Space of modular forms of level 2160, weight 2, and trivial character

• Label: 2160.2.a
• Level: $2160$
• Weight: $2$
• Character orbit: 2160.a
• Rep. character: $\chi_{2160}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $28$
• Sturm bound: $864$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(864\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2160))\).

	Total	New	Old
Modular forms	468	32	436
Cusp forms	397	32	365
Eisenstein series	71	0	71

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(18\)

Trace form

\( 32 q + 4 q^{7} + 32 q^{25} - 36 q^{31} - 32 q^{43} + 40 q^{49} + 8 q^{61} + 68 q^{67} + 8 q^{73} + 64 q^{79} + 8 q^{85} - 52 q^{91} + 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2160))\) 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	5
2160.2.a.a	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				270.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{11}-q^{13}+3q^{17}+\cdots\)
2160.2.a.b	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				270.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{11}+5q^{13}-3q^{17}+\cdots\)
2160.2.a.c	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+2q^{13}+3q^{17}-5q^{19}+\cdots\)
2160.2.a.d	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.f	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+q^{11}+q^{13}+q^{17}+\cdots\)
2160.2.a.e	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+4q^{11}-2q^{13}-5q^{17}+\cdots\)
2160.2.a.f	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{11}-3q^{17}+q^{19}+3q^{23}+\cdots\)
2160.2.a.g	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-2q^{11}-5q^{13}+4q^{17}+\cdots\)
2160.2.a.h	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}+6q^{11}-q^{13}+q^{19}+\cdots\)
2160.2.a.i	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-6q^{13}+7q^{17}-7q^{19}+\cdots\)
2160.2.a.j	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				135.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+2q^{11}-5q^{13}-8q^{17}+\cdots\)
2160.2.a.k	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-6q^{11}-4q^{13}-3q^{17}+\cdots\)
2160.2.a.l	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-2q^{11}+4q^{13}+q^{17}+\cdots\)
2160.2.a.m	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-4q^{11}-2q^{13}+5q^{17}+\cdots\)
2160.2.a.n	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.f	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-q^{11}+q^{13}-q^{17}+\cdots\)
2160.2.a.o	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+2q^{13}-3q^{17}-5q^{19}+\cdots\)
2160.2.a.p	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				270.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+3q^{11}-q^{13}-3q^{17}+\cdots\)
2160.2.a.q	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				270.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+3q^{11}+5q^{13}+3q^{17}+\cdots\)
2160.2.a.r	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{11}+3q^{17}+q^{19}-3q^{23}+\cdots\)
2160.2.a.s	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-6q^{11}-q^{13}+q^{19}+\cdots\)
2160.2.a.t	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+2q^{11}-5q^{13}-4q^{17}+\cdots\)
2160.2.a.u	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-6q^{13}-7q^{17}-7q^{19}+\cdots\)
2160.2.a.v	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				135.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-2q^{11}-5q^{13}+8q^{17}+\cdots\)
2160.2.a.w	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				1080.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}+2q^{11}+4q^{13}-q^{17}+\cdots\)
2160.2.a.x	$2160$	$2$	2160.a	1.a	$1$	$1$	$1$	$17.248$	\(\Q\)	None	✓				540.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}+6q^{11}-4q^{13}+3q^{17}+\cdots\)
2160.2.a.y	$2160$	$2$	2160.a	1.a	$1$	$2$	$2$	$17.248$	\(\Q(\sqrt{13}) \)	None	✓		✓		135.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1-\beta )q^{7}+(1-\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)
2160.2.a.z	$2160$	$2$	2160.a	1.a	$1$	$2$	$2$	$17.248$	\(\Q(\sqrt{73}) \)	None	✓		✓		1080.2.a.m	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-\beta q^{7}+(-1+\beta )q^{11}+3q^{13}+\cdots\)
2160.2.a.ba	$2160$	$2$	2160.a	1.a	$1$	$2$	$2$	$17.248$	\(\Q(\sqrt{13}) \)	None	✓		✓		135.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1-\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
2160.2.a.bb	$2160$	$2$	2160.a	1.a	$1$	$2$	$2$	$17.248$	\(\Q(\sqrt{73}) \)	None	✓		✓		1080.2.a.m	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-\beta q^{7}+(1-\beta )q^{11}+3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2160))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2160)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1080))\)\(^{\oplus 2}\)