Space of modular forms of level 315 and weight 2

• Label: 315.2
• Level: 315
• Weight: 2
• Dimension: 2268
• Nonzero newspaces: 30
• Newform subspaces: 93
• Sturm bound: 13824
• Trace bound: 9

Defining parameters

Level:	\( N \)	=	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 30 \)
Newform subspaces:			\( 93 \)
Sturm bound:			\(13824\)
Trace bound:			\(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(315))\).

	Total	New	Old
Modular forms	3840	2540	1300
Cusp forms	3073	2268	805
Eisenstein series	767	272	495

Trace form

2268 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 - 6 * q^5 - 40 * q^6 + 2 * q^7 - 12 * q^8 - 16 * q^9 - 36 * q^10 - 28 * q^11 - 64 * q^12 - 42 * q^14 - 92 * q^15 - 64 * q^16 - 64 * q^17 - 104 * q^18 - 36 * q^19 - 122 * q^20 - 108 * q^21 - 88 * q^22 - 72 * q^23 - 168 * q^24 - 30 * q^25 - 172 * q^26 - 92 * q^27 - 98 * q^28 - 104 * q^29 - 142 * q^30 - 60 * q^31 - 212 * q^32 - 88 * q^33 - 148 * q^34 - 88 * q^35 - 200 * q^36 - 120 * q^37 - 160 * q^38 - 32 * q^39 - 110 * q^40 - 116 * q^41 - 120 * q^42 - 16 * q^43 - 80 * q^44 - 28 * q^45 - 208 * q^46 - 4 * q^47 + 80 * q^48 - 52 * q^49 - 16 * q^50 - 40 * q^51 - 104 * q^52 + 44 * q^53 + 44 * q^54 - 108 * q^55 + 30 * q^56 + 16 * q^57 - 112 * q^58 + 20 * q^59 + 26 * q^60 - 152 * q^61 + 84 * q^63 - 204 * q^64 - 160 * q^65 - 164 * q^66 - 156 * q^67 + 4 * q^68 - 96 * q^69 - 300 * q^70 - 296 * q^71 + 36 * q^72 - 228 * q^73 - 188 * q^74 - 32 * q^75 - 364 * q^76 - 168 * q^77 - 40 * q^78 - 180 * q^79 + 34 * q^80 - 160 * q^81 - 204 * q^82 - 72 * q^83 - 96 * q^84 - 132 * q^85 - 100 * q^86 - 44 * q^87 - 72 * q^88 + 12 * q^89 + 202 * q^90 - 100 * q^91 + 288 * q^92 + 36 * q^93 + 80 * q^94 + 218 * q^95 + 16 * q^96 + 64 * q^97 + 268 * q^98 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(315))\)

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
315.2.a	\(\chi_{315}(1, \cdot)\)	315.2.a.a	1	1
		315.2.a.b	1
		315.2.a.c	2
		315.2.a.d	2
		315.2.a.e	2
		315.2.a.f	2
315.2.b	\(\chi_{315}(251, \cdot)\)	315.2.b.a	4	1
		315.2.b.b	4
315.2.d	\(\chi_{315}(64, \cdot)\)	315.2.d.a	2	1
		315.2.d.b	2
		315.2.d.c	2
		315.2.d.d	2
		315.2.d.e	6
315.2.g	\(\chi_{315}(314, \cdot)\)	315.2.g.a	16	1
315.2.i	\(\chi_{315}(106, \cdot)\)	315.2.i.a	2	2
		315.2.i.b	2
		315.2.i.c	8
		315.2.i.d	8
		315.2.i.e	12
		315.2.i.f	16
315.2.j	\(\chi_{315}(46, \cdot)\)	315.2.j.a	2	2
		315.2.j.b	2
		315.2.j.c	4
		315.2.j.d	4
		315.2.j.e	4
		315.2.j.f	6
		315.2.j.g	6
315.2.k	\(\chi_{315}(16, \cdot)\)	315.2.k.a	4	2
		315.2.k.b	24
		315.2.k.c	36
315.2.l	\(\chi_{315}(121, \cdot)\)	315.2.l.a	4	2
		315.2.l.b	24
		315.2.l.c	36
315.2.m	\(\chi_{315}(8, \cdot)\)	315.2.m.a	12	2
		315.2.m.b	12
315.2.p	\(\chi_{315}(118, \cdot)\)	315.2.p.a	4	2
		315.2.p.b	4
		315.2.p.c	4
		315.2.p.d	8
		315.2.p.e	16
315.2.r	\(\chi_{315}(184, \cdot)\)	315.2.r.a	4	2
		315.2.r.b	84
315.2.t	\(\chi_{315}(101, \cdot)\)	315.2.t.a	2	2
		315.2.t.b	30
		315.2.t.c	32
315.2.u	\(\chi_{315}(59, \cdot)\)	315.2.u.a	88	2
315.2.z	\(\chi_{315}(104, \cdot)\)	315.2.z.a	8	2
		315.2.z.b	80
315.2.bb	\(\chi_{315}(89, \cdot)\)	315.2.bb.a	8	2
		315.2.bb.b	24
315.2.be	\(\chi_{315}(236, \cdot)\)	315.2.be.a	2	2
		315.2.be.b	30
		315.2.be.c	32
315.2.bf	\(\chi_{315}(109, \cdot)\)	315.2.bf.a	4	2
		315.2.bf.b	16
		315.2.bf.c	16
315.2.bh	\(\chi_{315}(169, \cdot)\)	315.2.bh.a	4	2
		315.2.bh.b	4
		315.2.bh.c	64
315.2.bj	\(\chi_{315}(26, \cdot)\)	315.2.bj.a	12	2
		315.2.bj.b	12
315.2.bl	\(\chi_{315}(41, \cdot)\)	315.2.bl.a	2	2
		315.2.bl.b	2
		315.2.bl.c	2
		315.2.bl.d	2
		315.2.bl.e	2
		315.2.bl.f	2
		315.2.bl.g	2
		315.2.bl.h	2
		315.2.bl.i	24
		315.2.bl.j	24
315.2.bo	\(\chi_{315}(4, \cdot)\)	315.2.bo.a	4	2
		315.2.bo.b	84
315.2.bq	\(\chi_{315}(164, \cdot)\)	315.2.bq.a	88	2
315.2.bs	\(\chi_{315}(52, \cdot)\)	315.2.bs.a	4	4
		315.2.bs.b	4
		315.2.bs.c	4
		315.2.bs.d	4
		315.2.bs.e	160
315.2.bv	\(\chi_{315}(23, \cdot)\)	315.2.bv.a	176	4
315.2.bx	\(\chi_{315}(2, \cdot)\)	315.2.bx.a	176	4
315.2.bz	\(\chi_{315}(73, \cdot)\)	315.2.bz.a	4	4
		315.2.bz.b	4
		315.2.bz.c	32
		315.2.bz.d	32
315.2.cb	\(\chi_{315}(13, \cdot)\)	315.2.cb.a	176	4
315.2.cc	\(\chi_{315}(92, \cdot)\)	315.2.cc.a	144	4
315.2.ce	\(\chi_{315}(53, \cdot)\)	315.2.ce.a	64	4
315.2.cg	\(\chi_{315}(157, \cdot)\)	315.2.cg.a	4	4
		315.2.cg.b	4
		315.2.cg.c	4
		315.2.cg.d	4
		315.2.cg.e	160

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(315))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(315)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)