Atlas › Modular forms › Weight 4 › Level 124 › 124.4.a

Newspace 124.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(124))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
124.4.a.a	4	degree 4	$4q - 4q^{3} - 14q^{5} - 12q^{7} + O(q^{8})$	+
124.4.a.b	4	degree 4	$4q + 2q^{3} + 6q^{5} + 16q^{7} + O(q^{8})$	−

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 $T$
124.4.a.a	$x^{4} + 18x^{3} - 19x^{2} - 518x + 842$
124.4.a.b	$x^{4} - 8x^{3} - 561x^{2} + 2864x + 1196$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(124))$	8
Old forms	37
Cusp forms $S_{4}(\Gamma_0(124))$	45
Eisenstein series $E_{4}(\Gamma_0(124))$	6
Modular forms $M_{4}(\Gamma_0(124))$	51

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(124)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 31, 62, 124]: [1, 2, 3, 9, 26, 51] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(124, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 124:4 1000

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