Atlas › Modular forms › Weight 2 › Level 816 › 816.2.a

Newspace 816.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(816))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
816.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 4q^{7} + O(q^{8})$	+ −	816c1 rank 0
816.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+ −	816h1 rank 1
816.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ +	816a1 rank 1
816.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ +	816e1 rank 0
816.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$	+ −	816g1 rank 1
816.2.a.f	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	+ −	816b1 rank 0
816.2.a.g	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	+ +	816f1 rank 0
816.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$	− +	816j1 rank 1
816.2.a.i	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 4q^{7} + O(q^{8})$	− +	816i1 rank 1
816.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + O(q^{8})$	− +	816d1 rank 0
816.2.a.k	2	$\mathbb{Q}(\sqrt{57})$	$2q - 2q^{3} - q^{5} - 8q^{7} + O(q^{8})$	+ −
816.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{3} - q^{5} + 2q^{7} + O(q^{8})$	− +
816.2.a.m	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{3} + 3q^{5} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = 3697T_{5} + 8016T_{7} + 4295T_{11} + 3553T_{13} + 8636T_{19} + 1949T_{23} + 2396T_{29} + 8529T_{31}$ 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 $T$
816.2.a.a	$x - 113382$
816.2.a.b	$x + 124056$
816.2.a.c	$x + 98542$
816.2.a.d	$x - 122602$
816.2.a.e	$x + 50598$
816.2.a.f	$x - 36588$
816.2.a.g	$x - 40900$
816.2.a.h	$x + 24722$
816.2.a.i	$x + 127604$
816.2.a.j	$x - 35200$
816.2.a.k	$x^{2} - 7010x - 46669448$
816.2.a.l	$x^{2} - 73302x - 9704886032$
816.2.a.m	$x^{2} - 41842x + 239198128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(816))$	16
Old forms	117
Cusp forms $S_{2}(\Gamma_0(816))$	133
Eisenstein series $E_{2}(\Gamma_0(816))$	23
Modular forms $M_{2}(\Gamma_0(816))$	156

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.77 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, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816]: [0, 1, 1, 2, 3, 3, 5, 4, 2, 8, 6, 12, 8, 12, 22, 22, 42, 40, 80, 152] (certified at ell = 2147483489)
T = [(5, 3697), (7, 8016), (11, 4295), (13, 3553), (19, 8636), (23, 1949), (29, 2396), (31, 8529)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(816, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 816:2 1000

← 815.2.a · 817.2.a →