Space of modular forms of level 392 and weight 2

• Label: 392.2
• Level: 392
• Weight: 2
• Dimension: 2442
• Nonzero newspaces: 12
• Newform subspaces: 52
• Sturm bound: 18816
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 12 \)
Newform subspaces:			\( 52 \)
Sturm bound:			\(18816\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(392))\).

	Total	New	Old
Modular forms	5064	2636	2428
Cusp forms	4345	2442	1903
Eisenstein series	719	194	525

Trace form

2442 * q - 30 * q^2 - 30 * q^3 - 30 * q^4 - 30 * q^6 - 36 * q^7 - 54 * q^8 - 48 * q^9 - 30 * q^10 - 18 * q^11 - 30 * q^12 + 12 * q^13 - 36 * q^14 - 30 * q^15 - 30 * q^16 - 48 * q^17 - 66 * q^18 - 30 * q^19 - 66 * q^20 - 114 * q^22 - 66 * q^23 - 102 * q^24 - 96 * q^25 - 90 * q^26 - 90 * q^27 - 84 * q^28 - 126 * q^30 - 66 * q^31 - 90 * q^32 - 72 * q^33 - 102 * q^34 - 36 * q^35 - 114 * q^36 + 12 * q^37 - 66 * q^38 - 42 * q^39 - 90 * q^40 - 12 * q^41 - 54 * q^43 + 18 * q^44 + 12 * q^45 + 30 * q^46 - 66 * q^47 + 90 * q^48 - 78 * q^49 - 12 * q^50 - 114 * q^51 + 78 * q^52 - 12 * q^53 + 114 * q^54 - 90 * q^55 + 6 * q^56 - 156 * q^57 + 30 * q^58 - 102 * q^59 + 102 * q^60 + 78 * q^62 - 72 * q^63 + 30 * q^64 - 60 * q^65 + 18 * q^66 - 66 * q^67 - 30 * q^68 - 24 * q^69 - 60 * q^70 - 126 * q^71 - 66 * q^72 - 24 * q^73 - 198 * q^74 - 150 * q^75 - 174 * q^76 - 18 * q^77 - 222 * q^78 - 66 * q^79 - 258 * q^80 - 192 * q^81 - 174 * q^82 - 186 * q^83 - 144 * q^84 - 84 * q^85 - 246 * q^86 - 282 * q^87 - 174 * q^88 - 156 * q^89 - 186 * q^90 - 108 * q^91 - 210 * q^92 - 264 * q^93 - 102 * q^94 - 186 * q^95 - 78 * q^96 - 240 * q^97 - 60 * q^98 - 120 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.2.a	\(\chi_{392}(1, \cdot)\)	392.2.a.a	1	1
		392.2.a.b	1
		392.2.a.c	1
		392.2.a.d	1
		392.2.a.e	1
		392.2.a.f	1
		392.2.a.g	2
		392.2.a.h	2
392.2.b	\(\chi_{392}(197, \cdot)\)	392.2.b.a	2	1
		392.2.b.b	2
		392.2.b.c	4
		392.2.b.d	4
		392.2.b.e	6
		392.2.b.f	6
		392.2.b.g	12
392.2.e	\(\chi_{392}(195, \cdot)\)	392.2.e.a	4	1
		392.2.e.b	4
		392.2.e.c	8
		392.2.e.d	8
		392.2.e.e	12
392.2.f	\(\chi_{392}(391, \cdot)\)	None	0	1
392.2.i	\(\chi_{392}(177, \cdot)\)	392.2.i.a	2	2
		392.2.i.b	2
		392.2.i.c	2
		392.2.i.d	2
		392.2.i.e	2
		392.2.i.f	2
		392.2.i.g	4
		392.2.i.h	4
392.2.l	\(\chi_{392}(31, \cdot)\)	None	0	2
392.2.m	\(\chi_{392}(19, \cdot)\)	392.2.m.a	4	2
		392.2.m.b	8
		392.2.m.c	8
		392.2.m.d	8
		392.2.m.e	8
		392.2.m.f	8
		392.2.m.g	12
		392.2.m.h	16
392.2.p	\(\chi_{392}(165, \cdot)\)	392.2.p.a	4	2
		392.2.p.b	4
		392.2.p.c	4
		392.2.p.d	8
		392.2.p.e	8
		392.2.p.f	8
		392.2.p.g	12
		392.2.p.h	24
392.2.q	\(\chi_{392}(57, \cdot)\)	392.2.q.a	42	6
		392.2.q.b	42
392.2.t	\(\chi_{392}(55, \cdot)\)	None	0	6
392.2.u	\(\chi_{392}(27, \cdot)\)	392.2.u.a	324	6
392.2.x	\(\chi_{392}(29, \cdot)\)	392.2.x.a	324	6
392.2.y	\(\chi_{392}(9, \cdot)\)	392.2.y.a	84	12
		392.2.y.b	84
392.2.z	\(\chi_{392}(37, \cdot)\)	392.2.z.a	648	12
392.2.bc	\(\chi_{392}(3, \cdot)\)	392.2.bc.a	648	12
392.2.bd	\(\chi_{392}(47, \cdot)\)	None	0	12

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(392)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)