Atlas › Modular forms › Weight 2 › Level 872 › 872.2.a

Newspace 872.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(872))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
872.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + O(q^{8})$	−	872a1 rank 1
872.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + O(q^{8})$	+
872.2.a.c	3	degree 3	$3q - 2q^{3} - 2q^{5} - q^{7} + O(q^{8})$	+
872.2.a.d	5	degree 5	$5q - 2q^{3} - 9q^{7} + O(q^{8})$	−
872.2.a.e	7	degree 7	$7q + 3q^{3} - 4q^{5} + 9q^{7} + O(q^{8})$	+
872.2.a.f	9	degree 9	$9q + 5q^{3} + 7q^{5} + 5q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
872.2.a.a	$x - 1$
872.2.a.b	$x^{2} + 6x + 1$
872.2.a.c	$x^{3} + 8x^{2} - 48x - 64$
872.2.a.d	$x^{5} + 2x^{4} - 103x^{3} - 380x^{2} + 864x + 3008$
872.2.a.e	$x^{7} + 9x^{6} - 188x^{5} - 1095x^{4} + 11605x^{3} + 29660x^{2} - 248080x + 84544$
872.2.a.f	$x^{9} - 26x^{8} + 43x^{7} + 3879x^{6} - 30662x^{5} - 56057x^{4} + 1203174x^{3} - 2071872x^{2} - 10220000x + 27668608$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(872))$	27
Old forms	80
Cusp forms $S_{2}(\Gamma_0(872))$	107
Eisenstein series $E_{2}(\Gamma_0(872))$	7
Modular forms $M_{2}(\Gamma_0(872))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(872)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(218))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(436))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.368 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, 8, 109, 218, 436, 872]: [0, 1, 2, 3, 9, 29, 58, 114] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(872, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 872:2 1000

← 871.2.a · 873.2.a →