Space of modular forms of level 1968, weight 2, and trivial character

• Label: 1968.2.a
• Level: $1968$
• Weight: $2$
• Character orbit: 1968.a
• Rep. character: $\chi_{1968}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $24$
• Sturm bound: $672$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(672\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1968))\).

	Total	New	Old
Modular forms	348	40	308
Cusp forms	325	40	285
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.

\(2\)	\(3\)	\(41\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(38\)	\(4\)	\(34\)		\(36\)	\(4\)	\(32\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(49\)	\(7\)	\(42\)		\(46\)	\(7\)	\(39\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(41\)	\(6\)	\(35\)		\(38\)	\(6\)	\(32\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(46\)	\(3\)	\(43\)		\(43\)	\(3\)	\(40\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(49\)	\(5\)	\(44\)		\(46\)	\(5\)	\(41\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(38\)	\(5\)	\(33\)		\(35\)	\(5\)	\(30\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(46\)	\(5\)	\(41\)		\(43\)	\(5\)	\(38\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(41\)	\(5\)	\(36\)		\(38\)	\(5\)	\(33\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(168\)	\(17\)	\(151\)		\(157\)	\(17\)	\(140\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(180\)	\(23\)	\(157\)		\(168\)	\(23\)	\(145\)		\(12\)	\(0\)	\(12\)

Trace form

40 * q - 2 * q^3 + 40 * q^9 + 8 * q^11 + 4 * q^15 - 8 * q^19 - 16 * q^23 + 48 * q^25 - 2 * q^27 - 16 * q^29 + 4 * q^31 + 8 * q^33 - 24 * q^35 - 16 * q^37 - 4 * q^39 + 4 * q^43 + 40 * q^49 + 8 * q^51 - 8 * q^55 - 8 * q^57 + 32 * q^59 - 8 * q^67 - 16 * q^69 + 16 * q^71 - 14 * q^75 + 8 * q^79 + 40 * q^81 - 8 * q^83 - 16 * q^85 + 16 * q^89 + 40 * q^95 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1968))\) 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	41
1968.2.a.a	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				123.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+2q^{7}+q^{9}+3q^{11}+\cdots\)
1968.2.a.b	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
1968.2.a.c	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
1968.2.a.d	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				492.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}+5q^{11}+\cdots\)
1968.2.a.e	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				984.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+3q^{11}-6q^{13}+\cdots\)
1968.2.a.f	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
1968.2.a.g	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{7}+q^{9}+6q^{11}+\cdots\)
1968.2.a.h	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
1968.2.a.i	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				984.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-q^{11}+4q^{13}+\cdots\)
1968.2.a.j	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				123.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+4q^{7}+q^{9}-5q^{11}+\cdots\)
1968.2.a.k	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				984.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
1968.2.a.l	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				492.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
1968.2.a.m	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.e	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}-2q^{11}-7q^{13}+\cdots\)
1968.2.a.n	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				984.2.a.c	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-4q^{7}+q^{9}-5q^{11}+\cdots\)
1968.2.a.o	$1968$	$2$	1968.a	1.a	$1$	$1$	$1$	$15.715$	\(\Q\)	None	✓				246.2.a.b	$1$	$0$	\(0\)	\(1\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
1968.2.a.p	$1968$	$2$	1968.a	1.a	$1$	$2$	$2$	$15.715$	\(\Q(\sqrt{2}) \)	None	✓				984.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(-4\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+(2+\beta )q^{7}+q^{9}+\cdots\)
1968.2.a.q	$1968$	$2$	1968.a	1.a	$1$	$2$	$2$	$15.715$	\(\Q(\sqrt{3}) \)	None	✓		✓		492.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+(-1-\beta )q^{7}+q^{9}+\cdots\)
1968.2.a.r	$1968$	$2$	1968.a	1.a	$1$	$2$	$2$	$15.715$	\(\Q(\sqrt{2}) \)	None	✓		✓		123.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+(2+\beta )q^{7}+q^{9}+\cdots\)
1968.2.a.s	$1968$	$2$	1968.a	1.a	$1$	$2$	$2$	$15.715$	\(\Q(\sqrt{6}) \)	None	✓		✓		492.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
1968.2.a.t	$1968$	$2$	1968.a	1.a	$1$	$3$	$3$	$15.715$	3.3.961.1	None	✓		✓		984.2.a.h	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}-2q^{7}+q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
1968.2.a.u	$1968$	$2$	1968.a	1.a	$1$	$3$	$3$	$15.715$	3.3.892.1	None	✓				984.2.a.f	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{2}q^{5}-\beta _{2}q^{7}+q^{9}+(3-\beta _{1}+\cdots)q^{11}+\cdots\)
1968.2.a.v	$1968$	$2$	1968.a	1.a	$1$	$3$	$3$	$15.715$	3.3.1436.1	None	✓				984.2.a.g	$1$	$0$	\(0\)	\(3\)	\(3\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1968.2.a.w	$1968$	$2$	1968.a	1.a	$1$	$3$	$3$	$15.715$	3.3.316.1	None	✓		✓		123.2.a.d	$1$	$0$	\(0\)	\(3\)	\(4\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{1}-\beta _{2})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
1968.2.a.x	$1968$	$2$	1968.a	1.a	$1$	$5$	$5$	$15.715$	5.5.3858104.1	None	✓		✓		984.2.a.i	$1$	$0$	\(0\)	\(-5\)	\(1\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{4}q^{5}-\beta _{3}q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1968))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1968)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(123))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(246))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(328))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(492))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(656))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(984))\)\(^{\oplus 2}\)