Atlas › Modular forms › Weight 2 › Level 289 › 289.2.a

Newspace 289.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(289))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
289.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	289a1 rank 1
289.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - q^{3} + 3q^{4} - q^{5} - 6q^{6} - 3q^{7} + O(q^{8})$
289.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + q^{3} + 3q^{4} + q^{5} + 6q^{6} + 3q^{7} + O(q^{8})$
289.2.a.d	3	degree 3	$3q - 3q^{3} - 6q^{5} + 3q^{6} + O(q^{8})$
289.2.a.e	3	degree 3	$3q + 3q^{3} + 6q^{5} - 3q^{6} + O(q^{8})$
289.2.a.f	4	degree 4	$4q + 4q^{2} + 4q^{4} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
289.2.a.a	$x + 19$
289.2.a.b	$x^{2} + 30x + 173$
289.2.a.c	$x^{2} - 28x + 79$
289.2.a.d	$x^{3} + 39x^{2} + 363x - 251$
289.2.a.e	$x^{3} - 39x^{2} + 279x + 2071$
289.2.a.f	$x^{4} - 4x^{3} - 962x^{2} + 2644x + 212879$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(289))$	15
Old forms	2
Cusp forms $S_{2}(\Gamma_0(289))$	17
Eisenstein series $E_{2}(\Gamma_0(289))$	17
Modular forms $M_{2}(\Gamma_0(289))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(289)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 17, 289]: [0, 2, 26] (certified at ell = 2147483249)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(289, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 289:2 1000

← 288.2.a · 290.2.a →