Space of modular forms of level 4640, weight 2, and trivial character

• Label: 4640.2.a
• Level: $4640$
• Weight: $2$
• Character orbit: 4640.a
• Rep. character: $\chi_{4640}(1,\cdot)$
• Character field: $\Q$
• Dimension: $112$
• Newform subspaces: $27$
• Sturm bound: $1440$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(1440\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4640))\).

	Total	New	Old
Modular forms	736	112	624
Cusp forms	705	112	593
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.

\(2\)	\(5\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(87\)	\(12\)	\(75\)		\(84\)	\(12\)	\(72\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(97\)	\(16\)	\(81\)		\(93\)	\(16\)	\(77\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(97\)	\(16\)	\(81\)		\(93\)	\(16\)	\(77\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(87\)	\(10\)	\(77\)		\(83\)	\(10\)	\(73\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(97\)	\(16\)	\(81\)		\(93\)	\(16\)	\(77\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(87\)	\(12\)	\(75\)		\(83\)	\(12\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(87\)	\(12\)	\(75\)		\(83\)	\(12\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(97\)	\(18\)	\(79\)		\(93\)	\(18\)	\(75\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(348\)	\(46\)	\(302\)		\(333\)	\(46\)	\(287\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(388\)	\(66\)	\(322\)		\(372\)	\(66\)	\(306\)		\(16\)	\(0\)	\(16\)

Trace form

112 * q + 96 * q^9 + 32 * q^13 - 16 * q^17 + 32 * q^21 + 112 * q^25 + 32 * q^33 + 32 * q^37 + 32 * q^41 + 128 * q^49 + 64 * q^57 + 32 * q^61 - 16 * q^65 - 64 * q^69 - 16 * q^73 - 64 * q^77 + 112 * q^81 - 32 * q^89 - 32 * q^93 - 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4640))\) 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	5	29
4640.2.a.a	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
4640.2.a.b	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}+2q^{11}+\cdots\)
4640.2.a.c	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}+2q^{11}+2q^{13}+\cdots\)
4640.2.a.d	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}-2q^{11}+2q^{13}+\cdots\)
4640.2.a.e	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
4640.2.a.f	$4640$	$2$	4640.a	1.a	$1$	$1$	$1$	$37.051$	\(\Q\)	None	✓	✓			4640.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}-2q^{11}+\cdots\)
4640.2.a.g	$4640$	$2$	4640.a	1.a	$1$	$2$	$2$	$37.051$	\(\Q(\sqrt{5}) \)	None	✓	✓			4640.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{5}+(-1+3\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
4640.2.a.h	$4640$	$2$	4640.a	1.a	$1$	$2$	$2$	$37.051$	\(\Q(\sqrt{2}) \)	None	✓	✓			4640.2.a.h	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}-3q^{9}+\beta q^{11}-2q^{13}+\cdots\)
4640.2.a.i	$4640$	$2$	4640.a	1.a	$1$	$2$	$2$	$37.051$	\(\Q(\sqrt{5}) \)	None	✓	✓			4640.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+(1-3\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
4640.2.a.j	$4640$	$2$	4640.a	1.a	$1$	$3$	$3$	$37.051$	3.3.229.1	None	✓	✓			4640.2.a.j	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-q^{5}+(-2-2\beta _{1}+\cdots)q^{7}+\cdots\)
4640.2.a.k	$4640$	$2$	4640.a	1.a	$1$	$3$	$3$	$37.051$	3.3.229.1	None	✓	✓			4640.2.a.j	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}-q^{5}+(2+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
4640.2.a.l	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	4.4.11344.1	None	✓	✓			4640.2.a.l	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-q^{5}-\beta _{3}q^{7}+(2-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
4640.2.a.m	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4640.2.a.m	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+\beta _{2}q^{7}-q^{9}+\beta _{3}q^{11}+\cdots\)
4640.2.a.n	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	\(\Q(\sqrt{6}, \sqrt{10})\)	None	✓	✓			4640.2.a.n	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}-\beta _{3}q^{7}+3q^{9}+\beta _{3}q^{11}+\cdots\)
4640.2.a.o	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	\(\Q(\sqrt{6}, \sqrt{26})\)	None	✓	✓			4640.2.a.o	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}-\beta _{1}q^{7}+3q^{9}+\beta _{2}q^{11}+\cdots\)
4640.2.a.p	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4640.2.a.p	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}-\beta _{2}q^{7}-q^{9}+\beta _{2}q^{11}+\cdots\)
4640.2.a.q	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4640.2.a.q	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}+\beta _{3}q^{7}-q^{9}-\beta _{2}q^{11}+\cdots\)
4640.2.a.r	$4640$	$2$	4640.a	1.a	$1$	$4$	$4$	$37.051$	4.4.11344.1	None	✓	✓			4640.2.a.l	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}-q^{5}+\beta _{3}q^{7}+(2-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
4640.2.a.s	$4640$	$2$	4640.a	1.a	$1$	$6$	$6$	$37.051$	6.6.26118032.1	None	✓	✓	✓		4640.2.a.s	$1$	$1$	\(0\)	\(-1\)	\(-6\)	\(-3\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
4640.2.a.t	$4640$	$2$	4640.a	1.a	$1$	$6$	$6$	$37.051$	6.6.39643024.1	None	✓	✓	✓		4640.2.a.t	$1$	$1$	\(0\)	\(-1\)	\(6\)	\(-5\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-1-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
4640.2.a.u	$4640$	$2$	4640.a	1.a	$1$	$6$	$6$	$37.051$	6.6.26118032.1	None	✓	✓			4640.2.a.s	$1$	$0$	\(0\)	\(1\)	\(-6\)	\(3\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
4640.2.a.v	$4640$	$2$	4640.a	1.a	$1$	$6$	$6$	$37.051$	6.6.39643024.1	None	✓	✓			4640.2.a.t	$1$	$0$	\(0\)	\(1\)	\(6\)	\(5\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(1+\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
4640.2.a.w	$4640$	$2$	4640.a	1.a	$1$	$7$	$7$	$37.051$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		4640.2.a.w	$1$	$1$	\(0\)	\(-1\)	\(7\)	\(-1\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(\beta _{1}-\beta _{5})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
4640.2.a.x	$4640$	$2$	4640.a	1.a	$1$	$7$	$7$	$37.051$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			4640.2.a.w	$1$	$0$	\(0\)	\(1\)	\(7\)	\(1\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(-\beta _{1}+\beta _{5})q^{7}+(1+\cdots)q^{9}+\cdots\)
4640.2.a.y	$4640$	$2$	4640.a	1.a	$1$	$8$	$8$	$37.051$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		4640.2.a.y	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(\beta _{1}+\beta _{5})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
4640.2.a.z	$4640$	$2$	4640.a	1.a	$1$	$10$	$10$	$37.051$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		4640.2.a.z	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+\beta _{9}q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4640.2.a.ba	$4640$	$2$	4640.a	1.a	$1$	$10$	$10$	$37.051$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		4640.2.a.ba	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}-\beta _{7}q^{7}+(3+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4640))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4640)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(928))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1160))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2320))\)\(^{\oplus 2}\)