Space of modular forms of level 768 and weight 4

• Label: 768.4
• Level: 768
• Weight: 4
• Dimension: 20632
• Nonzero newspaces: 12
• Sturm bound: 131072
• Trace bound: 49

Defining parameters

Level:	\( N \)	=	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 12 \)
Sturm bound:			\(131072\)
Trace bound:			\(49\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(768))\).

	Total	New	Old
Modular forms	49856	20840	29016
Cusp forms	48448	20632	27816
Eisenstein series	1408	208	1200

Trace form

20632 * q - 24 * q^3 - 64 * q^4 - 32 * q^6 - 48 * q^7 - 40 * q^9 - 64 * q^10 - 32 * q^12 - 64 * q^13 - 24 * q^15 - 64 * q^16 - 32 * q^18 - 48 * q^19 - 32 * q^21 - 64 * q^22 - 32 * q^24 - 80 * q^25 - 24 * q^27 - 64 * q^28 - 32 * q^30 - 32 * q^31 - 56 * q^33 - 64 * q^34 - 32 * q^36 - 64 * q^37 - 24 * q^39 - 64 * q^40 - 32 * q^42 - 48 * q^43 - 32 * q^45 - 64 * q^46 - 32 * q^48 + 2648 * q^49 + 2952 * q^51 - 64 * q^52 + 3008 * q^53 - 32 * q^54 + 1104 * q^55 - 1384 * q^57 - 64 * q^58 - 5504 * q^59 - 32 * q^60 - 7360 * q^61 - 2048 * q^63 - 64 * q^64 - 7808 * q^65 - 32 * q^66 - 8208 * q^67 - 2144 * q^69 - 64 * q^70 - 896 * q^71 - 32 * q^72 + 3376 * q^73 + 4392 * q^75 - 64 * q^76 + 7616 * q^77 - 32 * q^78 + 11280 * q^79 - 48 * q^81 - 64 * q^82 - 32 * q^84 + 1936 * q^85 - 24 * q^87 - 64 * q^88 - 32 * q^90 - 48 * q^91 - 464 * q^93 - 64 * q^94 - 32 * q^96 - 112 * q^97 - 24 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(768))\)

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
768.4.a	\(\chi_{768}(1, \cdot)\)	768.4.a.a	1	1
		768.4.a.b	1
		768.4.a.c	1
		768.4.a.d	1
		768.4.a.e	2
		768.4.a.f	2
		768.4.a.g	2
		768.4.a.h	2
		768.4.a.i	2
		768.4.a.j	2
		768.4.a.k	2
		768.4.a.l	2
		768.4.a.m	2
		768.4.a.n	2
		768.4.a.o	2
		768.4.a.p	2
		768.4.a.q	3
		768.4.a.r	3
		768.4.a.s	3
		768.4.a.t	3
		768.4.a.u	4
		768.4.a.v	4
768.4.c	\(\chi_{768}(767, \cdot)\)	768.4.c.a	2	1
		768.4.c.b	2
		768.4.c.c	2
		768.4.c.d	2
		768.4.c.e	2
		768.4.c.f	2
		768.4.c.g	2
		768.4.c.h	2
		768.4.c.i	2
		768.4.c.j	2
		768.4.c.k	4
		768.4.c.l	4
		768.4.c.m	4
		768.4.c.n	4
		768.4.c.o	4
		768.4.c.p	4
		768.4.c.q	4
		768.4.c.r	4
		768.4.c.s	8
		768.4.c.t	8
		768.4.c.u	8
		768.4.c.v	16
768.4.d	\(\chi_{768}(385, \cdot)\)	768.4.d.a	2	1
		768.4.d.b	2
		768.4.d.c	2
		768.4.d.d	2
		768.4.d.e	2
		768.4.d.f	2
		768.4.d.g	2
		768.4.d.h	2
		768.4.d.i	2
		768.4.d.j	2
		768.4.d.k	2
		768.4.d.l	2
		768.4.d.m	2
		768.4.d.n	2
		768.4.d.o	2
		768.4.d.p	2
		768.4.d.q	4
		768.4.d.r	4
		768.4.d.s	4
		768.4.d.t	4
768.4.f	\(\chi_{768}(383, \cdot)\)	768.4.f.a	4	1
		768.4.f.b	8
		768.4.f.c	8
		768.4.f.d	12
		768.4.f.e	12
		768.4.f.f	12
		768.4.f.g	12
		768.4.f.h	12
		768.4.f.i	12
768.4.j	\(\chi_{768}(193, \cdot)\)	768.4.j.a	4	2
		768.4.j.b	4
		768.4.j.c	12
		768.4.j.d	12
		768.4.j.e	16
		768.4.j.f	16
		768.4.j.g	16
		768.4.j.h	16
768.4.k	\(\chi_{768}(191, \cdot)\)	n/a	192	2
768.4.n	\(\chi_{768}(97, \cdot)\)	n/a	192	4
768.4.o	\(\chi_{768}(95, \cdot)\)	n/a	368	4
768.4.r	\(\chi_{768}(49, \cdot)\)	n/a	384	8
768.4.s	\(\chi_{768}(47, \cdot)\)	n/a	752	8
768.4.v	\(\chi_{768}(25, \cdot)\)	None	0	16
768.4.w	\(\chi_{768}(23, \cdot)\)	None	0	16
768.4.z	\(\chi_{768}(13, \cdot)\)	n/a	6144	32
768.4.ba	\(\chi_{768}(11, \cdot)\)	n/a	12224	32

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(768))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(768)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(384))\)\(^{\oplus 2}\)