Space of modular forms of level 4732, weight 2, and trivial character

• Label: 4732.2.a
• Level: $4732$
• Weight: $2$
• Character orbit: 4732.a
• Rep. character: $\chi_{4732}(1,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $22$
• Sturm bound: $1456$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1456\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4732))\).

	Total	New	Old
Modular forms	770	78	692
Cusp forms	687	78	609
Eisenstein series	83	0	83

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

\(2\)	\(7\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(18\)
\(-\)	\(+\)	\(-\)	$+$	\(21\)
\(-\)	\(-\)	\(+\)	$+$	\(18\)
\(-\)	\(-\)	\(-\)	$-$	\(21\)
Plus space			\(+\)	\(39\)
Minus space			\(-\)	\(39\)

Trace form

\( 78 q + 4 q^{5} + 78 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\) 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	7	13
4732.2.a.a	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}+2q^{15}+\cdots\)
4732.2.a.b	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-q^{7}-3q^{9}+2q^{11}-7q^{17}+\cdots\)
4732.2.a.c	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{9}-2q^{11}-3q^{17}+\cdots\)
4732.2.a.d	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-3q^{9}+2q^{11}-3q^{17}+\cdots\)
4732.2.a.e	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-q^{7}-3q^{9}+2q^{11}-4q^{17}+\cdots\)
4732.2.a.f	$4732$	$2$	4732.a	1.a	$1$	$1$	$1$	$37.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}-3q^{9}-2q^{11}-7q^{17}+\cdots\)
4732.2.a.g	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}+q^{7}+(2+\beta )q^{9}+\cdots\)
4732.2.a.h	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}-q^{7}+(2+\beta )q^{9}+\cdots\)
4732.2.a.i	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+q^{7}+3q^{9}+(-4+\cdots)q^{11}+\cdots\)
4732.2.a.j	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+\beta )q^{5}+q^{7}+\beta q^{9}+\cdots\)
4732.2.a.k	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}-q^{7}+\beta q^{9}+(4+\cdots)q^{11}+\cdots\)
4732.2.a.l	$4732$	$2$	4732.a	1.a	$1$	$2$	$2$	$37.785$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}-q^{7}+(1+2\beta )q^{9}+\cdots\)
4732.2.a.m	$4732$	$2$	4732.a	1.a	$1$	$3$	$3$	$37.785$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2\beta _{1}q^{5}+q^{7}-3q^{9}+(2-\beta _{1}+4\beta _{2})q^{11}+\cdots\)
4732.2.a.n	$4732$	$2$	4732.a	1.a	$1$	$3$	$3$	$37.785$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta _{1}q^{5}-q^{7}-3q^{9}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
4732.2.a.o	$4732$	$2$	4732.a	1.a	$1$	$4$	$4$	$37.785$	4.4.25492.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{5}-q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
4732.2.a.p	$4732$	$2$	4732.a	1.a	$1$	$4$	$4$	$37.785$	4.4.25492.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}+q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
4732.2.a.q	$4732$	$2$	4732.a	1.a	$1$	$6$	$6$	$37.785$	6.6.2854789.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{2}-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\beta _{3}-\beta _{5})q^{5}+\cdots\)
4732.2.a.r	$4732$	$2$	4732.a	1.a	$1$	$6$	$6$	$37.785$	6.6.2854789.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{2}-\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
4732.2.a.s	$4732$	$2$	4732.a	1.a	$1$	$8$	$8$	$37.785$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-8\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}-q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
4732.2.a.t	$4732$	$2$	4732.a	1.a	$1$	$8$	$8$	$37.785$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{5})q^{5}+q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
4732.2.a.u	$4732$	$2$	4732.a	1.a	$1$	$9$	$9$	$37.785$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-4\)	\(9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{4})q^{3}+\beta _{7}q^{5}+q^{7}+\cdots\)
4732.2.a.v	$4732$	$2$	4732.a	1.a	$1$	$9$	$9$	$37.785$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(4\)	\(-9\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{4})q^{3}-\beta _{7}q^{5}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4732))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4732)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\)\(^{\oplus 2}\)