Atlas › Modular forms › Weight 2 › Level 729 › 729.2.a

Newspace 729.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(729))$

Newforms

Label	Dim	Coefficient field	Trace form
729.2.a.a	6	degree 6	$6q - 3q^{2} + 3q^{4} - 6q^{5} + O(q^{8})$
729.2.a.b	6	degree 6	$6q - 3q^{2} + 9q^{4} + 3q^{5} + 6q^{7} + O(q^{8})$
729.2.a.c	6	degree 6	$6q - 12q^{7} + O(q^{8})$
729.2.a.d	6	degree 6	$6q + 3q^{2} + 3q^{4} + 6q^{5} + O(q^{8})$
729.2.a.e	6	degree 6	$6q + 3q^{2} + 9q^{4} - 3q^{5} + 6q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
729.2.a.a	$x^{6} + 21x^{5} + 96x^{4} - 525x^{3} - 4941x^{2} - 11682x - 8691$
729.2.a.b	$x^{6} - 6x^{5} - 186x^{4} + 1113x^{3} + 7020x^{2} - 49203x + 68037$
729.2.a.c	$x^{6} - 105x^{4} + 1899x^{2} - 8427$
729.2.a.d	$x^{6} - 21x^{5} + 96x^{4} + 525x^{3} - 4941x^{2} + 11682x - 8691$
729.2.a.e	$x^{6} + 6x^{5} - 186x^{4} - 1113x^{3} + 7020x^{2} + 49203x + 68037$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(729))$	30
Old forms	34
Cusp forms $S_{2}(\Gamma_0(729))$	64
Eisenstein series $E_{2}(\Gamma_0(729))$	35
Modular forms $M_{2}(\Gamma_0(729))$	99

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(729)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(243))^{\oplus 2}$

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, 3, 9, 27, 81, 243, 729]: [0, 1, 2, 4, 10, 28, 82] (certified at ell = 2147481883)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (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(729, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 729:2 1000

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