Atlas › Modular forms › Weight 2 › Level 861 › 861.2.a

Newspace 861.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(861))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
861.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	861a1 rank 0
861.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	861d1 rank 1
861.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	861c1 rank 1
861.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	861b1 rank 1
861.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− − +	$-1$
861.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 2q^{3} + 5q^{4} + 4q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
861.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 2q^{3} + 5q^{4} - 3q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
861.2.a.h	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
861.2.a.i	4	degree 4	$4q + q^{2} - 4q^{3} + 3q^{4} - 3q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
861.2.a.j	5	degree 5	$5q - 3q^{2} - 5q^{3} + 7q^{4} + q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	+ + −	$+1$
861.2.a.k	5	degree 5	$5q + 3q^{2} - 5q^{3} + 11q^{4} - 9q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$
861.2.a.l	5	degree 5	$5q + 3q^{2} + 5q^{3} + 7q^{4} + 3q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	− + +	$+1$
861.2.a.m	7	degree 7	$7q + 4q^{2} + 7q^{3} + 8q^{4} + q^{5} + 4q^{6} + 7q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
861.2.a.a	$x - 5$
861.2.a.b	$x + 10$
861.2.a.c	$x - 8$
861.2.a.d	$x + 2$
861.2.a.e	$x^{2} + 8x + 14$
861.2.a.f	$x^{2} - 11x + 26$
861.2.a.g	$x^{2} + 10x + 8$
861.2.a.h	$x^{3} - 2x^{2} - 36x + 74$
861.2.a.i	$x^{4} + 8x^{3} - 26x^{2} - 174x + 388$
861.2.a.j	$x^{5} - 154x^{3} - 320x^{2} + 6064x + 22322$
861.2.a.k	$x^{5} + 24x^{4} + 48x^{3} - 2344x^{2} - 17860x - 36546$
861.2.a.l	$x^{5} - 12x^{4} - 28x^{3} + 574x^{2} - 760x - 1798$
861.2.a.m	$x^{7} - 7x^{6} - 144x^{5} + 768x^{4} + 3686x^{3} - 13498x^{2} - 23274x + 70348$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(861))$	39
Old forms	70
Cusp forms $S_{2}(\Gamma_0(861))$	109
Eisenstein series $E_{2}(\Gamma_0(861))$	7
Modular forms $M_{2}(\Gamma_0(861))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(861)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(287))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.417 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, 7, 21, 41, 123, 287, 861]: [0, 1, 1, 4, 4, 16, 30, 116] (certified at ell = 2147482921)
T = [(2, 1), (5, 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(861, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 861:2 1000

← 860.2.a · 862.2.a →