Space of modular forms of level 156, weight 4, and trivial character

• Label: 156.4.a
• Level: $156$
• Weight: $4$
• Character orbit: 156.a
• Rep. character: $\chi_{156}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	156.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(156))\).

	Total	New	Old
Modular forms	90	6	84
Cusp forms	78	6	72
Eisenstein series	12	0	12

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	Dim
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(3\)

Trace form

\( 6 q + 16 q^{5} - 20 q^{7} + 54 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(156))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
156.4.a.a	$156$	$4$	156.a	1.a	$1$	$1$	$1$	$9.204$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-6\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}-4q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
156.4.a.b	$156$	$4$	156.a	1.a	$1$	$1$	$1$	$9.204$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-2\)	\(-32\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}-2^{5}q^{7}+9q^{9}-68q^{11}+\cdots\)
156.4.a.c	$156$	$4$	156.a	1.a	$1$	$2$	$2$	$9.204$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(8\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta q^{5}+(4-3\beta )q^{7}+9q^{9}+\cdots\)
156.4.a.d	$156$	$4$	156.a	1.a	$1$	$2$	$2$	$9.204$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(24\)	\(8\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(12+\beta )q^{5}+(4-3\beta )q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(156))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(156)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)