Space of modular forms of level 210 and weight 2

• Label: 210.2
• Level: 210
• Weight: 2
• Dimension: 249
• Nonzero newspaces: 12
• Newform subspaces: 32
• Sturm bound: 4608
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 12 \)
Newform subspaces:			\( 32 \)
Sturm bound:			\(4608\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(210))\).

	Total	New	Old
Modular forms	1344	249	1095
Cusp forms	961	249	712
Eisenstein series	383	0	383

Trace form

249 * q + q^2 + 9 * q^3 + 9 * q^4 + 21 * q^5 + 17 * q^6 + 33 * q^7 + q^8 + 17 * q^9 + 5 * q^10 + 28 * q^11 - 7 * q^12 + 14 * q^13 - 15 * q^14 - 19 * q^15 + q^16 - 6 * q^17 - 39 * q^18 + 20 * q^19 - 7 * q^20 - 27 * q^21 - 4 * q^22 - 7 * q^24 - 7 * q^25 - 10 * q^26 - 63 * q^27 - 23 * q^28 - 66 * q^29 - 31 * q^30 - 16 * q^31 + q^32 - 76 * q^33 - 62 * q^34 - 111 * q^35 - 31 * q^36 - 122 * q^37 - 52 * q^38 - 106 * q^39 - 11 * q^40 - 38 * q^41 - 27 * q^42 - 20 * q^43 - 4 * q^44 - 51 * q^45 - 24 * q^46 - 24 * q^47 - 7 * q^48 + 25 * q^49 + 33 * q^50 - 30 * q^51 - 34 * q^52 + 30 * q^53 - 31 * q^54 - 32 * q^55 + 17 * q^56 + 28 * q^57 - 26 * q^58 - 20 * q^59 + q^60 - 66 * q^61 + 32 * q^62 - 55 * q^63 + 9 * q^64 - 46 * q^65 - 36 * q^66 - 172 * q^67 - 6 * q^68 - 56 * q^69 + q^70 - 72 * q^71 + 41 * q^72 - 126 * q^73 + 30 * q^74 - 111 * q^75 - 12 * q^76 - 12 * q^77 + 38 * q^78 - 48 * q^79 + 21 * q^80 + 49 * q^81 + 122 * q^82 + 36 * q^83 + 45 * q^84 + 66 * q^85 + 52 * q^86 + 110 * q^87 + 20 * q^88 + 26 * q^89 + 85 * q^90 + 110 * q^91 + 24 * q^92 + 160 * q^93 + 128 * q^94 + 80 * q^95 + q^96 + 74 * q^97 + q^98 + 188 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(210))\)

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
210.2.a	\(\chi_{210}(1, \cdot)\)	210.2.a.a	1	1
		210.2.a.b	1
		210.2.a.c	1
		210.2.a.d	1
		210.2.a.e	1
210.2.b	\(\chi_{210}(41, \cdot)\)	210.2.b.a	4	1
		210.2.b.b	4
210.2.d	\(\chi_{210}(209, \cdot)\)	210.2.d.a	8	1
		210.2.d.b	8
210.2.g	\(\chi_{210}(169, \cdot)\)	210.2.g.a	2	1
		210.2.g.b	2
210.2.i	\(\chi_{210}(121, \cdot)\)	210.2.i.a	2	2
		210.2.i.b	2
		210.2.i.c	2
		210.2.i.d	2
210.2.j	\(\chi_{210}(113, \cdot)\)	210.2.j.a	12	2
		210.2.j.b	12
210.2.m	\(\chi_{210}(13, \cdot)\)	210.2.m.a	8	2
		210.2.m.b	8
210.2.n	\(\chi_{210}(79, \cdot)\)	210.2.n.a	4	2
		210.2.n.b	12
210.2.r	\(\chi_{210}(101, \cdot)\)	210.2.r.a	12	2
		210.2.r.b	12
210.2.t	\(\chi_{210}(59, \cdot)\)	210.2.t.a	4	2
		210.2.t.b	4
		210.2.t.c	4
		210.2.t.d	4
		210.2.t.e	8
		210.2.t.f	8
210.2.u	\(\chi_{210}(73, \cdot)\)	210.2.u.a	16	4
		210.2.u.b	16
210.2.x	\(\chi_{210}(23, \cdot)\)	210.2.x.a	64	4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(210))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(210)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)