Space of modular forms of level 252, weight 2, and trivial character

• Label: 252.2.a
• Level: $252$
• Weight: $2$
• Character orbit: 252.a
• Rep. character: $\chi_{252}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(252))\).

	Total	New	Old
Modular forms	60	2	58
Cusp forms	37	2	35
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(3\)	\(0\)	\(3\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(8\)	\(1\)	\(7\)		\(6\)	\(1\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(28\)	\(1\)	\(27\)		\(17\)	\(1\)	\(16\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(32\)	\(1\)	\(31\)		\(20\)	\(1\)	\(19\)		\(12\)	\(0\)	\(12\)

Trace form

2 * q - 4 * q^5 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 6 * q^25 - 4 * q^29 + 8 * q^31 + 4 * q^35 + 4 * q^37 - 12 * q^41 - 8 * q^43 - 24 * q^47 + 2 * q^49 + 12 * q^53 + 8 * q^55 + 8 * q^59 - 4 * q^61 + 24 * q^65 - 20 * q^71 - 12 * q^73 + 8 * q^77 + 8 * q^79 + 16 * q^83 - 16 * q^85 - 12 * q^89 + 8 * q^91 + 16 * q^95 - 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(252))\) 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	7
252.2.a.a	$252$	$2$	252.a	1.a	$1$	$1$	$1$	$2.012$	\(\Q\)	None	✓		✓	✓	84.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-q^{7}-2q^{11}-6q^{13}+4q^{17}+\cdots\)
252.2.a.b	$252$	$2$	252.a	1.a	$1$	$1$	$1$	$2.012$	\(\Q\)	None	✓		✓	✓	84.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+6q^{11}+2q^{13}-4q^{19}+6q^{23}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(252)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)