Atlas › Modular forms › Weight 6 › Level 81 › 81.6.a

Newspace 81.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(81))$

Newforms

Label	Dim	Coefficient field	Trace form
81.6.a.a	2	$\mathbb{Q}(\sqrt{129})$	$2q - 3q^{2} + 5q^{4} + 30q^{5} - 128q^{7} + O(q^{8})$
81.6.a.b	2	$\mathbb{Q}(\sqrt{129})$	$2q + 3q^{2} + 5q^{4} - 30q^{5} - 128q^{7} + O(q^{8})$
81.6.a.c	4	degree 4	$4q - 3q^{2} + 49q^{4} - 78q^{5} - 28q^{7} + O(q^{8})$
81.6.a.d	4	degree 4	$4q + 3q^{2} + 49q^{4} + 78q^{5} - 28q^{7} + O(q^{8})$
81.6.a.e	6	degree 6	$6q + 150q^{4} + 372q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
81.6.a.a	$x^{2} - 27x - 108$
81.6.a.b	$x^{2} + 27x - 108$
81.6.a.c	$x^{4} + 81x^{3} - 3915x^{2} - 365229x - 3623130$
81.6.a.d	$x^{4} - 81x^{3} - 3915x^{2} + 365229x - 3623130$
81.6.a.e	$x^{6} - 18306x^{4} + 87276609x^{2} - 108699367500$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(81))$	18
Old forms	21
Cusp forms $S_{6}(\Gamma_0(81))$	39
Eisenstein series $E_{6}(\Gamma_0(81))$	12
Modular forms $M_{6}(\Gamma_0(81))$	51

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(81)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 27, 81]: [1, 3, 6, 16, 46] (certified at ell = 2147483179)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(81, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 81:6 1000

← 80.6.a · 82.6.a →