Space of modular forms of level 500 and weight 4

• Label: 500.4
• Level: 500
• Weight: 4
• Dimension: 11968
• Nonzero newspaces: 9
• Sturm bound: 60000
• Trace bound: 9

Defining parameters

Level:	\( N \)	=	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 9 \)
Sturm bound:			\(60000\)
Trace bound:			\(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(500))\).

	Total	New	Old
Modular forms	22950	12224	10726
Cusp forms	22050	11968	10082
Eisenstein series	900	256	644

Trace form

11968 * q - 32 * q^2 + 4 * q^3 - 30 * q^4 - 80 * q^5 - 46 * q^6 - 16 * q^7 - 74 * q^8 - 169 * q^9 - 40 * q^10 - 20 * q^11 - 110 * q^12 + 68 * q^13 - 30 * q^14 + 130 * q^16 + 464 * q^17 + 276 * q^18 + 116 * q^19 - 40 * q^20 - 644 * q^21 + 330 * q^22 - 832 * q^23 - 50 * q^24 - 800 * q^25 - 534 * q^26 - 1232 * q^27 - 910 * q^28 - 394 * q^29 - 40 * q^30 + 448 * q^31 - 662 * q^32 + 1700 * q^33 - 30 * q^34 + 860 * q^35 - 186 * q^36 - 550 * q^37 - 3410 * q^38 - 3624 * q^39 - 2090 * q^40 - 22 * q^41 - 270 * q^42 + 460 * q^43 + 2630 * q^44 + 2240 * q^45 + 2954 * q^46 + 3608 * q^47 + 9470 * q^48 + 4757 * q^49 + 4370 * q^50 + 4496 * q^51 + 6186 * q^52 + 2074 * q^53 + 7510 * q^54 + 760 * q^55 + 2834 * q^56 - 1964 * q^57 - 538 * q^58 - 268 * q^59 - 6140 * q^60 - 634 * q^61 - 8290 * q^62 - 4664 * q^63 - 9690 * q^64 - 235 * q^65 - 5190 * q^66 - 2476 * q^67 - 2458 * q^68 - 7092 * q^69 - 40 * q^70 - 3976 * q^71 - 1662 * q^72 - 3372 * q^73 - 50 * q^74 - 5920 * q^75 + 1750 * q^76 - 6860 * q^77 + 3150 * q^78 - 3952 * q^79 - 40 * q^80 - 9415 * q^81 + 17066 * q^82 - 6252 * q^83 + 22530 * q^84 - 6965 * q^85 + 7354 * q^86 + 2896 * q^87 + 5490 * q^88 + 6504 * q^89 - 1390 * q^90 + 7792 * q^91 - 10470 * q^92 + 16404 * q^93 - 22110 * q^94 + 3160 * q^95 - 16366 * q^96 + 34100 * q^97 - 27024 * q^98 + 10580 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(500))\)

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
500.4.a	\(\chi_{500}(1, \cdot)\)	500.4.a.a	4	1
		500.4.a.b	6
		500.4.a.c	6
		500.4.a.d	8
500.4.c	\(\chi_{500}(249, \cdot)\)	500.4.c.a	4	1
		500.4.c.b	8
		500.4.c.c	12
500.4.e	\(\chi_{500}(307, \cdot)\)	n/a	288	2
500.4.g	\(\chi_{500}(101, \cdot)\)	500.4.g.a	28	4
		500.4.g.b	64
500.4.i	\(\chi_{500}(49, \cdot)\)	500.4.i.a	32	4
		500.4.i.b	56
500.4.l	\(\chi_{500}(7, \cdot)\)	n/a	1032	8
500.4.m	\(\chi_{500}(21, \cdot)\)	n/a	740	20
500.4.o	\(\chi_{500}(9, \cdot)\)	n/a	760	20
500.4.r	\(\chi_{500}(3, \cdot)\)	n/a	8920	40

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(500)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 2}\)