Atlas › Modular forms › Weight 6 › Level 28 › 28.6.a

Newspace 28.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(28))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
28.6.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 96q^{5} + 49q^{7} + O(q^{8})$	−
28.6.a.b	1	$\mathbb{Q}$	$q + 26q^{3} + 16q^{5} - 49q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
28.6.a.a	$x + 290$
28.6.a.b	$x - 74$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(28))$	2
Old forms	15
Cusp forms $S_{6}(\Gamma_0(28))$	17
Eisenstein series $E_{6}(\Gamma_0(28))$	6
Modular forms $M_{6}(\Gamma_0(28))$	23

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(28)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.114 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, 2, 4, 5, 12, 23] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 2 = 1 x 2 (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(28, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 28:6 1000

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