Space of modular forms of level 176, weight 6, and trivial character

• Label: 176.6.a
• Level: $176$
• Weight: $6$
• Character orbit: 176.a
• Rep. character: $\chi_{176}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $12$
• Sturm bound: $144$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	176.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(144\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(176))\).

	Total	New	Old
Modular forms	126	25	101
Cusp forms	114	25	89
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(6\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(11\)
Minus space		\(-\)	\(14\)

Trace form

\( 25 q + 18 q^{3} + 38 q^{5} - 124 q^{7} + 2025 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\) 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	11
176.6.a.a	$176$	$6$	176.a	1.a	$1$	$1$	$1$	$28.228$	\(\Q\)	None	✓				44.6.a.a	$1$	$0$	\(0\)	\(-7\)	\(-79\)	\(50\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-79q^{5}+50q^{7}-194q^{9}+\cdots\)
176.6.a.b	$176$	$6$	176.a	1.a	$1$	$1$	$1$	$28.228$	\(\Q\)	None	✓				22.6.a.b	$1$	$1$	\(0\)	\(-1\)	\(-51\)	\(166\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-51q^{5}+166q^{7}-242q^{9}+\cdots\)
176.6.a.c	$176$	$6$	176.a	1.a	$1$	$1$	$1$	$28.228$	\(\Q\)	None	✓				11.6.a.a	$1$	$1$	\(0\)	\(15\)	\(-19\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+15q^{3}-19q^{5}-10q^{7}-18q^{9}+\cdots\)
176.6.a.d	$176$	$6$	176.a	1.a	$1$	$1$	$1$	$28.228$	\(\Q\)	None	✓				22.6.a.a	$1$	$0$	\(0\)	\(21\)	\(81\)	\(-98\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+21q^{3}+3^{4}q^{5}-98q^{7}+198q^{9}+\cdots\)
176.6.a.e	$176$	$6$	176.a	1.a	$1$	$1$	$1$	$28.228$	\(\Q\)	None	✓				22.6.a.c	$1$	$0$	\(0\)	\(29\)	\(-31\)	\(230\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+29q^{3}-31q^{5}+230q^{7}+598q^{9}+\cdots\)
176.6.a.f	$176$	$6$	176.a	1.a	$1$	$2$	$2$	$28.228$	\(\Q(\sqrt{793}) \)	None	✓				22.6.a.d	$1$	$1$	\(0\)	\(-29\)	\(-13\)	\(14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-14-\beta )q^{3}+(-4-5\beta )q^{5}+(4+\cdots)q^{7}+\cdots\)
176.6.a.g	$176$	$6$	176.a	1.a	$1$	$2$	$2$	$28.228$	\(\Q(\sqrt{31}) \)	None	✓				44.6.a.b	$1$	$1$	\(0\)	\(6\)	\(-22\)	\(-268\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(-11-2\beta )q^{5}+(-134+\cdots)q^{7}+\cdots\)
176.6.a.h	$176$	$6$	176.a	1.a	$1$	$2$	$2$	$28.228$	\(\Q(\sqrt{37}) \)	None	✓				88.6.a.a	$1$	$1$	\(0\)	\(14\)	\(18\)	\(-48\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(7+\beta )q^{3}+9q^{5}+(-24-11\beta )q^{7}+\cdots\)
176.6.a.i	$176$	$6$	176.a	1.a	$1$	$3$	$3$	$28.228$	3.3.54492.1	None	✓		✓		11.6.a.b	$1$	$0$	\(0\)	\(-34\)	\(24\)	\(-84\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{3}+(7+4\beta _{1}+\beta _{2})q^{5}+\cdots\)
176.6.a.j	$176$	$6$	176.a	1.a	$1$	$3$	$3$	$28.228$	3.3.1784453.1	None	✓		✓		88.6.a.b	$1$	$1$	\(0\)	\(-14\)	\(56\)	\(-112\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{2})q^{3}+(19+\beta _{1}-\beta _{2})q^{5}+\cdots\)
176.6.a.k	$176$	$6$	176.a	1.a	$1$	$4$	$4$	$28.228$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.d	$1$	$0$	\(0\)	\(5\)	\(93\)	\(94\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(24-2\beta _{1}-\beta _{3})q^{5}+\cdots\)
176.6.a.l	$176$	$6$	176.a	1.a	$1$	$4$	$4$	$28.228$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.c	$1$	$0$	\(0\)	\(13\)	\(-19\)	\(-58\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-5+2\beta _{1}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(176))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(176)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)