Atlas › Modular forms › Weight 2 › Level 267 › 267.2.a

Newspace 267.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(267))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
267.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+ −	$+1$	267b1 rank 0
267.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 2q^{7} + O(q^{8})$	− +	$+1$	267a1 rank 0
267.2.a.c	3	degree 3	$3q - 4q^{2} + 3q^{3} + 4q^{4} - 7q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
267.2.a.d	3	degree 3	$3q - 3q^{3} - 3q^{5} - 6q^{7} + O(q^{8})$	+ +	$-1$
267.2.a.e	3	degree 3	$3q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$
267.2.a.f	4	degree 4	$4q + q^{2} - 4q^{3} + 7q^{4} + 3q^{5} - q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
267.2.a.a	$x - 4$
267.2.a.b	$x$
267.2.a.c	$x^{3} + 11x^{2} + 38x + 41$
267.2.a.d	$x^{3} + 3x^{2} - 1$
267.2.a.e	$x^{3} - 7x^{2} + 12x - 5$
267.2.a.f	$x^{4} - 4x^{3} - 11x^{2} + 35x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(267))$	15
Old forms	14
Cusp forms $S_{2}(\Gamma_0(267))$	29
Eisenstein series $E_{2}(\Gamma_0(267))$	3
Modular forms $M_{2}(\Gamma_0(267))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(267)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 89, 267]: [0, 1, 8, 32] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(267, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 267:2 1000

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