Space of modular forms of level 56 and weight 2

• Label: 56.2
• Level: 56
• Weight: 2
• Dimension: 42
• Nonzero newspaces: 6
• Newform subspaces: 10
• Sturm bound: 384
• Trace bound: 2

Defining parameters

Level:	\( N \)	=	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 6 \)
Newform subspaces:			\( 10 \)
Sturm bound:			\(384\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(56))\).

	Total	New	Old
Modular forms	132	62	70
Cusp forms	61	42	19
Eisenstein series	71	20	51

Trace form

42 * q - 6 * q^2 - 6 * q^3 - 6 * q^4 - 6 * q^6 - 6 * q^7 - 12 * q^8 - 18 * q^9 - 6 * q^10 - 12 * q^11 - 6 * q^12 - 6 * q^13 - 6 * q^14 - 24 * q^15 - 6 * q^16 - 18 * q^17 + 12 * q^18 - 6 * q^19 + 12 * q^20 + 18 * q^22 + 12 * q^23 + 30 * q^24 + 6 * q^25 + 24 * q^26 + 24 * q^27 + 42 * q^28 + 42 * q^30 + 12 * q^31 + 24 * q^32 - 6 * q^33 + 30 * q^34 - 6 * q^35 + 18 * q^36 - 6 * q^37 + 12 * q^38 + 24 * q^40 - 36 * q^41 - 42 * q^42 - 12 * q^43 - 30 * q^44 - 6 * q^45 - 36 * q^46 + 12 * q^47 - 66 * q^48 - 6 * q^49 - 54 * q^50 + 36 * q^51 - 60 * q^52 + 6 * q^53 - 78 * q^54 + 24 * q^55 - 48 * q^56 - 36 * q^58 + 30 * q^59 - 72 * q^60 - 60 * q^62 + 30 * q^63 - 54 * q^64 - 12 * q^65 - 30 * q^66 + 12 * q^67 - 6 * q^68 + 12 * q^69 + 18 * q^70 + 24 * q^71 + 12 * q^72 - 30 * q^73 + 78 * q^74 + 54 * q^75 + 66 * q^76 + 18 * q^77 + 72 * q^78 + 12 * q^79 + 108 * q^80 + 12 * q^81 + 66 * q^82 + 30 * q^83 + 102 * q^84 + 102 * q^86 + 36 * q^87 + 66 * q^88 - 6 * q^89 + 72 * q^90 - 18 * q^91 + 66 * q^92 + 6 * q^93 + 30 * q^94 - 12 * q^95 + 18 * q^96 + 36 * q^97 + 18 * q^98 - 84 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
56.2.a	\(\chi_{56}(1, \cdot)\)	56.2.a.a	1	1
		56.2.a.b	1
56.2.b	\(\chi_{56}(29, \cdot)\)	56.2.b.a	2	1
		56.2.b.b	4
56.2.e	\(\chi_{56}(27, \cdot)\)	56.2.e.a	2	1
		56.2.e.b	4
56.2.f	\(\chi_{56}(55, \cdot)\)	None	0	1
56.2.i	\(\chi_{56}(9, \cdot)\)	56.2.i.a	2	2
		56.2.i.b	2
56.2.l	\(\chi_{56}(31, \cdot)\)	None	0	2
56.2.m	\(\chi_{56}(3, \cdot)\)	56.2.m.a	12	2
56.2.p	\(\chi_{56}(37, \cdot)\)	56.2.p.a	12	2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(56)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)