Space of modular forms of level 416 and weight 4

• Label: 416.4
• Level: 416
• Weight: 4
• Dimension: 8854
• Nonzero newspaces: 20
• Sturm bound: 43008
• Trace bound: 7

Defining parameters

Level:	\( N \)	=	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 20 \)
Sturm bound:			\(43008\)
Trace bound:			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(416))\).

	Total	New	Old
Modular forms	16512	9074	7438
Cusp forms	15744	8854	6890
Eisenstein series	768	220	548

Trace form

8854 * q - 40 * q^2 - 28 * q^3 - 40 * q^4 - 44 * q^5 - 40 * q^6 - 60 * q^7 - 40 * q^8 - 154 * q^9 - 280 * q^10 - 28 * q^11 + 56 * q^12 + 74 * q^13 + 328 * q^14 + 188 * q^15 + 560 * q^16 + 340 * q^17 + 320 * q^18 - 28 * q^19 - 200 * q^20 - 552 * q^21 - 432 * q^22 - 1292 * q^23 + 48 * q^24 - 734 * q^25 - 64 * q^26 + 464 * q^27 - 800 * q^28 + 356 * q^29 - 2424 * q^30 + 2364 * q^31 - 1280 * q^32 + 840 * q^33 - 1104 * q^34 + 884 * q^35 - 2960 * q^36 - 1708 * q^37 - 2008 * q^38 - 1192 * q^39 + 976 * q^40 - 964 * q^41 + 4480 * q^42 - 1644 * q^43 + 4040 * q^44 + 2412 * q^45 + 2840 * q^46 + 1308 * q^47 + 4848 * q^48 + 2126 * q^49 + 7080 * q^50 + 2720 * q^51 + 2476 * q^52 - 988 * q^53 + 2128 * q^54 - 124 * q^55 - 2432 * q^56 - 2128 * q^57 - 6448 * q^58 - 2780 * q^59 - 11552 * q^60 + 1828 * q^61 - 6816 * q^62 - 5028 * q^63 - 9952 * q^64 - 1836 * q^65 - 12088 * q^66 - 4108 * q^67 - 5584 * q^68 + 536 * q^69 - 2848 * q^70 - 764 * q^71 + 3368 * q^72 + 1436 * q^73 + 6904 * q^74 + 3064 * q^75 + 10200 * q^76 - 1288 * q^77 + 11868 * q^78 - 1928 * q^79 + 20576 * q^80 + 2974 * q^81 + 12760 * q^82 - 4908 * q^83 + 13072 * q^84 - 3840 * q^85 + 1808 * q^86 - 6260 * q^87 - 2544 * q^88 - 6756 * q^89 - 14848 * q^90 - 1520 * q^91 - 20400 * q^92 - 11408 * q^93 - 25728 * q^94 + 14292 * q^95 - 35632 * q^96 + 1028 * q^97 - 22496 * q^98 + 26748 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(416))\)

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
416.4.a	\(\chi_{416}(1, \cdot)\)	416.4.a.a	1	1
		416.4.a.b	1
		416.4.a.c	1
		416.4.a.d	1
		416.4.a.e	3
		416.4.a.f	3
		416.4.a.g	4
		416.4.a.h	5
		416.4.a.i	5
		416.4.a.j	6
		416.4.a.k	6
416.4.b	\(\chi_{416}(209, \cdot)\)	416.4.b.a	2	1
		416.4.b.b	34
416.4.e	\(\chi_{416}(337, \cdot)\)	416.4.e.a	40	1
416.4.f	\(\chi_{416}(129, \cdot)\)	416.4.f.a	2	1
		416.4.f.b	4
		416.4.f.c	16
		416.4.f.d	20
416.4.i	\(\chi_{416}(289, \cdot)\)	416.4.i.a	4	2
		416.4.i.b	4
		416.4.i.c	4
		416.4.i.d	8
		416.4.i.e	8
		416.4.i.f	12
		416.4.i.g	22
		416.4.i.h	22
416.4.k	\(\chi_{416}(31, \cdot)\)	416.4.k.a	20	2
		416.4.k.b	20
		416.4.k.c	22
		416.4.k.d	22
416.4.l	\(\chi_{416}(343, \cdot)\)	None	0	2
416.4.n	\(\chi_{416}(105, \cdot)\)	None	0	2
416.4.p	\(\chi_{416}(25, \cdot)\)	None	0	2
416.4.s	\(\chi_{416}(135, \cdot)\)	None	0	2
416.4.u	\(\chi_{416}(47, \cdot)\)	416.4.u.a	80	2
416.4.w	\(\chi_{416}(225, \cdot)\)	416.4.w.a	4	2
		416.4.w.b	40
		416.4.w.c	40
416.4.z	\(\chi_{416}(81, \cdot)\)	416.4.z.a	80	2
416.4.ba	\(\chi_{416}(17, \cdot)\)	416.4.ba.a	80	2
416.4.bd	\(\chi_{416}(83, \cdot)\)	n/a	664	4
416.4.bf	\(\chi_{416}(53, \cdot)\)	n/a	576	4
416.4.bg	\(\chi_{416}(77, \cdot)\)	n/a	664	4
416.4.bi	\(\chi_{416}(99, \cdot)\)	n/a	664	4
416.4.bk	\(\chi_{416}(15, \cdot)\)	n/a	160	4
416.4.bn	\(\chi_{416}(7, \cdot)\)	None	0	4
416.4.bp	\(\chi_{416}(121, \cdot)\)	None	0	4
416.4.br	\(\chi_{416}(9, \cdot)\)	None	0	4
416.4.bs	\(\chi_{416}(71, \cdot)\)	None	0	4
416.4.bu	\(\chi_{416}(63, \cdot)\)	n/a	168	4
416.4.bx	\(\chi_{416}(115, \cdot)\)	n/a	1328	8
416.4.bz	\(\chi_{416}(69, \cdot)\)	n/a	1328	8
416.4.ca	\(\chi_{416}(29, \cdot)\)	n/a	1328	8
416.4.cc	\(\chi_{416}(11, \cdot)\)	n/a	1328	8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(416))\) into lower level spaces

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