Atlas › Modular forms › Weight 2 › Level 886 › 886.2.a

Newspace 886.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(886))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
886.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	886b1 rank 1
886.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{5} + 3q^{7} + O(q^{8})$	+ +	$-1$	886a1 rank 1
886.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 4q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$	886c1 rank 0
886.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} + O(q^{8})$	− −	$-1$	886e1 rank 1
886.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{7} + O(q^{8})$	− −	$-1$	886d1 rank 1
886.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
886.2.a.g	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 4q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
886.2.a.h	4	degree 4	$4q + 4q^{2} + 8q^{3} + 4q^{4} + 4q^{5} + 8q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
886.2.a.i	6	degree 6	$6q - 6q^{2} + 3q^{3} + 6q^{4} - 7q^{5} - 3q^{6} + O(q^{8})$	+ +	$-1$
886.2.a.j	6	degree 6	$6q + 6q^{2} + 6q^{4} + 7q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$
886.2.a.k	9	degree 9	$9q - 9q^{2} - 4q^{3} + 9q^{4} + 9q^{5} + 4q^{6} - 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
886.2.a.a	$x - 1$
886.2.a.b	$x + 12$
886.2.a.c	$x - 14$
886.2.a.d	$x + 9$
886.2.a.e	$x$
886.2.a.f	$x^{2} + 7x - 49$
886.2.a.g	$x^{4} + 19x^{3} + 81x^{2} - 241x - 1529$
886.2.a.h	$x^{4} - 20x^{3} + 33x^{2} + 805x - 2894$
886.2.a.i	$x^{6} + 18x^{5} + 17x^{4} - 713x^{3} - 879x^{2} + 464x + 311$
886.2.a.j	$x^{6} - 21x^{5} + 104x^{4} + 388x^{3} - 3336x^{2} - 1125x + 23741$
886.2.a.k	$x^{9} - 23x^{8} + 43x^{7} + 2252x^{6} - 13783x^{5} - 35541x^{4} + 413903x^{3} - 422113x^{2} - 2295014x + 4139063$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(886))$	36
Old forms	74
Cusp forms $S_{2}(\Gamma_0(886))$	110
Eisenstein series $E_{2}(\Gamma_0(886))$	3
Modular forms $M_{2}(\Gamma_0(886))$	113

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(886)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(443))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.394 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, 443, 886]: [0, 1, 38, 113] (certified at ell = 2147481311)
T = [(3, 1), (5, 3)]; h of degree 36 = 1 x 36 (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(886, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 886:2 1000

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