Atlas › Modular forms › Weight 2 › Level 286 › 286.2.a

Newspace 286.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(286))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
286.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 3q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	286a1 rank 0
286.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + +	$-1$	286c1 rank 1
286.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$	286b1 rank 1
286.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$	286d1 rank 0
286.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− + +	$+1$	286e1 rank 0
286.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	− − −	$+1$	286f1 rank 0
286.2.a.g	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} + 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{17} + 9T_{19}$ 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 $T$
286.2.a.a	$x - 116$
286.2.a.b	$x + 42$
286.2.a.c	$x - 14$
286.2.a.d	$x - 38$
286.2.a.e	$x + 18$
286.2.a.f	$x + 52$
286.2.a.g	$x^{3} + 100x^{2} + 2796x + 14864$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(286))$	9
Old forms	30
Cusp forms $S_{2}(\Gamma_0(286))$	39
Eisenstein series $E_{2}(\Gamma_0(286))$	7
Modular forms $M_{2}(\Gamma_0(286))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(286)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(143))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 11, 13, 22, 26, 143, 286]: [0, 1, 2, 1, 5, 5, 16, 46] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (17, 7), (19, 9)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(286, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 286:2 1000

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