Atlas › Modular forms › Weight 2 › Level 226 › 226.2.a

Newspace 226.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(226))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
226.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 4q^{5} - 2q^{6} + O(q^{8})$	− −	$-1$	226a1 rank 1
226.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$
226.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$
226.2.a.d	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} + 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$

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 $T$
226.2.a.a	$x + 14$
226.2.a.b	$x^{2} + 12x + 28$
226.2.a.c	$x^{2} - 14x + 46$
226.2.a.d	$x^{4} - 14x^{3} + 6x^{2} + 156x - 244$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(226))$	9
Old forms	18
Cusp forms $S_{2}(\Gamma_0(226))$	27
Eisenstein series $E_{2}(\Gamma_0(226))$	3
Modular forms $M_{2}(\Gamma_0(226))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(226)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 113, 226]: [0, 1, 10, 30] (certified at ell = 2147480161)
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(226, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 226:2 1000

← 225.2.a · 227.2.a →