Atlas › Modular forms › Weight 2 › Level 885 › 885.2.a

Newspace 885.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(885))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
885.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$	885d1 rank 1
885.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} + O(q^{8})$	− − +	$-1$	885c1 rank 1
885.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + O(q^{8})$	+ − −	$-1$	885b1 rank 1
885.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	885a1 rank 0
885.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− + −	$-1$
885.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{3} + 8q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− + +	$+1$
885.2.a.g	3	degree 3	$3q - q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$
885.2.a.h	3	degree 3	$3q + 2q^{2} - 3q^{3} + 4q^{4} - 3q^{5} - 2q^{6} + O(q^{8})$	+ + −	$+1$
885.2.a.i	4	degree 4	$4q + 2q^{2} - 4q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$
885.2.a.j	6	degree 6	$6q - 4q^{2} - 6q^{3} + 8q^{4} - 6q^{5} + 4q^{6} + O(q^{8})$	+ + +	$-1$
885.2.a.k	6	degree 6	$6q + 2q^{2} + 6q^{3} + 6q^{4} - 6q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	− + +	$+1$
885.2.a.l	9	degree 9	$9q + 3q^{2} + 9q^{3} + 13q^{4} + 9q^{5} + 3q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
885.2.a.a	$x + 8$
885.2.a.b	$x$
885.2.a.c	$x - 1$
885.2.a.d	$x - 8$
885.2.a.e	$x^{2} + 12x + 28$
885.2.a.f	$x^{2} - 12x - 60$
885.2.a.g	$x^{3} + 7x^{2} - 88x - 548$
885.2.a.h	$x^{3} - 2x^{2} - 54x + 176$
885.2.a.i	$x^{4} - 2x^{3} - 98x^{2} + 88x + 2248$
885.2.a.j	$x^{6} + 4x^{5} - 218x^{4} - 548x^{3} + 15159x^{2} + 16624x - 323104$
885.2.a.k	$x^{6} - 20x^{5} - 22x^{4} + 1830x^{3} - 1131x^{2} - 44890x - 16706$
885.2.a.l	$x^{9} - 15x^{8} - 278x^{7} + 4952x^{6} + 3765x^{5} - 249495x^{4} + 147032x^{3} + 3619154x^{2} - 1590424x - 16312736$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(885))$	39
Old forms	78
Cusp forms $S_{2}(\Gamma_0(885))$	117
Eisenstein series $E_{2}(\Gamma_0(885))$	7
Modular forms $M_{2}(\Gamma_0(885))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(885)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(177))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(295))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.438 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, 5, 15, 59, 177, 295, 885]: [0, 1, 1, 4, 6, 22, 32, 124] (certified at ell = 2147482481)
T = [(2, 1), (7, 3)]; h of degree 39 = 1 x 39 (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(885, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 885:2 1000

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