Space of modular forms of level 338 and weight 8

• Label: 338.8
• Level: 338
• Weight: 8
• Dimension: 8184
• Nonzero newspaces: 8
• Sturm bound: 56784
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 8 \)
Sturm bound:			\(56784\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(338))\).

	Total	New	Old
Modular forms	25071	8184	16887
Cusp forms	24615	8184	16431
Eisenstein series	456	0	456

Trace form

8184 * q - 8 * q^2 + 12 * q^3 + 64 * q^4 - 210 * q^5 - 96 * q^6 - 5040 * q^7 + 2560 * q^8 - 25371 * q^9 - 18240 * q^10 + 16908 * q^11 + 14592 * q^12 + 29928 * q^13 - 29824 * q^14 - 167112 * q^15 - 28672 * q^16 - 128064 * q^17 + 191304 * q^18 + 445860 * q^19 - 2304 * q^20 - 862584 * q^21 - 8736 * q^22 + 98640 * q^23 - 6144 * q^24 + 434725 * q^25 + 257616 * q^27 - 322560 * q^28 + 904956 * q^29 + 1635648 * q^30 - 239184 * q^31 - 32768 * q^32 - 1350696 * q^33 - 1665552 * q^34 - 3840 * q^35 + 750912 * q^36 + 855492 * q^37 + 4270304 * q^38 + 1629420 * q^39 + 107520 * q^40 + 98976 * q^41 - 3380160 * q^42 - 5444372 * q^43 - 4467456 * q^44 - 12408780 * q^45 - 2242752 * q^46 + 2146704 * q^47 + 49152 * q^48 + 11360153 * q^49 + 3653752 * q^50 + 1886088 * q^51 - 1693504 * q^52 + 11017926 * q^53 + 8172288 * q^54 - 5227680 * q^55 - 3973120 * q^56 - 26435616 * q^57 - 10105152 * q^58 - 12460428 * q^59 - 2892288 * q^60 + 4560236 * q^61 + 9049280 * q^62 + 55110528 * q^63 + 1835008 * q^64 + 26353695 * q^65 + 17959296 * q^66 - 5371308 * q^67 + 662016 * q^68 - 55472352 * q^69 - 20501952 * q^70 - 50589720 * q^71 - 18393600 * q^72 - 46543926 * q^73 - 23859904 * q^74 + 65186700 * q^75 + 28535040 * q^76 + 133309872 * q^77 + 35465760 * q^78 + 43034672 * q^79 - 147456 * q^80 - 106829775 * q^81 - 79929120 * q^82 - 86720004 * q^83 - 23508480 * q^84 - 117121878 * q^85 + 6583136 * q^86 - 11452200 * q^87 - 559104 * q^88 + 81535722 * q^89 + 84047760 * q^90 + 59782244 * q^91 + 32374272 * q^92 + 93959472 * q^93 + 8044992 * q^94 - 52385352 * q^95 - 393216 * q^96 - 226231734 * q^97 - 64300872 * q^98 - 72270756 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(338))\)

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
338.8.a	\(\chi_{338}(1, \cdot)\)	338.8.a.a	1	1
		338.8.a.b	1
		338.8.a.c	1
		338.8.a.d	1
		338.8.a.e	2
		338.8.a.f	2
		338.8.a.g	3
		338.8.a.h	3
		338.8.a.i	4
		338.8.a.j	4
		338.8.a.k	5
		338.8.a.l	5
		338.8.a.m	8
		338.8.a.n	8
		338.8.a.o	9
		338.8.a.p	9
		338.8.a.q	12
		338.8.a.r	12
338.8.b	\(\chi_{338}(337, \cdot)\)	338.8.b.a	2	1
		338.8.b.b	2
		338.8.b.c	2
		338.8.b.d	2
		338.8.b.e	4
		338.8.b.f	4
		338.8.b.g	6
		338.8.b.h	8
		338.8.b.i	16
		338.8.b.j	18
		338.8.b.k	24
338.8.c	\(\chi_{338}(191, \cdot)\)	n/a	182	2
338.8.e	\(\chi_{338}(23, \cdot)\)	n/a	180	2
338.8.g	\(\chi_{338}(27, \cdot)\)	n/a	1284	12
338.8.h	\(\chi_{338}(25, \cdot)\)	n/a	1296	12
338.8.i	\(\chi_{338}(3, \cdot)\)	n/a	2520	24
338.8.k	\(\chi_{338}(17, \cdot)\)	n/a	2544	24

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(338))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(338)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(338))\)\(^{\oplus 1}\)