Atlas › Modular forms › Weight 4 › Level 55 › 55.4.a

Newspace 55.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(55))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
55.4.a.a	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} - 7q^{4} - 5q^{5} - 3q^{6} - 9q^{7} + O(q^{8})$	+ −	$-1$
55.4.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - 7q^{2} - 3q^{3} + 17q^{4} + 10q^{5} + 19q^{6} - 25q^{7} + O(q^{8})$	− +	$-1$
55.4.a.c	3	degree 3	$3q + 5q^{2} - 3q^{3} + 23q^{4} - 15q^{5} + 59q^{6} - 15q^{7} + O(q^{8})$	+ +	$+1$
55.4.a.d	4	degree 4	$4q + q^{2} + 9q^{3} + 19q^{4} + 20q^{5} - 19q^{6} + 9q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
55.4.a.a	$x + 8$
55.4.a.b	$x^{2} + 16x - 4$
55.4.a.c	$x^{3} + 4x^{2} - 881x + 2558$
55.4.a.d	$x^{4} - 28x^{3} - 283x^{2} + 9726x - 14592$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(55))$	10
Old forms	6
Cusp forms $S_{4}(\Gamma_0(55))$	16
Eisenstein series $E_{4}(\Gamma_0(55))$	4
Modular forms $M_{4}(\Gamma_0(55))$	20

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(55)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 5, 11, 55]: [1, 3, 4, 20] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 10 = 1 x 10 (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(55, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 55:4 1000

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