Space of modular forms of level 338 and weight 2

• Label: 338.2
• Level: 338
• Weight: 2
• Dimension: 1170
• Nonzero newspaces: 8
• Newform subspaces: 32
• Sturm bound: 14196
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 8 \)
Newform subspaces:			\( 32 \)
Sturm bound:			\(14196\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(338))\).

	Total	New	Old
Modular forms	3777	1170	2607
Cusp forms	3322	1170	2152
Eisenstein series	455	0	455

Trace form

1170 * q + q^2 + 4 * q^3 + q^4 + 6 * q^5 + 4 * q^6 - 5 * q^8 - 19 * q^9 - 24 * q^10 - 12 * q^11 - 4 * q^12 - 24 * q^13 - 16 * q^14 - 24 * q^15 - 7 * q^16 - 12 * q^17 - 17 * q^18 - 36 * q^19 - 24 * q^21 + 12 * q^22 + 4 * q^24 + q^25 - 32 * q^27 - 48 * q^29 - 24 * q^30 - 48 * q^31 + q^32 - 72 * q^33 - 30 * q^34 - 96 * q^35 - 35 * q^36 - 24 * q^37 - 52 * q^38 - 52 * q^39 + 6 * q^40 - 60 * q^41 - 40 * q^42 - 44 * q^43 - 36 * q^44 - 72 * q^45 - 24 * q^46 - 48 * q^47 + 4 * q^48 - 7 * q^49 - 23 * q^50 - 72 * q^51 - 7 * q^52 - 18 * q^53 - 32 * q^54 - 48 * q^55 - 16 * q^56 - 48 * q^57 - 24 * q^58 - 36 * q^59 - 24 * q^60 - 16 * q^61 - 40 * q^62 - 144 * q^63 - 5 * q^64 - 87 * q^65 - 48 * q^66 - 84 * q^67 + 12 * q^68 - 144 * q^69 - 24 * q^70 - 72 * q^71 - 35 * q^72 - 102 * q^73 - 40 * q^74 - 172 * q^75 - 36 * q^76 - 144 * q^77 - 60 * q^78 - 112 * q^79 - 127 * q^81 - 60 * q^82 - 108 * q^83 - 24 * q^84 - 138 * q^85 - 4 * q^86 - 120 * q^87 + 12 * q^88 - 102 * q^89 - 42 * q^90 - 76 * q^91 - 24 * q^92 - 96 * q^93 - 72 * q^94 - 24 * q^95 + 4 * q^96 - 102 * q^97 - 39 * q^98 - 60 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(338))\)

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
338.2.a	\(\chi_{338}(1, \cdot)\)	338.2.a.a	1	1
		338.2.a.b	1
		338.2.a.c	1
		338.2.a.d	1
		338.2.a.e	1
		338.2.a.f	1
		338.2.a.g	3
		338.2.a.h	3
338.2.b	\(\chi_{338}(337, \cdot)\)	338.2.b.a	2	1
		338.2.b.b	2
		338.2.b.c	2
		338.2.b.d	6
338.2.c	\(\chi_{338}(191, \cdot)\)	338.2.c.a	2	2
		338.2.c.b	2
		338.2.c.c	2
		338.2.c.d	2
		338.2.c.e	2
		338.2.c.f	2
		338.2.c.g	2
		338.2.c.h	6
		338.2.c.i	6
338.2.e	\(\chi_{338}(23, \cdot)\)	338.2.e.a	4	2
		338.2.e.b	4
		338.2.e.c	4
		338.2.e.d	4
		338.2.e.e	12
338.2.g	\(\chi_{338}(27, \cdot)\)	338.2.g.a	96	12
		338.2.g.b	108
338.2.h	\(\chi_{338}(25, \cdot)\)	338.2.h.a	192	12
338.2.i	\(\chi_{338}(3, \cdot)\)	338.2.i.a	168	24
		338.2.i.b	192
338.2.k	\(\chi_{338}(17, \cdot)\)	338.2.k.a	336	24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(338)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 2}\)