Atlas › Modular forms › Weight 2 › Level 519 › 519.2.a

Newspace 519.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(519))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
519.2.a.a	3	degree 3	$3q - 2q^{2} + 3q^{3} - 5q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− −	$-1$
519.2.a.b	3	degree 3	$3q - 3q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
519.2.a.c	11	degree 11	$11q + 3q^{2} + 11q^{3} + 11q^{4} + 9q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
519.2.a.d	12	degree 12	$12q - 2q^{2} - 12q^{3} + 18q^{4} + q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
519.2.a.a	$x^{3} + 7x^{2} + 14x + 7$
519.2.a.b	$x^{3} + 3x^{2} - 1$
519.2.a.c	$x^{11} - 12x^{10} + 23x^{9} + 244x^{8} - 1324x^{7} + 1892x^{6} + 1700x^{5} - 6499x^{4} + 2993x^{3} + 4328x^{2} - 3936x + 603$
519.2.a.d	$x^{12} + x^{11} - 83x^{10} - 75x^{9} + 2388x^{8} + 2086x^{7} - 30608x^{6} - 28205x^{5} + 173642x^{4} + 184949x^{3} - 323994x^{2} - 450875x - 94231$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(519))$	29
Old forms	28
Cusp forms $S_{2}(\Gamma_0(519))$	57
Eisenstein series $E_{2}(\Gamma_0(519))$	3
Modular forms $M_{2}(\Gamma_0(519))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(519)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(173))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 173, 519]: [0, 1, 15, 60] (certified at ell = 2147482093)
T = [(2, 1), (5, 1)]; h of degree 29 = 1 x 29 (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(519, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 519:2 1000

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