Atlas › Modular forms › Weight 2 › Level 542 › 542.2.a

Newspace 542.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(542))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
542.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$	542b1 rank 1
542.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	− +	$+1$	542a1 rank 0
542.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
542.2.a.d	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
542.2.a.e	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
542.2.a.f	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + 4q^{5} + 3q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$
542.2.a.g	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - 4q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
542.2.a.h	6	degree 6	$6q - 6q^{2} + 6q^{4} + 2q^{5} + 10q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
542.2.a.a	$x + 1$
542.2.a.b	$x - 8$
542.2.a.c	$x^{2} + 14x + 41$
542.2.a.d	$x^{3} + x^{2} - 33x - 85$
542.2.a.e	$x^{3} + 5x^{2} - 77x - 313$
542.2.a.f	$x^{3} - 15x^{2} + 17x + 335$
542.2.a.g	$x^{4} + 16x^{3} + 8x^{2} - 268x - 233$
542.2.a.h	$x^{6} - 6x^{5} - 262x^{4} + 2156x^{3} + 12879x^{2} - 157280x + 348134$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(542))$	23
Old forms	44
Cusp forms $S_{2}(\Gamma_0(542))$	67
Eisenstein series $E_{2}(\Gamma_0(542))$	3
Modular forms $M_{2}(\Gamma_0(542))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(542)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(271))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 271, 542]: [0, 1, 23, 70] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 23 = 1 x 23 (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(542, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 542:2 1000

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