Atlas › Modular forms › Weight 2 › Level 718 › 718.2.a

Newspace 718.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(718))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
718.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$	718b1 rank 2
718.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{7} + O(q^{8})$	+ −	$+1$	718a1 rank 0
718.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} + q^{7} + O(q^{8})$	− −	$-1$	718c1 rank 1
718.2.a.d	5	degree 5	$5q + 5q^{2} - 8q^{3} + 5q^{4} - 5q^{5} - 8q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
718.2.a.e	6	degree 6	$6q - 6q^{2} - 2q^{3} + 6q^{4} - 4q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
718.2.a.f	7	degree 7	$7q - 7q^{2} + 4q^{3} + 7q^{4} + 9q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
718.2.a.g	8	degree 8	$8q + 8q^{2} + 4q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
718.2.a.a	$x + 5$
718.2.a.b	$x$
718.2.a.c	$x + 3$
718.2.a.d	$x^{5} + 13x^{4} + 57x^{3} + 108x^{2} + 87x + 23$
718.2.a.e	$x^{6} + 6x^{5} - 2x^{4} - 41x^{3} - 21x^{2} + 64x + 47$
718.2.a.f	$x^{7} - 13x^{6} + 47x^{5} - 175x^{3} - 35x^{2} + 102x + 32$
718.2.a.g	$x^{8} - 8x^{7} - 4x^{6} + 183x^{5} - 421x^{4} - 496x^{3} + 2941x^{2} - 3704x + 1520$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(718))$	29
Old forms	60
Cusp forms $S_{2}(\Gamma_0(718))$	89
Eisenstein series $E_{2}(\Gamma_0(718))$	3
Modular forms $M_{2}(\Gamma_0(718))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(718)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(359))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 2, 359, 718]: [0, 1, 31, 92] (certified at ell = 2147476963)
T = [(3, 1), (5, 1)]; h of degree 29 = 1 x 29 (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(718, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 718:2 1000

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