Space of modular forms of level 408 and weight 4

• Label: 408.4
• Level: 408
• Weight: 4
• Dimension: 5734
• Nonzero newspaces: 15
• Sturm bound: 36864
• Trace bound: 6

Defining parameters

Level:	\( N \)	=	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 15 \)
Sturm bound:			\(36864\)
Trace bound:			\(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(408))\).

	Total	New	Old
Modular forms	14208	5854	8354
Cusp forms	13440	5734	7706
Eisenstein series	768	120	648

Trace form

5734 * q - 4 * q^2 - 18 * q^3 - 72 * q^4 - 28 * q^5 + 12 * q^6 - 40 * q^7 + 152 * q^8 + 62 * q^9 - 104 * q^10 + 56 * q^11 + 96 * q^12 + 148 * q^13 + 200 * q^14 + 20 * q^15 + 160 * q^16 - 134 * q^17 - 364 * q^18 - 160 * q^19 - 112 * q^20 + 144 * q^21 - 928 * q^22 - 672 * q^23 - 1080 * q^24 + 1474 * q^25 - 112 * q^26 + 198 * q^27 - 384 * q^28 - 244 * q^29 + 488 * q^30 - 648 * q^31 + 496 * q^32 - 792 * q^33 + 1300 * q^34 - 1312 * q^35 + 2040 * q^36 - 1404 * q^37 + 1552 * q^38 - 292 * q^39 + 2000 * q^40 - 116 * q^41 - 1528 * q^42 + 1840 * q^43 - 2304 * q^44 + 540 * q^45 - 3568 * q^46 + 336 * q^47 - 5040 * q^48 - 1306 * q^49 - 3940 * q^50 - 1014 * q^51 - 3552 * q^52 - 3172 * q^53 - 4988 * q^54 - 9840 * q^55 - 6672 * q^56 - 1512 * q^57 - 232 * q^58 - 1672 * q^59 + 9856 * q^60 + 2836 * q^61 + 12504 * q^62 + 7688 * q^63 + 13632 * q^64 + 7024 * q^65 + 10800 * q^66 + 10512 * q^67 + 20608 * q^68 + 6192 * q^69 + 11344 * q^70 + 6976 * q^71 + 1832 * q^72 + 484 * q^73 + 1664 * q^74 + 5058 * q^75 - 4880 * q^76 - 3552 * q^77 - 4784 * q^78 - 6120 * q^79 - 15712 * q^80 - 4234 * q^81 - 12168 * q^82 - 13592 * q^83 - 3984 * q^84 - 6884 * q^85 + 1520 * q^86 - 6508 * q^87 + 5120 * q^88 + 932 * q^89 - 9144 * q^90 - 9920 * q^91 - 3456 * q^92 + 1376 * q^93 - 13808 * q^94 - 12784 * q^95 - 12864 * q^96 + 3196 * q^97 - 6708 * q^98 - 896 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(408))\)

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
408.4.a	\(\chi_{408}(1, \cdot)\)	408.4.a.a	1	1
		408.4.a.b	1
		408.4.a.c	2
		408.4.a.d	3
		408.4.a.e	3
		408.4.a.f	3
		408.4.a.g	3
		408.4.a.h	4
		408.4.a.i	4
408.4.c	\(\chi_{408}(169, \cdot)\)	408.4.c.a	2	1
		408.4.c.b	6
		408.4.c.c	6
		408.4.c.d	12
408.4.e	\(\chi_{408}(239, \cdot)\)	None	0	1
408.4.f	\(\chi_{408}(205, \cdot)\)	408.4.f.a	48	1
		408.4.f.b	48
408.4.h	\(\chi_{408}(203, \cdot)\)	n/a	212	1
408.4.j	\(\chi_{408}(35, \cdot)\)	n/a	192	1
408.4.l	\(\chi_{408}(373, \cdot)\)	n/a	108	1
408.4.o	\(\chi_{408}(407, \cdot)\)	None	0	1
408.4.q	\(\chi_{408}(251, \cdot)\)	n/a	424	2
408.4.s	\(\chi_{408}(13, \cdot)\)	n/a	216	2
408.4.v	\(\chi_{408}(217, \cdot)\)	408.4.v.a	24	2
		408.4.v.b	28
408.4.x	\(\chi_{408}(47, \cdot)\)	None	0	2
408.4.ba	\(\chi_{408}(25, \cdot)\)	n/a	112	4
408.4.bb	\(\chi_{408}(263, \cdot)\)	None	0	4
408.4.bc	\(\chi_{408}(229, \cdot)\)	n/a	432	4
408.4.bd	\(\chi_{408}(59, \cdot)\)	n/a	848	4
408.4.bh	\(\chi_{408}(41, \cdot)\)	n/a	432	8
408.4.bi	\(\chi_{408}(7, \cdot)\)	None	0	8
408.4.bl	\(\chi_{408}(91, \cdot)\)	n/a	864	8
408.4.bm	\(\chi_{408}(5, \cdot)\)	n/a	1696	8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(408))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(408)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(102))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 2}\)