Atlas › Modular forms › Weight 2 › Level 591 › 591.2.a

Newspace 591.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(591))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
591.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{7} + O(q^{8})$	+ +	$-1$	591a1 rank 1
591.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{7} + O(q^{8})$	+ +	$-1$
591.2.a.c	3	degree 3	$3q - 2q^{2} + 3q^{3} + 2q^{4} - 6q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− −	$-1$
591.2.a.d	13	degree 13	$13q + 5q^{2} + 13q^{3} + 17q^{4} + 12q^{5} + 5q^{6} - q^{7} + O(q^{8})$	− +	$+1$
591.2.a.e	14	degree 14	$14q - 14q^{3} + 20q^{4} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
591.2.a.a	$x$
591.2.a.b	$x^{2} - 2$
591.2.a.c	$x^{3} + 8x^{2} + 18x + 10$
591.2.a.d	$x^{13} - 17x^{12} + 86x^{11} + 59x^{10} - 1937x^{9} + 5844x^{8} + 79x^{7} - 37198x^{6} + 94464x^{5} - 115669x^{4} + 77153x^{3} - 25982x^{2} + 2816x + 300$
591.2.a.e	$x^{14} - 87x^{12} + 17x^{11} + 2890x^{10} - 1113x^{9} - 45391x^{8} + 25855x^{7} + 334840x^{6} - 250313x^{5} - 949754x^{4} + 865765x^{3} + 249868x^{2} - 282756x + 9566$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(591))$	33
Old forms	32
Cusp forms $S_{2}(\Gamma_0(591))$	65
Eisenstein series $E_{2}(\Gamma_0(591))$	3
Modular forms $M_{2}(\Gamma_0(591))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(591)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(197))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 197, 591]: [0, 1, 17, 68] (certified at ell = 2147479489)
T = [(2, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(591, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 591:2 1000

← 590.2.a · 592.2.a →