Atlas › Modular forms › Weight 2 › Level 496 › 496.2.a

Newspace 496.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(496))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
496.2.a.a	1	$\mathbb{Q}$	$q - 3q^{5} + 3q^{7} + O(q^{8})$	+	496a1 rank 1
496.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$	−	496f1 rank 1
496.2.a.c	1	$\mathbb{Q}$	$q + q^{5} - 3q^{7} + O(q^{8})$	−	496e1 rank 1
496.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} - 3q^{5} + q^{7} + O(q^{8})$	+	496d1 rank 0
496.2.a.e	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 3q^{7} + O(q^{8})$	−	496b1 rank 0
496.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + O(q^{8})$	−	496c1 rank 0
496.2.a.g	2	$\mathbb{Q}(\sqrt{33})$	$2q - 4q^{3} + 3q^{5} - q^{7} + O(q^{8})$	−
496.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 4q^{7} + O(q^{8})$	+
496.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	+
496.2.a.j	3	degree 3	$3q - 2q^{3} - 3q^{5} - 5q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23} + 9T_{29} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
496.2.a.a	$x + 21$
496.2.a.b	$x + 166$
496.2.a.c	$x + 109$
496.2.a.d	$x + 171$
496.2.a.e	$x - 61$
496.2.a.f	$x + 56$
496.2.a.g	$x^{2} - 133x - 2516$
496.2.a.h	$x^{2} - 258x + 16134$
496.2.a.i	$x^{2} - 272x + 5491$
496.2.a.j	$x^{3} + 443x^{2} + 21916x - 2614478$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(496))$	15
Old forms	44
Cusp forms $S_{2}(\Gamma_0(496))$	59
Eisenstein series $E_{2}(\Gamma_0(496))$	11
Modular forms $M_{2}(\Gamma_0(496))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(496)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(248))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.682 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, 8, 16, 31, 62, 124, 248, 496]: [0, 1, 2, 3, 4, 3, 10, 19, 36, 68] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8), (29, 9), (37, 10), (41, 11), (43, 12)]; h of degree 15 = 1 x 15 (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(496, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 496:2 1000

← 495.2.a · 497.2.a →