Atlas › Modular forms › Weight 2 › Level 789 › 789.2.a

Newspace 789.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(789))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
789.2.a.a	7	degree 7	$7q - 3q^{2} + 7q^{3} + q^{4} - 9q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
789.2.a.b	7	degree 7	$7q + q^{2} - 7q^{3} + q^{4} + q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
789.2.a.c	14	degree 14	$14q + 4q^{2} + 14q^{3} + 18q^{4} + 7q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− +	$+1$
789.2.a.d	15	degree 15	$15q - q^{2} - 15q^{3} + 25q^{4} - q^{5} + q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
789.2.a.a	$x^{7} + 12x^{6} + 48x^{5} + 54x^{4} - 76x^{3} - 136x^{2} + 33x + 3$
789.2.a.b	$x^{7} - 2x^{6} - 12x^{5} + 18x^{4} + 34x^{3} - 20x^{2} - 3x + 1$
789.2.a.c	$x^{14} - 11x^{13} + 5x^{12} + 323x^{11} - 997x^{10} - 1569x^{9} + 9770x^{8} - 5917x^{7} - 17230x^{6} + 13064x^{5} + 11876x^{4} - 3640x^{3} - 1312x^{2} + 364x - 13$
789.2.a.d	$x^{15} + 2x^{14} - 84x^{13} - 138x^{12} + 2574x^{11} + 2708x^{10} - 38401x^{9} - 16513x^{8} + 295363x^{7} - 32292x^{6} - 1101856x^{5} + 563092x^{4} + 1600662x^{3} - 1275396x^{2} - 201013x + 222995$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(789))$	43
Old forms	44
Cusp forms $S_{2}(\Gamma_0(789))$	87
Eisenstein series $E_{2}(\Gamma_0(789))$	3
Modular forms $M_{2}(\Gamma_0(789))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(789)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(263))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 263, 789]: [0, 1, 23, 90] (certified at ell = 2147482739)
T = [(2, 1), (5, 1)]; h of degree 43 = 1 x 43 (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(789, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 789:2 1000

← 788.2.a · 790.2.a →