Atlas › Modular forms › Weight 4 › Level 142 › 142.4.a

Newspace 142.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(142))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
142.4.a.a	4	degree 4	$4q - 8q^{2} - 8q^{3} + 16q^{4} - 11q^{5} + 16q^{6} + 19q^{7} + O(q^{8})$	+ −	$-1$
142.4.a.b	4	degree 4	$4q + 8q^{2} - 14q^{3} + 16q^{4} - 21q^{5} - 28q^{6} - 47q^{7} + O(q^{8})$	− +	$-1$
142.4.a.c	5	degree 5	$5q - 10q^{2} + 4q^{3} + 20q^{4} + 19q^{5} - 8q^{6} + 5q^{7} + O(q^{8})$	+ +	$+1$
142.4.a.d	5	degree 5	$5q + 10q^{2} + 10q^{3} + 20q^{4} + 29q^{5} + 20q^{6} + 23q^{7} + O(q^{8})$	− −	$+1$

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 $T$
142.4.a.a	$x^{4} + 19x^{3} + 53x^{2} - 660x - 2935$
142.4.a.b	$x^{4} + 35x^{3} + 293x^{2} - 328x - 5363$
142.4.a.c	$x^{5} - 23x^{4} - 293x^{3} + 6074x^{2} + 10077x - 11962$
142.4.a.d	$x^{5} - 39x^{4} + 259x^{3} + 5254x^{2} - 55215x - 14370$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(142))$	18
Old forms	34
Cusp forms $S_{4}(\Gamma_0(142))$	52
Eisenstein series $E_{4}(\Gamma_0(142))$	4
Modular forms $M_{4}(\Gamma_0(142))$	56

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(142)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 71, 142]: [1, 2, 19, 56] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(142, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 142:4 1000

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