Atlas › Modular forms › Weight 2 › Level 259 › 259.2.a

Newspace 259.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(259))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
259.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 4q^{5} + q^{7} + O(q^{8})$	− +	$+1$	259a1 rank 0
259.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{4} + 6q^{5} - 2q^{7} + O(q^{8})$	+ −	$+1$
259.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$
259.2.a.d	3	degree 3	$3q - 3q^{2} + 3q^{4} - 6q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$
259.2.a.e	3	degree 3	$3q + q^{2} - 2q^{3} - q^{4} - 6q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
259.2.a.f	4	degree 4	$4q + 2q^{3} + 10q^{4} + 6q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$
259.2.a.g	4	degree 4	$4q + q^{2} + 5q^{4} + q^{5} - 9q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
259.2.a.a	$x - 1$
259.2.a.b	$x^{2} - 72$
259.2.a.c	$x^{2} - x - 4$
259.2.a.d	$x^{3} + 3x^{2} - 18x - 57$
259.2.a.e	$x^{3} + 5x^{2} - 8x - 41$
259.2.a.f	$x^{4} - 6x^{3} - 63x^{2} + 445x - 367$
259.2.a.g	$x^{4} - x^{3} - 114x^{2} + 5x + 2740$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(259))$	19
Old forms	4
Cusp forms $S_{2}(\Gamma_0(259))$	23
Eisenstein series $E_{2}(\Gamma_0(259))$	3
Modular forms $M_{2}(\Gamma_0(259))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(259)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 7, 37, 259]: [0, 1, 3, 26] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (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(259, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 259:2 1000

← 258.2.a · 260.2.a →