Atlas › Modular forms › Weight 6 › Level 55 › 55.6.a

Newspace 55.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(55))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
55.6.a.a	3	degree 3	$3q - 7q^{2} - 36q^{3} + 41q^{4} + 75q^{5} + 101q^{6} - 102q^{7} + O(q^{8})$	− −	$-1$
55.6.a.b	4	degree 4	$4q - 5q^{2} + 61q^{4} - 100q^{5} - 157q^{6} - 90q^{7} + O(q^{8})$	+ +	$-1$
55.6.a.c	5	degree 5	$5q + 9q^{2} + 115q^{4} + 125q^{5} - 237q^{6} + 70q^{7} + O(q^{8})$	− +	$+1$
55.6.a.d	6	degree 6	$6q + 3q^{2} + 115q^{4} - 150q^{5} + 325q^{6} - 66q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
55.6.a.a	$x^{3} + 43x^{2} + 266x - 272$
55.6.a.b	$x^{4} + 5x^{3} - 422x^{2} - 180x + 26696$
55.6.a.c	$x^{5} - 9x^{4} - 997x^{3} + 3073x^{2} + 255296x + 991236$
55.6.a.d	$x^{6} - 3x^{5} - 1619x^{4} - 2391x^{3} + 703544x^{2} + 2119012x - 71685216$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(55))$	18
Old forms	10
Cusp forms $S_{6}(\Gamma_0(55))$	28
Eisenstein series $E_{6}(\Gamma_0(55))$	4
Modular forms $M_{6}(\Gamma_0(55))$	32

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(55)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 6, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 4 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 55:6 1000

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