Atlas › Modular forms › Weight 2 › Level 439 › 439.2.a

Newspace 439.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(439))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
439.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + q^{3} - 2q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
439.2.a.b	9	degree 9	$9q - q^{2} - 4q^{3} + 7q^{4} - 12q^{5} - 5q^{6} + O(q^{8})$	+	$-1$
439.2.a.c	25	degree 25	$25q + 4q^{2} + 3q^{3} + 28q^{4} + 15q^{5} + O(q^{8})$	−	$+1$

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 $T$
439.2.a.a	$x^{2} + x - 1$
439.2.a.b	$x^{9} + 5x^{8} - 5x^{7} - 38x^{6} + x^{5} + 51x^{4} - 19x^{2} + 2x + 1$
439.2.a.c	$x^{25} - 7x^{24} - 78x^{23} + 638x^{22} + 2078x^{21} - 22652x^{20} - 19720x^{19} + 416374x^{18} - 68307x^{17} - 4423461x^{16} + 2881060x^{15} + 28571527x^{14} - 23685149x^{13} - 114991163x^{12} + 89171181x^{11} + 291131720x^{10} - 156851208x^{9} - 451620211x^{8} + 93443707x^{7} + 384331295x^{6} + 38723209x^{5} - 140850851x^{4} - 46388365x^{3} + 9213266x^{2} + 5123490x + 431903$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(439))$	36
Old forms	0
Cusp forms $S_{2}(\Gamma_0(439))$	36
Eisenstein series $E_{2}(\Gamma_0(439))$	1
Modular forms $M_{2}(\Gamma_0(439))$	37

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 439]: [0, 37] (certified at ell = 2147481151)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 4 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(439, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 439:2 1000

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