Atlas › Modular forms › Weight 2 › Level 657 › 657.2.a

Newspace 657.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(657))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
657.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	−	657d1 rank 1
657.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 4q^{5} + 2q^{7} + O(q^{8})$	+	657a1 rank 0
657.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	−	657c1 rank 1
657.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	+	657b1 rank 0
657.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−
657.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 3q^{5} - 6q^{7} + O(q^{8})$	+
657.2.a.g	4	degree 4	$4q - q^{2} + 5q^{4} - 9q^{5} - 4q^{7} + O(q^{8})$	−
657.2.a.h	4	degree 4	$4q - 12q^{7} + O(q^{8})$	+
657.2.a.i	6	degree 6	$6q + q^{2} + 7q^{4} - 5q^{5} + 8q^{7} + O(q^{8})$	+
657.2.a.j	8	degree 8	$8q + 12q^{4} + 16q^{7} + O(q^{8})$	−

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 $T$
657.2.a.a	$x + 7$
657.2.a.b	$x - 11$
657.2.a.c	$x - 9$
657.2.a.d	$x - 5$
657.2.a.e	$x^{2} - 2x - 12$
657.2.a.f	$x^{2} - 12x + 16$
657.2.a.g	$x^{4} + 28x^{3} + 252x^{2} + 698x - 191$
657.2.a.h	$x^{4} - 28x^{2} + 169$
657.2.a.i	$x^{6} + 14x^{5} - 69x^{4} - 1306x^{3} + 125x^{2} + 20684x - 21961$
657.2.a.j	$x^{8} - 272x^{6} + 19984x^{4} - 284628x^{2} + 111747$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(657))$	30
Old forms	41
Cusp forms $S_{2}(\Gamma_0(657))$	71
Eisenstein series $E_{2}(\Gamma_0(657))$	7
Modular forms $M_{2}(\Gamma_0(657))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(657)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(219))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 73, 219, 657]: [0, 1, 2, 6, 26, 76] (certified at ell = 2147482873)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(657, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 657:2 1000

← 656.2.a · 658.2.a →