Space of modular forms of level 42 and weight 8

• Label: 42.8
• Level: 42
• Weight: 8
• Dimension: 82
• Nonzero newspaces: 4
• Newform subspaces: 13
• Sturm bound: 768
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 13 \)
Sturm bound:			\(768\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(42))\).

	Total	New	Old
Modular forms	360	82	278
Cusp forms	312	82	230
Eisenstein series	48	0	48

Trace form

82 * q - 16 * q^2 - 384 * q^4 + 216 * q^5 + 864 * q^6 + 2904 * q^7 - 1024 * q^8 - 654 * q^9 + 3072 * q^10 - 576 * q^11 - 2324 * q^13 + 12608 * q^14 - 15012 * q^15 - 8192 * q^16 - 48792 * q^17 - 4848 * q^18 - 4712 * q^19 + 14592 * q^20 - 24414 * q^21 - 117312 * q^22 - 242172 * q^23 + 61440 * q^24 + 167698 * q^25 + 400768 * q^26 - 50432 * q^28 - 559044 * q^29 - 324480 * q^30 - 743048 * q^31 - 65536 * q^32 - 79020 * q^33 + 1048992 * q^34 + 1431072 * q^35 + 1433472 * q^36 - 34140 * q^37 - 277088 * q^38 - 1579440 * q^39 + 196608 * q^40 - 1157220 * q^41 - 731664 * q^42 - 2674776 * q^43 - 36864 * q^44 + 4694220 * q^45 + 1771776 * q^46 + 33900 * q^47 + 4325398 * q^49 + 4350992 * q^50 - 5261184 * q^51 - 1157888 * q^52 - 3483864 * q^53 + 1818144 * q^54 - 6930612 * q^55 - 919552 * q^56 + 5421324 * q^57 + 4583616 * q^58 + 5529384 * q^59 + 18432 * q^60 - 16472564 * q^61 + 1802560 * q^62 - 3826848 * q^63 - 7864320 * q^64 + 4126692 * q^65 - 2586816 * q^66 + 19539824 * q^67 - 3122688 * q^68 + 5646024 * q^69 + 675456 * q^70 - 14826744 * q^71 - 1182720 * q^72 - 15579272 * q^73 + 877504 * q^74 + 9886536 * q^75 + 4945408 * q^76 + 30622956 * q^77 + 14567712 * q^78 + 1098152 * q^79 + 884736 * q^80 - 12114846 * q^81 - 29210976 * q^82 - 3440976 * q^83 - 11252352 * q^84 - 24310152 * q^85 - 14784416 * q^86 - 17714880 * q^87 - 5965824 * q^88 + 14311164 * q^89 + 10077696 * q^90 + 62670208 * q^91 + 8683008 * q^92 + 45078768 * q^93 + 612288 * q^94 - 19549164 * q^95 + 3932160 * q^96 - 24261008 * q^97 - 30737872 * q^98 - 17633160 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.8.a	\(\chi_{42}(1, \cdot)\)	42.8.a.a	1	1
		42.8.a.b	1
		42.8.a.c	1
		42.8.a.d	1
		42.8.a.e	1
		42.8.a.f	1
42.8.d	\(\chi_{42}(41, \cdot)\)	42.8.d.a	20	1
42.8.e	\(\chi_{42}(25, \cdot)\)	42.8.e.a	2	2
		42.8.e.b	2
		42.8.e.c	4
		42.8.e.d	6
		42.8.e.e	6
42.8.f	\(\chi_{42}(5, \cdot)\)	42.8.f.a	36	2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)