Space of modular forms of level 342, weight 2, and trivial character

• Label: 342.2.a
• Level: $342$
• Weight: $2$
• Character orbit: 342.a
• Rep. character: $\chi_{342}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $120$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(120\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(342))\).

	Total	New	Old
Modular forms	68	7	61
Cusp forms	53	7	46
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.

\(2\)	\(3\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(8\)	\(2\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(5\)	\(2\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(31\)	\(2\)	\(29\)		\(24\)	\(2\)	\(22\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(37\)	\(5\)	\(32\)		\(29\)	\(5\)	\(24\)		\(8\)	\(0\)	\(8\)

Trace form

7 * q - q^2 + 7 * q^4 + 2 * q^5 + 2 * q^7 - q^8 + 2 * q^10 + 4 * q^11 - 6 * q^13 + 4 * q^14 + 7 * q^16 - 4 * q^17 - q^19 + 2 * q^20 + 4 * q^22 + 10 * q^23 - 7 * q^25 + 8 * q^26 + 2 * q^28 - 2 * q^29 - 4 * q^31 - q^32 + 2 * q^34 + 12 * q^35 - 18 * q^37 + 3 * q^38 + 2 * q^40 - 18 * q^41 + 8 * q^43 + 4 * q^44 + 4 * q^46 - 20 * q^47 - 7 * q^49 - 15 * q^50 - 6 * q^52 + 6 * q^53 - 8 * q^55 + 4 * q^56 - 8 * q^58 - 28 * q^59 - 10 * q^61 + 4 * q^62 + 7 * q^64 - 8 * q^65 + 4 * q^67 - 4 * q^68 - 12 * q^70 + 12 * q^71 + 20 * q^73 - 10 * q^74 - q^76 - 28 * q^77 + 4 * q^79 + 2 * q^80 - 6 * q^82 + 28 * q^83 - 24 * q^85 - 8 * q^86 + 4 * q^88 + 26 * q^89 + 8 * q^91 + 10 * q^92 - 8 * q^94 - 2 * q^95 - 2 * q^97 - q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(342))\) 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	19
342.2.a.a	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓	✓	✓	✓	342.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{11}+\cdots\)
342.2.a.b	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓				114.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}+4q^{11}+\cdots\)
342.2.a.c	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓		✓	✓	114.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{13}+4q^{14}+\cdots\)
342.2.a.d	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓				38.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{5}+3q^{7}-q^{8}-4q^{10}+\cdots\)
342.2.a.e	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓				38.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+6q^{11}+5q^{13}+\cdots\)
342.2.a.f	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓				114.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}-4q^{11}+4q^{14}+\cdots\)
342.2.a.g	$342$	$2$	342.a	1.a	$1$	$1$	$1$	$2.731$	\(\Q\)	None	✓	✓	✓	✓	342.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+q^{8}+2q^{10}+2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(342)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)