Space of modular forms of level 532, weight 2, and trivial character

• Label: 532.2.a
• Level: $532$
• Weight: $2$
• Character orbit: 532.a
• Rep. character: $\chi_{532}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $160$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	532.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(160\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(532))\).

	Total	New	Old
Modular forms	86	10	76
Cusp forms	75	10	65
Eisenstein series	11	0	11

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

\(2\)	\(7\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(9\)	\(0\)	\(9\)		\(8\)	\(0\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(13\)	\(0\)	\(13\)		\(11\)	\(0\)	\(11\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(41\)	\(4\)	\(37\)		\(36\)	\(4\)	\(32\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(45\)	\(6\)	\(39\)		\(39\)	\(6\)	\(33\)		\(6\)	\(0\)	\(6\)

Trace form

10 * q - 4 * q^3 + 6 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^21 - 8 * q^23 + 14 * q^25 - 4 * q^27 + 8 * q^29 - 8 * q^31 - 12 * q^33 + 4 * q^35 - 8 * q^37 + 20 * q^39 + 4 * q^41 + 12 * q^45 + 12 * q^47 + 10 * q^49 + 4 * q^51 + 4 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 + 8 * q^61 - 4 * q^65 - 8 * q^67 - 20 * q^69 - 24 * q^71 - 12 * q^73 + 8 * q^75 + 8 * q^77 - 20 * q^79 - 14 * q^81 + 32 * q^83 + 28 * q^85 - 44 * q^87 - 20 * q^89 - 4 * q^91 - 8 * q^93 + 8 * q^97 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(532))\) 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	7	19
532.2.a.a	$532$	$2$	532.a	1.a	$1$	$1$	$1$	$4.248$	\(\Q\)	None	✓	✓			532.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-3q^{9}+4q^{11}+4q^{13}+\cdots\)
532.2.a.b	$532$	$2$	532.a	1.a	$1$	$2$	$2$	$4.248$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	532.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+q^{7}+(-1+3\beta )q^{9}+\cdots\)
532.2.a.c	$532$	$2$	532.a	1.a	$1$	$2$	$2$	$4.248$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓		532.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(6\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+3q^{5}+q^{7}+(2+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
532.2.a.d	$532$	$2$	532.a	1.a	$1$	$2$	$2$	$4.248$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	532.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-2\beta )q^{5}-q^{7}+(-2+\cdots)q^{9}+\cdots\)
532.2.a.e	$532$	$2$	532.a	1.a	$1$	$3$	$3$	$4.248$	3.3.733.1	None	✓	✓	✓	✓	532.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}-\beta _{2})q^{5}-q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(532))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(532)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 2}\)