Atlas › Modular forms › Weight 2 › Level 507 › 507.2.a

Newspace 507.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(507))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
507.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+	507c1 rank 1
507.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+	507b1 rank 1
507.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+	507a1 rank 1
507.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 2q^{3} + 5q^{4} + 3q^{5} - q^{6} + 3q^{7} + O(q^{8})$	−
507.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 4q^{4} + O(q^{8})$	+
507.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} + O(q^{8})$	+
507.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 2q^{3} + 5q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	−
507.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + O(q^{8})$	−
507.2.a.i	3	degree 3	$3q - 3q^{2} - 3q^{3} + 11q^{4} - 6q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+
507.2.a.j	3	degree 3	$3q - q^{2} + 3q^{3} - q^{4} - 4q^{5} - q^{6} - 10q^{7} + O(q^{8})$	−
507.2.a.k	3	degree 3	$3q + q^{2} + 3q^{3} - q^{4} + 4q^{5} + q^{6} + 10q^{7} + O(q^{8})$	−
507.2.a.l	3	degree 3	$3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+

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 $T$
507.2.a.a	$x + 7$
507.2.a.b	$x - 2$
507.2.a.c	$x + 2$
507.2.a.d	$x^{2} - 8x - 52$
507.2.a.e	$x^{2} - 108$
507.2.a.f	$x^{2} - 3$
507.2.a.g	$x^{2} + 8x - 52$
507.2.a.h	$x^{2} - 2x - 49$
507.2.a.i	$x^{3} + 21x^{2} + 98x - 49$
507.2.a.j	$x^{3} + 13x^{2} + 40x - 13$
507.2.a.k	$x^{3} - 13x^{2} + 40x + 13$
507.2.a.l	$x^{3} - 21x^{2} + 98x + 49$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(507))$	25
Old forms	22
Cusp forms $S_{2}(\Gamma_0(507))$	47
Eisenstein series $E_{2}(\Gamma_0(507))$	27
Modular forms $M_{2}(\Gamma_0(507))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(507)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(169))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.314 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, 13, 39, 169, 507]: [0, 1, 1, 6, 15, 62] (certified at ell = 2147483053)
T = [(2, 1), (5, 3)]; h of degree 25 = 1 x 25 (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(507, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 507:2 1000

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