Space of modular forms of level 588, weight 2, and trivial character

• Label: 588.2.a
• Level: $588$
• Weight: $2$
• Character orbit: 588.a
• Rep. character: $\chi_{588}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(588))\).

	Total	New	Old
Modular forms	136	6	130
Cusp forms	89	6	83
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)	\(+\)		\(16\)	\(0\)	\(16\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(19\)	\(0\)	\(19\)		\(11\)	\(0\)	\(11\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(20\)	\(0\)	\(20\)		\(12\)	\(0\)	\(12\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(17\)	\(0\)	\(17\)		\(9\)	\(0\)	\(9\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(16\)	\(1\)	\(15\)		\(12\)	\(1\)	\(11\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(16\)	\(2\)	\(14\)		\(12\)	\(2\)	\(10\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(16\)	\(0\)	\(16\)		\(12\)	\(0\)	\(12\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(16\)	\(3\)	\(13\)		\(12\)	\(3\)	\(9\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(65\)	\(2\)	\(63\)		\(42\)	\(2\)	\(40\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(71\)	\(4\)	\(67\)		\(47\)	\(4\)	\(43\)		\(24\)	\(0\)	\(24\)

Trace form

6 * q - 4 * q^5 + 6 * q^9 + 4 * q^11 + 4 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^25 - 8 * q^29 - 8 * q^31 + 8 * q^33 + 14 * q^37 + 6 * q^39 - 12 * q^41 + 22 * q^43 - 4 * q^45 - 24 * q^47 - 32 * q^53 - 8 * q^55 + 18 * q^57 + 8 * q^59 + 4 * q^61 - 28 * q^65 + 10 * q^67 + 8 * q^69 - 20 * q^71 + 12 * q^73 + 16 * q^75 + 10 * q^79 + 6 * q^81 + 16 * q^83 - 24 * q^85 - 8 * q^87 - 12 * q^89 + 10 * q^93 + 20 * q^95 + 12 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(588))\) 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
588.2.a.a	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓		✓	✓	84.2.i.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+2q^{11}-3q^{13}+\cdots\)
588.2.a.b	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓	✓			588.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+2q^{11}+4q^{13}+\cdots\)
588.2.a.c	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓				84.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-6q^{11}-2q^{13}+4q^{19}+\cdots\)
588.2.a.d	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓				84.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+q^{9}+2q^{11}+6q^{13}+\cdots\)
588.2.a.e	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓	✓			588.2.a.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+2q^{11}-4q^{13}+\cdots\)
588.2.a.f	$588$	$2$	588.a	1.a	$1$	$1$	$1$	$4.695$	\(\Q\)	None	✓				84.2.i.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+2q^{11}+3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(588))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(588)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)