Atlas › Modular forms › Weight 2 › Level 810 › 810.2.a

Newspace 810.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(810))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
810.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 4q^{7} + O(q^{8})$	+ −	810b1 rank 0
810.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ −	810d1 rank 1
810.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ −	810a1 rank 0
810.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + 5q^{7} + O(q^{8})$	+ −	810c1 rank 0
810.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− +	810h1 rank 1
810.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	− +	810e1 rank 0
810.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	− +	810f1 rank 0
810.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} + 5q^{7} + O(q^{8})$	− +	810g1 rank 0
810.2.a.i	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + q^{7} + O(q^{8})$	+ +
810.2.a.j	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ +
810.2.a.k	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− −
810.2.a.l	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
810.2.a.a	$x - 142$
810.2.a.b	$x + 473$
810.2.a.c	$x - 178$
810.2.a.d	$x - 118$
810.2.a.e	$x + 62$
810.2.a.f	$x + 272$
810.2.a.g	$x + 107$
810.2.a.h	$x + 20$
810.2.a.i	$x^{2} + 691x + 86626$
810.2.a.j	$x^{2} - 884x + 127864$
810.2.a.k	$x^{2} + 277x + 18778$
810.2.a.l	$x^{2} - 200x - 6428$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(810))$	16
Old forms	123
Cusp forms $S_{2}(\Gamma_0(810))$	139
Eisenstein series $E_{2}(\Gamma_0(810))$	47
Modular forms $M_{2}(\Gamma_0(810))$	186

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.654 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, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810]: [0, 1, 1, 1, 3, 2, 3, 4, 5, 4, 10, 8, 11, 10, 22, 20, 29, 58, 56, 166] (certified at ell = 2147482801)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 16 = 1 x 16 (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(810, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 810:2 1000

← 809.2.a · 811.2.a →