Atlas › Modular forms › Weight 2 › Level 711 › 711.2.a

Newspace 711.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(711))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
711.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{7} + O(q^{8})$	+	711b1 rank 1
711.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{7} + O(q^{8})$	+	711a1 rank 1
711.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 3q^{5} - q^{7} + O(q^{8})$	+	711c1 rank 0
711.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{7} + O(q^{8})$	−
711.2.a.e	4	degree 4	$4q + 3q^{2} + 3q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+
711.2.a.f	5	degree 5	$5q + 2q^{4} - 7q^{5} - 5q^{7} + O(q^{8})$	−
711.2.a.g	7	degree 7	$7q - 2q^{2} + 12q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+
711.2.a.h	12	degree 12	$12q + 18q^{4} + 6q^{7} + O(q^{8})$	−

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 $T$
711.2.a.a	$x + 1$
711.2.a.b	$x - 1$
711.2.a.c	$x - 4$
711.2.a.d	$x^{2} + 2x - 1$
711.2.a.e	$x^{4} - 7x^{3} + 8x^{2} + 30x - 53$
711.2.a.f	$x^{5} + 7x^{4} + 10x^{3} - 14x^{2} - 19x + 16$
711.2.a.g	$x^{7} - 36x^{5} - 50x^{4} + 104x^{3} - 41x^{2} + x + 1$
711.2.a.h	$x^{12} - 83x^{10} + 2469x^{8} - 34517x^{6} + 238391x^{4} - 749008x^{2} + 771147$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(711))$	33
Old forms	44
Cusp forms $S_{2}(\Gamma_0(711))$	77
Eisenstein series $E_{2}(\Gamma_0(711))$	7
Modular forms $M_{2}(\Gamma_0(711))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(711)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(237))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.429 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, 79, 237, 711]: [0, 1, 2, 7, 28, 82] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(711, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 711:2 1000

← 710.2.a · 712.2.a →