Space of modular forms of level 176, weight 4, and trivial character

• Label: 176.4.a
• Level: $176$
• Weight: $4$
• Character orbit: 176.a
• Rep. character: $\chi_{176}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $10$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	176.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(176))\).

	Total	New	Old
Modular forms	78	15	63
Cusp forms	66	15	51
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
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(4\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(6\)

Trace form

\( 15 q - 6 q^{3} + 2 q^{5} + 36 q^{7} + 135 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(176))\) 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
176.4.a.a	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				88.4.a.b	$1$	$1$	\(0\)	\(-7\)	\(9\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+9q^{5}-2q^{7}+22q^{9}+11q^{11}+\cdots\)
176.4.a.b	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				22.4.a.b	$1$	$0$	\(0\)	\(-4\)	\(14\)	\(8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+14q^{5}+8q^{7}-11q^{9}+11q^{11}+\cdots\)
176.4.a.c	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				22.4.a.c	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+10q^{7}-26q^{9}-11q^{11}+\cdots\)
176.4.a.d	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				88.4.a.a	$1$	$1$	\(0\)	\(1\)	\(-7\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-7q^{5}+6q^{7}-26q^{9}+11q^{11}+\cdots\)
176.4.a.e	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				44.4.a.a	$1$	$0$	\(0\)	\(5\)	\(-7\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-7q^{5}+26q^{7}-2q^{9}+11q^{11}+\cdots\)
176.4.a.f	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	\(\Q\)	None	✓				22.4.a.a	$1$	$1$	\(0\)	\(7\)	\(-19\)	\(-14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-19q^{5}-14q^{7}+22q^{9}-11q^{11}+\cdots\)
176.4.a.g	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	\(\Q(\sqrt{97}) \)	None	✓		✓		44.4.a.b	$1$	$1$	\(0\)	\(-9\)	\(11\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(6-\beta )q^{5}+(-8+6\beta )q^{7}+\cdots\)
176.4.a.h	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	\(\Q(\sqrt{5}) \)	None	✓				88.4.a.c	$1$	$0$	\(0\)	\(2\)	\(-6\)	\(56\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-3+4\beta )q^{5}+(28+\beta )q^{7}+\cdots\)
176.4.a.i	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	\(\Q(\sqrt{3}) \)	None	✓		✓		11.4.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-20\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+(-10+\beta )q^{7}+\cdots\)
176.4.a.j	$176$	$4$	176.a	1.a	$1$	$3$	$3$	$10.384$	3.3.11109.1	None	✓		✓		88.4.a.d	$1$	$0$	\(0\)	\(-2\)	\(8\)	\(-24\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(176))\) into lower level spaces

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