Atlas › Modular forms › Weight 4 › Level 81 › 81.4.a

Newspace 81.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(81))$

Newforms

Label	Dim	Coefficient field	Trace form
81.4.a.a	2	$\mathbb{Q}(\sqrt{33})$	$2q - 3q^{2} + 5q^{4} - 15q^{5} + 7q^{7} + O(q^{8})$
81.4.a.b	2	$\mathbb{Q}(\sqrt{57})$	$2q - 3q^{2} + 17q^{4} + 12q^{5} + 10q^{7} + O(q^{8})$
81.4.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 10q^{4} - 44q^{7} + O(q^{8})$
81.4.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q + 3q^{2} + 5q^{4} + 15q^{5} + 7q^{7} + O(q^{8})$
81.4.a.e	2	$\mathbb{Q}(\sqrt{57})$	$2q + 3q^{2} + 17q^{4} - 12q^{5} + 10q^{7} + O(q^{8})$

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 $T$
81.4.a.a	$x^{2} + 48x + 543$
81.4.a.b	$x^{2} - 33x - 426$
81.4.a.c	$x^{2} - 1200$
81.4.a.d	$x^{2} - 48x + 543$
81.4.a.e	$x^{2} + 33x - 426$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(81))$	10
Old forms	11
Cusp forms $S_{4}(\Gamma_0(81))$	21
Eisenstein series $E_{4}(\Gamma_0(81))$	12
Modular forms $M_{4}(\Gamma_0(81))$	33

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.107 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, 2, 4, 10, 28] (certified at ell = 2147483179)
T = [(2, 1), (5, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 81:4 1000

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