Atlas › Modular forms › Weight 2 › Level 624 › 624.2.a

Newspace 624.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(624))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
624.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} + 4q^{7} + O(q^{8})$	+ +	624b1 rank 1
624.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ −	624g1 rank 1
624.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+ +	624a1 rank 1
624.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+ −	624c1 rank 0
624.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$	− −	624j1 rank 0
624.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− −	624f1 rank 1
624.2.a.g	1	$\mathbb{Q}$	$q + q^{3} + 4q^{7} + O(q^{8})$	− +	624d1 rank 0
624.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	− −	624i1 rank 0
624.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	− −	624h1 rank 0
624.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} + O(q^{8})$	− +	624e1 rank 0
624.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
624.2.a.a	$x + 104$
624.2.a.b	$x + 66$
624.2.a.c	$x + 60$
624.2.a.d	$x - 52$
624.2.a.e	$x - 54$
624.2.a.f	$x + 160$
624.2.a.g	$x + 28$
624.2.a.h	$x - 96$
624.2.a.i	$x - 8$
624.2.a.j	$x + 88$
624.2.a.k	$x^{2} - 136x + 1736$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(624))$	12
Old forms	89
Cusp forms $S_{2}(\Gamma_0(624))$	101
Eisenstein series $E_{2}(\Gamma_0(624))$	23
Modular forms $M_{2}(\Gamma_0(624))$	124

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.59 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, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624]: [0, 1, 1, 2, 3, 3, 5, 1, 4, 8, 5, 6, 12, 10, 18, 18, 34, 32, 64, 120] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 12 = 1 x 12 (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(624, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 624:2 1000

← 623.2.a · 625.2.a →