Space of modular forms of level 56 and weight 6

• Label: 56.6
• Level: 56
• Weight: 6
• Dimension: 248
• Nonzero newspaces: 6
• Newform subspaces: 13
• Sturm bound: 1152
• Trace bound: 2

Defining parameters

Level:	\( N \)	=	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	=	\( 6 \)
Nonzero newspaces:			\( 6 \)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1152\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(56))\).

	Total	New	Old
Modular forms	516	268	248
Cusp forms	444	248	196
Eisenstein series	72	20	52

Trace form

248 * q - 2 * q^2 - 46 * q^3 - 46 * q^4 + 148 * q^5 + 226 * q^6 - 78 * q^7 + 484 * q^8 + 572 * q^9 - 1270 * q^10 - 284 * q^11 - 3158 * q^12 - 1970 * q^13 + 2378 * q^14 + 3048 * q^15 + 6618 * q^16 + 3610 * q^17 - 8056 * q^18 + 1130 * q^19 - 11756 * q^20 - 5736 * q^21 - 4970 * q^22 - 10788 * q^23 + 15902 * q^24 - 1688 * q^25 + 6748 * q^26 + 20936 * q^27 + 28490 * q^28 + 6564 * q^29 + 12458 * q^30 + 20836 * q^31 - 33352 * q^32 - 7678 * q^33 - 42338 * q^34 - 4302 * q^35 - 6414 * q^36 - 17226 * q^37 + 9272 * q^38 - 74664 * q^39 - 13320 * q^40 - 13448 * q^41 - 20518 * q^42 + 50284 * q^43 + 43698 * q^44 - 6218 * q^45 + 16624 * q^46 - 7500 * q^47 + 69822 * q^48 + 96648 * q^49 + 121234 * q^50 + 121036 * q^51 - 1460 * q^52 + 8242 * q^53 - 277810 * q^54 - 116616 * q^55 - 165472 * q^56 - 17136 * q^57 - 86300 * q^58 + 60310 * q^59 + 19368 * q^60 + 78796 * q^61 + 85928 * q^62 - 47722 * q^63 + 289418 * q^64 - 9676 * q^65 + 257862 * q^66 - 80236 * q^67 + 298506 * q^68 - 27844 * q^69 - 227070 * q^70 - 383560 * q^71 - 446596 * q^72 - 159562 * q^73 - 534314 * q^74 - 299666 * q^75 - 415534 * q^76 + 49362 * q^77 + 420272 * q^78 + 699244 * q^79 + 563628 * q^80 + 209370 * q^81 + 144218 * q^82 + 321254 * q^83 + 538262 * q^84 + 167096 * q^85 + 593734 * q^86 - 249108 * q^87 + 461794 * q^88 + 79006 * q^89 - 642328 * q^90 - 300354 * q^91 - 752598 * q^92 - 124418 * q^93 - 959802 * q^94 + 113668 * q^95 - 386350 * q^96 - 564160 * q^97 + 176518 * q^98 + 255700 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(56))\)

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
56.6.a	\(\chi_{56}(1, \cdot)\)	56.6.a.a	1	1
		56.6.a.b	1
		56.6.a.c	2
		56.6.a.d	2
		56.6.a.e	2
56.6.b	\(\chi_{56}(29, \cdot)\)	56.6.b.a	14	1
		56.6.b.b	16
56.6.e	\(\chi_{56}(27, \cdot)\)	56.6.e.a	2	1
		56.6.e.b	36
56.6.f	\(\chi_{56}(55, \cdot)\)	None	0	1
56.6.i	\(\chi_{56}(9, \cdot)\)	56.6.i.a	10	2
		56.6.i.b	10
56.6.l	\(\chi_{56}(31, \cdot)\)	None	0	2
56.6.m	\(\chi_{56}(3, \cdot)\)	56.6.m.a	76	2
56.6.p	\(\chi_{56}(37, \cdot)\)	56.6.p.a	76	2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

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