Space of modular forms of level 238, weight 2, and trivial character

• Label: 238.2.a
• Level: $238$
• Weight: $2$
• Character orbit: 238.a
• Rep. character: $\chi_{238}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $6$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	238.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(238))\).

	Total	New	Old
Modular forms	40	7	33
Cusp forms	33	7	26
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.

\(2\)	\(7\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(1\)	\(1\)	\(0\)		\(1\)	\(1\)	\(0\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(18\)	\(2\)	\(16\)		\(15\)	\(2\)	\(13\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(22\)	\(5\)	\(17\)		\(18\)	\(5\)	\(13\)		\(4\)	\(0\)	\(4\)

Trace form

7 * q - q^2 + 4 * q^3 + 7 * q^4 + 2 * q^5 - 4 * q^6 - q^7 - q^8 + 3 * q^9 - 6 * q^10 - 8 * q^11 + 4 * q^12 - 6 * q^13 + 3 * q^14 + 8 * q^15 + 7 * q^16 - q^17 - 5 * q^18 - 12 * q^19 + 2 * q^20 - 4 * q^21 - 8 * q^22 + 8 * q^23 - 4 * q^24 + 17 * q^25 - 6 * q^26 + 16 * q^27 - q^28 + 14 * q^29 + 8 * q^30 - q^32 + 16 * q^33 - q^34 + 2 * q^35 + 3 * q^36 - 10 * q^37 + 20 * q^38 - 24 * q^39 - 6 * q^40 - 26 * q^41 - 4 * q^42 - 20 * q^43 - 8 * q^44 - 14 * q^45 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 7 * q^49 - 7 * q^50 - 6 * q^52 + 2 * q^53 - 16 * q^54 + 8 * q^55 + 3 * q^56 - 40 * q^57 - 2 * q^58 - 4 * q^59 + 8 * q^60 - 22 * q^61 - 5 * q^63 + 7 * q^64 + 12 * q^65 - 12 * q^67 - q^68 + 32 * q^69 - 6 * q^70 + 16 * q^71 - 5 * q^72 - 10 * q^73 + 22 * q^74 - 28 * q^75 - 12 * q^76 - 12 * q^77 + 24 * q^78 - 24 * q^79 + 2 * q^80 + 7 * q^81 + 6 * q^82 + 36 * q^83 - 4 * q^84 + 6 * q^85 - 20 * q^86 - 24 * q^87 - 8 * q^88 + 22 * q^89 - 6 * q^90 - 10 * q^91 + 8 * q^92 + 16 * q^93 - 16 * q^94 - 4 * q^96 - 18 * q^97 - q^98 + 32 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(238))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7	17
238.2.a.a	$238$	$2$	238.a	1.a	$1$	$1$	$1$	$1.900$	\(\Q\)	None	✓	✓	✓	✓	238.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
238.2.a.b	$238$	$2$	238.a	1.a	$1$	$1$	$1$	$1.900$	\(\Q\)	None	✓	✓	✓	✓	238.2.a.b	$1$	$0$	\(-1\)	\(2\)	\(4\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}+4q^{5}-2q^{6}+q^{7}+\cdots\)
238.2.a.c	$238$	$2$	238.a	1.a	$1$	$1$	$1$	$1.900$	\(\Q\)	None	✓	✓	✓	✓	238.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(-4\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-4q^{5}-2q^{6}+q^{7}+\cdots\)
238.2.a.d	$238$	$2$	238.a	1.a	$1$	$1$	$1$	$1.900$	\(\Q\)	None	✓	✓	✓	✓	238.2.a.d	$1$	$0$	\(1\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
238.2.a.e	$238$	$2$	238.a	1.a	$1$	$1$	$1$	$1.900$	\(\Q\)	None	✓	✓	✓	✓	238.2.a.e	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
238.2.a.f	$238$	$2$	238.a	1.a	$1$	$2$	$2$	$1.900$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	238.2.a.f	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(1-\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(238))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(238)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 2}\)