Atlas › Modular forms › Weight 2 › Level 858 › 858.2.a

Newspace 858.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(858))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
858.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+ + − +	$+1$	858a1 rank 0
858.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − − −	$+1$	858c1 rank 0
858.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	+ − + −	$-1$	858b1 rank 1
858.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − − −	$+1$	858d1 rank 0
858.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	− + + −	$-1$	858f1 rank 1
858.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + O(q^{8})$	− + − −	$+1$	858h1 rank 0
858.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + − +	$-1$	858g1 rank 1
858.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + + +	$+1$	858e1 rank 0
858.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 4q^{5} - q^{6} + O(q^{8})$	− + − −	$+1$	858i1 rank 0
858.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− − + −	$+1$	858j1 rank 0
858.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	858k1 rank 0
858.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− − − +	$+1$	858l1 rank 0
858.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 4q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− − − +	$+1$	858m1 rank 0
858.2.a.n	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + − −	$-1$
858.2.a.o	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$
858.2.a.p	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ − + +	$+1$
858.2.a.q	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + O(q^{8})$	− − + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{23} + 4T_{29} + 5T_{31} + 6T_{17} + 7T_{19} + 8T_{37}$ 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 $T$
858.2.a.a	$x + 26$
858.2.a.b	$x - 108$
858.2.a.c	$x + 70$
858.2.a.d	$x - 68$
858.2.a.e	$x + 42$
858.2.a.f	$x - 106$
858.2.a.g	$x + 152$
858.2.a.h	$x + 18$
858.2.a.i	$x + 14$
858.2.a.j	$x + 22$
858.2.a.k	$x - 156$
858.2.a.l	$x + 84$
858.2.a.m	$x + 94$
858.2.a.n	$x^{2} + 22x - 1936$
858.2.a.o	$x^{2} + 72x + 108$
858.2.a.p	$x^{2} - 94x - 7016$
858.2.a.q	$x^{2} - 34x - 3032$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(858))$	21
Old forms	140
Cusp forms $S_{2}(\Gamma_0(858))$	161
Eisenstein series $E_{2}(\Gamma_0(858))$	15
Modular forms $M_{2}(\Gamma_0(858))$	176

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.259 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, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858]: [0, 1, 1, 3, 2, 1, 5, 5, 6, 6, 16, 18, 16, 46, 60, 176] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (23, 3), (29, 4), (31, 5), (17, 6), (19, 7), (37, 8)]; h of degree 21 = 1 x 21 (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(858, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 858:2 1000

← 857.2.a · 859.2.a →