Atlas › Modular forms › Weight 8 › Level 28 › 28.8.a

Newspace 28.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(28))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
28.8.a.a	2	$\mathbb{Q}(\sqrt{3529})$	$2q - 14q^{3} + 42q^{5} + 686q^{7} + O(q^{8})$	−
28.8.a.b	2	$\mathbb{Q}(\sqrt{1009})$	$2q + 14q^{3} - 294q^{5} - 686q^{7} + O(q^{8})$	+

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 $T$
28.8.a.a	$x^{2} - 28x - 56268$
28.8.a.b	$x^{2} + 280x - 81300$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(28))$	4
Old forms	21
Cusp forms $S_{8}(\Gamma_0(28))$	25
Eisenstein series $E_{8}(\Gamma_0(28))$	6
Modular forms $M_{8}(\Gamma_0(28))$	31

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(28)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$

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, 2, 4, 7, 14, 28]: [1, 3, 5, 5, 16, 31] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(28, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 28:8 1000

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