Atlas › Modular forms › Weight 2 › Level 996 › 996.2.a

Newspace 996.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(996))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
996.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + 4q^{7} + O(q^{8})$	+ +	996a1 rank 0
996.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	− +	996c1 rank 1
996.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 2q^{7} + O(q^{8})$	− +	996b1 rank 1
996.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ −
996.2.a.e	4	degree 4	$4q - 4q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	+ +
996.2.a.f	5	degree 5	$5q + 5q^{3} + 4q^{5} + O(q^{8})$	− −

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 $T$
996.2.a.a	$x - 16$
996.2.a.b	$x - 3$
996.2.a.c	$x + 7$
996.2.a.d	$x^{2} + 2x - 1$
996.2.a.e	$x^{4} + 10x^{3} - 184x^{2} - 1772x + 1675$
996.2.a.f	$x^{5} - 4x^{4} - 172x^{3} + 1314x^{2} - 1145x - 3064$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(996))$	14
Old forms	149
Cusp forms $S_{2}(\Gamma_0(996))$	163
Eisenstein series $E_{2}(\Gamma_0(996))$	11
Modular forms $M_{2}(\Gamma_0(996))$	174

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(996)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(166))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(249))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(332))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(498))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.389 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, 83, 166, 249, 332, 498, 996]: [0, 1, 1, 2, 3, 5, 8, 23, 30, 45, 88, 174] (certified at ell = 2147483323)
T = [(5, 1), (7, 3)]; h of degree 14 = 1 x 14 (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(996, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 996:2 1000

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