Atlas › Modular forms › Weight 2 › Level 548 › 548.2.a

Newspace 548.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(548))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
548.2.a.a	4	degree 4	$4q - 3q^{3} - 4q^{5} - q^{7} + O(q^{8})$	−
548.2.a.b	8	degree 8	$8q + 3q^{3} + 2q^{5} - q^{7} + O(q^{8})$	+

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 $T$
548.2.a.a	$x^{4} + 7x^{3} + 10x^{2} - 2x - 7$
548.2.a.b	$x^{8} - 5x^{7} - 38x^{6} + 212x^{5} + 221x^{4} - 1778x^{3} + 226x^{2} + 2034x + 402$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(548))$	12
Old forms	55
Cusp forms $S_{2}(\Gamma_0(548))$	67
Eisenstein series $E_{2}(\Gamma_0(548))$	5
Modular forms $M_{2}(\Gamma_0(548))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(548)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(137))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(274))^{\oplus 2}$

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, 2, 4, 137, 274, 548]: [0, 1, 2, 12, 36, 72] (certified at ell = 2147483249)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(548, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 548:2 1000

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