Space of modular forms of level 504, weight 2, and trivial character

• Label: 504.2.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.a
• Rep. character: $\chi_{504}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(504))\).

	Total	New	Old
Modular forms	112	8	104
Cusp forms	81	8	73
Eisenstein series	31	0	31

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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(5\)

Trace form

\( 8 q - 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(504))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
504.2.a.a	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-2q^{11}+2q^{13}-6q^{17}+\cdots\)
504.2.a.b	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
504.2.a.c	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.d	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-6q^{11}-6q^{13}+2q^{17}+\cdots\)
504.2.a.e	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}+6q^{13}+2q^{17}+4q^{19}+\cdots\)
504.2.a.f	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}+2q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.g	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+6q^{11}-6q^{13}-2q^{17}+\cdots\)
504.2.a.h	$504$	$2$	504.a	1.a	$1$	$1$	$1$	$4.024$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+q^{7}+2q^{17}-2q^{19}-8q^{23}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces

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