Atlas › Modular forms › Weight 2 › Level 721 › 721.2.a

Newspace 721.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(721))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
721.2.a.a	3	degree 3	$3q + q^{2} + 3q^{3} + 7q^{4} + 2q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
721.2.a.b	4	degree 4	$4q - 2q^{3} - 8q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− −	$-1$
721.2.a.c	7	degree 7	$7q - 3q^{2} - 2q^{3} + 15q^{4} - 11q^{5} - 10q^{6} + 7q^{7} + O(q^{8})$	− −	$-1$
721.2.a.d	11	degree 11	$11q - 3q^{2} - 2q^{3} + 7q^{4} - 5q^{5} - 6q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
721.2.a.e	12	degree 12	$12q + 2q^{2} - 3q^{3} + 12q^{4} + q^{5} + 8q^{6} - 12q^{7} + O(q^{8})$	+ −	$+1$
721.2.a.f	14	degree 14	$14q + 6q^{3} + 12q^{4} + 13q^{5} + 14q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
721.2.a.a	$x^{3} - 4x^{2} + 1$
721.2.a.b	$x^{4} + 2x^{3} - 9x^{2} - 16x - 5$
721.2.a.c	$x^{7} + 5x^{6} - 21x^{4} - 20x^{3} - x^{2} + 4x + 1$
721.2.a.d	$x^{11} + 5x^{10} - 15x^{9} - 72x^{8} + 119x^{7} + 307x^{6} - 522x^{5} - 195x^{4} + 628x^{3} - 230x^{2} - 58x + 31$
721.2.a.e	$x^{12} + x^{11} - 55x^{10} - 51x^{9} + 955x^{8} + 1135x^{7} - 6478x^{6} - 9936x^{5} + 14180x^{4} + 27011x^{3} - 4536x^{2} - 18904x - 4855$
721.2.a.f	$x^{14} - 6x^{13} - 29x^{12} + 221x^{11} + 145x^{10} - 2579x^{9} + 1086x^{8} + 12985x^{7} - 10794x^{6} - 30775x^{5} + 30577x^{4} + 34265x^{3} - 35599x^{2} - 14563x + 14709$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(721))$	51
Old forms	16
Cusp forms $S_{2}(\Gamma_0(721))$	67
Eisenstein series $E_{2}(\Gamma_0(721))$	3
Modular forms $M_{2}(\Gamma_0(721))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(721)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 7, 103, 721]: [0, 1, 9, 70] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(721, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 721:2 1000

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