Atlas › Modular forms › Weight 2 › Level 538 › 538.2.a

Newspace 538.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(538))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
538.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 5q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
538.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
538.2.a.c	4	degree 4	$4q - 4q^{2} + 3q^{3} + 4q^{4} + 5q^{5} - 3q^{6} - q^{7} + O(q^{8})$	+ −	$+1$
538.2.a.d	7	degree 7	$7q - 7q^{2} - 4q^{3} + 7q^{4} - 6q^{5} + 4q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
538.2.a.e	7	degree 7	$7q + 7q^{2} + q^{3} + 7q^{4} + 7q^{5} + q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
538.2.a.a	$x^{2} + 16x + 59$
538.2.a.b	$x^{2} - 4x - 9$
538.2.a.c	$x^{4} - 18x^{3} + 45x^{2} + 256x - 16$
538.2.a.d	$x^{7} + 22x^{6} + 48x^{5} - 1842x^{4} - 11679x^{3} + 14023x^{2} + 202484x + 189675$
538.2.a.e	$x^{7} - 22x^{6} - 55x^{5} + 4150x^{4} - 28680x^{3} + 37221x^{2} + 175952x - 398928$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(538))$	22
Old forms	44
Cusp forms $S_{2}(\Gamma_0(538))$	66
Eisenstein series $E_{2}(\Gamma_0(538))$	3
Modular forms $M_{2}(\Gamma_0(538))$	69

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(538)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(269))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 269, 538]: [0, 1, 23, 69] (certified at ell = 2147482661)
T = [(3, 1), (5, 3)]; h of degree 22 = 1 x 22 (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(538, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 538:2 1000

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