Atlas › Modular forms › Weight 2 › Level 518 › 518.2.a

Newspace 518.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(518))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
518.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{7} + O(q^{8})$	+ + −	$+1$
518.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
518.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
518.2.a.d	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
518.2.a.e	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} + 3q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
518.2.a.f	5	degree 5	$5q + 5q^{2} + q^{3} + 5q^{4} - 3q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
518.2.a.a	$x^{2} - 98$
518.2.a.b	$x^{2} + 12x + 31$
518.2.a.c	$x^{2} - 2x - 2$
518.2.a.d	$x^{3} + 10x^{2} + x - 124$
518.2.a.e	$x^{3} - 10x^{2} - 43x + 452$
518.2.a.f	$x^{5} + 8x^{4} - 173x^{3} - 420x^{2} + 6744x - 6388$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(518))$	17
Old forms	56
Cusp forms $S_{2}(\Gamma_0(518))$	73
Eisenstein series $E_{2}(\Gamma_0(518))$	7
Modular forms $M_{2}(\Gamma_0(518))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(518)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(259))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 7, 14, 37, 74, 259, 518]: [0, 1, 1, 4, 3, 11, 26, 80] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(518, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 518:2 1000

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