Space of modular forms of level 690 and weight 2

• Label: 690.2
• Level: 690
• Weight: 2
• Dimension: 2889
• Nonzero newspaces: 12
• Newform subspaces: 45
• Sturm bound: 50688
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 12 \)
Newform subspaces:			\( 45 \)
Sturm bound:			\(50688\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(690))\).

	Total	New	Old
Modular forms	13376	2889	10487
Cusp forms	11969	2889	9080
Eisenstein series	1407	0	1407

Trace form

2889 * q + q^2 + 5 * q^3 + q^4 + 9 * q^5 + 5 * q^6 + 8 * q^7 + q^8 + q^9 - 7 * q^10 - 20 * q^11 - 11 * q^12 - 18 * q^13 - 24 * q^14 + 3 * q^15 + q^16 + 58 * q^17 + 73 * q^18 + 76 * q^19 + 37 * q^20 + 124 * q^21 + 84 * q^22 + 153 * q^23 + 5 * q^24 + 105 * q^25 + 102 * q^26 + 137 * q^27 + 96 * q^28 + 70 * q^29 + 73 * q^30 + 88 * q^31 + q^32 + 40 * q^33 - 14 * q^34 + 20 * q^35 + q^36 + 86 * q^37 - 28 * q^38 + 46 * q^39 - 23 * q^40 + 50 * q^41 - 24 * q^42 + 92 * q^43 - 4 * q^44 + 9 * q^45 - 47 * q^46 + 128 * q^47 - 11 * q^48 + 177 * q^49 - 15 * q^50 + 82 * q^51 - 18 * q^52 + 46 * q^53 - 39 * q^54 + 68 * q^55 - 8 * q^56 - 24 * q^57 - 2 * q^58 + 68 * q^59 - 9 * q^60 - 2 * q^61 - 16 * q^62 - 212 * q^63 + q^64 + 14 * q^65 - 212 * q^66 - 28 * q^67 - 30 * q^68 - 235 * q^69 + 8 * q^70 - 120 * q^71 - 71 * q^72 - 38 * q^73 - 42 * q^74 - 175 * q^75 - 44 * q^76 - 96 * q^77 - 142 * q^78 - 8 * q^79 + 9 * q^80 - 151 * q^81 - 22 * q^82 - 20 * q^83 - 52 * q^84 + 14 * q^85 - 68 * q^86 + 74 * q^87 - 4 * q^88 - 78 * q^89 - 23 * q^90 + 32 * q^91 - 23 * q^92 - 64 * q^93 - 16 * q^94 - 72 * q^95 - 11 * q^96 - 302 * q^97 - 215 * q^98 + 40 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(690))\)

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
690.2.a	\(\chi_{690}(1, \cdot)\)	690.2.a.a	1	1
		690.2.a.b	1
		690.2.a.c	1
		690.2.a.d	1
		690.2.a.e	1
		690.2.a.f	1
		690.2.a.g	1
		690.2.a.h	1
		690.2.a.i	1
		690.2.a.j	1
		690.2.a.k	1
		690.2.a.l	2
690.2.d	\(\chi_{690}(139, \cdot)\)	690.2.d.a	4	1
		690.2.d.b	4
		690.2.d.c	6
		690.2.d.d	6
690.2.e	\(\chi_{690}(551, \cdot)\)	690.2.e.a	16	1
		690.2.e.b	16
690.2.h	\(\chi_{690}(689, \cdot)\)	690.2.h.a	24	1
		690.2.h.b	24
690.2.i	\(\chi_{690}(47, \cdot)\)	690.2.i.a	4	2
		690.2.i.b	4
		690.2.i.c	8
		690.2.i.d	8
		690.2.i.e	32
		690.2.i.f	32
690.2.j	\(\chi_{690}(367, \cdot)\)	690.2.j.a	24	2
		690.2.j.b	24
690.2.m	\(\chi_{690}(31, \cdot)\)	690.2.m.a	10	10
		690.2.m.b	10
		690.2.m.c	20
		690.2.m.d	20
		690.2.m.e	20
		690.2.m.f	20
		690.2.m.g	30
		690.2.m.h	30
690.2.n	\(\chi_{690}(89, \cdot)\)	690.2.n.a	240	10
		690.2.n.b	240
690.2.q	\(\chi_{690}(11, \cdot)\)	690.2.q.a	160	10
		690.2.q.b	160
690.2.r	\(\chi_{690}(49, \cdot)\)	690.2.r.a	120	10
		690.2.r.b	120
690.2.w	\(\chi_{690}(7, \cdot)\)	690.2.w.a	240	20
		690.2.w.b	240
690.2.x	\(\chi_{690}(77, \cdot)\)	690.2.x.a	960	20

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(690))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(690)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(230))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(345))\)\(^{\oplus 2}\)