Atlas › Modular forms › Weight 12 › Level 11 › 11.12.a

Newspace 11.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(11))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
11.12.a.a	3	degree 3	$3q - 393q^{3} + 2856q^{4} - 7305q^{5} + 6864q^{6} - 5082q^{7} + O(q^{8})$	−	$-1$
11.12.a.b	5	degree 5	$5q + 32q^{2} + 160q^{3} + 6220q^{4} - 8398q^{5} + 4078q^{6} + 79040q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
11.12.a.a	$x^{3} + 393x^{2} - 100761x - 40559681$
11.12.a.b	$x^{5} - 192x^{4} - 689694x^{3} + 87529598x^{2} + 118411313409x - 7587187401978$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(11))$	8
Old forms	2
Cusp forms $S_{12}(\Gamma_0(11))$	10
Eisenstein series $E_{12}(\Gamma_0(11))$	2
Modular forms $M_{12}(\Gamma_0(11))$	12

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(11)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.484 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, 11]: [2, 12] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (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(11, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 11:12 1000

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