Atlas › Modular forms › Weight 2 › Level 527 › 527.2.a

Newspace 527.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(527))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
527.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} + 5q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
527.2.a.b	5	degree 5	$5q - 2q^{2} - 3q^{3} + 4q^{4} - 2q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
527.2.a.c	7	degree 7	$7q - q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 6q^{6} - 10q^{7} + O(q^{8})$	+ +	$-1$
527.2.a.d	11	degree 11	$11q - 2q^{2} + 7q^{3} + 16q^{4} + 6q^{5} - q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
527.2.a.e	16	degree 16	$16q + 3q^{2} + 3q^{3} + 21q^{4} + 4q^{5} + 2q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$

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 $T$
527.2.a.a	$x^{2} - x - 11$
527.2.a.b	$x^{5} + 5x^{4} - 6x^{3} - 35x^{2} - 3x + 29$
527.2.a.c	$x^{7} + 4x^{6} - 6x^{5} - 18x^{4} + 12x^{3} + 7x^{2} - 6x + 1$
527.2.a.d	$x^{11} - 5x^{10} - 27x^{9} + 138x^{8} + 112x^{7} - 840x^{6} + 203x^{5} + 1059x^{4} - 306x^{3} - 361x^{2} + 52x + 31$
527.2.a.e	$x^{16} - 6x^{15} - 49x^{14} + 345x^{13} + 699x^{12} - 6845x^{11} - 2997x^{10} + 65227x^{9} - 12270x^{8} - 323313x^{7} + 154161x^{6} + 795532x^{5} - 507943x^{4} - 769812x^{3} + 581190x^{2} + 35441x - 19863$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(527))$	41
Old forms	6
Cusp forms $S_{2}(\Gamma_0(527))$	47
Eisenstein series $E_{2}(\Gamma_0(527))$	3
Modular forms $M_{2}(\Gamma_0(527))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(527)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 31, 527]: [0, 2, 3, 50] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(527, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 527:2 1000

← 526.2.a · 528.2.a →