Atlas › Modular forms › Weight 8 › Level 39 › 39.8.a

Newspace 39.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(39))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
39.8.a.a	2	$\mathbb{Q}(\sqrt{29})$	$2q - 8q^{2} + 54q^{3} + 8q^{4} - 132q^{5} - 216q^{6} - 1116q^{7} + O(q^{8})$	− +	$-1$
39.8.a.b	3	degree 3	$3q - 14q^{2} - 81q^{3} + 340q^{4} - 370q^{5} + 378q^{6} + 48q^{7} + O(q^{8})$	+ −	$-1$
39.8.a.c	4	degree 4	$4q - 6q^{2} - 108q^{3} + 406q^{4} - 276q^{5} + 162q^{6} - 1116q^{7} + O(q^{8})$	+ +	$+1$
39.8.a.d	5	degree 5	$5q + 135q^{3} + 454q^{4} + 774q^{5} + 1420q^{7} + O(q^{8})$	− −	$+1$

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 $T$
39.8.a.a	$x^{2} + 140x + 4436$
39.8.a.b	$x^{3} + 384x^{2} - 64344x - 11044800$
39.8.a.c	$x^{4} + 282x^{3} - 109713x^{2} - 20865612x + 954770284$
39.8.a.d	$x^{5} - 774x^{4} + 5793x^{3} + 103763376x^{2} - 22000980456x + 1017490921920$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(39))$	14
Old forms	16
Cusp forms $S_{8}(\Gamma_0(39))$	30
Eisenstein series $E_{8}(\Gamma_0(39))$	4
Modular forms $M_{8}(\Gamma_0(39))$	34

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(39)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 13, 39]: [1, 3, 9, 34] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 5 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(39, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 39:8 1000

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