Atlas › Modular forms › Weight 2 › Level 261 › 261.2.a

Newspace 261.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(261))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
261.2.a.a	$x^{2} + 5x + 5$
261.2.a.b	$x^{2} + 3x + 1$
261.2.a.c	$x^{2} - 5x + 5$
261.2.a.d	$x^{2} - 4x + 2$
261.2.a.e	$x^{3} + 2x^{2} - 14x + 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(261))$	11
Old forms	16
Cusp forms $S_{2}(\Gamma_0(261))$	27
Eisenstein series $E_{2}(\Gamma_0(261))$	7
Modular forms $M_{2}(\Gamma_0(261))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(261)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 9, 29, 87, 261]: [0, 1, 2, 3, 12, 32] (certified at ell = 2147483353)
T = [(2, 1), (5, 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(261, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 261:2 1000

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