Atlas › Modular forms › Weight 8 › Level 55 › 55.8.a

Newspace 55.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(55))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
55.8.a.a	4	degree 4	$4q - 21q^{2} + 189q^{4} - 500q^{5} + 411q^{6} + 1262q^{7} + O(q^{8})$	+ −	$-1$
55.8.a.b	5	degree 5	$5q - 3q^{2} + 105q^{4} + 625q^{5} - 1187q^{6} - 890q^{7} + O(q^{8})$	− +	$-1$
55.8.a.c	6	degree 6	$6q - 5q^{2} + 251q^{4} - 750q^{5} - 547q^{6} + 550q^{7} + O(q^{8})$	+ +	$+1$
55.8.a.d	7	degree 7	$7q + 29q^{2} + 108q^{3} + 555q^{4} + 875q^{5} + 635q^{6} + 1194q^{7} + O(q^{8})$	− −	$+1$

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 $T$
55.8.a.a	$x^{4} + 21x^{3} - 3562x^{2} + 67756x - 191848$
55.8.a.b	$x^{5} + 3x^{4} - 6014x^{3} + 122628x^{2} + 3963224x - 86900768$
55.8.a.c	$x^{6} + 5x^{5} - 8803x^{4} - 119035x^{3} + 17449820x^{2} + 495618500x + 2429642064$
55.8.a.d	$x^{7} - 137x^{6} - 4641x^{5} + 1037177x^{4} - 6526288x^{3} - 1828067104x^{2} + 16049594864x + 777518022512$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(55))$	22
Old forms	18
Cusp forms $S_{8}(\Gamma_0(55))$	40
Eisenstein series $E_{8}(\Gamma_0(55))$	4
Modular forms $M_{8}(\Gamma_0(55))$	44

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(55)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 5, 8, 44] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(55, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 55:8 1000

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