Atlas › Modular forms › Weight 2 › Level 211 › 211.2.a

Newspace 211.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(211))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
211.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} + 2q^{5} + 4q^{6} + q^{7} + O(q^{8})$	−	$+1$
211.2.a.b	3	degree 3	$3q - 2q^{2} - q^{3} - 8q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+	$-1$
211.2.a.c	3	degree 3	$3q - 3q^{3} + 2q^{4} - 5q^{5} - 8q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
211.2.a.d	9	degree 9	$9q - q^{2} - q^{3} + 11q^{4} + 15q^{5} - 5q^{6} - 2q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
211.2.a.a	$x^{2} - 10x + 5$
211.2.a.b	$x^{3} + 5x^{2} - 22x - 13$
211.2.a.c	$x^{3} + 9x^{2} + 11x - 29$
211.2.a.d	$x^{9} + 4x^{8} - 176x^{7} - 395x^{6} + 10908x^{5} + 3867x^{4} - 246603x^{3} + 347656x^{2} + 536892x - 786568$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(211))$	17
Old forms	0
Cusp forms $S_{2}(\Gamma_0(211))$	17
Eisenstein series $E_{2}(\Gamma_0(211))$	1
Modular forms $M_{2}(\Gamma_0(211))$	18

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.166 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, 211]: [0, 18] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 17 = 1 x 17 (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(211, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 211:2 1000

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