Space of modular forms of level 1, weight 116, and trivial character

• Label: 1.116.a
• Level: $1$
• Weight: $116$
• Character orbit: 1.a
• Rep. character: $\chi_{1}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $1$
• Sturm bound: $9$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 116 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(9\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{116}(\Gamma_0(1))\).

	Total	New	Old
Modular forms	10	10	0
Cusp forms	9	9	0
Eisenstein series	1	1	0

Trace form

\( 9 q - 116180770089455544 q^{2} + 344843759214607763941669692 q^{3} + 104670059680921154232978400984552512 q^{4} - 9416265734776133053944609096571667575890 q^{5} + 256991120598421923060797165626503750361356768 q^{6} + 1823100316304157772850498188908022382936745628056 q^{7} - 219342532030837467786318171319354148041440220761600 q^{8} + 26072637303355254030743970272505117374453788742960594013 q^{9} + O(q^{10}) \)

Decomposition of \(S_{116}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1.116.a.a	$1$	$116$	1.a	1.a	$1$	$9$	$9$	$83.750$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(-11\!\cdots\!44\)	\(34\!\cdots\!92\)	\(-94\!\cdots\!90\)	\(18\!\cdots\!56\)	$+$	$2^{142}\cdot 3^{52}\cdot 5^{17}\cdot 7^{8}\cdot 11^{3}\cdot 13^{3}\cdot 17\cdot 19^{3}\cdot 23^{3}\cdot 29^{2}$	$\mathrm{SU}(2)$	\(q+(-12908974454383949-\beta _{1}+\cdots)q^{2}+\cdots\)