Space of modular forms of level 592, weight 8, and trivial character

• Label: 592.8.a
• Level: $592$
• Weight: $8$
• Character orbit: 592.a
• Rep. character: $\chi_{592}(1,\cdot)$
• Character field: $\Q$
• Dimension: $126$
• Newform subspaces: $12$
• Sturm bound: $608$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(608\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(592))\).

	Total	New	Old
Modular forms	538	126	412
Cusp forms	526	126	400
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)		\(136\)	\(32\)	\(104\)		\(133\)	\(32\)	\(101\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(134\)	\(31\)	\(103\)		\(131\)	\(31\)	\(100\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(133\)	\(31\)	\(102\)		\(130\)	\(31\)	\(99\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(135\)	\(32\)	\(103\)		\(132\)	\(32\)	\(100\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(271\)	\(64\)	\(207\)		\(265\)	\(64\)	\(201\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(267\)	\(62\)	\(205\)		\(261\)	\(62\)	\(199\)		\(6\)	\(0\)	\(6\)

Trace form

126 * q + 89618 * q^9 + 13500 * q^15 + 23972 * q^17 - 101738 * q^19 + 2344 * q^21 + 232278 * q^23 + 1989238 * q^25 - 238044 * q^27 - 128024 * q^29 + 446770 * q^31 - 176184 * q^33 - 1248324 * q^35 + 1597476 * q^39 + 300604 * q^41 - 1229890 * q^43 + 1009000 * q^45 + 97920 * q^47 + 15840166 * q^49 + 1915980 * q^51 + 685096 * q^53 - 2609492 * q^55 + 577808 * q^57 + 1554170 * q^59 - 4104936 * q^61 + 1959588 * q^63 - 4214896 * q^65 + 12106044 * q^67 + 11934120 * q^69 - 15241144 * q^71 + 8118100 * q^73 + 22343940 * q^75 - 7956168 * q^77 + 1394714 * q^79 + 49967150 * q^81 - 4674344 * q^83 + 7534000 * q^85 + 13390080 * q^87 + 17156 * q^89 - 6576428 * q^91 - 26260456 * q^93 - 2172644 * q^95 - 20451284 * q^97 - 37897608 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(592))\) 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
592.8.a.a	$592$	$8$	592.a	1.a	$1$	$4$	$4$	$184.932$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				74.8.a.b	$1$	$1$	\(0\)	\(41\)	\(-363\)	\(774\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(10-\beta _{2})q^{3}+(-89-6\beta _{1}-4\beta _{2}+\cdots)q^{5}+\cdots\)
592.8.a.b	$592$	$8$	592.a	1.a	$1$	$4$	$4$	$184.932$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				74.8.a.a	$1$	$0$	\(0\)	\(53\)	\(111\)	\(1666\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(14-2\beta _{1}-\beta _{3})q^{3}+(2^{5}-6\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
592.8.a.c	$592$	$8$	592.a	1.a	$1$	$6$	$6$	$184.932$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				74.8.a.c	$1$	$1$	\(0\)	\(-28\)	\(-14\)	\(980\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{1})q^{3}+(-4+2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
592.8.a.d	$592$	$8$	592.a	1.a	$1$	$7$	$7$	$184.932$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				74.8.a.d	$1$	$0$	\(0\)	\(-40\)	\(512\)	\(-1284\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-6+\beta _{1})q^{3}+(74-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
592.8.a.e	$592$	$8$	592.a	1.a	$1$	$10$	$10$	$184.932$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				148.8.a.a	$1$	$1$	\(0\)	\(13\)	\(-374\)	\(1163\)		$-$	$+$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-37-\beta _{1}-\beta _{3})q^{5}+\cdots\)
592.8.a.f	$592$	$8$	592.a	1.a	$1$	$10$	$10$	$184.932$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				37.8.a.a	$1$	$0$	\(0\)	\(95\)	\(-624\)	\(501\)		$-$	$-$	$+$	$2^{12}\cdot 3$	$\mathrm{SU}(2)$	\(q+(10+\beta _{3})q^{3}+(-62+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
592.8.a.g	$592$	$8$	592.a	1.a	$1$	$11$	$11$	$184.932$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		37.8.a.b	$1$	$1$	\(0\)	\(-121\)	\(376\)	\(-2243\)		$-$	$+$	$-$	$2^{16}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{2})q^{3}+(34-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
592.8.a.h	$592$	$8$	592.a	1.a	$1$	$11$	$11$	$184.932$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		148.8.a.b	$1$	$0$	\(0\)	\(13\)	\(626\)	\(-209\)		$-$	$-$	$+$	$2^{13}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(57+\beta _{2})q^{5}+(-19+\cdots)q^{7}+\cdots\)
592.8.a.i	$592$	$8$	592.a	1.a	$1$	$14$	$14$	$184.932$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				296.8.a.a	$1$	$1$	\(0\)	\(98\)	\(-513\)	\(-783\)		$+$	$-$	$-$	$2^{28}\cdot 3$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{3}+(-37-\beta _{4})q^{5}+(-56+\cdots)q^{7}+\cdots\)
592.8.a.j	$592$	$8$	592.a	1.a	$1$	$15$	$15$	$184.932$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				296.8.a.b	$1$	$0$	\(0\)	\(-30\)	\(13\)	\(1481\)		$+$	$+$	$+$	$2^{31}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+(1+\beta _{4})q^{5}+(99+2\beta _{1}+\cdots)q^{7}+\cdots\)
592.8.a.k	$592$	$8$	592.a	1.a	$1$	$17$	$17$	$184.932$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		296.8.a.d	$1$	$1$	\(0\)	\(-111\)	\(-112\)	\(-1949\)		$+$	$-$	$-$	$2^{38}$	$\mathrm{SU}(2)$	\(q+(-7+\beta _{1})q^{3}+(-7-\beta _{4})q^{5}+(-115+\cdots)q^{7}+\cdots\)
592.8.a.l	$592$	$8$	592.a	1.a	$1$	$17$	$17$	$184.932$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		296.8.a.c	$1$	$0$	\(0\)	\(17\)	\(362\)	\(-97\)		$+$	$+$	$+$	$2^{39}\cdot 7$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(21-\beta _{3})q^{5}+(-6+3\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(592))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(592)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 2}\)