Space of modular forms of level 252, weight 6, and Character 252.k

• Label: 252.6.k
• Level: $252$
• Weight: $6$
• Character orbit: 252.k
• Rep. character: $\chi_{252}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $34$
• Newform subspaces: $7$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(252, [\chi])\).

	Total	New	Old
Modular forms	504	34	470
Cusp forms	456	34	422
Eisenstein series	48	0	48

Trace form

34 * q + 39 * q^5 - 206 * q^7 + 279 * q^11 + 1352 * q^13 + 729 * q^17 - 1381 * q^19 - 2385 * q^23 - 14882 * q^25 + 5820 * q^29 + 2933 * q^31 + 15 * q^35 - 15079 * q^37 + 8376 * q^41 + 6212 * q^43 - 15447 * q^47 + 15160 * q^49 - 28263 * q^53 - 100098 * q^55 - 19539 * q^59 - 40705 * q^61 + 50772 * q^65 - 19027 * q^67 - 23484 * q^71 - 47575 * q^73 - 100989 * q^77 - 28789 * q^79 - 83916 * q^83 - 386226 * q^85 + 61989 * q^89 - 176914 * q^91 + 54747 * q^95 + 195524 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(252, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.6.k.a	$252$	$6$	252.k	7.c	$3$	$2$	$1$	$40.417$	\(\Q(\sqrt{-3}) \)	None					84.6.i.a	$2$	$0$	\(0\)	\(0\)	\(-69\)	\(245\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-69+69\zeta_{6})q^{5}+(98+7^{2}\zeta_{6})q^{7}+\cdots\)
252.6.k.b	$252$	$6$	252.k	7.c	$3$	$2$	$1$	$40.417$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			252.6.k.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-25\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-87+149\zeta_{6})q^{7}-775q^{13}+(1711+\cdots)q^{19}+\cdots\)
252.6.k.c	$252$	$6$	252.k	7.c	$3$	$2$	$1$	$40.417$	\(\Q(\sqrt{-3}) \)	None					28.6.e.a	$2$	$0$	\(0\)	\(0\)	\(19\)	\(-140\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(19-19\zeta_{6})q^{5}+(-133+126\zeta_{6})q^{7}+\cdots\)
252.6.k.d	$252$	$6$	252.k	7.c	$3$	$4$	$2$	$40.417$	\(\Q(\sqrt{-3}, \sqrt{109})\)	None					28.6.e.b	$2$	$0$	\(0\)	\(0\)	\(42\)	\(112\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(21\beta _{1}-2\beta _{2}+2\beta _{3})q^{5}+(28-7\beta _{3})q^{7}+\cdots\)
252.6.k.e	$252$	$6$	252.k	7.c	$3$	$4$	$2$	$40.417$	\(\Q(\sqrt{-3}, \sqrt{7081})\)	None					84.6.i.b	$2$	$0$	\(0\)	\(0\)	\(47\)	\(-174\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+24\beta _{2})q^{5}+(-73-2\beta _{1}+\cdots)q^{7}+\cdots\)
252.6.k.f	$252$	$6$	252.k	7.c	$3$	$8$	$4$	$40.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					84.6.i.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-42\)	$2^{6}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{3})q^{5}+(10+30\beta _{1}-\beta _{6})q^{7}+\cdots\)
252.6.k.g	$252$	$6$	252.k	7.c	$3$	$12$	$6$	$40.417$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		252.6.k.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-182\)	$2^{12}\cdot 3^{9}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{8}q^{5}+(-8-2^{4}\beta _{1}+3\beta _{4}-\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

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