Atlas › Modular forms › Weight 2 › Level 680 › 680.2.a

Newspace 680.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(680))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
680.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + 2q^{7} + O(q^{8})$	− −	680b1 rank 0
680.2.a.b	1	$\mathbb{Q}$	$q - q^{5} + O(q^{8})$	+ −	680a1 rank 1
680.2.a.c	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 2q^{7} + O(q^{8})$	− −	680c1 rank 0
680.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{5} - 6q^{7} + O(q^{8})$	− +
680.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + O(q^{8})$	+ +
680.2.a.f	3	degree 3	$3q + q^{3} - 3q^{5} + O(q^{8})$	+ −
680.2.a.g	3	degree 3	$3q + q^{3} - 3q^{5} + 6q^{7} + O(q^{8})$	+ +
680.2.a.h	3	degree 3	$3q + 3q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
680.2.a.a	$x - 5$
680.2.a.b	$x$
680.2.a.c	$x - 8$
680.2.a.d	$x^{2} + 20x + 88$
680.2.a.e	$x^{2} - 8$
680.2.a.f	$x^{3} - x^{2} - 180x + 208$
680.2.a.g	$x^{3} - 19x^{2} + 1016$
680.2.a.h	$x^{3} - 15x^{2} - 28x - 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(680))$	16
Old forms	85
Cusp forms $S_{2}(\Gamma_0(680))$	101
Eisenstein series $E_{2}(\Gamma_0(680))$	15
Modular forms $M_{2}(\Gamma_0(680))$	116

Decomposition of the old subspace

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, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680]: [0, 1, 2, 1, 3, 3, 2, 6, 6, 10, 12, 10, 22, 30, 60, 116] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(680, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 680:2 1000

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