Atlas › Modular forms › Weight 4 › Level 226 › 226.4.a

Newspace 226.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(226))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
226.4.a.a	5	degree 5	$5q + 10q^{2} - 11q^{3} + 20q^{4} - 29q^{5} - 22q^{6} - 26q^{7} + O(q^{8})$	− +	$-1$
226.4.a.b	7	degree 7	$7q - 14q^{2} - 7q^{3} + 28q^{4} - 13q^{5} + 14q^{6} - 2q^{7} + O(q^{8})$	+ −	$-1$
226.4.a.c	7	degree 7	$7q - 14q^{2} - q^{3} + 28q^{4} + 27q^{5} + 2q^{6} + 26q^{7} + O(q^{8})$	+ +	$+1$
226.4.a.d	9	degree 9	$9q + 18q^{2} + 13q^{3} + 36q^{4} + 41q^{5} + 26q^{6} + 2q^{7} + O(q^{8})$	− −	$+1$

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 $T$
226.4.a.a	$x^{5} + 40x^{4} + 535x^{3} + 2583x^{2} + 1920x - 8724$
226.4.a.b	$x^{7} + 20x^{6} - 389x^{5} - 7477x^{4} + 24274x^{3} + 534500x^{2} - 233004x - 7137136$
226.4.a.c	$x^{7} - 26x^{6} - 339x^{5} + 9225x^{4} + 20748x^{3} - 739674x^{2} + 1976472x - 807948$
226.4.a.d	$x^{9} - 54x^{8} + 397x^{7} + 22153x^{6} - 397638x^{5} - 762974x^{4} + 60074668x^{3} - 372132516x^{2} - 289941656x + 5709790152$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(226))$	28
Old forms	56
Cusp forms $S_{4}(\Gamma_0(226))$	84
Eisenstein series $E_{4}(\Gamma_0(226))$	4
Modular forms $M_{4}(\Gamma_0(226))$	88

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(226)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.142 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]: [1, 2, 30, 88] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 5 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 226:4 1000

← 225.4.a · 227.4.a →