Space of modular forms of level 237, weight 2, and trivial character

• Label: 237.2.a
• Level: $237$
• Weight: $2$
• Character orbit: 237.a
• Rep. character: $\chi_{237}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $3$
• Sturm bound: $53$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 237 = 3 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	237.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(53\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(237))\).

	Total	New	Old
Modular forms	28	13	15
Cusp forms	25	13	12
Eisenstein series	3	0	3

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

\(3\)	\(79\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(4\)	\(1\)		\(5\)	\(4\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(9\)	\(7\)	\(2\)		\(8\)	\(7\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(11\)	\(4\)	\(7\)		\(10\)	\(4\)	\(6\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(17\)	\(9\)	\(8\)		\(15\)	\(9\)	\(6\)		\(2\)	\(0\)	\(2\)

Trace form

13 * q + q^2 + q^3 + 17 * q^4 - 6 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 13 * q^9 - 2 * q^10 + 7 * q^12 - 2 * q^13 - 20 * q^14 + 2 * q^15 + 21 * q^16 - 14 * q^17 + q^18 - 4 * q^19 - 32 * q^20 + 4 * q^21 - 6 * q^22 - 4 * q^23 + 3 * q^24 + 7 * q^25 - 12 * q^26 + q^27 - 6 * q^29 - 6 * q^30 + 15 * q^32 + 4 * q^33 - 2 * q^34 - 12 * q^35 + 17 * q^36 + 14 * q^37 + 4 * q^38 + 14 * q^39 - 32 * q^40 - 18 * q^41 - 12 * q^42 + 20 * q^43 - 6 * q^45 + 4 * q^46 - 12 * q^47 - q^48 + 17 * q^49 - 3 * q^50 - 2 * q^51 + 2 * q^52 + 2 * q^53 + 3 * q^54 - 28 * q^56 + 12 * q^57 - 58 * q^58 + 8 * q^59 - 18 * q^60 + 10 * q^61 + 10 * q^62 + 4 * q^63 - 3 * q^64 - 20 * q^65 - 4 * q^66 - 4 * q^67 - 46 * q^68 + 20 * q^69 + 24 * q^70 - 40 * q^71 - 3 * q^72 + 10 * q^73 + 38 * q^74 + 31 * q^75 - 26 * q^76 - 24 * q^77 - 2 * q^78 - 9 * q^79 - 14 * q^80 + 13 * q^81 + 10 * q^82 + 16 * q^83 - 20 * q^84 - 40 * q^85 + 80 * q^86 - 14 * q^87 + 12 * q^88 - 18 * q^89 - 2 * q^90 + 8 * q^91 + 18 * q^92 + 8 * q^93 + 4 * q^94 + 40 * q^95 + 19 * q^96 + 34 * q^97 + 45 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(237))\) 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	79
237.2.a.a	$237$	$2$	237.a	1.a	$1$	$2$	$2$	$1.892$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	237.2.a.a	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
237.2.a.b	$237$	$2$	237.a	1.a	$1$	$4$	$4$	$1.892$	4.4.1957.1	None	✓	✓	✓	✓	237.2.a.b	$1$	$1$	\(-3\)	\(-4\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}-q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
237.2.a.c	$237$	$2$	237.a	1.a	$1$	$7$	$7$	$1.892$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	237.2.a.c	$1$	$0$	\(2\)	\(7\)	\(-2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(237))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(237)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(79))\)\(^{\oplus 2}\)