Atlas › Modular forms › Weight 2 › Level 717 › 717.2.a

Newspace 717.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(717))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
717.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + O(q^{8})$	+ +	$-1$
717.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} + 6q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
717.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{6} + O(q^{8})$	− +	$+1$
717.2.a.d	6	degree 6	$6q - 2q^{2} - 6q^{3} + 4q^{4} + 5q^{5} + 2q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
717.2.a.e	7	degree 7	$7q + 7q^{3} + 10q^{4} + 3q^{5} + 3q^{7} + O(q^{8})$	− +	$+1$
717.2.a.f	8	degree 8	$8q - 5q^{2} + 8q^{3} + 7q^{4} - 13q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
717.2.a.g	12	degree 12	$12q + 3q^{2} - 12q^{3} + 15q^{4} - q^{5} - 3q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
717.2.a.a	$x^{2} + x - 1$
717.2.a.b	$x^{2} - 7x + 11$
717.2.a.c	$x^{2} - 3x + 1$
717.2.a.d	$x^{6} - 3x^{5} - 7x^{4} + 13x^{3} + 2x^{2} - 4x - 1$
717.2.a.e	$x^{7} - 3x^{6} - 23x^{5} + 32x^{4} + 148x^{3} - 34x^{2} - 150x + 61$
717.2.a.f	$x^{8} + 18x^{7} + 123x^{6} + 379x^{5} + 374x^{4} - 715x^{3} - 2159x^{2} - 1839x - 443$
717.2.a.g	$x^{12} - 2x^{11} - 57x^{10} + 97x^{9} + 1113x^{8} - 1479x^{7} - 9750x^{6} + 9178x^{5} + 40089x^{4} - 24805x^{3} - 70705x^{2} + 25107x + 36605$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(717))$	39
Old forms	40
Cusp forms $S_{2}(\Gamma_0(717))$	79
Eisenstein series $E_{2}(\Gamma_0(717))$	3
Modular forms $M_{2}(\Gamma_0(717))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(717)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(239))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.231 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, 239, 717]: [0, 1, 21, 82] (certified at ell = 2147482093)
T = [(2, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 4 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(717, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 717:2 1000

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