Atlas › Modular forms › Weight 2 › Level 837 › 837.2.a

Newspace 837.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(837))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
837.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	−
837.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{4} + 4q^{5} + 5q^{7} + O(q^{8})$	−
837.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{4} - 2q^{7} + O(q^{8})$	−
837.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} - 4q^{5} + 5q^{7} + O(q^{8})$	−
837.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{5} + 2q^{7} + O(q^{8})$	−
837.2.a.f	3	degree 3	$3q - q^{2} + 3q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−
837.2.a.g	3	degree 3	$3q + q^{2} + 3q^{4} - q^{5} - 2q^{7} + O(q^{8})$	−
837.2.a.h	4	degree 4	$4q - 8q^{7} + O(q^{8})$	+
837.2.a.i	4	degree 4	$4q + 4q^{4} - 8q^{7} + O(q^{8})$	−
837.2.a.j	5	degree 5	$5q - 3q^{2} + 5q^{4} - 5q^{5} + O(q^{8})$	+
837.2.a.k	5	degree 5	$5q + 3q^{2} + 5q^{4} + 5q^{5} + O(q^{8})$	+
837.2.a.l	6	degree 6	$6q + 10q^{4} + 8q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
837.2.a.a	$x^{2} + 14x + 47$
837.2.a.b	$x^{2} - 11x + 26$
837.2.a.c	$x^{2} - 108$
837.2.a.d	$x^{2} + 11x + 26$
837.2.a.e	$x^{2} - 14x + 47$
837.2.a.f	$x^{3} - 2x^{2} - 55x - 57$
837.2.a.g	$x^{3} + 2x^{2} - 55x + 57$
837.2.a.h	$x^{4} - 46x^{2} + 484$
837.2.a.i	$x^{4} - 84x^{2} + 484$
837.2.a.j	$x^{5} + 18x^{4} - 1141x^{2} - 5707x - 8161$
837.2.a.k	$x^{5} - 18x^{4} + 1141x^{2} - 5707x + 8161$
837.2.a.l	$x^{6} - 215x^{4} + 14374x^{2} - 285628$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(837))$	40
Old forms	51
Cusp forms $S_{2}(\Gamma_0(837))$	91
Eisenstein series $E_{2}(\Gamma_0(837))$	11
Modular forms $M_{2}(\Gamma_0(837))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(837)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(279))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 9, 27, 31, 93, 279, 837]: [0, 1, 2, 4, 3, 12, 34, 98] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 40 = 1 x 40 (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(837, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 837:2 1000

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