Space of modular forms of level 59, weight 9, and Character 59.b

• Label: 59.9.b
• Level: $59$
• Weight: $9$
• Character orbit: 59.b
• Rep. character: $\chi_{59}(58,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $2$
• Sturm bound: $45$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	59.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 59 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(45\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(59, [\chi])\).

	Total	New	Old
Modular forms	41	41	0
Cusp forms	39	39	0
Eisenstein series	2	2	0

Trace form

\( 39 q + 40 q^{3} - 4866 q^{4} - 296 q^{5} - 162 q^{7} + 67295 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(59, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
59.9.b.a	$59$	$9$	59.b	59.b	$2$	$3$	$3$	$24.035$	3.3.1593.1	\(\Q(\sqrt{-59}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-41\beta _{1}+22\beta _{2})q^{3}+2^{8}q^{4}+(-316\beta _{1}+\cdots)q^{5}+\cdots\)
59.9.b.b	$59$	$9$	59.b	59.b	$2$	$36$	$36$	$24.035$		None		$2$	$0$	\(0\)	\(40\)	\(-296\)	\(-162\)		$\mathrm{SU}(2)[C_{2}]$