Atlas › Modular forms › Weight 6 › Level 39 › 39.6.a

Newspace 39.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(39))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
39.6.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 9q^{3} - 28q^{4} - 74q^{5} + 18q^{6} - 112q^{7} + O(q^{8})$	− −	$-1$
39.6.a.b	2	$\mathbb{Q}(\sqrt{14})$	$2q - 4q^{2} - 18q^{3} + 56q^{4} - 48q^{5} + 36q^{6} + 72q^{7} + O(q^{8})$	+ +	$-1$
39.6.a.c	3	degree 3	$3q - 27q^{3} + 14q^{4} + 54q^{5} + 84q^{7} + O(q^{8})$	+ −	$+1$
39.6.a.d	4	degree 4	$4q + 6q^{2} + 36q^{3} + 142q^{4} + 24q^{5} + 54q^{6} + 72q^{7} + O(q^{8})$	− +	$+1$

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 $T$
39.6.a.a	$x + 72$
39.6.a.b	$x^{2} + 52x - 220$
39.6.a.c	$x^{3} - 54x^{2} - 3483x + 17496$
39.6.a.d	$x^{4} - 30x^{3} - 7773x^{2} + 206724x + 7573084$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(39))$	10
Old forms	12
Cusp forms $S_{6}(\Gamma_0(39))$	22
Eisenstein series $E_{6}(\Gamma_0(39))$	4
Modular forms $M_{6}(\Gamma_0(39))$	26

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(39)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 7, 26] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 39:6 1000

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