Space of modular forms of level 231, weight 2, and trivial character

• Label: 231.2.a
• Level: $231$
• Weight: $2$
• Character orbit: 231.a
• Rep. character: $\chi_{231}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $5$
• Sturm bound: $64$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(64\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(231))\).

	Total	New	Old
Modular forms	36	11	25
Cusp forms	29	11	18
Eisenstein series	7	0	7

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(12\)	\(0\)	\(12\)		\(9\)	\(0\)	\(9\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(24\)	\(11\)	\(13\)		\(20\)	\(11\)	\(9\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q + q^2 - q^3 + 17 * q^4 + 10 * q^5 + 5 * q^6 - q^7 - 3 * q^8 + 11 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + 2 * q^13 - 3 * q^14 + 2 * q^15 + 17 * q^16 + 22 * q^17 + q^18 + 4 * q^19 - 2 * q^20 - q^21 + q^22 + 9 * q^24 + 29 * q^25 - 10 * q^26 - q^27 - 7 * q^28 + 18 * q^29 - 18 * q^30 - 8 * q^31 - 35 * q^32 - q^33 - 14 * q^34 + 2 * q^35 + 17 * q^36 - 6 * q^37 - 36 * q^38 - 14 * q^39 - 42 * q^40 + 30 * q^41 + q^42 - 20 * q^43 - 7 * q^44 + 10 * q^45 - 32 * q^46 - 24 * q^47 - 31 * q^48 + 11 * q^49 - 65 * q^50 + 6 * q^51 - 58 * q^52 - 6 * q^53 + 5 * q^54 - 6 * q^55 - 15 * q^56 - 20 * q^57 - 46 * q^58 - 20 * q^59 - 6 * q^60 + 26 * q^61 - 16 * q^62 - q^63 + 29 * q^64 + 36 * q^65 - 3 * q^66 - 20 * q^67 + 26 * q^68 + 16 * q^69 + 2 * q^70 + 8 * q^71 - 3 * q^72 + 38 * q^73 + 54 * q^74 - 15 * q^75 + 4 * q^76 - q^77 - 14 * q^78 + 16 * q^79 + 6 * q^80 + 11 * q^81 - 14 * q^82 - 4 * q^83 - 7 * q^84 - 4 * q^85 + 12 * q^86 - 6 * q^87 + 9 * q^88 + 62 * q^89 - 2 * q^90 + 10 * q^91 + 8 * q^92 - 8 * q^93 + 24 * q^94 - 16 * q^95 + 33 * q^96 + 14 * q^97 + q^98 - q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
231.2.a.a	$231$	$2$	231.a	1.a	$1$	$1$	$1$	$1.845$	\(\Q\)	None	✓	✓			231.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
231.2.a.b	$231$	$2$	231.a	1.a	$1$	$2$	$2$	$1.845$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓		231.2.a.b	$1$	$0$	\(-1\)	\(-2\)	\(6\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(3+\beta )q^{4}+3q^{5}+\beta q^{6}+\cdots\)
231.2.a.c	$231$	$2$	231.a	1.a	$1$	$2$	$2$	$1.845$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	231.2.a.c	$1$	$0$	\(1\)	\(2\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
231.2.a.d	$231$	$2$	231.a	1.a	$1$	$3$	$3$	$1.845$	3.3.837.1	None	✓	✓	✓	✓	231.2.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
231.2.a.e	$231$	$2$	231.a	1.a	$1$	$3$	$3$	$1.845$	3.3.229.1	None	✓	✓	✓	✓	231.2.a.e	$1$	$0$	\(2\)	\(3\)	\(4\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(231))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(231)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)