Space of modular forms of level 192, weight 8, and trivial character

• Label: 192.8.a
• Level: $192$
• Weight: $8$
• Character orbit: 192.a
• Rep. character: $\chi_{192}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $22$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(192))\).

	Total	New	Old
Modular forms	236	28	208
Cusp forms	212	28	184
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
\(+\)	\(+\)	\(+\)		\(60\)	\(7\)	\(53\)		\(54\)	\(7\)	\(47\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(58\)	\(6\)	\(52\)		\(52\)	\(6\)	\(46\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(58\)	\(7\)	\(51\)		\(52\)	\(7\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(60\)	\(8\)	\(52\)		\(54\)	\(8\)	\(46\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(120\)	\(15\)	\(105\)		\(108\)	\(15\)	\(93\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(116\)	\(13\)	\(103\)		\(104\)	\(13\)	\(91\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q + 20412 * q^9 + 7064 * q^13 - 5816 * q^17 - 27432 * q^21 + 405252 * q^25 + 103376 * q^29 - 1465176 * q^37 + 882568 * q^41 + 3294172 * q^49 + 1815632 * q^53 + 1972456 * q^61 + 666624 * q^65 - 4790448 * q^69 + 2685304 * q^73 - 3461536 * q^77 + 14880348 * q^81 + 13697888 * q^85 + 4781048 * q^89 + 9661464 * q^93 + 7294552 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(192))\) 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
192.8.a.a	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				3.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-390\)	\(64\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-390q^{5}+2^{6}q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.b	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				12.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(-270\)	\(1112\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-270q^{5}+1112q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.c	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.c	$1$	$0$	\(0\)	\(-27\)	\(-110\)	\(504\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-110q^{5}+504q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.d	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(26\)	\(-1056\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+26q^{5}-1056q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.e	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				96.8.a.a	$1$	$0$	\(0\)	\(-27\)	\(70\)	\(-92\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+70q^{5}-92q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.f	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				6.8.a.a	$1$	$0$	\(0\)	\(-27\)	\(114\)	\(-1576\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+114q^{5}-1576q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.g	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				12.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(378\)	\(832\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+378q^{5}+832q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.h	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(530\)	\(120\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+530q^{5}+120q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.i	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				3.8.a.a	$1$	$1$	\(0\)	\(27\)	\(-390\)	\(-64\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-390q^{5}-2^{6}q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.j	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				12.8.a.b	$1$	$0$	\(0\)	\(27\)	\(-270\)	\(-1112\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-270q^{5}-1112q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.k	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.c	$1$	$0$	\(0\)	\(27\)	\(-110\)	\(-504\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-110q^{5}-504q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.l	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.a	$1$	$1$	\(0\)	\(27\)	\(26\)	\(1056\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+26q^{5}+1056q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.m	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				96.8.a.a	$1$	$1$	\(0\)	\(27\)	\(70\)	\(92\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+70q^{5}+92q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.n	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				6.8.a.a	$1$	$0$	\(0\)	\(27\)	\(114\)	\(1576\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+114q^{5}+1576q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.o	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				12.8.a.a	$1$	$1$	\(0\)	\(27\)	\(378\)	\(-832\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+378q^{5}-832q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.p	$192$	$8$	192.a	1.a	$1$	$1$	$1$	$59.978$	\(\Q\)	None	✓				24.8.a.b	$1$	$0$	\(0\)	\(27\)	\(530\)	\(-120\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+530q^{5}-120q^{7}+3^{6}q^{9}+\cdots\)
192.8.a.q	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{46}) \)	None	✓				96.8.a.e	$1$	$1$	\(0\)	\(-54\)	\(-196\)	\(504\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-98+\beta )q^{5}+(252-3\beta )q^{7}+\cdots\)
192.8.a.r	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{235}) \)	None	✓		✓		96.8.a.d	$1$	$0$	\(0\)	\(-54\)	\(-180\)	\(1032\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-90+\beta )q^{5}+(516+5\beta )q^{7}+\cdots\)
192.8.a.s	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{6}) \)	None	✓				96.8.a.c	$1$	$1$	\(0\)	\(-54\)	\(28\)	\(-936\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(14+3\beta )q^{5}+(-468+7\beta )q^{7}+\cdots\)
192.8.a.t	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{46}) \)	None	✓				96.8.a.e	$1$	$0$	\(0\)	\(54\)	\(-196\)	\(-504\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-98+\beta )q^{5}+(-252+3\beta )q^{7}+\cdots\)
192.8.a.u	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{235}) \)	None	✓		✓		96.8.a.d	$1$	$1$	\(0\)	\(54\)	\(-180\)	\(-1032\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-90+\beta )q^{5}+(-516-5\beta )q^{7}+\cdots\)
192.8.a.v	$192$	$8$	192.a	1.a	$1$	$2$	$2$	$59.978$	\(\Q(\sqrt{6}) \)	None	✓				96.8.a.c	$1$	$0$	\(0\)	\(54\)	\(28\)	\(936\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(14+3\beta )q^{5}+(468-7\beta )q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(192))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(192)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)