Atlas › Modular forms › Weight 2 › Level 841 › 841.2.a

Newspace 841.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(841))$

Newforms

Label	Dim	Coefficient field	Trace form
841.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + q^{3} - q^{4} + q^{5} - 3q^{6} + O(q^{8})$
841.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{4} + 6q^{5} - 10q^{6} + 4q^{7} + O(q^{8})$
841.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{3} - q^{4} + q^{5} - 3q^{6} + O(q^{8})$
841.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 6q^{6} + O(q^{8})$
841.2.a.e	3	degree 3	$3q - q^{2} + q^{3} - q^{4} - 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$
841.2.a.f	3	degree 3	$3q + q^{2} - q^{3} - q^{4} - 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$
841.2.a.g	6	degree 6	$6q - 2q^{2} - 2q^{3} + 8q^{4} - 2q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$
841.2.a.h	6	degree 6	$6q + 2q^{2} + 2q^{3} + 8q^{4} - 2q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$
841.2.a.i	8	degree 8	$8q - 4q^{2} - 6q^{3} + 6q^{4} - q^{5} + 3q^{6} + O(q^{8})$
841.2.a.j	8	degree 8	$8q + 4q^{2} + 6q^{3} + 6q^{4} - q^{5} + 3q^{6} + O(q^{8})$
841.2.a.k	12	degree 12	$12q + 8q^{4} + 8q^{5} + 24q^{6} + 10q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
841.2.a.a	$x^{2} - 2x - 4$
841.2.a.b	$x^{2} - 20$
841.2.a.c	$x^{2} + 2x - 4$
841.2.a.d	$x^{2} + 4x - 4$
841.2.a.e	$x^{3} - 2x^{2} - 29x - 13$
841.2.a.f	$x^{3} + 2x^{2} - 29x + 13$
841.2.a.g	$x^{6} + 8x^{5} - 77x^{4} - 477x^{3} + 1591x^{2} + 2490x - 6705$
841.2.a.h	$x^{6} - 8x^{5} - 77x^{4} + 477x^{3} + 1591x^{2} - 2490x - 6705$
841.2.a.i	$x^{8} + 22x^{7} + 78x^{6} - 1256x^{5} - 8625x^{4} + 12979x^{3} + 177803x^{2} + 144192x - 493949$
841.2.a.j	$x^{8} - 22x^{7} + 78x^{6} + 1256x^{5} - 8625x^{4} - 12979x^{3} + 177803x^{2} - 144192x - 493949$
841.2.a.k	$x^{12} - 295x^{10} + 30262x^{8} - 1317401x^{6} + 27188372x^{4} - 263453436x^{2} + 960814009$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(841))$	54
Old forms	4
Cusp forms $S_{2}(\Gamma_0(841))$	58
Eisenstein series $E_{2}(\Gamma_0(841))$	29
Modular forms $M_{2}(\Gamma_0(841))$	87

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(841)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 29, 841]: [0, 3, 73] (certified at ell = 2147480161)
T = [(2, 1), (3, 3)]; h of degree 54 = 1 x 54 (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(841, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 841:2 1000

← 840.2.a · 842.2.a →