Atlas › Modular forms › Weight 2 › Level 686 › 686.2.a

Newspace 686.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(686))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
686.2.a.a	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} + O(q^{8})$	+
686.2.a.b	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - q^{6} + O(q^{8})$	+
686.2.a.c	3	degree 3	$3q + 3q^{2} - 5q^{3} + 3q^{4} - 4q^{5} - 5q^{6} + O(q^{8})$	−
686.2.a.d	3	degree 3	$3q + 3q^{2} + 5q^{3} + 3q^{4} + 4q^{5} + 5q^{6} + O(q^{8})$	−
686.2.a.e	6	degree 6	$6q - 6q^{2} + 6q^{4} + O(q^{8})$	+
686.2.a.f	6	degree 6	$6q + 6q^{2} + 6q^{4} + O(q^{8})$	−

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 $T$
686.2.a.a	$x^{3} + x^{2} - 65x - 169$
686.2.a.b	$x^{3} - x^{2} - 65x + 169$
686.2.a.c	$x^{3} + 17x^{2} + 87x + 127$
686.2.a.d	$x^{3} - 17x^{2} + 87x - 127$
686.2.a.e	$x^{6} - 334x^{4} + 32871x^{2} - 808781$
686.2.a.f	$x^{6} - 178x^{4} + 8403x^{2} - 119351$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(686))$	24
Old forms	61
Cusp forms $S_{2}(\Gamma_0(686))$	85
Eisenstein series $E_{2}(\Gamma_0(686))$	27
Modular forms $M_{2}(\Gamma_0(686))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(686)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(343))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.354 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, 7, 14, 49, 98, 343, 686]: [0, 1, 1, 4, 5, 16, 33, 100] (certified at ell = 2147479489)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (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(686, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 686:2 1000

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