Atlas › Modular forms › Weight 2 › Level 489 › 489.2.a

Newspace 489.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(489))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
489.2.a.a	4	degree 4	$4q - 2q^{2} + 4q^{3} - 3q^{5} - 2q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
489.2.a.b	5	degree 5	$5q - 2q^{2} - 5q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
489.2.a.c	8	degree 8	$8q + 4q^{2} - 8q^{3} + 12q^{4} + 5q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
489.2.a.d	10	degree 10	$10q + q^{2} + 10q^{3} + 13q^{4} - q^{5} + q^{6} + 14q^{7} + O(q^{8})$	− +	$+1$

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 $T$
489.2.a.a	$x^{4} + 5x^{3} + 6x^{2} - 1$
489.2.a.b	$x^{5} + 5x^{4} - 8x^{3} - 34x^{2} + 53x - 16$
489.2.a.c	$x^{8} - 9x^{7} + 5x^{6} + 141x^{5} - 336x^{4} - 59x^{3} + 491x^{2} - 96x - 113$
489.2.a.d	$x^{10} - 42x^{8} + 14x^{7} + 595x^{6} - 281x^{5} - 3518x^{4} + 1821x^{3} + 8375x^{2} - 3659x - 5768$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(489))$	27
Old forms	26
Cusp forms $S_{2}(\Gamma_0(489))$	53
Eisenstein series $E_{2}(\Gamma_0(489))$	3
Modular forms $M_{2}(\Gamma_0(489))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(489)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(163))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 163, 489]: [0, 1, 14, 56] (certified at ell = 2147483179)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(489, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 489:2 1000

← 488.2.a · 490.2.a →