Atlas › Modular forms › Weight 2 › Level 856 › 856.2.a

Newspace 856.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(856))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
856.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	−	856c1 rank 1
856.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$	+	856b1 rank 1
856.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 4q^{7} + O(q^{8})$	+	856a1 rank 1
856.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} - 3q^{5} + O(q^{8})$	−	856d1 rank 1
856.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - q^{5} + O(q^{8})$	−
856.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{7} + O(q^{8})$	−
856.2.a.g	3	degree 3	$3q + 2q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	−
856.2.a.h	6	degree 6	$6q - 7q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+
856.2.a.i	10	degree 10	$10q - q^{3} + 5q^{5} - 4q^{7} + O(q^{8})$	+

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 $T$
856.2.a.a	$x + 61$
856.2.a.b	$x + 69$
856.2.a.c	$x - 43$
856.2.a.d	$x + 67$
856.2.a.e	$x^{2} - x - 911$
856.2.a.f	$x^{2} - 70x + 1033$
856.2.a.g	$x^{3} + 4x^{2} - 2280x - 12247$
856.2.a.h	$x^{6} + 150x^{5} - 1422x^{4} - 1122155x^{3} - 62814092x^{2} - 1327605172x - 9522731688$
856.2.a.i	$x^{10} - 189x^{9} - 87x^{8} + 2097560x^{7} - 124345253x^{6} - 2997812037x^{5} + 493432809735x^{4} - 15305624411606x^{3} + 109006634830372x^{2} + 1988306051323104x - 25009124866828048$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(856))$	27
Old forms	78
Cusp forms $S_{2}(\Gamma_0(856))$	105
Eisenstein series $E_{2}(\Gamma_0(856))$	7
Modular forms $M_{2}(\Gamma_0(856))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(856)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(214))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(428))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.53 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, 107, 214, 428, 856]: [0, 1, 2, 3, 10, 29, 57, 112] (certified at ell = 2147482621)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(856, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 856:2 1000

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