Space of modular forms of level 252, weight 4, and trivial character

• Label: 252.4.a
• Level: $252$
• Weight: $4$
• Character orbit: 252.a
• Rep. character: $\chi_{252}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(252))\).

	Total	New	Old
Modular forms	156	8	148
Cusp forms	132	8	124
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)	\(+\)		\(22\)	\(0\)	\(22\)		\(18\)	\(0\)	\(18\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(18\)	\(0\)	\(18\)		\(14\)	\(0\)	\(14\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(18\)	\(0\)	\(18\)		\(14\)	\(0\)	\(14\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(22\)	\(0\)	\(22\)		\(18\)	\(0\)	\(18\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(20\)	\(2\)	\(18\)		\(18\)	\(2\)	\(16\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(18\)	\(2\)	\(16\)		\(16\)	\(2\)	\(14\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(18\)	\(2\)	\(16\)		\(16\)	\(2\)	\(14\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(20\)	\(2\)	\(18\)		\(18\)	\(2\)	\(16\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(80\)	\(4\)	\(76\)		\(68\)	\(4\)	\(64\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(76\)	\(4\)	\(72\)		\(64\)	\(4\)	\(60\)		\(12\)	\(0\)	\(12\)

Trace form

8 * q - 18 * q^5 + 12 * q^11 + 14 * q^13 - 12 * q^17 + 46 * q^19 + 24 * q^23 - 108 * q^25 - 132 * q^29 - 52 * q^31 - 42 * q^35 + 316 * q^37 + 252 * q^41 - 740 * q^43 + 924 * q^47 + 392 * q^49 - 312 * q^53 + 296 * q^55 - 846 * q^59 - 934 * q^61 - 732 * q^65 + 696 * q^67 + 456 * q^71 + 648 * q^73 - 420 * q^77 + 632 * q^79 - 390 * q^83 - 460 * q^85 + 2232 * q^89 + 350 * q^91 - 1032 * q^95 - 204 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(252))\) 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
252.4.a.a	$252$	$4$	252.a	1.a	$1$	$1$	$1$	$14.868$	\(\Q\)	None	✓				84.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}-7q^{7}-4q^{11}+54q^{13}+\cdots\)
252.4.a.b	$252$	$4$	252.a	1.a	$1$	$1$	$1$	$14.868$	\(\Q\)	None	✓				84.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}+7q^{7}-6^{2}q^{11}+62q^{13}+\cdots\)
252.4.a.c	$252$	$4$	252.a	1.a	$1$	$1$	$1$	$14.868$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}+7q^{7}+12q^{11}-82q^{13}+\cdots\)
252.4.a.d	$252$	$4$	252.a	1.a	$1$	$1$	$1$	$14.868$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}-7q^{7}+40q^{11}-12q^{13}+\cdots\)
252.4.a.e	$252$	$4$	252.a	1.a	$1$	$2$	$2$	$14.868$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	252.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-7q^{7}-13\beta q^{11}-30q^{13}+\cdots\)
252.4.a.f	$252$	$4$	252.a	1.a	$1$	$2$	$2$	$14.868$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	252.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+7q^{7}+\beta q^{11}+26q^{13}-5\beta q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(252)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)