Space of modular forms of level 5994, weight 2, and trivial character

• Label: 5994.2.a
• Level: $5994$
• Weight: $2$
• Character orbit: 5994.a
• Rep. character: $\chi_{5994}(1,\cdot)$
• Character field: $\Q$
• Dimension: $144$
• Newform subspaces: $30$
• Sturm bound: $2052$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5994 = 2 \cdot 3^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5994.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(2052\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5994))\).

	Total	New	Old
Modular forms	1050	144	906
Cusp forms	1003	144	859
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.

\(2\)	\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(124\)	\(17\)	\(107\)		\(119\)	\(17\)	\(102\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(137\)	\(20\)	\(117\)		\(131\)	\(20\)	\(111\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(136\)	\(19\)	\(117\)		\(130\)	\(19\)	\(111\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(128\)	\(16\)	\(112\)		\(122\)	\(16\)	\(106\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(134\)	\(21\)	\(113\)		\(128\)	\(21\)	\(107\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(127\)	\(14\)	\(113\)		\(121\)	\(14\)	\(107\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(131\)	\(15\)	\(116\)		\(125\)	\(15\)	\(110\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(133\)	\(22\)	\(111\)		\(127\)	\(22\)	\(105\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(510\)	\(62\)	\(448\)		\(487\)	\(62\)	\(425\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(540\)	\(82\)	\(458\)		\(516\)	\(82\)	\(434\)		\(24\)	\(0\)	\(24\)

Trace form

144 * q + 144 * q^4 + 144 * q^16 + 12 * q^19 + 12 * q^22 + 144 * q^25 + 12 * q^34 - 36 * q^43 + 24 * q^46 + 120 * q^49 - 24 * q^55 + 24 * q^58 + 24 * q^61 + 144 * q^64 + 12 * q^67 + 48 * q^70 - 36 * q^73 + 12 * q^76 + 72 * q^79 + 12 * q^82 + 12 * q^88 + 48 * q^91 + 24 * q^94 - 36 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5994))\) 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	37
5994.2.a.a	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓				666.2.e.b	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}-q^{8}+4q^{10}-q^{11}+\cdots\)
5994.2.a.b	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓				666.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+2q^{7}-q^{8}+2q^{10}+\cdots\)
5994.2.a.c	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓	✓			5994.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+2q^{7}-q^{8}+2q^{10}+\cdots\)
5994.2.a.d	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓	✓			5994.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}+2q^{11}+\cdots\)
5994.2.a.e	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓	✓			5994.2.a.d	$1$	$1$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}-2q^{11}+\cdots\)
5994.2.a.f	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓	✓			5994.2.a.c	$1$	$0$	\(1\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}+2q^{10}+\cdots\)
5994.2.a.g	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓				666.2.e.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}+2q^{10}+\cdots\)
5994.2.a.h	$5994$	$2$	5994.a	1.a	$1$	$1$	$1$	$47.862$	\(\Q\)	None	✓				666.2.e.b	$1$	$0$	\(1\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}+q^{8}+4q^{10}+q^{11}+\cdots\)
5994.2.a.i	$5994$	$2$	5994.a	1.a	$1$	$2$	$2$	$47.862$	\(\Q(\sqrt{3}) \)	None	✓	✓			5994.2.a.i	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}+2q^{7}-q^{8}-\beta q^{10}+\cdots\)
5994.2.a.j	$5994$	$2$	5994.a	1.a	$1$	$2$	$2$	$47.862$	\(\Q(\sqrt{3}) \)	None	✓	✓			5994.2.a.i	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+2q^{7}+q^{8}+\beta q^{10}+\cdots\)
5994.2.a.k	$5994$	$2$	5994.a	1.a	$1$	$3$	$3$	$47.862$	3.3.321.1	None	✓	✓			5994.2.a.k	$1$	$0$	\(-3\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5994.2.a.l	$5994$	$2$	5994.a	1.a	$1$	$3$	$3$	$47.862$	3.3.321.1	None	✓	✓			5994.2.a.k	$1$	$1$	\(3\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{1})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.m	$5994$	$2$	5994.a	1.a	$1$	$4$	$4$	$47.862$	4.4.43569.1	None	✓	✓			5994.2.a.m	$1$	$1$	\(-4\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{3}q^{5}-\beta _{2}q^{7}-q^{8}+\cdots\)
5994.2.a.n	$5994$	$2$	5994.a	1.a	$1$	$4$	$4$	$47.862$	4.4.22545.1	None	✓	✓			5994.2.a.n	$1$	$0$	\(-4\)	\(0\)	\(5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{1})q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
5994.2.a.o	$5994$	$2$	5994.a	1.a	$1$	$4$	$4$	$47.862$	4.4.22545.1	None	✓	✓			5994.2.a.n	$1$	$1$	\(4\)	\(0\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{1})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.p	$5994$	$2$	5994.a	1.a	$1$	$4$	$4$	$47.862$	4.4.43569.1	None	✓	✓			5994.2.a.m	$1$	$0$	\(4\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{3}q^{5}-\beta _{2}q^{7}+q^{8}+\cdots\)
5994.2.a.q	$5994$	$2$	5994.a	1.a	$1$	$5$	$5$	$47.862$	5.5.9204129.1	None	✓	✓			5994.2.a.q	$1$	$1$	\(-5\)	\(0\)	\(-5\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta _{2})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.r	$5994$	$2$	5994.a	1.a	$1$	$5$	$5$	$47.862$	5.5.9204129.1	None	✓	✓			5994.2.a.q	$1$	$0$	\(5\)	\(0\)	\(5\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1-\beta _{2})q^{5}+(1-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
5994.2.a.s	$5994$	$2$	5994.a	1.a	$1$	$6$	$6$	$47.862$	6.6.17271549.1	None	✓				666.2.e.c	$1$	$1$	\(-6\)	\(0\)	\(5\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.t	$5994$	$2$	5994.a	1.a	$1$	$6$	$6$	$47.862$	6.6.17271549.1	None	✓		✓		666.2.e.c	$1$	$1$	\(6\)	\(0\)	\(-5\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.u	$5994$	$2$	5994.a	1.a	$1$	$7$	$7$	$47.862$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		666.2.e.d	$1$	$1$	\(-7\)	\(0\)	\(3\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{6}q^{5}+(\beta _{1}+\beta _{5})q^{7}+\cdots\)
5994.2.a.v	$5994$	$2$	5994.a	1.a	$1$	$7$	$7$	$47.862$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				666.2.e.d	$1$	$1$	\(7\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{6}q^{5}+(\beta _{1}+\beta _{5})q^{7}+\cdots\)
5994.2.a.w	$5994$	$2$	5994.a	1.a	$1$	$8$	$8$	$47.862$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		5994.2.a.w	$1$	$1$	\(-8\)	\(0\)	\(-6\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta _{1})q^{5}-\beta _{4}q^{7}+\cdots\)
5994.2.a.x	$5994$	$2$	5994.a	1.a	$1$	$8$	$8$	$47.862$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			5994.2.a.x	$1$	$0$	\(-8\)	\(0\)	\(6\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{6})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.y	$5994$	$2$	5994.a	1.a	$1$	$8$	$8$	$47.862$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		5994.2.a.x	$1$	$1$	\(8\)	\(0\)	\(-6\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{6})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5994.2.a.z	$5994$	$2$	5994.a	1.a	$1$	$8$	$8$	$47.862$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			5994.2.a.w	$1$	$0$	\(8\)	\(0\)	\(6\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1-\beta _{1})q^{5}-\beta _{4}q^{7}+q^{8}+\cdots\)
5994.2.a.ba	$5994$	$2$	5994.a	1.a	$1$	$10$	$10$	$47.862$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		666.2.e.e	$1$	$0$	\(-10\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{6}q^{5}-\beta _{9}q^{7}-q^{8}+\cdots\)
5994.2.a.bb	$5994$	$2$	5994.a	1.a	$1$	$10$	$10$	$47.862$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		666.2.e.e	$1$	$0$	\(10\)	\(0\)	\(1\)	\(1\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{6}q^{5}-\beta _{9}q^{7}+q^{8}+\cdots\)
5994.2.a.bc	$5994$	$2$	5994.a	1.a	$1$	$11$	$11$	$47.862$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		666.2.e.f	$1$	$0$	\(-11\)	\(0\)	\(-1\)	\(5\)		$+$	$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{8}q^{5}-\beta _{2}q^{7}-q^{8}+\cdots\)
5994.2.a.bd	$5994$	$2$	5994.a	1.a	$1$	$11$	$11$	$47.862$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		666.2.e.f	$1$	$0$	\(11\)	\(0\)	\(1\)	\(5\)		$-$	$-$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{8}q^{5}-\beta _{2}q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5994))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5994)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(333))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(666))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(999))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1998))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2997))\)\(^{\oplus 2}\)