Space of modular forms of level 4, weight 16, and trivial character

• Label: 4.16.a
• Level: $4$
• Weight: $16$
• Character orbit: 4.a
• Rep. character: $\chi_{4}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $8$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(8\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(4))\).

	Total	New	Old
Modular forms	9	1	8
Cusp forms	6	1	5
Eisenstein series	3	0	3

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
\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)

Trace form

q - 276 * q^3 - 132210 * q^5 - 3585736 * q^7 - 14272731 * q^9 + 47801700 * q^11 + 247784966 * q^13 + 36489960 * q^15 - 2127682062 * q^17 - 1074862756 * q^19 + 989663136 * q^21 + 24982896168 * q^23 - 13038094025 * q^25 + 7899572088 * q^27 - 165099671946 * q^29 + 100736332256 * q^31 - 13193269200 * q^33 + 474070156560 * q^35 + 42490420334 * q^37 - 68388650616 * q^39 - 1388779245414 * q^41 - 1168783477180 * q^43 + 1886997765510 * q^45 - 1645655322672 * q^47 + 8109941151753 * q^49 + 587240249112 * q^51 - 4469627500578 * q^53 - 6319862757000 * q^55 + 296662120656 * q^57 - 28794808426572 * q^59 + 15719941145942 * q^61 + 51178245365016 * q^63 - 32759650354860 * q^65 + 61627103890604 * q^67 - 6895279342368 * q^69 - 66780412989192 * q^71 - 57749646345094 * q^73 + 3598513950900 * q^75 - 171404276551200 * q^77 + 198700138788272 * q^79 + 202617807858729 * q^81 - 113345193514212 * q^83 + 281300845417020 * q^85 + 45567509457096 * q^87 - 48230883277974 * q^89 - 888491472844976 * q^91 - 27803227702656 * q^93 + 142107604970760 * q^95 + 95121696327074 * q^97 - 682260805442700 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(4))\) 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
4.16.a.a	$4$	$16$	4.a	1.a	$1$	$1$	$1$	$5.708$	\(\Q\)	None	✓	✓	✓	✓	4.16.a.a	$1$	$1$	\(0\)	\(-276\)	\(-132210\)	\(-3585736\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-276q^{3}-132210q^{5}-3585736q^{7}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(4)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)