Space of modular forms of level 400 and weight 5

• Label: 400.5
• Level: 400
• Weight: 5
• Dimension: 9862
• Nonzero newspaces: 14
• Sturm bound: 48000
• Trace bound: 5

Defining parameters

Level:	\( N \)	=	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 5 \)
Nonzero newspaces:			\( 14 \)
Sturm bound:			\(48000\)
Trace bound:			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(400))\).

	Total	New	Old
Modular forms	19592	10046	9546
Cusp forms	18808	9862	8946
Eisenstein series	784	184	600

Trace form

9862 * q - 26 * q^2 - 20 * q^3 - 32 * q^4 - 40 * q^5 + 24 * q^6 - 22 * q^7 - 116 * q^8 - 232 * q^9 - 32 * q^10 + 64 * q^11 - 356 * q^12 - 156 * q^13 + 20 * q^14 + 648 * q^15 - 208 * q^16 + 650 * q^17 + 1366 * q^18 - 720 * q^19 - 32 * q^20 - 1878 * q^21 + 876 * q^22 - 1750 * q^23 - 1896 * q^24 - 680 * q^25 - 3496 * q^26 + 562 * q^27 + 288 * q^28 + 8064 * q^29 + 6016 * q^30 + 226 * q^31 - 5216 * q^32 - 7798 * q^33 - 20036 * q^34 - 2616 * q^35 - 12764 * q^36 - 11996 * q^37 - 4856 * q^38 + 2686 * q^39 + 6288 * q^40 + 10298 * q^41 + 18912 * q^42 + 10636 * q^43 + 20620 * q^44 + 11670 * q^45 + 33172 * q^46 - 8658 * q^47 + 24144 * q^48 + 1980 * q^49 - 9272 * q^50 + 3456 * q^51 - 33652 * q^52 + 676 * q^53 - 9976 * q^54 + 7750 * q^55 - 6992 * q^56 + 2614 * q^57 - 20480 * q^58 - 2800 * q^59 - 2080 * q^60 - 10128 * q^61 - 11496 * q^62 + 15998 * q^63 - 54056 * q^64 - 14424 * q^65 - 31460 * q^66 - 4500 * q^67 + 36488 * q^68 - 6150 * q^69 + 38200 * q^70 - 78754 * q^71 + 141868 * q^72 + 14302 * q^73 + 84436 * q^74 - 54712 * q^75 + 30836 * q^76 - 21246 * q^77 + 39300 * q^78 - 30 * q^79 - 12752 * q^80 + 5014 * q^81 - 58664 * q^82 + 154540 * q^83 - 197760 * q^84 + 48376 * q^85 - 115044 * q^86 + 171502 * q^87 - 143216 * q^88 - 27238 * q^89 - 77912 * q^90 + 22238 * q^91 - 38640 * q^92 - 93146 * q^93 + 62568 * q^94 - 128950 * q^95 + 217096 * q^96 - 58454 * q^97 + 176626 * q^98 - 55786 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.5.b	\(\chi_{400}(351, \cdot)\)	400.5.b.a	1	1
		400.5.b.b	1
		400.5.b.c	2
		400.5.b.d	2
		400.5.b.e	2
		400.5.b.f	2
		400.5.b.g	4
		400.5.b.h	4
		400.5.b.i	6
		400.5.b.j	6
		400.5.b.k	8
400.5.e	\(\chi_{400}(199, \cdot)\)	None	0	1
400.5.g	\(\chi_{400}(151, \cdot)\)	None	0	1
400.5.h	\(\chi_{400}(399, \cdot)\)	400.5.h.a	4	1
		400.5.h.b	4
		400.5.h.c	8
		400.5.h.d	8
		400.5.h.e	12
400.5.i	\(\chi_{400}(93, \cdot)\)	n/a	284	2
400.5.k	\(\chi_{400}(99, \cdot)\)	n/a	284	2
400.5.m	\(\chi_{400}(57, \cdot)\)	None	0	2
400.5.p	\(\chi_{400}(193, \cdot)\)	400.5.p.a	2	2
		400.5.p.b	2
		400.5.p.c	2
		400.5.p.d	2
		400.5.p.e	4
		400.5.p.f	4
		400.5.p.g	4
		400.5.p.h	4
		400.5.p.i	4
		400.5.p.j	4
		400.5.p.k	4
		400.5.p.l	4
		400.5.p.m	4
		400.5.p.n	4
		400.5.p.o	6
		400.5.p.p	8
		400.5.p.q	8
400.5.r	\(\chi_{400}(51, \cdot)\)	n/a	298	2
400.5.t	\(\chi_{400}(157, \cdot)\)	n/a	284	2
400.5.v	\(\chi_{400}(71, \cdot)\)	None	0	4
400.5.x	\(\chi_{400}(79, \cdot)\)	n/a	240	4
400.5.z	\(\chi_{400}(31, \cdot)\)	n/a	240	4
400.5.ba	\(\chi_{400}(39, \cdot)\)	None	0	4
400.5.bc	\(\chi_{400}(53, \cdot)\)	n/a	1904	8
400.5.bf	\(\chi_{400}(19, \cdot)\)	n/a	1904	8
400.5.bg	\(\chi_{400}(17, \cdot)\)	n/a	472	8
400.5.bj	\(\chi_{400}(73, \cdot)\)	None	0	8
400.5.bk	\(\chi_{400}(11, \cdot)\)	n/a	1904	8
400.5.bn	\(\chi_{400}(13, \cdot)\)	n/a	1904	8

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

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(400)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 15}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)