Space of modular forms of level 252, weight 8, and trivial character

• Label: 252.8.a
• Level: $252$
• Weight: $8$
• Character orbit: 252.a
• Rep. character: $\chi_{252}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $8$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(252))\).

	Total	New	Old
Modular forms	348	18	330
Cusp forms	324	18	306
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
\(+\)	\(+\)	\(+\)	\(+\)		\(46\)	\(0\)	\(46\)		\(42\)	\(0\)	\(42\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(42\)	\(0\)	\(42\)		\(38\)	\(0\)	\(38\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(42\)	\(0\)	\(42\)		\(38\)	\(0\)	\(38\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(46\)	\(0\)	\(46\)		\(42\)	\(0\)	\(42\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(44\)	\(4\)	\(40\)		\(42\)	\(4\)	\(38\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(42\)	\(4\)	\(38\)		\(40\)	\(4\)	\(36\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(42\)	\(5\)	\(37\)		\(40\)	\(5\)	\(35\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(44\)	\(5\)	\(39\)		\(42\)	\(5\)	\(37\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(176\)	\(9\)	\(167\)		\(164\)	\(9\)	\(155\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(172\)	\(9\)	\(163\)		\(160\)	\(9\)	\(151\)		\(12\)	\(0\)	\(12\)

Trace form

18 * q + 32 * q^5 - 6572 * q^11 + 11568 * q^13 - 37412 * q^17 - 4920 * q^19 - 66428 * q^23 + 402702 * q^25 - 243820 * q^29 + 31464 * q^31 - 56252 * q^35 - 444612 * q^37 - 233508 * q^41 + 804216 * q^43 + 538632 * q^47 + 2117682 * q^49 - 2161524 * q^53 - 1570584 * q^55 - 241480 * q^59 + 1899048 * q^61 - 9090672 * q^65 - 140160 * q^67 + 2557372 * q^71 + 11339844 * q^73 + 93296 * q^77 - 15423384 * q^79 + 16091248 * q^83 - 9278040 * q^85 + 7392516 * q^89 + 1885128 * q^91 + 30310048 * q^95 - 33749844 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\) 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
252.8.a.a	$252$	$8$	252.a	1.a	$1$	$1$	$1$	$78.721$	\(\Q\)	None	✓				84.8.a.a	$1$	$0$	\(0\)	\(0\)	\(-100\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10^{2}q^{5}-7^{3}q^{7}-2774q^{11}-3294q^{13}+\cdots\)
252.8.a.b	$252$	$8$	252.a	1.a	$1$	$1$	$1$	$78.721$	\(\Q\)	None	✓				84.8.a.b	$1$	$1$	\(0\)	\(0\)	\(240\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+240q^{5}+7^{3}q^{7}-702q^{11}-3958q^{13}+\cdots\)
252.8.a.c	$252$	$8$	252.a	1.a	$1$	$2$	$2$	$78.721$	\(\Q(\sqrt{3649}) \)	None	✓				84.8.a.c	$1$	$1$	\(0\)	\(0\)	\(-264\)	\(686\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-132-\beta )q^{5}+7^{3}q^{7}+(2490-7\beta )q^{11}+\cdots\)
252.8.a.d	$252$	$8$	252.a	1.a	$1$	$2$	$2$	$78.721$	\(\Q(\sqrt{21961}) \)	None	✓				84.8.a.d	$1$	$0$	\(0\)	\(0\)	\(-96\)	\(-686\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-48-\beta )q^{5}-7^{3}q^{7}+(-2070+\cdots)q^{11}+\cdots\)
252.8.a.e	$252$	$8$	252.a	1.a	$1$	$2$	$2$	$78.721$	\(\Q(\sqrt{3529}) \)	None	✓				28.8.a.a	$1$	$1$	\(0\)	\(0\)	\(-42\)	\(686\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-21-\beta )q^{5}+7^{3}q^{7}+(-3714+\cdots)q^{11}+\cdots\)
252.8.a.f	$252$	$8$	252.a	1.a	$1$	$2$	$2$	$78.721$	\(\Q(\sqrt{1009}) \)	None	✓				28.8.a.b	$1$	$0$	\(0\)	\(0\)	\(294\)	\(-686\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(147-11\beta )q^{5}-7^{3}q^{7}+(1746+182\beta )q^{11}+\cdots\)
252.8.a.g	$252$	$8$	252.a	1.a	$1$	$4$	$4$	$78.721$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	252.8.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1372\)		$-$	$+$	$+$	$-$	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-7^{3}q^{7}+(-5\beta _{1}+\beta _{3})q^{11}+\cdots\)
252.8.a.h	$252$	$8$	252.a	1.a	$1$	$4$	$4$	$78.721$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	252.8.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1372\)		$-$	$+$	$-$	$+$	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+7^{3}q^{7}+(3\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

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