Atlas › Modular forms › Weight 2 › Level 714 › 714.2.a

Newspace 714.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(714))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
714.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	714d1 rank 1
714.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + −	$+1$	714b1 rank 0
714.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − +	$+1$	714c1 rank 0
714.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$	714a1 rank 1
714.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− + + −	$-1$	714f1 rank 1
714.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	714g1 rank 0
714.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− + + +	$+1$	714e1 rank 0
714.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	714h1 rank 0
714.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	714i1 rank 0
714.2.a.j	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$
714.2.a.k	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ − + +	$+1$
714.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− − + −	$+1$
714.2.a.m	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{31} + 7T_{37} + 9T_{41}$ 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 $T$
714.2.a.a	$x + 82$
714.2.a.b	$x - 61$
714.2.a.c	$x - 113$
714.2.a.d	$x - 58$
714.2.a.e	$x + 26$
714.2.a.f	$x - 86$
714.2.a.g	$x - 77$
714.2.a.h	$x - 21$
714.2.a.i	$x + 45$
714.2.a.j	$x^{2} - 81x - 4374$
714.2.a.k	$x^{2} - 73x + 830$
714.2.a.l	$x^{2} - 65x + 1018$
714.2.a.m	$x^{2} + 60x - 8700$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(714))$	17
Old forms	120
Cusp forms $S_{2}(\Gamma_0(714))$	137
Eisenstein series $E_{2}(\Gamma_0(714))$	15
Modular forms $M_{2}(\Gamma_0(714))$	152

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.963 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, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714]: [0, 1, 1, 3, 1, 4, 2, 4, 6, 12, 8, 22, 14, 40, 52, 152] (certified at ell = 2147483137)
T = [(5, 1), (11, 3), (31, 5), (37, 7), (41, 9)]; h of degree 17 = 1 x 17 (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(714, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 714:2 1000

← 713.2.a · 715.2.a →