Atlas › Modular forms › Weight 2 › Level 998 › 998.2.a

Newspace 998.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(998))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
998.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} + q^{5} + 3q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
998.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} - 2q^{7} + O(q^{8})$	+ +	$-1$
998.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 3q^{3} + 2q^{4} - 3q^{5} + 3q^{6} + q^{7} + O(q^{8})$	− +	$+1$
998.2.a.d	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
998.2.a.e	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
998.2.a.f	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} - 5q^{5} + q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
998.2.a.g	4	degree 4	$4q + 4q^{2} - 5q^{3} + 4q^{4} - 3q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
998.2.a.h	9	degree 9	$9q + 9q^{2} + 4q^{3} + 9q^{4} + 2q^{5} + 4q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$
998.2.a.i	13	degree 13	$13q - 13q^{2} + 7q^{3} + 13q^{4} + 7q^{5} - 7q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
998.2.a.a	$x^{2} + 2x - 44$
998.2.a.b	$x^{2} - 11x - 121$
998.2.a.c	$x^{2} - 62x + 836$
998.2.a.d	$x^{3} + 75x^{2} + 234x - 46693$
998.2.a.e	$x^{3} - 24x^{2} - 118x + 1601$
998.2.a.f	$x^{4} + 14x^{3} - 683x^{2} - 3981x + 25209$
998.2.a.g	$x^{4} + 63x^{3} + 749x^{2} + 72x - 12464$
998.2.a.h	$x^{9} - 81x^{8} + 471x^{7} + 78128x^{6} - 653639x^{5} - 30606429x^{4} + 112106634x^{3} + 4731294682x^{2} + 2362179321x - 161566687953$
998.2.a.i	$x^{13} - 62x^{12} - 5565x^{11} + 373354x^{10} + 6482956x^{9} - 531625402x^{8} - 3067516900x^{7} + 240975345347x^{6} + 1501898309403x^{5} - 25801001048706x^{4} - 86327602178306x^{3} + 1061698389139995x^{2} - 1172036516129236x - 1402478628185252$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(998))$	42
Old forms	82
Cusp forms $S_{2}(\Gamma_0(998))$	124
Eisenstein series $E_{2}(\Gamma_0(998))$	3
Modular forms $M_{2}(\Gamma_0(998))$	127

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(998)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(499))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.502 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, 499, 998]: [0, 1, 42, 127] (certified at ell = 2147475601)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 10 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(998, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 998:2 1000

← 997.2.a · 999.2.a →