Atlas › Modular forms › Weight 2 › Level 517 › 517.2.a

Newspace 517.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(517))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
517.2.a.a	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{4} + 3q^{5} - 2q^{7} + O(q^{8})$	− +	$+1$	517b1 rank 0
517.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	517c1 rank 1
517.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + 3q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	517a1 rank 0
517.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
517.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 6q^{3} - 2q^{5} + O(q^{8})$	+ −	$+1$
517.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{6} - 6q^{7} + O(q^{8})$	+ −	$+1$
517.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} - 4q^{5} + 8q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
517.2.a.h	3	degree 3	$3q - 3q^{2} - 4q^{3} - 3q^{4} - 2q^{5} + 4q^{6} - q^{7} + O(q^{8})$	− −	$-1$
517.2.a.i	3	degree 3	$3q + q^{2} + 5q^{4} + 2q^{5} + 6q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
517.2.a.j	10	degree 10	$10q - 6q^{2} - 7q^{3} + 10q^{4} - 3q^{5} + 2q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
517.2.a.k	10	degree 10	$10q + 3q^{2} + 2q^{3} + 15q^{4} + 4q^{5} - 10q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{3} + 3T_{5} + 4T_{7} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
517.2.a.a	$x + 7$
517.2.a.b	$x + 37$
517.2.a.c	$x - 47$
517.2.a.d	$x^{2} + 74x - 2963$
517.2.a.e	$x^{2} - 102x - 2807$
517.2.a.f	$x^{2} - 56x + 712$
517.2.a.g	$x^{2} - 44x - 4124$
517.2.a.h	$x^{3} + 98x^{2} - 2676x - 124312$
517.2.a.i	$x^{3} - 52x^{2} + 288x + 8224$
517.2.a.j	$x^{10} + 404x^{9} + 68039x^{8} + 6165230x^{7} + 322401708x^{6} + 9580653501x^{5} + 141041092828x^{4} + 379341287045x^{3} - 12796440275116x^{2} - 92327568074040x + 135195116269648$
517.2.a.k	$x^{10} - 192x^{9} + 605x^{8} + 1343688x^{7} - 17838964x^{6} - 3429238455x^{5} + 9281823568x^{4} + 2537190724103x^{3} + 38120889786x^{2} - 454841772377452x + 1264259755702072$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(517))$	37
Old forms	10
Cusp forms $S_{2}(\Gamma_0(517))$	47
Eisenstein series $E_{2}(\Gamma_0(517))$	3
Modular forms $M_{2}(\Gamma_0(517))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(517)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 11, 47, 517]: [0, 2, 5, 50] (certified at ell = 2147480791)
T = [(2, 1), (3, 2), (5, 3), (7, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 10 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(517, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 517:2 1000

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