Space of modular forms of level 90 and weight 12

• Label: 90.12
• Level: 90
• Weight: 12
• Dimension: 573
• Nonzero newspaces: 6
• Newform subspaces: 26
• Sturm bound: 5184
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 12 \)
Nonzero newspaces:			\( 6 \)
Newform subspaces:			\( 26 \)
Sturm bound:			\(5184\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(90))\).

	Total	New	Old
Modular forms	2440	573	1867
Cusp forms	2312	573	1739
Eisenstein series	128	0	128

Trace form

573 * q - 96 * q^2 + 486 * q^3 + 11264 * q^4 - 6887 * q^5 + 6720 * q^6 + 74188 * q^7 + 98304 * q^8 - 166250 * q^9 + 398816 * q^10 - 122242 * q^11 + 40960 * q^12 + 1069830 * q^13 + 6789248 * q^14 + 737976 * q^15 + 24117248 * q^16 + 38932026 * q^17 + 37080704 * q^18 - 29836752 * q^19 - 24177664 * q^20 + 100691460 * q^21 + 104043712 * q^22 + 57901164 * q^23 + 15925248 * q^24 - 350410469 * q^25 - 99800256 * q^26 - 200561784 * q^27 - 28639232 * q^28 + 236991402 * q^29 - 101279232 * q^30 - 312081960 * q^31 - 100663296 * q^32 - 201740834 * q^33 + 116413568 * q^34 + 4009027568 * q^35 + 210368512 * q^36 + 3482028258 * q^37 - 4275453120 * q^38 - 4631878832 * q^39 - 266895360 * q^40 + 8036622056 * q^41 + 2698290688 * q^42 - 449218110 * q^43 - 4952956928 * q^44 - 5038036408 * q^45 + 1629122304 * q^46 + 12173740812 * q^47 + 1570766848 * q^48 - 10981652853 * q^49 + 1145045536 * q^50 - 18208585602 * q^51 + 4192155648 * q^52 + 30979603770 * q^53 + 23372554944 * q^54 - 48332844748 * q^55 - 1224605696 * q^56 - 55167025474 * q^57 + 10891557824 * q^58 - 2595245806 * q^59 + 12968255488 * q^60 + 17382562382 * q^61 + 40979132160 * q^62 + 34108577720 * q^63 - 59055800320 * q^64 + 17954302962 * q^65 + 13880504832 * q^66 + 42493789174 * q^67 - 38905540608 * q^68 - 32762701024 * q^69 - 47677683072 * q^70 - 13604437848 * q^71 + 29538582528 * q^72 + 151708963778 * q^73 + 51374755904 * q^74 + 138190432830 * q^75 - 26086217728 * q^76 - 227097483828 * q^77 + 73003571840 * q^78 + 132819360816 * q^79 + 18598592512 * q^80 + 271844198630 * q^81 + 82114203712 * q^82 + 277526911452 * q^83 - 117713375232 * q^84 - 312235958322 * q^85 - 64290981824 * q^86 + 62952998236 * q^87 + 106540761088 * q^88 - 163567978478 * q^89 - 59930791424 * q^90 + 1068689388864 * q^91 + 59290791936 * q^92 - 420626727424 * q^93 - 126910984832 * q^94 + 450492556428 * q^95 - 29527900160 * q^96 + 103601221596 * q^97 + 513484972896 * q^98 + 44389066396 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(90))\)

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
90.12.a	\(\chi_{90}(1, \cdot)\)	90.12.a.a	1	1
		90.12.a.b	1
		90.12.a.c	1
		90.12.a.d	1
		90.12.a.e	1
		90.12.a.f	1
		90.12.a.g	1
		90.12.a.h	1
		90.12.a.i	1
		90.12.a.j	1
		90.12.a.k	1
		90.12.a.l	2
		90.12.a.m	2
		90.12.a.n	2
90.12.c	\(\chi_{90}(19, \cdot)\)	90.12.c.a	4	1
		90.12.c.b	6
		90.12.c.c	6
		90.12.c.d	12
90.12.e	\(\chi_{90}(31, \cdot)\)	90.12.e.a	20	2
		90.12.e.b	22
		90.12.e.c	22
		90.12.e.d	24
90.12.f	\(\chi_{90}(17, \cdot)\)	90.12.f.a	20	2
		90.12.f.b	24
90.12.i	\(\chi_{90}(49, \cdot)\)	90.12.i.a	132	2
90.12.l	\(\chi_{90}(23, \cdot)\)	90.12.l.a	264	4

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(90)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)