Space of modular forms of level 3872, weight 2, and trivial character

• Label: 3872.2.a
• Level: $3872$
• Weight: $2$
• Character orbit: 3872.a
• Rep. character: $\chi_{3872}(1,\cdot)$
• Character field: $\Q$
• Dimension: $109$
• Newform subspaces: $43$
• Sturm bound: $1056$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 43 \)
Sturm bound:			\(1056\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3872))\).

	Total	New	Old
Modular forms	576	109	467
Cusp forms	481	109	372
Eisenstein series	95	0	95

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

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(25\)
\(+\)	\(-\)	\(-\)	\(30\)
\(-\)	\(+\)	\(-\)	\(29\)
\(-\)	\(-\)	\(+\)	\(25\)
Plus space		\(+\)	\(50\)
Minus space		\(-\)	\(59\)

Trace form

\( 109 q - 2 q^{5} + 113 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3872))\) 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	11
3872.2.a.a	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+6q^{9}+6q^{13}-3q^{15}+\cdots\)
3872.2.a.b	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓	✓			3872.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-2q^{7}+q^{9}+q^{13}-2q^{15}+\cdots\)
3872.2.a.c	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓	✓			3872.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+2q^{7}+q^{9}-q^{13}-2q^{15}+\cdots\)
3872.2.a.d	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-4q^{7}-2q^{9}+2q^{13}+\cdots\)
3872.2.a.e	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+4q^{7}-2q^{9}+2q^{13}+\cdots\)
3872.2.a.f	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{5}-3q^{9}-6q^{13}-2q^{17}-q^{25}+\cdots\)
3872.2.a.g	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.g	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}-3q^{9}-4q^{13}+8q^{17}-q^{25}+\cdots\)
3872.2.a.h	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.g	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}-3q^{9}+4q^{13}-8q^{17}-q^{25}+\cdots\)
3872.2.a.i	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+4q^{7}-2q^{9}+2q^{13}+\cdots\)
3872.2.a.j	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.c	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}-2q^{9}+2q^{13}+\cdots\)
3872.2.a.k	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓	✓			3872.2.a.b	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-2q^{7}+q^{9}-q^{13}+2q^{15}+\cdots\)
3872.2.a.l	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓	✓			3872.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+2q^{7}+q^{9}+q^{13}+2q^{15}+\cdots\)
3872.2.a.m	$3872$	$2$	3872.a	1.a	$1$	$1$	$1$	$30.918$	\(\Q\)	None	✓				352.2.a.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+6q^{9}+6q^{13}+3q^{15}+\cdots\)
3872.2.a.n	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓				352.2.m.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{5}+(2-4\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
3872.2.a.o	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓				352.2.m.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{5}+(-2+4\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
3872.2.a.p	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}-2q^{13}+\cdots\)
3872.2.a.q	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓	✓			3872.2.a.q	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-3q^{5}-2\beta q^{7}+2q^{9}-6q^{13}+\cdots\)
3872.2.a.r	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓	✓			3872.2.a.q	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-3q^{5}+2\beta q^{7}+2q^{9}+6q^{13}+\cdots\)
3872.2.a.s	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.s	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+2\beta )q^{5}-3q^{9}+(-2-3\beta )q^{13}+\cdots\)
3872.2.a.t	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.s	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+2\beta )q^{5}-3q^{9}+(2+3\beta )q^{13}+\cdots\)
3872.2.a.u	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓	✓			3872.2.a.u	$2$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}-2q^{7}+2q^{9}+2\beta q^{13}+\cdots\)
3872.2.a.v	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.v	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+(1-\beta )q^{5}-3q^{9}+(-3+\beta )q^{13}+\cdots\)
3872.2.a.w	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			3872.2.a.v	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{5}-3q^{9}+(3+\beta )q^{13}+(1+\cdots)q^{17}+\cdots\)
3872.2.a.x	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓	✓			3872.2.a.u	$2$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+2q^{7}+2q^{9}-2\beta q^{13}+\cdots\)
3872.2.a.y	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓				352.2.m.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+(-2+4\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
3872.2.a.z	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{5}) \)	None	✓				352.2.m.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+(2-4\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
3872.2.a.ba	$3872$	$2$	3872.a	1.a	$1$	$2$	$2$	$30.918$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$1$	\(0\)	\(1\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}-2q^{13}+\cdots\)
3872.2.a.bb	$3872$	$2$	3872.a	1.a	$1$	$3$	$3$	$30.918$	3.3.404.1	None	✓	✓			3872.2.a.bb	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
3872.2.a.bc	$3872$	$2$	3872.a	1.a	$1$	$3$	$3$	$30.918$	3.3.404.1	None	✓	✓			3872.2.a.bb	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
3872.2.a.bd	$3872$	$2$	3872.a	1.a	$1$	$3$	$3$	$30.918$	3.3.404.1	None	✓	✓			3872.2.a.bb	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+(1-\beta _{1})q^{7}+\cdots\)
3872.2.a.be	$3872$	$2$	3872.a	1.a	$1$	$3$	$3$	$30.918$	3.3.404.1	None	✓	✓			3872.2.a.bb	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+(1-\beta _{1})q^{7}+\cdots\)
3872.2.a.bf	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.4400.1	None	✓				352.2.m.c	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}+\beta _{1}q^{7}+(1+\cdots)q^{9}+\cdots\)
3872.2.a.bg	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.4400.1	None	✓				352.2.m.c	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}-\beta _{1}q^{7}+(1+\cdots)q^{9}+\cdots\)
3872.2.a.bh	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.7488.1	None	✓	✓			3872.2.a.bh	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-\beta _{1}q^{5}-\beta _{3}q^{7}+(5+2\beta _{1}+\cdots)q^{9}+\cdots\)
3872.2.a.bi	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.7488.1	None	✓	✓			3872.2.a.bh	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+(5+2\beta _{1}+\cdots)q^{9}+\cdots\)
3872.2.a.bj	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.22000.1	None	✓				352.2.m.d	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+3\beta _{2})q^{5}-\beta _{1}q^{7}+(5+\cdots)q^{9}+\cdots\)
3872.2.a.bk	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	4.4.22000.1	None	✓				352.2.m.d	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+3\beta _{2})q^{5}+\beta _{1}q^{7}+(5+\cdots)q^{9}+\cdots\)
3872.2.a.bl	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	\(\Q(\zeta_{24})^+\)	None	✓	✓			3872.2.a.bl	$2$	$1$	\(0\)	\(0\)	\(4\)	\(-8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(1-\beta _{3})q^{5}+(-2+\beta _{3})q^{7}+\cdots\)
3872.2.a.bm	$3872$	$2$	3872.a	1.a	$1$	$4$	$4$	$30.918$	\(\Q(\zeta_{24})^+\)	None	✓	✓			3872.2.a.bl	$2$	$0$	\(0\)	\(0\)	\(4\)	\(8\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(1+\beta _{3})q^{5}+(2+\beta _{3})q^{7}+\cdots\)
3872.2.a.bn	$3872$	$2$	3872.a	1.a	$1$	$6$	$6$	$30.918$	6.6.19898000.1	None	✓		✓		352.2.m.e	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{5}q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
3872.2.a.bo	$3872$	$2$	3872.a	1.a	$1$	$6$	$6$	$30.918$	6.6.19898000.1	None	✓		✓		352.2.m.e	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{5}q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3872.2.a.bp	$3872$	$2$	3872.a	1.a	$1$	$6$	$6$	$30.918$	6.6.19898000.1	None	✓		✓		352.2.m.e	$1$	$0$	\(0\)	\(5\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{5}q^{5}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3872.2.a.bq	$3872$	$2$	3872.a	1.a	$1$	$6$	$6$	$30.918$	6.6.19898000.1	None	✓		✓		352.2.m.e	$1$	$0$	\(0\)	\(5\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{5}q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3872))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3872)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(968))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1936))\)\(^{\oplus 2}\)