Atlas › Modular forms › Weight 2 › Level 391 › 391.2.a

Newspace 391.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(391))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
391.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
391.2.a.b	3	degree 3	$3q - q^{2} - 6q^{3} + 3q^{4} - 3q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
391.2.a.c	3	degree 3	$3q - q^{2} + 3q^{4} - q^{5} - 5q^{7} + O(q^{8})$	− −	$-1$
391.2.a.d	9	degree 9	$9q + 2q^{2} + 2q^{3} + 10q^{4} + 7q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
391.2.a.e	12	degree 12	$12q + 4q^{2} + 2q^{3} + 16q^{4} + 5q^{5} - 3q^{6} - 9q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
391.2.a.a	$x^{2} - x - 1$
391.2.a.b	$x^{3} + 7x^{2} + 12x + 1$
391.2.a.c	$x^{3} + x^{2} - 4x + 1$
391.2.a.d	$x^{9} - 4x^{8} - 32x^{7} + 157x^{6} + 113x^{5} - 1105x^{4} + 155x^{3} + 2554x^{2} - 455x - 1651$
391.2.a.e	$x^{12} - 6x^{11} - 32x^{10} + 206x^{9} + 353x^{8} - 2571x^{7} - 1212x^{6} + 13891x^{5} - 3342x^{4} - 28980x^{3} + 21150x^{2} + 7957x - 7213$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(391))$	29
Old forms	6
Cusp forms $S_{2}(\Gamma_0(391))$	35
Eisenstein series $E_{2}(\Gamma_0(391))$	3
Modular forms $M_{2}(\Gamma_0(391))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(391)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 23, 391]: [0, 2, 3, 38] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(391, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 391:2 1000

← 390.2.a · 392.2.a →