Atlas › Modular forms › Weight 2 › Level 543 › 543.2.a

Newspace 543.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(543))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
543.2.a.a	3	degree 3	$3q - 2q^{2} + 3q^{3} - q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
543.2.a.b	5	degree 5	$5q - q^{2} + 5q^{3} + 3q^{4} + 3q^{5} - q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$
543.2.a.c	7	degree 7	$7q - 4q^{2} - 7q^{3} + 8q^{4} - 3q^{5} + 4q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
543.2.a.d	8	degree 8	$8q + 3q^{2} - 8q^{3} + 5q^{4} + 5q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
543.2.a.e	8	degree 8	$8q + 3q^{2} + 8q^{3} + 17q^{4} - 2q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
543.2.a.a	$x^{3} + 5x^{2} - x - 13$
543.2.a.b	$x^{5} - 8x^{4} - 99x^{3} + 667x^{2} + 1290x - 6083$
543.2.a.c	$x^{7} + 13x^{6} - 48x^{5} - 800x^{4} + 652x^{3} + 10226x^{2} - 1797x - 23311$
543.2.a.d	$x^{8} - 18x^{7} - 93x^{6} + 2970x^{5} - 8872x^{4} - 54734x^{3} + 199387x^{2} + 273646x - 1006255$
543.2.a.e	$x^{8} + 3x^{7} - 315x^{6} - 146x^{5} + 27972x^{4} - 47236x^{3} - 661641x^{2} + 2121659x - 1682491$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(543))$	31
Old forms	28
Cusp forms $S_{2}(\Gamma_0(543))$	59
Eisenstein series $E_{2}(\Gamma_0(543))$	3
Modular forms $M_{2}(\Gamma_0(543))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(543)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(181))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 181, 543]: [0, 1, 15, 62] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 5 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(543, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 543:2 1000

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