Space of modular forms of level 312 and weight 2

• Label: 312.2
• Level: 312
• Weight: 2
• Dimension: 1076
• Nonzero newspaces: 18
• Newform subspaces: 46
• Sturm bound: 10752
• Trace bound: 10

Defining parameters

Level:	\( N \)	=	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 18 \)
Newform subspaces:			\( 46 \)
Sturm bound:			\(10752\)
Trace bound:			\(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(312))\).

	Total	New	Old
Modular forms	2976	1164	1812
Cusp forms	2401	1076	1325
Eisenstein series	575	88	487

Trace form

1076 * q + 4 * q^2 - 6 * q^3 - 16 * q^4 + 4 * q^5 - 16 * q^6 - 16 * q^7 - 8 * q^8 - 18 * q^9 - 32 * q^10 - 8 * q^11 - 28 * q^12 + 2 * q^13 - 8 * q^14 - 24 * q^15 - 24 * q^16 + 10 * q^17 + 16 * q^20 + 24 * q^21 - 8 * q^22 + 24 * q^23 + 12 * q^24 + 8 * q^26 - 6 * q^27 - 24 * q^28 + 18 * q^29 - 4 * q^30 - 24 * q^31 - 16 * q^32 - 8 * q^33 - 64 * q^34 + 24 * q^35 - 20 * q^36 - 6 * q^37 - 16 * q^38 - 18 * q^39 - 160 * q^40 - 14 * q^41 - 92 * q^42 - 88 * q^43 - 120 * q^44 - 14 * q^45 - 248 * q^46 - 24 * q^47 - 92 * q^48 - 74 * q^49 - 212 * q^50 - 60 * q^51 - 256 * q^52 - 68 * q^53 - 24 * q^54 - 152 * q^55 - 200 * q^56 - 52 * q^57 - 192 * q^58 - 80 * q^59 - 132 * q^60 - 86 * q^61 - 112 * q^62 - 56 * q^63 - 136 * q^64 - 38 * q^65 - 88 * q^66 - 96 * q^67 - 52 * q^69 - 128 * q^70 - 64 * q^71 - 36 * q^72 - 52 * q^73 - 32 * q^74 - 130 * q^75 + 24 * q^76 - 20 * q^78 - 72 * q^79 - 50 * q^81 + 48 * q^82 + 8 * q^83 + 88 * q^84 - 10 * q^85 + 112 * q^86 - 96 * q^87 + 64 * q^88 - 20 * q^89 + 152 * q^90 - 216 * q^91 + 144 * q^92 - 92 * q^93 + 120 * q^94 - 224 * q^95 + 156 * q^96 - 76 * q^97 + 204 * q^98 - 180 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(312))\)

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
312.2.a	\(\chi_{312}(1, \cdot)\)	312.2.a.a	1	1
		312.2.a.b	1
		312.2.a.c	1
		312.2.a.d	1
		312.2.a.e	1
		312.2.a.f	1
312.2.c	\(\chi_{312}(25, \cdot)\)	312.2.c.a	2	1
		312.2.c.b	2
		312.2.c.c	2
312.2.d	\(\chi_{312}(287, \cdot)\)	None	0	1
312.2.g	\(\chi_{312}(157, \cdot)\)	312.2.g.a	8	1
		312.2.g.b	16
312.2.h	\(\chi_{312}(155, \cdot)\)	312.2.h.a	8	1
		312.2.h.b	12
		312.2.h.c	32
312.2.j	\(\chi_{312}(131, \cdot)\)	312.2.j.a	48	1
312.2.m	\(\chi_{312}(181, \cdot)\)	312.2.m.a	2	1
		312.2.m.b	2
		312.2.m.c	24
312.2.n	\(\chi_{312}(311, \cdot)\)	None	0	1
312.2.q	\(\chi_{312}(217, \cdot)\)	312.2.q.a	2	2
		312.2.q.b	2
		312.2.q.c	2
		312.2.q.d	4
		312.2.q.e	6
312.2.t	\(\chi_{312}(187, \cdot)\)	312.2.t.a	2	2
		312.2.t.b	2
		312.2.t.c	2
		312.2.t.d	2
		312.2.t.e	24
		312.2.t.f	24
312.2.u	\(\chi_{312}(31, \cdot)\)	None	0	2
312.2.x	\(\chi_{312}(161, \cdot)\)	312.2.x.a	4	2
		312.2.x.b	8
		312.2.x.c	16
312.2.y	\(\chi_{312}(5, \cdot)\)	312.2.y.a	104	2
312.2.ba	\(\chi_{312}(179, \cdot)\)	312.2.ba.a	104	2
312.2.bb	\(\chi_{312}(61, \cdot)\)	312.2.bb.a	56	2
312.2.be	\(\chi_{312}(191, \cdot)\)	None	0	2
312.2.bf	\(\chi_{312}(49, \cdot)\)	312.2.bf.a	4	2
		312.2.bf.b	8
312.2.bj	\(\chi_{312}(23, \cdot)\)	None	0	2
312.2.bk	\(\chi_{312}(205, \cdot)\)	312.2.bk.a	8	2
		312.2.bk.b	48
312.2.bn	\(\chi_{312}(35, \cdot)\)	312.2.bn.a	104	2
312.2.bo	\(\chi_{312}(149, \cdot)\)	312.2.bo.a	208	4
312.2.bp	\(\chi_{312}(41, \cdot)\)	312.2.bp.a	56	4
312.2.bs	\(\chi_{312}(7, \cdot)\)	None	0	4
312.2.bt	\(\chi_{312}(19, \cdot)\)	312.2.bt.a	4	4
		312.2.bt.b	4
		312.2.bt.c	48
		312.2.bt.d	56

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(312))\) into lower level spaces

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