Atlas › Modular forms › Weight 2 › Level 141 › 141.2.a

Newspace 141.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(141))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
141.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$	141a1 rank 1
141.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	141b1 rank 0
141.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + O(q^{8})$	− +	$+1$	141c1 rank 0
141.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$	141d1 rank 1
141.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} - q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$	141e1 rank 0
141.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 2q^{3} + 5q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
141.2.a.a	$x + 61$
141.2.a.b	$x - 19$
141.2.a.c	$x - 33$
141.2.a.d	$x + 39$
141.2.a.e	$x + 9$
141.2.a.f	$x^{2} - 56x + 767$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(141))$	7
Old forms	8
Cusp forms $S_{2}(\Gamma_0(141))$	15
Eisenstein series $E_{2}(\Gamma_0(141))$	3
Modular forms $M_{2}(\Gamma_0(141))$	18

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(141)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 3, 47, 141]: [0, 1, 5, 18] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 7 = 1 x 7 (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(141, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 141:2 1000

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