Atlas › Modular forms › Weight 2 › Level 993 › 993.2.a

Newspace 993.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(993))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
993.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} + 4q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
993.2.a.b	3	degree 3	$3q + 6q^{2} + 3q^{3} + 6q^{4} + 4q^{5} + 6q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
993.2.a.c	10	degree 10	$10q - 4q^{2} + 10q^{3} + 6q^{4} - 4q^{5} - 4q^{6} - 21q^{7} + O(q^{8})$	− −	$-1$
993.2.a.d	10	degree 10	$10q + 7q^{2} - 10q^{3} + 13q^{4} - 7q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
993.2.a.e	15	degree 15	$15q - 6q^{2} - 15q^{3} + 18q^{4} - 8q^{5} + 6q^{6} - 15q^{7} + O(q^{8})$	+ +	$-1$
993.2.a.f	15	degree 15	$15q - q^{2} + 15q^{3} + 17q^{4} - 2q^{5} - q^{6} + 17q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
993.2.a.a	$x^{2} - 3x + 1$
993.2.a.b	$x^{3} - 10x^{2} + 29x - 21$
993.2.a.c	$x^{10} + 8x^{9} + 10x^{8} - 74x^{7} - 258x^{6} - 220x^{5} + 169x^{4} + 350x^{3} + 163x^{2} + 23x + 1$
993.2.a.d	$x^{10} - 7x^{9} - 23x^{8} + 192x^{7} + 275x^{6} - 1889x^{5} - 2583x^{4} + 6432x^{3} + 12095x^{2} + 3306x - 1109$
993.2.a.e	$x^{15} + 14x^{14} + 27x^{13} - 369x^{12} - 1394x^{11} + 2960x^{10} + 16287x^{9} - 4834x^{8} - 71128x^{7} - 28464x^{6} + 104696x^{5} + 88496x^{4} - 15536x^{3} - 34135x^{2} - 9500x - 562$
993.2.a.f	$x^{15} + 3x^{14} - 59x^{13} - 149x^{12} + 1306x^{11} + 2616x^{10} - 14035x^{9} - 19087x^{8} + 81240x^{7} + 50527x^{6} - 249379x^{5} + 22215x^{4} + 314612x^{3} - 215109x^{2} + 17880x + 7342$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(993))$	55
Old forms	54
Cusp forms $S_{2}(\Gamma_0(993))$	109
Eisenstein series $E_{2}(\Gamma_0(993))$	3
Modular forms $M_{2}(\Gamma_0(993))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(993)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(331))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 3, 331, 993]: [0, 1, 28, 112] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 55 = 1 x 55 (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(993, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 993:2 1000

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