Atlas › Modular forms › Weight 2 › Level 111 › 111.2.a

Newspace 111.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(111))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
111.2.a.a	3	degree 3	$3q + 3q^{2} - 3q^{3} + 5q^{4} + 4q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$
111.2.a.b	4	degree 4	$4q + 4q^{3} + 4q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− +	$+1$

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 $T$
111.2.a.a	$x^{3} - 7x^{2} + 13x - 5$
111.2.a.b	$x^{4} + 2x^{3} - 10x^{2} - 20x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(111))$	7
Old forms	4
Cusp forms $S_{2}(\Gamma_0(111))$	11
Eisenstein series $E_{2}(\Gamma_0(111))$	3
Modular forms $M_{2}(\Gamma_0(111))$	14

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(111)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.083 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, 37, 111]: [0, 1, 3, 14] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(111, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 111:2 1000

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