Atlas › Modular forms › Weight 2 › Level 825 › 825.2.a

Newspace 825.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(825))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
825.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - 4q^{7} + O(q^{8})$	− −	825b1 rank 1
825.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{7} + O(q^{8})$	+ +	825a1 rank 1
825.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	− +	825c1 rank 1
825.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+ +
825.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 4q^{7} + O(q^{8})$	+ +
825.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	− +
825.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	− +
825.2.a.h	3	degree 3	$3q - 3q^{2} + 3q^{3} + 5q^{4} - 3q^{6} - 8q^{7} + O(q^{8})$	− +
825.2.a.i	3	degree 3	$3q - 2q^{2} - 3q^{3} + 8q^{4} + 2q^{6} + 3q^{7} + O(q^{8})$	+ −
825.2.a.j	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} + q^{6} - 4q^{7} + O(q^{8})$	+ −
825.2.a.k	3	degree 3	$3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} + O(q^{8})$	+ −
825.2.a.l	3	degree 3	$3q + q^{2} + 3q^{3} + q^{4} + q^{6} + 4q^{7} + O(q^{8})$	− −
825.2.a.m	3	degree 3	$3q + 2q^{2} + 3q^{3} + 8q^{4} + 2q^{6} - 3q^{7} + O(q^{8})$	− −
825.2.a.n	3	degree 3	$3q + 3q^{2} - 3q^{3} + 5q^{4} - 3q^{6} + 8q^{7} + O(q^{8})$	+ +

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 $T$
825.2.a.a	$x + 13$
825.2.a.b	$x - 3$
825.2.a.c	$x + 3$
825.2.a.d	$x^{2} - 4x - 28$
825.2.a.e	$x^{2} + 12x + 33$
825.2.a.f	$x^{2} + 4x - 28$
825.2.a.g	$x^{2} - 14x - 1$
825.2.a.h	$x^{3} + 27x^{2} + 209x + 361$
825.2.a.i	$x^{3} - 7x^{2} - 104x + 328$
825.2.a.j	$x^{3} + 13x^{2} - 9x - 67$
825.2.a.k	$x^{3} - x^{2} - 113x - 59$
825.2.a.l	$x^{3} - 13x^{2} - 9x + 67$
825.2.a.m	$x^{3} + 7x^{2} - 104x - 328$
825.2.a.n	$x^{3} - 27x^{2} + 209x - 361$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(825))$	32
Old forms	77
Cusp forms $S_{2}(\Gamma_0(825))$	109
Eisenstein series $E_{2}(\Gamma_0(825))$	23
Modular forms $M_{2}(\Gamma_0(825))$	132

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(825)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(165))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(275))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.45 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, 11, 15, 25, 33, 55, 75, 165, 275, 825]: [0, 1, 1, 2, 4, 3, 6, 8, 12, 28, 32, 124] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(825, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 825:2 1000

← 824.2.a · 826.2.a →