Atlas › Modular forms › Weight 6 › Level 15 › 15.6.a

Newspace 15.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(15))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
15.6.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 9q^{3} - 28q^{4} - 25q^{5} + 18q^{6} - 132q^{7} + O(q^{8})$	+ +	$-1$
15.6.a.b	1	$\mathbb{Q}$	$q + 7q^{2} + 9q^{3} + 17q^{4} - 25q^{5} + 63q^{6} + 12q^{7} + O(q^{8})$	− +	$+1$
15.6.a.c	2	$\mathbb{Q}(\sqrt{409})$	$2q - q^{2} - 18q^{3} + 141q^{4} + 50q^{5} + 9q^{6} - 112q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
15.6.a.a	$x + 134$
15.6.a.b	$x - 19$
15.6.a.c	$x^{2} + 113x - 19814$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(15))$	4
Old forms	4
Cusp forms $S_{6}(\Gamma_0(15))$	8
Eisenstein series $E_{6}(\Gamma_0(15))$	4
Modular forms $M_{6}(\Gamma_0(15))$	12

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(15)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 3, 5, 15]: [1, 3, 3, 12] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(15, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 15:6 1000

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