Space of modular forms of level 416 and weight 2

• Label: 416.2
• Level: 416
• Weight: 2
• Dimension: 2878
• Nonzero newspaces: 20
• Newform subspaces: 55
• Sturm bound: 21504
• Trace bound: 9

Defining parameters

Level:	\( N \)	=	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 20 \)
Newform subspaces:			\( 55 \)
Sturm bound:			\(21504\)
Trace bound:			\(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(416))\).

	Total	New	Old
Modular forms	5760	3098	2662
Cusp forms	4993	2878	2115
Eisenstein series	767	220	547

Trace form

2878 * q - 40 * q^2 - 28 * q^3 - 40 * q^4 - 36 * q^5 - 40 * q^6 - 28 * q^7 - 40 * q^8 - 58 * q^9 - 56 * q^10 - 28 * q^11 - 72 * q^12 - 50 * q^13 - 120 * q^14 - 36 * q^15 - 80 * q^16 - 28 * q^17 - 80 * q^18 - 28 * q^19 - 72 * q^20 - 40 * q^21 - 64 * q^22 - 44 * q^23 - 16 * q^24 - 62 * q^25 - 24 * q^26 - 112 * q^27 - 20 * q^29 + 24 * q^30 - 68 * q^31 - 104 * q^33 - 16 * q^34 - 76 * q^35 + 16 * q^36 - 36 * q^37 - 56 * q^38 - 56 * q^39 - 112 * q^40 - 84 * q^41 - 80 * q^42 - 44 * q^43 - 120 * q^44 - 76 * q^45 - 104 * q^46 - 36 * q^47 - 144 * q^48 - 10 * q^49 - 120 * q^50 - 32 * q^51 - 52 * q^52 - 148 * q^53 - 64 * q^54 + 36 * q^55 - 64 * q^56 - 64 * q^57 - 32 * q^58 + 36 * q^59 - 32 * q^60 - 84 * q^61 + 60 * q^63 + 32 * q^64 - 100 * q^65 - 40 * q^66 + 52 * q^67 - 80 * q^68 - 104 * q^69 - 16 * q^70 + 36 * q^71 - 88 * q^72 - 52 * q^73 - 72 * q^74 - 40 * q^75 - 40 * q^76 - 72 * q^77 - 84 * q^78 - 72 * q^79 - 32 * q^80 - 42 * q^81 - 40 * q^82 - 108 * q^83 - 80 * q^84 - 96 * q^85 - 212 * q^87 - 48 * q^88 - 180 * q^89 - 64 * q^90 - 176 * q^91 - 16 * q^92 - 208 * q^93 - 64 * q^94 - 268 * q^95 - 48 * q^96 - 236 * q^97 - 324 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(416))\)

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
416.2.a	\(\chi_{416}(1, \cdot)\)	416.2.a.a	1	1
		416.2.a.b	1
		416.2.a.c	2
		416.2.a.d	2
		416.2.a.e	2
		416.2.a.f	4
416.2.b	\(\chi_{416}(209, \cdot)\)	416.2.b.a	2	1
		416.2.b.b	4
		416.2.b.c	6
416.2.e	\(\chi_{416}(337, \cdot)\)	416.2.e.a	2	1
		416.2.e.b	2
		416.2.e.c	8
416.2.f	\(\chi_{416}(129, \cdot)\)	416.2.f.a	2	1
		416.2.f.b	2
		416.2.f.c	2
		416.2.f.d	4
		416.2.f.e	4
416.2.i	\(\chi_{416}(289, \cdot)\)	416.2.i.a	2	2
		416.2.i.b	2
		416.2.i.c	4
		416.2.i.d	4
		416.2.i.e	4
		416.2.i.f	4
		416.2.i.g	8
416.2.k	\(\chi_{416}(31, \cdot)\)	416.2.k.a	2	2
		416.2.k.b	2
		416.2.k.c	4
		416.2.k.d	4
		416.2.k.e	8
		416.2.k.f	8
416.2.l	\(\chi_{416}(343, \cdot)\)	None	0	2
416.2.n	\(\chi_{416}(105, \cdot)\)	None	0	2
416.2.p	\(\chi_{416}(25, \cdot)\)	None	0	2
416.2.s	\(\chi_{416}(135, \cdot)\)	None	0	2
416.2.u	\(\chi_{416}(47, \cdot)\)	416.2.u.a	4	2
		416.2.u.b	20
416.2.w	\(\chi_{416}(225, \cdot)\)	416.2.w.a	4	2
		416.2.w.b	4
		416.2.w.c	8
		416.2.w.d	12
416.2.z	\(\chi_{416}(81, \cdot)\)	416.2.z.a	24	2
416.2.ba	\(\chi_{416}(17, \cdot)\)	416.2.ba.a	4	2
		416.2.ba.b	4
		416.2.ba.c	16
416.2.bd	\(\chi_{416}(83, \cdot)\)	416.2.bd.a	216	4
416.2.bf	\(\chi_{416}(53, \cdot)\)	416.2.bf.a	192	4
416.2.bg	\(\chi_{416}(77, \cdot)\)	416.2.bg.a	216	4
416.2.bi	\(\chi_{416}(99, \cdot)\)	416.2.bi.a	216	4
416.2.bk	\(\chi_{416}(15, \cdot)\)	416.2.bk.a	48	4
416.2.bn	\(\chi_{416}(7, \cdot)\)	None	0	4
416.2.bp	\(\chi_{416}(121, \cdot)\)	None	0	4
416.2.br	\(\chi_{416}(9, \cdot)\)	None	0	4
416.2.bs	\(\chi_{416}(71, \cdot)\)	None	0	4
416.2.bu	\(\chi_{416}(63, \cdot)\)	416.2.bu.a	4	4
		416.2.bu.b	4
		416.2.bu.c	8
		416.2.bu.d	8
		416.2.bu.e	16
		416.2.bu.f	16
416.2.bx	\(\chi_{416}(115, \cdot)\)	416.2.bx.a	432	8
416.2.bz	\(\chi_{416}(69, \cdot)\)	416.2.bz.a	432	8
416.2.ca	\(\chi_{416}(29, \cdot)\)	416.2.ca.a	432	8
416.2.cc	\(\chi_{416}(11, \cdot)\)	416.2.cc.a	432	8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(416))\) into lower level spaces

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