Space of modular forms of level 592, weight 2, and trivial character

• Label: 592.2.a
• Level: $592$
• Weight: $2$
• Character orbit: 592.a
• Rep. character: $\chi_{592}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $10$
• Sturm bound: $152$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(152\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(592))\).

	Total	New	Old
Modular forms	82	18	64
Cusp forms	71	18	53
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.

\(2\)	\(37\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(19\)	\(4\)	\(15\)		\(17\)	\(4\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(21\)	\(5\)	\(16\)		\(18\)	\(5\)	\(13\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(22\)	\(5\)	\(17\)		\(19\)	\(5\)	\(14\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(20\)	\(4\)	\(16\)		\(17\)	\(4\)	\(13\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(39\)	\(8\)	\(31\)		\(34\)	\(8\)	\(26\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(43\)	\(10\)	\(33\)		\(37\)	\(10\)	\(27\)		\(6\)	\(0\)	\(6\)

Trace form

18 * q + 14 * q^9 - 4 * q^15 - 4 * q^17 - 2 * q^19 - 8 * q^21 + 6 * q^23 + 10 * q^25 - 12 * q^27 - 8 * q^29 - 14 * q^31 - 8 * q^33 + 12 * q^35 + 4 * q^39 + 4 * q^41 - 10 * q^43 - 8 * q^45 + 10 * q^49 + 28 * q^51 - 8 * q^53 + 12 * q^55 - 16 * q^57 + 2 * q^59 + 8 * q^61 + 36 * q^63 - 16 * q^65 + 12 * q^67 - 8 * q^69 + 8 * q^71 - 20 * q^73 + 20 * q^75 - 24 * q^77 - 6 * q^79 + 2 * q^81 - 8 * q^83 + 16 * q^85 - 4 * q^89 - 28 * q^91 + 8 * q^93 + 28 * q^95 - 12 * q^97 + 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(592))\) 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	37
592.2.a.a	$592$	$2$	592.a	1.a	$1$	$1$	$1$	$4.727$	\(\Q\)	None	✓				37.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-3q^{11}-4q^{13}+\cdots\)
592.2.a.b	$592$	$2$	592.a	1.a	$1$	$1$	$1$	$4.727$	\(\Q\)	None	✓				148.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+3q^{7}-2q^{9}-5q^{11}+\cdots\)
592.2.a.c	$592$	$2$	592.a	1.a	$1$	$1$	$1$	$4.727$	\(\Q\)	None	✓				296.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots\)
592.2.a.d	$592$	$2$	592.a	1.a	$1$	$1$	$1$	$4.727$	\(\Q\)	None	✓				296.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}-2q^{9}+3q^{11}+2q^{17}+\cdots\)
592.2.a.e	$592$	$2$	592.a	1.a	$1$	$1$	$1$	$4.727$	\(\Q\)	None	✓				37.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}+q^{7}+6q^{9}+5q^{11}+\cdots\)
592.2.a.f	$592$	$2$	592.a	1.a	$1$	$2$	$2$	$4.727$	\(\Q(\sqrt{13}) \)	None	✓		✓		74.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-\beta q^{5}+(-2+2\beta )q^{7}+\cdots\)
592.2.a.g	$592$	$2$	592.a	1.a	$1$	$2$	$2$	$4.727$	\(\Q(\sqrt{5}) \)	None	✓				74.2.a.b	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+3\beta )q^{5}+2\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
592.2.a.h	$592$	$2$	592.a	1.a	$1$	$2$	$2$	$4.727$	\(\Q(\sqrt{17}) \)	None	✓				148.2.a.b	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}-\beta q^{7}+(1+\beta )q^{9}-\beta q^{11}+\cdots\)
592.2.a.i	$592$	$2$	592.a	1.a	$1$	$3$	$3$	$4.727$	3.3.229.1	None	✓		✓		296.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+\beta _{2}q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
592.2.a.j	$592$	$2$	592.a	1.a	$1$	$4$	$4$	$4.727$	4.4.48389.1	None	✓		✓		296.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(5\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{3})q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(592))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(592)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 2}\)