Space of modular forms of level 128, weight 4, and trivial character

• Label: 128.4.a
• Level: $128$
• Weight: $4$
• Character orbit: 128.a
• Rep. character: $\chi_{128}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $64$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(64\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(128))\).

	Total	New	Old
Modular forms	56	12	44
Cusp forms	40	12	28
Eisenstein series	16	0	16

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

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(30\)	\(7\)	\(23\)		\(22\)	\(7\)	\(15\)		\(8\)	\(0\)	\(8\)
\(-\)		\(26\)	\(5\)	\(21\)		\(18\)	\(5\)	\(13\)		\(8\)	\(0\)	\(8\)

Trace form

\( 12 q + 108 q^{9} + 104 q^{17} + 212 q^{25} + 464 q^{33} - 392 q^{41} - 852 q^{49} - 1360 q^{57} - 2512 q^{65} - 24 q^{73} + 1596 q^{81} + 1960 q^{89} + 4328 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(128))\) 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
128.4.a.a	$128$	$4$	128.a	1.a	$1$	$1$	$1$	$7.552$	\(\Q\)	None	✓	✓			128.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(-6\)	\(20\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-6q^{5}+20q^{7}-23q^{9}-14q^{11}+\cdots\)
128.4.a.b	$128$	$4$	128.a	1.a	$1$	$1$	$1$	$7.552$	\(\Q\)	None	✓	✓			128.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(6\)	\(-20\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+6q^{5}-20q^{7}-23q^{9}-14q^{11}+\cdots\)
128.4.a.c	$128$	$4$	128.a	1.a	$1$	$1$	$1$	$7.552$	\(\Q\)	None	✓	✓			128.4.a.a	$1$	$1$	\(0\)	\(2\)	\(-6\)	\(-20\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-6q^{5}-20q^{7}-23q^{9}+14q^{11}+\cdots\)
128.4.a.d	$128$	$4$	128.a	1.a	$1$	$1$	$1$	$7.552$	\(\Q\)	None	✓	✓			128.4.a.a	$1$	$0$	\(0\)	\(2\)	\(6\)	\(20\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+6q^{5}+20q^{7}-23q^{9}+14q^{11}+\cdots\)
128.4.a.e	$128$	$4$	128.a	1.a	$1$	$2$	$2$	$7.552$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		128.4.a.e	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(-8\)		$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-2-2\beta )q^{5}+(-4+\cdots)q^{7}+\cdots\)
128.4.a.f	$128$	$4$	128.a	1.a	$1$	$2$	$2$	$7.552$	\(\Q(\sqrt{3}) \)	None	✓	✓			128.4.a.e	$1$	$0$	\(0\)	\(-4\)	\(4\)	\(8\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(2+2\beta )q^{5}+(4+2\beta )q^{7}+\cdots\)
128.4.a.g	$128$	$4$	128.a	1.a	$1$	$2$	$2$	$7.552$	\(\Q(\sqrt{3}) \)	None	✓	✓			128.4.a.e	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(8\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(-2+2\beta )q^{5}+(4-2\beta )q^{7}+\cdots\)
128.4.a.h	$128$	$4$	128.a	1.a	$1$	$2$	$2$	$7.552$	\(\Q(\sqrt{3}) \)	None	✓	✓			128.4.a.e	$1$	$0$	\(0\)	\(4\)	\(4\)	\(-8\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(2-2\beta )q^{5}+(-4+2\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(128))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(128)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)