Atlas › Modular forms › Weight 4 › Level 156 › 156.4.a

Newspace 156.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(156))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
156.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 6q^{5} - 4q^{7} + O(q^{8})$	+ −
156.4.a.b	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} - 32q^{7} + O(q^{8})$	− −
156.4.a.c	2	$\mathbb{Q}(\sqrt{22})$	$2q - 6q^{3} + 8q^{7} + O(q^{8})$	+ +
156.4.a.d	2	$\mathbb{Q}(\sqrt{10})$	$2q + 6q^{3} + 24q^{5} + 8q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
156.4.a.a	$x + 18$
156.4.a.b	$x + 98$
156.4.a.c	$x^{2} - 24x - 5488$
156.4.a.d	$x^{2} - 48x - 1984$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(156))$	6
Old forms	72
Cusp forms $S_{4}(\Gamma_0(156))$	78
Eisenstein series $E_{4}(\Gamma_0(156))$	12
Modular forms $M_{4}(\Gamma_0(156))$	90

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(156)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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]: [1, 2, 2, 3, 5, 9, 5, 13, 16, 24, 46, 90] (certified at ell = 2147483629)
T = [(5, 1), (7, 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(156, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 156:4 1000

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