Space of modular forms of level 416 and weight 3

• Label: 416.3
• Level: 416
• Weight: 3
• Dimension: 5866
• Nonzero newspaces: 20
• Newform subspaces: 42
• Sturm bound: 32256
• Trace bound: 11

Defining parameters

Level:	\( N \)	=	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	=	\( 3 \)
Nonzero newspaces:			\( 20 \)
Newform subspaces:			\( 42 \)
Sturm bound:			\(32256\)
Trace bound:			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(416))\).

	Total	New	Old
Modular forms	11136	6086	5050
Cusp forms	10368	5866	4502
Eisenstein series	768	220	548

Trace form

5866 * q - 40 * q^2 - 32 * q^3 - 40 * q^4 - 48 * q^5 - 40 * q^6 - 28 * q^7 - 40 * q^8 - 26 * q^9 + 40 * q^10 + 56 * q^12 - 16 * q^13 - 56 * q^14 - 20 * q^15 - 80 * q^16 - 100 * q^17 - 160 * q^18 - 96 * q^19 - 200 * q^20 - 168 * q^21 - 336 * q^22 + 100 * q^23 - 464 * q^24 - 30 * q^25 - 144 * q^26 + 184 * q^27 - 160 * q^28 + 16 * q^29 - 88 * q^30 - 36 * q^31 - 112 * q^33 + 48 * q^34 - 220 * q^35 + 624 * q^36 + 80 * q^37 + 744 * q^38 - 224 * q^39 + 720 * q^40 + 84 * q^41 + 800 * q^42 - 192 * q^43 + 328 * q^44 + 88 * q^45 + 24 * q^46 - 20 * q^47 - 144 * q^48 - 62 * q^49 - 664 * q^50 - 480 * q^51 - 468 * q^52 - 32 * q^53 - 1232 * q^54 - 540 * q^55 - 1152 * q^56 - 392 * q^57 - 1104 * q^58 - 448 * q^59 - 928 * q^60 - 240 * q^61 - 480 * q^62 - 36 * q^63 + 416 * q^64 - 56 * q^65 + 1064 * q^66 + 416 * q^67 + 688 * q^68 + 216 * q^69 + 1280 * q^70 + 484 * q^71 + 1448 * q^72 - 44 * q^73 + 1016 * q^74 + 948 * q^75 + 984 * q^76 - 360 * q^77 + 188 * q^78 + 968 * q^79 - 672 * q^80 + 378 * q^81 - 1320 * q^82 + 1248 * q^83 - 2032 * q^84 + 48 * q^85 - 1424 * q^86 + 2020 * q^87 - 1392 * q^88 + 1108 * q^89 - 1792 * q^90 + 1312 * q^91 - 1328 * q^92 + 1296 * q^93 - 128 * q^94 + 924 * q^95 + 208 * q^96 + 716 * q^97 + 608 * q^98 + 160 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(416))\)

We only show spaces with odd 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
416.3.c	\(\chi_{416}(415, \cdot)\)	416.3.c.a	14	1
		416.3.c.b	14
416.3.d	\(\chi_{416}(287, \cdot)\)	416.3.d.a	12	1
		416.3.d.b	12
416.3.g	\(\chi_{416}(79, \cdot)\)	416.3.g.a	24	1
416.3.h	\(\chi_{416}(207, \cdot)\)	416.3.h.a	3	1
		416.3.h.b	3
		416.3.h.c	20
416.3.j	\(\chi_{416}(177, \cdot)\)	416.3.j.a	52	2
416.3.m	\(\chi_{416}(265, \cdot)\)	None	0	2
416.3.o	\(\chi_{416}(103, \cdot)\)	None	0	2
416.3.q	\(\chi_{416}(183, \cdot)\)	None	0	2
416.3.r	\(\chi_{416}(57, \cdot)\)	None	0	2
416.3.t	\(\chi_{416}(161, \cdot)\)	416.3.t.a	2	2
		416.3.t.b	2
		416.3.t.c	12
		416.3.t.d	12
		416.3.t.e	14
		416.3.t.f	14
416.3.v	\(\chi_{416}(367, \cdot)\)	416.3.v.a	4	2
		416.3.v.b	48
416.3.x	\(\chi_{416}(303, \cdot)\)	416.3.x.a	2	2
		416.3.x.b	2
		416.3.x.c	4
		416.3.x.d	44
416.3.y	\(\chi_{416}(95, \cdot)\)	416.3.y.a	28	2
		416.3.y.b	28
416.3.bb	\(\chi_{416}(159, \cdot)\)	416.3.bb.a	12	2
		416.3.bb.b	12
		416.3.bb.c	32
416.3.bc	\(\chi_{416}(5, \cdot)\)	416.3.bc.a	440	4
416.3.be	\(\chi_{416}(51, \cdot)\)	416.3.be.a	440	4
416.3.bh	\(\chi_{416}(27, \cdot)\)	416.3.bh.a	384	4
416.3.bj	\(\chi_{416}(21, \cdot)\)	416.3.bj.a	440	4
416.3.bl	\(\chi_{416}(33, \cdot)\)	416.3.bl.a	4	4
		416.3.bl.b	4
		416.3.bl.c	8
		416.3.bl.d	16
		416.3.bl.e	24
		416.3.bl.f	28
		416.3.bl.g	28
416.3.bm	\(\chi_{416}(41, \cdot)\)	None	0	4
416.3.bo	\(\chi_{416}(55, \cdot)\)	None	0	4
416.3.bq	\(\chi_{416}(23, \cdot)\)	None	0	4
416.3.bt	\(\chi_{416}(137, \cdot)\)	None	0	4
416.3.bv	\(\chi_{416}(145, \cdot)\)	416.3.bv.a	104	4
416.3.bw	\(\chi_{416}(37, \cdot)\)	416.3.bw.a	880	8
416.3.by	\(\chi_{416}(3, \cdot)\)	416.3.by.a	880	8
416.3.cb	\(\chi_{416}(43, \cdot)\)	416.3.cb.a	880	8
416.3.cd	\(\chi_{416}(141, \cdot)\)	416.3.cd.a	880	8

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(416))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(416)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(104))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(208))\)\(^{\oplus 2}\)