Atlas › Modular forms › Weight 8 › Level 52 › 52.8.a

Newspace 52.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(52))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
52.8.a.a	1	$\mathbb{Q}$	$q - 28q^{3} - 418q^{5} + 124q^{7} + O(q^{8})$	−
52.8.a.b	3	degree 3	$3q + 8q^{5} - 342q^{7} + O(q^{8})$	+
52.8.a.c	3	degree 3	$3q + 28q^{3} + 520q^{5} - 736q^{7} + O(q^{8})$	−

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 $T$
52.8.a.a	$x + 446$
52.8.a.b	$x^{3} - 8x^{2} - 34420x + 1851200$
52.8.a.c	$x^{3} - 548x^{2} + 38308x - 536640$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(52))$	7
Old forms	39
Cusp forms $S_{8}(\Gamma_0(52))$	46
Eisenstein series $E_{8}(\Gamma_0(52))$	6
Modular forms $M_{8}(\Gamma_0(52))$	52

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(52)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 13, 26, 52]: [1, 3, 5, 9, 27, 52] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(52, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 52:8 1000

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