Atlas › Modular forms › Weight 2 › Level 984 › 984.2.a

Newspace 984.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(984))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
984.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+ −	984c1 rank 1
984.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ −	984a1 rank 0
984.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	+ −	984b1 rank 0
984.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{7} + O(q^{8})$	− +	984d1 rank 1
984.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 4q^{7} + O(q^{8})$	− −
984.2.a.f	3	degree 3	$3q - 3q^{3} + O(q^{8})$	+ +
984.2.a.g	3	degree 3	$3q - 3q^{3} + 3q^{5} - 4q^{7} + O(q^{8})$	+ +
984.2.a.h	3	degree 3	$3q + 3q^{3} + q^{5} + 6q^{7} + O(q^{8})$	− +
984.2.a.i	5	degree 5	$5q + 5q^{3} + q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{19} + 9T_{23} + 11T_{29}$ 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 $T$
984.2.a.a	$x + 31$
984.2.a.b	$x - 81$
984.2.a.c	$x - 33$
984.2.a.d	$x - 15$
984.2.a.e	$x^{2} + 202x + 9809$
984.2.a.f	$x^{3} + 150x^{2} + 4205x - 112028$
984.2.a.g	$x^{3} - 201x^{2} + 6255x + 417289$
984.2.a.h	$x^{3} - 132x^{2} - 13753x + 1094428$
984.2.a.i	$x^{5} - 89x^{4} - 5569x^{3} + 484897x^{2} + 3745508x - 231532676$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(984))$	20
Old forms	141
Cusp forms $S_{2}(\Gamma_0(984))$	161
Eisenstein series $E_{2}(\Gamma_0(984))$	15
Modular forms $M_{2}(\Gamma_0(984))$	176

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(984)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(164))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(246))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(328))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(492))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.051 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, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984]: [0, 1, 1, 2, 3, 3, 5, 8, 4, 12, 16, 24, 46, 46, 90, 176] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (13, 5), (19, 7), (23, 9), (29, 11)]; h of degree 20 = 1 x 20 (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(984, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 984:2 1000

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