Space of modular forms of level 83, weight 13, and Character 83.b

• Label: 83.13.b
• Level: $83$
• Weight: $13$
• Character orbit: 83.b
• Rep. character: $\chi_{83}(82,\cdot)$
• Character field: $\Q$
• Dimension: $83$
• Newform subspaces: $3$
• Sturm bound: $91$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	83.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 83 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(91\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(83, [\chi])\).

	Total	New	Old
Modular forms	85	85	0
Cusp forms	83	83	0
Eisenstein series	2	2	0

Trace form

\( 83 q + 918 q^{3} - 168862 q^{4} + 103918 q^{7} + 16496465 q^{9} + O(q^{10}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(83, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
83.13.b.a	$83$	$13$	83.b	83.b	$2$	$1$	$1$	$75.861$	\(\Q\)	\(\Q(\sqrt{-83}) \)	✓	$2$	$0$	\(0\)	\(-617\)	\(0\)	\(-231577\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-617q^{3}+2^{12}q^{4}-231577q^{7}+\cdots\)
83.13.b.b	$83$	$13$	83.b	83.b	$2$	$2$	$2$	$75.861$	\(\Q(\sqrt{249}) \)	\(\Q(\sqrt{-83}) \)	✓	$2$	$0$	\(0\)	\(617\)	\(0\)	\(231577\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+(294-29\beta )q^{3}+2^{12}q^{4}+(116246+\cdots)q^{7}+\cdots\)
83.13.b.c	$83$	$13$	83.b	83.b	$2$	$80$	$80$	$75.861$		None		$2$	$0$	\(0\)	\(918\)	\(0\)	\(103918\)		$\mathrm{SU}(2)[C_{2}]$