Atlas › Modular forms › Weight 2 › Level 722 › 722.2.a

Newspace 722.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(722))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
722.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + 2q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	+	722b1 rank 1
722.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+	722c1 rank 0
722.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	+	722a1 rank 1
722.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	−	722f1 rank 1
722.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	−	722e1 rank 1
722.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} + 2q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	−	722d1 rank 0
722.2.a.g	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	+
722.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 5q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+
722.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 5q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	−
722.2.a.j	2	$\mathbb{Q}(\sqrt{7})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	−
722.2.a.k	3	degree 3	$3q - 3q^{2} + 3q^{4} + 6q^{5} + 6q^{7} + O(q^{8})$	+
722.2.a.l	3	degree 3	$3q + 3q^{2} + 3q^{4} + 6q^{5} + 6q^{7} + O(q^{8})$	−
722.2.a.m	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+
722.2.a.n	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
722.2.a.a	$x + 26$
722.2.a.b	$x - 18$
722.2.a.c	$x - 2$
722.2.a.d	$x$
722.2.a.e	$x + 48$
722.2.a.f	$x + 20$
722.2.a.g	$x^{2} + 12x - 216$
722.2.a.h	$x^{2} + 37x - 709$
722.2.a.i	$x^{2} + 41x - 361$
722.2.a.j	$x^{2} + 12x - 76$
722.2.a.k	$x^{3} - 90x^{2} + 2619x - 24813$
722.2.a.l	$x^{3} - 90x^{2} + 2553x - 22987$
722.2.a.m	$x^{4} + 4x^{3} - 1229x^{2} - 1256x + 350881$
722.2.a.n	$x^{4} - 1525x^{2} + 3720x + 457805$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(722))$	28
Old forms	48
Cusp forms $S_{2}(\Gamma_0(722))$	76
Eisenstein series $E_{2}(\Gamma_0(722))$	39
Modular forms $M_{2}(\Gamma_0(722))$	115

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(722)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(361))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.403 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, 19, 38, 361, 722]: [0, 1, 2, 7, 32, 97] (certified at ell = 2147483323)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(722, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 722:2 1000

← 721.2.a · 723.2.a →