Atlas › Modular forms › Weight 4 › Level 172 › 172.4.a

Newspace 172.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(172))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
172.4.a.a	5	degree 5	$5q - 11q^{3} - q^{5} - 8q^{7} + O(q^{8})$	+
172.4.a.b	6	degree 6	$6q + 7q^{3} - q^{5} + 20q^{7} + O(q^{8})$	−

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 $T$
172.4.a.a	$x^{5} + 12x^{4} - 63x^{3} - 794x^{2} + 804x + 7896$
172.4.a.b	$x^{6} - 6x^{5} - 831x^{4} + 3392x^{3} + 140952x^{2} - 950880x + 373968$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(172))$	11
Old forms	52
Cusp forms $S_{4}(\Gamma_0(172))$	63
Eisenstein series $E_{4}(\Gamma_0(172))$	6
Modular forms $M_{4}(\Gamma_0(172))$	69

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(172)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.118 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, 43, 86, 172]: [1, 2, 3, 12, 35, 69] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (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(172, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 172:4 1000

← 171.4.a · 173.4.a →