Atlas › Modular forms › Weight 2 › Level 318 › 318.2.a

Newspace 318.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(318))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
318.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	+ + +	$-1$	318c1 rank 1
318.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + −	$+1$	318e1 rank 0
318.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$	318b1 rank 0
318.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$	318d1 rank 1
318.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	− + +	$+1$	318a1 rank 0
318.2.a.f	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$
318.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− − −	$+1$

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 $T$
318.2.a.a	$x + 1$
318.2.a.b	$x - 7$
318.2.a.c	$x - 15$
318.2.a.d	$x + 15$
318.2.a.e	$x - 3$
318.2.a.f	$x^{2} - x - 10$
318.2.a.g	$x^{2} - 4x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(318))$	9
Old forms	42
Cusp forms $S_{2}(\Gamma_0(318))$	51
Eisenstein series $E_{2}(\Gamma_0(318))$	7
Modular forms $M_{2}(\Gamma_0(318))$	58

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.331 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, 6, 53, 106, 159, 318]: [0, 1, 1, 3, 5, 15, 20, 58] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 9 = 1 x 9 (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(318, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 318:2 1000

← 317.2.a · 319.2.a →