Atlas › Modular forms › Weight 2 › Level 621 › 621.2.a

Newspace 621.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(621))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
621.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	−	621b1 rank 1
621.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 3q^{5} + 4q^{7} + O(q^{8})$	+	621a1 rank 0
621.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	+
621.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	−
621.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{4} + q^{5} - q^{7} + O(q^{8})$	−
621.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{5} + 2q^{7} + O(q^{8})$	+
621.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{5} + 2q^{7} + O(q^{8})$	−
621.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - q^{5} - 4q^{7} + O(q^{8})$	+
621.2.a.i	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} - q^{5} - q^{7} + O(q^{8})$	+
621.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	−
621.2.a.k	6	degree 6	$6q - q^{2} + 11q^{4} + q^{5} + 4q^{7} + O(q^{8})$	+
621.2.a.l	6	degree 6	$6q + q^{2} + 11q^{4} - q^{5} + 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
621.2.a.a	$x + 10$
621.2.a.b	$x - 10$
621.2.a.c	$x^{2} - 4x - 94$
621.2.a.d	$x^{2} - 2x - 4$
621.2.a.e	$x^{2} - 2x - 16$
621.2.a.f	$x^{2} + 12x + 28$
621.2.a.g	$x^{2} - 12x + 28$
621.2.a.h	$x^{2} + 2x - 4$
621.2.a.i	$x^{2} + 2x - 16$
621.2.a.j	$x^{2} + 4x - 94$
621.2.a.k	$x^{6} - 2x^{5} - 200x^{4} + 674x^{3} + 8511x^{2} - 34304x + 27984$
621.2.a.l	$x^{6} + 2x^{5} - 200x^{4} - 674x^{3} + 8511x^{2} + 34304x + 27984$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(621))$	30
Old forms	37
Cusp forms $S_{2}(\Gamma_0(621))$	67
Eisenstein series $E_{2}(\Gamma_0(621))$	11
Modular forms $M_{2}(\Gamma_0(621))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(621)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(207))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.343 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, 3, 9, 23, 27, 69, 207, 621]: [0, 1, 2, 3, 4, 10, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (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(621, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 621:2 1000

← 620.2.a · 622.2.a →