Atlas › Modular forms › Weight 2 › Level 514 › 514.2.a

Newspace 514.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(514))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
514.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	514b1 rank 1
514.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	514a1 rank 1
514.2.a.c	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{4} - 4q^{7} + O(q^{8})$	+ −	$+1$
514.2.a.d	3	degree 3	$3q - 3q^{2} + 2q^{3} + 3q^{4} + 4q^{5} - 2q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
514.2.a.e	5	degree 5	$5q - 5q^{2} + 5q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$
514.2.a.f	9	degree 9	$9q + 9q^{2} + 2q^{3} + 9q^{4} + 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
514.2.a.a	$x + 4$
514.2.a.b	$x + 2$
514.2.a.c	$x^{2} - 24$
514.2.a.d	$x^{3} - 6x^{2} + 8x - 2$
514.2.a.e	$x^{5} + 4x^{4} - 2x^{3} - 22x^{2} - 22x - 6$
514.2.a.f	$x^{9} - 6x^{8} - 38x^{7} + 288x^{6} - 94x^{5} - 2726x^{4} + 7204x^{3} - 7472x^{2} + 3380x - 548$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(514))$	21
Old forms	42
Cusp forms $S_{2}(\Gamma_0(514))$	63
Eisenstein series $E_{2}(\Gamma_0(514))$	3
Modular forms $M_{2}(\Gamma_0(514))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(514)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(257))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 257, 514]: [0, 1, 22, 66] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(514, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 514:2 1000

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