Atlas › Modular forms › Weight 6 › Level 51 › 51.6.a

Newspace 51.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(51))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
51.6.a.a	2	$\mathbb{Q}(\sqrt{145})$	$2q - 7q^{2} + 18q^{3} + 33q^{4} - 113q^{5} - 63q^{6} + O(q^{8})$	− −	$-1$
51.6.a.b	3	degree 3	$3q + 5q^{2} - 27q^{3} + 37q^{4} - 37q^{5} - 45q^{6} - 176q^{7} + O(q^{8})$	+ +	$-1$
51.6.a.c	4	degree 4	$4q + 5q^{2} + 36q^{3} + 113q^{4} + 146q^{5} + 45q^{6} + 60q^{7} + O(q^{8})$	− +	$+1$
51.6.a.d	5	degree 5	$5q + q^{2} - 45q^{3} + 117q^{4} + 4q^{5} - 9q^{6} - 40q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
51.6.a.a	$x^{2} + 120x + 2295$
51.6.a.b	$x^{3} + 32x^{2} - 4031x + 65634$
51.6.a.c	$x^{4} - 151x^{3} + 3991x^{2} + 123177x + 672282$
51.6.a.d	$x^{5} - 5x^{4} - 10079x^{3} + 135327x^{2} + 24597432x - 623316384$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(51))$	14
Old forms	14
Cusp forms $S_{6}(\Gamma_0(51))$	28
Eisenstein series $E_{6}(\Gamma_0(51))$	4
Modular forms $M_{6}(\Gamma_0(51))$	32

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(51)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 3, 8, 32] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 51:6 1000

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