Atlas › Modular forms › Weight 2 › Level 788 › 788.2.a

Newspace 788.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(788))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
788.2.a.a	$x^{2} - 9x - 11$
788.2.a.b	$x^{6} + 18x^{5} + 71x^{4} - 263x^{3} - 1361x^{2} + 1567x + 1864$
788.2.a.c	$x^{9} - 9x^{8} - 320x^{7} + 3058x^{6} + 29701x^{5} - 347161x^{4} - 355130x^{3} + 12716705x^{2} - 41087976x + 35024256$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(788))$	17
Old forms	80
Cusp forms $S_{2}(\Gamma_0(788))$	97
Eisenstein series $E_{2}(\Gamma_0(788))$	5
Modular forms $M_{2}(\Gamma_0(788))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(788)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(197))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(394))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 4, 197, 394, 788]: [0, 1, 2, 17, 51, 102] (certified at ell = 2147479489)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(788, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 788:2 1000

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