Space of modular forms of level 400 and weight 8

• Label: 400.8
• Level: 400
• Weight: 8
• Dimension: 17327
• Nonzero newspaces: 14
• Sturm bound: 76800
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 14 \)
Sturm bound:			\(76800\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(400))\).

	Total	New	Old
Modular forms	33992	17512	16480
Cusp forms	33208	17327	15881
Eisenstein series	784	185	599

Trace form

17327 * q - 26 * q^2 + 8 * q^3 - 208 * q^4 - 40 * q^5 - 216 * q^6 + 646 * q^7 - 1028 * q^8 + 2573 * q^9 - 32 * q^10 + 5768 * q^11 - 27380 * q^12 - 29526 * q^13 + 22244 * q^14 + 37428 * q^15 + 13296 * q^16 - 1140 * q^17 - 29370 * q^18 - 62764 * q^19 - 32 * q^20 - 8414 * q^21 - 44372 * q^22 + 166542 * q^23 - 59240 * q^24 + 32240 * q^25 - 36040 * q^26 - 592630 * q^27 - 41856 * q^28 - 173338 * q^29 + 1353856 * q^30 + 455074 * q^31 - 390816 * q^32 - 1959998 * q^33 - 4639652 * q^34 - 408276 * q^35 - 2228604 * q^36 + 1759730 * q^37 + 5190968 * q^38 + 707042 * q^39 + 4367888 * q^40 + 1867588 * q^41 - 657344 * q^42 - 325572 * q^43 - 10092836 * q^44 - 3897030 * q^45 - 9208316 * q^46 - 5180978 * q^47 + 8499280 * q^48 + 2686157 * q^49 + 7947528 * q^50 + 9227840 * q^51 + 269692 * q^52 + 1194082 * q^53 - 11883512 * q^54 + 1229030 * q^55 - 4093200 * q^56 - 8207530 * q^57 - 1860736 * q^58 - 20429536 * q^59 + 131040 * q^60 - 449378 * q^61 + 6274072 * q^62 - 276614 * q^63 + 32440088 * q^64 + 9476476 * q^65 - 49072852 * q^66 + 47900984 * q^67 - 25747928 * q^68 - 10755350 * q^69 + 19691320 * q^70 - 73203382 * q^71 + 71698908 * q^72 - 23346944 * q^73 + 22119908 * q^74 + 1361128 * q^75 - 2297564 * q^76 + 32407586 * q^77 - 95293004 * q^78 + 93661794 * q^79 - 37919952 * q^80 + 14821651 * q^81 - 12464312 * q^82 + 23103260 * q^83 + 36361856 * q^84 + 62567136 * q^85 + 25689628 * q^86 - 188724790 * q^87 + 68685072 * q^88 - 30723804 * q^89 + 40165288 * q^90 - 8408674 * q^91 - 51523024 * q^92 + 360206 * q^93 - 68976120 * q^94 + 91099250 * q^95 + 7282440 * q^96 + 76406740 * q^97 + 155030978 * q^98 + 22542178 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.8.a	\(\chi_{400}(1, \cdot)\)	400.8.a.a	1	1
		400.8.a.b	1
		400.8.a.c	1
		400.8.a.d	1
		400.8.a.e	1
		400.8.a.f	1
		400.8.a.g	1
		400.8.a.h	1
		400.8.a.i	1
		400.8.a.j	1
		400.8.a.k	1
		400.8.a.l	1
		400.8.a.m	1
		400.8.a.n	1
		400.8.a.o	1
		400.8.a.p	1
		400.8.a.q	1
		400.8.a.r	1
		400.8.a.s	1
		400.8.a.t	2
		400.8.a.u	2
		400.8.a.v	2
		400.8.a.w	2
		400.8.a.x	2
		400.8.a.y	2
		400.8.a.z	2
		400.8.a.ba	2
		400.8.a.bb	2
		400.8.a.bc	2
		400.8.a.bd	2
		400.8.a.be	2
		400.8.a.bf	2
		400.8.a.bg	2
		400.8.a.bh	3
		400.8.a.bi	3
		400.8.a.bj	4
		400.8.a.bk	4
		400.8.a.bl	4
400.8.c	\(\chi_{400}(49, \cdot)\)	400.8.c.a	2	1
		400.8.c.b	2
		400.8.c.c	2
		400.8.c.d	2
		400.8.c.e	2
		400.8.c.f	2
		400.8.c.g	2
		400.8.c.h	2
		400.8.c.i	2
		400.8.c.j	2
		400.8.c.k	2
		400.8.c.l	2
		400.8.c.m	4
		400.8.c.n	4
		400.8.c.o	4
		400.8.c.p	4
		400.8.c.q	4
		400.8.c.r	4
		400.8.c.s	4
		400.8.c.t	4
		400.8.c.u	6
400.8.d	\(\chi_{400}(201, \cdot)\)	None	0	1
400.8.f	\(\chi_{400}(249, \cdot)\)	None	0	1
400.8.j	\(\chi_{400}(43, \cdot)\)	n/a	500	2
400.8.l	\(\chi_{400}(101, \cdot)\)	n/a	526	2
400.8.n	\(\chi_{400}(143, \cdot)\)	n/a	126	2
400.8.o	\(\chi_{400}(7, \cdot)\)	None	0	2
400.8.q	\(\chi_{400}(149, \cdot)\)	n/a	500	2
400.8.s	\(\chi_{400}(107, \cdot)\)	n/a	500	2
400.8.u	\(\chi_{400}(81, \cdot)\)	n/a	416	4
400.8.w	\(\chi_{400}(9, \cdot)\)	None	0	4
400.8.y	\(\chi_{400}(129, \cdot)\)	n/a	416	4
400.8.bb	\(\chi_{400}(41, \cdot)\)	None	0	4
400.8.bd	\(\chi_{400}(3, \cdot)\)	n/a	3344	8
400.8.be	\(\chi_{400}(21, \cdot)\)	n/a	3344	8
400.8.bh	\(\chi_{400}(23, \cdot)\)	None	0	8
400.8.bi	\(\chi_{400}(47, \cdot)\)	n/a	840	8
400.8.bl	\(\chi_{400}(29, \cdot)\)	n/a	3344	8
400.8.bm	\(\chi_{400}(67, \cdot)\)	n/a	3344	8

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

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