Atlas › Modular forms › Weight 2 › Level 140 › 140.2.a

Newspace 140.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(140))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
140.2.a.a	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + q^{7} + O(q^{8})$	− −			140a1 rank 0
140.2.a.b	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} - q^{7} + O(q^{8})$	+ +			140b1 rank 0

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 $T$
140.2.a.a	$x - 10$
140.2.a.b	$x + 12$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(140))$	2
Old forms	17
Cusp forms $S_{2}(\Gamma_0(140))$	19
Eisenstein series $E_{2}(\Gamma_0(140))$	11
Modular forms $M_{2}(\Gamma_0(140))$	30

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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]: [0, 1, 2, 1, 1, 3, 4, 6, 7, 6, 16, 30] (certified at ell = 2147483629)
T = [(3, 1), (11, 3)]; h of degree 2 = 1 x 2 (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(140, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 140:2 1000

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