Atlas › Modular forms › Weight 2 › Level 339 › 339.2.a

Newspace 339.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(339))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
339.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− −	$-1$	339c1 rank 1
339.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} - 3q^{7} + O(q^{8})$	− −	$-1$	339a1 rank 1
339.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$	339b1 rank 0
339.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
339.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$
339.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + 4q^{2} + 2q^{3} + 4q^{4} + 3q^{5} + 4q^{6} - q^{7} + O(q^{8})$	− +	$+1$
339.2.a.g	5	degree 5	$5q + 5q^{3} + 4q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	− +	$+1$
339.2.a.h	5	degree 5	$5q + q^{2} - 5q^{3} + 11q^{4} - 2q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
339.2.a.a	$x + 11$
339.2.a.b	$x + 3$
339.2.a.c	$x - 8$
339.2.a.d	$x^{2} - 4x - 46$
339.2.a.e	$x^{2} + 6x + 1$
339.2.a.f	$x^{2} - 13x + 4$
339.2.a.g	$x^{5} - 9x^{4} - 31x^{3} + 227x^{2} + 570x + 326$
339.2.a.h	$x^{5} + 5x^{4} - 226x^{3} - 1086x^{2} + 12721x + 59401$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(339))$	19
Old forms	18
Cusp forms $S_{2}(\Gamma_0(339))$	37
Eisenstein series $E_{2}(\Gamma_0(339))$	3
Modular forms $M_{2}(\Gamma_0(339))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(339)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 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, 113, 339]: [0, 1, 10, 40] (certified at ell = 2147480161)
T = [(2, 1), (5, 3)]; h of degree 19 = 1 x 19 (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(339, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 339:2 1000

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