Atlas › Modular forms › Weight 2 › Level 337 › 337.2.a

Newspace 337.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(337))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
337.2.a.a	12	degree 12	$12q - 6q^{2} - 11q^{3} + 10q^{4} - 10q^{5} + q^{6} - 13q^{7} + O(q^{8})$	+	$-1$
337.2.a.b	15	degree 15	$15q + 3q^{2} + 9q^{3} + 15q^{4} + 10q^{5} + q^{6} + 7q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
337.2.a.a	$x^{12} + 17x^{11} + 104x^{10} + 241x^{9} - 96x^{8} - 1347x^{7} - 1705x^{6} + 774x^{5} + 3017x^{4} + 1928x^{3} + 82x^{2} - 270x - 61$
337.2.a.b	$x^{15} - 12x^{14} + 18x^{13} + 297x^{12} - 1192x^{11} - 1356x^{10} + 13591x^{9} - 11498x^{8} - 46643x^{7} + 80176x^{6} + 34247x^{5} - 124275x^{4} + 18463x^{3} + 57441x^{2} - 11485x - 8029$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(337))$	27
Old forms	0
Cusp forms $S_{2}(\Gamma_0(337))$	27
Eisenstein series $E_{2}(\Gamma_0(337))$	1
Modular forms $M_{2}(\Gamma_0(337))$	28

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.173 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, 337]: [0, 28] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(337, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 337:2 1000

← 336.2.a · 338.2.a →