Space of modular forms of level 392 and weight 4

• Label: 392.4
• Level: 392
• Weight: 4
• Dimension: 7521
• Nonzero newspaces: 12
• Sturm bound: 37632
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 12 \)
Sturm bound:			\(37632\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(392))\).

	Total	New	Old
Modular forms	14472	7715	6757
Cusp forms	13752	7521	6231
Eisenstein series	720	194	526

Trace form

7521 * q - 32 * q^2 - 34 * q^3 - 42 * q^4 - 2 * q^5 - 2 * q^6 - 36 * q^7 - 14 * q^8 + 11 * q^9 - 86 * q^10 - 158 * q^11 - 86 * q^12 - 134 * q^13 - 36 * q^14 - 198 * q^15 - 14 * q^16 - 50 * q^17 - 352 * q^18 - 322 * q^19 - 962 * q^20 + 168 * q^21 - 318 * q^22 - 138 * q^23 + 938 * q^24 + 601 * q^25 + 1750 * q^26 + 1406 * q^27 + 1260 * q^28 + 198 * q^29 + 1874 * q^30 + 1806 * q^31 + 38 * q^32 + 920 * q^33 - 1082 * q^34 - 540 * q^35 - 2742 * q^36 - 1422 * q^37 - 3214 * q^38 - 3522 * q^39 - 1306 * q^40 - 2366 * q^41 + 1260 * q^42 - 3362 * q^43 + 3834 * q^44 - 3350 * q^45 + 2374 * q^46 - 450 * q^47 + 762 * q^48 + 342 * q^49 - 2222 * q^50 + 5734 * q^51 - 5234 * q^52 + 3706 * q^53 - 8302 * q^54 + 8398 * q^55 - 2346 * q^56 + 6012 * q^57 - 5082 * q^58 + 4414 * q^59 - 6906 * q^60 + 1558 * q^61 - 1882 * q^62 - 744 * q^63 + 1950 * q^64 - 2008 * q^65 + 3810 * q^66 - 10678 * q^67 - 3222 * q^68 - 12232 * q^69 - 1068 * q^70 - 10774 * q^71 - 394 * q^72 - 3554 * q^73 + 82 * q^74 - 1730 * q^75 + 4250 * q^76 + 1242 * q^77 + 9410 * q^78 + 4686 * q^79 + 14814 * q^80 - 7989 * q^81 + 11774 * q^82 + 5090 * q^83 + 8928 * q^84 + 7712 * q^85 + 14278 * q^86 + 9486 * q^87 + 9378 * q^88 + 4594 * q^89 + 11966 * q^90 + 7116 * q^91 + 6846 * q^92 + 11224 * q^93 + 9450 * q^94 + 16318 * q^95 + 2738 * q^96 - 4610 * q^97 - 5436 * q^98 + 1324 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.4.a	\(\chi_{392}(1, \cdot)\)	392.4.a.a	1	1
		392.4.a.b	1
		392.4.a.c	1
		392.4.a.d	1
		392.4.a.e	1
		392.4.a.f	2
		392.4.a.g	2
		392.4.a.h	2
		392.4.a.i	3
		392.4.a.j	3
		392.4.a.k	3
		392.4.a.l	3
		392.4.a.m	4
		392.4.a.n	4
392.4.b	\(\chi_{392}(197, \cdot)\)	n/a	118	1
392.4.e	\(\chi_{392}(195, \cdot)\)	n/a	116	1
392.4.f	\(\chi_{392}(391, \cdot)\)	None	0	1
392.4.i	\(\chi_{392}(177, \cdot)\)	392.4.i.a	2	2
		392.4.i.b	2
		392.4.i.c	2
		392.4.i.d	2
		392.4.i.e	2
		392.4.i.f	2
		392.4.i.g	2
		392.4.i.h	2
		392.4.i.i	4
		392.4.i.j	4
		392.4.i.k	4
		392.4.i.l	4
		392.4.i.m	6
		392.4.i.n	6
		392.4.i.o	8
		392.4.i.p	8
392.4.l	\(\chi_{392}(31, \cdot)\)	None	0	2
392.4.m	\(\chi_{392}(19, \cdot)\)	n/a	232	2
392.4.p	\(\chi_{392}(165, \cdot)\)	n/a	232	2
392.4.q	\(\chi_{392}(57, \cdot)\)	n/a	252	6
392.4.t	\(\chi_{392}(55, \cdot)\)	None	0	6
392.4.u	\(\chi_{392}(27, \cdot)\)	n/a	996	6
392.4.x	\(\chi_{392}(29, \cdot)\)	n/a	996	6
392.4.y	\(\chi_{392}(9, \cdot)\)	n/a	504	12
392.4.z	\(\chi_{392}(37, \cdot)\)	n/a	1992	12
392.4.bc	\(\chi_{392}(3, \cdot)\)	n/a	1992	12
392.4.bd	\(\chi_{392}(47, \cdot)\)	None	0	12

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(392)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)