Atlas › Modular forms › Weight 2 › Level 636 › 636.2.a

Newspace 636.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(636))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
636.2.a.a	2	$\mathbb{Q}(\sqrt{37})$	$2q - 2q^{3} - q^{5} + q^{7} + O(q^{8})$	+ +
636.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{3} + q^{5} - 3q^{7} + O(q^{8})$	+ −
636.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 5q^{5} - 3q^{7} + O(q^{8})$	− +
636.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + q^{5} + q^{7} + O(q^{8})$	− −

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 $T$
636.2.a.a	$x^{2} - 2x - 147$
636.2.a.b	$x^{2} + 8x + 3$
636.2.a.c	$x^{2} + 14x - 31$
636.2.a.d	$x^{2} - 4x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(636))$	8
Old forms	95
Cusp forms $S_{2}(\Gamma_0(636))$	103
Eisenstein series $E_{2}(\Gamma_0(636))$	11
Modular forms $M_{2}(\Gamma_0(636))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(636)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(159))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(212))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(318))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 53, 106, 159, 212, 318, 636]: [0, 1, 1, 2, 3, 5, 5, 15, 20, 30, 58, 114] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (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(636, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 636:2 1000

← 635.2.a · 637.2.a →