Atlas › Modular forms › Weight 4 › Level 51 › 51.4.a

Newspace 51.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(51))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
51.4.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - 7q^{4} + 16q^{5} + 3q^{6} + 34q^{7} + O(q^{8})$	+ +	$+1$
51.4.a.b	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} - 7q^{4} - 20q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− +	$-1$
51.4.a.c	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 8q^{7} + O(q^{8})$	+ −	$-1$
51.4.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{3} + 20q^{4} + 6q^{5} - 8q^{7} + O(q^{8})$	+ +	$+1$
51.4.a.e	3	degree 3	$3q + 5q^{2} + 9q^{3} + 13q^{4} + 8q^{5} + 15q^{6} - 8q^{7} + O(q^{8})$	− −	$+1$

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 $T$
51.4.a.a	$x - 15$
51.4.a.b	$x + 21$
51.4.a.c	$x + 9$
51.4.a.d	$x^{2} - 6x - 441$
51.4.a.e	$x^{3} - 13x^{2} + 13x + 171$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(51))$	8
Old forms	8
Cusp forms $S_{4}(\Gamma_0(51))$	16
Eisenstein series $E_{4}(\Gamma_0(51))$	4
Modular forms $M_{4}(\Gamma_0(51))$	20

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(51)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 17, 51]: [1, 2, 6, 20] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(51, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 51:4 1000

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