Atlas › Modular forms › Weight 2 › Level 532 › 532.2.a

Newspace 532.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(532))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
532.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} + q^{7} + O(q^{8})$	− −	532a1 rank 0
532.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	− +
532.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{3} + 6q^{5} + 2q^{7} + O(q^{8})$	− −
532.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} - 4q^{5} - 2q^{7} + O(q^{8})$	+ −
532.2.a.e	3	degree 3	$3q - q^{3} + 2q^{5} - 3q^{7} + O(q^{8})$	+ +

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 $T$
532.2.a.a	$x + 6$
532.2.a.b	$x^{2} + 9x + 19$
532.2.a.c	$x^{2} - 17x + 67$
532.2.a.d	$x^{2} + 11x - 1$
532.2.a.e	$x^{3} - 5x^{2} - 115x + 706$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(532))$	10
Old forms	65
Cusp forms $S_{2}(\Gamma_0(532))$	75
Eisenstein series $E_{2}(\Gamma_0(532))$	11
Modular forms $M_{2}(\Gamma_0(532))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(532)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(133))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(266))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.235 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, 7, 14, 19, 28, 38, 76, 133, 266, 532]: [0, 1, 2, 1, 4, 2, 7, 7, 13, 14, 44, 86] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 10 = 1 x 10 (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(532, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 532:2 1000

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