Atlas › Modular forms › Weight 2 › Level 800 › 800.2.a

Newspace 800.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(800))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
800.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{7} + O(q^{8})$		800c1 rank 1
800.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 2q^{7} + O(q^{8})$		800i1 rank 1
800.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 2q^{7} + O(q^{8})$		800f1 rank 0
800.2.a.d	1	$\mathbb{Q}$	$q + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	800a1 rank 1
800.2.a.e	1	$\mathbb{Q}$	$q + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	800h1 rank 1
800.2.a.f	1	$\mathbb{Q}$	$q + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$	800d1 rank 0
800.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - 2q^{7} + O(q^{8})$		800b1 rank 1
800.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - 2q^{7} + O(q^{8})$		800e1 rank 0
800.2.a.i	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{7} + O(q^{8})$		800g1 rank 0
800.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 6q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-20})$
800.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + O(q^{8})$
800.2.a.l	2	$\mathbb{Q}(\sqrt{5})$	$2q + O(q^{8})$
800.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
800.2.a.n	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-20})$

Hecke characteristic polynomials

The eigenvalue of $T = 3990T_{3} + 2855T_{7} + 5427T_{11} + 9524T_{13} + 7538T_{17} + 6229T_{19} + 7256T_{23} + 6862T_{29}$ 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 $T$
800.2.a.a	$x + 57006$
800.2.a.b	$x + 34958$
800.2.a.c	$x - 6222$
800.2.a.d	$x + 140840$
800.2.a.e	$x + 29780$
800.2.a.f	$x - 167020$
800.2.a.g	$x + 26706$
800.2.a.h	$x - 165234$
800.2.a.i	$x - 113694$
800.2.a.j	$x^{2} + 121966x + 1590322844$
800.2.a.k	$x^{2} - 29340x - 141634620$
800.2.a.l	$x^{2} + 29340x - 1417777020$
800.2.a.m	$x^{2} - 90288x - 924979464$
800.2.a.n	$x^{2} + 42722x - 1672311124$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(800))$	19
Old forms	78
Cusp forms $S_{2}(\Gamma_0(800))$	97
Eisenstein series $E_{2}(\Gamma_0(800))$	47
Modular forms $M_{2}(\Gamma_0(800))$	144

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.312 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, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800]: [0, 1, 2, 1, 3, 3, 4, 6, 3, 6, 10, 9, 16, 18, 28, 34, 64, 124] (certified at ell = 2147482921)
T = [(3, 3990), (7, 2855), (11, 5427), (13, 9524), (17, 7538), (19, 6229), (23, 7256), (29, 6862)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 12 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(800, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 800:2 1000

← 799.2.a · 801.2.a →