Atlas › Modular forms › Weight 2 › Level 315 › 315.2.a

Newspace 315.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(315))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
315.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} + q^{7} + O(q^{8})$	+ −	315b1 rank 1
315.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} + q^{7} + O(q^{8})$	− −	315a1 rank 0
315.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
315.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− −
315.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
315.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{11}$ 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 $T$
315.2.a.a	$x + 1$
315.2.a.b	$x - 9$
315.2.a.c	$x^{2} + 14x - 1$
315.2.a.d	$x^{2} + 12x - 89$
315.2.a.e	$x^{2} + 2x - 67$
315.2.a.f	$x^{2} - 14x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(315))$	10
Old forms	31
Cusp forms $S_{2}(\Gamma_0(315))$	41
Eisenstein series $E_{2}(\Gamma_0(315))$	15
Modular forms $M_{2}(\Gamma_0(315))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(315)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(105))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.346 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, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315]: [0, 1, 1, 1, 2, 4, 4, 6, 8, 10, 20, 52] (certified at ell = 2147483629)
T = [(2, 1), (11, 3)]; h of degree 10 = 1 x 10 (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(315, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 315:2 1000

← 314.2.a · 316.2.a →