Atlas › Modular forms › Weight 4 › Level 244 › 244.4.a

Newspace 244.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(244))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
244.4.a.a	1	$\mathbb{Q}$	$q + 2q^{3} + 5q^{5} - 7q^{7} + O(q^{8})$	+
244.4.a.b	5	degree 5	$5q - 5q^{3} - 21q^{5} + 6q^{7} + O(q^{8})$	+
244.4.a.c	9	degree 9	$9q - 3q^{3} + 24q^{5} + 13q^{7} + O(q^{8})$	−

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 $T$
244.4.a.a	$x - 7$
244.4.a.b	$x^{5} + 26x^{4} + 141x^{3} - 1207x^{2} - 13202x - 31367$
244.4.a.c	$x^{9} - 21x^{8} - 892x^{7} + 23543x^{6} + 78008x^{5} - 5401249x^{4} + 24726784x^{3} + 271213889x^{2} - 2313081501x + 4093495326$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(244))$	15
Old forms	75
Cusp forms $S_{4}(\Gamma_0(244))$	90
Eisenstein series $E_{4}(\Gamma_0(244))$	6
Modular forms $M_{4}(\Gamma_0(244))$	96

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(244)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 61, 122, 244]: [1, 2, 3, 17, 49, 96] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(244, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 244:4 1000

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