Space of modular forms of level 2368, weight 2, and trivial character

• Label: 2368.2.a
• Level: $2368$
• Weight: $2$
• Character orbit: 2368.a
• Rep. character: $\chi_{2368}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $36$
• Sturm bound: $608$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(608\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2368))\).

	Total	New	Old
Modular forms	316	72	244
Cusp forms	293	72	221
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(+\)		\(76\)	\(17\)	\(59\)		\(71\)	\(17\)	\(54\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(80\)	\(19\)	\(61\)		\(74\)	\(19\)	\(55\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(82\)	\(19\)	\(63\)		\(76\)	\(19\)	\(57\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(78\)	\(17\)	\(61\)		\(72\)	\(17\)	\(55\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(154\)	\(34\)	\(120\)		\(143\)	\(34\)	\(109\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(162\)	\(38\)	\(124\)		\(150\)	\(38\)	\(112\)		\(12\)	\(0\)	\(12\)

Trace form

\( 72 q + 72 q^{9} + 72 q^{25} - 16 q^{33} - 16 q^{41} + 72 q^{49} - 16 q^{57} - 32 q^{61} - 32 q^{69} + 48 q^{77} + 24 q^{81} + 16 q^{85} - 32 q^{89} - 48 q^{93} - 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2368))\) 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
2368.2.a.a	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}-3q^{7}+6q^{9}+3q^{11}+\cdots\)
2368.2.a.b	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				37.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots\)
2368.2.a.c	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}+q^{7}+6q^{9}+3q^{11}+\cdots\)
2368.2.a.d	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				37.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}-3q^{11}+4q^{13}+\cdots\)
2368.2.a.e	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
2368.2.a.f	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				296.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-3q^{11}+2q^{17}+\cdots\)
2368.2.a.g	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				296.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
2368.2.a.h	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				148.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+3q^{7}-2q^{9}+5q^{11}+\cdots\)
2368.2.a.i	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-3q^{9}-2q^{13}+2q^{17}+\cdots\)
2368.2.a.j	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}-3q^{9}-2q^{13}+2q^{17}+\cdots\)
2368.2.a.k	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				296.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}-2q^{9}+3q^{11}+2q^{17}+\cdots\)
2368.2.a.l	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
2368.2.a.m	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				37.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
2368.2.a.n	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				296.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
2368.2.a.o	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				148.2.a.a	$1$	$1$	\(0\)	\(1\)	\(4\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-3q^{7}-2q^{9}-5q^{11}+\cdots\)
2368.2.a.p	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.b	$1$	$1$	\(0\)	\(3\)	\(-4\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{5}+3q^{7}+6q^{9}-3q^{11}+\cdots\)
2368.2.a.q	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				37.2.a.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots\)
2368.2.a.r	$2368$	$2$	2368.a	1.a	$1$	$1$	$1$	$18.909$	\(\Q\)	None	✓				1184.2.a.a	$1$	$0$	\(0\)	\(3\)	\(4\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{5}-q^{7}+6q^{9}-3q^{11}+\cdots\)
2368.2.a.s	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{13}) \)	None	✓				74.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+\beta q^{5}+(2-2\beta )q^{7}+\cdots\)
2368.2.a.t	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{17}) \)	None	✓				148.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2q^{5}-\beta q^{7}+(1+\beta )q^{9}+\beta q^{11}+\cdots\)
2368.2.a.u	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{5}) \)	None	✓				74.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-3\beta )q^{5}+2\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
2368.2.a.v	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{13}) \)	None	✓				1184.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}+2\beta q^{7}+\beta q^{9}+\cdots\)
2368.2.a.w	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{5}) \)	None	✓				1184.2.a.j	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{5}-\beta q^{7}+2q^{9}+\beta q^{11}+\cdots\)
2368.2.a.x	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{17}) \)	None	✓				148.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{5}+\beta q^{7}+(1+\beta )q^{9}-\beta q^{11}+\cdots\)
2368.2.a.y	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{5}) \)	None	✓				74.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-3\beta )q^{5}-2\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
2368.2.a.z	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{13}) \)	None	✓				1184.2.a.i	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-2\beta q^{7}+\beta q^{9}+\cdots\)
2368.2.a.ba	$2368$	$2$	2368.a	1.a	$1$	$2$	$2$	$18.909$	\(\Q(\sqrt{13}) \)	None	✓				74.2.a.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(-2+2\beta )q^{7}+\cdots\)
2368.2.a.bb	$2368$	$2$	2368.a	1.a	$1$	$3$	$3$	$18.909$	3.3.229.1	None	✓				296.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-\beta _{2}q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
2368.2.a.bc	$2368$	$2$	2368.a	1.a	$1$	$3$	$3$	$18.909$	3.3.621.1	None	✓				1184.2.a.l	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
2368.2.a.bd	$2368$	$2$	2368.a	1.a	$1$	$3$	$3$	$18.909$	3.3.621.1	None	✓				1184.2.a.l	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+\cdots\)
2368.2.a.be	$2368$	$2$	2368.a	1.a	$1$	$3$	$3$	$18.909$	3.3.229.1	None	✓		✓		296.2.a.c	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-\beta _{2}q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2368.2.a.bf	$2368$	$2$	2368.a	1.a	$1$	$4$	$4$	$18.909$	4.4.2225.1	None	✓				1184.2.a.n	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{3})q^{5}+\cdots\)
2368.2.a.bg	$2368$	$2$	2368.a	1.a	$1$	$4$	$4$	$18.909$	4.4.48389.1	None	✓				296.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(-5\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{3})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
2368.2.a.bh	$2368$	$2$	2368.a	1.a	$1$	$4$	$4$	$18.909$	4.4.48389.1	None	✓				296.2.a.d	$1$	$0$	\(0\)	\(2\)	\(-5\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{3})q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
2368.2.a.bi	$2368$	$2$	2368.a	1.a	$1$	$4$	$4$	$18.909$	4.4.2225.1	None	✓				1184.2.a.n	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{3})q^{5}+\cdots\)
2368.2.a.bj	$2368$	$2$	2368.a	1.a	$1$	$8$	$8$	$18.909$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		1184.2.a.p	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{6}q^{5}-\beta _{4}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2368))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2368)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(592))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1184))\)\(^{\oplus 2}\)