Atlas › Modular forms › Weight 2 › Level 723 › 723.2.a

Newspace 723.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(723))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
723.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+ +	$-1$	723a1 rank 1
723.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 2q^{7} + O(q^{8})$	− −	$-1$	723b1 rank 1
723.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
723.2.a.d	5	degree 5	$5q - 2q^{2} + 5q^{3} + 2q^{4} - 8q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
723.2.a.e	9	degree 9	$9q + 3q^{2} - 9q^{3} + 7q^{4} + 12q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
723.2.a.f	10	degree 10	$10q - 4q^{2} - 10q^{3} + 16q^{4} - 10q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
723.2.a.g	13	degree 13	$13q + q^{2} + 13q^{3} + 23q^{4} + 6q^{5} + q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
723.2.a.a	$x - 25$
723.2.a.b	$x + 17$
723.2.a.c	$x^{2} - 66x + 909$
723.2.a.d	$x^{5} + 227x^{4} + 18651x^{3} + 665148x^{2} + 9333043x + 23354797$
723.2.a.e	$x^{9} - 197x^{8} + 11571x^{7} + 59132x^{6} - 30971018x^{5} + 1190886332x^{4} - 17275386675x^{3} + 64041758946x^{2} + 363769872095x - 896208129239$
723.2.a.f	$x^{10} + 280x^{9} + 28394x^{8} + 1094661x^{7} - 6501778x^{6} - 1598022262x^{5} - 33021533003x^{4} + 156629151695x^{3} + 10777615458247x^{2} + 87416307008964x + 69051227879549$
723.2.a.g	$x^{13} - 231x^{12} + 14765x^{11} + 194102x^{10} - 47427643x^{9} + 964463455x^{8} + 24002195436x^{7} - 601737896830x^{6} - 6878329783763x^{5} + 90003161575363x^{4} + 1275862536974915x^{3} + 3791120417983516x^{2} - 563201831489955x - 3667891028810539$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(723))$	41
Old forms	38
Cusp forms $S_{2}(\Gamma_0(723))$	79
Eisenstein series $E_{2}(\Gamma_0(723))$	3
Modular forms $M_{2}(\Gamma_0(723))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(723)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(241))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 241, 723]: [0, 1, 20, 82] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 10 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(723, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 723:2 1000

← 722.2.a · 724.2.a →