Atlas › Modular forms › Weight 2 › Level 516 › 516.2.a

Newspace 516.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(516))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
516.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+ −	516b1 rank 1
516.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} - q^{7} + O(q^{8})$	+ +	516a1 rank 0
516.2.a.c	1	$\mathbb{Q}$	$q + q^{3} + O(q^{8})$	− −	516c1 rank 0
516.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + 5q^{7} + O(q^{8})$	− −	516d1 rank 0
516.2.a.e	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{3} - q^{5} - q^{7} + O(q^{8})$	+ +
516.2.a.f	2	$\mathbb{Q}(\sqrt{57})$	$2q + 2q^{3} + q^{5} - q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
516.2.a.a	$x + 52$
516.2.a.b	$x - 27$
516.2.a.c	$x - 76$
516.2.a.d	$x + 61$
516.2.a.e	$x^{2} - 87x + 1488$
516.2.a.f	$x^{2} + 57x - 1596$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(516))$	8
Old forms	75
Cusp forms $S_{2}(\Gamma_0(516))$	83
Eisenstein series $E_{2}(\Gamma_0(516))$	11
Modular forms $M_{2}(\Gamma_0(516))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(516)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(129))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(172))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(258))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516]: [0, 1, 1, 2, 3, 5, 4, 13, 16, 25, 48, 94] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; h of degree 8 = 1 x 8 (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(516, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 516:2 1000

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