Space of modular forms of level 2112, weight 2, and trivial character

• Label: 2112.2.a
• Level: $2112$
• Weight: $2$
• Character orbit: 2112.a
• Rep. character: $\chi_{2112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $34$
• Sturm bound: $768$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(768\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2112))\).

	Total	New	Old
Modular forms	408	40	368
Cusp forms	361	40	321
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.

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(46\)	\(4\)	\(42\)		\(41\)	\(4\)	\(37\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(56\)	\(6\)	\(50\)		\(50\)	\(6\)	\(44\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(50\)	\(5\)	\(45\)		\(44\)	\(5\)	\(39\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(52\)	\(3\)	\(49\)		\(46\)	\(3\)	\(43\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(56\)	\(6\)	\(50\)		\(50\)	\(6\)	\(44\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(46\)	\(4\)	\(42\)		\(40\)	\(4\)	\(36\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(52\)	\(5\)	\(47\)		\(46\)	\(5\)	\(41\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(50\)	\(7\)	\(43\)		\(44\)	\(7\)	\(37\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(196\)	\(16\)	\(180\)		\(173\)	\(16\)	\(157\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(212\)	\(24\)	\(188\)		\(188\)	\(24\)	\(164\)		\(24\)	\(0\)	\(24\)

Trace form

\( 40 q + 40 q^{9} - 16 q^{13} + 16 q^{17} - 16 q^{21} + 56 q^{25} - 16 q^{37} + 16 q^{41} + 40 q^{49} + 16 q^{61} + 32 q^{69} - 16 q^{73} + 40 q^{81} + 32 q^{85} - 16 q^{89} - 16 q^{93} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2112))\) 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
2112.2.a.a	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}-2q^{7}+q^{9}+q^{11}+4q^{15}+\cdots\)
2112.2.a.b	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				132.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-2q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
2112.2.a.c	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				132.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-2q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
2112.2.a.d	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+q^{11}-2q^{13}+\cdots\)
2112.2.a.e	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}-q^{11}+6q^{13}+\cdots\)
2112.2.a.f	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
2112.2.a.g	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2112.2.a.h	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-q^{11}-2q^{17}+\cdots\)
2112.2.a.i	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
2112.2.a.j	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				33.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-4q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
2112.2.a.k	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
2112.2.a.l	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+q^{11}+6q^{13}+\cdots\)
2112.2.a.m	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+4q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
2112.2.a.n	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2112.2.a.o	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
2112.2.a.p	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+2q^{7}+q^{9}-q^{11}-4q^{15}+\cdots\)
2112.2.a.q	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.e	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
2112.2.a.r	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}+q^{9}+q^{11}+6q^{13}+\cdots\)
2112.2.a.s	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-q^{11}-2q^{13}+\cdots\)
2112.2.a.t	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				132.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+2q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
2112.2.a.u	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				132.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
2112.2.a.v	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
2112.2.a.w	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.d	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
2112.2.a.x	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+q^{11}-2q^{17}+\cdots\)
2112.2.a.y	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				264.2.a.b	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-4q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
2112.2.a.z	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}-q^{11}+6q^{13}+\cdots\)
2112.2.a.ba	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.c	$1$	$0$	\(0\)	\(1\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
2112.2.a.bb	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				33.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
2112.2.a.bc	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				1056.2.a.a	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2112.2.a.bd	$2112$	$2$	2112.a	1.a	$1$	$1$	$1$	$16.864$	\(\Q\)	None	✓				66.2.a.c	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+2q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
2112.2.a.be	$2112$	$2$	2112.a	1.a	$1$	$2$	$2$	$16.864$	\(\Q(\sqrt{5}) \)	None	✓		✓		1056.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
2112.2.a.bf	$2112$	$2$	2112.a	1.a	$1$	$2$	$2$	$16.864$	\(\Q(\sqrt{5}) \)	None	✓		✓		1056.2.a.k	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
2112.2.a.bg	$2112$	$2$	2112.a	1.a	$1$	$3$	$3$	$16.864$	3.3.229.1	None	✓		✓		1056.2.a.m	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}-q^{11}+\cdots\)
2112.2.a.bh	$2112$	$2$	2112.a	1.a	$1$	$3$	$3$	$16.864$	3.3.229.1	None	✓		✓		1056.2.a.m	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2112))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2112)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(704))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1056))\)\(^{\oplus 2}\)