Atlas › Modular forms › Weight 2 › Level 268 › 268.2.a

Newspace 268.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(268))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
268.2.a.a	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	+	268a1 rank 0
268.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - 5q^{7} + O(q^{8})$	−
268.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q + q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
268.2.a.a	$x - 4$
268.2.a.b	$x^{2} + 3x + 1$
268.2.a.c	$x^{2} + x - 5$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(268))$	5
Old forms	27
Cusp forms $S_{2}(\Gamma_0(268))$	32
Eisenstein series $E_{2}(\Gamma_0(268))$	5
Modular forms $M_{2}(\Gamma_0(268))$	37

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(268)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(134))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 67, 134, 268]: [0, 1, 2, 6, 19, 37] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(268, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 268:2 1000

← 267.2.a · 269.2.a →