Space of modular forms of level 240 and weight 4

• Label: 240.4
• Level: 240
• Weight: 4
• Dimension: 1754
• Nonzero newspaces: 14
• Newform subspaces: 43
• Sturm bound: 12288
• Trace bound: 13

Defining parameters

Level:	\( N \)	=	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 14 \)
Newform subspaces:			\( 43 \)
Sturm bound:			\(12288\)
Trace bound:			\(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(240))\).

	Total	New	Old
Modular forms	4832	1810	3022
Cusp forms	4384	1754	2630
Eisenstein series	448	56	392

Trace form

1754 * q + 4 * q^3 + 32 * q^4 - 2 * q^5 - 72 * q^6 - 60 * q^7 - 168 * q^8 - 114 * q^9 - 144 * q^10 + 120 * q^11 + 200 * q^12 + 192 * q^13 + 696 * q^14 - 144 * q^15 + 576 * q^16 - 156 * q^17 - 40 * q^18 - 536 * q^19 - 80 * q^20 + 228 * q^21 - 1344 * q^22 + 328 * q^23 - 224 * q^24 + 18 * q^25 + 40 * q^26 + 760 * q^27 + 240 * q^28 - 684 * q^29 - 900 * q^30 + 888 * q^31 + 560 * q^32 - 1180 * q^33 + 1504 * q^34 + 336 * q^35 + 1408 * q^36 + 392 * q^37 + 1632 * q^38 - 1316 * q^39 + 3848 * q^40 + 1540 * q^41 - 80 * q^42 + 52 * q^43 + 2528 * q^44 + 1270 * q^45 + 1120 * q^46 + 744 * q^47 - 3464 * q^48 - 450 * q^49 - 5888 * q^50 + 96 * q^51 - 10160 * q^52 - 4756 * q^53 - 3408 * q^54 - 1760 * q^55 - 2688 * q^56 - 2812 * q^57 - 3040 * q^58 + 1256 * q^59 + 928 * q^60 + 1332 * q^61 + 1992 * q^62 - 1312 * q^63 + 11936 * q^64 - 1396 * q^65 + 10648 * q^66 + 92 * q^67 + 9072 * q^68 + 6896 * q^69 + 8520 * q^70 + 3696 * q^71 + 5864 * q^72 + 9344 * q^73 - 3144 * q^74 + 3976 * q^75 - 3712 * q^76 - 1840 * q^77 - 5168 * q^78 + 3736 * q^79 - 13352 * q^80 + 466 * q^81 - 13280 * q^82 + 5816 * q^83 - 8384 * q^84 + 1600 * q^85 - 1312 * q^86 - 344 * q^87 - 11104 * q^88 + 572 * q^89 + 392 * q^90 - 2776 * q^91 + 496 * q^92 + 1704 * q^93 - 7120 * q^94 - 12176 * q^95 - 9440 * q^96 - 6480 * q^97 - 896 * q^98 - 7152 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(240))\)

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
240.4.a	\(\chi_{240}(1, \cdot)\)	240.4.a.a	1	1
		240.4.a.b	1
		240.4.a.c	1
		240.4.a.d	1
		240.4.a.e	1
		240.4.a.f	1
		240.4.a.g	1
		240.4.a.h	1
		240.4.a.i	1
		240.4.a.j	1
		240.4.a.k	1
		240.4.a.l	1
240.4.b	\(\chi_{240}(71, \cdot)\)	None	0	1
240.4.d	\(\chi_{240}(169, \cdot)\)	None	0	1
240.4.f	\(\chi_{240}(49, \cdot)\)	240.4.f.a	2	1
		240.4.f.b	2
		240.4.f.c	2
		240.4.f.d	2
		240.4.f.e	2
		240.4.f.f	4
		240.4.f.g	4
240.4.h	\(\chi_{240}(191, \cdot)\)	240.4.h.a	8	1
		240.4.h.b	16
240.4.k	\(\chi_{240}(121, \cdot)\)	None	0	1
240.4.m	\(\chi_{240}(119, \cdot)\)	None	0	1
240.4.o	\(\chi_{240}(239, \cdot)\)	240.4.o.a	4	1
		240.4.o.b	8
		240.4.o.c	24
240.4.s	\(\chi_{240}(61, \cdot)\)	240.4.s.a	44	2
		240.4.s.b	52
240.4.t	\(\chi_{240}(59, \cdot)\)	240.4.t.a	8	2
		240.4.t.b	272
240.4.v	\(\chi_{240}(17, \cdot)\)	240.4.v.a	4	2
		240.4.v.b	8
		240.4.v.c	8
		240.4.v.d	12
		240.4.v.e	36
240.4.w	\(\chi_{240}(127, \cdot)\)	240.4.w.a	12	2
		240.4.w.b	24
240.4.y	\(\chi_{240}(163, \cdot)\)	240.4.y.a	72	2
		240.4.y.b	72
240.4.bb	\(\chi_{240}(173, \cdot)\)	240.4.bb.a	280	2
240.4.bc	\(\chi_{240}(43, \cdot)\)	240.4.bc.a	72	2
		240.4.bc.b	72
240.4.bf	\(\chi_{240}(53, \cdot)\)	240.4.bf.a	280	2
240.4.bh	\(\chi_{240}(7, \cdot)\)	None	0	2
240.4.bi	\(\chi_{240}(137, \cdot)\)	None	0	2
240.4.bk	\(\chi_{240}(11, \cdot)\)	240.4.bk.a	192	2
240.4.bl	\(\chi_{240}(109, \cdot)\)	240.4.bl.a	144	2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(240))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(240)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)