Space of modular forms of level 504, weight 6, and trivial character

• Label: 504.6.a
• Level: $504$
• Weight: $6$
• Character orbit: 504.a
• Rep. character: $\chi_{504}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $23$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(504))\).

	Total	New	Old
Modular forms	496	38	458
Cusp forms	464	38	426
Eisenstein series	32	0	32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(58\)	\(4\)	\(54\)		\(54\)	\(4\)	\(50\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(66\)	\(4\)	\(62\)		\(62\)	\(4\)	\(58\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(64\)	\(6\)	\(58\)		\(60\)	\(6\)	\(54\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(60\)	\(5\)	\(55\)		\(56\)	\(5\)	\(51\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(62\)	\(4\)	\(58\)		\(58\)	\(4\)	\(54\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(62\)	\(4\)	\(58\)		\(58\)	\(4\)	\(54\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(64\)	\(5\)	\(59\)		\(60\)	\(5\)	\(55\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(60\)	\(6\)	\(54\)		\(56\)	\(6\)	\(50\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(244\)	\(18\)	\(226\)		\(228\)	\(18\)	\(210\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(252\)	\(20\)	\(232\)		\(236\)	\(20\)	\(216\)		\(16\)	\(0\)	\(16\)

Trace form

38 * q + 98 * q^5 + 268 * q^11 - 382 * q^13 - 688 * q^17 + 894 * q^19 + 3712 * q^23 + 28534 * q^25 - 12944 * q^29 - 9204 * q^31 + 7350 * q^35 + 12592 * q^37 - 17448 * q^41 - 22796 * q^43 + 34908 * q^47 + 91238 * q^49 - 79028 * q^53 + 30232 * q^55 + 78498 * q^59 - 4130 * q^61 - 22324 * q^65 - 91936 * q^67 - 107824 * q^71 - 47020 * q^73 - 23716 * q^77 + 53720 * q^79 - 19430 * q^83 + 33356 * q^85 + 276500 * q^89 - 22442 * q^91 - 249768 * q^95 + 4768 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(504))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
504.6.a.a	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.c	$1$	$0$	\(0\)	\(0\)	\(-74\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-74q^{5}+7^{2}q^{7}-6^{3}q^{11}-186q^{13}+\cdots\)
504.6.a.b	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				56.6.a.b	$1$	$0$	\(0\)	\(0\)	\(-32\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{5}+7^{2}q^{7}+624q^{11}-708q^{13}+\cdots\)
504.6.a.c	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.f	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}-7^{2}q^{7}+700q^{11}+158q^{13}+\cdots\)
504.6.a.d	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-7^{2}q^{7}-370q^{11}+122q^{13}+\cdots\)
504.6.a.e	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				56.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+7^{2}q^{7}+240q^{11}-744q^{13}+\cdots\)
504.6.a.f	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.a	$1$	$0$	\(0\)	\(0\)	\(34\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+34q^{5}+7^{2}q^{7}+756q^{11}+678q^{13}+\cdots\)
504.6.a.g	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.e	$1$	$0$	\(0\)	\(0\)	\(38\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+38q^{5}-7^{2}q^{7}-600q^{11}-674q^{13}+\cdots\)
504.6.a.h	$504$	$6$	504.a	1.a	$1$	$1$	$1$	$80.833$	\(\Q\)	None	✓				168.6.a.d	$1$	$0$	\(0\)	\(0\)	\(64\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{6}q^{5}+7^{2}q^{7}+54q^{11}+738q^{13}+\cdots\)
504.6.a.i	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{345}) \)	None	✓				56.6.a.e	$1$	$0$	\(0\)	\(0\)	\(-82\)	\(-98\)		$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-41-\beta )q^{5}-7^{2}q^{7}+(-170-2\beta )q^{11}+\cdots\)
504.6.a.j	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{429}) \)	None	✓	✓			504.6.a.j	$1$	$0$	\(0\)	\(0\)	\(-80\)	\(98\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-40-\beta )q^{5}+7^{2}q^{7}+(-6^{3}-3\beta )q^{11}+\cdots\)
504.6.a.k	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{106}) \)	None	✓	✓			504.6.a.k	$1$	$1$	\(0\)	\(0\)	\(-76\)	\(98\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-38+\beta )q^{5}+7^{2}q^{7}+(-282-5\beta )q^{11}+\cdots\)
504.6.a.l	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{37}) \)	None	✓	✓			504.6.a.l	$1$	$1$	\(0\)	\(0\)	\(-48\)	\(-98\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-24-5\beta )q^{5}-7^{2}q^{7}+(-184+\cdots)q^{11}+\cdots\)
504.6.a.m	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{193}) \)	None	✓				56.6.a.d	$1$	$1$	\(0\)	\(0\)	\(-42\)	\(-98\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-21-5\beta )q^{5}-7^{2}q^{7}+(358+14\beta )q^{11}+\cdots\)
504.6.a.n	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{114}) \)	None	✓	✓			504.6.a.n	$1$	$1$	\(0\)	\(0\)	\(-28\)	\(-98\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-14+\beta )q^{5}-7^{2}q^{7}+(298-7\beta )q^{11}+\cdots\)
504.6.a.o	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{1129}) \)	None	✓				168.6.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-98\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-7^{2}q^{7}+(-50+\beta )q^{11}+(270+\cdots)q^{13}+\cdots\)
504.6.a.p	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{4281}) \)	None	✓				168.6.a.i	$1$	$1$	\(0\)	\(0\)	\(10\)	\(98\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{5}+7^{2}q^{7}+(-3^{3}-3\beta )q^{11}+\cdots\)
504.6.a.q	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{114}) \)	None	✓	✓			504.6.a.n	$1$	$0$	\(0\)	\(0\)	\(28\)	\(-98\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(14+\beta )q^{5}-7^{2}q^{7}+(-298-7\beta )q^{11}+\cdots\)
504.6.a.r	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{37}) \)	None	✓	✓			504.6.a.l	$1$	$0$	\(0\)	\(0\)	\(48\)	\(-98\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(24+5\beta )q^{5}-7^{2}q^{7}+(184+37\beta )q^{11}+\cdots\)
504.6.a.s	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{177}) \)	None	✓		✓		56.6.a.c	$1$	$0$	\(0\)	\(0\)	\(62\)	\(98\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(31-5\beta )q^{5}+7^{2}q^{7}+(-486-6\beta )q^{11}+\cdots\)
504.6.a.t	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{249}) \)	None	✓				168.6.a.h	$1$	$1$	\(0\)	\(0\)	\(64\)	\(98\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2^{5}-\beta )q^{5}+7^{2}q^{7}+(-270+15\beta )q^{11}+\cdots\)
504.6.a.u	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{106}) \)	None	✓	✓			504.6.a.k	$1$	$0$	\(0\)	\(0\)	\(76\)	\(98\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(38+\beta )q^{5}+7^{2}q^{7}+(282-5\beta )q^{11}+\cdots\)
504.6.a.v	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{193}) \)	None	✓				168.6.a.g	$1$	$0$	\(0\)	\(0\)	\(78\)	\(-98\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(39-5\beta )q^{5}-7^{2}q^{7}+(185-37\beta )q^{11}+\cdots\)
504.6.a.w	$504$	$6$	504.a	1.a	$1$	$2$	$2$	$80.833$	\(\Q(\sqrt{429}) \)	None	✓	✓			504.6.a.j	$1$	$1$	\(0\)	\(0\)	\(80\)	\(98\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(40-\beta )q^{5}+7^{2}q^{7}+(6^{3}-3\beta )q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(504)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)