Atlas › Modular forms › Weight 6 › Level 40 › 40.6.a

Newspace 40.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(40))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
40.6.a.a	1	$\mathbb{Q}$	$q - 18q^{3} - 25q^{5} + 242q^{7} + O(q^{8})$	+
40.6.a.b	1	$\mathbb{Q}$	$q - 8q^{3} + 25q^{5} - 108q^{7} + O(q^{8})$	−
40.6.a.c	1	$\mathbb{Q}$	$q - 2q^{3} - 25q^{5} - 62q^{7} + O(q^{8})$	+
40.6.a.d	2	$\mathbb{Q}(\sqrt{129})$	$2q - 12q^{3} + 50q^{5} + 52q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
40.6.a.a	$x - 224$
40.6.a.b	$x + 116$
40.6.a.c	$x + 64$
40.6.a.d	$x^{2} - 40x - 7856$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(40))$	5
Old forms	21
Cusp forms $S_{6}(\Gamma_0(40))$	26
Eisenstein series $E_{6}(\Gamma_0(40))$	8
Modular forms $M_{6}(\Gamma_0(40))$	34

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(40)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 5, 8, 10, 20, 40]: [1, 2, 4, 3, 7, 9, 18, 34] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (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(40, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 40:6 1000

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