Space of modular forms of level 252, weight 3, and Character 252.z

• Label: 252.3.z
• Level: $252$
• Weight: $3$
• Character orbit: 252.z
• Rep. character: $\chi_{252}(73,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(252, [\chi])\).

	Total	New	Old
Modular forms	216	14	202
Cusp forms	168	14	154
Eisenstein series	48	0	48

Trace form

14 * q + 3 * q^5 + 4 * q^7 - 3 * q^11 - 3 * q^17 + 45 * q^19 - 21 * q^23 + 68 * q^25 + 108 * q^29 + 105 * q^31 + 99 * q^35 + 35 * q^37 - 136 * q^43 - 141 * q^47 - 100 * q^49 - 105 * q^53 - 231 * q^59 + 39 * q^61 - 90 * q^65 - 77 * q^67 + 456 * q^71 - 99 * q^73 + 147 * q^77 + 115 * q^79 - 522 * q^85 - 423 * q^89 - 318 * q^91 - 69 * q^95

Decomposition of \(S_{3}^{\mathrm{new}}(252, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.3.z.a	$252$	$3$	252.z	7.d	$6$	$2$	$1$	$6.867$	\(\Q(\sqrt{-3}) \)	None					28.3.h.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}-7q^{7}+(15-15\zeta_{6})q^{11}+\cdots\)
252.3.z.b	$252$	$3$	252.z	7.d	$6$	$2$	$1$	$6.867$	\(\Q(\sqrt{-3}) \)	None					84.3.m.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(5+3\zeta_{6})q^{7}+(-3+\cdots)q^{11}+\cdots\)
252.3.z.c	$252$	$3$	252.z	7.d	$6$	$2$	$1$	$6.867$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			252.3.z.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-8+5\zeta_{6})q^{7}+(-15+30\zeta_{6})q^{13}+\cdots\)
252.3.z.d	$252$	$3$	252.z	7.d	$6$	$4$	$2$	$6.867$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			252.3.z.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{5}+(7+7\beta _{1})q^{7}+(-2\beta _{2}+\beta _{3})q^{11}+\cdots\)
252.3.z.e	$252$	$3$	252.z	7.d	$6$	$4$	$2$	$6.867$	\(\Q(\sqrt{-3}, \sqrt{65})\)	None					84.3.m.b	$2$	$0$	\(0\)	\(0\)	\(9\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+\beta _{1}-\beta _{3})q^{5}+(1-\beta _{2})q^{7}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

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