Space of modular forms of level 252, weight 4, and Character 252.k

• Label: 252.4.k
• Level: $252$
• Weight: $4$
• Character orbit: 252.k
• Rep. character: $\chi_{252}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(252, [\chi])\).

	Total	New	Old
Modular forms	312	20	292
Cusp forms	264	20	244
Eisenstein series	48	0	48

Trace form

\( 20 q - 6 q^{5} + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(252, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.4.k.a	$252$	$4$	252.k	7.c	$3$	$2$	$1$	$14.868$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-37\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-19+\zeta_{6})q^{7}+89q^{13}+(163-163\zeta_{6})q^{19}+\cdots\)
252.4.k.b	$252$	$4$	252.k	7.c	$3$	$2$	$1$	$14.868$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(17\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-1+19\zeta_{6})q^{7}-19q^{13}+(-107+\cdots)q^{19}+\cdots\)
252.4.k.c	$252$	$4$	252.k	7.c	$3$	$4$	$2$	$14.868$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(24\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-7\beta _{1}-2\beta _{2})q^{5}+(10-8\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
252.4.k.d	$252$	$4$	252.k	7.c	$3$	$4$	$2$	$14.868$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-20\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\beta _{1}-\beta _{2}-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
252.4.k.e	$252$	$4$	252.k	7.c	$3$	$4$	$2$	$14.868$	\(\Q(\sqrt{-3}, \sqrt{385})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{5}+(-7+21\beta _{1})q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
252.4.k.f	$252$	$4$	252.k	7.c	$3$	$4$	$2$	$14.868$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		$2$	$0$	\(0\)	\(0\)	\(11\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+6\beta _{2})q^{5}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

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