Atlas › Modular forms › Weight 2 › Level 681 › 681.2.a

Newspace 681.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(681))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
681.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$	681c1 rank 2
681.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{7} + O(q^{8})$	+ +	$-1$	681a1 rank 1
681.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 2q^{5} + q^{7} + O(q^{8})$	− −	$-1$	681e1 rank 1
681.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 4q^{5} + q^{7} + O(q^{8})$	− +	$+1$	681d1 rank 0
681.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + O(q^{8})$	+ −	$+1$	681b1 rank 0
681.2.a.f	6	degree 6	$6q - 5q^{2} + 6q^{3} + 7q^{4} - 10q^{5} - 5q^{6} - 8q^{7} + O(q^{8})$	− −	$-1$
681.2.a.g	6	degree 6	$6q - q^{2} - 6q^{3} + 3q^{4} + q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
681.2.a.h	10	degree 10	$10q - q^{2} - 10q^{3} + 15q^{4} - 4q^{5} + q^{6} + 12q^{7} + O(q^{8})$	+ −	$+1$
681.2.a.i	10	degree 10	$10q + 5q^{2} + 10q^{3} + 15q^{4} + 8q^{5} + 5q^{6} + O(q^{8})$	− +	$+1$

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 $T$
681.2.a.a	$x + 107$
681.2.a.b	$x - 5$
681.2.a.c	$x + 69$
681.2.a.d	$x + 27$
681.2.a.e	$x - 67$
681.2.a.f	$x^{6} + 90x^{5} - 4662x^{4} - 608383x^{3} - 12147443x^{2} + 104458612x + 3015654707$
681.2.a.g	$x^{6} + 78x^{5} - 3350x^{4} - 299275x^{3} - 2445633x^{2} + 72331268x + 807655327$
681.2.a.h	$x^{10} + 30x^{9} - 11937x^{8} - 718059x^{7} + 5470496x^{6} + 909155003x^{5} + 2946518012x^{4} - 410210615465x^{3} - 1263728691306x^{2} + 66792247025182x - 97212942723497$
681.2.a.i	$x^{10} - 152x^{9} + 3253x^{8} + 630513x^{7} - 48811776x^{6} + 1398992257x^{5} - 14024468770x^{4} - 104885143221x^{3} + 2845035225462x^{2} - 7153683395348x - 87507231055599$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(681))$	37
Old forms	38
Cusp forms $S_{2}(\Gamma_0(681))$	75
Eisenstein series $E_{2}(\Gamma_0(681))$	3
Modular forms $M_{2}(\Gamma_0(681))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(681)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(227))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.404 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, 227, 681]: [0, 1, 20, 78] (certified at ell = 2147479573)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 10 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(681, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 681:2 1000

← 680.2.a · 682.2.a →