Atlas › Modular forms › Weight 8 › Level 51 › 51.8.a

Newspace 51.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(51))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
51.8.a.a	2	$\mathbb{Q}(\sqrt{1177})$	$2q + 7q^{2} + 54q^{3} + 357q^{4} + 320q^{5} + 189q^{6} - 1500q^{7} + O(q^{8})$	− −	$+1$
51.8.a.b	3	degree 3	$3q - 7q^{2} - 81q^{3} + 233q^{4} - 440q^{5} + 189q^{6} + 808q^{7} + O(q^{8})$	+ −	$-1$
51.8.a.c	4	degree 4	$4q - 15q^{2} + 108q^{3} + 141q^{4} - 310q^{5} - 405q^{6} - 1036q^{7} + O(q^{8})$	− +	$-1$
51.8.a.d	4	degree 4	$4q + 2q^{2} + 108q^{3} + 104q^{4} + 257q^{5} + 54q^{6} + 2244q^{7} + O(q^{8})$	− −	$+1$
51.8.a.e	5	degree 5	$5q - 15q^{2} - 135q^{3} + 553q^{4} + 173q^{5} + 405q^{6} + 400q^{7} + O(q^{8})$	+ +	$+1$

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 $T$
51.8.a.a	$x^{2} - 327x - 103032$
51.8.a.b	$x^{3} + 447x^{2} + 35091x - 1252395$
51.8.a.c	$x^{4} + 325x^{3} - 48119x^{2} - 21143301x - 1301279706$
51.8.a.d	$x^{4} - 259x^{3} - 263303x^{2} + 69197553x + 2890266480$
51.8.a.e	$x^{5} - 158x^{4} - 172640x^{3} + 9979260x^{2} + 6607989171x - 80591922378$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(51))$	18
Old forms	22
Cusp forms $S_{8}(\Gamma_0(51))$	40
Eisenstein series $E_{8}(\Gamma_0(51))$	4
Modular forms $M_{8}(\Gamma_0(51))$	44

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(51)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 12, 44] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 51:8 1000

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