Space of modular forms of level 400 and weight 4

• Label: 400.4
• Level: 400
• Weight: 4
• Dimension: 7373
• Nonzero newspaces: 14
• Sturm bound: 38400
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 14 \)
Sturm bound:			\(38400\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(400))\).

	Total	New	Old
Modular forms	14792	7558	7234
Cusp forms	14008	7373	6635
Eisenstein series	784	185	599

Trace form

7373 * q - 26 * q^2 - 16 * q^3 - 16 * q^4 - 40 * q^5 - 72 * q^6 - 42 * q^7 - 68 * q^8 - 13 * q^9 - 32 * q^10 + 32 * q^11 + 76 * q^12 + 174 * q^13 + 164 * q^14 - 12 * q^15 + 240 * q^16 - 216 * q^17 + 150 * q^18 - 548 * q^19 - 32 * q^20 - 638 * q^21 - 612 * q^22 - 130 * q^23 - 872 * q^24 + 80 * q^25 - 344 * q^26 + 794 * q^27 - 768 * q^28 - 598 * q^29 - 1664 * q^30 - 30 * q^31 - 416 * q^32 + 34 * q^33 + 1564 * q^34 + 204 * q^35 + 2820 * q^36 + 2438 * q^37 + 2888 * q^38 + 1298 * q^39 + 3088 * q^40 + 2160 * q^41 + 4576 * q^42 - 44 * q^43 + 2972 * q^44 + 570 * q^45 + 708 * q^46 + 430 * q^47 - 2480 * q^48 - 1365 * q^49 - 2872 * q^50 - 2224 * q^51 - 7748 * q^52 - 3754 * q^53 - 4856 * q^54 - 1210 * q^55 - 528 * q^56 - 3466 * q^57 - 16 * q^58 - 1896 * q^59 + 480 * q^60 - 414 * q^61 - 104 * q^62 - 7478 * q^63 + 8216 * q^64 - 1524 * q^65 + 12716 * q^66 - 7648 * q^67 + 5032 * q^68 + 778 * q^69 + 760 * q^70 + 3866 * q^71 - 4452 * q^72 + 3788 * q^73 - 9100 * q^74 + 10408 * q^75 - 9884 * q^76 + 3522 * q^77 - 24332 * q^78 + 12034 * q^79 - 6352 * q^80 + 2545 * q^81 - 12536 * q^82 + 4356 * q^83 - 16000 * q^84 - 7744 * q^85 - 1012 * q^86 + 3002 * q^87 + 272 * q^88 + 272 * q^89 + 3688 * q^90 - 5058 * q^91 + 10672 * q^92 - 946 * q^93 + 11400 * q^94 - 2350 * q^95 + 20232 * q^96 + 6576 * q^97 + 7426 * q^98 - 2342 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.4.a	\(\chi_{400}(1, \cdot)\)	400.4.a.a	1	1
		400.4.a.b	1
		400.4.a.c	1
		400.4.a.d	1
		400.4.a.e	1
		400.4.a.f	1
		400.4.a.g	1
		400.4.a.h	1
		400.4.a.i	1
		400.4.a.j	1
		400.4.a.k	1
		400.4.a.l	1
		400.4.a.m	1
		400.4.a.n	1
		400.4.a.o	1
		400.4.a.p	1
		400.4.a.q	1
		400.4.a.r	1
		400.4.a.s	1
		400.4.a.t	1
		400.4.a.u	1
		400.4.a.v	2
		400.4.a.w	2
		400.4.a.x	2
400.4.c	\(\chi_{400}(49, \cdot)\)	400.4.c.a	2	1
		400.4.c.b	2
		400.4.c.c	2
		400.4.c.d	2
		400.4.c.e	2
		400.4.c.f	2
		400.4.c.g	2
		400.4.c.h	2
		400.4.c.i	2
		400.4.c.j	2
		400.4.c.k	2
		400.4.c.l	2
		400.4.c.m	2
400.4.d	\(\chi_{400}(201, \cdot)\)	None	0	1
400.4.f	\(\chi_{400}(249, \cdot)\)	None	0	1
400.4.j	\(\chi_{400}(43, \cdot)\)	n/a	212	2
400.4.l	\(\chi_{400}(101, \cdot)\)	n/a	222	2
400.4.n	\(\chi_{400}(143, \cdot)\)	400.4.n.a	2	2
		400.4.n.b	4
		400.4.n.c	4
		400.4.n.d	8
		400.4.n.e	8
		400.4.n.f	12
		400.4.n.g	16
400.4.o	\(\chi_{400}(7, \cdot)\)	None	0	2
400.4.q	\(\chi_{400}(149, \cdot)\)	n/a	212	2
400.4.s	\(\chi_{400}(107, \cdot)\)	n/a	212	2
400.4.u	\(\chi_{400}(81, \cdot)\)	n/a	176	4
400.4.w	\(\chi_{400}(9, \cdot)\)	None	0	4
400.4.y	\(\chi_{400}(129, \cdot)\)	n/a	176	4
400.4.bb	\(\chi_{400}(41, \cdot)\)	None	0	4
400.4.bd	\(\chi_{400}(3, \cdot)\)	n/a	1424	8
400.4.be	\(\chi_{400}(21, \cdot)\)	n/a	1424	8
400.4.bh	\(\chi_{400}(23, \cdot)\)	None	0	8
400.4.bi	\(\chi_{400}(47, \cdot)\)	n/a	360	8
400.4.bl	\(\chi_{400}(29, \cdot)\)	n/a	1424	8
400.4.bm	\(\chi_{400}(67, \cdot)\)	n/a	1424	8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

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