Atlas › Modular forms › Weight 2 › Level 651 › 651.2.a

Newspace 651.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(651))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
651.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	651d1 rank 1
651.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 3q^{5} + q^{7} + O(q^{8})$	− − −	$+1$	651e1 rank 0
651.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	651a1 rank 0
651.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 4q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	651b1 rank 0
651.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	651c1 rank 1
651.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{3} - 4q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	+ + −	$+1$
651.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 6q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+ + −	$+1$
651.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - 2q^{3} + 5q^{4} + q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
651.2.a.i	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - 2q^{3} + 5q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
651.2.a.j	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} - 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
651.2.a.k	3	degree 3	$3q + q^{2} - 3q^{3} + q^{4} - 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
651.2.a.l	6	degree 6	$6q + 6q^{3} + 12q^{4} + 8q^{5} - 6q^{7} + O(q^{8})$	− + +	$+1$
651.2.a.m	6	degree 6	$6q + q^{2} + 6q^{3} + 13q^{4} + 5q^{5} + q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
651.2.a.a	$x - 5$
651.2.a.b	$x - 26$
651.2.a.c	$x - 105$
651.2.a.d	$x - 27$
651.2.a.e	$x + 43$
651.2.a.f	$x^{2} - 6$
651.2.a.g	$x^{2} - 138x + 4041$
651.2.a.h	$x^{2} - 103x + 1424$
651.2.a.i	$x^{2} + 135x + 4552$
651.2.a.j	$x^{3} + 53x^{2} - 1517x - 60233$
651.2.a.k	$x^{3} + 33x^{2} - 117x + 67$
651.2.a.l	$x^{6} - 136x^{5} + 2076x^{4} + 402380x^{3} - 19418315x^{2} + 278689892x - 1214928634$
651.2.a.m	$x^{6} - 7x^{5} - 3800x^{4} - 41298x^{3} + 2493933x^{2} + 31109657x - 235467926$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(651))$	31
Old forms	50
Cusp forms $S_{2}(\Gamma_0(651))$	81
Eisenstein series $E_{2}(\Gamma_0(651))$	7
Modular forms $M_{2}(\Gamma_0(651))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(651)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(217))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 7, 21, 31, 93, 217, 651]: [0, 1, 1, 4, 3, 12, 22, 88] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(651, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 651:2 1000

← 650.2.a · 652.2.a →