Atlas › Modular forms › Weight 2 › Level 876 › 876.2.a

Newspace 876.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(876))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
876.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 4q^{7} + O(q^{8})$	+ −	876a1 rank 1
876.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 2q^{7} + O(q^{8})$	− +	876b1 rank 1
876.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + O(q^{8})$	+ −
876.2.a.d	3	degree 3	$3q - 3q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+ +
876.2.a.e	5	degree 5	$5q + 5q^{3} - q^{5} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
876.2.a.a	$x + 11$
876.2.a.b	$x + 7$
876.2.a.c	$x^{2} - 2$
876.2.a.d	$x^{3} - 7x^{2} - 205x + 129$
876.2.a.e	$x^{5} - 11x^{4} - 177x^{3} + 2697x^{2} - 6294x - 19638$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(876))$	12
Old forms	131
Cusp forms $S_{2}(\Gamma_0(876))$	143
Eisenstein series $E_{2}(\Gamma_0(876))$	11
Modular forms $M_{2}(\Gamma_0(876))$	154

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(876)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(146))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(219))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(292))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(438))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 4, 6, 12, 73, 146, 219, 292, 438, 876]: [0, 1, 1, 2, 3, 5, 6, 20, 26, 40, 78, 154] (certified at ell = 2147482873)
T = [(5, 1), (7, 3)]; h of degree 12 = 1 x 12 (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(876, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 876:2 1000

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