series	8	2	6

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

\( 2 q + 2 q^{3} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(416, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
416.1.h.a	$416$	$1$	416.h	104.h	$2$	$1$	$1$	$0.208$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-26}) \)	None	✓	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)	$1$		\(q+q^{3}-q^{5}+q^{7}+q^{13}-q^{15}-q^{17}+\cdots\)
416.1.h.b	$416$	$1$	416.h	104.h	$2$	$1$	$1$	$0.208$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-26}) \)	None	✓	$2$	$0$	\(0\)	\(1\)	\(1\)	\(-1\)	$1$		\(q+q^{3}+q^{5}-q^{7}-q^{13}+q^{15}-q^{17}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(416, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(416, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 3}\)