Space of modular forms of level 504, weight 6, and Character 504.s

• Label: 504.6.s
• Level: $504$
• Weight: $6$
• Character orbit: 504.s
• Rep. character: $\chi_{504}(289,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $100$
• Newform subspaces: $8$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(504, [\chi])\).

	Total	New	Old
Modular forms	992	100	892
Cusp forms	928	100	828
Eisenstein series	64	0	64

Trace form

100 * q - 50 * q^5 - 24 * q^7 - 10 * q^11 - 560 * q^13 - 614 * q^17 - 2994 * q^19 + 722 * q^23 - 32020 * q^25 - 14464 * q^29 + 2238 * q^31 - 27162 * q^35 + 5414 * q^37 - 6384 * q^41 + 17288 * q^43 - 9426 * q^47 - 46796 * q^49 + 19898 * q^53 + 91844 * q^55 + 108354 * q^59 + 61346 * q^61 - 91076 * q^65 + 14914 * q^67 + 105136 * q^71 + 60958 * q^73 + 183598 * q^77 - 12014 * q^79 - 248920 * q^83 + 138580 * q^85 - 458 * q^89 + 265124 * q^91 - 41382 * q^95 - 47152 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(504, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
504.6.s.a	$504$	$6$	504.s	7.c	$3$	$8$	$4$	$80.833$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					168.6.q.a	$2$	$0$	\(0\)	\(0\)	\(-64\)	\(42\)	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2^{4}\beta _{1}-\beta _{2})q^{5}+(22-31\beta _{1}-3\beta _{3}+\cdots)q^{7}+\cdots\)
504.6.s.b	$504$	$6$	504.s	7.c	$3$	$10$	$5$	$80.833$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					56.6.i.b	$2$	$0$	\(0\)	\(0\)	\(-81\)	\(116\)	$2^{18}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2^{4}-2^{4}\beta _{3}-\beta _{4}-\beta _{5})q^{5}+(15+\cdots)q^{7}+\cdots\)
504.6.s.c	$504$	$6$	504.s	7.c	$3$	$10$	$5$	$80.833$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					168.6.q.c	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-97\)	$2^{12}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{5})q^{5}+(-5-9\beta _{1}+\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)
504.6.s.d	$504$	$6$	504.s	7.c	$3$	$10$	$5$	$80.833$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					56.6.i.a	$2$	$0$	\(0\)	\(0\)	\(31\)	\(-92\)	$2^{18}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6\beta _{1}+\beta _{5}+\beta _{7})q^{5}+(-2^{4}-13\beta _{1}+\cdots)q^{7}+\cdots\)
504.6.s.e	$504$	$6$	504.s	7.c	$3$	$10$	$5$	$80.833$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					168.6.q.b	$2$	$0$	\(0\)	\(0\)	\(75\)	\(-113\)	$2^{12}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-15\beta _{1}-\beta _{2}+\beta _{4})q^{5}+(-13-3\beta _{1}+\cdots)q^{7}+\cdots\)
504.6.s.f	$504$	$6$	504.s	7.c	$3$	$12$	$6$	$80.833$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					168.6.q.d	$2$	$0$	\(0\)	\(0\)	\(-17\)	\(144\)	$2^{18}\cdot 3\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3\beta _{1}-\beta _{2}+\beta _{4})q^{5}+(8+9\beta _{1}+\cdots)q^{7}+\cdots\)
504.6.s.g	$504$	$6$	504.s	7.c	$3$	$20$	$10$	$80.833$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			504.6.s.g	$2$	$0$	\(0\)	\(0\)	\(-19\)	\(-12\)	$2^{32}\cdot 3^{10}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{5}+(2-5\beta _{2}+\cdots)q^{7}+\cdots\)
504.6.s.h	$504$	$6$	504.s	7.c	$3$	$20$	$10$	$80.833$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			504.6.s.g	$2$	$0$	\(0\)	\(0\)	\(19\)	\(-12\)	$2^{32}\cdot 3^{10}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+2\beta _{2}+\beta _{3})q^{5}+(2-5\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(504, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)