Atlas › Modular forms › Weight 2 › Level 639 › 639.2.a

Newspace 639.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(639))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
639.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	−	639a1 rank 1
639.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{7} + O(q^{8})$	+
639.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−
639.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - q^{5} - 6q^{7} + O(q^{8})$	−
639.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 4q^{7} + O(q^{8})$	−
639.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 5q^{5} - 4q^{7} + O(q^{8})$	+
639.2.a.g	3	degree 3	$3q + 4q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	+
639.2.a.h	3	degree 3	$3q + q^{2} + 3q^{4} - 5q^{5} + 2q^{7} + O(q^{8})$	+
639.2.a.i	4	degree 4	$4q - 3q^{2} + 5q^{4} + 3q^{5} + 6q^{7} + O(q^{8})$	+
639.2.a.j	4	degree 4	$4q + 2q^{4} - 8q^{5} + 2q^{7} + O(q^{8})$	+
639.2.a.k	4	degree 4	$4q + 2q^{4} + 8q^{5} + 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
639.2.a.a	$x + 7$
639.2.a.b	$x^{2} + 2x - 97$
639.2.a.c	$x^{2} - 2x - 12$
639.2.a.d	$x^{2} + 2x - 4$
639.2.a.e	$x^{2} - 2x - 97$
639.2.a.f	$x^{2} - 18x + 76$
639.2.a.g	$x^{3} - 9x^{2} + 7x + 57$
639.2.a.h	$x^{3} + 14x^{2} - 61x - 1029$
639.2.a.i	$x^{4} - 6x^{3} - 89x^{2} - 106x + 76$
639.2.a.j	$x^{4} + 24x^{3} + 196x^{2} + 624x + 624$
639.2.a.k	$x^{4} - 24x^{3} + 196x^{2} - 624x + 624$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(639))$	29
Old forms	40
Cusp forms $S_{2}(\Gamma_0(639))$	69
Eisenstein series $E_{2}(\Gamma_0(639))$	7
Modular forms $M_{2}(\Gamma_0(639))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(639)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(213))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 9, 71, 213, 639]: [0, 1, 2, 7, 26, 74] (certified at ell = 2147482681)
T = [(2, 1), (5, 3)]; h of degree 29 = 1 x 29 (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(639, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 639:2 1000

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