Atlas › Modular forms › Weight 2 › Level 255 › 255.2.a

Newspace 255.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(255))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
255.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} - 2q^{3} + 3q^{4} - 2q^{5} - q^{6} + O(q^{8})$	+ + −	$+1$
255.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 2q^{3} + 3q^{4} + 2q^{5} - 3q^{6} + O(q^{8})$	+ − +	$+1$
255.2.a.c	3	degree 3	$3q + 3q^{3} + 2q^{4} + 3q^{5} + 4q^{7} + O(q^{8})$	− − −	$+1$
255.2.a.d	4	degree 4	$4q + q^{2} + 4q^{3} + 9q^{4} - 4q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
255.2.a.a	$x^{2} - x - 29$
255.2.a.b	$x^{2} - 3x + 1$
255.2.a.c	$x^{3} - 4x^{2} + 1$
255.2.a.d	$x^{4} - 5x^{3} - 10x^{2} + 7x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(255))$	11
Old forms	22
Cusp forms $S_{2}(\Gamma_0(255))$	33
Eisenstein series $E_{2}(\Gamma_0(255))$	7
Modular forms $M_{2}(\Gamma_0(255))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(255)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.099 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, 5, 15, 17, 51, 85, 255]: [0, 1, 1, 4, 2, 8, 10, 40] (certified at ell = 2147483489)
T = [(2, 1), (7, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(255, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 255:2 1000

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