Atlas › Modular forms › Weight 2 › Level 530 › 530.2.a

Newspace 530.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(530))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
530.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	530c1 rank 1
530.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 2q^{7} + O(q^{8})$	+ − −	$-1$	530b1 rank 1
530.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	530a1 rank 0
530.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	530d1 rank 1
530.2.a.e	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$
530.2.a.f	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$
530.2.a.g	4	degree 4	$4q + 4q^{2} + q^{3} + 4q^{4} - 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$
530.2.a.h	5	degree 5	$5q + 5q^{2} + 4q^{3} + 5q^{4} + 5q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
530.2.a.a	$x + 9$
530.2.a.b	$x + 6$
530.2.a.c	$x - 7$
530.2.a.d	$x + 7$
530.2.a.e	$x^{3} - 8x^{2} - 223x + 1043$
530.2.a.f	$x^{3} - 18x^{2} + 43x + 367$
530.2.a.g	$x^{4} - 10x^{3} - 115x^{2} + 771x - 122$
530.2.a.h	$x^{5} - 7x^{4} - 123x^{3} + 482x^{2} + 3839x + 1936$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(530))$	19
Old forms	58
Cusp forms $S_{2}(\Gamma_0(530))$	77
Eisenstein series $E_{2}(\Gamma_0(530))$	7
Modular forms $M_{2}(\Gamma_0(530))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(530)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(265))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 5, 10, 53, 106, 265, 530]: [0, 1, 1, 3, 5, 15, 28, 84] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(530, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 530:2 1000

← 529.2.a · 531.2.a →