Space of modular forms of level 324 and weight 8

• Label: 324.8
• Level: 324
• Weight: 8
• Dimension: 9352
• Nonzero newspaces: 8
• Sturm bound: 46656
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 8 \)
Sturm bound:			\(46656\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(324))\).

	Total	New	Old
Modular forms	20682	9464	11218
Cusp forms	20142	9352	10790
Eisenstein series	540	112	428

Trace form

9352 * q - 12 * q^2 - 20 * q^4 + 189 * q^5 - 18 * q^6 + 249 * q^7 - 9 * q^8 - 36 * q^9 + 227 * q^10 - 8319 * q^11 - 18 * q^12 + 14399 * q^13 + 43473 * q^14 - 31688 * q^16 - 58974 * q^17 - 18 * q^18 - 131958 * q^19 + 130797 * q^20 + 185967 * q^21 + 105648 * q^22 + 348141 * q^23 - 18 * q^24 - 248655 * q^25 - 27 * q^26 - 416961 * q^27 + 152103 * q^28 + 738435 * q^29 - 18 * q^30 + 856575 * q^31 - 308922 * q^32 + 154215 * q^33 + 584750 * q^34 - 2827824 * q^35 - 18 * q^36 - 449710 * q^37 - 252918 * q^38 - 1746145 * q^40 - 1148775 * q^41 + 4341267 * q^42 + 3417849 * q^43 + 8102439 * q^44 - 1459062 * q^45 - 4609881 * q^46 - 10838223 * q^47 - 9962685 * q^48 - 2329319 * q^49 + 1622208 * q^50 + 4677309 * q^51 + 7819721 * q^52 + 13126686 * q^53 + 17669844 * q^54 + 6249438 * q^55 + 3456189 * q^56 - 3123774 * q^57 - 6184369 * q^58 - 24381663 * q^59 - 28050651 * q^60 - 6908497 * q^61 - 25222635 * q^62 + 1006938 * q^63 + 9299113 * q^64 + 23522985 * q^65 + 33755958 * q^66 - 4308453 * q^67 - 4337502 * q^68 - 9814626 * q^69 + 5694489 * q^70 - 17658060 * q^71 - 18 * q^72 + 5372567 * q^73 - 1590663 * q^74 - 18281250 * q^75 - 18467166 * q^76 + 8574435 * q^77 + 19665 * q^78 - 18150243 * q^79 + 35458452 * q^81 + 18332246 * q^82 + 31933089 * q^83 - 19701 * q^84 + 20360164 * q^85 + 16265676 * q^86 - 92159568 * q^87 - 9815484 * q^88 - 177696609 * q^89 - 8220321 * q^90 + 2745957 * q^91 - 207789753 * q^92 + 157006170 * q^93 - 21667083 * q^94 + 82418124 * q^95 + 64345419 * q^96 - 58005493 * q^97 + 356672817 * q^98 - 156150594 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(324))\)

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
324.8.a	\(\chi_{324}(1, \cdot)\)	324.8.a.a	3	1
		324.8.a.b	3
		324.8.a.c	7
		324.8.a.d	7
		324.8.a.e	8
324.8.b	\(\chi_{324}(323, \cdot)\)	n/a	164	1
324.8.e	\(\chi_{324}(109, \cdot)\)	324.8.e.a	2	2
		324.8.e.b	2
		324.8.e.c	2
		324.8.e.d	2
		324.8.e.e	2
		324.8.e.f	2
		324.8.e.g	4
		324.8.e.h	4
		324.8.e.i	4
		324.8.e.j	4
		324.8.e.k	6
		324.8.e.l	6
		324.8.e.m	16
324.8.h	\(\chi_{324}(107, \cdot)\)	n/a	332	2
324.8.i	\(\chi_{324}(37, \cdot)\)	n/a	126	6
324.8.l	\(\chi_{324}(35, \cdot)\)	n/a	744	6
324.8.m	\(\chi_{324}(13, \cdot)\)	n/a	1134	18
324.8.p	\(\chi_{324}(11, \cdot)\)	n/a	6768	18

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 15}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)