Space of modular forms of level 693 and weight 2

• Label: 693.2
• Level: 693
• Weight: 2
• Dimension: 12486
• Nonzero newspaces: 40
• Newform subspaces: 104
• Sturm bound: 69120
• Trace bound: 9

Defining parameters

Level:	\( N \)	=	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 40 \)
Newform subspaces:			\( 104 \)
Sturm bound:			\(69120\)
Trace bound:			\(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(693))\).

	Total	New	Old
Modular forms	18240	13298	4942
Cusp forms	16321	12486	3835
Eisenstein series	1919	812	1107

Trace form

12486 * q - 42 * q^2 - 56 * q^3 - 34 * q^4 - 36 * q^5 - 56 * q^6 - 44 * q^7 - 76 * q^8 - 56 * q^9 - 66 * q^10 - 32 * q^11 - 160 * q^12 - 18 * q^13 - 54 * q^14 - 176 * q^15 - 6 * q^16 - 54 * q^17 - 104 * q^18 - 118 * q^19 - 142 * q^20 - 112 * q^21 - 90 * q^22 - 142 * q^23 - 188 * q^24 - 62 * q^25 - 178 * q^26 - 152 * q^27 - 224 * q^28 - 184 * q^29 - 260 * q^30 - 80 * q^31 - 358 * q^32 - 192 * q^33 - 224 * q^34 - 178 * q^35 - 412 * q^36 - 172 * q^37 - 298 * q^38 - 188 * q^39 - 226 * q^40 - 210 * q^41 - 268 * q^42 - 124 * q^43 - 106 * q^44 - 252 * q^45 - 98 * q^46 - 26 * q^47 - 112 * q^48 - 26 * q^49 - 172 * q^50 - 76 * q^51 - 66 * q^52 - 26 * q^53 - 156 * q^54 - 172 * q^55 - 66 * q^56 - 216 * q^57 - 138 * q^58 - 48 * q^59 - 132 * q^60 - 116 * q^61 - 198 * q^62 + 12 * q^63 - 616 * q^64 - 216 * q^65 - 286 * q^66 - 346 * q^67 - 350 * q^68 - 168 * q^69 - 408 * q^70 - 426 * q^71 - 248 * q^72 - 328 * q^73 - 534 * q^74 - 220 * q^75 - 516 * q^76 - 276 * q^77 - 624 * q^78 - 212 * q^79 - 222 * q^80 - 224 * q^81 - 406 * q^82 - 92 * q^83 - 274 * q^84 - 188 * q^85 - 202 * q^86 - 36 * q^87 - 158 * q^88 - 214 * q^89 + 20 * q^90 - 396 * q^91 - 124 * q^92 - 84 * q^93 - 150 * q^94 + 72 * q^95 + 164 * q^96 - 52 * q^97 - 166 * q^98 + 48 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(693))\)

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
693.2.a	\(\chi_{693}(1, \cdot)\)	693.2.a.a	1	1
		693.2.a.b	1
		693.2.a.c	1
		693.2.a.d	1
		693.2.a.e	2
		693.2.a.f	2
		693.2.a.g	2
		693.2.a.h	2
		693.2.a.i	2
		693.2.a.j	2
		693.2.a.k	2
		693.2.a.l	3
		693.2.a.m	3
693.2.c	\(\chi_{693}(307, \cdot)\)	693.2.c.a	2	1
		693.2.c.b	4
		693.2.c.c	4
		693.2.c.d	12
		693.2.c.e	16
693.2.e	\(\chi_{693}(188, \cdot)\)	693.2.e.a	24	1
693.2.g	\(\chi_{693}(197, \cdot)\)	693.2.g.a	24	1
693.2.i	\(\chi_{693}(100, \cdot)\)	693.2.i.a	2	2
		693.2.i.b	2
		693.2.i.c	2
		693.2.i.d	2
		693.2.i.e	2
		693.2.i.f	4
		693.2.i.g	6
		693.2.i.h	6
		693.2.i.i	8
		693.2.i.j	10
		693.2.i.k	12
		693.2.i.l	12
693.2.j	\(\chi_{693}(232, \cdot)\)	693.2.j.a	2	2
		693.2.j.b	2
		693.2.j.c	2
		693.2.j.d	6
		693.2.j.e	12
		693.2.j.f	18
		693.2.j.g	20
		693.2.j.h	28
		693.2.j.i	30
693.2.k	\(\chi_{693}(67, \cdot)\)	693.2.k.a	6	2
		693.2.k.b	74
		693.2.k.c	80
693.2.l	\(\chi_{693}(529, \cdot)\)	693.2.l.a	6	2
		693.2.l.b	74
		693.2.l.c	80
693.2.m	\(\chi_{693}(64, \cdot)\)	693.2.m.a	4	4
		693.2.m.b	4
		693.2.m.c	4
		693.2.m.d	4
		693.2.m.e	4
		693.2.m.f	8
		693.2.m.g	8
		693.2.m.h	16
		693.2.m.i	16
		693.2.m.j	20
		693.2.m.k	32
693.2.n	\(\chi_{693}(320, \cdot)\)	693.2.n.a	160	2
693.2.p	\(\chi_{693}(241, \cdot)\)	693.2.p.a	184	2
693.2.r	\(\chi_{693}(32, \cdot)\)	693.2.r.a	184	2
693.2.w	\(\chi_{693}(428, \cdot)\)	693.2.w.a	144	2
693.2.x	\(\chi_{693}(296, \cdot)\)	693.2.x.a	64	2
693.2.ba	\(\chi_{693}(439, \cdot)\)	693.2.ba.a	184	2
693.2.bd	\(\chi_{693}(419, \cdot)\)	693.2.bd.a	160	2
693.2.be	\(\chi_{693}(89, \cdot)\)	693.2.be.a	56	2
693.2.bg	\(\chi_{693}(10, \cdot)\)	693.2.bg.a	12	2
		693.2.bg.b	32
		693.2.bg.c	32
693.2.bj	\(\chi_{693}(76, \cdot)\)	693.2.bj.a	184	2
693.2.bk	\(\chi_{693}(122, \cdot)\)	693.2.bk.a	160	2
693.2.bn	\(\chi_{693}(263, \cdot)\)	693.2.bn.a	184	2
693.2.bq	\(\chi_{693}(8, \cdot)\)	693.2.bq.a	96	4
693.2.bs	\(\chi_{693}(125, \cdot)\)	693.2.bs.a	32	4
		693.2.bs.b	96
693.2.bu	\(\chi_{693}(118, \cdot)\)	693.2.bu.a	8	4
		693.2.bu.b	16
		693.2.bu.c	16
		693.2.bu.d	16
		693.2.bu.e	32
		693.2.bu.f	64
693.2.bw	\(\chi_{693}(25, \cdot)\)	693.2.bw.a	736	8
693.2.bx	\(\chi_{693}(4, \cdot)\)	693.2.bx.a	736	8
693.2.by	\(\chi_{693}(37, \cdot)\)	693.2.by.a	8	8
		693.2.by.b	40
		693.2.by.c	64
		693.2.by.d	64
		693.2.by.e	128
693.2.bz	\(\chi_{693}(148, \cdot)\)	693.2.bz.a	288	8
		693.2.bz.b	288
693.2.cb	\(\chi_{693}(74, \cdot)\)	693.2.cb.a	736	8
693.2.ce	\(\chi_{693}(47, \cdot)\)	693.2.ce.a	736	8
693.2.cg	\(\chi_{693}(19, \cdot)\)	693.2.cg.a	48	8
		693.2.cg.b	128
		693.2.cg.c	128
693.2.ch	\(\chi_{693}(13, \cdot)\)	693.2.ch.a	736	8
693.2.cj	\(\chi_{693}(20, \cdot)\)	693.2.cj.a	736	8
693.2.cm	\(\chi_{693}(26, \cdot)\)	693.2.cm.a	256	8
693.2.co	\(\chi_{693}(61, \cdot)\)	693.2.co.a	736	8
693.2.cq	\(\chi_{693}(29, \cdot)\)	693.2.cq.a	576	8
693.2.ct	\(\chi_{693}(107, \cdot)\)	693.2.ct.a	256	8
693.2.cx	\(\chi_{693}(2, \cdot)\)	693.2.cx.a	736	8
693.2.cz	\(\chi_{693}(40, \cdot)\)	693.2.cz.a	736	8
693.2.db	\(\chi_{693}(5, \cdot)\)	693.2.db.a	736	8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(693))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(693)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(231))\)\(^{\oplus 2}\)