Atlas › Modular forms › Weight 2 › Level 320 › 320.2.a

Newspace 320.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(320))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
320.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} - 2q^{7} + O(q^{8})$	−	320f1 rank 1
320.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + 2q^{7} + O(q^{8})$	−	320e1 rank 0
320.2.a.c	1	$\mathbb{Q}$	$q - q^{5} - 4q^{7} + O(q^{8})$	+	320b1 rank 1
320.2.a.d	1	$\mathbb{Q}$	$q - q^{5} + 4q^{7} + O(q^{8})$	+	320a1 rank 0
320.2.a.e	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} - 2q^{7} + O(q^{8})$	−	320d1 rank 0
320.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 2q^{7} + O(q^{8})$	−	320c1 rank 0
320.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
320.2.a.a	$x + 76$
320.2.a.b	$x - 44$
320.2.a.c	$x$
320.2.a.d	$x - 64$
320.2.a.e	$x - 76$
320.2.a.f	$x + 60$
320.2.a.g	$x^{2} - 64x + 736$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(320))$	8
Old forms	29
Cusp forms $S_{2}(\Gamma_0(320))$	37
Eisenstein series $E_{2}(\Gamma_0(320))$	23
Modular forms $M_{2}(\Gamma_0(320))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(320)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(160))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.262 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, 5, 8, 10, 16, 20, 32, 40, 64, 80, 160, 320]: [0, 1, 2, 1, 3, 3, 4, 6, 6, 10, 10, 16, 28, 52] (certified at ell = 2147483489)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(320, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 320:2 1000

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