Atlas › Modular forms › Weight 2 › Level 565 › 565.2.a

Newspace 565.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(565))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
565.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ −	$+1$	565a1 rank 0
565.2.a.b	6	degree 6	$6q - 3q^{2} - 3q^{3} + q^{4} + 6q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
565.2.a.c	8	degree 8	$8q + 5q^{2} + 7q^{3} + 11q^{4} - 8q^{5} - 2q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
565.2.a.d	10	degree 10	$10q - 5q^{2} - 5q^{3} + 13q^{4} - 10q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
565.2.a.e	12	degree 12	$12q + 5q^{2} + 3q^{3} + 15q^{4} + 12q^{5} + 4q^{6} + O(q^{8})$	− +	$+1$

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 $T$
565.2.a.a	$x + 1$
565.2.a.b	$x^{6} + 6x^{5} + 5x^{4} - 20x^{3} - 16x^{2} + 24x - 1$
565.2.a.c	$x^{8} - 12x^{7} + 42x^{6} - 237x^{4} + 260x^{3} + 281x^{2} - 452x + 109$
565.2.a.d	$x^{10} + 10x^{9} + 15x^{8} - 90x^{7} - 167x^{6} + 292x^{5} + 348x^{4} - 290x^{3} - 138x^{2} + 120x - 19$
565.2.a.e	$x^{12} - 8x^{11} - 16x^{10} + 270x^{9} - 362x^{8} - 1968x^{7} + 4303x^{6} + 3720x^{5} - 10258x^{4} - 3078x^{3} + 7761x^{2} + 1492x - 929$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(565))$	37
Old forms	18
Cusp forms $S_{2}(\Gamma_0(565))$	55
Eisenstein series $E_{2}(\Gamma_0(565))$	3
Modular forms $M_{2}(\Gamma_0(565))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(565)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 5, 113, 565]: [0, 1, 10, 58] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (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(565, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 565:2 1000

← 564.2.a · 566.2.a →