Space of modular forms of level 2320, weight 4, and trivial character

• Label: 2320.4.a
• Level: $2320$
• Weight: $4$
• Character orbit: 2320.a
• Rep. character: $\chi_{2320}(1,\cdot)$
• Character field: $\Q$
• Dimension: $168$
• Newform subspaces: $29$
• Sturm bound: $1440$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1440\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2320))\).

	Total	New	Old
Modular forms	1092	168	924
Cusp forms	1068	168	900
Eisenstein series	24	0	24

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(22\)
\(+\)	\(+\)	\(-\)	\(-\)	\(20\)
\(+\)	\(-\)	\(+\)	\(-\)	\(20\)
\(+\)	\(-\)	\(-\)	\(+\)	\(22\)
\(-\)	\(+\)	\(+\)	\(-\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)	\(22\)
\(-\)	\(-\)	\(+\)	\(+\)	\(22\)
\(-\)	\(-\)	\(-\)	\(-\)	\(20\)
Plus space			\(+\)	\(88\)
Minus space			\(-\)	\(80\)

Trace form

\( 168 q + 12 q^{3} - 72 q^{7} + 1472 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2320))\) 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
2320.4.a.a	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				290.4.a.d	$1$	$1$	\(0\)	\(-2\)	\(-5\)	\(24\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-5q^{5}+24q^{7}-23q^{9}-12q^{11}+\cdots\)
2320.4.a.b	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				290.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(-5\)	\(32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-5q^{5}+2^{5}q^{7}-23q^{9}-12q^{11}+\cdots\)
2320.4.a.c	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				290.4.a.c	$1$	$1$	\(0\)	\(0\)	\(5\)	\(20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+20q^{7}-3^{3}q^{9}+52q^{11}+\cdots\)
2320.4.a.d	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				290.4.a.b	$1$	$1$	\(0\)	\(2\)	\(-5\)	\(-12\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-5q^{5}-12q^{7}-23q^{9}+48q^{11}+\cdots\)
2320.4.a.e	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				1160.4.a.a	$1$	$0$	\(0\)	\(8\)	\(-5\)	\(6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}+6q^{7}+37q^{9}+58q^{11}+\cdots\)
2320.4.a.f	$2320$	$4$	2320.a	1.a	$1$	$1$	$1$	$136.884$	\(\Q\)	None	✓				145.4.a.a	$1$	$1$	\(0\)	\(8\)	\(-5\)	\(14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}+14q^{7}+37q^{9}-62q^{11}+\cdots\)
2320.4.a.g	$2320$	$4$	2320.a	1.a	$1$	$3$	$3$	$136.884$	3.3.6133.1	None	✓				290.4.a.g	$1$	$1$	\(0\)	\(1\)	\(-15\)	\(-47\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2})q^{3}-5q^{5}+(-17+4\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.h	$2320$	$4$	2320.a	1.a	$1$	$3$	$3$	$136.884$	3.3.48244.1	None	✓				290.4.a.e	$1$	$0$	\(0\)	\(2\)	\(-15\)	\(34\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-5q^{5}+(11-\beta _{1}-\beta _{2})q^{7}+\cdots\)
2320.4.a.i	$2320$	$4$	2320.a	1.a	$1$	$3$	$3$	$136.884$	3.3.2804.1	None	✓				290.4.a.f	$1$	$0$	\(0\)	\(2\)	\(15\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+5q^{5}+(-1-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
2320.4.a.j	$2320$	$4$	2320.a	1.a	$1$	$3$	$3$	$136.884$	3.3.1556.1	None	✓				1160.4.a.b	$1$	$0$	\(0\)	\(8\)	\(-15\)	\(42\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{2})q^{3}-5q^{5}+(15+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
2320.4.a.k	$2320$	$4$	2320.a	1.a	$1$	$5$	$5$	$136.884$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				290.4.a.i	$1$	$0$	\(0\)	\(-9\)	\(-25\)	\(-9\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}-5q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.l	$2320$	$4$	2320.a	1.a	$1$	$5$	$5$	$136.884$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				290.4.a.j	$1$	$0$	\(0\)	\(-9\)	\(25\)	\(-29\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+5q^{5}+(-6-\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.m	$2320$	$4$	2320.a	1.a	$1$	$5$	$5$	$136.884$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				290.4.a.h	$1$	$1$	\(0\)	\(-1\)	\(25\)	\(-21\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5q^{5}+(-4-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
2320.4.a.n	$2320$	$4$	2320.a	1.a	$1$	$6$	$6$	$136.884$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				145.4.a.c	$1$	$0$	\(0\)	\(1\)	\(-30\)	\(-3\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-5q^{5}+(-\beta _{1}-\beta _{2}-2\beta _{3}+\cdots)q^{7}+\cdots\)
2320.4.a.o	$2320$	$4$	2320.a	1.a	$1$	$6$	$6$	$136.884$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				580.4.a.a	$1$	$1$	\(0\)	\(5\)	\(-30\)	\(-9\)		$-$	$+$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{3}-5q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
2320.4.a.p	$2320$	$4$	2320.a	1.a	$1$	$6$	$6$	$136.884$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				580.4.a.b	$1$	$1$	\(0\)	\(5\)	\(30\)	\(-7\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+5q^{5}+(-1+2\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
2320.4.a.q	$2320$	$4$	2320.a	1.a	$1$	$6$	$6$	$136.884$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				145.4.a.b	$1$	$0$	\(0\)	\(13\)	\(30\)	\(79\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{4})q^{3}+5q^{5}+(12-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
2320.4.a.r	$2320$	$4$	2320.a	1.a	$1$	$7$	$7$	$136.884$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1160.4.a.c	$1$	$0$	\(0\)	\(-1\)	\(-35\)	\(-65\)		$+$	$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5q^{5}+(-9+\beta _{5})q^{7}+(7+\cdots)q^{9}+\cdots\)
2320.4.a.s	$2320$	$4$	2320.a	1.a	$1$	$7$	$7$	$136.884$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		145.4.a.d	$1$	$1$	\(0\)	\(-1\)	\(-35\)	\(-17\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-5q^{5}+(-3+\beta _{4}-2\beta _{5}+\cdots)q^{7}+\cdots\)
2320.4.a.t	$2320$	$4$	2320.a	1.a	$1$	$8$	$8$	$136.884$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		145.4.a.e	$1$	$1$	\(0\)	\(-17\)	\(40\)	\(-33\)		$-$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{3}+5q^{5}+(-4-\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.u	$2320$	$4$	2320.a	1.a	$1$	$8$	$8$	$136.884$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		580.4.a.c	$1$	$0$	\(0\)	\(-1\)	\(-40\)	\(19\)		$-$	$+$	$-$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5q^{5}+(2-\beta _{7})q^{7}+(7+\beta _{2}+\cdots)q^{9}+\cdots\)
2320.4.a.v	$2320$	$4$	2320.a	1.a	$1$	$8$	$8$	$136.884$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		580.4.a.d	$1$	$0$	\(0\)	\(-1\)	\(40\)	\(-35\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5q^{5}+(-4-\beta _{2})q^{7}+(19+\cdots)q^{9}+\cdots\)
2320.4.a.w	$2320$	$4$	2320.a	1.a	$1$	$9$	$9$	$136.884$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1160.4.a.e	$1$	$1$	\(0\)	\(3\)	\(-45\)	\(27\)		$+$	$+$	$-$	$-$	$2^{7}\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(3+\beta _{1}+\beta _{4})q^{7}+\cdots\)
2320.4.a.x	$2320$	$4$	2320.a	1.a	$1$	$9$	$9$	$136.884$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1160.4.a.f	$1$	$1$	\(0\)	\(3\)	\(45\)	\(-9\)		$+$	$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5q^{5}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
2320.4.a.y	$2320$	$4$	2320.a	1.a	$1$	$9$	$9$	$136.884$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1160.4.a.d	$1$	$0$	\(0\)	\(7\)	\(45\)	\(39\)		$+$	$-$	$-$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+5q^{5}+(4+\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
2320.4.a.z	$2320$	$4$	2320.a	1.a	$1$	$11$	$11$	$136.884$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1160.4.a.h	$1$	$1$	\(0\)	\(-8\)	\(-55\)	\(-16\)		$+$	$+$	$-$	$-$	$2^{9}\cdot 3^{3}\cdot 43$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-5q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.ba	$2320$	$4$	2320.a	1.a	$1$	$11$	$11$	$136.884$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1160.4.a.i	$1$	$1$	\(0\)	\(-8\)	\(55\)	\(-38\)		$+$	$-$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+5q^{5}+(-4+\beta _{1}+\cdots)q^{7}+\cdots\)
2320.4.a.bb	$2320$	$4$	2320.a	1.a	$1$	$11$	$11$	$136.884$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1160.4.a.g	$1$	$0$	\(0\)	\(-2\)	\(-55\)	\(-56\)		$+$	$+$	$+$	$+$	$2^{12}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5q^{5}+(-5-\beta _{7})q^{7}+(6+\cdots)q^{9}+\cdots\)
2320.4.a.bc	$2320$	$4$	2320.a	1.a	$1$	$13$	$13$	$136.884$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		1160.4.a.j	$1$	$0$	\(0\)	\(6\)	\(65\)	\(-2\)		$+$	$-$	$-$	$+$	$2^{15}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5q^{5}+(-1+\beta _{1}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2320))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2320)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1160))\)\(^{\oplus 2}\)