Space of modular forms of level 192, weight 4, and trivial character

• Label: 192.4.a
• Level: $192$
• Weight: $4$
• Character orbit: 192.a
• Rep. character: $\chi_{192}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $12$
• Sturm bound: $128$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(128\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(192))\).

	Total	New	Old
Modular forms	108	12	96
Cusp forms	84	12	72
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(28\)	\(3\)	\(25\)		\(22\)	\(3\)	\(19\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(26\)	\(2\)	\(24\)		\(20\)	\(2\)	\(18\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(26\)	\(3\)	\(23\)		\(20\)	\(3\)	\(17\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(28\)	\(4\)	\(24\)		\(22\)	\(4\)	\(18\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(56\)	\(7\)	\(49\)		\(44\)	\(7\)	\(37\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(52\)	\(5\)	\(47\)		\(40\)	\(5\)	\(35\)		\(12\)	\(0\)	\(12\)

Trace form

12 * q + 108 * q^9 - 72 * q^13 + 104 * q^17 + 120 * q^21 + 212 * q^25 + 400 * q^29 + 520 * q^37 - 472 * q^41 + 588 * q^49 - 752 * q^53 + 1992 * q^61 - 1536 * q^65 + 528 * q^69 - 296 * q^73 - 5408 * q^77 + 972 * q^81 - 2592 * q^85 + 88 * q^89 + 1848 * q^93 - 328 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(192))\) 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	3
192.4.a.a	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				24.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-14\)	\(-24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-14q^{5}-24q^{7}+9q^{9}+28q^{11}+\cdots\)
192.4.a.b	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(-10\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-10q^{5}+4q^{7}+9q^{9}+20q^{11}+\cdots\)
192.4.a.c	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+2^{4}q^{7}+9q^{9}+12q^{11}+\cdots\)
192.4.a.d	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}+12q^{7}+9q^{9}-60q^{11}+\cdots\)
192.4.a.e	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(14\)	\(-36\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+14q^{5}-6^{2}q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
192.4.a.f	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(18\)	\(8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{5}+8q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
192.4.a.g	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				24.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-14\)	\(24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-14q^{5}+24q^{7}+9q^{9}-28q^{11}+\cdots\)
192.4.a.h	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.c	$1$	$1$	\(0\)	\(3\)	\(-10\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-10q^{5}-4q^{7}+9q^{9}-20q^{11}+\cdots\)
192.4.a.i	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-6\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-6q^{5}-2^{4}q^{7}+9q^{9}-12q^{11}+\cdots\)
192.4.a.j	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.b	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(-12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}-12q^{7}+9q^{9}+60q^{11}+\cdots\)
192.4.a.k	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				96.4.a.a	$1$	$0$	\(0\)	\(3\)	\(14\)	\(36\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+14q^{5}+6^{2}q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
192.4.a.l	$192$	$4$	192.a	1.a	$1$	$1$	$1$	$11.328$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(3\)	\(18\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+18q^{5}-8q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(192))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(192)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)