Atlas › Modular forms › Weight 2 › Level 436 › 436.2.a

Newspace 436.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(436))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
436.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{5} - 4q^{7} + O(q^{8})$	+
436.2.a.b	3	degree 3	$3q - 6q^{5} - 3q^{7} + O(q^{8})$	−
436.2.a.c	4	degree 4	$4q + 8q^{5} + 5q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
436.2.a.a	$x^{2} - 6x - 119$
436.2.a.b	$x^{3} + 18x^{2} + 96x + 152$
436.2.a.c	$x^{4} - 24x^{3} + 188x^{2} - 520x + 368$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(436))$	9
Old forms	44
Cusp forms $S_{2}(\Gamma_0(436))$	53
Eisenstein series $E_{2}(\Gamma_0(436))$	5
Modular forms $M_{2}(\Gamma_0(436))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(436)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(218))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 109, 218, 436]: [0, 1, 2, 9, 29, 58] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(436, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 436:2 1000

← 435.2.a · 437.2.a →