Atlas › Modular forms › Weight 2 › Level 844 › 844.2.a

Newspace 844.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(844))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
844.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{5} - 4q^{7} + O(q^{8})$	−
844.2.a.b	5	degree 5	$5q - 4q^{3} - 2q^{5} - q^{7} + O(q^{8})$	−
844.2.a.c	10	degree 10	$10q + 6q^{3} + 3q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
844.2.a.a	$x^{2} - 2x - 1$
844.2.a.b	$x^{5} + 6x^{4} + 5x^{3} - 21x^{2} - 30x - 1$
844.2.a.c	$x^{10} - 6x^{9} - 41x^{8} + 266x^{7} + 375x^{6} - 3145x^{5} + 302x^{4} + 8310x^{3} - 352x^{2} - 6609x - 1933$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(844))$	17
Old forms	87
Cusp forms $S_{2}(\Gamma_0(844))$	104
Eisenstein series $E_{2}(\Gamma_0(844))$	5
Modular forms $M_{2}(\Gamma_0(844))$	109

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(844)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(211))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(422))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 211, 422, 844]: [0, 1, 2, 18, 55, 109] (certified at ell = 2147482681)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 2 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(844, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 844:2 1000

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