Space of modular forms of level 3864, weight 2, and trivial character

• Label: 3864.2.a
• Level: $3864$
• Weight: $2$
• Character orbit: 3864.a
• Rep. character: $\chi_{3864}(1,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $24$
• Sturm bound: $1536$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1536\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3864))\).

	Total	New	Old
Modular forms	784	68	716
Cusp forms	753	68	685
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(7\)	\(23\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(2\)
Plus space				\(+\)	\(28\)
Minus space				\(-\)	\(40\)

Trace form

\( 68 q - 8 q^{5} + 68 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	7	23
3864.2.a.a	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-q^{7}+q^{9}-q^{13}+3q^{15}+\cdots\)
3864.2.a.b	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
3864.2.a.c	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+q^{11}-6q^{13}+7q^{19}+\cdots\)
3864.2.a.d	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}+q^{9}+6q^{13}+2q^{15}+\cdots\)
3864.2.a.e	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
3864.2.a.f	$3864$	$2$	3864.a	1.a	$1$	$1$	$1$	$30.854$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
3864.2.a.g	$3864$	$2$	3864.a	1.a	$1$	$2$	$2$	$30.854$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-\beta q^{11}+2q^{13}+\cdots\)
3864.2.a.h	$3864$	$2$	3864.a	1.a	$1$	$2$	$2$	$30.854$	\(\Q(\sqrt{29}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-q^{7}+q^{9}+5q^{11}+(1+\cdots)q^{13}+\cdots\)
3864.2.a.i	$3864$	$2$	3864.a	1.a	$1$	$2$	$2$	$30.854$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
3864.2.a.j	$3864$	$2$	3864.a	1.a	$1$	$2$	$2$	$30.854$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{5}+q^{7}+q^{9}+(-3+2\beta )q^{11}+\cdots\)
3864.2.a.k	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.621.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+q^{7}+q^{9}+\cdots\)
3864.2.a.l	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{2}q^{5}-q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.m	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.837.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.n	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.1229.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.o	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.733.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{2})q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.p	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(3\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1})q^{5}+q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.q	$3864$	$2$	3864.a	1.a	$1$	$3$	$3$	$30.854$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{2}q^{5}-q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.r	$3864$	$2$	3864.a	1.a	$1$	$4$	$4$	$30.854$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-3\)	\(4\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{3})q^{5}+q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
3864.2.a.s	$3864$	$2$	3864.a	1.a	$1$	$4$	$4$	$30.854$	4.4.75645.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(1\)	\(-4\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.t	$3864$	$2$	3864.a	1.a	$1$	$4$	$4$	$30.854$	4.4.39605.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1})q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.u	$3864$	$2$	3864.a	1.a	$1$	$4$	$4$	$30.854$	4.4.256549.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(2\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(-1-\beta _{2}+\cdots)q^{11}+\cdots\)
3864.2.a.v	$3864$	$2$	3864.a	1.a	$1$	$5$	$5$	$30.854$	5.5.17679757.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(-6\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{1})q^{5}+q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.w	$3864$	$2$	3864.a	1.a	$1$	$6$	$6$	$30.854$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(2\)	\(-6\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{3}q^{5}-q^{7}+q^{9}+\beta _{4}q^{11}+\cdots\)
3864.2.a.x	$3864$	$2$	3864.a	1.a	$1$	$6$	$6$	$30.854$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-6\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{5}q^{5}-q^{7}+q^{9}+(1+\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3864))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3864)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(644))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(966))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1932))\)\(^{\oplus 2}\)