Atlas › Modular forms › Weight 2 › Level 927 › 927.2.a

Newspace 927.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(927))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
927.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−	927a1 rank 1
927.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 3q^{5} - 2q^{7} + O(q^{8})$	+
927.2.a.c	3	degree 3	$3q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	−
927.2.a.d	4	degree 4	$4q - 4q^{7} + O(q^{8})$	+
927.2.a.e	5	degree 5	$5q + 2q^{2} + 2q^{4} + 5q^{5} - 2q^{7} + O(q^{8})$	+
927.2.a.f	6	degree 6	$6q - 4q^{2} + 6q^{4} - 3q^{5} - 2q^{7} + O(q^{8})$	−
927.2.a.g	8	degree 8	$8q + q^{2} + 11q^{4} + q^{5} + 6q^{7} + O(q^{8})$	+
927.2.a.h	14	degree 14	$14q + 20q^{4} + 12q^{7} + O(q^{8})$	−

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 $T$
927.2.a.a	$x - 4$
927.2.a.b	$x^{2} - 12x + 31$
927.2.a.c	$x^{3} + 4x^{2} - 48x - 64$
927.2.a.d	$x^{4} - 16x^{2} + 16$
927.2.a.e	$x^{5} - 17x^{4} - 25x^{3} + 1593x^{2} - 7451x + 9722$
927.2.a.f	$x^{6} + 13x^{5} - 34x^{4} - 770x^{3} - 1446x^{2} + 769x + 71$
927.2.a.g	$x^{8} - 4x^{7} - 262x^{6} + 344x^{5} + 16474x^{4} + 20179x^{3} - 229418x^{2} - 478500x + 35656$
927.2.a.h	$x^{14} - 612x^{12} + 149264x^{10} - 18545888x^{8} + 1245007368x^{6} - 44371534183x^{4} + 778258451200x^{2} - 5239879680000$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(927))$	43
Old forms	58
Cusp forms $S_{2}(\Gamma_0(927))$	101
Eisenstein series $E_{2}(\Gamma_0(927))$	7
Modular forms $M_{2}(\Gamma_0(927))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(927)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(309))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 103, 309, 927]: [0, 1, 2, 9, 36, 106] (certified at ell = 2147482501)
T = [(2, 1), (5, 3)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 6 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(927, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 927:2 1000

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