Atlas › Modular forms › Weight 2 › Level 266 › 266.2.a

Newspace 266.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(266))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
266.2.a.a	2	$\mathbb{Q}(\sqrt{29})$	$2q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
266.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 3q^{3} + 2q^{4} + q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
266.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
266.2.a.d	3	degree 3	$3q + 3q^{2} - q^{3} + 3q^{4} + 5q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
266.2.a.a	$x^{2} + 2x - 115$
266.2.a.b	$x^{2} - 6x - 71$
266.2.a.c	$x^{2} - 4x - 9$
266.2.a.d	$x^{3} - 14x^{2} + 11x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(266))$	9
Old forms	28
Cusp forms $S_{2}(\Gamma_0(266))$	37
Eisenstein series $E_{2}(\Gamma_0(266))$	7
Modular forms $M_{2}(\Gamma_0(266))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(266)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(133))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 7, 14, 19, 38, 133, 266]: [0, 1, 1, 4, 2, 7, 14, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(266, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 266:2 1000

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