Atlas › Modular forms › Weight 4 › Level 50 › 50.4.a

Newspace 50.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(50))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
50.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 7q^{3} + 4q^{4} + 14q^{6} + 34q^{7} + O(q^{8})$	+
50.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 4q^{4} + 4q^{6} - 26q^{7} + O(q^{8})$	+
50.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 8q^{3} + 4q^{4} - 16q^{6} + 4q^{7} + O(q^{8})$	+
50.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 4q^{4} + 4q^{6} + 26q^{7} + O(q^{8})$	−
50.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 7q^{3} + 4q^{4} + 14q^{6} - 34q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
50.4.a.a	$x - 27$
50.4.a.b	$x + 28$
50.4.a.c	$x - 12$
50.4.a.d	$x - 28$
50.4.a.e	$x + 27$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(50))$	5
Old forms	12
Cusp forms $S_{4}(\Gamma_0(50))$	17
Eisenstein series $E_{4}(\Gamma_0(50))$	12
Modular forms $M_{4}(\Gamma_0(50))$	29

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(50)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 5, 10, 25, 50]: [1, 2, 3, 7, 9, 25] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(50, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 50:4 1000

← 49.4.a · 51.4.a →