Space of modular forms of level 231, weight 6, and trivial character

• Label: 231.6.a
• Level: $231$
• Weight: $6$
• Character orbit: 231.a
• Rep. character: $\chi_{231}(1,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $9$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(231))\).

	Total	New	Old
Modular forms	164	52	112
Cusp forms	156	52	104
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(20\)
Minus space			\(-\)	\(32\)

Trace form

\( 52 q - 16 q^{2} + 936 q^{4} - 64 q^{5} - 360 q^{6} - 768 q^{8} + 4212 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(231))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	11
231.6.a.a	$231$	$6$	231.a	1.a	$1$	$1$	$1$	$37.049$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-9\)	\(-76\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}-76q^{5}+18q^{6}+\cdots\)
231.6.a.b	$231$	$6$	231.a	1.a	$1$	$4$	$4$	$37.049$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(7\)	\(-36\)	\(-38\)	\(196\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-9q^{3}+(24+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
231.6.a.c	$231$	$6$	231.a	1.a	$1$	$5$	$5$	$37.049$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-13\)	\(45\)	\(-114\)	\(245\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+9q^{3}+(15-3\beta _{1}+\cdots)q^{4}+\cdots\)
231.6.a.d	$231$	$6$	231.a	1.a	$1$	$5$	$5$	$37.049$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-13\)	\(45\)	\(-2\)	\(-245\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+9q^{3}+(15-4\beta _{1}+\cdots)q^{4}+\cdots\)
231.6.a.e	$231$	$6$	231.a	1.a	$1$	$5$	$5$	$37.049$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(5\)	\(-45\)	\(-2\)	\(-245\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(14-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
231.6.a.f	$231$	$6$	231.a	1.a	$1$	$8$	$8$	$37.049$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-5\)	\(72\)	\(98\)	\(-392\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(21+\beta _{2})q^{4}+\cdots\)
231.6.a.g	$231$	$6$	231.a	1.a	$1$	$8$	$8$	$37.049$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(-72\)	\(-14\)	\(392\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
231.6.a.h	$231$	$6$	231.a	1.a	$1$	$8$	$8$	$37.049$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(72\)	\(86\)	\(392\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
231.6.a.i	$231$	$6$	231.a	1.a	$1$	$8$	$8$	$37.049$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(-72\)	\(-2\)	\(-392\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(21+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(231))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(231)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)