Atlas › Modular forms › Weight 2 › Level 656 › 656.2.a

Newspace 656.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(656))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
656.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−	656b1 rank 0
656.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	+	656a1 rank 1
656.2.a.c	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	+	656c1 rank 0
656.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{7} + O(q^{8})$	+
656.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{7} + O(q^{8})$	+
656.2.a.f	3	degree 3	$3q - 2q^{5} - 6q^{7} + O(q^{8})$	−
656.2.a.g	3	degree 3	$3q + 2q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	−
656.2.a.h	3	degree 3	$3q + 4q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−
656.2.a.i	4	degree 4	$4q - 2q^{3} + 4q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
656.2.a.a	$x - 10$
656.2.a.b	$x + 76$
656.2.a.c	$x + 52$
656.2.a.d	$x^{2} + 28x + 4$
656.2.a.e	$x^{2} - 100x + 700$
656.2.a.f	$x^{3} + 94x^{2} + 2192x + 15100$
656.2.a.g	$x^{3} - 178x^{2} + 8332x - 83116$
656.2.a.h	$x^{3} - 54x^{2} - 3036x + 150244$
656.2.a.i	$x^{4} + 72x^{3} - 11160x^{2} - 1019012x - 20847732$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(656))$	20
Old forms	59
Cusp forms $S_{2}(\Gamma_0(656))$	79
Eisenstein series $E_{2}(\Gamma_0(656))$	11
Modular forms $M_{2}(\Gamma_0(656))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(656)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(164))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(328))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 16, 41, 82, 164, 328, 656]: [0, 1, 2, 3, 4, 4, 12, 24, 46, 88] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 6 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(656, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 656:2 1000

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