Space of modular forms of level 234, weight 2, and trivial character

• Label: 234.2.a
• Level: $234$
• Weight: $2$
• Character orbit: 234.a
• Rep. character: $\chi_{234}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $84$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(84\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(234))\).

	Total	New	Old
Modular forms	50	5	45
Cusp forms	35	5	30
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(2\)	\(4\)		\(4\)	\(2\)	\(2\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(22\)	\(1\)	\(21\)		\(15\)	\(1\)	\(14\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(28\)	\(4\)	\(24\)		\(20\)	\(4\)	\(16\)		\(8\)	\(0\)	\(8\)

Trace form

5 * q + q^2 + 5 * q^4 + 2 * q^5 + q^8 + 4 * q^10 - q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 - 12 * q^19 + 2 * q^20 + 4 * q^22 + 4 * q^23 - 3 * q^25 + 3 * q^26 - 14 * q^29 - 8 * q^31 + q^32 - 2 * q^34 - 10 * q^35 + 6 * q^37 - 12 * q^38 + 4 * q^40 + 10 * q^41 - 18 * q^43 - 12 * q^46 - 24 * q^47 - 9 * q^49 + 7 * q^50 - q^52 - 2 * q^53 - 8 * q^55 + 2 * q^56 + 6 * q^58 + 12 * q^59 + 18 * q^61 - 12 * q^62 + 5 * q^64 - 8 * q^67 + 4 * q^68 - 20 * q^70 + 16 * q^71 + 22 * q^73 - 12 * q^74 - 12 * q^76 + 24 * q^77 + 4 * q^79 + 2 * q^80 - 2 * q^82 - 24 * q^83 + 16 * q^85 + 8 * q^86 + 4 * q^88 - 14 * q^89 + 6 * q^91 + 4 * q^92 - 14 * q^94 + 28 * q^95 - 6 * q^97 + 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(234))\) 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	3	13
234.2.a.a	$234$	$2$	234.a	1.a	$1$	$1$	$1$	$1.868$	\(\Q\)	None	✓	✓	✓	✓	234.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-2q^{7}-q^{8}+2q^{10}+\cdots\)
234.2.a.b	$234$	$2$	234.a	1.a	$1$	$1$	$1$	$1.868$	\(\Q\)	None	✓		✓	✓	26.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
234.2.a.c	$234$	$2$	234.a	1.a	$1$	$1$	$1$	$1.868$	\(\Q\)	None	✓				78.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+4q^{7}+q^{8}-2q^{10}+\cdots\)
234.2.a.d	$234$	$2$	234.a	1.a	$1$	$1$	$1$	$1.868$	\(\Q\)	None	✓	✓	✓	✓	234.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
234.2.a.e	$234$	$2$	234.a	1.a	$1$	$1$	$1$	$1.868$	\(\Q\)	None	✓				26.2.a.a	$1$	$0$	\(1\)	\(0\)	\(3\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}-q^{7}+q^{8}+3q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(234))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(234)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)