Atlas › Modular forms › Weight 2 › Level 992 › 992.2.a

Newspace 992.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(992))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
992.2.a.a	$x^{2} + 60x + 868$
992.2.a.b	$x^{2} + 36x - 644$
992.2.a.c	$x^{3} + 166x^{2} + 6124x - 42424$
992.2.a.d	$x^{3} - 78x^{2} - 2044x + 62024$
992.2.a.e	$x^{4} + 112x^{3} + 1268x^{2} - 82432x - 620288$
992.2.a.f	$x^{4} - 144x^{3} - 1684x^{2} + 866736x - 25386656$
992.2.a.g	$x^{6} - 60x^{5} - 9024x^{4} + 296080x^{3} + 20688544x^{2} + 236996544x + 147094848$
992.2.a.h	$x^{6} - 96x^{5} - 2336x^{4} + 303232x^{3} - 2620544x^{2} - 35100224x + 337577280$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(992))$	30
Old forms	91
Cusp forms $S_{2}(\Gamma_0(992))$	121
Eisenstein series $E_{2}(\Gamma_0(992))$	15
Modular forms $M_{2}(\Gamma_0(992))$	136

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(992)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(248))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(496))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.819 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, 8, 16, 31, 32, 62, 124, 248, 496, 992]: [0, 1, 2, 3, 4, 3, 6, 10, 19, 36, 68, 132] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 9 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(992, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 992:2 1000

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