Atlas › Modular forms › Weight 2 › Level 298 › 298.2.a

Newspace 298.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(298))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
298.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{5} + 4q^{7} + O(q^{8})$	+ +	$-1$	298b1 rank 1
298.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	298a1 rank 1
298.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
298.2.a.d	3	degree 3	$3q - 3q^{2} - 5q^{3} + 3q^{4} + q^{5} + 5q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
298.2.a.e	5	degree 5	$5q + 5q^{2} + q^{3} + 5q^{4} + 5q^{5} + q^{6} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
298.2.a.a	$x + 12$
298.2.a.b	$x + 8$
298.2.a.c	$x^{2} - 8x + 4$
298.2.a.d	$x^{3} + 2x^{2} - 29x - 83$
298.2.a.e	$x^{5} - 16x^{4} + 41x^{3} + 125x^{2} - 336x - 212$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(298))$	12
Old forms	24
Cusp forms $S_{2}(\Gamma_0(298))$	36
Eisenstein series $E_{2}(\Gamma_0(298))$	3
Modular forms $M_{2}(\Gamma_0(298))$	39

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(298)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(149))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 149, 298]: [0, 1, 13, 39] (certified at ell = 2147481629)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(298, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 298:2 1000

← 297.2.a · 299.2.a →