Atlas › Modular forms › Weight 2 › Level 908 › 908.2.a

Newspace 908.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(908))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
908.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} + 2q^{5} - 5q^{7} + O(q^{8})$	−
908.2.a.b	5	degree 5	$5q - 5q^{3} - q^{5} - 6q^{7} + O(q^{8})$	−
908.2.a.c	12	degree 12	$12q + 6q^{3} + q^{5} + 15q^{7} + O(q^{8})$	+

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 $T$
908.2.a.a	$x^{2} - 3x + 1$
908.2.a.b	$x^{5} + 6x^{4} + 7x^{3} - 5x^{2} - 8x - 2$
908.2.a.c	$x^{12} - 7x^{11} - 59x^{10} + 485x^{9} + 839x^{8} - 10748x^{7} + 3720x^{6} + 77227x^{5} - 81226x^{4} - 79309x^{3} - 11160x^{2} + 2940x + 586$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(908))$	19
Old forms	93
Cusp forms $S_{2}(\Gamma_0(908))$	112
Eisenstein series $E_{2}(\Gamma_0(908))$	5
Modular forms $M_{2}(\Gamma_0(908))$	117

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(908)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(227))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(454))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.344 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, 4, 227, 454, 908]: [0, 1, 2, 20, 59, 117] (certified at ell = 2147479573)
T = [(3, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(908, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 908:2 1000

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