Space of modular forms of level 49 and weight 16

• Label: 49.16
• Level: 49
• Weight: 16
• Dimension: 1387
• Nonzero newspaces: 4
• Sturm bound: 3136
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 49 = 7^{2} \)
Weight:	\( k \)	=	\( 16 \)
Nonzero newspaces:			\( 4 \)
Sturm bound:			\(3136\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_1(49))\).

	Total	New	Old
Modular forms	1500	1436	64
Cusp forms	1440	1387	53
Eisenstein series	60	49	11

Trace form

1387 * q + 201 * q^2 + 5385 * q^3 + 13873 * q^4 + 519111 * q^5 - 4082409 * q^6 + 3214806 * q^7 + 1309419 * q^8 + 682602 * q^9 + 26736159 * q^10 - 123292059 * q^11 + 1142059527 * q^12 - 1441164655 * q^13 + 350643762 * q^14 + 4072523229 * q^15 - 7376275511 * q^16 - 5861585721 * q^17 + 35890333005 * q^18 - 4890632875 * q^19 - 90397545525 * q^20 + 29602424922 * q^21 + 65477808129 * q^22 - 82989760779 * q^23 - 152897101929 * q^24 + 233770894900 * q^25 + 84747890115 * q^26 - 321949699353 * q^27 - 64223981850 * q^28 + 469322763555 * q^29 + 73535058831 * q^30 + 240591366185 * q^31 - 557521257603 * q^32 - 477254342427 * q^33 + 3696239242911 * q^34 - 155596957278 * q^35 - 11688404535369 * q^36 - 684209333509 * q^37 + 8466362996331 * q^38 + 2510345706981 * q^39 - 19200026130021 * q^40 - 10768880440815 * q^41 + 13945605345357 * q^42 + 5686719937313 * q^43 + 13087580101923 * q^44 - 41777327558487 * q^45 - 33437252862993 * q^46 - 1208184976119 * q^47 + 82021356843354 * q^48 + 16230546778992 * q^49 + 23765062426674 * q^50 - 83501108790579 * q^51 - 75372185823277 * q^52 + 103421963310867 * q^53 + 201257199769719 * q^54 - 140419538043195 * q^55 - 145458540560004 * q^56 - 24987420092091 * q^57 - 22769446949001 * q^58 + 286446242987949 * q^59 + 523539468135075 * q^60 - 302481241126549 * q^61 - 601761012048933 * q^62 + 162191235005016 * q^63 - 254381006084873 * q^64 + 489364370502987 * q^65 + 804720258107439 * q^66 - 201008482285459 * q^67 - 809609659370841 * q^68 - 89271924484017 * q^69 - 523682049092223 * q^70 - 469303495044879 * q^71 + 2815523176705389 * q^72 + 1623571819790783 * q^73 - 33951258528453 * q^74 - 1585946211230301 * q^75 - 3391693976826169 * q^76 - 575127477805356 * q^77 + 1253047720326969 * q^78 + 2309993527758173 * q^79 + 8710504254543996 * q^80 - 1627161703314720 * q^81 - 5624882233127808 * q^82 - 1943810342445825 * q^83 + 4029171631474374 * q^84 + 1715877250722885 * q^85 - 1010018724687606 * q^86 + 1004076598322991 * q^87 + 8401653629238012 * q^88 + 1186056979226355 * q^89 - 2469385702207470 * q^90 - 2296067074212093 * q^91 + 2686677215805474 * q^92 - 5980566876628383 * q^93 + 6863656037118624 * q^94 + 5694996965166393 * q^95 + 2759075268060930 * q^96 - 6401282842843672 * q^97 + 11524614735482940 * q^98 + 3317344320574104 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_1(49))\)

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
49.16.a	\(\chi_{49}(1, \cdot)\)	49.16.a.a	1	1
		49.16.a.b	1
		49.16.a.c	3
		49.16.a.d	4
		49.16.a.e	6
		49.16.a.f	9
		49.16.a.g	9
		49.16.a.h	16
49.16.c	\(\chi_{49}(18, \cdot)\)	49.16.c.a	2	2
		49.16.c.b	2
		49.16.c.c	2
		49.16.c.d	6
		49.16.c.e	6
		49.16.c.f	8
		49.16.c.g	8
		49.16.c.h	12
		49.16.c.i	18
		49.16.c.j	32
49.16.e	\(\chi_{49}(8, \cdot)\)	n/a	414	6
49.16.g	\(\chi_{49}(2, \cdot)\)	n/a	828	12

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

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_1(49))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_1(49)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)