Atlas › Modular forms › Weight 6 › Level 44 › 44.6.a

Newspace 44.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(44))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
44.6.a.a	1	$\mathbb{Q}$	$q + 7q^{3} - 79q^{5} - 50q^{7} + O(q^{8})$	−
44.6.a.b	2	$\mathbb{Q}(\sqrt{31})$	$2q - 6q^{3} - 22q^{5} + 268q^{7} + O(q^{8})$	+

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 $T$
44.6.a.a	$x + 72$
44.6.a.b	$x^{2} + 28x - 4268$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(44))$	3
Old forms	24
Cusp forms $S_{6}(\Gamma_0(44))$	27
Eisenstein series $E_{6}(\Gamma_0(44))$	6
Modular forms $M_{6}(\Gamma_0(44))$	33

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(44)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 2, 4, 6, 17, 33] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 44:6 1000

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