Atlas › Modular forms › Weight 4 › Level 39 › 39.4.a

Newspace 39.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(39))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
39.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 8q^{4} - 12q^{5} + 2q^{7} + O(q^{8})$	+ −	$-1$
39.4.a.b	2	$\mathbb{Q}(\sqrt{14})$	$2q + 2q^{2} - 6q^{3} + 14q^{4} + 24q^{5} - 6q^{6} + O(q^{8})$	+ +	$+1$
39.4.a.c	3	degree 3	$3q + 2q^{2} + 9q^{3} + 10q^{4} + 4q^{5} + 6q^{6} + 30q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
39.4.a.a	$x + 36$
39.4.a.b	$x^{2} - 74x + 1019$
39.4.a.c	$x^{3} - 14x^{2} - 2247x - 27696$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(39))$	6
Old forms	6
Cusp forms $S_{4}(\Gamma_0(39))$	12
Eisenstein series $E_{4}(\Gamma_0(39))$	4
Modular forms $M_{4}(\Gamma_0(39))$	16

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(39)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.099 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, 2, 5, 16] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 6 = 1 x 6 (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(39, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 39:4 1000

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