Space of modular forms of level 312 and weight 4

• Label: 312.4
• Level: 312
• Weight: 4
• Dimension: 3320
• Nonzero newspaces: 18
• Sturm bound: 21504
• Trace bound: 10

Defining parameters

Level:	\( N \)	=	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 18 \)
Sturm bound:			\(21504\)
Trace bound:			\(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(312))\).

	Total	New	Old
Modular forms	8352	3408	4944
Cusp forms	7776	3320	4456
Eisenstein series	576	88	488

Trace form

3320 * q - 4 * q^2 - 14 * q^3 - 64 * q^4 - 28 * q^5 + 16 * q^6 - 32 * q^7 + 152 * q^8 + 70 * q^9 - 96 * q^10 + 56 * q^11 + 100 * q^12 + 74 * q^13 + 200 * q^14 + 24 * q^15 + 168 * q^16 + 14 * q^17 - 344 * q^18 + 256 * q^19 - 112 * q^20 + 24 * q^21 - 920 * q^22 - 1128 * q^23 - 1076 * q^24 - 820 * q^25 - 56 * q^26 - 638 * q^27 - 376 * q^28 - 114 * q^29 + 492 * q^30 + 1208 * q^31 + 496 * q^32 + 1016 * q^33 + 2640 * q^34 + 2232 * q^35 + 2044 * q^36 + 582 * q^37 + 1552 * q^38 - 1362 * q^39 - 5792 * q^40 - 2098 * q^41 - 4332 * q^42 - 2904 * q^43 - 2904 * q^44 - 90 * q^45 + 1480 * q^46 + 2568 * q^47 - 860 * q^48 - 210 * q^49 + 6644 * q^50 + 1652 * q^51 + 9248 * q^52 + 3788 * q^53 + 4216 * q^54 + 4328 * q^55 + 10424 * q^56 - 220 * q^57 + 6656 * q^58 - 1552 * q^59 + 5052 * q^60 - 3074 * q^61 - 2864 * q^62 - 3240 * q^63 - 12040 * q^64 + 422 * q^65 - 10360 * q^66 - 416 * q^67 - 5952 * q^68 + 132 * q^69 - 13120 * q^70 + 1696 * q^71 - 7940 * q^72 - 388 * q^73 - 3136 * q^74 + 7414 * q^75 - 3752 * q^76 - 1344 * q^77 + 2076 * q^78 + 104 * q^79 + 4224 * q^80 + 1654 * q^81 + 10720 * q^82 + 3016 * q^83 + 16536 * q^84 - 6598 * q^85 + 21008 * q^86 - 14064 * q^87 + 21568 * q^88 - 6772 * q^89 + 5184 * q^90 - 15144 * q^91 + 720 * q^92 - 6852 * q^93 - 16296 * q^94 - 5344 * q^95 - 23012 * q^96 + 7604 * q^97 - 22476 * q^98 + 8204 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(312))\)

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
312.4.a	\(\chi_{312}(1, \cdot)\)	312.4.a.a	2	1
		312.4.a.b	2
		312.4.a.c	2
		312.4.a.d	2
		312.4.a.e	2
		312.4.a.f	2
		312.4.a.g	3
		312.4.a.h	3
312.4.c	\(\chi_{312}(25, \cdot)\)	312.4.c.a	10	1
		312.4.c.b	12
312.4.d	\(\chi_{312}(287, \cdot)\)	None	0	1
312.4.g	\(\chi_{312}(157, \cdot)\)	312.4.g.a	32	1
		312.4.g.b	40
312.4.h	\(\chi_{312}(155, \cdot)\)	n/a	164	1
312.4.j	\(\chi_{312}(131, \cdot)\)	n/a	144	1
312.4.m	\(\chi_{312}(181, \cdot)\)	312.4.m.a	84	1
312.4.n	\(\chi_{312}(311, \cdot)\)	None	0	1
312.4.q	\(\chi_{312}(217, \cdot)\)	312.4.q.a	4	2
		312.4.q.b	6
		312.4.q.c	8
		312.4.q.d	10
		312.4.q.e	12
312.4.t	\(\chi_{312}(187, \cdot)\)	n/a	168	2
312.4.u	\(\chi_{312}(31, \cdot)\)	None	0	2
312.4.x	\(\chi_{312}(161, \cdot)\)	312.4.x.a	84	2
312.4.y	\(\chi_{312}(5, \cdot)\)	n/a	328	2
312.4.ba	\(\chi_{312}(179, \cdot)\)	n/a	328	2
312.4.bb	\(\chi_{312}(61, \cdot)\)	n/a	168	2
312.4.be	\(\chi_{312}(191, \cdot)\)	None	0	2
312.4.bf	\(\chi_{312}(49, \cdot)\)	312.4.bf.a	20	2
		312.4.bf.b	24
312.4.bj	\(\chi_{312}(23, \cdot)\)	None	0	2
312.4.bk	\(\chi_{312}(205, \cdot)\)	n/a	168	2
312.4.bn	\(\chi_{312}(35, \cdot)\)	n/a	328	2
312.4.bo	\(\chi_{312}(149, \cdot)\)	n/a	656	4
312.4.bp	\(\chi_{312}(41, \cdot)\)	n/a	168	4
312.4.bs	\(\chi_{312}(7, \cdot)\)	None	0	4
312.4.bt	\(\chi_{312}(19, \cdot)\)	n/a	336	4

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(312))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(312)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(156))\)\(^{\oplus 2}\)