Atlas › Modular forms › Weight 4 › Level 129 › 129.4.a

Newspace 129.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(129))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
129.4.a.a	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} - 7q^{4} - 2q^{5} - 3q^{6} + 6q^{7} + O(q^{8})$	− +	$-1$
129.4.a.b	1	$\mathbb{Q}$	$q + 4q^{2} - 3q^{3} + 8q^{4} + 11q^{5} - 12q^{6} + 9q^{7} + O(q^{8})$	+ +	$+1$
129.4.a.c	2	$\mathbb{Q}(\sqrt{10})$	$2q - 2q^{2} + 6q^{3} + 6q^{4} - 14q^{5} - 6q^{6} - 38q^{7} + O(q^{8})$	− +	$-1$
129.4.a.d	5	degree 5	$5q - 3q^{2} - 15q^{3} + 19q^{4} - 12q^{5} + 9q^{6} - 18q^{7} + O(q^{8})$	+ −	$-1$
129.4.a.e	5	degree 5	$5q - 3q^{2} - 15q^{3} + 27q^{4} - 3q^{5} + 9q^{6} + q^{7} + O(q^{8})$	+ +	$+1$
129.4.a.f	8	degree 8	$8q + q^{2} + 24q^{3} + 55q^{4} + 4q^{5} + 3q^{6} + 52q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
129.4.a.a	$x + 3$
129.4.a.b	$x - 15$
129.4.a.c	$x^{2} + 16x + 24$
129.4.a.d	$x^{5} + 15x^{4} - 242x^{3} - 3618x^{2} + 8656x + 128496$
129.4.a.e	$x^{5} + 6x^{4} - 327x^{3} - 1532x^{2} + 22272x + 36864$
129.4.a.f	$x^{8} - 5x^{7} - 883x^{6} + 6247x^{5} + 205478x^{4} - 1851974x^{3} - 6016856x^{2} + 43224384x + 108957696$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(129))$	22
Old forms	20
Cusp forms $S_{4}(\Gamma_0(129))$	42
Eisenstein series $E_{4}(\Gamma_0(129))$	4
Modular forms $M_{4}(\Gamma_0(129))$	46

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(129)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 3, 43, 129]: [1, 2, 12, 46] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; 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(129, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 129:4 1000

← 128.4.a · 130.4.a →