Atlas › Modular forms › Weight 2 › Level 562 › 562.2.a

Newspace 562.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(562))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
562.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	562a1 rank 0
562.2.a.b	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
562.2.a.c	3	degree 3	$3q + 3q^{2} - 5q^{3} + 3q^{4} - 5q^{5} - 5q^{6} - q^{7} + O(q^{8})$	− −	$-1$
562.2.a.d	7	degree 7	$7q - 7q^{2} - 5q^{3} + 7q^{4} - 3q^{5} + 5q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
562.2.a.e	9	degree 9	$9q + 9q^{2} + 5q^{3} + 9q^{4} + 5q^{5} + 5q^{6} - q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
562.2.a.a	$x - 4$
562.2.a.b	$x^{3} - 13x + 13$
562.2.a.c	$x^{3} + 10x^{2} + 31x + 29$
562.2.a.d	$x^{7} + 8x^{6} + 5x^{5} - 59x^{4} - 128x^{3} - 80x^{2} - 4x + 4$
562.2.a.e	$x^{9} - 10x^{8} + 7x^{7} + 209x^{6} - 690x^{5} + 222x^{4} + 1352x^{3} - 828x^{2} - 480x + 128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(562))$	23
Old forms	46
Cusp forms $S_{2}(\Gamma_0(562))$	69
Eisenstein series $E_{2}(\Gamma_0(562))$	3
Modular forms $M_{2}(\Gamma_0(562))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(562)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(281))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 2, 281, 562]: [0, 1, 24, 72] (certified at ell = 2147482681)
T = [(3, 1), (5, 1)]; h of degree 23 = 1 x 23 (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(562, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 562:2 1000

← 561.2.a · 563.2.a →