Atlas › Modular forms › Weight 2 › Level 236 › 236.2.a

Newspace 236.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(236))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
236.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - 3q^{7} + O(q^{8})$	−	236a1 rank 1
236.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} - q^{7} + O(q^{8})$	+	236b1 rank 0
236.2.a.c	3	degree 3	$3q - 4q^{5} + 8q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
236.2.a.a	$x - 4$
236.2.a.b	$x + 42$
236.2.a.c	$x^{3} - 53x^{2} + 666x + 189$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(236))$	5
Old forms	23
Cusp forms $S_{2}(\Gamma_0(236))$	28
Eisenstein series $E_{2}(\Gamma_0(236))$	5
Modular forms $M_{2}(\Gamma_0(236))$	33

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(236)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(118))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 59, 118, 236]: [0, 1, 2, 6, 17, 33] (certified at ell = 2147482481)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 5 = 1 x 5 (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(236, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 236:2 1000

← 235.2.a · 237.2.a →