Space of modular forms of level 3888, weight 2, and trivial character

• Label: 3888.2.a
• Level: $3888$
• Weight: $2$
• Character orbit: 3888.a
• Rep. character: $\chi_{3888}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $39$
• Sturm bound: $1296$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 3888 = 2^{4} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3888.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 39 \)
Sturm bound:			\(1296\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3888))\).

	Total	New	Old
Modular forms	702	72	630
Cusp forms	595	72	523
Eisenstein series	107	0	107

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

\(2\)	\(3\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(15\)
\(+\)	\(-\)	\(-\)	\(21\)
\(-\)	\(+\)	\(-\)	\(18\)
\(-\)	\(-\)	\(+\)	\(18\)
Plus space		\(+\)	\(33\)
Minus space		\(-\)	\(39\)

Trace form

\( 72 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3888))\) 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
3888.2.a.a	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.a	$1$	$2$	\(0\)	\(0\)	\(-4\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}-2q^{11}-5q^{13}-2q^{17}+\cdots\)
3888.2.a.b	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-2q^{7}-4q^{13}+6q^{17}+7q^{19}+\cdots\)
3888.2.a.c	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+4q^{7}+6q^{11}+2q^{13}+q^{19}+\cdots\)
3888.2.a.d	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-3q^{7}+2q^{11}+q^{13}-4q^{17}+\cdots\)
3888.2.a.e	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{11}+q^{13}+2q^{17}-5q^{19}+\cdots\)
3888.2.a.f	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+3q^{7}+2q^{11}+q^{13}+8q^{17}+\cdots\)
3888.2.a.g	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{11}-2q^{13}+4q^{17}+q^{19}+\cdots\)
3888.2.a.h	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				243.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{7}+2q^{13}-8q^{19}-5q^{25}+7q^{31}+\cdots\)
3888.2.a.i	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				972.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{7}+5q^{13}+7q^{19}-5q^{25}-11q^{31}+\cdots\)
3888.2.a.j	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-6q^{11}-q^{13}+6q^{17}-5q^{19}+\cdots\)
3888.2.a.k	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				972.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+q^{7}-7q^{13}+q^{19}-5q^{25}+7q^{31}+\cdots\)
3888.2.a.l	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				972.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+q^{7}+2q^{13}-8q^{19}-5q^{25}-11q^{31}+\cdots\)
3888.2.a.m	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+6q^{11}-q^{13}-6q^{17}-5q^{19}+\cdots\)
3888.2.a.n	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				243.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}-7q^{13}+q^{19}-5q^{25}-11q^{31}+\cdots\)
3888.2.a.o	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				972.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}+5q^{13}+7q^{19}-5q^{25}+7q^{31}+\cdots\)
3888.2.a.p	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-2q^{13}-4q^{17}+q^{19}+\cdots\)
3888.2.a.q	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.d	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{7}-2q^{11}+q^{13}+4q^{17}+\cdots\)
3888.2.a.r	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{11}+q^{13}-2q^{17}-5q^{19}+\cdots\)
3888.2.a.s	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+3q^{7}-2q^{11}+q^{13}-8q^{17}+\cdots\)
3888.2.a.t	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.a	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-2q^{7}-4q^{13}-6q^{17}+7q^{19}+\cdots\)
3888.2.a.u	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				486.2.a.c	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}-6q^{11}+2q^{13}+q^{19}+\cdots\)
3888.2.a.v	$3888$	$2$	3888.a	1.a	$1$	$1$	$1$	$31.046$	\(\Q\)	None	✓				1944.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}+2q^{11}-5q^{13}+2q^{17}+\cdots\)
3888.2.a.w	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{3}) \)	None	✓				1944.2.a.k	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-2\beta q^{7}+(1+3\beta )q^{11}+\cdots\)
3888.2.a.x	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{33}) \)	None	✓				1944.2.a.l	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(1+\beta )q^{7}-4q^{11}+(2-2\beta )q^{13}+\cdots\)
3888.2.a.y	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{2}) \)	None	✓				972.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}-\beta q^{11}-q^{13}-\beta q^{17}+\cdots\)
3888.2.a.z	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{6}) \)	None	✓				243.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}+\beta q^{11}-q^{13}+3\beta q^{17}+\cdots\)
3888.2.a.ba	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{3}) \)	None	✓				243.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+q^{7}+\beta q^{11}+5q^{13}+q^{19}+\cdots\)
3888.2.a.bb	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{33}) \)	None	✓				1944.2.a.l	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(1+\beta )q^{7}+4q^{11}+(2-2\beta )q^{13}+\cdots\)
3888.2.a.bc	$3888$	$2$	3888.a	1.a	$1$	$2$	$2$	$31.046$	\(\Q(\sqrt{3}) \)	None	✓				1944.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+2\beta q^{7}+(-1+3\beta )q^{11}+\cdots\)
3888.2.a.bd	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				243.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{5}+(1-\beta _{1}+2\beta _{2})q^{7}+\cdots\)
3888.2.a.be	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				486.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(-2-\beta _{2})q^{7}+(-\beta _{1}+\beta _{2})q^{11}+\cdots\)
3888.2.a.bf	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				486.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}-\beta _{1}q^{11}+\cdots\)
3888.2.a.bg	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				972.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(1+\beta _{1})q^{7}+(-3-\beta _{1}+\beta _{2})q^{11}+\cdots\)
3888.2.a.bh	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				1944.2.a.o	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(1-\beta _{1})q^{7}+(-1+2\beta _{1}+\cdots)q^{11}+\cdots\)
3888.2.a.bi	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				1944.2.a.o	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(1-\beta _{1})q^{7}+(1-2\beta _{1}+\beta _{2})q^{11}+\cdots\)
3888.2.a.bj	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				972.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(1-\beta _{2})q^{7}+(3-\beta _{1})q^{11}+\cdots\)
3888.2.a.bk	$3888$	$2$	3888.a	1.a	$1$	$3$	$3$	$31.046$	\(\Q(\zeta_{18})^+\)	None	✓				243.2.a.e	$1$	$0$	\(0\)	\(0\)	\(6\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{5}+(1-\beta _{1}+2\beta _{2})q^{7}+(-1+\cdots)q^{11}+\cdots\)
3888.2.a.bl	$3888$	$2$	3888.a	1.a	$1$	$6$	$6$	$31.046$	6.6.9926793.1	None	✓		✓		1944.2.a.q	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+(-1+\beta _{4})q^{7}+(-\beta _{1}-\beta _{3}+\cdots)q^{11}+\cdots\)
3888.2.a.bm	$3888$	$2$	3888.a	1.a	$1$	$6$	$6$	$31.046$	6.6.9926793.1	None	✓		✓		1944.2.a.q	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(-1+\beta _{4})q^{7}+(\beta _{1}+\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3888))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3888)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(243))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(486))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(972))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1944))\)\(^{\oplus 2}\)