Space of modular forms of level 370 and weight 2

• Label: 370.2
• Level: 370
• Weight: 2
• Dimension: 1255
• Nonzero newspaces: 18
• Newform subspaces: 70
• Sturm bound: 16416
• Trace bound: 7

Defining parameters

Level:	\( N \)	=	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 18 \)
Newform subspaces:			\( 70 \)
Sturm bound:			\(16416\)
Trace bound:			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(370))\).

	Total	New	Old
Modular forms	4392	1255	3137
Cusp forms	3817	1255	2562
Eisenstein series	575	0	575

Trace form

1255 * q + q^2 + 4 * q^3 + q^4 + q^5 + 4 * q^6 + 8 * q^7 + q^8 + 13 * q^9 + q^10 + 12 * q^11 + 4 * q^12 + 14 * q^13 + 8 * q^14 + 4 * q^15 + q^16 + 18 * q^17 + 13 * q^18 + 20 * q^19 + q^20 + 32 * q^21 + 12 * q^22 + 24 * q^23 + 4 * q^24 + q^25 - 4 * q^26 - 56 * q^27 - 16 * q^28 - 42 * q^29 - 68 * q^30 - 184 * q^31 + q^32 - 96 * q^33 - 126 * q^34 - 100 * q^35 - 77 * q^36 - 131 * q^37 - 52 * q^38 - 112 * q^39 - 44 * q^40 - 174 * q^41 - 112 * q^42 - 100 * q^43 + 12 * q^44 - 95 * q^45 - 120 * q^46 - 24 * q^47 - 20 * q^48 - 39 * q^49 - 8 * q^50 + 72 * q^51 + 14 * q^52 + 54 * q^53 + 40 * q^54 + 12 * q^55 + 8 * q^56 + 80 * q^57 + 30 * q^58 - 12 * q^59 + 4 * q^60 - 28 * q^61 + 32 * q^62 - 112 * q^63 + q^64 - 67 * q^65 + 48 * q^66 - 4 * q^67 + 18 * q^68 - 192 * q^69 + 8 * q^70 - 72 * q^71 + 13 * q^72 - 70 * q^73 + 37 * q^74 - 176 * q^75 + 20 * q^76 - 48 * q^77 + 56 * q^78 - 64 * q^79 + q^80 - 167 * q^81 + 42 * q^82 + 12 * q^83 + 32 * q^84 - 63 * q^85 + 44 * q^86 - 96 * q^87 + 12 * q^88 + 13 * q^90 + 16 * q^91 - 48 * q^92 - 88 * q^93 - 96 * q^94 - 124 * q^95 + 4 * q^96 - 190 * q^97 - 231 * q^98 - 204 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(370))\)

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
370.2.a	\(\chi_{370}(1, \cdot)\)	370.2.a.a	1	1
		370.2.a.b	1
		370.2.a.c	1
		370.2.a.d	1
		370.2.a.e	2
		370.2.a.f	2
		370.2.a.g	3
370.2.b	\(\chi_{370}(149, \cdot)\)	370.2.b.a	2	1
		370.2.b.b	2
		370.2.b.c	4
		370.2.b.d	10
370.2.c	\(\chi_{370}(369, \cdot)\)	370.2.c.a	10	1
		370.2.c.b	10
370.2.d	\(\chi_{370}(221, \cdot)\)	370.2.d.a	2	1
		370.2.d.b	2
		370.2.d.c	6
370.2.e	\(\chi_{370}(121, \cdot)\)	370.2.e.a	2	2
		370.2.e.b	2
		370.2.e.c	2
		370.2.e.d	4
		370.2.e.e	4
		370.2.e.f	6
370.2.g	\(\chi_{370}(43, \cdot)\)	370.2.g.a	2	2
		370.2.g.b	2
		370.2.g.c	4
		370.2.g.d	10
		370.2.g.e	20
370.2.h	\(\chi_{370}(117, \cdot)\)	370.2.h.a	2	2
		370.2.h.b	2
		370.2.h.c	4
		370.2.h.d	10
		370.2.h.e	20
370.2.l	\(\chi_{370}(11, \cdot)\)	370.2.l.a	4	2
		370.2.l.b	4
		370.2.l.c	12
370.2.m	\(\chi_{370}(159, \cdot)\)	370.2.m.a	4	2
		370.2.m.b	4
		370.2.m.c	16
		370.2.m.d	16
370.2.n	\(\chi_{370}(269, \cdot)\)	370.2.n.a	4	2
		370.2.n.b	4
		370.2.n.c	4
		370.2.n.d	4
		370.2.n.e	8
		370.2.n.f	12
370.2.o	\(\chi_{370}(71, \cdot)\)	370.2.o.a	18	6
		370.2.o.b	18
		370.2.o.c	24
		370.2.o.d	24
370.2.q	\(\chi_{370}(97, \cdot)\)	370.2.q.a	4	4
		370.2.q.b	4
		370.2.q.c	8
		370.2.q.d	12
		370.2.q.e	16
		370.2.q.f	32
370.2.r	\(\chi_{370}(23, \cdot)\)	370.2.r.a	4	4
		370.2.r.b	4
		370.2.r.c	8
		370.2.r.d	12
		370.2.r.e	16
		370.2.r.f	32
370.2.v	\(\chi_{370}(99, \cdot)\)	370.2.v.a	60	6
		370.2.v.b	60
370.2.w	\(\chi_{370}(21, \cdot)\)	370.2.w.a	36	6
		370.2.w.b	48
370.2.x	\(\chi_{370}(9, \cdot)\)	370.2.x.a	108	6
370.2.ba	\(\chi_{370}(17, \cdot)\)	370.2.ba.a	108	12
		370.2.ba.b	120
370.2.bd	\(\chi_{370}(13, \cdot)\)	370.2.bd.a	108	12
		370.2.bd.b	120

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(370))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(370)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(185))\)\(^{\oplus 2}\)