Space of modular forms of level 1984, weight 2, and trivial character

• Label: 1984.2.a
• Level: $1984$
• Weight: $2$
• Character orbit: 1984.a
• Rep. character: $\chi_{1984}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $28$
• Sturm bound: $512$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(512\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1984))\).

	Total	New	Old
Modular forms	268	60	208
Cusp forms	245	60	185
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.

\(2\)	\(31\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(61\)	\(14\)	\(47\)		\(56\)	\(14\)	\(42\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(73\)	\(17\)	\(56\)		\(67\)	\(17\)	\(50\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(67\)	\(16\)	\(51\)		\(61\)	\(16\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(67\)	\(13\)	\(54\)		\(61\)	\(13\)	\(48\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(128\)	\(27\)	\(101\)		\(117\)	\(27\)	\(90\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(140\)	\(33\)	\(107\)		\(128\)	\(33\)	\(95\)		\(12\)	\(0\)	\(12\)

Trace form

\( 60 q + 60 q^{9} - 8 q^{17} + 52 q^{25} - 16 q^{29} - 16 q^{33} - 16 q^{37} - 24 q^{41} + 60 q^{49} - 16 q^{53} - 16 q^{57} + 8 q^{73} + 32 q^{77} + 12 q^{81} + 48 q^{85} - 24 q^{89} - 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1984))\) 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	31
1984.2.a.a	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+q^{9}+2q^{11}-4q^{13}+\cdots\)
1984.2.a.b	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+3q^{7}+q^{9}-2q^{11}+\cdots\)
1984.2.a.c	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				124.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+q^{7}+q^{9}-6q^{11}+\cdots\)
1984.2.a.d	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				124.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}-3q^{9}+6q^{11}+4q^{13}+\cdots\)
1984.2.a.e	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				124.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}-3q^{9}-6q^{11}+4q^{13}+\cdots\)
1984.2.a.f	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				62.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
1984.2.a.g	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				62.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
1984.2.a.h	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.c	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-3q^{7}-3q^{9}-2q^{11}+4q^{13}+\cdots\)
1984.2.a.i	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.c	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+3q^{7}-3q^{9}+2q^{11}+4q^{13}+\cdots\)
1984.2.a.j	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}+q^{9}-2q^{11}-4q^{13}+\cdots\)
1984.2.a.k	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				248.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-3q^{7}+q^{9}+2q^{11}+\cdots\)
1984.2.a.l	$1984$	$2$	1984.a	1.a	$1$	$1$	$1$	$15.842$	\(\Q\)	None	✓				124.2.a.a	$1$	$0$	\(0\)	\(2\)	\(3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}-q^{7}+q^{9}+6q^{11}+\cdots\)
1984.2.a.m	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{33}) \)	None	✓				248.2.a.d	$1$	$1$	\(0\)	\(-4\)	\(-3\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+(-1-\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
1984.2.a.n	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{5}) \)	None	✓				31.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(4\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+(2-\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
1984.2.a.o	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{3}) \)	None	✓				62.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{5}+2q^{7}+(1-2\beta )q^{9}+\cdots\)
1984.2.a.p	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{2}) \)	None	✓				992.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+(-1+\beta )q^{7}-q^{9}+(-2+\cdots)q^{11}+\cdots\)
1984.2.a.q	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{2}) \)	None	✓				992.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+(1+\beta )q^{7}-q^{9}+(2-2\beta )q^{11}+\cdots\)
1984.2.a.r	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{5}) \)	None	✓				31.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(-2+\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
1984.2.a.s	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{3}) \)	None	✓				62.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{5}-2q^{7}+(1+2\beta )q^{9}+\cdots\)
1984.2.a.t	$1984$	$2$	1984.a	1.a	$1$	$2$	$2$	$15.842$	\(\Q(\sqrt{33}) \)	None	✓				248.2.a.d	$1$	$1$	\(0\)	\(4\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+(-2+\beta )q^{5}-\beta q^{7}+q^{9}+\cdots\)
1984.2.a.u	$1984$	$2$	1984.a	1.a	$1$	$3$	$3$	$15.842$	3.3.316.1	None	✓				992.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}-\beta _{2}q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
1984.2.a.v	$1984$	$2$	1984.a	1.a	$1$	$3$	$3$	$15.842$	3.3.316.1	None	✓				248.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(1-\beta _{2})q^{5}+\cdots\)
1984.2.a.w	$1984$	$2$	1984.a	1.a	$1$	$3$	$3$	$15.842$	3.3.316.1	None	✓				992.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}-\beta _{2}q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
1984.2.a.x	$1984$	$2$	1984.a	1.a	$1$	$3$	$3$	$15.842$	3.3.316.1	None	✓				248.2.a.e	$1$	$0$	\(0\)	\(2\)	\(3\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}+(1-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1984.2.a.y	$1984$	$2$	1984.a	1.a	$1$	$4$	$4$	$15.842$	4.4.13968.1	None	✓		✓		992.2.a.e	$1$	$1$	\(0\)	\(-4\)	\(-2\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-\beta _{1}q^{5}+(1+\beta _{1}-\beta _{3})q^{7}+\cdots\)
1984.2.a.z	$1984$	$2$	1984.a	1.a	$1$	$4$	$4$	$15.842$	4.4.13968.1	None	✓		✓		992.2.a.e	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1984.2.a.ba	$1984$	$2$	1984.a	1.a	$1$	$6$	$6$	$15.842$	6.6.66862976.1	None	✓		✓		992.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{5}-\beta _{4}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
1984.2.a.bb	$1984$	$2$	1984.a	1.a	$1$	$6$	$6$	$15.842$	6.6.66862976.1	None	✓		✓		992.2.a.g	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{3}q^{5}+\beta _{4}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1984))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1984)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(496))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(992))\)\(^{\oplus 2}\)