Atlas › Modular forms › Weight 4 › Level 140 › 140.4.a

Newspace 140.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(140))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
140.4.a.a	1	$\mathbb{Q}$	$q - 5q^{3} - 5q^{5} + 7q^{7} + O(q^{8})$	+ −
140.4.a.b	1	$\mathbb{Q}$	$q - 5q^{3} + 5q^{5} + 7q^{7} + O(q^{8})$	− −
140.4.a.c	1	$\mathbb{Q}$	$q - 4q^{3} + 5q^{5} - 7q^{7} + O(q^{8})$	− +
140.4.a.d	1	$\mathbb{Q}$	$q + q^{3} - 5q^{5} - 7q^{7} + O(q^{8})$	+ +
140.4.a.e	1	$\mathbb{Q}$	$q + 8q^{3} - 5q^{5} + 7q^{7} + O(q^{8})$	+ −
140.4.a.f	1	$\mathbb{Q}$	$q + 9q^{3} + 5q^{5} - 7q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{11}$ 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 $T$
140.4.a.a	$x - 40$
140.4.a.b	$x + 50$
140.4.a.c	$x - 200$
140.4.a.d	$x + 20$
140.4.a.e	$x - 92$
140.4.a.f	$x - 174$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(140))$	6
Old forms	60
Cusp forms $S_{4}(\Gamma_0(140))$	66
Eisenstein series $E_{4}(\Gamma_0(140))$	12
Modular forms $M_{4}(\Gamma_0(140))$	78

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(140)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 7, 10, 14, 20, 28, 35, 70, 140]: [1, 2, 3, 3, 3, 7, 8, 12, 15, 14, 40, 78] (certified at ell = 2147483629)
T = [(3, 1), (11, 3)]; h of degree 6 = 1 x 6 (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(140, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 140:4 1000

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