Atlas › Modular forms › Weight 2 › Level 336 › 336.2.a

Newspace 336.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(336))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
336.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+ −	336e1 rank 1
336.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{7} + O(q^{8})$	+ +	336a1 rank 0
336.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + q^{7} + O(q^{8})$	+ −	336b1 rank 0
336.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + q^{7} + O(q^{8})$	− −	336d1 rank 0
336.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− +	336c1 rank 0
336.2.a.f	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} + q^{7} + O(q^{8})$	− −	336f1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
336.2.a.a	$x + 74$
336.2.a.b	$x - 52$
336.2.a.c	$x - 170$
336.2.a.d	$x + 10$
336.2.a.e	$x - 10$
336.2.a.f	$x + 40$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(336))$	6
Old forms	47
Cusp forms $S_{2}(\Gamma_0(336))$	53
Eisenstein series $E_{2}(\Gamma_0(336))$	23
Modular forms $M_{2}(\Gamma_0(336))$	76

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.37 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, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336]: [0, 1, 1, 2, 3, 1, 3, 5, 4, 4, 4, 8, 7, 12, 12, 12, 22, 20, 40, 72] (certified at ell = 2147483629)
T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; 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(336, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 336:2 1000

← 335.2.a · 337.2.a →