Space of modular forms of level 56, weight 4, and trivial character

• Label: 56.4.a
• Level: $56$
• Weight: $4$
• Character orbit: 56.a
• Rep. character: $\chi_{56}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(56))\).

	Total	New	Old
Modular forms	28	4	24
Cusp forms	20	4	16
Eisenstein series	8	0	8

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(7\)	\(2\)	\(5\)		\(5\)	\(2\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(16\)	\(3\)	\(13\)		\(12\)	\(3\)	\(9\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)

Trace form

4 * q + 2 * q^3 + 14 * q^5 + 48 * q^9 + 116 * q^11 - 54 * q^13 - 56 * q^15 - 28 * q^17 - 90 * q^19 - 42 * q^21 - 24 * q^23 + 176 * q^25 - 244 * q^27 - 452 * q^29 - 196 * q^31 - 432 * q^33 + 210 * q^35 + 636 * q^37 + 496 * q^39 - 132 * q^41 + 444 * q^43 + 1350 * q^45 - 924 * q^47 + 196 * q^49 + 28 * q^51 - 536 * q^53 + 1144 * q^55 - 828 * q^57 - 602 * q^59 - 530 * q^61 + 532 * q^63 + 756 * q^65 + 1256 * q^67 - 1280 * q^69 + 248 * q^71 + 2480 * q^73 - 3242 * q^75 - 308 * q^77 - 616 * q^79 - 1224 * q^81 - 162 * q^83 - 2180 * q^85 - 3436 * q^87 + 488 * q^89 + 966 * q^91 + 1208 * q^93 + 1512 * q^95 - 1212 * q^97 + 2436 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(56))\) 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	7
56.4.a.a	$56$	$4$	56.a	1.a	$1$	$1$	$1$	$3.304$	\(\Q\)	None	✓	✓	✓	✓	56.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(-16\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+24q^{11}+\cdots\)
56.4.a.b	$56$	$4$	56.a	1.a	$1$	$1$	$1$	$3.304$	\(\Q\)	None	✓	✓	✓	✓	56.4.a.b	$1$	$0$	\(0\)	\(6\)	\(8\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+8q^{5}-7q^{7}+9q^{9}+56q^{11}+\cdots\)
56.4.a.c	$56$	$4$	56.a	1.a	$1$	$2$	$2$	$3.304$	\(\Q(\sqrt{57}) \)	None	✓	✓	✓	✓	56.4.a.c	$1$	$0$	\(0\)	\(-2\)	\(22\)	\(14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(11+\beta )q^{5}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(56)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)