Atlas › Modular forms › Weight 2 › Level 826 › 826.2.a

Newspace 826.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(826))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
826.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 3q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	826a1 rank 0
826.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 3q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	826b1 rank 0
826.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} + 2q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
826.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + 3q^{3} + 2q^{4} - 3q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
826.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 4q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	− − +	$-1$
826.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
826.2.a.g	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 6q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ − −	$-1$
826.2.a.h	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} + 2q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
826.2.a.i	5	degree 5	$5q + 5q^{2} + 2q^{3} + 5q^{4} + 5q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$
826.2.a.j	6	degree 6	$6q + 6q^{2} - 2q^{3} + 6q^{4} + 3q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
826.2.a.a	$x + 1$
826.2.a.b	$x - 5$
826.2.a.c	$x^{2} + x - 11$
826.2.a.d	$x^{2} - 3x - 1$
826.2.a.e	$x^{2} + 7x + 11$
826.2.a.f	$x^{2} + 3x + 1$
826.2.a.g	$x^{4} + 8x^{3} + 5x^{2} - 14x - 11$
826.2.a.h	$x^{4} + 2x^{3} - 11x^{2} + 2x + 7$
826.2.a.i	$x^{5} - 7x^{4} + 9x^{3} + 5x^{2} - 3x - 1$
826.2.a.j	$x^{6} - x^{5} - 47x^{4} + 93x^{3} + 465x^{2} - 1551x + 1210$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(826))$	29
Old forms	88
Cusp forms $S_{2}(\Gamma_0(826))$	117
Eisenstein series $E_{2}(\Gamma_0(826))$	7
Modular forms $M_{2}(\Gamma_0(826))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(826)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(118))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(413))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 7, 14, 59, 118, 413, 826]: [0, 1, 1, 4, 6, 17, 42, 124] (certified at ell = 2147481901)
T = [(3, 1), (5, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(826, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 826:2 1000

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