Space of modular forms of level 40 and weight 4

• Label: 40.4
• Level: 40
• Weight: 4
• Dimension: 67
• Nonzero newspaces: 5
• Newform subspaces: 7
• Sturm bound: 384
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 5 \)
Newform subspaces:			\( 7 \)
Sturm bound:			\(384\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(40))\).

	Total	New	Old
Modular forms	168	79	89
Cusp forms	120	67	53
Eisenstein series	48	12	36

Trace form

67 * q + 4 * q^3 + 20 * q^4 - 9 * q^5 - 64 * q^6 - 8 * q^7 - 84 * q^8 + 63 * q^9 + 52 * q^10 + 108 * q^11 + 216 * q^12 + 10 * q^13 + 96 * q^14 - 196 * q^15 - 152 * q^16 - 138 * q^17 - 484 * q^18 - 316 * q^19 - 496 * q^20 - 184 * q^21 - 96 * q^22 + 528 * q^23 + 248 * q^24 + 67 * q^25 - 200 * q^26 + 520 * q^27 + 184 * q^28 + 378 * q^29 - 208 * q^30 + 160 * q^31 + 360 * q^32 - 456 * q^33 + 228 * q^34 - 424 * q^35 + 832 * q^36 - 494 * q^37 + 1456 * q^38 - 936 * q^39 + 1332 * q^40 - 426 * q^41 + 1904 * q^42 - 916 * q^43 + 448 * q^44 + 307 * q^45 + 272 * q^46 - 488 * q^47 - 360 * q^48 + 2307 * q^49 + 92 * q^50 + 3320 * q^51 - 876 * q^52 + 1042 * q^53 - 3832 * q^54 - 604 * q^55 - 2000 * q^56 - 2304 * q^57 - 1060 * q^58 - 1444 * q^59 - 3128 * q^60 - 1966 * q^61 - 2288 * q^62 - 3096 * q^63 - 4960 * q^64 + 58 * q^65 - 3920 * q^66 - 1292 * q^67 - 1148 * q^68 + 1400 * q^69 + 1192 * q^70 + 1336 * q^71 + 5444 * q^72 + 742 * q^73 + 3804 * q^74 + 3660 * q^75 + 3568 * q^76 + 320 * q^77 + 5560 * q^78 + 5648 * q^79 + 5000 * q^80 + 483 * q^81 + 5916 * q^82 + 4316 * q^83 + 8352 * q^84 + 982 * q^85 + 9808 * q^86 + 1696 * q^87 + 1216 * q^88 + 142 * q^89 - 2460 * q^90 - 3248 * q^91 - 8400 * q^92 - 1520 * q^93 - 6744 * q^94 - 4020 * q^95 - 12944 * q^96 - 1650 * q^97 - 11080 * q^98 - 4572 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.4.a	\(\chi_{40}(1, \cdot)\)	40.4.a.a	1	1
		40.4.a.b	1
		40.4.a.c	1
40.4.c	\(\chi_{40}(9, \cdot)\)	40.4.c.a	4	1
40.4.d	\(\chi_{40}(21, \cdot)\)	40.4.d.a	12	1
40.4.f	\(\chi_{40}(29, \cdot)\)	40.4.f.a	16	1
40.4.j	\(\chi_{40}(7, \cdot)\)	None	0	2
40.4.k	\(\chi_{40}(3, \cdot)\)	40.4.k.a	32	2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 1}\)