Atlas › Modular forms › Weight 2 › Level 838 › 838.2.a

Newspace 838.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(838))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
838.2.a.a	5	degree 5	$5q + 5q^{2} - 5q^{3} + 5q^{4} - 5q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
838.2.a.b	8	degree 8	$8q - 8q^{2} + 3q^{3} + 8q^{4} + 9q^{5} - 3q^{6} + O(q^{8})$	+ −	$+1$
838.2.a.c	9	degree 9	$9q - 9q^{2} - 2q^{3} + 9q^{4} - 8q^{5} + 2q^{6} + O(q^{8})$	+ +	$-1$
838.2.a.d	12	degree 12	$12q + 12q^{2} + 6q^{3} + 12q^{4} + 6q^{5} + 6q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
838.2.a.a	$x^{5} + 10x^{4} + 34x^{3} + 47x^{2} + 22x - 1$
838.2.a.b	$x^{8} - 12x^{7} + 24x^{6} + 179x^{5} - 790x^{4} + 805x^{3} + 220x^{2} - 428x - 40$
838.2.a.c	$x^{9} + 10x^{8} + 13x^{7} - 90x^{6} - 105x^{5} + 272x^{4} + 89x^{3} - 204x^{2} - 9x + 37$
838.2.a.d	$x^{12} - 12x^{11} + 3x^{10} + 484x^{9} - 1897x^{8} - 1266x^{7} + 18487x^{6} - 22034x^{5} - 37479x^{4} + 90051x^{3} - 45220x^{2} - 804x + 200$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(838))$	34
Old forms	70
Cusp forms $S_{2}(\Gamma_0(838))$	104
Eisenstein series $E_{2}(\Gamma_0(838))$	3
Modular forms $M_{2}(\Gamma_0(838))$	107

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(838)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(419))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 2, 419, 838]: [0, 1, 36, 107] (certified at ell = 2147482943)
T = [(3, 1), (5, 1)]; h of degree 34 = 1 x 34 (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(838, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 838:2 1000

← 837.2.a · 839.2.a →