Atlas › Modular forms › Weight 2 › Level 884 › 884.2.a

Newspace 884.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(884))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
884.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+ −
884.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - 2q^{5} + q^{7} + O(q^{8})$	− +
884.2.a.c	6	degree 6	$6q - q^{3} + 5q^{7} + O(q^{8})$	+ +
884.2.a.d	6	degree 6	$6q - q^{3} + 4q^{5} + 5q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
884.2.a.a	$x^{2} + 7x + 9$
884.2.a.b	$x^{2} + 7x + 11$
884.2.a.c	$x^{6} + x^{5} - 221x^{4} + 512x^{3} + 10920x^{2} - 42544x - 10800$
884.2.a.d	$x^{6} - 11x^{5} - 155x^{4} + 1564x^{3} + 2872x^{2} - 17536x - 32608$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(884))$	16
Old forms	105
Cusp forms $S_{2}(\Gamma_0(884))$	121
Eisenstein series $E_{2}(\Gamma_0(884))$	11
Modular forms $M_{2}(\Gamma_0(884))$	132

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(884)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(221))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(442))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 13, 17, 26, 34, 52, 68, 221, 442, 884]: [0, 1, 2, 1, 2, 5, 6, 10, 12, 22, 66, 132] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(884, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 884:2 1000

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