Space of modular forms of level 414 and weight 4

• Label: 414.4
• Level: 414
• Weight: 4
• Dimension: 3748
• Nonzero newspaces: 8
• Sturm bound: 38016
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 8 \)
Sturm bound:			\(38016\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(414))\).

	Total	New	Old
Modular forms	14608	3748	10860
Cusp forms	13904	3748	10156
Eisenstein series	704	0	704

Trace form

3748 * q - 8 * q^2 - 6 * q^3 + 16 * q^4 - 24 * q^5 + 36 * q^6 + 8 * q^7 + 16 * q^8 + 210 * q^9 + 24 * q^10 - 54 * q^11 - 48 * q^12 - 28 * q^13 - 136 * q^14 - 180 * q^15 + 64 * q^16 + 80 * q^17 - 312 * q^18 - 680 * q^19 - 448 * q^20 + 72 * q^21 - 400 * q^22 - 536 * q^23 + 48 * q^24 - 88 * q^25 + 412 * q^26 + 496 * q^28 + 1196 * q^29 + 576 * q^30 + 1288 * q^31 - 128 * q^32 - 1530 * q^33 - 684 * q^34 - 1426 * q^35 - 1032 * q^36 + 296 * q^37 - 724 * q^38 + 1164 * q^39 + 96 * q^40 - 52 * q^41 + 1632 * q^42 - 1870 * q^43 + 720 * q^44 + 1404 * q^45 - 168 * q^46 - 764 * q^47 + 288 * q^48 - 834 * q^49 - 872 * q^50 + 306 * q^51 - 112 * q^52 - 418 * q^53 - 8156 * q^54 - 10404 * q^55 - 6352 * q^56 - 10422 * q^57 - 5928 * q^58 - 3460 * q^59 + 1696 * q^60 + 2528 * q^61 + 8828 * q^62 + 11948 * q^63 - 896 * q^64 + 25500 * q^65 + 10336 * q^66 + 3770 * q^67 + 7656 * q^68 + 13466 * q^69 + 16608 * q^70 + 22668 * q^71 + 4976 * q^72 + 6416 * q^73 + 7304 * q^74 + 1546 * q^75 + 680 * q^76 - 3120 * q^77 - 3168 * q^78 - 5654 * q^79 - 2880 * q^80 - 16118 * q^81 - 9048 * q^82 - 29242 * q^83 - 9056 * q^84 - 26710 * q^85 - 15328 * q^86 + 6408 * q^87 + 336 * q^88 + 5918 * q^89 + 864 * q^90 - 7804 * q^91 + 1728 * q^92 + 5268 * q^93 - 1608 * q^94 + 6648 * q^95 - 768 * q^96 + 6964 * q^97 + 2092 * q^98 - 2304 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(414))\)

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
414.4.a	\(\chi_{414}(1, \cdot)\)	414.4.a.a	1	1
		414.4.a.b	1
		414.4.a.c	1
		414.4.a.d	1
		414.4.a.e	1
		414.4.a.f	2
		414.4.a.g	2
		414.4.a.h	2
		414.4.a.i	2
		414.4.a.j	2
		414.4.a.k	2
		414.4.a.l	3
		414.4.a.m	4
		414.4.a.n	4
414.4.d	\(\chi_{414}(413, \cdot)\)	414.4.d.a	24	1
414.4.e	\(\chi_{414}(139, \cdot)\)	n/a	132	2
414.4.f	\(\chi_{414}(137, \cdot)\)	n/a	144	2
414.4.i	\(\chi_{414}(55, \cdot)\)	n/a	300	10
414.4.j	\(\chi_{414}(17, \cdot)\)	n/a	240	10
414.4.m	\(\chi_{414}(13, \cdot)\)	n/a	1440	20
414.4.p	\(\chi_{414}(5, \cdot)\)	n/a	1440	20

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(414))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(414)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(207))\)\(^{\oplus 2}\)