Atlas › Modular forms › Weight 2 › Level 868 › 868.2.a

Newspace 868.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(868))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
868.2.a.a	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{7} + O(q^{8})$	+ +
868.2.a.b	3	degree 3	$3q - 3q^{3} - 4q^{5} + 3q^{7} + O(q^{8})$	− +
868.2.a.c	3	degree 3	$3q + q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+ −
868.2.a.d	3	degree 3	$3q + 3q^{3} + 2q^{5} - 3q^{7} + O(q^{8})$	+ +
868.2.a.e	5	degree 5	$5q + 3q^{3} + 8q^{5} + 5q^{7} + O(q^{8})$	− −

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 $T$
868.2.a.a	$x^{2} - 12$
868.2.a.b	$x^{3} + 7x^{2} + 6x - 9$
868.2.a.c	$x^{3} + x^{2} - 2x - 1$
868.2.a.d	$x^{3} - 5x^{2} - 4x + 11$
868.2.a.e	$x^{5} - 11x^{4} + 28x^{3} + 27x^{2} - 144x + 108$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(868))$	16
Old forms	107
Cusp forms $S_{2}(\Gamma_0(868))$	123
Eisenstein series $E_{2}(\Gamma_0(868))$	11
Modular forms $M_{2}(\Gamma_0(868))$	134

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(868)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(217))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(434))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 4, 7, 14, 28, 31, 62, 124, 217, 434, 868]: [0, 1, 2, 1, 4, 7, 3, 10, 19, 22, 68, 134] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 2 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(868, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 868:2 1000

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