Space of modular forms of level 400 and weight 6

• Label: 400.6
• Level: 400
• Weight: 6
• Dimension: 12350
• Nonzero newspaces: 14
• Sturm bound: 57600
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 6 \)
Nonzero newspaces:			\( 14 \)
Sturm bound:			\(57600\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(400))\).

	Total	New	Old
Modular forms	24392	12535	11857
Cusp forms	23608	12350	11258
Eisenstein series	784	185	599

Trace form

12350 * q - 26 * q^2 - 28 * q^3 - 48 * q^4 - 40 * q^5 + 72 * q^6 + 94 * q^7 + 220 * q^8 + 56 * q^9 - 32 * q^10 - 1300 * q^11 - 20 * q^12 + 516 * q^13 - 124 * q^14 - 2892 * q^15 - 912 * q^16 + 954 * q^17 - 3162 * q^18 + 2400 * q^19 - 32 * q^20 + 7570 * q^21 + 4396 * q^22 - 3754 * q^23 + 8344 * q^24 - 6240 * q^25 + 7288 * q^26 - 7774 * q^27 - 23744 * q^28 - 16960 * q^29 - 5504 * q^30 + 4226 * q^31 + 78304 * q^32 + 72370 * q^33 + 96668 * q^34 + 2364 * q^35 - 44988 * q^36 - 75932 * q^37 - 95848 * q^38 - 63286 * q^39 - 118512 * q^40 - 68790 * q^41 - 80384 * q^42 + 4576 * q^43 + 42620 * q^44 + 67770 * q^45 + 194788 * q^46 + 150430 * q^47 + 210256 * q^48 + 33804 * q^49 + 40328 * q^50 - 141160 * q^51 - 222116 * q^52 - 182812 * q^53 + 84232 * q^54 + 9110 * q^55 + 191088 * q^56 + 216998 * q^57 + 106752 * q^58 + 211668 * q^59 + 8160 * q^60 - 5192 * q^61 - 273896 * q^62 - 267758 * q^63 + 269208 * q^64 - 65724 * q^65 - 223156 * q^66 + 159380 * q^67 - 473240 * q^68 + 75706 * q^69 - 399560 * q^70 + 563042 * q^71 - 249540 * q^72 + 78870 * q^73 + 244100 * q^74 - 156632 * q^75 + 399492 * q^76 - 183246 * q^77 + 1146388 * q^78 - 1268718 * q^79 + 492848 * q^80 - 290318 * q^81 + 640808 * q^82 - 851976 * q^83 + 610688 * q^84 + 362896 * q^85 + 106268 * q^86 + 2136530 * q^87 - 655600 * q^88 + 452906 * q^89 - 1191512 * q^90 + 1512974 * q^91 - 1623440 * q^92 + 391214 * q^93 - 907704 * q^94 - 555550 * q^95 + 619272 * q^96 - 1027454 * q^97 + 908386 * q^98 - 2738474 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.6.a	\(\chi_{400}(1, \cdot)\)	400.6.a.a	1	1
		400.6.a.b	1
		400.6.a.c	1
		400.6.a.d	1
		400.6.a.e	1
		400.6.a.f	1
		400.6.a.g	1
		400.6.a.h	1
		400.6.a.i	1
		400.6.a.j	1
		400.6.a.k	1
		400.6.a.l	1
		400.6.a.m	1
		400.6.a.n	1
		400.6.a.o	2
		400.6.a.p	2
		400.6.a.q	2
		400.6.a.r	2
		400.6.a.s	2
		400.6.a.t	2
		400.6.a.u	2
		400.6.a.v	2
		400.6.a.w	2
		400.6.a.x	3
		400.6.a.y	3
		400.6.a.z	4
		400.6.a.ba	4
400.6.c	\(\chi_{400}(49, \cdot)\)	400.6.c.a	2	1
		400.6.c.b	2
		400.6.c.c	2
		400.6.c.d	2
		400.6.c.e	2
		400.6.c.f	2
		400.6.c.g	2
		400.6.c.h	2
		400.6.c.i	2
		400.6.c.j	2
		400.6.c.k	2
		400.6.c.l	4
		400.6.c.m	4
		400.6.c.n	4
		400.6.c.o	4
		400.6.c.p	6
400.6.d	\(\chi_{400}(201, \cdot)\)	None	0	1
400.6.f	\(\chi_{400}(249, \cdot)\)	None	0	1
400.6.j	\(\chi_{400}(43, \cdot)\)	n/a	356	2
400.6.l	\(\chi_{400}(101, \cdot)\)	n/a	374	2
400.6.n	\(\chi_{400}(143, \cdot)\)	400.6.n.a	2	2
		400.6.n.b	4
		400.6.n.c	4
		400.6.n.d	4
		400.6.n.e	16
		400.6.n.f	16
		400.6.n.g	20
		400.6.n.h	24
400.6.o	\(\chi_{400}(7, \cdot)\)	None	0	2
400.6.q	\(\chi_{400}(149, \cdot)\)	n/a	356	2
400.6.s	\(\chi_{400}(107, \cdot)\)	n/a	356	2
400.6.u	\(\chi_{400}(81, \cdot)\)	n/a	296	4
400.6.w	\(\chi_{400}(9, \cdot)\)	None	0	4
400.6.y	\(\chi_{400}(129, \cdot)\)	n/a	296	4
400.6.bb	\(\chi_{400}(41, \cdot)\)	None	0	4
400.6.bd	\(\chi_{400}(3, \cdot)\)	n/a	2384	8
400.6.be	\(\chi_{400}(21, \cdot)\)	n/a	2384	8
400.6.bh	\(\chi_{400}(23, \cdot)\)	None	0	8
400.6.bi	\(\chi_{400}(47, \cdot)\)	n/a	600	8
400.6.bl	\(\chi_{400}(29, \cdot)\)	n/a	2384	8
400.6.bm	\(\chi_{400}(67, \cdot)\)	n/a	2384	8

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

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

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