Space of modular forms of level 156, weight 2, and trivial character

• Label: 156.2.a
• Level: $156$
• Weight: $2$
• Character orbit: 156.a
• Rep. character: $\chi_{156}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $56$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	156.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(56\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(156))\).

	Total	New	Old
Modular forms	34	2	32
Cusp forms	23	2	21
Eisenstein series	11	0	11

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
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(15\)	\(1\)	\(14\)		\(10\)	\(1\)	\(9\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(19\)	\(1\)	\(18\)		\(13\)	\(1\)	\(12\)		\(6\)	\(0\)	\(6\)

Trace form

2 * q - 4 * q^5 + 2 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^15 - 4 * q^17 + 4 * q^21 + 6 * q^25 - 12 * q^29 - 8 * q^31 + 4 * q^33 + 8 * q^35 + 12 * q^37 - 4 * q^41 - 4 * q^45 - 4 * q^47 - 6 * q^49 - 8 * q^51 - 4 * q^53 + 16 * q^55 + 4 * q^57 + 4 * q^59 - 12 * q^61 - 4 * q^65 - 8 * q^67 + 28 * q^71 + 4 * q^73 - 16 * q^75 + 8 * q^77 - 8 * q^79 + 2 * q^81 + 12 * q^83 - 8 * q^85 - 4 * q^89 + 12 * q^93 + 8 * q^95 - 12 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(156))\) 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
156.2.a.a	$156$	$2$	156.a	1.a	$1$	$1$	$1$	$1.246$	\(\Q\)	None	✓	✓	✓	✓	156.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
156.2.a.b	$156$	$2$	156.a	1.a	$1$	$1$	$1$	$1.246$	\(\Q\)	None	✓	✓	✓	✓	156.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+q^{13}-6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(156))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(156)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)