Atlas › Modular forms › Weight 2 › Level 539 › 539.2.a

Newspace 539.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(539))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
539.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - 2q^{6} + O(q^{8})$	−	539d1 rank 1
539.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 3q^{5} + O(q^{8})$	+	539a1 rank 0
539.2.a.c	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{4} + q^{5} + O(q^{8})$	+	539b1 rank 0
539.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	−	539c1 rank 1
539.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{4} + O(q^{8})$	+
539.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 6q^{4} + 4q^{5} + 10q^{6} + O(q^{8})$	+
539.2.a.g	3	degree 3	$3q - 3q^{3} - 6q^{5} + 3q^{6} + O(q^{8})$	−
539.2.a.h	3	degree 3	$3q - q^{3} + 4q^{4} - 2q^{5} - q^{6} + O(q^{8})$	+
539.2.a.i	3	degree 3	$3q + q^{3} + 4q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+
539.2.a.j	3	degree 3	$3q + 3q^{3} + 6q^{5} - 3q^{6} + O(q^{8})$	−
539.2.a.k	4	degree 4	$4q + 2q^{2} + 10q^{4} + O(q^{8})$	+
539.2.a.l	10	degree 10	$10q + 2q^{2} + 18q^{4} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ 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 $T$
539.2.a.a	$x + 32$
539.2.a.b	$x - 10$
539.2.a.c	$x - 42$
539.2.a.d	$x + 23$
539.2.a.e	$x^{2} + 2x - 7$
539.2.a.f	$x^{2} - 45$
539.2.a.g	$x^{3} + 60x^{2} + 363x - 9253$
539.2.a.h	$x^{3} + 90x^{2} + 2305x + 18039$
539.2.a.i	$x^{3} - 90x^{2} + 2223x - 15255$
539.2.a.j	$x^{3} - 60x^{2} + 225x + 9125$
539.2.a.k	$x^{4} - 2x^{3} - 4471x^{2} + 21608x + 634912$
539.2.a.l	$x^{10} - 2x^{9} - 3293x^{8} + 2860x^{7} + 3306087x^{6} - 4520402x^{5} - 1181470159x^{4} + 1664985440x^{3} + 166303785424x^{2} - 142985156856x - 7867094169788$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(539))$	34
Old forms	15
Cusp forms $S_{2}(\Gamma_0(539))$	49
Eisenstein series $E_{2}(\Gamma_0(539))$	15
Modular forms $M_{2}(\Gamma_0(539))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(539)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.338 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, 7, 11, 49, 77, 539]: [0, 1, 2, 5, 10, 58] (certified at ell = 2147482681)
T = [(2, 1), (3, 3), (5, 5), (13, 7)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 8 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(539, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 539:2 1000

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