Atlas › Modular forms › Weight 2 › Level 493 › 493.2.a

Newspace 493.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(493))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
493.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - q^{4} + q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	493b1 rank 1
493.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{5} - 5q^{7} + O(q^{8})$	− +	$+1$	493a1 rank 0
493.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - q^{3} - 2q^{4} + 3q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
493.2.a.d	4	degree 4	$4q + 2q^{2} + 3q^{3} + 8q^{4} + 3q^{5} - 5q^{6} + q^{7} + O(q^{8})$	− +	$+1$
493.2.a.e	5	degree 5	$5q - 2q^{2} - 2q^{3} + 4q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
493.2.a.f	6	degree 6	$6q + 5q^{2} + 2q^{3} + 7q^{4} + q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
493.2.a.g	8	degree 8	$8q + 3q^{2} + 6q^{3} + 13q^{4} + q^{5} + 11q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
493.2.a.h	10	degree 10	$10q - 5q^{2} - 9q^{3} + 11q^{4} - 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
493.2.a.a	$x + 19$
493.2.a.b	$x + 46$
493.2.a.c	$x^{2} - 49x - 356$
493.2.a.d	$x^{4} - 33x^{3} + 313x^{2} - 963x + 746$
493.2.a.e	$x^{5} + 41x^{4} + 298x^{3} - 7315x^{2} - 121481x - 491409$
493.2.a.f	$x^{6} - 30x^{5} - 2315x^{4} + 80373x^{3} + 927404x^{2} - 54616180x + 480610157$
493.2.a.g	$x^{8} - 47x^{7} - 653x^{6} + 53470x^{5} - 508099x^{4} - 4910096x^{3} + 62496389x^{2} + 130902805x - 1483158668$
493.2.a.h	$x^{10} + 80x^{9} + 1271x^{8} - 41991x^{7} - 1405399x^{6} - 8666494x^{5} + 102306002x^{4} + 1439029351x^{3} + 4542783606x^{2} - 6214239818x - 37555501649$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(493))$	37
Old forms	6
Cusp forms $S_{2}(\Gamma_0(493))$	43
Eisenstein series $E_{2}(\Gamma_0(493))$	3
Modular forms $M_{2}(\Gamma_0(493))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(493)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.309 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, 17, 29, 493]: [0, 2, 3, 46] (certified at ell = 2147480161)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(493, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 493:2 1000

← 492.2.a · 494.2.a →