Space of modular forms of level 252, weight 8, and Character 252.k

• Label: 252.8.k
• Level: $252$
• Weight: $8$
• Character orbit: 252.k
• Rep. character: $\chi_{252}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $46$
• Newform subspaces: $5$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(252, [\chi])\).

	Total	New	Old
Modular forms	696	46	650
Cusp forms	648	46	602
Eisenstein series	48	0	48

Trace form

\( 46 q - 251 q^{5} + 394 q^{7} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(252, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.8.k.a	$252$	$8$	252.k	7.c	$3$	$2$	$1$	$78.721$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1255\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-249-757\zeta_{6})q^{7}+12605q^{13}+\cdots\)
252.8.k.b	$252$	$8$	252.k	7.c	$3$	$8$	$4$	$78.721$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(196\)	\(-434\)	$2^{6}\cdot 3^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+7^{2}\beta _{2}-\beta _{5})q^{5}+(47+\beta _{1}-206\beta _{2}+\cdots)q^{7}+\cdots\)
252.8.k.c	$252$	$8$	252.k	7.c	$3$	$10$	$5$	$78.721$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-249\)	\(332\)	$2^{18}\cdot 3^{8}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-50+50\beta _{1}-\beta _{4})q^{5}+(129-191\beta _{1}+\cdots)q^{7}+\cdots\)
252.8.k.d	$252$	$8$	252.k	7.c	$3$	$10$	$5$	$78.721$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-198\)	\(71\)	$2^{12}\cdot 3^{8}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(40\beta _{1}-\beta _{2})q^{5}+(40+65\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
252.8.k.e	$252$	$8$	252.k	7.c	$3$	$16$	$8$	$78.721$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(1680\)	$2^{20}\cdot 3^{13}\cdot 7^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{4})q^{5}+(35+140\beta _{1}-3\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)