Atlas › Modular forms › Weight 2 › Level 284 › 284.2.a

Newspace 284.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(284))$

Newforms

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

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 $T$
284.2.a.a	$x^{3} + 6x^{2} + 9x + 1$
284.2.a.b	$x^{3} - 2x^{2} - 11x + 21$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(284))$	6
Old forms	28
Cusp forms $S_{2}(\Gamma_0(284))$	34
Eisenstein series $E_{2}(\Gamma_0(284))$	5
Modular forms $M_{2}(\Gamma_0(284))$	39

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(284)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(142))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 71, 142, 284]: [0, 1, 2, 7, 20, 39] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 6 = 1 x 6 (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(284, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 284:2 1000

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