Space of modular forms of level 180 and weight 2

• Label: 180.2
• Level: 180
• Weight: 2
• Dimension: 337
• Nonzero newspaces: 12
• Newform subspaces: 21
• Sturm bound: 3456
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 12 \)
Newform subspaces:			\( 21 \)
Sturm bound:			\(3456\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(180))\).

	Total	New	Old
Modular forms	1024	393	631
Cusp forms	705	337	368
Eisenstein series	319	56	263

Trace form

337 * q + 6 * q^4 - 4 * q^5 - 10 * q^6 + 8 * q^7 + 12 * q^9 - 6 * q^10 + 22 * q^11 - 20 * q^12 + 22 * q^13 - 36 * q^14 + 3 * q^15 - 30 * q^16 - 8 * q^17 - 52 * q^18 + 12 * q^19 - 58 * q^20 - 26 * q^21 - 42 * q^22 - 24 * q^23 - 42 * q^24 - 14 * q^25 - 68 * q^26 - 24 * q^27 - 56 * q^28 - 72 * q^29 - 30 * q^30 - 18 * q^31 - 10 * q^32 - 78 * q^33 - 42 * q^34 - 56 * q^35 + 18 * q^36 - 32 * q^37 + 10 * q^38 - 50 * q^39 - 42 * q^40 - 100 * q^41 + 16 * q^42 - 12 * q^43 - 111 * q^45 - 40 * q^46 - 60 * q^47 - 34 * q^48 - 81 * q^49 + 26 * q^50 - 48 * q^51 + 8 * q^52 - 104 * q^53 - 34 * q^54 - 26 * q^55 + 68 * q^56 - 100 * q^57 + 20 * q^58 - 38 * q^59 + 46 * q^60 - 96 * q^61 + 100 * q^62 + 2 * q^63 + 60 * q^64 - 37 * q^65 + 88 * q^66 + 20 * q^67 + 134 * q^68 + 34 * q^69 + 42 * q^70 + 84 * q^71 + 138 * q^72 + 40 * q^73 + 132 * q^74 + 69 * q^75 - 26 * q^76 + 30 * q^77 + 152 * q^78 + 30 * q^79 + 112 * q^80 + 96 * q^81 - 76 * q^82 + 84 * q^83 + 100 * q^84 - 54 * q^85 + 22 * q^86 + 130 * q^87 - 42 * q^88 + 98 * q^89 + 138 * q^90 + 48 * q^91 + 56 * q^92 - 2 * q^93 - 48 * q^94 + 96 * q^95 + 104 * q^96 - 34 * q^97 + 126 * q^98 + 94 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)

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
180.2.a	\(\chi_{180}(1, \cdot)\)	180.2.a.a	1	1
180.2.d	\(\chi_{180}(109, \cdot)\)	180.2.d.a	2	1
180.2.e	\(\chi_{180}(71, \cdot)\)	180.2.e.a	8	1
180.2.h	\(\chi_{180}(179, \cdot)\)	180.2.h.a	4	1
		180.2.h.b	8
180.2.i	\(\chi_{180}(61, \cdot)\)	180.2.i.a	2	2
		180.2.i.b	6
180.2.j	\(\chi_{180}(17, \cdot)\)	180.2.j.a	4	2
180.2.k	\(\chi_{180}(127, \cdot)\)	180.2.k.a	2	2
		180.2.k.b	2
		180.2.k.c	2
		180.2.k.d	8
		180.2.k.e	12
180.2.n	\(\chi_{180}(59, \cdot)\)	180.2.n.a	4	2
		180.2.n.b	4
		180.2.n.c	8
		180.2.n.d	48
180.2.q	\(\chi_{180}(11, \cdot)\)	180.2.q.a	48	2
180.2.r	\(\chi_{180}(49, \cdot)\)	180.2.r.a	12	2
180.2.w	\(\chi_{180}(77, \cdot)\)	180.2.w.a	24	4
180.2.x	\(\chi_{180}(7, \cdot)\)	180.2.x.a	128	4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(180))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(180)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 1}\)