Space of modular forms of level 90 and weight 16

• Label: 90.16
• Level: 90
• Weight: 16
• Dimension: 783
• Nonzero newspaces: 6
• Sturm bound: 6912
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 16 \)
Nonzero newspaces:			\( 6 \)
Sturm bound:			\(6912\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_1(90))\).

	Total	New	Old
Modular forms	3304	783	2521
Cusp forms	3176	783	2393
Eisenstein series	128	0	128

Trace form

783 * q - 384 * q^2 + 8130 * q^3 + 278528 * q^4 + 9833 * q^5 + 2638080 * q^6 - 10097692 * q^7 + 6291456 * q^8 - 85912694 * q^9 - 90699904 * q^10 + 328928398 * q^11 - 219348992 * q^12 - 33669910 * q^13 + 714884608 * q^14 - 2169883914 * q^15 + 8858370048 * q^16 - 331559574 * q^17 + 14011822592 * q^18 + 2157113504 * q^19 - 7134101504 * q^20 + 5424725664 * q^21 + 34521198336 * q^22 - 79910452716 * q^23 - 17049845760 * q^24 - 74937105089 * q^25 + 449825190144 * q^26 + 82304308776 * q^27 - 323455614976 * q^28 + 853658990466 * q^29 + 223191318528 * q^30 + 944791041460 * q^31 - 103079215104 * q^32 - 2950008379130 * q^33 + 236932226560 * q^34 - 654870673952 * q^35 - 183745544192 * q^36 - 4201034513518 * q^37 - 3767905711872 * q^38 - 883222429748 * q^39 + 770357329920 * q^40 + 5696573387524 * q^41 + 5812389093376 * q^42 + 14815717878278 * q^43 - 9609090826240 * q^44 - 13835011504618 * q^45 + 1286363734016 * q^46 + 31122658359660 * q^47 + 2953326886912 * q^48 - 12811945510803 * q^49 - 43753344402304 * q^50 - 22943713558422 * q^51 + 7708822765568 * q^52 + 91297638979866 * q^53 - 36375300807936 * q^54 - 4602431511728 * q^55 - 31400396849152 * q^56 + 26408006136578 * q^57 + 68422030105344 * q^58 + 217866389659042 * q^59 - 9117488218112 * q^60 - 220053091668810 * q^61 - 86263054983168 * q^62 + 90527777462684 * q^63 - 321057395310592 * q^64 - 66180058379268 * q^65 - 140770257954816 * q^66 + 204145413032546 * q^67 + 87644460417024 * q^68 + 300554479241948 * q^69 - 438512982469632 * q^70 - 717337369009392 * q^71 + 105127826423808 * q^72 + 1486184909422826 * q^73 + 484806248752384 * q^74 + 815698128920100 * q^75 - 70131422887936 * q^76 - 2047874623986312 * q^77 - 732289064680960 * q^78 + 389232871197068 * q^79 + 122676882440192 * q^80 + 2881339043933714 * q^81 + 1240192042210560 * q^82 + 196444440397812 * q^83 + 139475060195328 * q^84 - 4129085640490862 * q^85 - 3112784595236608 * q^86 + 999595903187656 * q^87 + 565595313537024 * q^88 + 5952993239027054 * q^89 + 1531045270137856 * q^90 - 9296710606211744 * q^91 - 1309252857298944 * q^92 - 1572540870270892 * q^93 + 797748269445632 * q^94 + 954212610625428 * q^95 + 229523052298240 * q^96 + 7618168624457132 * q^97 + 7203498007938432 * q^98 + 5294851579386928 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_1(90))\)

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
90.16.a	\(\chi_{90}(1, \cdot)\)	90.16.a.a	1	1
		90.16.a.b	1
		90.16.a.c	1
		90.16.a.d	1
		90.16.a.e	1
		90.16.a.f	1
		90.16.a.g	1
		90.16.a.h	1
		90.16.a.i	1
		90.16.a.j	2
		90.16.a.k	2
		90.16.a.l	2
		90.16.a.m	2
		90.16.a.n	2
		90.16.a.o	3
		90.16.a.p	3
90.16.c	\(\chi_{90}(19, \cdot)\)	90.16.c.a	2	1
		90.16.c.b	4
		90.16.c.c	8
		90.16.c.d	8
		90.16.c.e	16
90.16.e	\(\chi_{90}(31, \cdot)\)	n/a	120	2
90.16.f	\(\chi_{90}(17, \cdot)\)	90.16.f.a	28	2
		90.16.f.b	32
90.16.i	\(\chi_{90}(49, \cdot)\)	n/a	180	2
90.16.l	\(\chi_{90}(23, \cdot)\)	n/a	360	4

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

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_1(90)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)