Space of modular forms of level 3168, weight 2, and trivial character

• Label: 3168.2.a
• Level: $3168$
• Weight: $2$
• Character orbit: 3168.a
• Rep. character: $\chi_{3168}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $36$
• Sturm bound: $1152$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1152\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\), \(17\), \(19\), \(47\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3168))\).

	Total	New	Old
Modular forms	608	50	558
Cusp forms	545	50	495
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(72\)	\(4\)	\(68\)		\(65\)	\(4\)	\(61\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(76\)	\(6\)	\(70\)		\(68\)	\(6\)	\(62\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(78\)	\(8\)	\(70\)		\(70\)	\(8\)	\(62\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(74\)	\(7\)	\(67\)		\(66\)	\(7\)	\(59\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(80\)	\(6\)	\(74\)		\(72\)	\(6\)	\(66\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)		\(76\)	\(4\)	\(72\)		\(68\)	\(4\)	\(64\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)		\(74\)	\(7\)	\(67\)		\(66\)	\(7\)	\(59\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)		\(78\)	\(8\)	\(70\)		\(70\)	\(8\)	\(62\)		\(8\)	\(0\)	\(8\)
Plus space			\(+\)		\(296\)	\(22\)	\(274\)		\(265\)	\(22\)	\(243\)		\(31\)	\(0\)	\(31\)
Minus space			\(-\)		\(312\)	\(28\)	\(284\)		\(280\)	\(28\)	\(252\)		\(32\)	\(0\)	\(32\)

Trace form

\( 50 q + 4 q^{5} + 20 q^{13} - 12 q^{17} + 38 q^{25} - 12 q^{29} + 20 q^{37} - 12 q^{41} + 66 q^{49} + 28 q^{53} + 20 q^{61} - 56 q^{65} + 4 q^{73} + 8 q^{85} + 44 q^{89} + 12 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3168))\) 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	11
3168.2.a.a	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-q^{11}-2q^{13}-2q^{17}+\cdots\)
3168.2.a.b	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-q^{11}+4q^{13}+6q^{17}+\cdots\)
3168.2.a.c	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{11}+2q^{17}+6q^{19}-6q^{23}+\cdots\)
3168.2.a.d	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{11}+2q^{17}-6q^{19}+6q^{23}+\cdots\)
3168.2.a.e	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}+q^{11}-2q^{13}-2q^{17}+\cdots\)
3168.2.a.f	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}+q^{11}+4q^{13}+6q^{17}+\cdots\)
3168.2.a.g	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}+q^{11}-2q^{13}+2q^{19}+\cdots\)
3168.2.a.h	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{11}-6q^{13}+4q^{17}-6q^{19}+\cdots\)
3168.2.a.i	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{11}-6q^{13}+4q^{17}+6q^{19}+\cdots\)
3168.2.a.j	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-q^{11}-2q^{13}-2q^{19}+\cdots\)
3168.2.a.k	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-2q^{13}-2q^{17}+6q^{19}+\cdots\)
3168.2.a.l	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}-2q^{13}+2q^{17}+6q^{19}+\cdots\)
3168.2.a.m	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}+4q^{13}+2q^{17}+2q^{23}+\cdots\)
3168.2.a.n	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}-2q^{13}+2q^{17}-6q^{19}+\cdots\)
3168.2.a.o	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}+4q^{13}+2q^{17}-2q^{23}+\cdots\)
3168.2.a.p	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{11}-2q^{13}-2q^{17}-6q^{19}+\cdots\)
3168.2.a.q	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}+q^{11}+4q^{13}-6q^{17}+\cdots\)
3168.2.a.r	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-q^{11}-2q^{13}-2q^{19}+\cdots\)
3168.2.a.s	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{11}-6q^{13}-2q^{17}+4q^{19}+\cdots\)
3168.2.a.t	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{11}-2q^{17}-6q^{19}-6q^{23}+\cdots\)
3168.2.a.u	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{11}-6q^{13}-2q^{17}-4q^{19}+\cdots\)
3168.2.a.v	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{11}-2q^{17}+6q^{19}+6q^{23}+\cdots\)
3168.2.a.w	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}+q^{11}-2q^{13}+2q^{19}+\cdots\)
3168.2.a.x	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓	✓			3168.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}-q^{11}+4q^{13}-6q^{17}+\cdots\)
3168.2.a.y	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-q^{11}-2q^{13}+8q^{17}+\cdots\)
3168.2.a.z	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}+q^{11}-2q^{13}+8q^{17}+\cdots\)
3168.2.a.ba	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-2q^{7}+q^{11}+4q^{13}-2q^{17}+\cdots\)
3168.2.a.bb	$3168$	$2$	3168.a	1.a	$1$	$1$	$1$	$25.297$	\(\Q\)	None	✓				1056.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+2q^{7}-q^{11}+4q^{13}-2q^{17}+\cdots\)
3168.2.a.bc	$3168$	$2$	3168.a	1.a	$1$	$2$	$2$	$25.297$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}-q^{11}+2q^{13}+(-4+\cdots)q^{17}+\cdots\)
3168.2.a.bd	$3168$	$2$	3168.a	1.a	$1$	$2$	$2$	$25.297$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+q^{11}+2q^{13}+(-4+\cdots)q^{17}+\cdots\)
3168.2.a.be	$3168$	$2$	3168.a	1.a	$1$	$2$	$2$	$25.297$	\(\Q(\sqrt{5}) \)	None	✓				1056.2.a.k	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(-1+\beta )q^{7}-q^{11}+\cdots\)
3168.2.a.bf	$3168$	$2$	3168.a	1.a	$1$	$2$	$2$	$25.297$	\(\Q(\sqrt{5}) \)	None	✓		✓		1056.2.a.k	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(1-\beta )q^{7}+q^{11}+(3+\cdots)q^{13}+\cdots\)
3168.2.a.bg	$3168$	$2$	3168.a	1.a	$1$	$3$	$3$	$25.297$	3.3.229.1	None	✓		✓		1056.2.a.m	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+q^{11}-\beta _{1}q^{13}+\cdots\)
3168.2.a.bh	$3168$	$2$	3168.a	1.a	$1$	$3$	$3$	$25.297$	3.3.229.1	None	✓		✓		1056.2.a.m	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(1+\beta _{2})q^{7}-q^{11}-\beta _{1}q^{13}+\cdots\)
3168.2.a.bi	$3168$	$2$	3168.a	1.a	$1$	$4$	$4$	$25.297$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		3168.2.a.bi	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}-q^{11}+(2-\beta _{2})q^{13}+\cdots\)
3168.2.a.bj	$3168$	$2$	3168.a	1.a	$1$	$4$	$4$	$25.297$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		3168.2.a.bi	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}+q^{11}+(2+\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3168))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3168)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1056))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1584))\)\(^{\oplus 2}\)