Atlas › Modular forms › Weight 2 › Level 731 › 731.2.a

Newspace 731.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(731))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
731.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + O(q^{8})$	+ +	$-1$	731a1 rank 1
731.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + q^{3} - 2q^{4} - q^{5} - q^{6} + O(q^{8})$	+ +	$-1$
731.2.a.c	6	degree 6	$6q - q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - 7q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
731.2.a.d	8	degree 8	$8q + q^{2} - 3q^{3} + 3q^{4} - 7q^{5} - q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
731.2.a.e	19	degree 19	$19q + 2q^{2} + 5q^{3} + 26q^{4} + 11q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
731.2.a.f	21	degree 21	$21q + 2q^{2} - q^{3} + 32q^{4} - 3q^{5} + q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
731.2.a.a	$x - 2$
731.2.a.b	$x^{2} + x - 4$
731.2.a.c	$x^{6} + 4x^{5} - x^{4} - 16x^{3} - 13x^{2} - x + 1$
731.2.a.d	$x^{8} + 2x^{7} - 21x^{6} - 24x^{5} + 131x^{4} + 69x^{3} - 215x^{2} - 50x + 46$
731.2.a.e	$x^{19} - 7x^{18} - 53x^{17} + 435x^{16} + 894x^{15} - 10047x^{14} - 4548x^{13} + 113803x^{12} - 25108x^{11} - 683939x^{10} + 362811x^{9} + 2172759x^{8} - 1383186x^{7} - 3412797x^{6} + 1968634x^{5} + 2310386x^{4} - 872528x^{3} - 438020x^{2} + 97184x + 18664$
731.2.a.f	$x^{21} - x^{20} - 86x^{19} + 100x^{18} + 2897x^{17} - 3306x^{16} - 50238x^{15} + 46482x^{14} + 502944x^{13} - 286856x^{12} - 3005123x^{11} + 444674x^{10} + 10310917x^{9} + 2738738x^{8} - 17778836x^{7} - 10981847x^{6} + 11063844x^{5} + 11299712x^{4} + 1516088x^{3} - 1019600x^{2} - 232064x + 8336$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(731))$	57
Old forms	8
Cusp forms $S_{2}(\Gamma_0(731))$	65
Eisenstein series $E_{2}(\Gamma_0(731))$	3
Modular forms $M_{2}(\Gamma_0(731))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(731)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 17, 43, 731]: [0, 2, 4, 68] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(731, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 731:2 1000

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