Space of modular forms of level 169 and weight 4

• Label: 169.4
• Level: 169
• Weight: 4
• Dimension: 3405
• Nonzero newspaces: 8
• Newform subspaces: 43
• Sturm bound: 9464
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 169 = 13^{2} \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 8 \)
Newform subspaces:			\( 43 \)
Sturm bound:			\(9464\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(169))\).

	Total	New	Old
Modular forms	3663	3610	53
Cusp forms	3435	3405	30
Eisenstein series	228	205	23

Trace form

3405 * q - 66 * q^2 - 66 * q^3 - 66 * q^4 - 66 * q^5 - 66 * q^6 + 6 * q^7 + 78 * q^8 - 66 * q^9 - 246 * q^10 - 186 * q^11 - 534 * q^12 - 216 * q^13 - 366 * q^14 - 210 * q^15 - 66 * q^16 + 312 * q^17 + 1590 * q^18 + 822 * q^19 + 966 * q^20 - 66 * q^21 - 846 * q^22 - 522 * q^23 - 2178 * q^24 - 804 * q^25 - 822 * q^26 - 1782 * q^27 + 42 * q^28 + 792 * q^29 + 1086 * q^30 + 558 * q^31 + 1242 * q^32 + 1134 * q^33 + 1326 * q^34 - 66 * q^35 + 246 * q^36 - 432 * q^37 - 2790 * q^38 - 144 * q^39 + 306 * q^40 + 876 * q^41 + 1194 * q^42 + 774 * q^43 - 534 * q^44 - 636 * q^45 - 1722 * q^46 - 330 * q^47 + 702 * q^48 - 2010 * q^49 + 126 * q^50 + 714 * q^51 + 258 * q^52 - 2022 * q^53 - 1518 * q^54 - 2874 * q^55 - 2070 * q^56 - 2394 * q^57 - 2118 * q^58 - 1050 * q^59 - 2382 * q^60 + 2196 * q^61 + 5166 * q^62 + 6294 * q^63 + 9306 * q^64 + 3987 * q^65 + 8154 * q^66 + 1758 * q^67 - 2718 * q^68 - 4122 * q^69 - 4038 * q^70 - 3954 * q^71 - 5958 * q^72 - 4386 * q^73 - 5214 * q^74 - 1302 * q^75 + 3606 * q^76 - 822 * q^77 - 3018 * q^78 + 1794 * q^79 - 6846 * q^80 - 990 * q^81 - 7182 * q^82 - 6930 * q^83 - 4614 * q^84 - 2100 * q^85 + 5538 * q^86 + 7734 * q^87 + 810 * q^88 + 7374 * q^89 + 13794 * q^90 + 2652 * q^91 + 9474 * q^92 + 8310 * q^93 + 3366 * q^94 + 3846 * q^95 - 3174 * q^96 - 3690 * q^97 + 5382 * q^98 - 10458 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(169))\)

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
169.4.a	\(\chi_{169}(1, \cdot)\)	169.4.a.a	1	1
		169.4.a.b	1
		169.4.a.c	1
		169.4.a.d	1
		169.4.a.e	1
		169.4.a.f	2
		169.4.a.g	2
		169.4.a.h	2
		169.4.a.i	2
		169.4.a.j	2
		169.4.a.k	9
		169.4.a.l	9
169.4.b	\(\chi_{169}(168, \cdot)\)	169.4.b.a	2	1
		169.4.b.b	2
		169.4.b.c	2
		169.4.b.d	2
		169.4.b.e	4
		169.4.b.f	4
		169.4.b.g	18
169.4.c	\(\chi_{169}(22, \cdot)\)	169.4.c.a	2	2
		169.4.c.b	2
		169.4.c.c	2
		169.4.c.d	2
		169.4.c.e	2
		169.4.c.f	4
		169.4.c.g	4
		169.4.c.h	4
		169.4.c.i	4
		169.4.c.j	4
		169.4.c.k	18
		169.4.c.l	18
169.4.e	\(\chi_{169}(23, \cdot)\)	169.4.e.a	2	2
		169.4.e.b	2
		169.4.e.c	4
		169.4.e.d	4
		169.4.e.e	4
		169.4.e.f	8
		169.4.e.g	8
		169.4.e.h	36
169.4.g	\(\chi_{169}(14, \cdot)\)	169.4.g.a	540	12
169.4.h	\(\chi_{169}(12, \cdot)\)	169.4.h.a	528	12
169.4.i	\(\chi_{169}(3, \cdot)\)	169.4.i.a	1080	24
169.4.k	\(\chi_{169}(4, \cdot)\)	169.4.k.a	1056	24

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(169))\) into lower level spaces

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