Atlas › Modular forms › Weight 2 › Level 692 › 692.2.a

Newspace 692.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(692))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
692.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	+	692a1 rank 0
692.2.a.b	4	degree 4	$4q - 3q^{5} - 5q^{7} + O(q^{8})$	−
692.2.a.c	10	degree 10	$10q + 2q^{3} + 3q^{5} + 11q^{7} + O(q^{8})$	+

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 $T$
692.2.a.a	$x - 4$
692.2.a.b	$x^{4} + 9x^{3} - x^{2} - 155x - 293$
692.2.a.c	$x^{10} - 11x^{9} - 408x^{8} + 4738x^{7} + 48966x^{6} - 686551x^{5} - 812380x^{4} + 32809077x^{3} - 111231597x^{2} + 76970779x - 14192597$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(692))$	15
Old forms	70
Cusp forms $S_{2}(\Gamma_0(692))$	85
Eisenstein series $E_{2}(\Gamma_0(692))$	5
Modular forms $M_{2}(\Gamma_0(692))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(692)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(173))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(346))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 173, 346, 692]: [0, 1, 2, 15, 45, 90] (certified at ell = 2147482093)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (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(692, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 692:2 1000

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