Space of modular forms of level 12996, weight 2, and trivial character

• Label: 12996.2.a
• Level: $12996$
• Weight: $2$
• Character orbit: 12996.a
• Rep. character: $\chi_{12996}(1,\cdot)$
• Character field: $\Q$
• Dimension: $142$
• Newform subspaces: $48$
• Sturm bound: $4560$

Defining parameters

Level:	\( N \)	\(=\)	\( 12996 = 2^{2} \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	12996.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 48 \)
Sturm bound:			\(4560\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(12996))\).

	Total	New	Old
Modular forms	2400	142	2258
Cusp forms	2161	142	2019
Eisenstein series	239	0	239

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

\(2\)	\(3\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(300\)	\(0\)	\(300\)		\(261\)	\(0\)	\(261\)		\(39\)	\(0\)	\(39\)
\(+\)	\(+\)	\(-\)	\(-\)		\(310\)	\(0\)	\(310\)		\(270\)	\(0\)	\(270\)		\(40\)	\(0\)	\(40\)
\(+\)	\(-\)	\(+\)	\(-\)		\(310\)	\(0\)	\(310\)		\(270\)	\(0\)	\(270\)		\(40\)	\(0\)	\(40\)
\(+\)	\(-\)	\(-\)	\(+\)		\(300\)	\(0\)	\(300\)		\(260\)	\(0\)	\(260\)		\(40\)	\(0\)	\(40\)
\(-\)	\(+\)	\(+\)	\(-\)		\(300\)	\(33\)	\(267\)		\(280\)	\(33\)	\(247\)		\(20\)	\(0\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)		\(290\)	\(24\)	\(266\)		\(270\)	\(24\)	\(246\)		\(20\)	\(0\)	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)		\(290\)	\(40\)	\(250\)		\(270\)	\(40\)	\(230\)		\(20\)	\(0\)	\(20\)
\(-\)	\(-\)	\(-\)	\(-\)		\(300\)	\(45\)	\(255\)		\(280\)	\(45\)	\(235\)		\(20\)	\(0\)	\(20\)
Plus space			\(+\)		\(1180\)	\(64\)	\(1116\)		\(1061\)	\(64\)	\(997\)		\(119\)	\(0\)	\(119\)
Minus space			\(-\)		\(1220\)	\(78\)	\(1142\)		\(1100\)	\(78\)	\(1022\)		\(120\)	\(0\)	\(120\)

Trace form

142 * q + 2 * q^5 - 2 * q^13 - 4 * q^17 + 6 * q^23 + 142 * q^25 + 2 * q^29 - 16 * q^31 - 16 * q^35 + 4 * q^37 - 6 * q^41 - 12 * q^43 - 12 * q^47 + 146 * q^49 - 8 * q^55 - 2 * q^59 + 16 * q^61 + 32 * q^65 - 12 * q^67 - 30 * q^71 + 22 * q^73 - 6 * q^77 + 12 * q^79 - 10 * q^83 + 32 * q^85 + 40 * q^89 - 12 * q^91 + 6 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(12996))\) 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	19
12996.2.a.a	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}-4q^{11}-5q^{13}-4q^{23}+\cdots\)
12996.2.a.b	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-4\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}-4q^{11}+5q^{13}-4q^{23}+\cdots\)
12996.2.a.c	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{11}-2q^{13}-6q^{17}-2q^{23}+\cdots\)
12996.2.a.d	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$		\(q-4q^{7}-2q^{13}-5q^{25}+4q^{31}+10q^{37}+\cdots\)
12996.2.a.e	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$		\(q-q^{7}-7q^{13}-5q^{25}+11q^{31}-q^{37}+\cdots\)
12996.2.a.f	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$		\(q-q^{7}+7q^{13}-5q^{25}-11q^{31}+q^{37}+\cdots\)
12996.2.a.g	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$+$	$1$		\(q+5q^{7}-5q^{13}-5q^{25}+7q^{31}-11q^{37}+\cdots\)
12996.2.a.h	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$		\(q+5q^{7}+5q^{13}-5q^{25}-7q^{31}+11q^{37}+\cdots\)
12996.2.a.i	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}-5q^{11}+4q^{13}+3q^{17}+\cdots\)
12996.2.a.j	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}-q^{13}-3q^{17}-5q^{23}+\cdots\)
12996.2.a.k	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}+q^{13}-3q^{17}-5q^{23}+\cdots\)
12996.2.a.l	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-q^{11}-4q^{13}+3q^{17}+\cdots\)
12996.2.a.m	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-q^{11}+4q^{13}+3q^{17}+\cdots\)
12996.2.a.n	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}+5q^{11}+6q^{13}+5q^{17}+\cdots\)
12996.2.a.o	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}+4q^{11}-5q^{13}+4q^{23}+\cdots\)
12996.2.a.p	$12996$	$2$	12996.a	1.a	$1$	$1$	$1$	$103.774$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}+4q^{11}+5q^{13}+4q^{23}+\cdots\)
12996.2.a.q	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{33}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.r	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{7}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.s	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{7}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.t	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{57}) \)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$
12996.2.a.u	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{6}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.v	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{6}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.w	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{7}) \)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$+$
12996.2.a.x	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.y	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.z	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.ba	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bb	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bc	$12996$	$2$	12996.a	1.a	$1$	$2$	$2$	$103.774$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bd	$12996$	$2$	12996.a	1.a	$1$	$3$	$3$	$103.774$	\(\Q(\zeta_{18})^+\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.be	$12996$	$2$	12996.a	1.a	$1$	$3$	$3$	$103.774$	\(\Q(\zeta_{18})^+\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bf	$12996$	$2$	12996.a	1.a	$1$	$3$	$3$	$103.774$	\(\Q(\zeta_{18})^+\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$
12996.2.a.bg	$12996$	$2$	12996.a	1.a	$1$	$3$	$3$	$103.774$	\(\Q(\zeta_{18})^+\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$
12996.2.a.bh	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	4.4.7600.1	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$
12996.2.a.bi	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	4.4.7600.1	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$
12996.2.a.bj	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	\(\Q(\sqrt{3}, \sqrt{19})\)	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$
12996.2.a.bk	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	\(\Q(\zeta_{20})^+\)	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-6\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.bl	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	\(\Q(\zeta_{20})^+\)	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-6\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.bm	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	4.4.2225.1	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bn	$12996$	$2$	12996.a	1.a	$1$	$4$	$4$	$103.774$	4.4.2225.1	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bo	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.73227321.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(9\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.bp	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.73227321.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(9\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bq	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.53327808.1	not computed	✓	?	✓			$-1$		\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$
12996.2.a.br	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.53327808.1	not computed	✓	?				$-1$		\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$
12996.2.a.bs	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.20319417.1	None	✓	?				$-1$		\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
12996.2.a.bt	$12996$	$2$	12996.a	1.a	$1$	$6$	$6$	$103.774$	6.6.20319417.1	None	✓	?				$-1$		\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
12996.2.a.bu	$12996$	$2$	12996.a	1.a	$1$	$8$	$8$	$103.774$	8.8.\(\cdots\).1	not computed	✓	?	✓			$-1$		\(0\)	\(0\)	\(-14\)	\(8\)		$-$	$-$	$+$	$+$
12996.2.a.bv	$12996$	$2$	12996.a	1.a	$1$	$16$	$16$	$103.774$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	not computed	✓	?	✓			$-1$		\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(12996))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(12996)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(722))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1083))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1444))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4332))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(6498))\)\(^{\oplus 2}\)