Space of modular forms of level 216 and weight 3

• Label: 216.3
• Level: 216
• Weight: 3
• Dimension: 1120
• Nonzero newspaces: 9
• Newform subspaces: 20
• Sturm bound: 7776
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	=	\( 3 \)
Nonzero newspaces:			\( 9 \)
Newform subspaces:			\( 20 \)
Sturm bound:			\(7776\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(216))\).

	Total	New	Old
Modular forms	2772	1184	1588
Cusp forms	2412	1120	1292
Eisenstein series	360	64	296

Trace form

1120 * q - 8 * q^2 - 12 * q^3 - 14 * q^4 - 12 * q^6 - 26 * q^7 - 26 * q^8 - 24 * q^9 - 22 * q^10 - 46 * q^11 - 12 * q^12 - 28 * q^13 + 30 * q^14 - 36 * q^15 + 54 * q^16 - 16 * q^17 - 12 * q^18 - 18 * q^19 + 66 * q^20 + 48 * q^21 + 14 * q^22 + 138 * q^23 + 42 * q^24 + 54 * q^25 + 222 * q^26 + 162 * q^27 + 132 * q^28 + 216 * q^29 + 228 * q^30 + 150 * q^31 + 202 * q^32 - 6 * q^33 - 14 * q^34 + 102 * q^35 - 72 * q^36 - 84 * q^37 - 190 * q^38 - 156 * q^39 - 122 * q^40 - 100 * q^41 - 432 * q^42 - 194 * q^43 - 694 * q^44 + 108 * q^45 - 418 * q^46 - 330 * q^47 - 246 * q^48 - 118 * q^49 - 524 * q^50 + 126 * q^51 - 298 * q^52 - 12 * q^54 - 88 * q^55 - 762 * q^56 - 42 * q^57 - 630 * q^58 + 440 * q^59 - 672 * q^60 + 380 * q^61 - 990 * q^62 + 108 * q^63 - 146 * q^64 + 300 * q^65 - 462 * q^66 + 414 * q^67 - 346 * q^68 - 120 * q^69 - 82 * q^70 - 18 * q^71 + 72 * q^72 - 128 * q^73 + 330 * q^74 - 600 * q^75 + 366 * q^76 - 1008 * q^77 + 426 * q^78 - 738 * q^79 + 1518 * q^80 - 576 * q^81 + 600 * q^82 - 1330 * q^83 + 846 * q^84 - 392 * q^85 + 1826 * q^86 - 648 * q^87 + 1046 * q^88 - 406 * q^89 - 954 * q^90 + 150 * q^91 - 810 * q^92 - 360 * q^93 + 726 * q^94 - 594 * q^95 - 1728 * q^96 + 400 * q^97 - 1058 * q^98 - 576 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(216))\)

We only show spaces with odd 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
216.3.b	\(\chi_{216}(163, \cdot)\)	216.3.b.a	16	1
		216.3.b.b	16
216.3.e	\(\chi_{216}(161, \cdot)\)	216.3.e.a	2	1
		216.3.e.b	2
		216.3.e.c	4
216.3.g	\(\chi_{216}(55, \cdot)\)	None	0	1
216.3.h	\(\chi_{216}(53, \cdot)\)	216.3.h.a	2	1
		216.3.h.b	2
		216.3.h.c	6
		216.3.h.d	6
		216.3.h.e	8
		216.3.h.f	8
216.3.j	\(\chi_{216}(125, \cdot)\)	216.3.j.a	44	2
216.3.k	\(\chi_{216}(127, \cdot)\)	None	0	2
216.3.m	\(\chi_{216}(17, \cdot)\)	216.3.m.a	4	2
		216.3.m.b	8
216.3.p	\(\chi_{216}(19, \cdot)\)	216.3.p.a	4	2
		216.3.p.b	40
216.3.r	\(\chi_{216}(43, \cdot)\)	216.3.r.a	12	6
		216.3.r.b	408
216.3.s	\(\chi_{216}(7, \cdot)\)	None	0	6
216.3.u	\(\chi_{216}(41, \cdot)\)	216.3.u.a	108	6
216.3.x	\(\chi_{216}(5, \cdot)\)	216.3.x.a	420	6

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(216))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(216)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)