Atlas › Modular forms › Weight 2 › Level 488 › 488.2.a

Newspace 488.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(488))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
488.2.a.a	$x^{2} + 84x - 548$
488.2.a.b	$x^{3} + 106x^{2} + 1788x + 6452$
488.2.a.c	$x^{4} - 80x^{3} + 1163x^{2} + 19292x - 51220$
488.2.a.d	$x^{6} - 132x^{5} + 4517x^{4} + 19926x^{3} - 3645580x^{2} + 61991780x - 302320544$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(488))$	15
Old forms	44
Cusp forms $S_{2}(\Gamma_0(488))$	59
Eisenstein series $E_{2}(\Gamma_0(488))$	7
Modular forms $M_{2}(\Gamma_0(488))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(488)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(244))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 8, 61, 122, 244, 488]: [0, 1, 2, 3, 5, 17, 34, 66] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(488, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 488:2 1000

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