Space of modular forms of level 320, weight 4, and trivial character

• Label: 320.4.a
• Level: $320$
• Weight: $4$
• Character orbit: 320.a
• Rep. character: $\chi_{320}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $19$
• Sturm bound: $192$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(192\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(320))\).

	Total	New	Old
Modular forms	156	24	132
Cusp forms	132	24	108
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.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(42\)	\(7\)	\(35\)		\(36\)	\(7\)	\(29\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(38\)	\(5\)	\(33\)		\(32\)	\(5\)	\(27\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(36\)	\(5\)	\(31\)		\(30\)	\(5\)	\(25\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(40\)	\(7\)	\(33\)		\(34\)	\(7\)	\(27\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(82\)	\(14\)	\(68\)		\(70\)	\(14\)	\(56\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(74\)	\(10\)	\(64\)		\(62\)	\(10\)	\(52\)		\(12\)	\(0\)	\(12\)

Trace form

24 * q + 216 * q^9 - 144 * q^13 + 240 * q^21 + 600 * q^25 + 400 * q^29 + 464 * q^33 + 1024 * q^37 + 80 * q^41 + 1176 * q^49 - 752 * q^53 - 688 * q^57 + 3072 * q^61 + 528 * q^69 - 1904 * q^77 + 1336 * q^81 - 240 * q^85 + 848 * q^89 + 7968 * q^93

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(320))\) 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
320.4.a.a	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.c	$1$	$1$	\(0\)	\(-10\)	\(5\)	\(-18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+5q^{5}-18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
320.4.a.b	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				10.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(-5\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-5q^{5}+4q^{7}+37q^{9}+12q^{11}+\cdots\)
320.4.a.c	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(-6\)	\(5\)	\(34\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}+5q^{5}+34q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
320.4.a.d	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				20.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(-5\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-5q^{5}-2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
320.4.a.e	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.b	$1$	$0$	\(0\)	\(-4\)	\(-5\)	\(16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-5q^{5}+2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
320.4.a.f	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				160.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(5\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+5q^{5}-6q^{7}-23q^{9}+60q^{11}+\cdots\)
320.4.a.g	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(5\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+5q^{5}+6q^{7}-23q^{9}-2^{5}q^{11}+\cdots\)
320.4.a.h	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				5.4.a.a	$1$	$0$	\(0\)	\(2\)	\(5\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+5q^{5}-6q^{7}-23q^{9}+2^{5}q^{11}+\cdots\)
320.4.a.i	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				160.4.a.a	$1$	$1$	\(0\)	\(2\)	\(5\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+5q^{5}+6q^{7}-23q^{9}-60q^{11}+\cdots\)
320.4.a.j	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.b	$1$	$1$	\(0\)	\(4\)	\(-5\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-5q^{5}-2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
320.4.a.k	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				20.4.a.a	$1$	$1$	\(0\)	\(4\)	\(-5\)	\(16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-5q^{5}+2^{4}q^{7}-11q^{9}-60q^{11}+\cdots\)
320.4.a.l	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.a	$1$	$1$	\(0\)	\(6\)	\(5\)	\(-34\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+5q^{5}-34q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
320.4.a.m	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(0\)	\(8\)	\(-5\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}-4q^{7}+37q^{9}-12q^{11}+\cdots\)
320.4.a.n	$320$	$4$	320.a	1.a	$1$	$1$	$1$	$18.881$	\(\Q\)	None	✓				40.4.a.c	$1$	$0$	\(0\)	\(10\)	\(5\)	\(18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}+5q^{5}+18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)
320.4.a.o	$320$	$4$	320.a	1.a	$1$	$2$	$2$	$18.881$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(-8\)	\(-10\)	\(8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}-5q^{5}+(4+5\beta )q^{7}+\cdots\)
320.4.a.p	$320$	$4$	320.a	1.a	$1$	$2$	$2$	$18.881$	\(\Q(\sqrt{10}) \)	None	✓		✓		160.4.a.f	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}-3\beta q^{7}+13q^{9}-2\beta q^{11}+\cdots\)
320.4.a.q	$320$	$4$	320.a	1.a	$1$	$2$	$2$	$18.881$	\(\Q(\sqrt{5}) \)	None	✓				160.4.a.d	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+5q^{5}-7\beta q^{7}-7q^{9}+2\beta q^{11}+\cdots\)
320.4.a.r	$320$	$4$	320.a	1.a	$1$	$2$	$2$	$18.881$	\(\Q(\sqrt{13}) \)	None	✓				160.4.a.e	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+5q^{5}+\beta q^{7}+5^{2}q^{9}-6\beta q^{11}+\cdots\)
320.4.a.s	$320$	$4$	320.a	1.a	$1$	$2$	$2$	$18.881$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(8\)	\(-10\)	\(-8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}-5q^{5}+(-4+5\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(320))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(320)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)