Atlas › Modular forms › Weight 2 › Level 509 › 509.2.a

Newspace 509.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(509))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
509.2.a.a	14	degree 14	$14q - 3q^{2} - 5q^{3} + 3q^{4} - 5q^{5} - 8q^{6} - 17q^{7} + O(q^{8})$	+	$-1$
509.2.a.b	28	degree 28	$28q + 3q^{2} + 5q^{3} + 41q^{4} + q^{5} + 2q^{6} + 19q^{7} + O(q^{8})$	−	$+1$

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 $T$
509.2.a.a	$x^{14} + 8x^{13} + x^{12} - 109x^{11} - 117x^{10} + 508x^{9} + 504x^{8} - 1193x^{7} - 621x^{6} + 1445x^{5} + 12x^{4} - 682x^{3} + 232x^{2} + 21x - 13$
509.2.a.b	$x^{28} - 8x^{27} - 79x^{26} + 775x^{25} + 2020x^{24} - 29956x^{23} - 9270x^{22} + 607348x^{21} - 451466x^{20} - 7233548x^{19} + 9489672x^{18} + 53890768x^{17} - 89144249x^{16} - 258670457x^{15} + 486611251x^{14} + 798783953x^{13} - 1647657054x^{12} - 1521661510x^{11} + 3479175501x^{10} + 1574898974x^{9} - 4402183614x^{8} - 508093292x^{7} + 2997640229x^{6} - 394588162x^{5} - 840597920x^{4} + 243169571x^{3} + 43794823x^{2} - 17566912x + 839924$

Orbit

Characteristic polynomial of

T

509.2.a.a

x^{14} + 8x^{13} + x^{12} - 109x^{11} - 117x^{10} + 508x^{9} + 504x^{8} - 1193x^{7} - 621x^{6} + 1445x^{5} + 12x^{4} - 682x^{3} + 232x^{2} + 21x - 13

509.2.a.b

x^{28} - 8x^{27} - 79x^{26} + 775x^{25} + 2020x^{24} - 29956x^{23} - 9270x^{22} + 607348x^{21} - 451466x^{20} - 7233548x^{19} + 9489672x^{18} + 53890768x^{17} - 89144249x^{16} - 258670457x^{15} + 486611251x^{14} + 798783953x^{13} - 1647657054x^{12} - 1521661510x^{11} + 3479175501x^{10} + 1574898974x^{9} - 4402183614x^{8} - 508093292x^{7} + 2997640229x^{6} - 394588162x^{5} - 840597920x^{4} + 243169571x^{3} + 43794823x^{2} - 17566912x + 839924

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(509))$	42
Old forms	0
Cusp forms $S_{2}(\Gamma_0(509))$	42
Eisenstein series $E_{2}(\Gamma_0(509))$	1
Modular forms $M_{2}(\Gamma_0(509))$	43

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 509]: [0, 43] (certified at ell = 2147468909)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 5 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(509, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 509:2 1000

← 508.2.a · 510.2.a →