Space of modular forms of level 400 and weight 9

• Label: 400.9
• Level: 400
• Weight: 9
• Dimension: 19816
• Nonzero newspaces: 14
• Sturm bound: 86400
• Trace bound: 5

Defining parameters

Level:	\( N \)	=	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 9 \)
Nonzero newspaces:			\( 14 \)
Sturm bound:			\(86400\)
Trace bound:			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(400))\).

	Total	New	Old
Modular forms	38792	20000	18792
Cusp forms	38008	19816	18192
Eisenstein series	784	184	600

Trace form

19816 * q - 26 * q^2 - 20 * q^3 + 160 * q^4 - 40 * q^5 + 3192 * q^6 - 22 * q^7 - 8756 * q^8 - 14470 * q^9 - 32 * q^10 - 19808 * q^11 + 14044 * q^12 + 97064 * q^13 - 58924 * q^14 - 170712 * q^15 + 245296 * q^16 + 292430 * q^17 - 223754 * q^18 + 167536 * q^19 - 32 * q^20 - 745590 * q^21 + 985676 * q^22 - 390550 * q^23 - 800616 * q^24 + 708280 * q^25 - 184840 * q^26 - 2542478 * q^27 + 1142688 * q^28 + 3687012 * q^29 - 8219264 * q^30 - 491038 * q^31 + 11917984 * q^32 + 8663882 * q^33 + 14376636 * q^34 - 1207896 * q^35 - 31435100 * q^36 - 15169496 * q^37 - 26001176 * q^38 - 7650050 * q^39 + 19219088 * q^40 + 18376958 * q^41 + 71760672 * q^42 + 17084716 * q^43 - 6938804 * q^44 - 15477930 * q^45 - 90301484 * q^46 + 7355502 * q^47 - 71146416 * q^48 + 12280110 * q^49 + 42125128 * q^50 - 56082624 * q^51 + 68312268 * q^52 - 37115864 * q^53 - 8206840 * q^54 + 24359590 * q^55 - 36346448 * q^56 + 52233334 * q^57 - 1034720 * q^58 - 46004176 * q^59 - 524320 * q^60 - 104251820 * q^61 - 64043496 * q^62 - 16582402 * q^63 - 34167848 * q^64 - 22098024 * q^65 + 517129564 * q^66 + 41247980 * q^67 - 92405752 * q^68 + 260717082 * q^69 - 241588040 * q^70 + 307718366 * q^71 - 213413012 * q^72 - 136620478 * q^73 + 284074996 * q^74 - 503503192 * q^75 + 322332596 * q^76 - 148971486 * q^77 + 563765700 * q^78 - 30 * q^79 - 158758352 * q^80 + 236719624 * q^81 - 525627624 * q^82 + 717820780 * q^83 - 920628864 * q^84 - 333447144 * q^85 - 27298692 * q^86 - 229775378 * q^87 + 798776464 * q^88 + 566215358 * q^89 + 990555688 * q^90 - 1119671842 * q^91 + 20898960 * q^92 - 558568346 * q^93 - 1252672536 * q^94 + 51540650 * q^95 - 1526142968 * q^96 - 445893234 * q^97 + 1928675506 * q^98 - 503909290 * q^99

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.9.b	\(\chi_{400}(351, \cdot)\)	400.9.b.a	1	1
		400.9.b.b	1
		400.9.b.c	2
		400.9.b.d	2
		400.9.b.e	2
		400.9.b.f	4
		400.9.b.g	4
		400.9.b.h	6
		400.9.b.i	6
		400.9.b.j	10
		400.9.b.k	10
		400.9.b.l	12
		400.9.b.m	16
400.9.e	\(\chi_{400}(199, \cdot)\)	None	0	1
400.9.g	\(\chi_{400}(151, \cdot)\)	None	0	1
400.9.h	\(\chi_{400}(399, \cdot)\)	400.9.h.a	4	1
		400.9.h.b	4
		400.9.h.c	8
		400.9.h.d	12
		400.9.h.e	20
		400.9.h.f	24
400.9.i	\(\chi_{400}(93, \cdot)\)	n/a	572	2
400.9.k	\(\chi_{400}(99, \cdot)\)	n/a	572	2
400.9.m	\(\chi_{400}(57, \cdot)\)	None	0	2
400.9.p	\(\chi_{400}(193, \cdot)\)	n/a	142	2
400.9.r	\(\chi_{400}(51, \cdot)\)	n/a	602	2
400.9.t	\(\chi_{400}(157, \cdot)\)	n/a	572	2
400.9.v	\(\chi_{400}(71, \cdot)\)	None	0	4
400.9.x	\(\chi_{400}(79, \cdot)\)	n/a	480	4
400.9.z	\(\chi_{400}(31, \cdot)\)	n/a	480	4
400.9.ba	\(\chi_{400}(39, \cdot)\)	None	0	4
400.9.bc	\(\chi_{400}(53, \cdot)\)	n/a	3824	8
400.9.bf	\(\chi_{400}(19, \cdot)\)	n/a	3824	8
400.9.bg	\(\chi_{400}(17, \cdot)\)	n/a	952	8
400.9.bj	\(\chi_{400}(73, \cdot)\)	None	0	8
400.9.bk	\(\chi_{400}(11, \cdot)\)	n/a	3824	8
400.9.bn	\(\chi_{400}(13, \cdot)\)	n/a	3824	8

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

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

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