Atlas › Modular forms › Weight 2 › Level 156 › 156.2.a

Newspace 156.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(156))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
156.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} - 2q^{7} + O(q^{8})$	+ −			156a1 rank 1
156.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + 2q^{7} + O(q^{8})$	− −			156b1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
156.2.a.a	$x + 6$
156.2.a.b	$x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(156))$	2
Old forms	21
Cusp forms $S_{2}(\Gamma_0(156))$	23
Eisenstein series $E_{2}(\Gamma_0(156))$	11
Modular forms $M_{2}(\Gamma_0(156))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(156)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.215 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, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156]: [0, 1, 1, 2, 3, 5, 1, 5, 6, 10, 18, 34] (certified at ell = 2147483629)
T = [(5, 1), (7, 1)]; 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(156, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 156:2 1000

← 155.2.a · 157.2.a →