Atlas › Modular forms › Weight 2 › Level 316 › 316.2.a

Newspace 316.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(316))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
316.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} + q^{7} + O(q^{8})$	−	316b1 rank 1
316.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + 3q^{7} + O(q^{8})$	+	316a1 rank 0
316.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 5q^{5} - 2q^{7} + O(q^{8})$	−
316.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q + 4q^{3} + 3q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
316.2.a.a	$x + 73$
316.2.a.b	$x - 41$
316.2.a.c	$x^{2} + 43x - 1257$
316.2.a.d	$x^{2} + 19x - 5373$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(316))$	6
Old forms	32
Cusp forms $S_{2}(\Gamma_0(316))$	38
Eisenstein series $E_{2}(\Gamma_0(316))$	5
Modular forms $M_{2}(\Gamma_0(316))$	43

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(316)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(158))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 4, 79, 158, 316]: [0, 1, 2, 7, 22, 43] (certified at ell = 2147483053)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; 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(316, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 316:2 1000

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