Atlas › Modular forms › Weight 4 › Level 40 › 40.4.a

Newspace 40.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(40))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
40.4.a.a	1	$\mathbb{Q}$	$q - 6q^{3} - 5q^{5} - 34q^{7} + O(q^{8})$	+
40.4.a.b	1	$\mathbb{Q}$	$q + 4q^{3} + 5q^{5} + 16q^{7} + O(q^{8})$	−
40.4.a.c	1	$\mathbb{Q}$	$q + 10q^{3} - 5q^{5} - 18q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
40.4.a.a	$x + 252$
40.4.a.b	$x + 1052$
40.4.a.c	$x + 220$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(40))$	3
Old forms	11
Cusp forms $S_{4}(\Gamma_0(40))$	14
Eisenstein series $E_{4}(\Gamma_0(40))$	8
Modular forms $M_{4}(\Gamma_0(40))$	22

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 3, 3, 5, 7, 12, 22] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 3 = 1 x 3 (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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 40:4 1000

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