Atlas › Modular forms › Weight 2 › Level 536 › 536.2.a

Newspace 536.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(536))$

Newforms

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

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 $T$
536.2.a.a	$x^{2} + 7x + 11$
536.2.a.b	$x^{3} + 9x^{2} - 19x - 164$
536.2.a.c	$x^{3} - 9x^{2} - 37x + 362$
536.2.a.d	$x^{4} + 6x^{3} - 185x^{2} - 655x + 8212$
536.2.a.e	$x^{5} - 7x^{4} - 152x^{3} + 1198x^{2} - 2194x + 669$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(536))$	17
Old forms	48
Cusp forms $S_{2}(\Gamma_0(536))$	65
Eisenstein series $E_{2}(\Gamma_0(536))$	7
Modular forms $M_{2}(\Gamma_0(536))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(536)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(134))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(268))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.21 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, 67, 134, 268, 536]: [0, 1, 2, 3, 6, 19, 37, 72] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(536, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 536:2 1000

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