Space of modular forms of level 338 and weight 10

• Label: 338.10
• Level: 338
• Weight: 10
• Dimension: 10522
• Nonzero newspaces: 8
• Sturm bound: 70980
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	=	\( 10 \)
Nonzero newspaces:			\( 8 \)
Sturm bound:			\(70980\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(338))\).

	Total	New	Old
Modular forms	32169	10522	21647
Cusp forms	31713	10522	21191
Eisenstein series	456	0	456

Trace form

10522 * q + 16 * q^2 - 156 * q^3 + 256 * q^4 + 870 * q^5 - 2496 * q^6 - 9160 * q^7 - 20480 * q^8 + 4653 * q^9 + 320640 * q^10 - 129828 * q^11 - 205824 * q^12 - 216864 * q^13 + 712448 * q^14 + 2251512 * q^15 + 65536 * q^16 - 4213812 * q^17 - 3074832 * q^18 + 210524 * q^19 + 2064384 * q^20 - 1743408 * q^21 - 898368 * q^22 + 2362752 * q^23 - 638976 * q^24 - 12914975 * q^25 + 9023328 * q^27 - 2344960 * q^28 + 71700288 * q^29 - 28714368 * q^30 - 77363752 * q^31 + 1048576 * q^32 + 3728832 * q^33 + 64321056 * q^34 + 187172040 * q^35 - 4018944 * q^36 - 141572416 * q^37 - 145083712 * q^38 - 102830052 * q^39 + 3563520 * q^40 + 154771572 * q^41 + 226102656 * q^42 + 97810652 * q^43 + 80317440 * q^44 - 55503840 * q^45 - 215430528 * q^46 - 361503480 * q^47 - 10223616 * q^48 + 92204913 * q^49 + 267661456 * q^50 + 726481560 * q^51 - 66558720 * q^52 - 129814146 * q^53 - 714097152 * q^54 - 453799800 * q^55 - 103088128 * q^56 + 552271176 * q^57 + 521233536 * q^58 + 950011668 * q^59 + 417933312 * q^60 - 460066960 * q^61 - 462904960 * q^62 - 813493368 * q^63 - 83886080 * q^64 - 904349247 * q^65 - 674992896 * q^66 - 1804260820 * q^67 + 14932992 * q^68 + 4013955768 * q^69 + 1815247488 * q^70 + 3213554760 * q^71 + 443535360 * q^72 - 704773078 * q^73 - 974495872 * q^74 - 3094985340 * q^75 + 53894144 * q^76 - 5879467584 * q^77 - 282276288 * q^78 - 1849791232 * q^79 + 528482304 * q^80 + 571559769 * q^81 + 673058880 * q^82 + 8927768268 * q^83 + 2959638528 * q^84 + 8042361078 * q^85 + 1549982144 * q^86 - 2790832200 * q^87 - 229982208 * q^88 - 11952040710 * q^89 - 7808430240 * q^90 - 4209919920 * q^91 - 1986791424 * q^92 + 5953378536 * q^93 + 2652855168 * q^94 + 9876985032 * q^95 - 163577856 * q^96 + 7710514826 * q^97 + 10404891024 * q^98 + 20039913900 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(338))\)

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
338.10.a	\(\chi_{338}(1, \cdot)\)	338.10.a.a	1	1
		338.10.a.b	1
		338.10.a.c	1
		338.10.a.d	1
		338.10.a.e	3
		338.10.a.f	3
		338.10.a.g	5
		338.10.a.h	5
		338.10.a.i	5
		338.10.a.j	5
		338.10.a.k	6
		338.10.a.l	6
		338.10.a.m	10
		338.10.a.n	10
		338.10.a.o	12
		338.10.a.p	12
		338.10.a.q	15
		338.10.a.r	15
338.10.b	\(\chi_{338}(337, \cdot)\)	n/a	116	1
338.10.c	\(\chi_{338}(191, \cdot)\)	n/a	230	2
338.10.e	\(\chi_{338}(23, \cdot)\)	n/a	232	2
338.10.g	\(\chi_{338}(27, \cdot)\)	n/a	1644	12
338.10.h	\(\chi_{338}(25, \cdot)\)	n/a	1632	12
338.10.i	\(\chi_{338}(3, \cdot)\)	n/a	3288	24
338.10.k	\(\chi_{338}(17, \cdot)\)	n/a	3264	24

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

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(338))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(338)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 2}\)