Space of modular forms of level 49 and weight 8

• Label: 49.8
• Level: 49
• Weight: 8
• Dimension: 634
• Nonzero newspaces: 4
• Newform subspaces: 17
• Sturm bound: 1568
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 49 = 7^{2} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 17 \)
Sturm bound:			\(1568\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(49))\).

	Total	New	Old
Modular forms	716	683	33
Cusp forms	656	634	22
Eisenstein series	60	49	11

Trace form

634 * q - 15 * q^2 - 69 * q^3 + 497 * q^4 - 9 * q^5 - 2601 * q^6 - 350 * q^7 + 3627 * q^8 + 1293 * q^9 + 18159 * q^10 - 2517 * q^11 - 40761 * q^12 + 9919 * q^13 - 46494 * q^14 + 10809 * q^15 + 178121 * q^16 + 143655 * q^17 + 91029 * q^18 - 92885 * q^19 - 750645 * q^20 - 141897 * q^21 + 453153 * q^22 + 459171 * q^23 + 944343 * q^24 + 123185 * q^25 - 589869 * q^26 - 1056213 * q^27 - 587930 * q^28 - 260763 * q^29 + 2017551 * q^30 + 1285411 * q^31 - 795459 * q^32 - 486285 * q^33 - 220497 * q^34 + 101577 * q^35 - 1965897 * q^36 - 646941 * q^37 + 3800091 * q^38 + 4919838 * q^39 + 278619 * q^40 - 4221483 * q^41 - 7030947 * q^42 - 2203025 * q^43 - 10274397 * q^44 - 4778562 * q^45 + 8006079 * q^46 + 9970161 * q^47 + 28507482 * q^48 + 11133724 * q^49 + 2034474 * q^50 - 4026753 * q^51 - 18360237 * q^52 - 12027915 * q^53 - 23613945 * q^54 - 7553850 * q^55 - 12791940 * q^56 - 12710865 * q^57 - 5053113 * q^58 + 6705273 * q^59 + 41339235 * q^60 + 26929278 * q^61 + 18945867 * q^62 - 8285718 * q^63 - 13885321 * q^64 - 3810093 * q^65 - 11376273 * q^66 - 4743365 * q^67 + 1762215 * q^68 - 26792805 * q^69 - 13492143 * q^70 - 1519671 * q^71 + 6967917 * q^72 + 8452711 * q^73 + 30210219 * q^74 + 8272659 * q^75 - 5216057 * q^76 + 8195859 * q^77 + 27112953 * q^78 + 24047827 * q^79 - 50397444 * q^80 - 4569855 * q^81 + 28597296 * q^82 + 11327967 * q^83 + 104549382 * q^84 - 16008075 * q^85 + 7323594 * q^86 - 36070557 * q^87 - 162576516 * q^88 - 63525093 * q^89 - 250564350 * q^90 - 36285487 * q^91 + 29843682 * q^92 + 92654559 * q^93 + 107721216 * q^94 + 108275883 * q^95 + 227294466 * q^96 + 29253070 * q^97 + 200716572 * q^98 + 73064580 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(49))\)

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
49.8.a	\(\chi_{49}(1, \cdot)\)	49.8.a.a	1	1
		49.8.a.b	1
		49.8.a.c	2
		49.8.a.d	2
		49.8.a.e	4
		49.8.a.f	4
		49.8.a.g	8
49.8.c	\(\chi_{49}(18, \cdot)\)	49.8.c.a	2	2
		49.8.c.b	2
		49.8.c.c	2
		49.8.c.d	4
		49.8.c.e	4
		49.8.c.f	4
		49.8.c.g	8
		49.8.c.h	16
49.8.e	\(\chi_{49}(8, \cdot)\)	49.8.e.a	186	6
49.8.g	\(\chi_{49}(2, \cdot)\)	49.8.g.a	384	12

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(49))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(49)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)