Atlas › Modular forms › Weight 8 › Level 44 › 44.8.a

Newspace 44.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(44))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
44.8.a.a	3	degree 3	$3q - 62q^{3} + 288q^{5} - 196q^{7} + O(q^{8})$	+
44.8.a.b	4	degree 4	$4q + 75q^{3} - 219q^{5} + 1150q^{7} + O(q^{8})$	−

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 $T$
44.8.a.a	$x^{3} - 226x^{2} - 53616x - 2265120$
44.8.a.b	$x^{4} + 144x^{3} - 332904x^{2} - 55729600x + 14396461200$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(44))$	7
Old forms	32
Cusp forms $S_{8}(\Gamma_0(44))$	39
Eisenstein series $E_{8}(\Gamma_0(44))$	6
Modular forms $M_{8}(\Gamma_0(44))$	45

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(44)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 11, 22, 44]: [1, 3, 5, 8, 23, 45] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(44, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 44:8 1000

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